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Investigation of DNP Mechanisms: The Solid Effect
by
Albert Andrew Smith
B.S., Mount Union College (2007)
Submitted to the Department of Chemistry in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2012
© 2012 Massachusetts Institute of Technology. All rights reserved
Signature of Author............................................................................................................................ Department of Chemistry
May 17, 2012
Certified by ........................................................................................................................................
Robert G. Griffin Professor of Chemistry
Thesis Supervisor Accepted by .......................................................................................................................................
Robert W. Field Professor of Chemistry
Chairman, Departmental Committee on Graduate Students
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Chairman, Departmental Committee on Graduate Students This doctoral thesis has been examined by a Committee of the Department of Chemistry as follows: Professor Robert W. Field.................................................................................................................. Chairman Professor Troy Van Voorhis .............................................................................................................. Professor Robert G. Griffin................................................................................................................ Thesis Supervisor
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Investigation of DNP Mechanisms: The Solid Effect
by
Albert Andrew Smith
Submitted to the Department of Chemistry on May 17, 2012 in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy in Chemistry
ABSTRACT Dynamic Nuclear Polarization (DNP) enhances signal to noise in NMR experiments, by transferring the large electron Boltzmann polarization to nuclear polarization, via application of pulsed or continuous-wave microwave irradiation. This results in increases in NMR sensitivity of 2-3 orders of magnitude. DNP greatly reduces experimental times and makes some experiments possible that are otherwise unfeasible due to lack of sensitivity. DNP methods have undergone vast improvements in recent years. However, continued advancement of DNP methods will rely on having a clear understanding of the underlying mechanisms. We develop instrumentation and software intended for the study of DNP mechanisms. This includes a three-channel (e-, 13C, 1H) probe for observing both electrons and nuclei, and a 140 GHz pulsed-EPR spectrometer. We also have developed DNPsim, a program designed for easy quantum-mechanical simulation of basic DNP experiments, combined with the flexibility to customize simulations for more advanced experiments and mechanistic studies. Using these tools, we develop a theoretical framework for the solid effect DNP mechanism, which considers the roles of quantum mechanical and relaxation processes in many-spin systems. NMR experiments under static conditions that monitor nuclear polarization buildup were fit to models of electron-nuclear polarization transfer; the results show that nuclei near the electron and the observed (bulk) nuclei compete for electron polarization. Therefore bulk nuclear enhancements are reduced, since nuclei near the electron deplete electron polarization. This result is also reproduced for magic angle spinning NMR experiments. EPR experiments that monitor electron polarization as a function of microwave frequency can be used to measure DNP ‘matching conditions’. Experiments utilizing the solid effect show DNP matching conditions that are a result of polarization transfer through many spin, high-order coherences. Previously, it was thought that transfers involving high-order coherences should be highly forbidden, whereas these experiments present strong evidence of their presence. Simulations using DNPsim also show that high-order coherences can play a significant role in DNP polarization transfers in strongly coupled, many-spin systems. Thesis Supervisor: Robert G. Griffin Title: Professor of Chemistry Director of the Francis Bitter Magnet Laboratory
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Acknowledgements
The research in this thesis would not be possible without the support of Bob
Griffin. I learned many fundamentals of magnetic resonance through NMR and DNP
classes in the lab, by designing and constructing much of the instrumentation needed for
my experiments, and through pursuing some of my own questions about DNP processes.
I would not have been successful in this without the resources, collaborations, and
guidance provided by Bob.
Throughout my Ph.D. research, I have worked most closely with Björn Corzilius.
No one else in the lab has taken the same level of interest in working out DNP
mechanisms, and I appreciate having someone willing to spend the time to consider a
problem and work out the best solution. The DNP subgroup has also usually been a very
collaborative group of people, and I have also learned quite a bit from them. This
includes Vlad Michaelis, TC Ong, Xander Barnes, Thorsten Maly, Loren Andreas,
Eugenio Daviso, Evgeny Markhasin, and Marcel Reese. Although I have not worked on
research projects with them, I appreciate the advice and guidance of other members of the
lab, including Galia Debelouchina, Marvin Bayro, Matt Eddy, and Marc Caporini. I
would also like to thank collaborators Ken Yokoyama and Joanne Stubbe, and also Matt
Kiesewetter, Olesya Haze, Joe Wallish, and Tim Swager. Finally, thanks to the Bitter
staff including Jeff Bryant, Dave Ruben, Ron DeRocher, Ajay Thakkar, Mike Mullins,
and Chris Turner for their technical expertise and advice.
Wednesday night wingers has been essential for surviving grad school. It has
always been helpful to be able to relax and blow off steam in the middle of the week.
Thanks to the original wingers group, Marc, Marvin, Xander, Galia, Ziad, Becky, and
Leo for inviting me out when I first joined the group, and to Vlad, Björn, Eugenio and
Susanne for keeping up the tradition. Also thanks to Loren and TC for being good friends
throughout grad school, and Tim and Dan for being great housemates.
My biggest thanks goes to my Mom and Dad, and my sister Bonnie. I always look
forward to our weekly phone calls; whether I am venting over some frustration,
celebrating a success in lab, failing to string words together into a sentence due to
exhaustion, or just listening to what’s happening at home, I always am recharged and
better prepared for the next week. I would not have made it to MIT without their
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continuous support, and certainly would not have finished without their encouragement
and understanding.
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Table of Contents
Investigation of DNP Mechanisms: The Solid Effect..........................................................1 Abstract ................................................................................................................................5 Acknowledgements..............................................................................................................7 Chapter 1 Introduction......................................................................................................13 1.1 Background ............................................................................................................14 1.1.1 Solid-State NMR...........................................................................................14 1.1.2 Electron Paramagnetic Resonance................................................................15 1.2 Theory of DNP Mechanism...................................................................................23 1.2.1 Magnetic Resonance Theory.........................................................................23 1.2.1.1 Introduction to Density Matrices .........................................................23 1.2.1.2 Polarization and Coherence .................................................................25 1.2.1.3 Rotating Frame Transformation...........................................................26 1.2.2 DNP Mechanisms .........................................................................................27 1.2.2.1 Solid Effect ..........................................................................................27 1.2.2.2 Cross Effect..........................................................................................28 1.2.2.3 Thermal Mixing ...................................................................................31 1.2.3 Many Spin Mechanisms................................................................................31 1.3 Solid Effect Studies................................................................................................32 1.3.1 Chapter 2: A 140 GHz Pulse EPR/212 MHz NMR Spectrometer for DNP Studies..............................................................................................................32 1.3.2 Chapter 3: Solid Effect DNP and Polarization Pathways .............................33 1.3.3 Chapter 4: Solid Effect in MAS DNP...........................................................33 1.3.4 Chapter 5: Observation of Strongly Forbidden Transitions via Electron-Detected Solid Effect DNP ...............................................................34 1.3.5 Chapter 6: DNPsim: A Flexible Program for DNP Simulations ..................34 1.4 Outlook ..................................................................................................................35 1.5 Bibliography ..........................................................................................................35 Chapter 2 A 140 GHz Pulsed EPR/212 MHz NMR Spectrometer for DNP Studies.......38 2.1 Motivation..............................................................................................................40 2.2 Instrument Design..................................................................................................42 2.2.1 140 GHz EPR Bridge....................................................................................42 2.2.2 EPR Control and Detection...........................................................................46 2.2.3 DNP Probe ....................................................................................................47 2.2.4 Full System Control ......................................................................................51
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2.3 Experimental Results .............................................................................................54 2.4 Conclusions............................................................................................................62 2.5 Appendix................................................................................................................63 2.6 Bibliography ..........................................................................................................64 Chapter 3 Solid Effect Dynamic Nuclear Polarization and Polarization Pathways .........67 3.1 Motivation..............................................................................................................69 3.2 Theory ....................................................................................................................73 3.2.1 Rate equations...............................................................................................73 3.2.1.1 Relaxation ............................................................................................78 3.2.1.2 Spin-Diffusion .....................................................................................78 3.2.1.3 Off-Resonant Electron Saturation........................................................80 3.2.1.4 Solid Effect DNP .................................................................................80 3.2.1.5 Higher Order Processes........................................................................81 3.2.1.6 Rate Equations .....................................................................................82 3.2.2 Implications of the Rate Equations ...............................................................82 3.2.2.1 Case (A): No diffusion barrier .............................................................85 3.2.2.2 Case (B): Two-Step Bulk Polarization ................................................88 3.2.2.3 Case (C): Direct bulk polarization .......................................................90 3.3 Experimental ..........................................................................................................92 3.4 Results and Discussion ..........................................................................................93 3.4.1 Case (A) ........................................................................................................96 3.4.2 Case (B) ........................................................................................................98 3.4.3 Case (C) ........................................................................................................99 3.5 Conclusions..........................................................................................................104 3.6 Appendix..............................................................................................................104 3.6.1 Solving One-Step Transfer Equations without Fast Equilibrium ...............104 3.6.2 Two-step DNP Transfer..............................................................................106 3.7 Bibliography ........................................................................................................109 Chapter 4 Highly Efficient Solid Effect in Magic Angle Spinning Dynamic Nuclear Polarization at High Field ................................................................................................113 4.1 Motivation............................................................................................................115 4.2 Theory ..................................................................................................................117 4.2.1 Diagonalization of the Static Hamiltonian for the Solid Effect..................117 4.2.2 Transition Moments of the Solid Effect......................................................121 4.2.3 DNP Kinetics Based on Rate Equations .....................................................123
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4.3 Experimental ........................................................................................................132 4.4 Results and Discussion ........................................................................................134 4.4.1 Analysis of SE DNP Matching Conditions at 140 GHz .............................134 4.4.2 Polarization Dynamics ................................................................................137 4.4.3 Influence of Spin-Diffusion Efficiency on DNP ........................................145 4.4.4 The Solid Effect at Fields >5 T...................................................................149 4.5 Conclusions..........................................................................................................152 4.6 Appendix..............................................................................................................153 4.7 Bibliography ........................................................................................................158 Chapter 5 Observation of Strongly Forbidden Transitions via Electron-Detected Solid Effect Dynamic Nuclear Polarization ..............................................................................161 5.1 Motivation............................................................................................................163 5.2 Theory ..................................................................................................................164 5.3 Experimental ........................................................................................................167 5.4 Results and Discussion ........................................................................................168 5.5 Conclusions..........................................................................................................175 5.6 Bibliography ........................................................................................................175 Chapter 6 DNP Simulator...............................................................................................178 6.1 Motivation............................................................................................................180 6.2 Theory ..................................................................................................................182 6.2.1 Calculation of the Hamiltonian...................................................................183 6.2.2 Relaxation ...................................................................................................187 6.2.1.1 Construction of the Relaxation Matrix ..............................................187 6.2.1.2 Recovery to Thermal Equilibrium .....................................................198 6.2.3 Propagation of the Spin System..................................................................200 6.2.3.1 Propagation in Hilbert Space .............................................................200 6.2.3.2 Propagation in Liouville Space..........................................................202 6.3 Simulator Usage...................................................................................................208 6.3.1 Spin System ................................................................................................209 6.3.1.1 Direct Input ........................................................................................209 6.3.1.2 Orientation-Dependent Input .............................................................212 6.3.1.3 Relaxation Parameters .......................................................................217 6.3.2 Experimental Parameters ............................................................................220 6.3.2.1 Sweep Parameters ..............................................................................220 6.3.2.2 Propagation Parameters .....................................................................221
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6.3.2.3 Initiation and Detection Parameters...................................................225 6.3.2.4 General Parameters ............................................................................228 6.3.3 Liouville Space Basis Set............................................................................230 6.3.4 DNPsim Programs ......................................................................................233 6.3.4.1 GyroRatio...........................................................................................233 6.3.4.2 PowderOrientations............................................................................233 6.3.4.3 n_spin_system....................................................................................234 6.3.4.4 Verify .................................................................................................236 6.3.4.4 State_list_gen.....................................................................................236 6.4 Examples..............................................................................................................237 6.4.1 Solid Effect Field Profile ............................................................................238 6.4.2 Nuclear Orientation via Electron Spin-Locking .........................................239 6.4.3 Cross Effect.................................................................................................241 6.4.4 Dressed-State Soli Effect ............................................................................243 6.4.5 Nuclear Rotating Frame DNP.....................................................................245 6.5 Investigating DNP via Simulation: The Solid Effect...........................................247 6.6 Conclusions..........................................................................................................253 6.7 Appendix..............................................................................................................254 6.8 Bibliography ........................................................................................................255
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Chapter 1
Introduction
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1.1 Background Dynamic Nuclear Polarization (DNP) is a method of enhancing the signal to noise
of nuclear magnetic resonance (NMR) experiments [1; 2]. This method relies on the
transfer of the electron Boltzmann polarization to nuclear polarization. Because the
electron Boltzmann polarization is several orders of magnitude larger than the nuclear
polarization (660 for 1H), this transfer results in significant enhancements: currently as
high as ~300 for 1H [3-5], and higher for other nuclei [6]. This enhancement in signal to
noise corresponds with a reduction in experiment time by a factor of ~90,000, which is an
incredible improvement. This is for a model DNP sample, whereas enhancements may be
lower for less ideal samples. However, the reduction in experimental time is rarely
negligible, making DNP an attractive method of improving signal to noise and reducing
experimental times. We begin by discussing the motivation behind solid-state NMR
(ssNMR) and electron paramagnetic resonance (EPR), and the drawbacks of each that
can be addressed with DNP.
1.1.1 Solid-State NMR NMR experiments measure the interactions in a system of nuclear spins. These
interactions can provide a wealth of information about the environment of those spins-
especially information about the structure and dynamics of a spin system. To understand
this information, in the context of ssNMR, we begin with a basic explanation of the
evolution of the spin system in an NMR experiment. We begin with an isolated spin.
When an isolated nuclear spin is placed in a magnetic field, that spin will oscillate
around the direction of the magnetic field, with a frequency given by
ω0 = −γ B0 . (1)
In (1), ω0 is the oscillation frequency of the spin, known as the Larmor frequency. γ is
the gyromagnetic ratio of the spin, and B0 is the strength of the static magnetic field. The
gyromagnetic ratio gives the ratio of the magnetic dipole moment of the spin to its
angular momentum.
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Much more information is available for a system of spins. Although the dynamics
are considerably more complex in the many spin case, the same basic concept applies.
For a large spin system, each spin responds to the net magnetic field it experiences.
However, it is not possible to isolate the evolution of a single spin, because neighboring
spins affect each other; additionally, the spins are coupled to other quantum mechanical
(QM) processes. Therefore, one must consider the evolution of the full QM system. This
evolution is given by
ddt
ρ(t) = −i H f (t),ρ(t)⎡⎣ ⎤⎦ , (2)
where we give the Liouville-von Neumann equation for a QM system. We will discuss
the density matrix formalism more in Section 1.2.1, and go into great detail on how to
solve (2) numerically in Chapter 6. We are brief in our discussion here: ρ(t) is a density
matrix that describes the evolution of the expectation values of the full QM system. The
spin system, which is later defined as σ (t) , is a subset of the full system. Then, H f (t) is
the Hamiltonian, for which the subscript f denotes that this governs the dynamics of the
full QM system. If we assume that the spin system, σ (t) , does not affect the rest of the
QM system significantly, one can rewrite (2) as follows [7]:
ddtσ (t) = −i H0 ,σ (t)⎡⎣ ⎤⎦ − Γ{σ (t)−σ eq}. (3)
In (3), H0 is the Hamiltonian describing interactions of the spin system, σ (t) .
We will assume that H0 does not include any oscillating fields, so that the only
interactions are the spin-spin couplings, and the couplings to the static magnetic field,
B0 . By assuming that the spin system does not affect the rest of the QM system, one may
neglect all terms in ρ(t) besides those in σ (t) . However, one must still consider the
actions of the rest of the QM system on σ (t) . These actions are incorporated in the term
−Γ{σ (t)−σ eq} , where Γ is a function that is referred to as the relaxation superoperator.
If (3) is valid, which is usually the case in NMR experiments, then H0 can be given in
general by:
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H I
z = γ mB0 1−σ m( ) Imm=1
NI
∑ Nuclear Zeeman with Chemical Shift
HII = Imdm,n
II Inn>m
NS
∑m=1
NI
∑ Nuclear-Nuclear Coupling (4)
HQ = ImQmIm
m=1
NI
∑ Nuclear Quadrupole Coupling
H0 = Hz + HII + HQ Static Hamiltonian
The power of ssNMR comes from the information content in H zI , HII ,
HQ , and
Γ , and the ability to manipulate the spin system in such away that this information can
be extracted. We begin with a discussion of the terms in (4), including the information
content of each term. HzI is the Zeeman Hamiltonian, where the term γ mB0(1)Im gives
the Larmor frequency of the mth nucleus. Traditionally, B0 = 0,0, B0( ) such that the
external magnetic field is in the z-direction with a strength of B0 . Then, γ mB0(1)Im
reduces to γ mB0Sz , and since γ m varies significantly for different types of heteronuclei
(1H, 13C, 15N, etc.), they can easily be observed and manipulated independently. Hz also
contains the chemical shift tensor, σ m . The chemical shift is a result of shielding of the
nucleus by the electron density around it. Therefore the chemical shift contains
information about the electronic environment, which can be used to help identify the
chemical groups (e.g. carbonyl, aliphatic) that the nucleus is part of. HII is the nuclear-
nuclear coupling Hamiltonian. The coupling tensor, dm,n
II , contains the scalar and dipolar
couplings between neighboring nuclei. The scalar coupling is a result of through-bond
interactions between nuclei. The dipole coupling usually dominates in ssNMR
experiments, and its magnitude is proportional to r −3 . Therefore, measurement of the
coupling tensor can provide significant structural information (short and medium range
distances, <10 Å). For a high-spin nucleus (I>1/2), the quadrupolar Hamiltonian, HQ ,
gives the self-interaction of that nucleus. The nuclear quadrupole coupling is a result of
an electric field gradient, which distorts the shape of the nucleus. Therefore, the
quadrupole coupling gives information about the size of the electric field gradient at the
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nucleus, which is in part a function of the symmetry of surroundings of the nucleus.
Finally, Γ couples the spin system to the surrounding environment. This term brings
about relaxation of the spin system, and can provide information about the dynamics of
the system.
Although the NMR Hamiltonian contains a lot of information, it is necessary to be
able to extract that information. We start by describing one of the most basic NMR
experiments. A one-pulse, free induction decay experiment consists of applying a single
radio-frequency (RF) pulse matched to the Larmor frequency of a particular type of
nucleus (e.g. 13C). If the pulse is set so that it rotates all spins 90º from the external
magnetic field, then those spins will produce their own magnetic field that oscillates
around the external magnetic field with frequencies given by the combined effects of the
terms in (4). This oscillating field may be observed and Fourier transformed in order to
obtain a frequency spectrum of the spin system. For a single crystal, which contains
repeated instances of the same spin system with the same orientation, it may be possible
to separate the terms in (4) with only a one-pulse experiment. However, many ssNMR
samples are powders- a collection of near-identical spin systems, but that have many
randomly distributed orientations relative to the external magnetic field. Therefore, one
will observe resonances resulting from all possible orientations of the tensors in (4), and
each orientation will contain the combined effects of the Zeeman, nuclear coupling, and
quadrupole Hamiltonians (if high-spin nuclei are present). Fourier transformation of the
oscillating signal will result in a spectrum that is typically very broad, and for which it is
difficult to separate the various effects, except in the case of very small spin systems.
Clearly, one must deconvolute this information in order for it to be useful. Fortunately,
there are many options to do just that. Common methods in ssNMR include magic angle
spinning (MAS), 1H decoupling, cross polarization, and correlation experiments.
Magic angle spinning (MAS) NMR is typically used in ssNMR experiments when
a powder sample is used (a sample for which near identical spin system occur in many,
random orientations). MAS spins a sample around an axis that is ~54.7º away from the
external magnetic field, at frequencies from ~2-70 kHz. This method averages the
chemical shift tensor to its isotropic value, and removes the dipole couplings [8; 9]. Note
that spinning at 54.7º will not fully average quadrupole tensors; we will assume spin-1/2
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nuclei for the remainder of this discussion, and therefore will not have quadrupolar
couplings.
MAS typically only partially removes dipole couplings between 1H nuclei and
low-γ nuclei such as 13C and 15N, which are often observed in ssNMR experiments. To
further reduce these couplings, 1H decoupling is used. An RF field is applied to the 1H to
further eliminate the dipole coupling [10; 11]. Additionally, it is common when observing
low-γ nuclei to transfer high polarization from 1H nuclei to enhance the signal to noise,
via a cross polarization experiment [12].
A combination of MAS, 1H decoupling, and cross polarization can yield high
quality, one-dimensional spectra of low-γ nuclei such as 13C and 15N. In this case,
resonances in a one-dimensional spectrum are primarily determined by the isotropic
(average) chemical shift of the nuclei being observed. Of course, imperfect decoupling
and MAS, in addition to relaxation processes brought about by Γ , will add broadening to
the spectrum. The chemical shift alone will not usually give enough information about
the spin system. For more than a few spins, it is not always possible to determine which
resonance corresponds to which spin based solely on chemical shift- in other words, one
cannot assign the spectrum. Additionally, a one-dimensional spectrum may not have
sufficient resolution to separate all resonances with only the chemical shift. Finally, most
of the structural information available from ssNMR is obtained by observation of
couplings. Therefore, one needs to reintroduce other terms in the Hamiltonian- in
particular the dipole couplings.
There are a variety of methods to reintroduce dipole couplings. Some examples
include Rotational-Echo DOuble Resonance NMR (REDOR) [13] and Transferred-Echo
DOuble-Resonance NMR (TEDOR) [14]. These experiments actively reintroduce
heteronuclear dipole couplings by applying RF pulses to one of the heteronuclei in sync
with the rotor cycle, in order to reintroduce the coupling, and either dephase or transfer
coherence. Proton-Driven Spin-Diffusion (PDSD) uses the residual homonuclear dipole
couplings that remain even with MAS to perform non-coherent polarization transfer
between spins, and is aided by the strong coupling to nearby protons. Proton Assisted
Recoupling (PAR) [15] and Proton Assisted Insensitive Nuclei Cross Polarization (PAIN-
CP) [16] use a coherent transfer of polarization between homonuclear or heternuclear
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spin pairs, with the assistance of a nearby proton. These are just some of many
experiments that allow the transfer of population between spins.
Recoupling experiments can be combined with multi-dimensional methods to
provide correlations between spins of the spin-system. Multi-dimensional experiments
provide correlations between the resonances seen in one-dimensional experiments. These
correlations are indicative of a coupling between spins. By observing the couplings of
neighboring spins, one may determine which pairs of neighboring spins correspond to
which pairs of resonances. This additional information allows assignment of spectra, can
provide additional resolution, and provides structural information.
Our brief discussion of the use of MAS, decoupling, cross polarization,
recoupling, and multi-dimensional experiments to elucidate structure only begins to cover
the many experiments and types of information available via ssNMR. Although a
powerful technique, ssNMR suffers from a major drawback- lack of sensitivity. It is
possible to observe NMR signal because spins in thermal equilibrium slightly favor one
orientation relative to the magnetic field, so that the signals resulting from individual
spins do not fully cancel each other out. The favoring of one orientation is known as
polarization, and is calculated as
P =
N + − N −
N + + N − , (5)
where N + is the number of spins aligned parallel to the magnetic field, and N − is the
number of spins aligned anti-parallel to the magnetic field. Thermal polarization can be
calculated using a Boltzmann distribution as
Pthermal =
exp[hγ B0 / kbT ]− exp[−hγ B0 / kbT ]exp[hγ B0 / kbT ]+ exp[−hγ B0 / kbT ]
= tanhhγ B0
kBT⎡
⎣⎢
⎤
⎦⎥ , (6)
where h is Planck’s constant, γ is the gyromagnetic ratio, B0 is the external magnetic
field, kB is the Boltzmann constant, and T is the temperature. Figure 1 shows the thermal
polarization for several nuclei and an electron at a field strength of 5 T (213 MHz 1H
frequency) at temperatures ranging from .1 K to 300 K.
One sees that the thermal polarizations of the nuclei are quite low even at liquid
nitrogen temperature (.006 % for 1H at 5T). This results in very low signal to noise for
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ssNMR experiments. This is compounded by the fact that many molecules of interest are
very large, such as proteins, resulting in a low sample concentration. Additionally, multi-
dimensional experiments usually lose signal during population transfer steps. Finally, it
usually requires seconds for thermal polarization to recover after an NMR experiment, so
that polarization recovery consumes most experimental time. These are some major
factors that make many ssNMR experiments prohibitively long.
1.1.2 Electron Paramagnetic Resonance Electron Paramagnetic Resonance (EPR) follows the same principles as NMR,
and is also governed by (3). However, the usual Hamiltonian contains additional terms,
which are given in (7).
Hz
S = B0g jS jj=1
NS
∑ Electron Zeeman
Figure 1: Polarization versus Temperature
Thermal polarization for e–, 1H, 13C, and 15N at 5 T is shown for temperatures ranging
from .1 K to 300 K.
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HSS = S jd j ,k
SS Skk> j
NS
∑j=1
NS
∑ Electron-Electron Coupling (7)
HIS = S j Aj ,mSm
m=1
NI
∑j=1
NS
∑ Electron-Nuclear Coupling
HD = S j DjS j
j=1
NS
∑ Electron Zero-Field Coupling
Then the full static Hamiltonian is given by the sum of all terms in (4) and (7).
H0 = Hz
I + HzS + HSS + HII + HIS + HD + HQ Static Hamiltonian (8)
The additional terms in (7) do appear in ssNMR experiments in some cases, but
usually ssNMR is not performed for systems that are very close to a paramagnetic
electron, and so the electronic terms are ignored. The terms in (7) have considerable
information content, as was the case for ssNMR. HzS is the Zeeman interaction for the
electron, where the tensor g j gives the strength of interaction of the electron with the
external field. This term is analogous to the chemical shift in NMR. However, whereas
chemical shift results from shielding of the nucleus by the electron, the g-tensor is a result
of a non-vanishing spin-orbit coupling of the electron. Therefore, g j contains
information about the hybridization of the paramagnetic electrons’ orbitals.
The electron-electron coupling, given by the tensor d j ,k
SS found in HSS , can
contain structural information about paramagnetic systems. For example, pulsed Double
Electron-Electron Resonance (DEER) experiments have become a very common means
of gaining long-distance contacts in biologically relevant paramagnetic species [17].
Because electrons have much larger interactions, EPR can be used to measure
significantly longer distances than is possible with NMR. For example, our collaboration
with Yokoyama, et al. utilized this method in studies of radical propagation in
ribonucleotide reductase [18]. The electron-nuclear coupling, Aj ,m , found in HIS may be
measured to determine electron distribution over magnetic nuclei and to determine
distances to nearby nuclei. This may be done using Electron-Nuclear DOuble-Resonance
(ENDOR) [19; 20], or in the case of large couplings, can often be measured directly in
field-swept EPR experiments.
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Finally, the zero-field coupling in EPR, given by the tensor Dj that is found in
HD , occurs when a single paramagnetic center has multiple electrons around it. These
electrons experience nearly identical environments, but interact with each other, resulting
in the zero-field coupling. As with the nuclear quadrupole coupling, if the paramagnetic
center is relatively symmetric in its electronic distribution, then the zero-field splitting
will be small. For example, Gadolinium DOTA has a relatively small zero-field splitting
due to the symmetry of the DOTA ligands around the Gadolinium center- making it an
ideal candidate for solid effect DNP experiments [21].
Although the information provided by EPR is similar to that from NMR, there are
significant differences. The electron gyromagnetic ratio is ~660 times larger than the 1H
gyromagnetic ratio, and ~2600 times larger than 13C. As a result, most EPR interactions
are about 3 orders of magnitude faster than NMR interactions. This causes the methods to
vary greatly. MAS serves little purpose in EPR, because samples cannot be rotated fast
enough to achieve any significant averaging of the EPR interactions. As a result, it is
difficult to measure at large spin-systems with EPR, because multiple electrons will
usually have many overlapping spectral components. Also, EPR spectra at high magnetic
fields (>3 T) are usually too broad to perform a one-pulse experiment as is possible in
NMR, and so the magnetic field must be swept to obtain the full spectrum. EPR has a
significant advantage in sensitivity, though. Because of the higher gyromagnetic ratio,
EPR polarization is higher by a factor of 660 when compared to 1H, as shown in Figure 1.
Additionally, recovery of polarization for electrons occurs ~1000 times faster for
electrons, making repetition times in EPR much shorter. The result is that it is possible to
quickly obtain EPR signal from low numbers of electron spins.
ssNMR is a powerful method to extract information about nuclear spin systems,
but suffers from low sensitivity due to low Boltzmann polarization, and slow polarization
recovery times. EPR, on the other hand, has much higher thermal polarization and fast
polarization recovery times. We are thus motivated to attempt to combine the benefits of
both methods. If one includes paramagnetic centers in a ssNMR sample, it is possible to
transfer the higher polarization to nuclei, and as a result greatly increase the signal to
noise in ssNMR experiments. Because of fast polarization recovery times for electrons, a
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single electron can polarize on the order of 1000 nuclei. The set of experiments that
utilize polarization transfer from electrons to nuclei are known collectively as Dynamic
Nuclear Polarization (DNP), the basic theory of which is presented next.
1.2 Theory of DNP Mechanisms Solid-state DNP mechanisms can be categorized into continuous-wave (cw) and
pulsed methods. cw-DNP methods use non-coherent methods of polarization transfer
from electrons to nuclei, whereas pulsed DNP uses some form of coherent mechanism,
usually involving a spin-lock on either the electrons or nuclei involved. cw-DNP methods
are much more common, and considerably better developed. We will focus on these
mechanisms, although Chapter 6 shows some simulations of various pulsed-DNP
methods. However, before discussing these mechanisms in detail, we will begin with a
discussion of basic magnetic resonance theory.
1.2.1 Magnetic Resonance Theory To understand basic theory of DNP, we will need to understand density matrix
formalism, and some basic mechanics that are commonly used. We will begin by
discussing the interpretation of the density matrix and of the magnetic resonance
Hamiltonian. Additionally, we will introduce concepts of polarization and coherence, and
the rotating frame transformation.
1.2.1.1 Introduction to Density Matrices
We begin by giving the evolution of the spin system using density matrix
formalism, which is discussed in more detail in Spin Dynamics, chapter 11 [22].
ddtσ (t) = −i H (t),σ (t)⎡⎣ ⎤⎦ . (9)
In (9), σ (t) is a density matrix that describes the spin system. For this discussion, we
will not consider other quantum states which lead to relaxation, although these are taken
into consideration in Chapter 6. H (t) describes the interactions of the spin system with
itself and applied magnetic fields, including the static external magnetic field and any
24
microwave or RF fields. The brackets represent a commutator, for which
A, B⎡⎣ ⎤⎦ = AB − BA .
For a single spin system, with spin-1/2, the density matrix can be written as
σ (t) = x(t)Sx + y(t)Sy + z(t)Sz . (10)
Sx , Sy , and Sz are known as spin operators, and represent the magnetization in the x, y,
or z direction, respectively, that results from the spin. Then x(t) , y(t) , and z(t) give the
magnitude and time dependence of the magnetization in each direction, and are often
represented as Sx (t) ,
Sy (t) , and Sz (t) . Similarly, the Hamiltonian for this system
can be written generally as
H (t) = X (t)Sx +Y (t)Sy + Z(t)Sz . (11)
In this case, X (t) , Y (t) , and Z(t) are the magnitude fields that are applied to the spin
system in the x-, y-, and z-directions. Then, we have the following relationships so that
the spin system evolves appropriately:
Sx ,Sy⎡⎣ ⎤⎦ = − Sy ,Sz
⎡⎣ ⎤⎦ = iSz
Sy ,Sz⎡⎣ ⎤⎦ = − Sz ,Sy
⎡⎣ ⎤⎦ = iSx
Sz ,Sx⎡⎣ ⎤⎦ = − Sx ,Sz⎡⎣ ⎤⎦ = iSy
Sx ,Sx⎡⎣ ⎤⎦ = Sy ,Sy⎡⎣ ⎤⎦ = Sz ,Sz⎡⎣ ⎤⎦ = 0
. (12)
This means that a field in the z-direction will cause magnetization to rotate from the x to
y, and similar results are obtained for fields in the y- and z-directions. Note that these
relationships can be used more easily for numerical evaluation by defining
Sx =
12
0 11 0
⎛
⎝⎜⎞
⎠⎟, Sy =
i2
0 −11 0
⎛
⎝⎜⎞
⎠⎟, Sz =
12
1 00 −1
⎛
⎝⎜⎞
⎠⎟, (13)
which are known as the Pauli matrices, and have the relationships given in (12).
For a multiple-spin system, the density matrix becomes a linear combination of
products of the spin operators. The Hamiltonian will also contain products of the spin
operators. Whereas terms in the Hamiltonian that only have one spin operator represent
external fields applied to the spin system, terms with multiple spin operators represent
25
couplings between spins. For example, S jxSkz represents a field applied to the jth spin in
the x-direction, for which the sign of the field depends on the z component of the kth spin.
Similarly, the kth spin experiences a field in the z-direction, for which the sign of the field
depends on the x component of the jth spin.
1.2.1.2 Polarization and Coherence
Traditionally, in magnetic resonance experiments, we assume that a large, static
magnetic field is applied in the z-direction of the axis frame. The Zeeman Hamiltonian
resulting from this field is given approximately by
Hz = γ j B0S jz
j=1
N
∑ , (14)
where B0 is the size of the field, γ j is the gyromagnetic ratio of the jth spin, and
S jz
indicates that the field is applied in the z-direction. Note that we have used S generically
for all spins in the system; we will later use S and I to indicate electrons and nuclei,
respectively.
Although there are exceptions, this field is usually by far the largest term in the
Hamiltonian of a magnetic resonance experiment. As a result, spins align themselves with
this field, resulting in non-zero expectation values for
S jz when the system is at
thermal equilibrium. We will use the term “polarization” to refer to the alignment of a
spin with the external field, which means that if there is polarization on the jth spin, then
S jz ≠ 0 . Note that polarization does not evolve under (14).
If polarization is rotated away from the external field by 90º, it will then be
aligned in the x- or y- direction. This results in non-zero expectation values for
S jx and
S jy , which are known as single-quantum coherences. This name is used because
observable magnetization in the x- and y- directions results from superpositions of the
spin-up and spin-down states of a spin. Note that these coherences will rotate around the
z-axis under the Hamiltonian in (14). In a many spin system, it is possible to have states
that result from linear combinations of the spin operators, and these are also known as
26
coherences. The number of spins involved in a coherence gives the order of a coherence.
For example, if the expectation value of
S jxSkx is nonzero, then we would say that we
have a 2nd-order coherence. Additionally, high order coherences refer to those states
involving at least three spins.
1.2.1.3 Rotating Frame Transformation
In many cases in magnetic resonance experiments, we will apply an oscillating
field to the spin system. For example for a one-spin system, which is experiencing a static
and an oscillating magnetic field, the total Hamiltonian would be given by
H (t) = 2ω1S cos ω MWt⎡⎣ ⎤⎦Sx +ω0S Sz , (15)
ω0S gives the Larmor frequency resulting from the static field (replacing γ B0 with ω0S ).
An oscillating field with a strength of 2ω1S and a frequency of ω MW is also applied. It is
difficult to evaluate (9) with a time-dependent Hamiltonian, so it is useful to eliminate the
oscillating field. We do this by evaluating the derivative of
exp[−iω MWtSz ]σ (t)exp[iω MWtSz ] .
σ r (t) = exp[−iω MWtSz ]σ (t)exp[iω MWtSz ]ddtσ r (t) = −iω MW Szσ
r (t)+ iω MWσr (t)Sz
− iexp[−iω MWtSz ] H (t),σ (t)⎡⎣ ⎤⎦exp[iω MWtSz ]
= iω MW Sz ,σr (t)⎡⎣ ⎤⎦ − i exp[−iω MWtSz ]H (t)exp[iω MWtSz ],σ
r (t)⎡⎣ ⎤⎦
, (16)
We may define Hr (t) in order to write this in the same form as (9).
H r (t) = exp[−iω MWtSz ]H (t)exp[iω MWtSz ]−ω MW Sz
ddtσ r (t) = −i H r (t),σ r (t)⎡⎣ ⎤⎦
(17)
However, note that we can further rearrange Hr (t) , to obtain
27
H r (t) = exp[−iω MWtSz ]ω0S Sz exp[iω MWtSz ]
exp[−iω MWtSz ]2ω1S cos ω MWt⎡⎣ ⎤⎦Sx exp[iω MWtSz ]
−ω MW Sz
= ω0S −ω MW( )Sz +ω1S Sx +ω1S cos 2ω MWt⎡⎣ ⎤⎦Sx + sin 2ω MWt⎡⎣ ⎤⎦Sy( ) (18)
We note that the terms oscillating at 2ω MW will have little effect on the evolution of the
system, and can be dropped. Then, the remaining terms in Hr (t) are static. Defining
Δω0S =ω0S −ω MW , it can be written as
Hr = Δω0S Sz +ω1S Sx . (19)
In the rotating frame, the Zeeman field is significantly reduced, and additionally
there is now a static field in the x-direction. In this frame, we can see that the applied
field will change the behavior of the system significantly, since the energy levels are
considerably different.
The rotating frame transformation will also be useful in instances where the spin
system is much larger. The treatment is the same, except that nonsecular couplings
(couplings that include fields in the x- and y- directions) may be fast-oscillating in some
cases and static in others, so care must be taken when truncating the Hamiltonian.
1.2.2 DNP Mechanisms Continuous-wave DNP can be categorized into three mechanisms: solid effect
(SE), cross effect (CE), and thermal mixing (TM). We briefly discuss each of these
mechanisms here.
1.2.2.1 Solid Effect
SE is a DNP mechanism that can be understood in basic form using a two-spin,
electron-nuclear system [23-25]. SE is the dominant DNP mechanism when ω0 I < δ ,Δ ,
where ω0 I is the nuclear Larmor frequency, δ is the homogenous EPR linewidth, and Δ
is the inhomogeneous EPR linewidth. The Hamiltonian for the two-spin SE is given in
the rotating frame by
28
H0r = Δω0SSz +ω0 I Iz + ASz Iz + BSz Ix
H MWr =ω1SSx
H r = H0r + H MW
r
. (20)
ω0S and ω0 I are the electron nuclear Larmor frequencies. A and B are the secular and
pseudo-secular couplings between the electron and nucleus. ω1S is the strength of an
applied microwave field, and ω MW is the frequency of the applied microwave field. For
SE, the electron and nucleus must have a pseudo-secular coupling, B, between them.
Without this term, the static Hamiltonian, H0 , is diagonal and therefore no transfer is
allowed between the electron and the nucleus. With B, the nuclear states are mixed, and
the microwave Hamiltonian can then drive an electron-nuclear polarization transfer. This
occurs if microwave irradiation is applied such that ωmw =ω0S ±ω0 I . Under microwave
irradiation, the electron and nucleus undergo a flip-flop, which transfers the electron
polarization to the nucleus. Figure 2 illustrates this process with an energy level diagram.
In Figure 2, energy levels are shown for an electron-nuclear system, where the
nuclear states are separated by a small energy difference, and the electronic states are
Figure 2: Solid Effect Energy Levels and Populations
When a microwave field is applied at the difference of the electron and nuclear Larmor
frequencies, the double-quantum electron-nuclear transition is energy conserving, and so
the electron and nuclear polarizations are equalized, leading to additional nuclear
polarization.
29
separated by a large energy difference. This is accompanied by an initially larger electron
polarization. However when driving the system with a microwave field given by
ωmw =ω0S −ω0 I , the energy levels of the double quantum transition are equalized in
population (saturated). This brings about polarization on the nucleus. Hu et al. provides
an in-depth quantum mechanical description of this process [26].
1.2.2.2 Cross Effect
The cross effect may be understood as a three-spin, electron-electron-nuclear
DNP mechanism [27-31]. CE is the dominant mechanism when δ <ω0 I < Δ . The
Hamiltonian for CE is given in the rotating frame by
H0r =ω0S1
S1z +ω0S2S2z +ω0 I Iz + d 3S1zS2z − S1S2( )
+ A1S1z Iz + A2S2z Iz + B1S1z Ix + B2S2z Ix
H MWr =ω1S S1x + S2x( )
H r = H0r + H MW
r
. (21)
30
ω0S1
, ω0S2
, and ω0 I are the resonance frequencies of the first and second electron and the
nucleus, respectively. d is the electron-electron dipolar coupling, A1 and B1 are the
secular and pseudo-secular couplings between the first electron and the nucleus. A2 and
B2 correspond to the second electron and the nucleus. As before, ω1S and ω MW are the
strength and frequency of the applied microwave field.
Figure 3: Cross Effect Energy Levels and Populations
We show the energy levels for an electron-electron-nuclear system. When microwaves
are applied at the resonance frequency of the first or second electron ( ω0S1
or ω0S2
), that
electron saturates. If the central energy levels are degenerate, as is the case when
±ω0 I =ω0S1
−ω0S2, then the central transition also saturates, leading to nuclear
polarization.
31
In this case, microwave irradiation is applied such that ω MW =ω0S1
, so that the
first electron becomes saturated. Then, if ±ω0 I =ω0S1
−ω0S2, an energy conserving, three-
spin flip-flip-flop mechanism allows the first electron to recover polarization, while
simultaneously polarizing the nucleus. The mechanism is allowed because the couplings
appearing in (21) mix the electronic and nuclear states efficiently, as long as the matching
condition, ±ω0 I =ω0S1
−ω0S2, is satisfied. This is illustrated in Figure 3, where
microwave irradiation is applied on-resonant with either the first or second electron,
saturating the electronic transitions. Then, equilibration of the central energy levels leads
to nuclear polarization. Hu et al. also describes cross effect using a full quantum
mechanical treatment [26].
1.2.2.3 Thermal Mixing
Thermal mixing can be described with a multi-electron system, coupled to a
nucleus or nuclei. In this mechanism, a strongly coupled electron spin system is irradiated
with microwaves [32-35]. Then in the rotating frame, this spin system is cooled. This
reduced spin temperature is subsequently transmitted to nuclear spins, which results in
nuclear polarization enhancement. Because the electron spin system must be strongly
coupled, this mechanism is only dominant when ω0 I < δ ,Δ , where the large
homogeneous linewidth, δ , results from the strong electron couplings.
1.2.3 Many Spin Mechanisms In the preceding discussion, we have briefly described three methods of cw-DNP.
Aside from TM, we discussed each in terms of a few spins. This ignores the fact that
DNP is inherently a many-spin mechanism: Significant enhancements to signal to noise
in ssNMR are a result of each electron polarizing many surrounding nuclei- usually on
the order of 1000 nuclei. This is possible because of several factors. First, the electron
recovers its thermal polarization about 1000-10,000 times faster than a nucleus. Second,
strong electron-nuclear couplings allow direct polarization transfer to many surrounding
nuclei. Third, many nuclear-nuclear couplings allow efficient spin-diffusion, which is
32
polarization transfer between the same type of nuclei. Finally, strongly coupled spin
systems may utilize high order coherence to efficiently transfer polarization. Therefore,
to fully understand DNP mechanisms, theoretical models must account for large numbers
of spins, include relaxation, and use QM methods that allow for high order coherences.
1.3 Solid Effect Studies The goal of this thesis is to improve models of DNP polarization transfer by
developing the theoretical framework to describe DNP mechanisms using a many-spin
system, which accounts for QM and relaxation processes. We focus on the 1H SE DNP
mechanism, and this is done for several reasons: (1) SE can be simplified into a two-spin,
cw-DNP mechanism, which is the simplest of the solid-state DNP mechanisms. (2) SE is
fully microwave driven, and therefore can be started and stopped by turning microwaves
on and off; on the other hand, a sample that is optimized for CE and TM will always have
an active DNP mechanism- turning microwaves on and off only affects saturation of the
electron, but does not change the DNP transfer. (3) SE uses paramagnetic centers with
narrow EPR linewidths, so DNP conditions are spectrally resolved and can therefore be
more easily controlled and probed. The goals and outcomes of the SE investigations are
summarized here.
1.3.1 Chapter 2: A 140 GHz Pulsed EPR/212 MHz NMR Spectrometer for DNP Studies
DNP, by definition, is an electron-nuclear process. Therefore, direct observation
of both electrons and nuclei is essential. Measurement of electron and nuclear relaxation
rates, microwave field strengths, and observation of polarization enhancement on nuclei
and polarization loss on electrons during DNP experiments is necessary for
understanding many DNP processes. This chapter describes a pulsed-EPR spectrometer
coupled with a two-channel NMR spectrometer that we designed and constructed. This
allows measurement of parameters that are relevant to DNP in a single experimental set
up. Basic EPR, ENDOR, and DNP experiments are demonstrated, including a SE DNP
33
enhancement of 144, which is the highest SE enhancement demonstrated at high fields
(>3 T).
1.3.2 Chapter 3: Solid Effect DNP and Polarization Pathways Considerable work has been done to understand transfer of polarization between
like-spins in NMR experiments, a process known as spin diffusion. DNP experiments
rely on spin-diffusion to spread bulk nuclear polarization throughout the sample.
However, spin diffusion rates are attenuated for nuclei near the electron. Since the
electrons are the source of polarization in DNP, this raises the question of what role these
nearby nuclei play in the DNP process. We start from a QM, Louiville space treatment,
and show that with reasonable assumptions, one can model the electron-nuclear DNP
transfer and nuclear spin-diffusion using simple rate equations. With models based on
rate-equations, we show that nuclei nearby to the electron compete with bulk nuclei
polarization, rather than the nearby nuclei helping to transfer polarization to the bulk
nuclei. The implications of this result are discussed, including the role of relaxation rates
in determining the impact of attenuated spin-diffusion.
1.3.3 Chapter 4: Highly Efficient Solid Effect in MAS DNP at High Field
Chapter 3 shows that nearby nuclei hinder the transfer of electron polarization to
bulk nuclei under static DNP conditions. These experiments are replicated under MAS
conditions, and the results are shown to still apply. Additionally, the role of 1H
concentration in DNP enhancement is examined. High 1H concentration accelerates spin
diffusion, but also can reduce signal enhancement because the total polarization must be
shared among more spins. The SE enhancement was measured as a function of 1H
concentration for both the trityl and Gd-DOTA paramagnetic centers. Trityl has a spin-
1/2 paramagnetic center, and a polarization recovery rate of ~1 ms, whereas Gd-DOTA
has a spin-7/2 paramagnetic center, and a polarization recovery rate of ~10 μs. It was
found that 1H spin-diffusion is the rate limiting process for SE-DNP with Gd-DOTA,
whereas the initial electron-nuclear DNP transfer is rate limiting for trityl.
34
1.3.4 Chapter 5: Observation of Strongly Forbidden Transitions via Electron-Detected Solid Effect DNP
Chapter 5 uses detection of electron polarization under SE-DNP conditions to
investigate the SE mechanism. Loss of electron polarization is observed at the usual SE
conditions, ω MW =ω0S ±ω0 I , but additionally at ω MW =ω0S ± nω0 I , where n is an
integer. This indicates that SE is active for both electron-nuclear pairs, but also for one
electron and multiple nuclei. This requires electron-nuclear coherences involving 3 or
more spins to play a major role. A simple QM treatment shows that this process must
involve strong nuclear-nuclear couplings to bring about the observed electron saturation.
This suggests that the nuclear-nuclear couplings also cause high order coherences to play
a role in the n=1 SE condition.
1.3.5 Chapter 6: DNPsim: A Flexible Program for DNP Simulations
In the final chapter, we present MATLAB software capable of simulating a
variety of DNP experiments. Considerable work has been done recently to simulate a
DNP mechanisms. However, the theory underlying those simulations is not addressed in
detail, and the software for replication of those simulations is not available. We discuss
theoretical aspects of simulation, and introduce usage of our simulator. Our program
allows easy setup of basic DNP experiments; additionally, a variety of advanced options
allow a very flexible interface for more complicated experiments and in-depth
examination of the mechanisms. We demonstrate several DNP experiments, and conclude
with a study of the role of high-order coherences in SE DNP. We are able to show that
for a seven spin system (6 nuclei, 1 electron), high order coherences play a significant
role in accelerating the electron-nuclear DNP polarization transfer, but do not make
major contributions to the nuclear-nuclear spin-diffusion rates.
35
1.4 Outlook At present, the field of DNP is in a state of rapid expansion. The availability of a
variety of radicals for solid effect (SA-BDPA, trityl, Gd-DOTA [21; 36; 37]) and cross
effect (TOTAPOL, bTbk, sbTs, bTbtk-py [3-5; 38]) give flexibility in experimental
conditions. New sample types are being used for experiments such as surface DNP [39]
and solvent-free DNP [40], thus moving away from more common glassy samples.
Finally, the emergence of high frequency, tunable gyrotrons at higher fields [41-43] and
availability of commercial spectrometers for DNP at 1H frequencies up to 800 MHz
allows DNP to be used in new experimental regimes.
Intelligent adaptation of methods to new experimental regimes will require a
strong understanding of the underlying mechanisms. This thesis includes the construction
of a versatile EPR/NMR spectrometer, explains models of polarization transfer, explores
the role of high order coherence, and presents DNP simulation software. These tools will
allow future researchers to have the framework with which to understand why certain
DNP experiments work, and will guide the development of new DNP methods.
1.5 Bibliography [1] T.R. Carver, and C.P. Slichter, Physical Review 92 (1953) 212. [2] A.W. Overhauser, Phys. Rev. 92 (1953) 411. [3] C. Song, K.-N. Hu, C.-G. Joo, T.M. Swager, and R.G. Griffin, J. Am Chem. Soc 128
(2006) 11385-90. [4] Y. Matsuki, T. Maly, O. Ouari, H. Karoui, F. Le Moigne, E. Rizzato, S. Lyubenova, J.
Herzfeld, T.F. Prisner, P. Tordo, and R.G. Griffin, Angewandte Chemie 48 (2009) 4996-5000.
[5] M.K. Kiesewetter, B. Corzilius, A.A. Smith, R.G. Griffin, and T.M. Swager, J. Am. Chem. Soc. 134 (2012).
[6] T. Maly, A.-F. Miller, and R.G. Griffin, ChemPhysChem 11 (2010) 999-1001. [7] R.R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of nuclear magnetic
resonance in one and two dimensions, Clarendon, Oxford, 1987. [8] E.R. Andrew, A. Bradbury, and R.G. Eades, Nature 183 (1959) 1802-1803. [9] J. Herzfeld, and A.E. Berger, Journal of Chemical Physics 73 (1980) 6021-6030. [10] A.E. Bennett, C.M. Rienstra, M. Auger, K.V. Lakshmi, and R.G. Griffin, The
Journal of Chemical Physics 103 (1995) 6951-6958. [11] B.M. Fung, A.K. Khitrin, and K. Ermolaev, J. Magn. Reson. 142 (2000) 97-101. [12] A. Pines, M.G. Gibby, and J.S. Waugh, J. Chem. Phys. 56 (1972) 1776-1777. [13] T. Gullion, and J. Schaefer, Journal of Magnetic Resonance 81 (1989) 196-200.
36
[14] A.W. Hing, S. Vega, and J. Schaefer, Journal of Magnetic Resonance A 103 (1993) 151-162.
[15] G. De Paepe, J. Lewandowski, A. Loquet, A. Bockmann, and R.G. Griffin, Journal of Chemical Physics 129 (2008).
[16] J. Lewandowski, G. De Paepe, and R. Griffin, J Am Chem Soc 129 (2007) 728-9. [17] A.D. Milov, A.B. Ponomarev, and Y.D. Tsvetkov, Chemical Physics Letters 110
(1984) 67-72. [18] K. Yokoyama, A.A. Smith, B. Corzilius, R.G. Griffin, and J. Stubbe, J. Am. Chem.
Soc. 133 (2011) 18420-18432. [19] W.B. Mims, Proc. R. Soc. Lond. A. 283 (1965) 452-457. [20] E.R. Davies, Physics Letters A 47A (1974) 1-2. [21] B. Corzilius, A.A. Smith, A.B. Barnes, C. Luchinat, I. Bertini, and R.G. Griffin,
Journal of the American Chemical Society 133 (2011) 5648-5651. [22] M.H. Levitt, D. Suter, and R.R. Ernst, The Journal of Chemical Physics The Journal
of Chemical Physics J. Chem. Phys. 84 (1986) 4243-4255. [23] A. Abragam, and W. Proctor, G., C. R. Acad. Sci. 246 (1958) 2253. [24] C. Jeffries, D., Physical Review 106 (1957) 164. [25] C.D. Jeffries, Physical Review Phys. Rev. PR 117 (1960) 1056. [26] K.-N. Hu, G.T. Debelouchina, A.A. Smith, and R.G. Griffin, Journal of Chemical
Physics 134 (2011). [27] A. Kessenikh, V., V. Lushchikov, I., A. Manenkov, A., and Y. Taran, V., Soviet
Physics - Solid State 5 (1963) 321-329. [28] A. Kessenikh, V., A. Manenkov, A., and G. Pyatnitskii, I., Soviet Physics - Solid
State 6 (1964) 641-643. [29] C. Hwang, F., and D. Hill, A., Physical Review Letters 18 (1967) 110. [30] C. Hwang, F., and D. Hill, A., Physical Review Letters 19 (1967) 1011. [31] D. Wollan, S., Physical Review B: Condensed Matter 13 (1976) 3671. [32] R.A. Wind, M.J. Duijvestijn, d.L. van, C., A. Manenschijn, and J. Vriend, Progress
in Nuclear Magnetic Resonance Spectroscopy 17 (1985) 33-67. [33] M. Goldman, Spin temperature and nuclear magnetic resonance in solids, Clarendon
Press, Oxford, 1970. [34] M. Duijvestijn, J., R. Wind, A., and J. Smidt, Physica B+C 138 (1986) 147-170. [35] W. Wenckebach, Th, T. Swanenburg, J. B., and N. Poulis, J., Physics Reports 14
(1974) 181-255. [36] O. Haze, B. Corzilius, A.A. Smith, R.G. Griffin, and T.M. Swager, J. Am. Chem.
Soc. In Preparation (2012). [37] J. Ardenkjaer-Larsen, I. Laursen, I. Leunbach, G. Ehnholm, L. Wistrand, J.
Petersson, and K. Golman, J Magn Reson 133 (1998) 1-12. [38] E.L. Dane, B. Corzilius, E. Rizzato, P. Stocker, T. Maly, A.A. Smith, R.G. Griffin,
O. Ouari, P. Tordo, and T.M. Swager, J. Org. Chem. 75 (2012) 3533-3536. [39] A. Lesage, M. Lelli, D. Gajan, M.A. Caporini, V. Vitzthum, P. Mieville, J. Alauzun,
A. Roussey, C. Thieuleux, A. Mehdi, G. Bodenhausen, C. Coperet, and L. Emsley, Journal of the American Chemical Society 132 (2010) 15459-15461.
[40] V. Vitzthum, F. Borcard, S. Jannin, M. Morin, P. Miéville, M.A. Caporini, A. Sienkiewicz, S. Gerber-Lemaire, and G. Bodenhausen, ChemPhysChem 12 (2011) 2929-2932.
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[41] V.S. Bajaj, M.K. Hornstein, K.E. Kreischer, J.R. Sirigiri, P.P. Woskov, M.L. Mak-Jurkauskas, J. Herzfeld, R.J. Temkin, and R.G. Griffin, Journal of Magnetic Resonance 190 (2007) 86-114.
[42] M. Hornstein, V. Bajaj, R. Griffin, and R. Temkin, IEEE Transactions on Plasma Science 34 (2006) 524-533.
[43] A.C. Torrezan, S.-T. Han, I. Mastovsky, M.A. Shapiro, J.R. Sirigiri, R.J. Temkin, A.B. Barnes, and R.G. Griffin, IEEE Transactions on Plasma Science (2010).
38
Chapter 2
A 140 GHz Pulsed EPR / 212 MHz NMR Spectrometer for DNP Studies
Contributing: Björn Corzilius, Jeffrey Bryant, Ron DeRocher, Paul
Woskov, Rick Temkin
39
Abstract A versatile spectrometer designed for the study of Dynamic Nuclear Polarization
at low temperatures and high fields is described. The spectrometer functions both as an
NMR spectrometer operating at 212 MHz (1H frequency) with DNP capabilities, and as a
pulsed-EPR spectrometer operating at 140 GHz. A coiled TE011 resonator acts as both an
NMR coil and microwave resonator, and a double balanced (1H, 13C) radio frequency
circuit greatly stabilizes the NMR performance. A new 140 GHz microwave bridge has
also been developed, which utilizes a four-phase network and ELDOR channel at 8.75
GHz, that is then multiplied and mixed to obtain 140 GHz microwave pulses with an
output power of 120 mW. Nutation frequencies obtained are as follows: 6 MHz on S = ½
electron spins, 100 kHz on 1H, and 50 kHz on 13C. We demonstrate basic EPR, ELDOR,
ENDOR, and DNP experiments here. The DNP results include a solid effect
enhancement of 144 and sensitivity gain of 310, and a cross effect enhancement of 118.
40
2.1 Motivation Dynamic Nuclear Polarization (DNP) is a method of enhancing signals in nuclear
magnetic resonance (NMR) experiments, by transferring the large spin-polarization of
paramagnetic electrons to the surrounding nuclear spins [1-5]. The DNP phenomenon
was initially postulated and demonstrated in the 1950s [6; 7], but because DNP efficiency
scales unfavorably with increases in the magnetic field, it had not seen much use at high
fields (>5T) until recently. However, when the gyrotron was introduced as a means of
providing high microwave field strengths, DNP at high magnetic field and under magic
angle spinning (MAS) conditions became a viable experiment [8-10]. Biradical
polarizing agents have further contributed to the efficiency of cross-effect DNP at high
magnetic fields [11-13]. Our recent experiments have shown that the solid effect can also
be efficient at high field [14; 15] and can be performed with high-spin paramagnetic
metal centers [16]. With these advances, DNP has seen a variety of applications in solid-
state NMR [17-21], and additional advances in liquid-state DNP [22; 23] and dissolution
DNP [24-26] have also expanded the scope of DNP applications.
Although much work has been done to understand the DNP mechanism, a
complete model of the process is only possible with the characterization of the electron
and the nuclear spins, in addition to accurate measurement of experimental parameters.
This requires measurement of electron and nuclear spin-lattice (T1) and spin-spin (T2)
relaxation times, the electron nutation frequency and the EPR lineshape. It can also be
very useful to know the frequency or field dependence of both the nuclear polarization
enhancement, and the electron polarization depletion.
Additionally, there has been interest in developing pulsed-DNP techniques,
including polarization transfers performed in the rotating frame, and experiments that
increase the excitation bandwidth. The dressed-state solid effect (DSSE) and nuclear
spin-orientation via electron spin locking (NOVEL) both eliminate the unfavorable field
dependence of laboratory-frame DNP techniques such as cross effect and solid effect, by
using electron spin-locking to perform transfers in the rotating frame [27; 28]. However,
calibration of parameters to optimize both NOVEL and DSSE require measurement of
the electronic nutation frequency, and can benefit from indirect measurement of DNP via
41
electron detection. However, to determine the effectiveness of each method for enhancing
bulk nuclear polarization, one must use direct observation via NMR detection. Another
approach to improving DNP efficiency via pulsed methods exchanges continuous wave
(cw) microwave irradiation for short, strong pulses as a means of increasing the
bandwidth of a DNP experiment without increasing average power [29; 30].
A pulsed-EPR spectrometer integrated with an NMR spectrometer is the ideal
instrument for in-depth studies of DNP mechanisms and implementation of pulsed-DNP:
EPR detection capabilities allow characterization of the paramagnetic electrons including
measurement of the EPR spectrum as well as T1 and T2 of the electron. Also, direct
measurement of the electron nutation frequency during DNP experiments is only possible
with EPR detection. Multiple EPR channels with different relative phases allow for spin
locking of the electrons, which is required for DSSE and NOVEL. A frequency-
sweepable electron-electron double resonance (ELDOR) channel can allow for indirect
measurement of DNP conditions, and provide additional flexibility when performing
DNP experiments. Finally, the NMR spectrometer allows direct measurement of the
nuclear T1, the DNP buildup time (TB), and the nuclear enhancement (ε∞); the knowledge
of those parameters is essential to DNP studies.
Several DNP spectrometers equipped with EPR detection capabilities have been
recently described in the literature. The groups of Vega and Goldfarb have developed an
EPR/DNP system operating at 95 GHz and have recently shown DNP enhancements and
also double resonance experiments detecting DNP via EPR observation [31; 32]. The
Prisner group demonstrated liquid-state DNP, also using an EPR/DNP system [22; 33].
The Köckenberger group has also implemented an EPR setup for use in conjunction with
a dissolution DNP experiment [34]. Finally, the HIPER EPR system in the Smith lab,
which is designed around a ~1 kW extended interaction klystron (EIK), has been used to
show DNP enhancements via pulsed cross effect [30].
We describe a 140 GHz/212 MHz EPR/DNP spectrometer designed for the study
of DNP mechanisms. Our system is unique in its flexibility for performing EPR and DNP
experiments in the solid-state, and shows superior performance in obtaining DNP
enhancements in a static sample. Preliminary results show EPR, ELDOR, and electron-
nuclear double resonance (ENDOR) data, and also DNP enhancements observed via
42
NMR. We obtain an enhancement of 144 and sensitivity gain of 310 under solid effect
conditions using OX063 trityl, where the microwave field strength is critical to
performance. This is the highest and enhancement and sensitivity gain to date reported
using 1H solid effect at high fields (>3T). We also achieve an enhancement of 118 using
TOTAPOL [12]. We discuss the reasons that this enhancement is lower than
enhancements obtained with TOTAPOL under magic angle spinning conditions, but also
note that our enhancement is obtained with only a fraction of the microwave power
available with our system.
2.2 Instrument Design The 140 GHz EPR/212 MHz NMR can be grouped into a few systems: the
magnet and field control, temperature control, the EPR spectrometer, the NMR
spectrometer, and the DNP probe. Of these systems, the EPR spectrometer and the DNP
probe have undergone major changes that will be described in detail here, in addition to a
description of the spectrometer control. We briefly describe the magnet and field control,
temperature control, and NMR spectrometer before detailing the novel components of the
spectrometer.
The 140 GHz EPR/212 MHz NMR spectrometer is based around a Magnex 5
T/130 mm bore magnet with a ±0.4 T superconducting sweep coil. An NMR probe
containing a small water sample sits just below the sample space in the magnet, and is
used in conjunction with a Resonance Research field-mapping unit (FMU). This system
both measures the magnetic field and sweeps it to the desired position, as described in
detail by Maly, et al. [35]. An Oxford SpectrostatCF cryostat and ITC502 temperature
controller allow precise temperature regulation down to 1.4 K. A 2-channel RNMR
console (courtesy of David Ruben) allows flexible radio frequency (RF) pulse creation
and NMR detection.
2.2.1 EPR Spectrometer: 140 GHz EPR Bridge We have built a five-channel (four phases and 1 sweepable ELDOR channel) EPR
bridge operating at 140 GHz with 120 mW of power. The EPR bridge can be broken into
43
three major sections as shown in Figure 1: microwave pulse generation (red), which
includes the five channel network, signal down-mixing (blue), and quadrature detection
(green). We describe each section, and also discuss phase-locking of the three sections,
which is essential for EPR signal detection.
Pulse generation is based around a Virginia Diodes Inc. active multiplier chain
(AMC), which requires either pulsed or cw input at 34.5-35.5 GHz and multiplies that
input frequency by 4 to give 120 mW of power at 138-142 GHz. Therefore we generate
phase-controlled pulses at 35 GHz to feed into the AMC. Microwaves are generated at
8.75 GHz via multiplication of a 2.1875 GHz source by 4. The microwave power at 8.75
GHz is split: half is directed into the five-channel network where microwave pulses are
generated and a quarter is fed into a ×3 multiplier giving 26.25 GHz (the last quarter is
used as reference frequency for quadrature detection). Pulses from the output of the five-
channel network and cw microwaves from the ×3 multiplier are mixed together,
providing pulses at 35 GHz that can be fed into the AMC. We have designed the
generation of microwave pulses at 35 GHz to minimize phase error. In our configuration,
the phase error at 35 GHz is the same as the phase error at 8.75 GHz, because the mixing
step (26.25 GHz + 8.75 GHz) does not introduce any new error. Although it would be
simpler to create pulses at 8.75 GHz and use a ×4 multiplier to create 35 GHz pulses, this
would add an additional factor of 4 in the phase error.
44
The five-channel network consists of four phases and one sweepable ELDOR
channel. 8.75 GHz microwaves are fed into the four channels, where they are first
attenuated, and then phase shifted to the appropriate phase (usually 0º, 90º, 180º, 270º).
One may also use the channels for pulses with different power levels in the same
experiment. Finally, pulses are generated using a double-balanced mixer (Marki M1-
0412), by feeding an attenuated transistor-transistor logic (TTL) pulse from a pulse
programmer into the intermediate frequency port of the mixer. We use mixers to create a
~φ dB
x3
x4To Probe
900
φ
8.75 GHz
8.75 GHz
140 GHz
26.25 GHz
35 GHz
x4
x5 x12
2.1875 GHz
10.9375 GHz 131.25 GHz
dB
φ
φ
φ
φ
Pulse Programmer
~8.3-9.3 GHz
RE IM
dB
dB
dB
dB
dB
dB
dB
To Pulse Programmer
Figure 1: 140 GHz EPR Bridge Design
Microwave pulses are generated at 8.75 GHz (8.25 GHz-9.25 GHz for ELDOR channel),
and multiplied and mixed to 140 GHz for 120 mW pulses. Signal is referenced to 131.25
GHz, and detected using a heterodyne quadrature detector operating at 8.75 GHz which
down-converts signal for detection. Red indicates the pulse generation, blue indicates
down-mixing, and green indicates the quadrature detection.
45
very fast, but cheap microwave switch. On its own, a mixer makes a very poor
microwave switch, because it is prone to a large amount of power leakage. However,
when the output of the mixer is fed into a multiplier, as done with the AMC in our
system, the nonlinear behavior of the multiplier cuts off any low power leaking through
the mixer in the off-state [36]. In fact, it is possible to generate sub-nanosecond pulses
with this method. Although not yet implemented, this EPR bridge is designed to drive a
gyro-traveling-wave-tube-amplifier with 40 dB gain, making very short pulses necessary
to produce π/2 pulses with the high microwave powers that would be available [37; 38].
One should note that because the ×4 multiplier only cuts off microwave power if the
power is below some threshold, when one channel is activated, power from the other
channels will leak through the multiplier. This is remedied by having the channels paired
such that if one channel of the pair has a given power level and phase, the second channel
of the pair has the same power level and an 180º phase shift. As a result, the power
leakage is cancelled out for the four phases.
The ELDOR channel of the five-channel network is a voltage controlled oscillator
(VCO) operating between 8.25 GHz and 9.25 GHz, to give a 4 GHz sweep range when
multiplied to 140 GHz. Like the other channels of the five-channel network, this channel
is first fed into an attenuator, and gated with a mixer. As mentioned above, when one of
the other channels is activated, this will allow the ELDOR channel to leak through. Since
there is no channel with opposite phase of the ELDOR channel, we must use a standard
microwave switch in addition to the mixer on this channel to avoid power leakage when
other channels are in use. Short pulses can still be achieved by opening the microwave
switch before the mixer, as long as no other channel is activated during that period.
Down-mixing of the EPR signal to 8.75 GHz is achieved with a biasable mixer
(Millitech MXB-08) that mixes the EPR signal at 140 GHz with a reference frequency of
131.25 GHz. This allows for heterodyne quadrature detection at 8.75 GHz. In the
reference arm, multiplying the 2.1875 GHz source by 5 generates microwaves at 10.9375
GHz. 131.25 GHz microwaves are subsequently generated from a ×12 AMC (Millitech
AMC-08), which is fed into the biasable mixer.
Output from the biasable mixer is amplified and fed into the quadrature detection.
The EPR signal is split and enters two mixers. The reference signal for the two mixers is
46
obtained by taking cw 8.75 GHz microwaves from the ×4 multiplier in the pulse
generation arm. This is sent through a variable phase shifter, and then a 90º-hybrid power
splitter. The 90º phase shift between the two mixers allows one to obtain the real and
imaginary signals from the two mixers. The output from the mixers is amplified, and
finally the down-mixed signal enters an oscilloscope.
The phase of the EPR signal will be random if the reference frequencies of the
pulse generation, down-mixing, and quadrature arms are not phase-locked. Therefore
phase-locking is essential, and is achieved by generating all reference frequencies from
the same source. In our case, this is the 2.1875 GHz source. The pulse generation and
quadrature arms both require 8.75 GHz input, which is generated by ×4 multiplication of
the source. The down-mixing arm then requires 2.1875 GHz input, which is obtained
directly from the source. Therefore, the total multiplication factor for each bridge of the
arm is as follows: 64 × 2.1875 GHz for the pulse generation arm, 60 × 2.1875 GHz for
the down-mixing arm, and 4 × 2.1875 GHz for the quadrature arm. One may note that a
simpler scheme would be to use an 8.75 GHz source and factors of 16, 15, and 1,
respectively. However, this would require a ×5 multiplier operating with at least 43.75
GHz output frequency. The limited availability of such multipliers made it simpler to
start at lower frequency, and use a ×5 multiplier with 10.9375 GHz output.
2.2.2 EPR Spectrometer: Control and Detection In order to have precise control over the EPR spectrometer, we have implemented
SpecMan4EPR with much help from Boris Epel [39]. SpecMan allows for simple
integration of a variety of components by providing drivers for many different
components, and provides a convenient software interface to allow for coordination of all
components. We have used an AWG1000 (Chase Scientific) as a pulse programmer. The
AWG is a 12-bit arbitrary waveform generator, which has the waveform generation itself
replaced with 12 positive emitter-coupler logic (PECL) channels, which have a 1 ns pulse
resolution. The AWG triggers the five pulse channels (plus an additional switch on the
ELDOR channel), a protection switch on the receiver, detection with an oscilloscope, and
can also trigger RNMR in order to initiate an NMR pulse sequence. SpecMan allows the
47
AWG1000 to initiate a pulse sequence on either an internal trigger or an external trigger,
so that it is possible to perform an experiment on the EPR that begins concurrently with
an NMR experiment.
For detection, we use a WaveRunner 6200 (LeCroy) oscilloscope that can detect
frequencies up to 2 GHz, with a sampling rate of 10 Giga-samples per second. This is
also interfaced with SpecMan, and allows for a variety of detection modes, including
acquisition of the full waveform, Fourier transform of the waveform, and echo
integration. Finally, a TCP/IP driver in SpecMan allows us to pass parameters to
LabView programs, and the driver waits for a response from the program confirming that
the parameter has been set. We use this to communicate with the field controller, to set
the ENDOR and ELDOR frequencies, and to set an attenuator which is built into the 140
GHz AMC (the ELDOR VCO and AMC attenuator both require voltage inputs, for which
we communicate directly with a 0-5 V digital to analog converter that is attached to the
VCO and AMC).
2.2.3 DNP Probe In order to perform efficient DNP experiments with low power microwaves (120
mW), and have good radio frequency (RF) performance, it is essential to have both a
high-Q EPR resonator and an efficient NMR circuit. We use a coiled TE011 cylindrical
resonator for this purpose, which was first proposed and used for 9 GHz ENDOR
experiments [40; 41], and later implemented at 140 GHz by Weis, et al. [42]. The coiled
TE011 resonator, which is shown in Figure 2b, is a fundamental mode microwave
resonator- therefore having the full microwave field concentrated at a single node and
giving higher microwave field strengths and filling factors than higher mode resonators
(the field strengths will still depend on the coupling and loss of the resonator). At the
same time the sample is situated around a node of the electric field component, reducing
dielectric losses and sample heating. Because the resonator is also a solenoid shaped coil,
it can also act as an RF resonator when attached to a tuning circuit.
48
Figure 2: DNP Probe Design
(a) Full probe and removable circuit-box. (b) Coiled TE011 resonator that serves as both a
microwave resonator and an RF coil. (c) Resonator with microwave tuning plungers,
sample space, microwave entry, and RF connections.
49
Although this resonator has been in use in our lab for some time, the RF circuit
had inherent instability due to the proximity of the center of the RF coil to the waveguide,
as can be seen in Figure 2c where the waveguide meets the resonator iris. The waveguide
must either be very close to the coil, essentially acting as a capacitor, which makes it very
prone to arcing, or the waveguide must touch the coil, essentially grounding the coil and
causing a very inhomogeneous RF field. However, if the circuit is tuned so that the center
of the coil has the same electric potential as ground (a virtual ground), then both arcing
and RF inhomogeneity can be prevented. This may be achieved with a balanced RF
circuit [43]. Denysenkov and co-workers have demonstrated a one-channel balanced,
lumped-element circuit using the coiled TE011 resonator at 400 MHz/260 GHz in a liquid-
DNP probe [33]. We have taken a similar approach. However, for a solid-state DNP
probe, it is beneficial to have both a 1H channel and a 13C channel so that one may
polarize 1H and transfer that polarization via cross-polarization to 13C where there is
significantly less background signal [44]. Additionally, because the probe must operate at
low temperature, a transmission line circuit is used rather than a lumped-element circuit.
For this purpose, we have designed a double balanced (1H, 13C) transmission line circuit
to use in conjunction with the coiled TE011 resonator.
The circuit design is shown in Figure 3. For tuning and matching of both
channels, we use a parallel tune/series match configuration, and additionally use a
variable capacitor to ground on the opposite side of each channel to allow for balancing
of the circuit. For simple tuning of the circuit, we make the total transmission line length
on each side of the 1H channel equal to the 1H wavelength. Conveniently, for a Teflon
dielectric transmission line, this is ~1 m, which is approximately the length of the probe
itself. However, for the 13C channel, we add an additional 1 m of transmission line on
each side of the circuit (inside the circuit box), so that the total length on each side is
equal to half the 13C wavelength. This causes the circuit to behave as a lumped element
circuit, because half-integer wavelength transmission lines act similarly to electrical
shorts [45]. Isolation of the channels is achieved using 1H traps on the 13C channel and 13C traps on the 1H channel.
50
The full probe is shown in Figure 2a. We have used three metal rods for support
of the probe, with brass disks providing electrical contact between the three rods, the
waveguide, and the outer conductor of the transmission line. This was necessary to
establish a good ground on the outside of the transmission line. Without the use of the
brass disks to connect the transmission line to the waveguide and support rods, there are
significant standing waves on the outer conductor of the transmission line. The presence
of standing waves significantly changes the behavior of the circuit- making it highly
unstable and also difficult to model. Therefore, additional grounding is essential.
The upper half of the support rods are of stainless steel in order to reduce heat
conductivity at the top of the probe. However, even so-called non-magnetic stainless steel
may become significantly magnetic upon extended field- and thermocycling, so the lower
half of the rods are made of brass at the bottom. Therefore, magnetization of the stainless
steel will not affect the magnetic field homogeneity at the sample. A fourth rod made of
Figure 3: Balanced RF Circuit Design
A balanced NMR probe is achieved by having symmetric circuitry between the tune and
balance capacitors on each channel, placing a voltage node at the center of the resonator
coil. Polyflon variable capacitors are used. Transmission lines are 1 m in length with a
Teflon dielectric (1 m≈λ1H≈λ13C/4).
51
G10 composite material can be rotated from outside the probe, and is used to move one
plunger in and out of the cavity for microwave tuning. We also note that the tuning box
seen in Figure 2a is connected to the probe via N-type elbow joints that can be
disconnected from the probe itself. This allows one to easily remove the top-loading
probe from the magnet when changing samples. Also, the use of different circuit boxes
for different nuclei with the same probe is possible.
2.2.4 Full System Control In Figure 4, we show the full spectrometer configuration. The computer labeled
SpecMan controls the EPR. It is connected internally to the pulse programmer, which in
turn is connected to the EPR bridge. USB connectors attach to a digital-to-analog
converter that provides voltages for the AMC attenuator and ELDOR VCO. The
WaveRunner oscilloscope and the FMU communicate via TCP/IP with the SpecMan
computer, all being connected to an Ethernet network. Finally, the magnet power supply
and a water probe are connected to the FMU. The computer labeled RNMR is connected
to the RNMR console, which includes the pulse programmer, 1H and 13C channels, and
the receiver. The RNMR console connects to the probe (filters and directional couplers
are not shown in Figure 4), and an active duplexer (Warner Harrison) separates the RF
pulses on the 13C channel from the 13C NMR signal.
For DNP experiments, it is necessary to synchronize the EPR and NMR
spectrometers. In our configuration, the computers that control the EPR and NMR do not
communicate directly. Although experiments are queued on separate computers, the
experiments can be coordinated with triggers between the AWG, which acts as the EPR
pulse programmer, and RNMR, which controls the NMR pulse programmer. We show
the full spectrometer configuration in Figure 4. Note that an external trigger for each
pulse programmer is attached to a channel of the other pulse programmer. We describe
how this configuration may be used to execute pulsed-DNP experiments, ENDOR
experiments, and cw-DNP experiments for which electron and nuclear polarization is
observed simultaneously.
52
In order to do a pulsed-DNP experiment, one must execute the EPR pulse
sequence repeatedly to achieve significant buildup of nuclear polarization. Because the
EPR sequence requires precise (nanosecond) timing, it is necessary to use a high-
resolution pulse-programmer such as the AWG that we use for the EPR. However, the
AWG does not have enough memory to store a seconds-long buildup with nanosecond
resolution. Therefore, we load one or several instances of the pulse sequence onto the
AWG, but then use repeated triggering of the AWG via the RNMR pulse programmer to
Figure 4: Full Spectrometer Configuration
Separate computers and pulse programmers control EPR and NMR spectrometers.
However, both pulse programmers can be externally triggered, allowing simple
configuration of ENDOR and pulsed-DNP experiments, as well as simultaneous EPR and
NMR detection.
53
re-execute the sequence many times, allowing for an extended, seconds long buildup
period, with nanosecond resolution for the EPR pulses.
External triggering of the RNMR pulse programmer allows for simple setup of
ENDOR experiments. ENDOR requires that an RF pulse be inserted between microwave
pulses. RNMR can in fact be triggered both on rising and falling pulses, which allows us
to start and stop an RF pulse with the rising and falling edges of a single trigger from the
AWG. As a result, not only can the RF pulse be initiated by the EPR spectrometer, but its
length may also be controlled without adjustment of parameters in RNMR.
Finally, we may use external triggering of the AWG to measure the electron
polarization after a DNP experiment, as is shown in Figure 5, where both the electron
polarization and the nuclear polarization are measured together at the end of the
experiment. In this case, RNMR executes a DNP experiment, and at the conclusion of the
Figure 5: Simultaneous EPR and NMR Detection
Pulse sequence to perform DNP transfer and detect e- and 1H (detected on 13C)
polarizations after the DNP polarization period. We use a CP transfer to 13C to measure
the 1H polarization, and use a Hahn echo to measure the e- polarization.
54
experiment, initiates cross polarization (CP) to measure the nuclear polarization, while
simultaneously triggering a Hahn echo sequence to measure the electron polarization.
2.3 Experimental Results To demonstrate the capabilities of our EPR/DNP spectrometer, we will show the
results of several typical EPR and DNP experiments. First, we briefly discuss the mw and
rf performance of the instrument. 1H and 13C nutation frequencies were determined by
measuring transverse 13C magnetization via a CP experiment. The 1H nutation frequency
was measured by incrementing the length of the initial flip pulse on the 1H channel before
CP. In order to measure the 13C nutation frequency, a pulse on the 13C channel was
applied immediately after the CP period. The pulse was shifted 90º in phase from the spin
lock. Therefore, the length of the pulse could be incremented in order to measure the 13C
nutation frequency. Finally, a Hahn echo was applied to the electrons, where the initial
pulse was incremented to obtain the e- nutation frequency. The obtained frequencies were
100 kHz on 1H, 50 kHz on 13C, and 6 MHz on e-. The 6 MHz nutation frequency for
electrons is significantly higher than we have obtained previously [42], and leads to
excellent DNP performance, as we will show.
In Figure 6, we show several EPR experiments. Figure 6a shows an EPR
spectrum of bTbtk-py (0.7 mM in 60:40 glycerol:D2O), a water-soluble analogue of the
bTbk biradical [11; 46]. This was acquired with a Hahn echo, where the timing of the
π/2–τ–π pulse sequence is 55 ns–200 ns–110 ns followed by detection via integration of
the Hahn echo. 641 field points were acquired by repeating the pulse sequence 400 times
at each field point, using a 4 step phase cycle, with a repetition time of 5 ms. The
derivative spectrum in Figure 6a was calculated with the EasySpin fieldmod function,
with modulation amplitude of 1 mT [47]. Note that we can use a short delay between
pulses by phase cycling in order to eliminate any ring-down or free-induction decay that
would otherwise interfere with acquisition of the Hahn echo.
In Figure 6b, we show an ELDOR hole burning experiment on 40 mM trityl in
60:40 glycerol:D2O (a typical solid effect DNP sample). We apply a saturating pulse to
the center of the spectrum with the ELDOR channel, wait 20 μs in order to eliminate any
55
residual coherence, and then apply a Hahn echo pulse sequence (40 ns–400 ns–80 ns).
The integrated intensity of the Hahn echo is recorded for each field position. In order to
acquire this spectrum, the magnetic field and ELDOR frequency are swept together so
that the ELDOR frequency remains on resonance with the center of the trityl spectrum.
Note that after 5 ms, the trityl line is almost fully saturated. This is an important
observation: when measuring the T1 of electrons in a highly concentrated sample, such as
is necessary for DNP, the recovery time is a composite of electron-electron spin-
diffusion, spectral diffusion, and electron spin-lattice relaxation (T1). However, if it is
possible to saturate the full EPR spectrum, then the contribution from electron-electron
spin diffusion and spectral diffusion is greatly diminished, giving an accurate
measurement of the T1.
In Figure 6c, we show a Mims ENDOR of BDPA doped in polystyrene [48]. The
pulse sequence for Mims ENDOR is π/2–τ–π/2–T–π/2, which causes a stimulated echo at
a time τ after the final π/2 pulse. However, during the period T, an RF π-pulse is applied
to the 1H channel, which affects the refocusing of the stimulated echo. The timing of our
sequence was 45 ns–200 ns–45 ns–7.1 μs–45 ns, with a 7 μs RF pulse during the period
T. The magnetic field was set to 4994.5 mT, putting the center of the BDPA spectrum on
resonant with the 140 GHz source. The RF frequency was swept from 207.654 MHz to
217.654 MHz, with 181 frequency points recorded. 1600 shots were taken at each point,
using a 4-step phase cycle. By decreasing the matching of the RF circuit so that the
reflected power was similar across the full 10 MHz sweep width, we were able to record
the full ENDOR spectrum without any retuning of the circuit. Some optimization of the
tuning was required to obtain a symmetric ENDOR spectrum. Note that it is possible to
obtain a 5 μs π-pulse for narrower frequency sweeps, since one may use better matching
of the RF circuit.
56
4991 4993
tELDOR
0 μs79 μs
630 μs5000 μs
B0 [mT]4995 4997
4980 4990 5000B0 [mT]
(a)
(b)
ωRF/2π [MHz]
(c)
209 211 213 215 217
Figure 6: EPR Experiments
(a) Absorption and pseudo-modulated spectrum of bTbtk-py, acquired with a Hahn echo
(pseudo-modulated spectrum calculated with EasySpin[47]). (b) Field/frequency swept
ELDOR hole-burning experiment of 40 mM trityl in 60:40 glycerol:D2O. (c) Mims
ENDOR of BDPA in polystyrene.
57
In Figure 7, we show a frequency swept enhancement profile of 40 mM trityl in
60:40 13C-glycerol:D2O acquired at 80 K, where we polarize 1H for 10 s and observe the
enhancement via CP on 13C. The frequency given is the offset from 140 GHz, which is on
resonance with the center of the EPR spectrum (B0=4993.5 mT). We retune the
microwave cavity at each frequency position to obtain the optimal electron nutation
frequency, and acquire 8 shots at each frequency point. DNP enhancement is observed at
the DNP+ and DNP– matching conditions. Note that at each matching condition, the
shape of the frequency profile is very similar to the trityl frequency swept spectrum
−200 0 200μ-wave offset [MHz]
DNP+
DNP-
Figure 7: Solid effect DNP Enhancement Profile
DNP enhancement profile of 40 mM trityl in 60:40 13C glycerol:D2O that was obtained
by sweeping the microwave frequency rather than the magnetic field. The EPR spectrum
shown above the enhancement profile is a simulation of a frequency-swept spectrum of a
trityl. Note that the EPR cavity was retuned at each frequency.
58
(Figure 7, top). The maximum enhancement occurs at -212 MHz (DNP+). In Figure 8a,
we acquire a spectrum at the maximum enhancement with (on) and without (off)
microwave irradiation. Both on and off signals were acquired with a 10 s recycle delay,
and the electron nutation frequency was ~6 MHz for the on signal. The on signal is
acquired with 32 shots, and the off signal with 6464 shots. We also measure the DNP
buildup time ( TB ) by incrementing the polarization period, which is shown in Figure 8b.
Finally, the 1H is measured by irradiating the sample for 10 s, followed by a period
without irradiation, during which 1H T1 relaxation occurs. This is shown in Figure 8c. If
the on and off signals are compared directly, the on signal is found to be 350 times larger
than the off signal. However, both on and off signal were acquired with a 10 s recycle
delay, whereas the recovery times of the on and off signal are different. The off signal
recovers with the 1H T1 (21.7 s) and the on signal recovers with the DNP buildup time
(4.4 s); we must account for this difference in recovery time. We do so by applying a
scaling factor to the intensity of each spectrum. The scaling factor is given by
, where is the recycle delay and is the recovery time for a
particular experiment, in this case either or . This gives the enhancement that would
be obtained if both on and off signals were performed with an infinite recycle delay.
Therefore, the enhancement obtained is . However, because the DNP buildup
time is significantly shorter than the recovery time without DNP, a significant amount of
experimental time can be saved and therefore we should also take this into account. We
may do so by calculating the DNP sensitivity gain, which is given by . For our
experiment, we obtain a sensitivity gain of 310, which is comparable to enhancements
obtained with the cross effect [12; 46].
59
In Figure 9, we show a cross effect enhancement profile of 40 mM TOTAPOL in
60:25:15 13C-glycerol:D2O:H2O acquired at 80 K. In this experiment, the field was set to
B0=4991.5 mT, the microwave frequency was swept from 139.510 GHz to 140.686 GHz
over 99 points, with 40 shots at each point and a recycle delay of 4 s. In Figure 10, we
show the cross effect enhancement for 40 mM TOTAPOL in 60:40 13C-glycerol:D2O at
80 K. The field was set to B0=4993.0 mT, and the microwave frequency was 140 GHz.
Note that this corresponds to the maximum in Figure 9, although we have increased the
microwave frequency and B0 together to use the main, 140 GHz source. The on signal
and off signal shown in Figure 10a were both obtained with a 5 s recycle delay; 64 shots
were acquired for the on signal, and 16768 shots were acquired for the off signal.
Although not shown here, the DNP buildup time (TB) and 1H spin-lattice relaxation time
(T1) were measured, and both were found to be 1.5 s. Therefore unlike the solid effect, it
Figure 8: Solid effect DNP Sensitivity Gain
(a) On signal versus off signal, scaled to the relative intensities for infinite recycle delay,
giving an enhancement of 144 and an overall sensitivity gain of 310. The off-signal has
been multiplied by a factor 20 for improved visibility. (b) Buildup of nuclear
polarization, with a time constant . (c) Decay of nuclear polarization with a time
constant of .
60
is not necessary to scale the spectrum to account for differences in recovery time. The
enhancement may be obtained directly from the relative intensity of the on and off signal,
which in this case is . We also show the power dependence of the cross effect
enhancement in Figure 10b, where 16 shots were acquired at each microwave power,
with a 10 s repetition rate. The maximum enhancement is obtained well below the full
microwave power- at about 25% of the full power (full power corresponds to 4.8 MHz)
Therefore, the T1, TB and the on signal that are shown in Figure 10a are all acquired at
25% of the full power.
139.8140.2140.6μ-wave frequency [GHz]
Figure 9: Cross effect DNP Enhancement profile
DNP enhancement profile of 20 mM TOTAPOL in 60:25:15 13C-glycerol:D2O:H2O,
obtained by sweeping the microwave frequency while the magnetic field was fixed at
4991.5 mT. The EPR spectrum shown above the enhancement profile is a simulation of a
frequency-swept spectrum of a nitroxide radical, also with the magnetic field fixed at
4991.5 mT.
61
There are several significant differences between the cross effect results here, and
the results typically seen in MAS experiments [11; 12; 46; 49]. First, as the power is
increased in MAS experiments, the enhancement also increases, whereas enhancements
here decrease significantly at full microwave power. This is clearly a result of the higher
field strengths obtained in the microwave cavity. Nanni et al. have recently calculated
that for 5 W power incident on a sample in an MAS stator, there is an electron nutation
frequency of 0.5-1.0 MHz (calculated at 250 GHz) [50]. On the other hand, we have done
our cross effect experiments with electron nutation frequencies up to 4.8 MHz, allowing
us to over-saturate the cross effect. The second significant difference is that the field
profile has significantly fewer features; in particular, the 14N hyperfine couplings are not
visible in the static cross effect experiment, whereas they are clearly visible in an MAS
experiment. Finally, despite optimizing both the microwave power and microwave
frequency, the enhancement ( ) is significantly lower than the enhancement
reported for TOTAPOL under MAS conditions ( ). This difference likely
has to do with the modulation of the electron resonance frequencies due to MAS, which
allows saturation of more electrons with lower microwave power. Additionally, the cross
effect matching condition is not satisfied for many TOTAPOL electron pairs in the static
sample. However under MAS, an electron pair that does not initially satisfy the cross
effect matching condition may rotate through the matching condition at some point in the
rotor cycle, allowing many more electron pairs to be utilized in the cross effect under
MAS conditions. Hu et al. have recently presented a detailed analytical treatment of the
cross effect under static conditions [51], however it will be necessary to treat the cross
effect under MAS conditions to fully understand differences in the field profile and
enhancements.
62
2.4 Conclusions We have presented significant improvements to EPR and DNP instrumentation,
leading to improvements in experimental flexibility, RF stability, and DNP performance.
A double-balanced, transmission line probe design with a TE011 coiled resonator that acts
as both the NMR coil and EPR cavity was implemented. With this design, we achieve
high electron nutation frequencies and stable RF performance, which had previously been
difficult to obtain reliably due to arcing and RF inhomogeneities in a similar, but
unbalanced probe design. Also, we presented a novel design for a five-channel, 120 mW,
140 GHz EPR bridge. With this system, we have demonstrated highly efficient
performance of EPR and DNP. An enhancement of 144 and sensitivity gain of 310 for the
−100 −50 0 50 10013C Chemical Shift [kHz]
On Signal
Off Signal(x20)
ε∞=118
TB=T1=1.5 s
0 0.60.2 0.80.4 1.0Power [a.u.]
ε [a
.u]
(a) (b)
Figure 10: Cross effect DNP Sensitivity Gain
(a) On signal versus off signal, scaled to the relative intensities for infinite recycle delay,
giving an enhancement of 118. The off signal has been multiplied by a factor 20 for
improved visibility. The buildup time, TB, and nuclear T1 were the same as is common in
cross effect experiments (traces not shown). (b) Power dependence of ε, which shows
that we obtain the maximum enhancement at ~.27 times the full available microwave
power. Note that full power corresponds to ~4.7 MHz nutation frequency on electrons.
63
1H solid effect far exceeds any previous solid effect experiments at high field (≥3 T) and
liquid-nitrogen temperatures. We have also obtained an enhancement of 118 for the cross
effect and have shown that it is possible to saturate the cross effect with an excess of
microwave power. Both results demonstrate significant improvements in performance of
instrumentation for static DNP. With the integrated NMR and EPR spectrometers and
superior DNP performance, we hope to achieve a better understanding of various DNP
processes and further improve DNP methods.
2.5 Appendix Table 1: Microwave Components for 140 GHz Pulsed EPR Bridge
Component Manufacturer Part number 1 Princeton Microwave PmT-G3220-2.1875 2 Mini-Circuits ZX60-3011+ 3 MITEQ CD-202-402-10S-R 4 MITEQ MAX5H105115 5 Millitech AMC-08-RNH00 6 Millitech MXB-08 7 Marki Microwave AQA-1933
64
8 Mini-Circuits ZX10-2-98-S(+) 9 Arra G6684-30 10 MITEQ SYS3X2327 11 Mini-Circuits ZVA-183-S+ 12 MITEQ M2640W1 13 Virginia Diodes Inc. Spacek A369-3XWB-24 (35 GHz
amplifier) D70 (x2 multiplier to 70 GHz) D142 (x2 multiplier to 140 GHz)
14 American Microwave Corp. SW-2184-1A 15 MITEQ AMF-4F-08001200-15-10P 16 ATM Microwave P216 17 Anaren 1H0568-3 18 Marki Microwave M1-0412LA 19 DMP DMP200 20 Arra 9426A 21 ATM Microwave P416 22 ATM Microwave AV066-30 23 ATM Microwave P1306 24 Princeton Microwave PmT-VCO-0810-8.0
2.6 Bibliography [1] A. Abragam, and M. Goldman, Rep. Prog. Phys. 41 (1978) 395-467. [2] C.D. Jeffries, Dynamic Nuclear Orientation, Interscience, New York, 1963. [3] R.A. Wind, M.J. Duijvestijn, d.L. van, C., A. Manenschijn, and J. Vriend, Prog. Nucl.
Mag. Res. Sp. 17 (1985) 33-67. [4] A.B. Barnes, G. De Paëpe, P.C.A. van der Wel, K.N. Hu, C.G. Joo, V.S. Bajaj, M.L.
Mak-Jurkauskas, J.R. Sirigiri, J. Herzfeld, R.J. Temkin, and R.G. Griffin, Appl. Magn. Reson. 34 (2008) 237-263.
[5] T. Maly, G.T. Debelouchina, V.S. Bajaj, K.-N. Hu, C.-G. Joo, M.L. Mak-Jurkauskas, J.R. Sirigiri, P.C.A. van der Wel, J. Herzfeld, R.J. Temkin, and R.G. Griffin, J. Chem. Phys. 128 (2008) 052211-19.
[6] T.R. Carver, and C.P. Slichter, Phys. Rev. 102 (1956) 975. [7] A.W. Overhauser, Phys. Rev. 92 (1953) 411. [8] V. Bajaj, C. Farrar, M. Hornstein, I. Mastovsky, J. Vieregg, J. Bryant, B. Elena, K.
Kreischer, R. Temkin, and R. Griffin, J. Magn. Reson. 160 (2003) 85-90. [9] L. Becerra, G. Gerfen, R. Temkin, D. Singel, and R. Griffin, Phys. Rev. Lett. 71
(1993) 3561-3564. [10] G.J. Gerfen, L.R. Becerra, D.A. Hall, R.G. Griffin, R.J. Temkin, and D.J. Singel, J.
Chem. Phys. 102 (1995) 9494-9497.
65
[11] Y. Matsuki, T. Maly, O. Ouari, H. Karoui, F. Le Moigne, E. Rizzato, S. Lyubenova, J. Herzfeld, T.F. Prisner, P. Tordo, and R.G. Griffin, Angew. Chem. Int. Ed. 48 (2009) 4996-5000.
[12] C. Song, K.-N. Hu, T.M. Swager, and R.G. Griffin, J. Am. Chem. Soc 128 (2006) 11385-90.
[13] K. Hu, H. Yu, T. Swager, and R. Griffin, J. Am. Chem. Soc. 126 (2004) 10844-5. [14] A.A. Smith, B. Corzilius, A.B. Barnes, T. Maly, and R.G. Griffin, J. Chem. Phys.
136 (2012) 015101. [15] Corzilius, J. Chem. Phys. submitted (2012). [16] B. Corzilius, A.A. Smith, A.B. Barnes, C. Luchinat, I. Bertini, and R.G. Griffin, J.
Am. Chem. Soc. 133 (2011) 5648-5651. [17] P. van der Wel, K. Hu, J. Lewandowski, and R. Griffin, J. Am. Chem. Soc. 128
(2006) 10840-6. [18] V.S. Bajaj, M.L. Mak-Jurkauskas, M. Belenky, J. Herzfeld, and R.G. Griffin, Proc.
Natl. Acad. Sci. 106 (2009) 9244-9249. [19] G.T. Debelouchina, M.J. Bayro, P.C.A. van der Wel, M.A. Caporini, A.B. Barnes,
M. Rosay, W.E. Maas, and R.G. Griffin, Phys. Chem. Chem. Phys. 12 (2010) 5911-5919.
[20] A. Lesage, M. Lelli, D. Gajan, M.A. Caporini, V. Vitzthum, P. Mieville, J. Alauzun, A. Roussey, C. Thieuleux, A. Mehdi, G. Bodenhausen, C. Coperet, and L. Emsley, J. Am. Chem. Soc. 132 (2010) 15459-15461.
[21] M.J. Bayro, G.T. Debelouchina, M.T. Eddy, N.R. Birkett, C.E. MacPhee, M. Rosay, W.E. Maas, C.M. Dobson, and R.G. Griffin, J. Am. Chem. Soc. (2011).
[22] M.J. Prandolini, V.P. Denysenkov, M. Gafurov, S. Lyubenova, B. Endeward, M. Bennati, and T.F. Prisner, Appl. Magn. Reson. 34 (2008) 399-407.
[23] B.D. Armstrong, M.D. Lingwood, E.R. McCarney, E.R. Brown, P. Bl¸mler, and S. Han, J. Magn. Reson. 191 (2008) 273-281.
[24] S. Bowen, and C. Hilty, Angew. Chem. Int. Ed. Engl. 47 (2008) 5235-5237. [25] J. Leggett, R. Hunter, J. Granwehr, R. Panek, A.J. Perez-Linde, A.J. Horsewill, J.
McMaster, G. Smith, and W. Kockenberger, Phys. Chem. Chem. Phys. 12 (2010) 5883-5892.
[26] S. Jannin, A. Bornet, S. Colombo, and G. Bodenhausen, Chem. Phys. Lett. 517 (2011) 234-236.
[27] V. Weis, M. Bennati, M. Rosay, and R.G. Griffin, J. Chem. Phys. 113 (2000) 6795-6802.
[28] A. Henstra, P. Dirksen, J. Schmidt, and W.T. Wenckebach, J. Magn. Reson. 77 (1988) 389-393.
[29] S. Un, T. Prisner, R.T. Weber, M.J. Seaman, K.W. Fishbein, A.E. McDermott, D.J. Singel, and R.G. Griffin, Chem. Phys. Lett. 189 (1992) 54-59.
[30] R.I. Hunter, P.A.S. Cruickshank, D.R. Bolton, P.C. Riedi, and G.M. Smith, Phys. Chem. Chem. Phys. (2010).
[31] A. Feintuch, D. Shimon, Y. Hovav, D. Banerjee, I. Kaminker, Y. Lipkin, K. Zibzener, B. Epel, S. Vega, and D. Goldfarb, J. Magn. Reson. 209 (2011) 136-141.
[32] V. Nagarajan, Y. Hovav, A. Feintuch, S. Vega, and D. Goldfarb, J. Chem. Phys. 132 (2010) 214504-1 to 13.
66
[33] V.P. Denysenkov, M.J. Prandolini, A. Krahn, M. Gafurov, B. Endeward, and T.F. Prisner, Appl. Magn. Reson. 34 (2008) 289-299.
[34] J. Granwehr, J. Leggett, and W. Kockenberger, J. Magn. Reson. 187 (2007) 266-76. [35] T. Maly, J. Bryant, D. Ruben, and R. Griffin, J. Magn. Reson. 183 (2006) 303-7. [36] D.R. Bolton, P.A.S. Cruickshank, D.A. Robertson, and G.M. Smith, Electron. Lett.
43 (2007). [37] C.D. Joye, M.A. Shapiro, J.R. Sirigiri, and R.J. Temkin, IEEE T. Electron Dev. 56
(2009) 818-827. [38] H.J. Kim, E.A. Nanni, M.A. Shapiro, J.R. Sirigiri, P.P. Woskov, and R.J. Temkin,
Phys. Rev. Lett. 105 (2010) 135101 (1-4). [39] B. Epel, I. Gromov, S. Stoll, A. Schweiger, and D. Goldfarb, Concepts Magnetic
Res. B 26B (2005) 36-45. [40] J.S. Hyde, J. Chem. Phys. 43 (1965) 1806-1818. [41] K. Gruber, J. Forrer, A. Schweiger, and H.H. Gunthard, J. Phys. E: Sci. Inst. 7
(1974) 569. [42] V. Weis, M. Bennati, J.A. Bryant, and R.G. Griffin, J. Magn. Reson. 140 (1999)
293-299. [43] F. Engelke, Concepts Magnetic Res. 15 (2002) 129-155. [44] S.R. Hartmann, and E.L. Hahn, Phys. Rev. 128 (1962) 2042-2053. [45] D.D. Traficante, Concepts Magnetic Res. 5 (1992) 57-86. [46] M. Kiesewetter, J. Am. Chem. Soc. 134 (2012) 4537-4540. [47] S. Stoll, and A. Schweiger, J. Magn. Reson. 178 (2006) 42-55. [48] W.B. Mims, Phys. Rev. B 5 (1972) 2409. [49] E.L. Dane, B. Corzilius, E. Rizzato, P. Stocker, T. Maly, A.A. Smith, R.G. Griffin,
O. Ouari, P. Tordo, and T.M. Swager, J. Org. Chem. 75 (2012) 3533-3536. [50] E.A. Nanni, A.B. Barnes, Y. Matsuki, P.P. Woskov, B. Corzilius, R.G. Griffin, and
R.J. Temkin, J. Magn. Reson. 210 (2011) 16-23. [51] K.-N. Hu, G.T. Debelouchina, A.A. Smith, and R.G. Griffin, J. Chem. Phys. 134
(2011).
67
Chapter 3
Solid Effect Dynamic Nuclear Polarization and Polarization Pathways
Contributing: Björn Corzilius, Alexander B. Barnes, Thorsten Maly
68
Abstract Using dynamic nuclear polarization/nuclear magnetic resonance (DNP/NMR)
instrumentation that utilizes a microwave cavity and a balanced r.f. circuit, we observe a
solid effect DNP enhancement of 94 at 5 T and 80 K using trityl radical as the polarizing
agent. Because the buildup rate of the solid effect increases with microwave field
strength, we obtain a sensitivity gain of 128. The data suggest that higher microwave
field strengths would lead to further improvements in sensitivity. In addition, the
observation of microwave field dependent enhancements permits us to draw conclusions
about the path that polarization takes during the DNP process. By measuring the time
constant for the polarization buildup and enhancement as a function of the microwave
field strength, we are able to compare models of polarization transfer, and show that the
major contribution to the bulk polarization arises via direct transfer from electrons, rather
than transferring first to nearby nuclei and then transferring to bulk nuclei in a slow
diffusion step. In addition, the model predicts that nuclei near the electron receive
polarization that can relax, decrease the electron polarization, and attenuate the DNP
enhancement. The magnitude of this effect depends on the number of near nuclei
participating in the polarization transfer, hence the size of the diffusion barrier, their T1
and the transfer rate. Approaches to optimizing the DNP enhancement are discussed.
69
3.1 Motivation Dynamic nuclear polarization (DNP) is a method of enhancing NMR signals by
transferring the large Boltzmann polarization of unpaired electrons to the nuclear spin
reservoir, thereby enhancing the NMR signal intensities by several orders of magnitude
[1; 2]. Since DNP greatly enhances signal-to-noise, it permits otherwise prohibitively
long experiments to be performed and/or acquisition of enhanced information, both in
shorter periods of time. There are many situations that can benefit greatly from the
enhanced sensitivity of DNP. For instance, protein samples exhibit spectra that are
intractable without multidimensional experiments. Concurrently, the multiple dimensions
and the magnetization transfer steps involved in acquisition of these spectra often lead to
results with low sensitivity. Thus, there are now many examples where the quality of
spectra of biological samples improves dramatically with polarization enhancement [3-
13]. Another compelling illustration of the utility of DNP involved NMR studies of
surfaces where spectral acquisition required many days or weeks of signal averaging
without DNP, whereas with DNP excellent results were obtained in less than an hour
[14].
Continuous wave (CW) DNP in insulating solids generally proceeds via one of
three mechanisms, depending on the relative magnitude of the nuclear Larmor frequency
( ω0 I ), and the homogeneous (δ ) and inhomogeneous linewidths (Δ ) of the electron
paramagnetic resonance (EPR) spectrum. In the case ω0 I δ ,Δ (the EPR spectrum is
narrow compared to the nuclear Larmor frequency) the solid effect [15-17] governs the
DNP process. Since the SE utilizes forbidden electron-nuclear transitions, the transition
moments exhibit a ω0 I−2 dependence. Therefore, the experimentally observed SE
enhancements at high field (≥5 T) and liquid-N2 temperatures (80-90 K) in contemporary
DNP experiments have until recently been limited to ~5-15 [18-21], although higher
enhancements have been obtained at lower fields and liquid-helium temperatures [22;
23]. In contrast, in the regime Δ >ω0 I > δ the three spin cross effect (CE) becomes
dominant [24-30]. The CE utilizes biradicals, where two electrons are tethered together in
the correct relative orientation [28-30], as polarizing agents, and to date it has proven to
70
be the most efficient DNP mechanism for high field experiments, yielding 1H
enhancements of up to 250 [31; 32]. The third mechanism, thermal mixing (TM), is
important when the EPR spectrum is homogeneously broadened – when Δ,δ ≥ω0 I [33-
35]. However, at high fields (≥5 Tesla) the g-anisotropies of many polarizing agents are
typically larger than the homogeneous contributions to the linewidth, and therefore TM
has not been an important mechanism in most contemporary MAS DNP experiments.
However, at the lower temperatures (~1.5 K) used in dissolution DNP where the EPR line
behaves homogeneously, it could be dominant [36].
Considerable effort has been focused on understanding the steps involved in the
transfer of the large electronic polarization to nuclei. Quantum mechanical treatments
were used to describe the two-spin interaction in the SE [37-39], and the three-spin
interaction in the CE [30], and rate equations have been used to describe the buildup of
polarization in the bulk [27; 40]. More recently, simulations were used to understand
interactions in small systems of spins [41; 42], and finally experiments monitoring the
attenuation of electron polarization were performed to better understand the behavior of
the electron spin reservoir during DNP [39; 43; 44]. However, as we continue to optimize
DNP experiments via modifications to samples, development of polarizing agents,
instrumentation, and implementation, including pulsed-DNP techniques, it is important to
understand all the steps and limiting factors active in the DNP polarization transfer.
The goal of this paper is to provide a description of the transfer of polarization
from the electrons to the nuclei guided by experimental data, and to determine the role of
the “diffusion barrier” in this process. Interestingly, the “diffusion barrier” has been
defined in a variety of ways, and its size has been measured with both indirect and direct
experiments, resulting in conflicting estimates of the barrier radius. The initial discussion
of a spin-diffusion barrier was included in the pioneering work of Bloembergen [45] on
the effects of paramagnetic relaxation in crystals. Subsequently, Khutsishvili defined the
barrier size in terms of the shift in the resonance frequency due to electron-nuclear
coupling relative to the NMR line-width [46]. Blumberg proposed a similar definition,
stipulating the radius as the position where the electron nuclear coupling became larger
than the local dipolar field (giving a slightly larger radius than Khutsishvili) [47].
Goldman [48], and Schmugge and Jeffries [22], used the Blumberg definition together
71
with relaxation data to indirectly predict barriers with radii of 16 to 17 Å. In the 1970’s,
Wolfe, in a series of elegant experiments using single crystals, directly measured the
radius of the diffusion barrier in Yb3+/Nd3+-doped yttrium ethyl sulfate (YES) and Eu2+
doped crystals of CaF2 [49-51]. In particular at ~1.7 K the electron T1 (hereafter T1S )
becomes long and it is possible to observe resonances from nuclei adjacent to the
paramagnetic center as well as the large line due to the bulk resonance whose position is
unshifted by the paramagnet. Saturation of the bulk resonance with a weak field
subsequently saturates all of the lines in the spectrum except those from nuclei
immediately adjacent to the metal center, indicating that essentially all nuclei directly
communicate with the bulk. The radius of the diffusion barrier derived from these
measurements is ~5 Å which is significantly smaller than predicted by the Khutsishvili
and Blumberg definitions and estimated by Goldman and Schmugge and Jefferies.
Interestingly, this distance is comparable to the C• → -CH2- distance on trityl (OX063)-
suggesting it may be unnecessary to consider the spin-diffusion barrier at all. Finally, we
should mention that other efforts directed at measuring the size of the diffusion barrier
are discussed in greater detail in the recent review article by Ramanathan [52].
In this paper we consider the role of partial or total quenching of spin-diffusion on
the DNP process. With this goal in mind, we have focused on SE experiments, as this
mechanism is the simplest of the three CW DNP mechanisms, involving only two spins
in the initial polarization transfer. We have performed a series of experiments where we
measured (1) the time constant for the polarization buildup, (2) the polarization
enhancement, ε , and (3) the ω1S dependence of ε . In order to explain our experimental
observations, we consider three models based on differential equations describing the
polarization transfer from electrons to the bulk nuclei, each representing a different
pathway of polarization transfer, and we attempt to fit our data to each model. The
models, which are shown schematically in Figure 1, consider (A) transfer of electron
polarization to all the nuclei, (B) transfer first to the nuclei neighboring the electron
(inside the “diffusion barrier”) and then to the bulk and (C) finally to nuclei outside the
“diffusion barrier” and subsequently to the bulk via 1H spin-diffusion. Model (A) is the
null hypothesis, for which the spins inside the diffusion barrier do not consume much
72
electron polarization, either because they are few in number, or their T1 s are long enough
that they do not use much polarization. In view of the experimental results from Wolfe
showing a very thin barrier, it is important to consider this case. Model (B) follows the
mechanism discussed by of Blumberg and Khutsishvili where the barrier is passable due
to coupling of the spin-diffusion process to other interactions, albeit slowly, so that spins
near the electron are polarized first, and then polarization diffuses outward to the bulk
nuclei. This model has been used recently by Hovav et al., where calculations are done in
Liouville space with a single “core” nucleus and bulk nuclei to which polarization
diffuses [42]. Model (C) does not allow spin-diffusion across the barrier- in the spirit of
Wolfe’s experimental results where protons within 3-4 Å do not show any diffusion.
However, there will be polarization transfer from the electron to nuclei within the spin-
diffusion barrier and due to T1 relaxation, these nuclei can act as a polarization sink.
The model that best fits the experimental data – the magnitude of the
enhancement, the polarization build-up time, and the ω1S dependence – is case (C).
However, the protons of trityl are on the border of the 5 Å radius measured by Wolfe,
suggesting that our diffusion barrier may have a larger radius [53]. We also note that
when the microwave field strength is not limiting, the solid effect has the potential to
provide very large enhancements at high fields, and can further increase the sensitivity
gain per unit time by an acceleration factor, κ = T1I TB , since the solid effect generates
polarization on a time scale shorter than the nuclear T1 .
The paper is organized as follows. In Section II we outline three mathematical
models for solid effect polarization transfer and the rate equations that describe each.
Section III provides the experiments details, and Section IV describes the experimental
results including the pulse sequence used to acquire the data, and the experimental
polarization buildup times acquired as a function of the microwave field, ω1S . This is
followed by a discussion of the experimental results and includes a description of the
approximate size of the diffusion barrier based on data from the experiments of Wolfe
[49-51] and a structure of Finland trityl determined via EPR measurements and quantum
chemical calculations [53].
73
3.2 Theory
3.2.1 Rate equations To understand the transfer of electron polarization to bulk nuclear polarization,
we employ a slow (seconds) time-scale, and use linear differential equations to describe
this transfer. To obtain these expressions we first consider a Liouville space calculation
that leads to the differential equations given in equation (7) below.
We start by examining the Hamiltonian describing a single electron and many
nuclei. This Hamiltonian governs interactions leading to electron-nuclear polarization
transfer in the microwave rotating frame, where we will assume the microwave frequency
is near the condition for positive, double quantum ( ΔmS = ±1, ΔmI = ±1) DNP
enhancement ( ω MW =ω0S −ω0 I ). The terms of the Hamiltonian in (1) are the electron
Zeeman, nuclear Zeeman, electron-nuclear coupling, nuclear-nuclear coupling, and
microwave Hamiltonians, respectively. Δω0S is the microwave offset from the electron
Larmor frequency, the ω0 I j
are the nuclear Larmor frequencies, the Aj are the secular
electron-nuclear dipole couplings, Bj and
C j are the non-secular electron-nuclear dipole
couplings, Dj ,k is the nuclear-nuclear coupling tensor, and ω1S is the microwave field
strength. Note that, because we are in the rotating frame, rapidly oscillating electron-
nuclear dipole terms have been dropped.
74
H 0 = H S + H I + H IS + H II
H = H 0 + H M
H S = Δω0S Sz
H I = −ω0 I jI jz
j
NI
∑
H IS = AjSz I jz +Bj
2Sz I j
+ + I j−( ) + C j
2iSz I j
+ − I j−( )
j
NI
∑
H II = I j Dj ,k Ikk> j
NI
∑j
NI
∑
H M =ω1S
2S + + S −( )
(1)
Note that our Hamiltonian, H , only describes interactions between spins, and does not
include interaction of the spins with the lattice. In principle, it is possible to describe the
full system with
ddt
ρ(t) = −i H f (t),ρ(t)⎡⎣ ⎤⎦
(2)
where H f (t) and ρ(t) describe the full spin system and lattice. However, such an
approach is prohibitively difficult, and we are only interested in the state of the spin
system, so we will use a superoperator approach to include the relaxation brought about
by the lattice interaction with the spin system [54]. The superoperator equation is given in
(3), where σσ is a column vector describing the state of the spin system.
ddtσ = − iH +Γ( )σ + Γσ eq
(3)
In (3), the superoperator H is given by H = H ⊗ E − E ⊗H where H is the
Hamiltonian from (1), E is the identity operator, and H is the transpose of H . ΓΓ is the
relaxation superoperator, which describes population transfer between states of the spin
system brought about by the matrix. If we consider the basis set defined by
Sz , I1z , I2z ,…,S+ ,S − ,…( ) , then ΓΓ will contain the relaxation rates for individual states
along its diagonal, and any cross-relaxation between spins on the off-diagonal. σσ eq is the
equilibrium population, where σσ eq ∝ exp −H 0 kBT( ) (such that
H 0σ eq = 0 , where
75
H 0 = H 0 ⊗ E − E ⊗ H 0 ). We want to model observations made over the time scale of
the nuclear relaxation, which allows us to assume that many of the quantum-mechanical
states in the system have reached quasi-equilibrium. A state in quasi-equilbrium evolves
rapidly compared to the time-scale of observation (up to ~100 ms in our case). This
causes the state to react relatively quickly to changes in other states that are not in quasi-
equilibrium. Although the state in quasi-equilibrium may change on the time-scale of the
observation, it is still reasonable to use the following approximation for that state which
we denote by σσ j , allowing for a quasi-steady state solution of the value of the state
σσ j
in terms of the other states.
ddtσ j = − iH j ,k +Γ j ,k( )σ j +Γ j ,kσ eq, j
k∑ = 0
(4)
We will assume quasi-equilibrium for all states except the polarization
states
Sz , I1z , I2z ,…( ) . A similar assumption was made by Hovav, et al. when computing
the evolution of polarization [41]. In particular, they assume a quasi-equilibrium of the
coherences connecting eigenstates of the static Hamiltonian to accelerate their
computations. For our arguments, we do not go into the eigenframe, since this was only
necessary for the method of computation. We also go one step further, assuming quasi-
equilibrium for states without a transverse component, which we will refer to as spin-
order states ( I1z I2z for example). One may note that the results in Hovav, et al. do not
show any oscillations that would result from a coherence not in quasi-equilibrium,
suggesting this approach is reasonable [41].
For states involving a coherence ( Sx , I1x …), the assumption of quasi-equilibrium
is clearly reasonable since the lifetime is on the order of 1 μs for the electron, and is on
the order of 1 ms for a nucleus. Also, we may do this for states including a factor of Sz
Sz I1z ,Sz I2x ,…( ) , since the electron T1 is on the order of 1 ms (although we will not
initially make this assumption for the electron polarization itself). This leaves the states
describing nuclear spin-order ( I1z I2z for example). These spin-orders represent the
buildup of polarization on one nucleus that is dependent on the state of another. We do
expect some of this behavior, because the state of one nucleus will change the offset of
76
the DNP condition on another. However, this effect is small, and so the spin-orders
should also remain small. Nuclear spins that are distant from the electron should have
relatively uniform polarization due to rapid spin-diffusion, which will suppress spin-
order. Spins near to the electron will have less uniform polarization, although their T1 s
will be shorter so that the lifetime of spin-order adjacent to the electron will be reduced.
Thus, it should be a reasonable approximation to include spin-order in the quasi-
equilibrium assumption.
Under the quasi-equilibrium assumption, we set derivatives of all states to zero,
excepting the polarization states
Sz , I1z , I2z ,…( ) . We group the states in quasi-equilibrium
into two column vectors: σσ Q contains the states in quasi-equilbrium whose derivatives
are zero, and σσ P contains the polarization states. As such, we can rewrite (3) with the
quasi-equilibrium assumption( dσσ Q dt = 0 ), and divide the superoperator matrices H
and ΓΓ into sub-matrices H PP , H QP ,
H QQ , ΓΓPP ,
ΓΓQP , and
ΓΓQQ .
ddtσ P
0
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥= −
iH PP +ΓPP iH QP +ΓQP
iH QP +ΓQP iH QQ +ΓQQ
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥σ P
σ Q
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥+
ΓPP 0
0 ΓQQ
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥σ P,eq
σ Q ,eq
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
(5)
We will assume that we can omit all cross-relaxation, thus we can eliminate the two
matrices ( ΓΓQP ) that couple the polarization states to the states in quasi-equilibrium. It is
also possible to eliminate H PP from (5), which describes coherent transitions between
polarization states, because there are no terms in the Hamiltonian driving transitions
directly between these states. This results in (6), for which we discuss the remaining
terms.
ddtσ P
0
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥= −
Γ PP iH QP
iH QP iH QQ + ΓQQ
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
σσ P
σ Q
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥+
ΓΓ PP 0
0 ΓQQ
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
σ P,eq
σ Q ,eq
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
(6)
Omission of cross-relaxation implies that ΓΓPP and ΓΓQQ are diagonal matrices that
contain T1 relaxation rates of all the spins and relaxation rates of all the states in quasi-
77
equilibrium, respectively. The products −ΓΓPPσ P and
ΓΓPPσ P,eq lead to T1 loss and
recovery towards thermal equilibrium, respectively. Although their magnitudes are much
smaller, −ΓΓQQσ Q
and ΓΓQQσ Q ,eq lead to loss and recovery towards thermal equilibrium of
non-polarization states, respectively.
The matrix H QP contains terms that connect the polarization states to the states in
quasi-equilibrium, namely the non-secular electron-nuclear dipole couplings ( H IS
excluding zz terms), the non-secular parts of the nuclear-nuclear dipolar coupling ( H II
excluding zz terms), and the microwave Hamiltonian ( H M ).
All terms in the Hamiltonian appear in the matrix H QQ . In the particular basis set
we are using, the Zeeman terms ( H S and H I ) will appear on the diagonal of H QQ . The
diagonal element of H QQ corresponding to a particular state will give the frequency at
which a particular state oscillates from real to imaginary, which we will refer to as the
phase oscillation of that state. An important point: states with a slow phase oscillation are
likely to play a major role in the DNP and spin-diffusion processes. When there is
transfer to a state with slow phase-oscillation, that state does not rapidly invert its sign
and invert the transfer, and hence becomes populated (note that population transfer can
occur between two or more rapidly oscillating states if there is a match of the frequencies,
but in our case we match to the polarization states which are not oscillating). Of course,
high-order states that are less likely to be accessed will not play a major role even if their
phase-oscillation is slow.
This is a simplification of the resonance condition, since many additional states
may be involved in the resonance, complicating the situation. The electron-nuclear and
nuclear-nuclear dipole couplings ( H IS and H II ) will connect the states in H QQ , and as a
result resonance conditions will be a complex function of many of the couplings. H M
will also connect states in H QQ , although because the microwave is far off-resonant, its
effects will be less important.
78
We now rearrange (6) into (7) which shows that the polarization states are linearly
coupled under the quasi-equilibrium assumption, and discuss several processes that will
occur during DNP as described by (7). For each process, we will give a differential
equation that shows how the process will manifest itself in our model.
σ Q = −i iH QQ + ΓQQ( )−1H QPσ P + iH QQ + ΓQQ( )−1
ΓQQσ Q ,eq
dσ P
dt= − ΓΓ PP + H QP iH QQ + ΓQQ( )−1
H QP⎛⎝
⎞⎠ σ P
+ Γ PPσ P,eq + iH QP iH QQ + ΓQQ( )−1ΓQQσ Q ,eq
(7)
3.2.1.1 Relaxation
The matrix ΓΓPP is diagonal and contains the longitudinal relaxation rates of all
spins in the system. With some rearranging of (7), we see the term ΓΓPP σ P,eq −σ P( ) that
is responsible for all electron and nuclear T1 loss and recovery. Relaxation loss of
coherences occurs via the matrix ΓΓQQ . Additionally, although small, there is some
population of the states in quasi-equilibrium, which contributes to polarizations of the
spins via the term iH QP iH QQ +ΓQQ( )−1
ΓQQσ Q ,eq . Without this term, the loss due to some
conversion of polarization to coherence would cause calculated polarizations to fall short
of the thermal equilibrium. Although important for exact Liouville space calculations,
this term only makes small contributions here and will not be explicitly included in our
treatment. Therefore, the effects of T1 relaxation will appear in our model as
dPIj
dt= 1
T1IjPIeq − PI
j( )dPSdt
= 1
T1SPSeq − PS( ).
(8)
3.2.1.2 Spin-Diffusion
The non-secular terms in the nuclear-nuclear dipolar Hamiltonian will generate
double- and zero-quantum coherences between nuclei. This generation of the coherences
79
occurs via the matrix H QP , but once generated are governed by the matrix
iH QQ +ΓQQ .
The diagonal elements of H QQ , which result from H I , determine that double-quantum
coherences of spins j and k ( I j+Ik
+ + I j−Ik
− ) oscillate at a frequency ω0 I j
+ω0 Ik so they will
not become populated, whereas the zero-quantum ( I j+Ik
− + I j−Ik
+ ) are nearly static and thus
can be populated. Once the zero-quantum coherence is generated, polarization loss on
nucleus j and polarization gain on nucleus k will be proportional to the population of the
coherence, as is given by the matrix H QP . This inverse proportionality results in
conservation of polarization during spin-diffusion.
Transfer to many other states will affect the population of the coherence. This is
how the spin-diffusion barrier manifests itself in this picture. The secular coupling
( AjSz I jz + Ak Sz Ikz ), appearing in the matrix
H QQ , will transfer the zero-quantum
coherence, I j+Ik
− + I j−Ik
+ , to Sz I j
+Ik− + I j
−Ik+( ) with a rate proportional to
Aj − Ak , where the
net oscillation frequency is either higher or lower depending on the state of the electron.
As a result, it will be more difficult to transfer polarization into this coherence, and
therefore inhibit spin-diffusion leading to the spin-diffusion barrier. In fact, without
relaxation of the states involved, spin-diffusion would be completely quenched, but non-
zero relaxation rates of I j+Ik
− + I j−Ik
+ and Sz I j
+Ik− + I j
−Ik+( )
in ΓΓQQ give a finite rate of spin-
diffusion. Note that nuclear dipole couplings will also have the same offsetting effect,
although to a much smaller extent.
It is interesting to note that if the Sz state of the electron either has a short
lifetime or is rapidly modulated by the microwave field (near on-resonant radiation), then
it will be difficult to populate Sz I j
+Ik− + I j
−Ik+( ) . This will help accelerate spin-diffusion.
The former case is described by Horvitz [55]. The latter case is electron-nuclear
decoupling [22; 54; 56]. We note that our experiments are much too far off-resonant for
electron-nuclear decoupling to be significant. Therefore, (9) gives the rate of spin-
diffusion in our model, in a form that conserves polarization, and note that kSDn, j does not
change significantly with microwaves on or off.
80
dPIj
dt= kSD
n, j PIn − PI
j( )n=1
NI
∑
(9)
3.2.1.3 Off-Resonant Electron Saturation
The microwave Hamiltonian ( H M ) generates electron coherence ( i(S+ − S − ) ) via
the matrix H QP . Once there, terms in
H QQ
resulting from the electron-Zeeman ( H S )
and electron-nuclear couplings ( H IS ) will govern the electron coherence’s further
evolution. The Zeeman ( Δω0S Sz ) and secular couplings ( AjSz I jz ) will combine to give
an ensemble of oscillation frequencies of the electron. For the solid effect, this oscillation
is fast, so the electron coherence will be rapidly returned to electron polarization via H M
in H QP . However, there will be a small average population of the coherence, and this
will be subject to electron T2 relaxation, resulting in partial saturation of the electron. We
note that aside from transferring from i(S + − S − ) to various i(S+ − S − )I jz states (leading
to offsets on the oscillation frequency), transfer out of the electron coherence is slow
compared to equilibration of the electron coherence with the electron polarization. The
result is that off-resonant saturation of the electron is largely decoupled from other
processes. As such, we may then use a single loss term to describe the off-resonant
saturation of the electron, as shown in (10). Note that k0 will vary with the oscillation
frequency of the electron- however, for a relatively narrow EPR resonance the variation
in k0 will be small enough that it can be approximated with a single value.
dPSdt
= −k0PS
(10)
3.2.1.4 Solid Effect DNP
The initial step of the solid effect matches that of off-resonant microwave
irradiation, with a transfer of electron polarization to electron coherence via the H QP
matrix. However, in this case the non-secular electron-nuclear dipole coupling
81
( Bj
2 Sz (I j+ + I j
− ) ) in the H QQ matrix drives the electron coherence to electron-nuclear
zero- and double-quantum coherences. Assuming the double-quantum DNP condition is
satisfied ( ω MW ≈ω0S −ω0 I ), the double quantum coherence ( S + I j
+ + S − I j− ) has a phase
oscillation near zero, and therefore this state becomes populated. The microwave field
( ω1S
2(S+ + S − ) ) then converts this to
iSz (I j+ − I j
− ) , and finally the non-secular electron-
nuclear dipole coupling ( BjSz (I j
+ + I j− ) ) in the
H QP matrix generates nuclear
polarization. Of course, as with the other processes, this will be offset by secular
couplings to other spins.
We should note that the solid effect does not generally conserve polarization as
spin-diffusion does. First, we already noted there is off-resonant saturation of the
electron. Second, loss of nuclear polarization can occur when transferred to iSz (I j
+ − I j− ) .
This effect will occur without an applied microwave field, and as a result this will
manifest itself in the observed T1I so we do not need to further account for it. Therefore,
if we account for off-resonant saturation of the electron, we can then also consider the
solid effect to be polarization conserving, and can represent the solid effect process in our
model as
dPIj
dt= kDNP
j PS − PIj( )
dPSdt
= kDNPn PI
n − PS( )n=1
NI
∑ .
(11)
3.2.1.5 Higher Order Processes
One should note that under DNP conditions, the matrix H QQ is not block-
diagonal, meaning that every spin polarization is connected to every other spin
polarization, although in many cases very weakly. This results from transfers with more
steps inside the H QQ matrix, which can lead to interesting results. Hovav et al. have
recently shown an example of this with a chain of coupled spins [42]. We will describe
polarization transfer between spins via coupled differential equations. But, one should
82
note that although our formulas suggest polarization transfers between spin pairs, this
does not mean additional spins are not involved in those transfers, and the rate constants
driving polarization between spin-pairs may be larger or smaller due to these effects.
3.2.1.6 Rate Equations
We can now write rate equations describing the polarization transfer, as shown in
(12), and discuss why this formula is reasonable.
dPIj
dt= kDNP
j PS − PIj( ) + kSD
n, j PIn − PI
j( )n=1
NI
∑ + 1
T1Ij PI
eq − PIj( )
dPSdt
= −k0PS + kDNPn PI
n − PS( )n=1
NI
∑ + 1
T1SPSeq − PS( )
(12)
In (12), PS represents the electron polarization, and the PIj represent the nuclear
polarizations. T1S and T1Ij give the longitudinal relaxation.
kSD
n, j PIn − PI
j( ) describes
nuclear spin-diffusion, which in this form is polarization conserving ( kSDn, j = kSD
j ,n ). Off-
resonant saturation of the electron is given by −k0 PS . Finally, the solid effect is given by
kDNP
j PS − PIj( ) , which is also treated as polarization conserving, since we have already
accounted for off-resonant electron saturation.
We now examine (12) in limiting situations to calculate enhancements and time
constants for the polarization transfer.
3.2.2 Implications of the Rate Equations There are several unknown parameters in (12), including the rate constants of
polarization transfer and spin-lattice relaxation. The polarization transfer rate constants
are for DNP driven polarization transfer from the electron to the jth nucleus ( kDNPj ) and
spin-diffusion mediated transfer from the jth to the kth nucleus ( kSDj ,k ). The spin-lattice
relaxation rates constants are 1 T1Ij for the ith nucleus and 1 T1S for the electron. Finally,
the rate constant for partial saturation of the electron due to the off-resonant microwave
field is k0 . In the absence of simplifying assumptions, these equations are not particularly
83
useful. We therefore consider the nuclear polarization adjacent to the electron, and the
average nuclear polarization of bulk nuclei. The nuclei in the immediate vicinity of the
electron are few in number; in fact, for Finland trityl, 1H ENDOR performed by
Bowman, et al. showed the closest approach of a solvent proton to be 4.8 Å from the
electron,[53] and on the border of the ~5 Å diffusion barrier observed by Wolfe. One
may consider where the electron-nuclear dipolar coupling and nuclear-nuclear dipole
coupling become similar in magnitude- however if one considers protons ~3 Å apart, this
occurs at ~25 Å away from the electron. As a result, almost all nuclei in the sample
would fall inside this boundary. Clearly, the diffusion barrier must fall between these
limits, but for the moment we will not make assumptions about the distance. We will,
however, assume some nuclei are within the barrier. For simplicity, we also assume these
near neighbor protons (using the parlance of Wolfe, et al.) within the barrier are equal in
polarization, and describe them with a single polarization term, PI(n) where (n) refers to
the nearby nuclei; their DNP and spin-lattice relaxation rates are kDNP(n) and 1 T1I
(n) ,
respectively. Note we have dropped the superscript j because we are considering the near
neighbor nuclei as equivalent.
As one moves away from the electron to the more distant bulk nuclei, we
encounter a polarization gradient, resulting from a finite rate of spin-diffusion. This
gradient will attenuate the DNP rate, because transfer from electrons will be inhibited by
the higher nuclear polarization near the electron. However, in many cases it is not
difficult to account for this polarization gradient. We will argue in a forthcoming paper
that it is not usually necessary to explicitly include spin-diffusion. This is a result of the
fact that when spin-diffusion is sufficiently rapid, the polarization gradient (but not the
average polarization) equilibrates quickly relative to the total DNP buildup time, and the
ratio of nuclear polarization near the electron to the average nuclear polarization remains
approximately constant throughout most of the DNP buildup process. One may then use
an effective rate constant, kDNP(b) , which is some fraction of the average DNP rate constant,
and accounts for the attenuated rate of DNP. This constant gives the rate of polarization
transfer when multiplied by the difference of the electron polarization and the spatially
averaged bulk nuclear polarization, allowing us to forgo explicit inclusion of the
84
polarization gradient in our models. It is safe to use these assumptions if the initial DNP
polarization transfer is rate limiting and therefore has a strong influence on the DNP
enhancement, as one would see from a dependence of enhancement on microwave power.
Similar arguments can be used to describe the transfer of polarization from nearby nuclei
to bulk nuclei, the effective rate given here as kSD , where SD refers to the spin-diffusion
process between the nearby and bulk nuclei (this is different from kSDn,i because this rate
constant refers to a single nuclear pair, whereas kSD refers to the net diffusion between
nearby and bulk nuclei).
We will use the effective rate constants, kSD and kDNP(b) (and will see in our
experimental results that this is justified by a strong power dependence). We also
introduce N I(b) , N I
(n) , and NS which are the number of bulk and nearby nuclei in the
sample that are being treated equivalently, and the number of electrons in the sample,
respectively. Using these assumptions, (12) can be rewritten as:
dPI(b)
dt= kDNP
(b) PS − PI(b)( ) + kSD PI
(n) − PI(b)( )
+ 1
T1I(b) PI ,eq − PI
(b)( )dPI
(n)
dt= kDNP
(n) PS − PI(n)( ) + NI
(b)
NI(n) kSD PI
(b) − PI(n)( )
+ 1
T1I(n) PI ,eq − PI
(n)( )dPSdt
= −k0PS +NI(b)
NS
kDNP(b) PI
(b) − PS( ) + NI(n)
NS
kDNP(n) PI
(n) − PS( )
+ 1
T1SPS ,eq − PS( )
(13)
It is the case that off-resonant saturation in our experiments is not negligible, with
k0 being as large as 0.31 ms-1, which competes with 1 T1S of 0.71 ms-1. k0 can be
calculated via the Bloch equations[57] for off resonant irradiation as shown in (14), if the
field strength and T1S and T2S are known.
85
k0 =ω1S
2 T2S1+ Δω 0S
2 T2S2 (14)
It is then possible to eliminate k0 from the equations, and maintain the form of (13) via
the definitions in (15).
1
T1S∗ = k0 +
1
T1S
PS ,eq∗ = PS ,eq
T1S∗
T1S
dPSdt
= NI(b)
NS
kDNP(b) PI
(b) − PS( ) + NI(n)
NS
kDNP(n) PI
(n) − PS( ) + 1
T1S* PS ,eq
* − PS( )
(15)
One should note that this causes the amount of polarization available for DNP to be
decreased if k0 is on the order of T1S−1 , since
T1S
*( )−1≥ T1S
−1 in all cases.
Here we have separated the rate equations for the bulk and near neighbor nuclear
polarizations. Knowing the mechanism via which polarization transfers from electrons to
bulk nuclei is crucial in order to understand the primary processes of DNP. Therefore,
three cases of polarization transfer are tested here and illustrated in Figure 1.
3.2.2.1 Case (A): No Diffusion Barrier
86
As noted above the experimental evidence from Wolfe’s experiments suggest that
the spin-diffusion barrier is much thinner than is commonly assumed, and therefore it is
important to consider the limiting case where it vanishes. Thus, in case (A), we examine
the possibility of ignoring nearby nuclei, or equivalently treating them as part of the bulk,
and transferring polarization directly to the bulk. If nuclei adjacent to the electron are
omitted from consideration, then (13) simplifies to (16), which is commonly found in the
literature [27; 58-60]. Here we have dropped the superscript (b) from kDNP and T1I since
we are no longer differentiating between bulk, and near neighbor nuclei.
Figure 1: Three possible models for the transfer of polarization.
Transfer from electrons (red) to nearby nuclei (n) (yellow) and the bulk (b) nuclei
(green). In (A), we show a model where spin-diffusion is fast among all nuclei that
receive polarization. Due to the rate of this diffusion process, all nuclei maintain nearly
the same polarization, and thus one can consider only the total nuclear polarization. In
(B), spin-diffusion is fast among both the nearby and bulk nuclei. However, there is a
slow spin-diffusion step to transfer polarization between these two groups. In this case,
we allow electron-nuclear polarization transfer only to nearby nuclei and then to the bulk
via spin-diffusion. In (C), spin-diffusion is fast among the bulk nuclei. However, we do
not allow any transfer of polarization between the nearby and bulk nuclei, but rather
allow for a fast DNP step to the nearby nuclei, and a slow DNP step to the bulk nuclei.
We note these illustrations are not representative of the shape of the spin-diffusion
barrier, or the number of electrons inside it.
87
dPIdt
= kDNP PS − PI( ) + 1
T1IPI ,eq − PI( )
dPSdt
= NI
NS
kDNP PI − PS( ) + 1
T1S* PS ,eq
* − PS( ) (16)
Assuming that the electron rapidly reaches quasi-equilibrium ( dPS / dt = 0 ) leads to (17).
1
TB= 1
T1I+ kDNP1+ NI
NSkDNPT1S
*
⎛
⎝⎜⎞
⎠⎟
PI∞ = TB
kDNP1+ NI
NSkDNPT1S
* PS ,eq* + 1
T1IPI ,eq
⎛
⎝⎜⎞
⎠⎟
PI t( ) = PI∞ 1− exp − t TB( )( )
(17)
Here we introduce TB , which is the time constant for the appearance of polarization due
to microwave irradiation, and PI∞ , the polarization obtained on the nuclei in an infinitely
Figure 2: Maximum solid effect DNP enhancement.
The maximum DNP enhancement, ε∞ as a function of the product N I NS( )T1S at
different values of the nuclear spin lattice relaxation time T1I . Note that as the nuclear
T1I increases, the maximum DNP enhancement also increases.
88
long DNP experiment. Experimentally measuring T1I and TB then allows one to calculate
the expected enhancement in this model, as given in (18).
ε∞ = TB1
TB− 1
T1I
⎛⎝⎜
⎞⎠⎟PS ,eq*
PI ,eq+ 1
T1I
⎛
⎝⎜⎞
⎠⎟ (18)
Under a specific set of experimental conditions determined by the temperature
and sample characteristics, there is an upper bound on the enhancement that is less than
γ S γ I
. It is a function of the number of nuclei per electron N I NS( ) , and the electron
and nuclear spin-lattice relaxation times T1S , T1I( ) . In figure 2, we plot ε∞ calculated
when N I NS( )kDNPT1S 1 and k0 ≈ 0 . We see that increasing T1I increases the
maximum possible enhancement as does increasing the electron concentration, whereas
increasing T1S decreases enhancement. Although these parameters can be difficult to vary
independently, they may be optimized in a sample by changing the electron and proton
concentrations, or by varying the temperature or the paramagnetic center to alter T1S .
3.2.2.2 Case (B): Two-Step Bulk Polarization
In case (B), we examine a two-step model in which the major path of polarization
to the bulk is through the nearby nuclei adjacent to the electron spin via a slow spin-
diffusion step. By requiring polarization to proceed initially from the electron to the
nearby nuclei, and then to the bulk, we obtain (19), where we have dropped the
superscripts on kDNP since we only transfer to nearby nuclei.
dPI(b)
dt= kSD PI
(n) − PI(b)( ) + 1
T1I(b) PI ,eq − PI
(b)( )dPI
(n)
dt= kDNP PS − PI
(n)( ) + NI(b)
NI(n) kSD PI
(b) − PI(n)( )
+ 1
T1I(n) PI ,eq − PI
(n)( )dPSdt
= NI(n)
NS
kDNP PI(n) − PS( ) + 1
T1S* PS ,eq
* − PS( )
(19)
One may again assume a fast quasi-equilibrium of the electron with the nearby nuclei, but
it is not clear that quasi-equilibrium between the nearby and bulk nuclei is reasonable.
89
Instead, we assume that the derivatives of the nearby and bulk nuclear polarization have a
proportionality, α, and utilize this to solve for the buildup time (see Appendix for
derivation).
α =
− kDNP1+ NI
(n )
NSkDNPT1S
*+ kSD 1− NI
(b)
NI(n)
⎛⎝⎜
⎞⎠⎟+ 1
T1I(n) −
1T1I
(b)
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜
⎞
⎠⎟ +…
kDNP
1+ NI(n )
NSkDNPT1S
*+ kSD 1− NI
(b)
NI(n)
⎛⎝⎜
⎞⎠⎟+ 1
T1I(n) −
1T1I
(b)
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜
⎞
⎠⎟
2
+ 4NI
(b)
NI(n) kSD
2
2kSD1
TB= kSD 1−α( ) + 1
T1I(b)
(20)
Also, we may calculate the enhancement, given in (21).
ε∞ = 1
PI ,eq
kSDDT1I(b) + PI ,eq
1+ kSD 1−C( )T1I(b) (21)
The constants C and D are calculated, along with intermediate constants A and B, and are
given in (22).
A =NI(n )
NSkDNPT1S
*
1+ NI(n )
NSkDNPT1S
*
B =PS ,eq*
1+ NI(n )
NSkDNPT1S
*
C =NI(b )
NI(n ) kSDT1I
(n)
1+ NI(b )
NI(n ) kSDT1I
(n) + kDNP 1− A( )T1I(n)
D =kDNPBT1I
(b) + PI ,eq1+ NI
(b )
NI(n ) kSDT1I
(n) + kDNP 1− A( )T1I(n)
(22)
In this case, we must also calculate the observed nuclear longitudinal relaxation rate,
T1Iobs , because it will be different from T1I
(b) . This is because T1I(b) represents the bulk spin-
lattice relaxation, but does not include relaxation enhancement that results from spin-
diffusion to paramagnetically relaxed nuclei adjacent to the electron.
90
αT1=
− kSDNI(b)
NI(n) −1
⎛⎝⎜
⎞⎠⎟+ 1
T1I(n) −
1T1I(b)
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎞
⎠⎟
+ kSDNI(b)
NI(n) −1
⎛⎝⎜
⎞⎠⎟+ 1
T1I(n) −
1T1I(b)
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎞
⎠⎟
2
+ 4 NI(b)
NI(n) kSD
2
2kSD1
T1Iobs = kSD 1−αT1( ) + 1
T1I(b)
(23)
3.2.2.3 Case (C): Direct Bulk Polarization
In case (C) we consider the buildup of nuclear polarization if some electron
polarization transfers to nearby nuclei (which act as a polarization sink), and some
transfers directly to more distant bulk nuclei without being transferred to the nearby
nuclei first. In this case we assume that polarization transfer from the nearby nuclei to the
bulk via spin-diffusion is ineffective, a situation that may be created both by fast spin-
lattice relaxation of nearby nuclei, and by the spin-diffusion barrier [45; 47-51; 61-64].
dPI(b)
dt= kDNP
(b) PS − PI(b)( ) + 1
T1I(b) PI ,eq − PI
(b)( )dPI
(n)
dt= kDNP
(n) PS − PI(n)( ) + 1
T1I(n) PI ,eq − PI
(n)( )dPSdt
= NI(b)
NS
kDNP(b) PI
(b) − PS( ) + NI(n)
NS
kDNP(n) PI
(n) − PS( ) + 1
T1S* PS ,eq
* − PS( )
(24)
The nearby nuclei and electrons should both reach quasi-equilibrium, and so we may set
both of their derivatives to zero. However, for a clearer understanding of this process, we
start by simply setting dPI(n) dt = 0 . Writing PI
(n) as a function of PS then allows us to
rearrange dPS dt as shown in (25).
91
PI(n) = kDNP
(n)
1+ kDNP(n) T1I
(n) PS +PI ,eq
1+ kDNP(n) T1I
(n)
dPSdt
= NI(b)
NS
PI(b) − PS( ) + NI
(n)
NS
kDNP(n)
1+ kDNP(n) T1I
(n) +1
T1S*
⎛⎝⎜
⎞⎠⎟
*…
NI
(n)
NS
kDNP(n)
1+ kDNP(n) T1I
(n) +1
T1S*
⎛⎝⎜
⎞⎠⎟
−1PS ,eq
*
T1S* + NI
(n)
NS
kDNP(n) PI ,eq
1+ kDNP(n) T1I
(n)
⎛
⎝⎜⎞
⎠⎟− PS
⎛
⎝⎜
⎞
⎠⎟
(25)
Defining a new T1Seff and
PS ,eq
eff allows us to simplify the equations.
1
T1Seff =
NI(n)
NS
kDNP(n)
1+ kDNP(n) T1I
(n) +1
T1S*
⎛⎝⎜
⎞⎠⎟
PS ,eqeff = PS ,eq
* T1Seff
T1S* + NI
(n)
NS
kDNP(n) PI ,eqT1S
eff
1+ kDNP(n) T1I
(n)
⎛
⎝⎜⎞
⎠⎟
dPSdt
= NI(b)
NS
kDNP(b) PI
(b) − PS( ) + 1
T1Seff PS ,eq
eff − PS( )dPI
(b)
dt= kDNP
(b) PS − PI(b)( ) + 1
T1I(b) PI ,eq − PI( )
(26)
In this form, we see the transfer to the nearby nuclei simply depletes the overall amount
of polarization that is available from the electron, and also increases the effective electron
relaxation rate. One may then evaluate the differential equations in (26) the same manner
as was done for (16), and obtain the buildup time and enhancement in (27).
1TB
= 1T1I
+kDNP
(b)
1+ NI( b )
NSkDNP
(b) T1Seff
⎛
⎝⎜⎜
⎞
⎠⎟⎟
ε∞ = TB
kDNP(b)
1+ NI( b )
NSkDNP
(b) T1Seff
PS ,eqeff
PI ,eq
+ 1T1I
(b)
⎛
⎝⎜⎜
⎞
⎠⎟⎟
(27)
92
3.3 Experimental The power-dependence experiment seen in Figure 6 was performed with a 30 mW
microwave source operating at 139.5 GHz. All other experiments were recorded using a
120 mW source operating at 140.0 GHz. A coiled TE011 resonator (Q~1000) was used to
enhance the microwave field strengths to obtain electron nutation frequencies of up to 3.5
Figure 3: Pulse sequence and Spectrum
(a) Pulse sequence used for acquisition of DNP enhanced signals. Following saturation
of the 1H and 13C magnetization is a long CW microwave pulse that transfers electron
polarization to 1H that is subsequently transferred to 13C for observation. (b) Comparison
of enhanced 13C signal and the off-signal, recorded with recycle delays of 10 seconds and
18 seconds, respectively. We scale both spectra to the amplitude that would be obtained
with an infinitely long recycle delay, based on buildup and T1 data.
93
MHz, and to also act as a solenoid NMR coil [65]. Electron nutation frequencies were
determined using a two-pulse echo where the second pulse was set to approximately the
length of a π-pulse and the first pulse was incremented. A value of T1S = 1.43 ms was
measured for trityl radical using a saturation recovery experiment, with a 3 ms saturation
pulse and detected with a Hahn echo. T2S = 890 ns was measured by incrementing the
delay in a Hahn echo. This was performed at several pulse lengths and powers, and the
reported T2S was extrapolated to infinite pulse length, thus removing dephasing effects
from electron-electron couplings. A double-balanced 1H, 13C RF circuit was used for RF
irradiation and detection. Balancing of the circuit has greatly decreased arcing between
the iris of the microwave resonator and the waveguide. 40 mM OX063 trityl (a gift from
K. Golman and J.-H. Ardenkjær-Larsen of Nycomed Innovation AB, now GE Healthcare,
Malmo, Sweden) was used as a polarizing agent, being dissolved in a 60:25:15 (by
volume) 13C3-glycerol:D2O:H2O solution for experiments in Figure 6, and dissolved in a
60:40 (by volume) 13C-glycerol:D2O solution in all other experiments. All experiments
were performed at 80 K. The magnetic field was set to a position corresponding to the
positive solid effect matching condition, ω MW =ω0S −ω0 I for 1H polarization.
Experiments were performed by first applying a saturating train of pulses on both the 1H
and 13C channels, followed by microwave irradiation for some period, and finally 1H
polarization was transferred to 13C via cross-polarization [66; 67] and observed via echo-
detection [68]. For nuclear T1 measurements, a delay was placed between the microwave
irradiation period and the CP period, and the polarization decay was measured rather than
the buildup. Because our spectrometer can perform both the required NMR and EPR
measurements, all nutation frequencies, T1S and T2S , and DNP buildups and
enhancements were measured on the same sample, and the sample was not removed
between these measurements. The exception to this statement is the power dependence
illustrated in Figure 6, which was recorded earlier.
3.4 Results and Discussion
94
Figure 3 shows an enhancement of 94 obtained using the full microwave strength
ω1S 2π = 3.5 MHz( ) . We recorded the on-signal with a 10 s recycle delay (RD) and the
off-signal with an 18 s recycle delay, and scaled the peak amplitudes to give the relative
intensity that would be obtained for an infinite recycle delay on both on- and off-signal,
where the scaling factor is given by 1− exp(tRD / TB )( )−1 where tRD is the recycle delay,
and TB the characteristic buildup time. When the buildup time and T1 are different, as
seen in Figure 4 and Figure 5, a better measure of the improvement in signal-to-noise is
sensitivity (S/N*t-1/2) rather than enhancement, which in this case is a gain of 128. (One
may take more factors into account when calculating improvements from DNP, such as
dilution of the sample and bleaching due to the electron spin, as done recently by Jannin
et al. [69]) To our knowledge, this is the best gain in sensitivity reported for
contemporary DNP experiments using the solid effect with 1H enhancement at high fields
Figure 4: DNP Buildup
Microwave field dependence of the DNP enhancement and buildup times. The length of
polarization time is varied to observe the magnitude of the NMR signal, allowing one to
determine the DNP buildup time, TB . Measurement is taken at four power levels.
95
(5 T). Additionally, we see in Figure 6 that as we increase the microwave field strength,
we do not yet observe evidence of saturation of the solid effect.
We attribute our high enhancement to the use of a TE011 microwave cavity, which
is a high-Q (~1000), fundamental mode structure, and thus gives a large gain in the
microwave field strength. This suggests that if microwave field strength is not a limiting
factor, then DNP with a narrow line radical via the solid effect could perform very well at
high fields, since it can give both large enhancements and decrease the recycle delay. In
contrast, in most cases high microwave fields have not been shown to decrease the
buildup times when the cross effect is the dominant DNP mechanism (a recent exception
can be found in Feintuch, et al. [21]).
To gain more information about the DNP processes from these experiments, we
test each of the models discussed in the theory section and determine whether they fit the
observed buildup curves and enhancements. We first note that the strong power
dependence seen in Figure 6 shows that spin-diffusion is sufficiently fast to use effective
rate constants to account for polarization gradients due to finite rates of spin-diffusion.
Thus, the three models that we proposed that use this assumption are valid. Crucial to
testing these models are the buildup curves shown in Figure 4, where we have
incremented polarization times at increasing microwave field strengths to observe the
DNP buildup time, TB , and enhancements. In Figure 5 we show data where we have
Figure 5: Nuclear T1
Nuclear T1 ( T1Iobs ) measured by first polarizing 1H via DNP for 10 seconds, then turning
off the microwaves for some period of time and observing the magnetization decay with a
rate constant of 1 T1Iobs .
96
polarized the sample for 10 s, and then incremented a delay in order to observe T1
relaxation (T1Iobs ).
3.4.1 Case (A) In this case, we have neglected any important role of nearby nuclei. If this model
is correct, then we may calculate the value of ε∞ from TB and T1Iobs
using (18). Taking
the values for TB in Figure 4, T1Iobs = 13.7 s from Figure 5, and PS ,eq
* from Table I, we
calculate the value of ε∞ for each field strength and show this in Figure 7 according to
model (A), using (18).
Table I: Using the Bloch equations and T1S = 1.43 ms , T2S = 890 ns , (see the
experimental section for details) and Δω0S 2π = 212 MHz , we calculate the electron
polarization available for DNP, PS ,eq
* as defined in Eq(4). Note that Δω0S =ω0 I .
ω1S 2π [MHz] PS ,eq∗ PI ,eq
3.5 459 2.5 539 1.5 611 1.1 632
Figure 6: Power Dependence
Microwave field strength dependence of the solid effect DNP enhancement, ε, after 10
seconds (relative to off signal acquired with 10 s recycle delay).
97
One sees that for each microwave field strength, the observed enhancement is
lower than the enhancement predicted by case (A). This indicates that case (A) is not
sufficient to describe the polarization transfer, and some additional process must be
attenuating the total enhancement. Fits to cases (B) and (C) explore whether this
attenuation is due to an inefficient transfer of polarization to the bulk via a spin-diffusion
step, or due to depletion of the electron polarization by a transfer to isolated nearby
nuclei, respectively.
3.4.2 Case (B) We next consider the two-step model, for which we have an initial DNP step to
nearby nuclei, and then a slow spin-diffusion step to bulk nuclei. In this case, we are
confronted with many parameters, and more complicated formulas, so we refer to
Figure 7: Case (A) Fit
Calculated and experimentally observed solid effect DNP enhancements for the observed
buildup times shown in Figure 4 and using Eqn. (7). Note that the experimentally
observed enhancements are significantly lower than expected theoretically indicating that
the model discussed as Case (A) is not supported by the experimental data.
98
computer simulations to find a solution. We utilize the equations found in the theory and
appendix sections to quickly calculate accurate T1Iobs and TB rather than solving the
differential equations numerically. One may note that there are several unknown
parameters, including kDNP , kSD , T1I(b) , T1I
(n) , T1S , N I(b) N I
(n) , and N I(b) NS , whereas we
have three known parameters, TB , T1Iobs , and ε∞ . Therefore we must sample the space to
obtain the range of acceptable solutions. We fix T1S at 1.4 ms, because N I(n) NS and T1S
always appear together in our equations, thus it is redundant to vary both parameters. We
take TB , T1Iobs , and ε∞ for full microwave field strength (3.5 MHz) and use a simplex
routine to fit calculated values to these experimental measurements, using the fit function
given in (28). We weight the enhancements, observed T1I , and TB equally, and also apply
a penalty if the simulation uses a ratio of bulk nuclei to electrons that is greater than 2000
(for our sample, there are ~1640 protons per electron).
σ = ε∞
calc − ε∞obs( )2
+ TBcalc −TB
obs( )2+ T1I
calc −T1Iobs( )2
+1000 N I(b) NS ≥ 2000( )
(28)
We performed 1000 simplex fits using the MatLab[70] fminsearch function, with random
starting positions between the upper and lower starting bounds specified in Table II (the
simplex fit does not prevent solutions from being outside the bounds). The error, σ, is
evaluated for the three parameters and minimized. Of the 1000 simplex fits, 155 fits have
an RMS<0.5. We tabulate the range of each of the six parameters used for these 155 fits
in Table II, and also show one example fit. We see that it is possible to fit the
experimental data to this model, however we find for all solutions that T1I(n) > T1I
(b) which
is physically unreasonable (Table II two right columns), as proximity to the electron
causes paramagnetic relaxation and results in a short T1I . [45] Thus, the experimental
data are not explained satisfactorily with reasonable parameters by this model, and so we
discard it.
99
3.4.3 Case (C) We finally consider the case for which some polarization is transferred to nearby
nuclei that act as a polarization sink, and some polarization is transferred directly to the
bulk nuclei. Again, not all of the parameters are experimentally determined, however we
can group the parameters from (27) as kDNPeff and
PS ,eq
eff PI ,eq , and calculate these directly
from T1Iobs , TB , and ε∞ , as shown in (29).
kDNPeff =
kDNP(b)
1+ NI( b )
NSkDNP
(b) T1Seff
= 1TB
− 1T1I
obs
PS ,eqeff
PI ,eq
= 1kDNP
eff
ε∞
TB
− 1T1I
(b)
⎛
⎝⎜⎞
⎠⎟
(29)
Calculated values for kDNPeff and
PS ,eq
eff PI ,eq are plotted in Figure 8 for each of the four
microwave powers for which buildup curves were recorded, and also for the value of the
parameters at zero microwave power.
As seen in Figure 4, the enhancement increases and buildup time decreases with
an increase in the microwave field strength, ω1S . This is consistent with an increasing
Table II: Parameters used to simulate the experimental data to Case (B) model, and
some of the results. The simulation was performed only for the full microwave field
strength (3.5 MHz) with the measured parameters TB =7.4 s , ε∞=94 , and T1Iobs =13.7 s .
N I(n)
NS
N I(b)
N I(n) kDNP[s−1] kSD [s−1] T1I
(n)[s] T1I(b)[s]
Upper Starting Bound 1 1 0 0 0 0
Lower Starting Bound 1000 1000 1 2 30 300
Max Value for Fit 11169 3.9 17.8 0.21 214.2 10.2 Min Value for Fit 510 0.6 0.10 0.03 16.1 4.5
Example Fit 860 2.3 0.28 0.05 46.4 8.8
100
value of kDNP(b) , as one would expect. Furthermore, we see that the calculation of kDNP
eff
yields a rate constant that does not increase linearly, but in fact is accelerating upwards as
the microwave strength is increased. This additional gain in the magnitude of the rate
constant will lead to shorter buildup times; however, we note that it is accompanied by a
decreasing value of PS ,eq
eff , thus the actual enhancement may be attenuated. We also
calculate PS ,eq
eff from the experiments, which we show in Figure 8, and see that it is
decreasing as we increase the microwave field strength, towards a minimum value.
Again, this is what we would expect to see as kDNP(n) increases with the microwave field
strength, where eventually the polarization transfer to nearby nuclei becomes saturated.
Because all parameters calculated from the experimental data behave as expected when
the microwave field strength is changed, we believe this model is presently the optimal
description of the polarization transfer in the solid effect.
101
By comparing three different models to experimental data, we have shown
evidence that the primary mechanism for enhancement of bulk nuclear polarization
transfer is direct transfer of electron polarization to bulk nuclei, rather than transfer
through the nearby nuclei via a slow spin-diffusion step. However, we note that this does
Figure 8: Fit parameters for spin-diffusion barrier model.
a) Enhancement at infinite time from the four buildup curves shown in Figure 4, and also
for buildup with no microwave power. b) Values calculated for kDNPeff from experiment. c)
Values calculated for PS ,eqeff (solid line) from experiment and PS ,eq
* (dashed line) from
Bloch equations.
102
not exclude slow diffusion from the nearby nuclei to the bulk, but does suggest that there
is no major contribution from this process.
Our results have implications both on the consequence of the spin-diffusion
barrier, and the distance for which it is possible to perform direct solid effect DNP
transfers. We first point out that we do not expect there to be a sharp drop off in the rate
of spin-diffusion; rather the rate of diffusion varies continuously. This means that to
define a spin-diffusion barrier, we must assign some cutoff for the rate. The natural
choice for this would be that the diffusion rate from a particular spin to the bulk is equal
to that spin’s rate of polarization transfer to the lattice- therefore spins in the barrier
would contribute more towards polarization loss than towards polarization of the bulk.
This is essentially the definition which Wolfe proposed [49]. With this definition, Wolfe
showed that only about 12 protons or 19F nuclei near a paramagnetic impurity were
actually out of contact with the bulk via spin-diffusion in a paramagnetically doped
crystal (Y(C2H5SO4)3·9H2O:Yb3+) or in in CaF2, respectively [49-51]. However, in the
latter case, one proton at a distance of 5.2 Å was on the border of the barrier- where the
transfer rates to the bulk and lattice were about the same. This is an important result, as it
shows definitively that a spin-diffusion barrier exists, albeit much smaller than in many
previous treatments.
The experiments of Bowman et al. show that the nearest proton to the trityl center is at
least 4.8 Å away [53]. It seems unlikely that between radii of 4.8 Å and 5.2 Å there are
sufficiently many protons to account for the depletion of electron polarization seen in our
experiments. There are some important differences, however, that could allow the barrier
to be larger in our experiments. The first difference is temperature. Wolfe actually shows
that the proton found at 5.2 Å goes from being in strong contact with the bulk at 1.4 K to
being in strong contact with the lattice at 4 K, suggesting the diffusion barrier is getting
larger with higher temperature. This increase ceases at 4 K, however the reason for this is
that both 1 T1I and the diffusion rate constant to the bulk are linearly dependent on the
relaxation rate constant of the electron, 1 T1S [49]. At higher temperature, where there is
significantly more motion in the system, fluctuations in the dipolar field due to that
motion will contribute to nuclear spin-lattice relaxation. Near the electron, these
103
fluctuations in the field will be stronger, but will not depend on the electron T1 . As a
result, the nuclear relaxation rate becomes faster. If the spin-diffusion rate does not have
as large of an increase, which is possible since the rates no longer only depend on the
electron T1 , then the spin-diffusion barrier will expand.
The second difference is that the number of bulk spins in Wolfe’s experiment was
large enough, and the T1 of these spins long enough that the polarization of the bulk
could be treated as fixed. In our experiments, the bulk spins become polarized, and as a
result the rate of transfer from nearby to bulk decreases. Additionally, the polarization of
a nucleus under DNP conditions will be far from equilibrium, accelerating its T1
relaxation. Using the definition that the barrier occurs where the rate of diffusion to the
bulk equals the spin-lattice relaxation rate implies that with a decrease in the diffusion
rate, and increase in T1 relaxation, the diffusion barrier will also get larger. Because of
the large depletion of available polarization we observe and the differences between
experiments, we expect that our barrier is larger than the barrier observed by Wolfe.
To gain insight into how many “near neighbor” nuclei there are, we consider the
value of PS ,eq
eff in the case that kDNP(n) T1I
(n) 1 , which is shown in (30).
PS ,eq
eff = PS ,eq* 1+
T1S
T1I(n)
N I(n)
NS
⎛
⎝⎜⎞
⎠⎟+ PI ,eq 1+
T1I(n)
TS
NS
N I(n)
⎛
⎝⎜⎞
⎠⎟
(30)
To obtain PS ,eq
eff =211, which is the case for ω1S 2π = 3.5 MHz in Figure 8c, one needs
T1S T1I
(n)( ) N I(n) NS( ) ≈1.2 . This implies that if there are ~10 nuclei within the barrier,
then T1I(n) must be ~12 ms, whereas if there are ~100 nuclei within the barrier, then T1I
(n) is
~120 ms; in other words, if the T1I(n) is longer, more spins must be inside the diffusion
barrier to account for our observations. Since it is not clear what the rate of paramagnetic
relaxation is, though, it is difficult to determine the “diffusion barrier” radius and the
number of nuclei within this radius with these experiments.
Our results also highlight the ability of the solid effect to transfer polarization
over large distances. Afeworki and coworkers had shown that it is possible to transfer
104
polarization over 30-60 Å distances in 15.1 MHz 13C-DNP experiments [71]. Our results
support this finding since our model requires direct transfer of polarization to bulk nuclei,
and further demonstrate that distant transfers are also possible at much higher nuclear
Larmor frequencies.
3.5 Conclusions We demonstrate through fitting several models to experimental data that
polarization is primarily transported from the electron directly to bulk nuclei. This is
opposed to polarization being transferred to nearby nuclei and then to the bulk via a slow
spin-diffusion step. Also shown is that the polarization available from the electron is
decreased because of polarization transferred to nearby nuclei that is then rapidly relaxed
away, which is described by PS ,eq
eff . Finally, we see that it is necessary to take into account
experimental conditions when considering the spin-diffusion barrier, as its effective size
depends on relative rates that vary under different conditions.
Additionally, DNP via the solid-effect, using the narrow line trityl radical, has
shown a gain in sensitivity of 128. Enhancements are still increasing with microwave
field strength at our peak available power, suggesting that where higher field strengths
available, the solid effect can be a very useful DNP mechanism because it both leads to
large enhancements and further boosts sensitivity by decreasing the buildup time.
3.6 Appendix
3.6.1 Solving One-Step Transfer Equations without Fast Equilibrium
Here we present a solution of a general pair of differential equations at long times
without assuming a fast equilibrium. We will solve the following two equations.
105
dPadt
= k Pb − Pa( ) + 1
T1aPa,eq − Pa( )
dPbdt
= Na
Nb
k Pa − Pb( ) + 1
T1bPb,eq − Pb( )
(A. 1)
We begin by assuming that the first and second derivatives are proportional for Pa and
Pb , and have proportionality, α, that is constant in time. We can solve for the necessary
value of α for time independence by requiring the second derivatives have the same
proportionality as the first derivatives- a consequence of time independence of α.
dPbdt
=α dPadt
d 2Pbdt 2
=α d 2Padt 2
(A. 2)
One can take then take the second derivatives, and using the proportionality constant of
the first derivatives, solve for the value of α by satisfying the second part of (A. 2).
α d 2Padt 2
=α kdPbdt
− dPadt
⎛⎝⎜
⎞⎠⎟ −
1
T1a
dPadt
⎛⎝⎜
⎞⎠⎟= − k α −α 2( ) + 1
T1a
⎛⎝⎜
⎞⎠⎟dPadt
d 2Pbdt 2
= nk dPadt
− dPbdt
⎛⎝⎜
⎞⎠⎟ −
1
T1b
dPbdt
= − nk α −1( ) + 1
T1b
⎛⎝⎜
⎞⎠⎟dPadt
(A. 3)
Setting these two results equal, we obtain the following.
k α −α 2( ) + αT1a
= nk α −1( ) + 1
T1b
0 = kα 2 + k n −1( ) + 1
T1b− 1
T1a
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎞
⎠⎟α − nk
α =− n −1( )k + 1
T1b− 1T1a
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎞
⎠⎟+ n −1( )k + 1
T1b− 1T1a
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎞
⎠⎟
2
+ 4nk2
2k
(A. 4)
Given α, we may now solve for the buildup time, TB . Since we know the proportionality
of the first derivatives, we can say the following.
Pb t( ) = Pb0 −αPa
0( ) +αPa (A. 5)
106
We do not offer analytic solutions to Pa0 and Pb
0 , but note that these are not necessary to
solve to determine the buildup time, TB . Plugging this equation into (A. 1), we can obtain
the buildup rate.
dPadt
= − k 1−α( ) + 1
T1a
⎛⎝⎜
⎞⎠⎟Pa + k Pb
0 −αPa0( ) + 1
T1aPa,eq (A. 6)
Thus, the buildup rate is given by the coefficient of Pa .
1
Tb= k 1−α( ) + 1
T1a (A. 7)
We point out that once the time derivative of α goes to zero, it forces all derivatives to
have the same proportionality. Once this condition is met for one time, it will continue to
be met for all times.
Now that we know how to solve a case where fast equilibrium ( dPa dt = 0 ) is not
a valid assumption, we apply this technique to two systems of equations.
3.6.2 Two-Step DNP Transfer For the two step transfer, it should be possible to approach both steps by assuming
proportionality of the derivatives- however the solutions to the proportionality constants
will involve quartic equations, which do not have general solutions as do quadratic
equations. As an alternative, we take the fast equilibrium solution of the electrons, and
apply the assumption of proportionality of derivatives for the second (nearby to bulk
nuclei) transfer step. We note that assumption of fast equilibrium of the electrons in the
one-step transfer is in fact a very good solution, and we present the alternative here only
as example before presenting it in the more difficult case of a two-step transfer.
To begin, we present the rate equations governing the two-step transfer.
107
dPI(b)
dt= kSD PI
(n) − PI(b)( ) + 1
T1I(b) PI ,eq − PI
(b)( )dPI
(n)
dt= kDNP PS − PI
(n)( ) + NI(b)
NI(n) kSD PI
(b) − PI(n)( )
+ 1
T1I(n) PI ,eq − PI
(n)( )dPSdt
= NI(n)
NS
kDNP PI(n) − PS( ) + 1
T1S* PS ,eq
* − PS( )
(A. 8)
In this case, we assume the electron reaches a fast equilibrium, so that dPS dt = 0 , thus
PS (t) can be written as shown in (A. 9).
Ps =NI(n )
NSkDNPT1S
*
1+ NI(n )
NSkDNPT1S
*
A
PI(n) +
PS ,eq*
1+ NI(n )
NSkDNPT1S
*
B
(A. 9)
We may now substitute this into dPI(n) dt .
dPI(n)
dt= − kDNP
1+ NI(n )
NSkDNPT1S
*+ NI
(b)
NI(n) kSD PI
(b) − PI(n)( )
+ 1
T1I(n) PI ,eq − PI
(n)( ) + kDNPB (A. 10)
We now assume that dPI(n) dt =α dPI
(b) dt where α is constant in time. If we take the
second derivatives of PI(n) and αPI
(b) , and substitute dPI(n) dt =α dPI
(b) dt , we obtain
the following equations.
d 2PI(n)
dt 2− kDNP1+ NI
(n )
NSkDNPT1S
*− NI
(b)
NI(n) kSD 1−α( ) + α
T1I(n)
⎛
⎝⎜
⎞
⎠⎟dPI
(b)
dt
α d 2PI(b)
dt 2= − kSD 1−α( )α + α
T1I(b)
⎛⎝⎜
⎞⎠⎟dPI
(b)
dt
(A. 11)
Setting these equal, we obtain (A. 12).
108
kDNP1+ NI
(n )
NSkDNPT1S
*α − NI
(b)
NI(n) kSD 1−α( ) + α
T1I(n) = kSD 1−α( )α + α
T1I(b)
α =
− kDNP1+ NI
(n )
NSkDNPT1S
*+ kSD
NI(b)
NI(n) −1
⎛⎝⎜
⎞⎠⎟+ 1
T1I(n) −
1T1I
(b)
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜
⎞
⎠⎟ +…
kDNP
1+ NI(n )
NSkDNPT1S
*+ kSD
NI(b)
NI(n) −1
⎛⎝⎜
⎞⎠⎟+ 1
T1I(n) −
1T1I
(b)
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜
⎞
⎠⎟
2
+ 4NI
(b)
NI(n) kSD
2
2kSD
(A. 12)
Now that we have obtained α, we can calculate the buildup rate, TB , by substituting PI(n)
into dPI(b) dt .
dPI(b)
dt= kSD 1−α( ) + 1
T1I(b)
⎛⎝⎜
⎞⎠⎟PI(b) + kSD PI ,0
(n) −αPI ,0(b)( ) + PI ,eq
T1I(b) (A. 13)
Again we can obtain the buildup time from the coefficient to PI(b) .
1
TB= kSD 1−α( ) + 1
T1I(b) (A. 14)
Another important point here is that due to the spin-diffusion, the observed T1I will not
be equal to T1I(b) . Rather, it is a function of the spin-diffusion, T1I
(b) , and T1I(n) . We can
obtain this easily by setting kDNP = 0 in the above formulas, causing A=0 as well.
αT1=
− kSDNI(b)
NI(n) −1
⎛⎝⎜
⎞⎠⎟+ 1
T1I(n) −
1T1I(b)
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎞
⎠⎟+
kSDNI(b)
NI(n) −1
⎛⎝⎜
⎞⎠⎟+ 1
T1I(n) −
1T1I(b)
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎞
⎠⎟
2
+ 4 NI(b)
NI(n) kSD
2
2kSD1
T1Iobs = kSD 1−αT1( ) + 1
T1I(b)
(A. 15)
109
Finally, we calculate the equilibrium nuclear polarization, which is a trivial calculation,
as it only requires setting all derivatives to zero. We already have set dPS dt = 0 in (A.
9). Here, we show the result of setting dPI(n) dt = 0 .
PI(n) =
NI(b )
NI(n ) kSDT1I
(n)
1+ NI(b )
NI(n ) kSDT1I
(n) + kDNP 1− A( )T1I(n)C
PI(b)
+kDNPBT1I
(b) + PI ,eq1+ NI
(b )
NI(n ) kSDT1I
(n) + kDNP 1− A( )T1I(n)D
(A. 16)
Finally, we set dPI(b) dt = 0 to obtain the nuclear enhancement.
ε = PI(b) t = ∞( ) = kSDDT1I
(b) + PI ,eq1+ kSD 1−C( )T1I(b)
(A. 17)
Thus, we have obtained formulas for the observed buildup time, TB , the enhancement, ε,
and the observed nuclear relaxation time, T1Iobs .
3.7 Bibliography [1] A.W. Overhauser, Physical Review 92 (1953) 411. [2] T.R. Carver, and C.P. Slichter, Physical Review 92 (1953) 212. [3] D. Hall, D. Maus, G. Gerfen, S. Inati, L. Becerra, F. Dahlquist, and R. Griffin,
Science 276 (1997) 930-2. [4] M. Rosay, V. Weis, K.E. Kreischer, R.J. Temkin, and R.G. Griffin, Journal of the
American Chemical Society 124 (2002) 3214-3215. [5] M. Rosay, A. Zeri, N. Astrof, S. Opella, J. Herzfeld, and R. Griffin, J Am Chem Soc
123 (2001) 1010-1. [6] M. Rosay, J. Lansing, K. Haddad, W. Bachovchin, J. Herzfeld, R. Temkin, and R.
Griffin, J Am Chem Soc 125 (2003) 13626-7. [7] T. Maly, G.T. Debelouchina, V.S. Bajaj, K.-N. Hu, C.-G. Joo, M.L. Mak-Jurkauskas,
J.R. Sirigiri, P.C.A. van der Wel, J. Herzfeld, R.J. Temkin, and R.G. Griffin, J. Chem. Phys. 128 (2008) 052211-19.
[8] T. Maly, G.T. Debelouchina, V.S. Bajaj, K.-N. Hu, C.-G. Joo, M.L. Mak-Jurkauskas, J.R. Sirigiri, P.C.A. van der Wel, J. Herzfeld, R.J. Temkin, and R.G. Griffin, The Journal of Chemical Physics 128 (2008) 052211-19.
110
[9] A.B. Barnes, G. De Paëpe, P.C.A. van der Wel, K.N. Hu, C.G. Joo, V.S. Bajaj, M.L. Mak-Jurkauskas, J.R. Sirigiri, J. Herzfeld, R.J. Temkin, and R.G. Griffin, Applied Magnetic Resonance 34 (2008) 237-263.
[10] V.S. Bajaj, M.L. Mak-Jurkauskas, M. Belenky, J. Herzfeld, and R.G. Griffin, Proceedings of the National Academy of Sciences 106 (2009) 9244-9249.
[11] V.S. Bajaj, M.L. Mak-Jurkauskas, M. Belenky, J. Herzfeld, and R.G. Griffin, Journal of Magnetic Resonance 202 (2010) 9-13.
[12] G.T. Debelouchina, M.J. Bayro, P.C.A. van der Wel, M.A. Caporini, A.B. Barnes, M. Rosay, W.E. Maas, and R.G. Griffin, Physical Chemistry Chemical Physics 12 (2010) 5911-5919.
[13] Ü. Akbey, W.T. Franks, A. Linden, S. Lange, and R.G. Griffin, Angewandte Chemie International Edition 49 (2010) 7803-7806.
[14] A. Lesage, M. Lelli, D. Gajan, M.A. Caporini, V. Vitzthum, P. Mieville, J. Alauzun, A. Roussey, C. Thieuleux, A. Mehdi, G. Bodenhausen, C. Coperet, and L. Emsley, Journal of the American Chemical Society 132 (2010) 15459-15461.
[15] C. Jeffries, D., Physical Review 106 (1957) 164. [16] C.D. Jeffries, Physical Review Phys. Rev. PR 117 (1960) 1056. [17] A. Abragam, and W. Proctor, G., C. R. Acad. Sci. (1959). [18] L. Becerra, G. Gerfen, R. Temkin, D. Singel, and R. Griffin, Physical Review
Letters 71 (1993) 3561-3564. [19] K. Hu, V. Bajaj, M. Rosay, and R. Griffin, J Chem Phys 126 (2007) 044512. [20] B. Corzilius, A.A. Smith, A.B. Barnes, C. Luchinat, I. Bertini, and R.G. Griffin,
Journal of the American Chemical Society 133 (2011) 5648-5651. [21] A. Feintuch, D. Shimon, Y. Hovav, D. Banerjee, I. Kaminker, Y. Lipkin, K.
Zibzener, B. Epel, S. Vega, and D. Goldfarb, Journal of Magnetic Resonance 209 (2011) 136-141.
[22] T.J. Schmugge, and C.D. Jeffries, Physical Review 138 (1965) A1785. [23] W. de Boer, Journal of Low Temperature Physics 22 (1976) 185-212. [24] A. Kessenikh, V., V. Lushchikov, I., A. Manenkov, A., and Y. Taran, V., Soviet
Physics - Solid State 5 (1963) 321-329. [25] A. Kessenikh, V., A. Manenkov, A., and G. Pyatnitskii, I., Soviet Physics - Solid
State 6 (1964) 641-643. [26] C. Hwang, F., and D. Hill, A., Physical Review Letters 19 (1967) 1011. [27] D. Wollan, S., Physical Review B: Condensed Matter 13 (1976) 3671. [28] K. Hu, H. Yu, T. Swager, and R. Griffin, Journal of the American Chemical Society
126 (2004) 10844-5. [29] K.-N. Hu, C. Song, H.-h. Yu, T.M. Swager, and R.G. Griffin, The Journal of
Chemical Physics 128 (2008) 052302-17. [30] K.-N. Hu, G.T. Debelouchina, A.A. Smith, and R.G. Griffin, Journal of Chemical
Physics 134 (2011). [31] C. Song, K.-N. Hu, T.M. Swager, and R.G. Griffin, J. Am Chem. Soc 128 (2006)
11385-90. [32] Y. Matsuki, T. Maly, O. Ouari, H. Karoui, F. Le Moigne, E. Rizzato, S. Lyubenova,
J. Herzfeld, T.F. Prisner, P. Tordo, and R.G. Griffin, Angewandte Chemie 48 (2009) 4996-5000.
111
[33] M.J. Duijvestijn, A. Manenschijn, J. Smidt, and R.A. Wind, Journal of Magnetic Resonance 64 (1985).
[34] M. Goldman, Spin temperature and nuclear magnetic resonance in solids, Clarendon Press, Oxford, 1970.
[35] M. Duijvestijn, J., R. Wind, A., and J. Smidt, Physica B+C 138 (1986) 147-170. [36] S. Jannin, A. Comment, F. Kurdzesau, J.A. Konter, P. Hautle, B.v.d. Brandt, and
J.J.v.d. Klink, The Journal of Chemical Physics 128 (2008) 241102. [37] G. Jeschke, and A. Schweiger, Molecular Physics 88 (1996) 355-383. [38] W.T. Wenckebach, Applied Magnetic Resonance 34 (2008) 227-235. [39] V. Weis, M. Bennati, M. Rosay, and R.G. Griffin, The Journal of Chemical Physics
113 (2000) 6795-6802. [40] A. Abragam, and M. Goldman, Reports on Progress in Physics 41 (1978) 395-467. [41] Y. Hovav, A. Feintuch, and S. Vega, Journal of Magnetic Resonance 207 (2010)
176-189. [42] Y. Hovav, A. Feintuch, and S. Vega, Journal of Chemical Physics 134 (2011)
074509(1-20). [43] V. Nagarajan, Y. Hovav, A. Feintuch, S. Vega, and D. Goldfarb, The Journal of
Chemical Physics 132 (2010) 214504-1 to 13. [44] J. Granwehr, and W. Köckenberger, Applied Magnetic Resonance 34 (2008) 355-
378. [45] N. Bloembergen, Physica 15 (1949) 386-426. [46] G.R. Khutsishvili, Sov. Phys. JETP 15 (1962) 909. [47] W.E. Blumberg, Physical Review 119 (1960) 79-84. [48] M. Goldman, Physical Review 138 (1965). [49] J.P. Wolfe, Physical Review Letters 31 (1973) 907-910. [50] J.P. Wolfe, and R.S. Markiewicz, Physical Review Letters 30 (1973) 1105-1107. [51] A.D.A. Hansen, and J.P. Wolfe, Physics Letters 66A (1978) 320-322. [52] C. Ramanathan, Applied Magnetic Resonance 34 (2008) 409-421. [53] M. Bowman, C. Mailer, and H. Halpern, J Magn Reson 172 (2005) 254-67. [54] R.R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of nuclear magnetic
resonance in one and two dimensions, Clarendon, Oxford, 1987. [55] E.P. Horvitz, Physical Review B 3 (1971) 2868-2872. [56] A. Schweiger, and G. Jeschke, Principles of pulse electron paramagnetic resonance,
Oxford University Press, Oxford, UK, 2001. [57] F. Bloch, Physical Review 70 (1946) 460. [58] O.S. Leifson, and C.D. Jeffries, Physical Review 122 (1961) 1781-1795. [59] C.D. Jeffries, Dynamic Nuclear Orientation, Interscience, New York, 1963. [60] A. Abragam, and M. Goldman, Nuclear magnetism : order and disorder, Clarendon
Press Oxford University Press, Oxford New York, 1982. [61] N. Bloembergen, E.M. Purcell, and R.V. Pound, Physical Review 73 (1948) 679-
712. [62] J.I. Kaplan, Physical Review B 3 (1971). [63] H.E. Rorshach, Physica 30 (1964) 38. [64] P.G. de Gennes, Phys. Chem. Solids 7 (1958). [65] V. Weis, M. Bennati, J.A. Bryant, and R.G. Griffin, Journal of Magnetic Resonance
140 (1999) 293-299.
112
[66] S.R. Hartmann, and E.L. Hahn, Phys. Rev. 5 (1962) 2042-2053. [67] A. Pines, M.G. Gibby, and J.S. Waugh, The Journal of Chemical Physics 56 (1972)
1776-1777. [68] E.L. Hahn, Physical Review 80 (1950) 580-594. [69] V. Vitzthum, F. Borcard, S. Jannin, M. Morin, P. Miéville, M.A. Caporini, A.
Sienkiewicz, S. Gerber-Lemaire, and G. Bodenhausen, ChemPhysChem 12 (2011) 2929-2932.
[70] MathWorks, MatLab, MathWorks, Natick, 2009. [71] M. Afeworki, S. Vega, and J. Schaefer, Macromolecules 25 (1992) 4100-4105.
113
Chapter 4: Solid Effect in Magic Angle Spinning
Dynamic Nuclear Polarization
Contributing: Björn Corzilius
114
Abstract
For over five decades, the solid effect (SE) has been heavily utilized as a
mechanism for performing dynamic nuclear polarization (DNP). Nevertheless, it has not
found widespread application in contemporary, high magnetic field DNP experiments
because SE enhancements display an ω 0−2 field dependence. In particular, for nominally
forbidden zero and double quantum SE transitions to be partially allowed, it is necessary
for mixing of adjacent nuclear spin states to occur, and this leads to the observed field
dependence. However, recently we have improved our instrumentation and report here an
enhancement of ε = 91 obtained with the organic radical trityl (OX063) in magic angle
spinning (MAS) experiments. This is a factor of 6-7 higher than previous values in the
literature under similar conditions. Because the solid effect depends strongly on the
microwave field strength, we attribute this large enhancement to larger microwave field
strengths inside the sample volume, achieved with more efficient coupling of the
gyrotron to the sample chamber. In addition, we develop a theoretical model to explain
the dependence of the buildup rate of enhanced nuclear polarization and the steady-state
enhancement on the microwave power. Buildup times and enhancements were measured
as a function of 1H concentration for both trityl and Gd-DOTA. Comparison of the results
indicates that for trityl the initial polarization step is the slower, rate-determining step.
However, for Gd-DOTA the spread of nuclear polarization via homonuclear 1H spin
diffusion is the rate-limiting. Finally, we discuss the applicability of the solid effect at
fields > 5 T and the requirements to address the unfavorable field dependence of the solid
effect.
115
4.1 Motivation
Dynamic nuclear polarization (DNP) is a technique capable of enhancing the
sensitivity of nuclear magnetic resonance (NMR) by several orders of magnitude. DNP
was first proposed by Overhauser [1] and confirmed experimentally by Carver and
Slichter [2] for the case of a conducting solid with mobile electrons. Subsequently,
additional DNP mechanisms emerged for insulating solids, the first being solid effect
(SE) [3-5] which relies on the availability of a narrow line polarizing agent. Thereafter,
two DNP mechanisms which greatly differ from the solid effect were discovered;
namely, the cross effect (CE) [6-10] and thermal mixing (TM) [11]. Both the CE and TM
rely on multi-spin interactions and are active when the breadth of the EPR spectrum
matches or exceeds the nuclear Larmor frequency of the nucleus being polarized.
The SE utilizes mixing of electronic and nuclear spin states mediated by the non-
secular electron-nuclear coupling and microwave irradiation to excite nominally
“forbidden” zero or double quantum (ZQ/DQ) transitions leading to enhanced nuclear
polarization [12]. In order to selectively excite the transition leading to either positive or
negative nuclear enhancements, both forbidden transitions must be well separated in the
field/frequency domain [13]. This is the case when both the homogeneous and
inhomogeneous linewidth of the polarizing agent’s electron paramagnetic resonance
(EPR) signal is smaller than twice the Larmor frequency of the nucleus to be polarized.
Thus, “narrow-linewidth” radicals such as trityl [14] or BDPA [15] are effective
polarizing agents for the solid effect [16].
More than 50 years ago Jeffries et al. and Abragam [3-5] independently
performed the first experiments based on the SE. These early efforts were conducted at
116
low magnetic field (0.3-1.4 T) where the inherent efficiency of the solid effect is
relatively high due to the favorable mixing of electronic and nuclear spin states.
Furthermore, high-power microwave sources and cavities operating in the 10-40 GHz
frequency range are readily available, and considerable attention was focused on
understanding and implementing SE experiments. However, with the contemporary
transition of DNP to high magnetic fields, the SE has played a minor role in comparison
to the more generally efficient cross effect using biradical polarizing agents [16-18]. The
reason for this has been the understanding that, due to the ω 0−2 field dependence, the
enhancements will not be large which appears to have been confirmed experimentally.
For example, in-situ magic angle spinning (MAS) experiments performed by Hu et al. at
5 T (140 GHz) yielded ε ≈ 10-15 [19]. In contrast, with biradicals as polarizing agents,
enhancements of ~200 were achieved with relatively low radical concentrations of 10-20
mM [20]. Accordingly, these results stimulated development of methods focused on
optimizing DNP parameters and polarizing agents for the cross effect [16; 17; 19; 21-27].
The exceptions to this statement are the theoretical discussions by Vega and coworkers
[28-30].
Despite the modest enhancements, we have continued to investigate DNP
experiments with the SE, and have recently shown that enhancements of 94 are feasible
at 5 T and 80 K using trityl as the polarizing agent. These experiments were performed
with a microwave cavity in order to increase microwave field strength inside the sample
volume [31]. These results suggested that if we could efficiently couple the microwaves
to the sample in MAS experiments that similar enhancements should be observed. In this
paper we show that this is indeed the case for MAS experiments performed without a
117
resonant structure if the microwaves generated by a high-power gyrotron are efficiently
coupled into the MAS stator through overmoded corrugated waveguides. We analyze the
results and, based on experimental findings, suggest that significantly larger
enhancements can be obtained at high field given the availability of sufficient microwave
field strength.
4.2 Theory
4.2.1 Diagonalization of the static Hamiltonian for the solid effect
The simplest system allowing us to describe the SE consists of two spins: an
electron spin and a nuclear spin, with S = I = 12 . The static Hamiltonian describing this
system contains the electron and nuclear Zeeman interaction between the respective spins
and the external magnetic field as well as the electron-nuclear dipolar coupling between
the two spins:
H0 =ω0S Sz −ω0 I Iz + ASz Iz + BSz Ix . (1)
Si and Ii are vector elements of the electron and nuclear spin operators and ,
respectively, ω0S and ω0I denote the electron and nuclear Larmor frequencies, and A and
B represent the secular and non-secular part of the electron-nuclear dipolar coupling.
Note that the following treatment is in accordance with that by Schweiger and Jeschke
[32], except for the sign of the nuclear Zeeman interaction, which we explicitly take as
negative. Equation (1) is valid in the so-called “pseudo-high-field approximation”
meaning all non-secular are omitted, except those couplings that lead to tilting of the
118
nuclear eigenstates ( BSz Ix ). Although Hu et al. [33] recently published a detailed
description of the diagonalization of this Hamiltonian, we repeat some of the crucial steps
in this section since they are important in the discussion of our experimental
observations.
The Hamiltonian in (1) is block diagonal in the two-dimensional vector space
S ,mS = 12 ,± 1
2 with the kets and corresponding to mS = ±1 2 . As usual, the
electron polarization operators are
Sα /β = 1
2E ± Sz , (2)
where E is the unit operator. Inserting Sα /β leads to a new form of the Hamiltonian
H0 =ω0S Sz −ω0 I Sα Iz +
A2
Sα Iz +B2
Sα Ix
H0α
−ω0 I Sβ Iz −
A2
S β Iz −B2
S β Ix
H0β
(3)
that can be easily diagonalized analytically by a unitary transformation
H0D =UH0U
−1 (4)
119
with the transformation operator
. (5)
The diagonalized Hamiltonian has the form:
′H0 =ω0S Sz −ωαSα Iz −ω
βS β Iz (6)
with
ωα ,β = ω0 I ∓
A2
⎛⎝⎜
⎞⎠⎟
cos ηα ,β( ) ± B2
sin ηα ,β( ) . (7)
The branching angles are depicted in Figure 1 and are defined as
ηα ,β = arctan BA∓ 2ω0 I
⎛
⎝⎜⎞
⎠⎟ with − π
2<ηα ,β < π
2. (8)
Note that this definition of the ηα ,β differs from the one used by Schweiger and
Jeschke [32] since the second term in (1) , −ω0 Iz , is negative rather than positive.
Figure 1. Solid Effect Branching Angles
Derivation of the branching angles in the α (green) and β (red) electron spin subspaces.
120
We satisfy the matching condition for the solid effect if the states connected by the zero
or double quantum transitions are degenerate in the frame rotating with the frequency of
the microwave irradiation. In this case the matching condition is
Δω0S
0,2( ) = ∓ω0 I
2cosηα + cosηβ( ) ± A
4cosηα − cosηβ( )∓ B
4sinηα + sinηβ( ) , (9)
where and are the microwave frequency offsets required for matching the
zero and double quantum transition frequencies, respectively. We define Δω0S as
Δω0S =ωmw −ω0S (10)
so that if Δω0S = Δω0S(0) or Δω0S = Δω0S
(2) , then the solid effect matching condition is
satisfied.
In the case of we see that the absolute values of the branching angles
converge and we can define a common branching angle
η = arctan − B
2ω0 I
⎛
⎝⎜⎞
⎠⎟≈ηα ≈ −ηβ . (11)
In this limit eqn. (9) simplifies to the often-cited solid effect matching condition
Δω0S0,2( ) = ±ω0 I . (12)
Therefore, we satisfy the matching condition for the solid effect by irradiating the spin
system at the sum or difference of the electron and nuclear Larmor frequencies.
121
4.2.2 Transition moments of the solid effect
In the limit of small microwave fields the transition moments for the double and
zero quantum transitions can be obtained by transforming the rotating-frame microwave
Hamiltonian
Hmw =ω1S Sx (13)
into the eigenframe of the static spin Hamiltonian . We then obtain
′Hmw =ω1S Sx cosη− + 2Sy I y sinη−( ) (14)
with
η− = ηα −ηβ
2. (15)
can be expressed in the basis of raising and lowering operators
S ± = Sx ± iSy
I ± = Ix ± iI y
(16)
which yields
′Hmw =
ω1S
2S + + S −( )cosη− + S + I − + S − I +( )− S + I + + S − I −( )⎡
⎣⎤⎦sinη−{ } . (17)
While the first term describes single-quantum EPR coherences, we see that introduction
of an anisotropic electron-nuclear coupling with the non-secular component B, the second
term becomes non-zero; therefore, zero and double quantum coherences can be generated
which drive the solid effect transitions.
We make a further simplification by using the fact that ω0 I A 2 . Using
eqn. (11), we obtain:
122
′Hmw =ω1S
2S + + S −( )cosη + S + I − + S − I +( )− S + I + + S − I −( )⎡
⎣⎤⎦sinη{ } . (18)
Simple trigonometric rules allow us to express eqn. (18) as
′Hmw =
ω1S
2−2ω0 I
4ω0 I2 + B2
S + + S −( ) + B4ω0 I
2 + B2S + I − + S − I +( )− S + I + + S − I −( )⎡
⎣⎤⎦
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪.(19)
By further making the valid assumption of ω0 I B 2 , we can simplify (19) to:
′Hmw =ω1S
2S + + S −( ) + B
2ω0 I
S + I − + S − I +( )− S + I + + S − I −( )⎡⎣
⎤⎦
⎧⎨⎩⎪
⎫⎬⎭⎪
. (20)
Equation (20) is derived for ω0 I A, B , and, since A and B decrease with increasing
distance between the electron and the nuclear spins, it is only valid for nuclei which
reside a certain radius from the electron spin. However, in a high magnetic field, this
condition is satisfied for all protons of trityl and in the surrounding matrix.
In (20) we also see that the matrix elements driving the zero and double quantum
solid effect transitions scale as ω1S B ω0 I( ) . So there are two possibilities to increase the
transition moment at a given field for a given nucleus: (i) increase the microwave field
strength ω1S, or (ii) increase the electron-nuclear coupling constant. (i) can be achieved by
using higher power microwave sources and/or by improving the coupling of the
microwave power to the sample volume and is therefore an engineering problem. (ii) is
an inherent characteristic of the sample itself. For a given nucleus the electron-nuclear
dipolar coupling is modulated by the distance and orientation with respect to the external
magnetic field vector. The orientation plays a crucial role, since the non-secular
component vanishes, if one of the principal axes of the electron-nuclear coupling tensor is
parallel to the magnetic field vector.
123
4.2.3 DNP kinetics based on rate equations
All of the derivations in the preceding sections describe the evolution of the state
of a single electron-nucleus pair in the absence of any relaxation processes. However, SE
DNP is a non-coherent transfer of electron polarization to nuclei, and therefore to
describe the kinetics of polarization buildup in an ensemble of nuclei during a DNP
experiment, it is essential to include the relaxation processes that govern the behavior of
the system. In the next section we derive the rate equations describing the polarization
transfer. This approach, in combination with results obtained in the preceding sections
allows us to calculate a rate constant, , which describes the initial DNP polarization
transfer. This approach has been treated in detail by Smith et al. [31], and the validity of
this derivation discussed extensively. It was shown that this treatment is valid and
applicable if changes in the observed nuclear polarization occur on a timescale slow
compared to that of individual coherences, which typically occur on a timescale of a few
milliseconds or shorter. We show in a subsequent section that the experimental build-up
time constant is a few tens of seconds, so this assumption is valid for the case considered
here.
Here we present the equations describing the excitation of the double quantum
(DQ) electron-nuclear coherence. A similar treatment can be performed for the zero
quantum (ZQ) case, which is shown in the Appendix. We begin with the complete
electron-nuclear spin Hamiltonian including the microwave induced coupling terms in the
eigenframe of the static Hamiltonian:
124
′H = Δω0S Sz −ω0 I Iz + A+ B
2sinη
⎛⎝⎜
⎞⎠⎟
Sz Iz +ω1S
2S + + S −( ) + ω1S B
4ω0 I
S + I − + S − I +( )− S + I + + S − I −( )⎡⎣
⎤⎦
. (21)
By using the sum and difference relations
(22)
and
Δω0S Sz −ω0 I Iz =12
Δω0S −ω0 I( ) Sz + Iz( ) + Δω0S +ω0 I( ) Sz − Iz( )⎡⎣ ⎤⎦ (23)
we can express (21) as:
′H = 12
Δω0S −ω0 I( ) Sz + Iz( ) + A+ B2
sinη⎛⎝⎜
⎞⎠⎟
Sz + Iz( ) Iz −ω1S B2ω0 I
S + I + + S − I −( )⎡
⎣⎢
⎤
⎦⎥
+ 12
Δω0S +ω0 I( ) Sz − Iz( ) + A+ B2
sinη⎛⎝⎜
⎞⎠⎟
Sz − Iz( ) Iz +ω1S B2ω0 I
S + I − + S − I +( )⎡
⎣⎢
⎤
⎦⎥
+ω1S
2S + + S −( )
. (24)
Figure 2. Solid Effect Hamiltonian
The full Hamiltonian (left) in the eigenframe of the static Hamiltonian (see eqn. (21)) can
be separated into the DQ (green) and the ZQ (red) subspaces, if the SQ coherences (light
blue) are neglected. Coherence states are marked with light color, population states with
dark color.
125
In (24) we see that all operators in the first line correspond only to matrix
elements in the DQ subspace in the full Hamiltonian, with elements represented in green
in Figure 2. Similarly, the second line of eqn. (24) corresponds to the ZQ subspace
(marked in red in Figure 2). The third line, at last, represents the SQ coherences between
the electron-spin states (marked as blue in Figure 2). By selecting the DQ matching
condition for the solid effect (see eqn. (13)) and neglecting the additional oscillation
resulting from the secular electron-nuclear coupling, we see that the two polarization
states in the respective subframe become energetically degenerate (in the rotating frame),
allowing for the appearance of DQ coherence between these states. We may neglect the
ZQ coherence in this case, since we irradiate off-resonant by 2 ω0 I , and the transition
moment is small at the same time. Treatment of the SQ coherence term is more
complicated: Due to the relatively large ratio between T1S and T2S in combination with
the much larger SQ transition moment, electron spin polarization might be significantly
reduced by off-resonant excitation of the EPR (SQ) transition despite being far off-
resonance. However, by assuming that a quasi-equilibrium treatment is valid, we can
mathematically uncouple the SQ and the DQ excitation without introducing significant
errors, as long as transverse relaxation of the electron SQ coherence (described by T2S ) is
fast compared to the DNP transfer. This allows us to easily describe an effectively
reduced electron spin polarization and reduced electron by the well known steady-
state solution of the Bloch equations [34], and as detailed by Smith et al. [31] The DQ
excitation can then be treated neglecting the SQ coherences, while the influence of the
SQ coherence is represented by the reduced effective electron polarization and reduced
electron T1 . The same arguments hold for the ZQ case; we only have to substitute DQ
126
with ZQ and vice versa in the description above. However, we will only focus on the DQ
treatment from now on.
With the above conditions in mind, we can separate the DQ subspace from the
full Hamiltonian:
′HDQ = 1
2ΩDNP
DQ Sz + Iz( ) + A+ B2
sinη⎛⎝⎜
⎞⎠⎟
Sz + Iz( ) Iz −ω1S B2ω0 I
S + I + + S − I −( )⎡
⎣⎢
⎤
⎦⎥ . (25)
In (25) we have introduced ΩDNPDQ = Δω0S −ω0 I as the offset between the microwave
frequency and the DQ matching condition. Furthermore, we can drop the secular part of
the electron-nuclear coupling term, since within this subspace the corresponding operator
takes the form of an identity matrix and only leads to an offset of the eigenstates in the
DQ subspace with respect to the ZQ subspace. We are left with a truncated DQ
Hamiltonian of the form
TABLE 1. Spin-operators for a single spin (based on the Pauli spin operators ) and
the corresponding product spin operators for the electron-nuclear double quantum and
zero quantum subspace as described in eqn. (24) and depicted in Figure 2.
Single spin Double quantum subspace Zero quantum subspace
Sz + Iz( ) 2 Sz − Iz( ) 2
Sx Ix − Sy I y = S + I + + S − I −( ) 2
Sx Ix + Sy I y = S + I − + S − I +( ) 2
Sy Ix + Sx I y = S + I + − S − I −( ) 2i
Sy Ix − Sx I y = S + I − − S − I +( ) 2i
σ + =σ x + iσ y
S + I + = Sx Ix − Sy I y + i Sy Ix + Sx I y( )
S + I − = Sx Ix + Sy I y + i Sy Ix − Sx I y( )
σ − =σ x − iσ y
S − I − = Sx Ix − Sy I y − i Sy Ix + Sx I y( )
S − I + = Sx Ix + Sy I y − i Sy Ix − Sx I y( )
127
′HDQ(t) = 1
2ΩDNP
DQ Sz + Iz( )− ω1S B2ω0 I
S + I + + S − I −( )⎡
⎣⎢
⎤
⎦⎥ . (26)
If we consider this Hamiltonian in the DQ subspace (as depicted in Figure 2), we see that
it resembles a simple one-spin Hamiltonian consisting of a Zeeman term and a coupling
term, driven by microwave irradiation. Accordingly, we can redefine a basis set for this
subspace, which resembles the basis set of a single-spin in Hilbert-space. This basis set
(incl. the respective raising and lowering operators) is given in Table 1. We see that if the
DQ matching condition is fulfilled, magnetization stored in Sz + Iz 2 (corresponding
to in the single spin case) is transformed into
S + I + − S − I − 2i (corresponding to
) under influence of the microwave term
S + I + + S − I −( ) 2 (corresponding to ).
S + I + − S − I − 2i magnetization will further evolve to − Sz + Iz 2 and subsequently to
− S + I + − S − I − 2i , and will finally return to Sz + Iz 2 . If the microwave frequency is
applied with an offset to the actual DQ matching condition, magnetization in
S + I + − S − I − 2i will be in parallel evolving into and then into
− S + I + − S − I − 2i , from where the microwave field will drive it back to Sz + Iz 2 ;
note that the return to Sz + Iz 2 results in a reduction of the SE transfer efficiency. So
we see that this spin system can be described by the common Bloch equations in analogy
to a single spin system.
To do so, we define the following set of expectation values that fulfill the usual
commutation relations:
128
PS = MSz= Sz
PI = M Iz= Iz
Mx = S + I + + S − I − 2
M y = S + I + − S − I − 2i
. (27)
where PS and PI are the polarization of the electrons and nuclear spins respectively, and
Mx and My are the transverse DQ coherences.
We also define the effective transition moment for the solid effect transitions, according
to eqn. (18):
ωDNP =ω1S sinη . (28)
With these definitions at hand we formulate a set of Bloch-like differential
equations describing the evolution of the spin system during solid effect DNP. For the
sake of simplicity, we omit the effects caused by off-resonance excitation of the SQ EPR
transition as well as the ZQ DNP transition. The former (i.e. the SQ excitation) can easily
be solved with the usual Bloch equations, if line splitting due to secular electron-nuclear
coupling is ignored. We will discuss the impact of this excitation later in the course of
this paper.
First, we define a set of rate equations, describing the rate at which the
expectation values introduced above change:
129
(29)
Note that the superscript of has been dropped for simplicity. It is almost needless to
mention that the ZQ case can similarly be described if several signs are inverted (a
detailed derivation can be found in the Appendix).
We now assume that the coherences reach a quasi-steady state under microwave
irradiation, which requires the timescale of the observation to be much slower than the
evolution of the state, and therefore dMx dt = dM y dt = 0 . Note that this does not
prohibit changes in Mx and M y in response to changes in PS and PI . Using this
assumption, we find that
Mx = −ΩDNPT2,DNP M y . (30)
Inserting (30) into (29) yields
M y =ωDNPT2,DNP
1+ ΩDNPT2,DNP( )2
PS − PI( )2
, (31)
which in turn can be inserted into (29), giving us effective terms describing the buildup
and decay of electron and nuclear polarization during microwave irradiation of the solid
effect transition:
130
dPS
dt= −
ωDNP2 T2,DNP
2 1+ ΩDNPT2,DNP( )2⎛⎝
⎞⎠
PS − PI( ) + 1T1S
PS ,eq − PS( )
dPI
dt=
ωDNP2 T2,DNP
2 1+ ΩDNPT2,DNP( )2⎛⎝
⎞⎠
PS − PI( ) + 1T1I
PI ,eq − PI( ). (32)
Defining the rate constant
kDNP =ωDNP
2 T2,DNP
2 1+ ΩDNPT2,DNP( )2⎛⎝
⎞⎠
(33)
yields
dPS
dt= −kDNP PS − PI( ) + 1
T1S
PS ,eq − PS( )dPI
dt= kDNP PS − PI( ) + 1
T1I
PI ,eq − PI( ). (34)
These rate equations only describe a pair of one electron and one nuclear spin. If one
electron spin polarizes several nuclear spins we have to generalize (34) to:
dPS
dt= − kDNP
m PS − PIm( )
m=1
NI
∑ + 1T1S
PS ,eq − PS( )dPI ,m
dt= kDNP
m PS − PIm( ) + 1
T1Im PI ,eq
m − PIm( )
. (35)
If we now assume that spin-diffusion leads to a fast equilibration of all nuclei, we see that
dPS
dt= −
N I
NS
kDNP PS − PI( ) + 1T1S
PS ,eq − PS( )dPI
dt= kDNP PS − PI( ) + 1
T1I
PI ,eq − PI( ). (36)
Note that we have dropped all indices m because all nuclei are now indistinguishable.
131
During DNP, polarization will build up in the pool of nuclear spins and will eventually
reach a steady state polarization when
dPI
dt= kDNP PS
∞ − PI∞( ) + 1
T1I
PI ,eq − PI∞( ) = 0 . (37)
PS∞ and PI
∞ describe the electron and nuclear polarization in the steady state, respectively.
We now introduce the DNP enhancement factor at infinite polarization time:
ε∞ =
PI∞
PI ,eq
, (38)
so we can now rearrange (37) to obtain
kDNP
PS∞
PI ,eq
− ε∞
⎛
⎝⎜
⎞
⎠⎟ =
1T1I
ε∞ −1( ) . (39)
This allows us to calculate our enhancement, if the DNP rate constant, nuclear
longitudinal relaxation time and the steady-state electron spin polarization are known:
. (40)
Here, KDNP = kDNPT1I has been introduced, which can be considered as an effective DNP
equilibrium constant between DNP buildup and polarization decay by longitudinal
nuclear spin relaxation. If the steady-state enhancement and the steady-state electron
polarization are known, can be calculated by:
KDNP = kDNPT1I =ε∞ −1( )
PS∞
PI ,eq
− ε∞⎛
⎝⎜
⎞
⎠⎟
. (41)
132
4.3 Experimental
All experiments were performed using custom-built instrumentation including a
MAS DNP spectrometer operating at 212 MHz 1H frequency (courtesy of D. Ruben) and
triple resonance (1H, 13C, 15N) DNP probe. The probe is equipped with a sample eject
system which allows continuous operation for several weeks at cryogenic temperatures
and an arbitrary number of sample rotor ejections at temperatures of ~80 K [35].
139.64 GHz (for simplicity, we will refer to this frequency as 140 GHz in the following)
microwaves are generated with a gyrotron described elsewhere [36], and transmitted to
the MAS stator using a HE11, overmoded, corrugated waveguides. The output power of
the gyrotron was determined with a calorimeter immediately before the microwaves enter
the DNP probe.
The NMR magnet is centered at 4.98 T and is equipped with a ±0.075 T
superconducting sweep coil. The 140 GHz gyrotron is a fixed-frequency microwave
source, and therefore a magnet with a superconducting sweep coil is required to set the
field to the optimium position for polarizing agents with varying g-factors and/or
different DNP mechanisms. Frequency-tunable gyrotrons are under development [37].
The sample resides in a 4 mm outer diameter sapphire rotor with 0.7 mm wall
thickness. Sapphire is the rotor material of choice because it is transparent at microwave
frequencies. The sample is packed between Vespel spacers and has a height of ~4 mm.
The polarizing agent is dissolved in the appropriate volume of a cryoprotecting
mixture of d8-glycerol/D2O/H2O (60:30:10 vol.-%) containing 1 M 13C-urea. This
glycerol/water ratio and the 1H concentration of the matrix are known to provide
optimum enhancements under most experimental DNP conditions. We dissolve trityl
133
(OX063) in this solvent mixture to obtain a concentration of 40 mM; for comparison the
Gd-DOTA is used at 10 mM. We also prepare samples with varying 1H concentrations by
dissolving an appropriate amount of polarizing agent together with 13C-urea in pure d8-,
d5-, or h8-glycerol. This solution was then divided into equal parts and diluted with the
required volumes of H2O and D2O to yield 60:40 vol.-% mixtures of glycerol/water with
varying 1H concentrations.
To determine DNP enhancement profiles and build-up time constants, the nuclear
spin polarization was measured following a presaturation sequence and a period of
polarization buildup. Presaturation on both 1H and 13C channels consisted of 16 108°
pulses of 50 kHz rf strength (phase alternating along +x and +y) separated by 5 ms. This
was followed by a variable recovery time during which 1H polarization is allowed to
build-up. It was read-out via a ramped cross polarization (CP) step to 13C. A CP spin-
locking field of 83 kHz was applied on 1H while the 13C field was carefully optimized for
highest CP efficiency. Under these conditions a contact time of 2 ms resulted in an
optimal transfer of polarization from 1H to 13C for a sample of 13C-urea in 60:30:10 vol.-
% d8-glycerol/D2O/H2O. This contact time was used for all experiments in this study.
During acquisition of the FID an 83 kHz TPPM decoupling field was applied to 1H.
134
4.4 Results and Discussion
4.4.1 Analysis of SE DNP matching conditions at 140 GHz
In order to locate the field that optimally satisfies the DNP matching condition,
the 1H DNP enhancement is recorded with a field sweep covering both the positive as
well as the negative DNP condition (representing excitation of the ZQ and DQ
transitions, respectively). The field sweep is centered around the trityl EPR resonance
field at 4.982 T and spans a width that corresponds to at least twice the 1H Larmor
frequency of ~212 MHz. The sweep width is chosen to cover both DNP transitions (i.e.
the ZQ and the DQ transition), which includes broadening by inhomogeneity (e.g. g-
anisotropy and electron-nuclear coupling). At each field position, the probe’s 1H channel
was carefully retuned so that a constant level of reflected power was maintained
throughout the total range of the field sweep and the intensity of the 1H Bloch decay
recorded. Off-signals (i.e. non-DNP enhanced signals) were observed at both field
extrema of the sweep and show essentially no difference in intensity. This insures that the
sensitivity of the DNP probe does not change significantly across the field sweep which
would otherwise lead to distortions in the field profile (Figure 3). The 1H field profile is
normalized to unity, and, subsequently, the absolute 1H DNP enhancement was
determined via CP to 13C at the field of maximum DNP enhancement to avoid significant
1H background signals from probe components. This approach enables us to record high-
quality, high-resolution field profiles in a relatively short period of time as illustrated by
the data in Figure 3.
135
In Figure 3 we show the field dependent enhancement profile obtained with
40 mM trityl (OX063), which exhibits two well-resolved peaks separated by a field
equivalent to twice the 1H Larmor frequency, clearly illustrating that solid effect is the
dominant DNP mechanism. The shape of each DNP peak resembles the shape of the EPR
spectrum since there is no significant overlap of positive and negative solid effect
matching conditions. The maximum enhancement is observed by excitation of the DQ
condition at the optimal field value of 4.9894 T. The ZQ transitions at 4.9743 T yield a
relative enhancement of –0.88 with respect to the maximum (positive) enhancement.
Interestingly, DNP enhancements at fields corresponding to the allowed EPR transition
are observed. These enhancements show a negative and a positive region, with a zero
crossing close to the center of the EPR line. This is most probably caused by a low
Figure 3. Field Profile
140 GHz DNP enhancement field profile for a 40 mM solution of trityl (OX063) in a
mixture of 60:30:10 vol.-% d8-glycerol/D2O/H2O recorded with a gyrotron microwave
output power of 8 W at 85 K and 5 kHz MAS. Data points were obtained by recording
the amplitude of the enhanced 1H Bloch decay FID.
136
efficiency cross effect (CE) or thermal mixing (TM). In a study by Borghini et al. similar
observations were made using the BDPA radical at a field of 2.5 T and at temperatures as
low as 0.7 K and explained as thermal mixing [38].
The CE and TM require inhomogeneous or homogeneous broadening of the EPR
line with an effective breadth matching at least the 1H Larmor frequency (212 MHz).
Although the inherent EPR line width (FWHM) of trityl is ~50 MHz, there remains a
small non-vanishing EPR amplitude covering a breadth of >212 MHz. The high
concentration of 40 mM further allows for significant intermolecular electron-electron
coupling required for the CE or TM. Maly et al. have shown that monomeric trityl
radicals can be used as efficient polarizing agents for 2H DNP at 5 T, and have therefore
directly proved that the intermolecular electron-electron couplings between trityl
molecules are sufficient for efficient CE [39].
Figure 4. DNP Buildup Curves
140 GHz DNP build-up curves for a 40 mM solution of trityl (OX063) in a mixture of
60:30:10 vol.-% d8-glycerol/D2O/H2O recorded at various microwave power levels. For
details see text and Table 2.
137
4.4.2 Polarization dynamics
Studies of the time dependence of the DNP enhancement as a function of
microwave power performed at the field of maximum enhancement permit further
analysis of the SE DNP mechanism. Build-up curves were acquired for three different
microwave powers (3.3, 8.9, and 13.4 W). These results are compared to the respective
data obtained without microwave irradiation, yielding the nuclear T1I and off-signal
amplitude. The build-up curves are displayed in Figure 4, and the time constants are
extracted from least-square fits with exponential functions. In the case of the non-DNP-
enhanced signal (i.e. the “off-signal”) it was necessary to utilize a biexponential function.
The second component, which constitutes about 20% of the overall peak height, most
likely arises from spins in the spacer material and consistently gives a T1 = 5.8 s.
Therefore, it can be easily distinguished from the slower build-up of 13C-urea with
T1 = 26.2 s, and the DNP enhanced build-up curves were corrected for the background
signal by subtracting this component before fitting with a monoexponential function.
TABLE 2. Results of the analysis of 140 GHz DNP enhancements obtained with a
40 mM solution of trityl (OX063) in a mixture of 60:30:10 vol.-% d8-glycerol/D2O/H2O.
For definitions see eqns. (44)-(46) below.
Pmw (W) T (K) TB (s) (s-1)
0 82 26.2 1 0 660 0
3.26 85 24.9 26.6 2.03 × 10-3 509 0.053
8.90 88 22.8 64.2 5.71 × 10-3 486 0.150
13.41 92 21.0 90.6 9.39 × 10-3 454 0.246
138
DNP enhancements were determined by dividing the amplitude of the infinite-time urea
signal recorded with mw irradiation by that without mw irradiation. Results are listed in
Table 2.
As can be seen in Figure 5, enhancements clearly show a near-linear behavior as
function of incident microwave power. This is expected since in the theory above kDNP
scales with ω1S2 , and ω1S
2 scales linearly with microwave power. A slight deviation from
linearity is seen, however, leading to a slight decrease in slope with higher microwave
power. The build-up time constants are reduced compared to the spin-lattice relaxation
constant under microwave irradiation and show a monotonic trend of faster build-up with
higher microwave power.
Figure 5. Enhancements and Buildup Times
Obtained 140 GHz DNP enhancements (filled blue circles) and build-up time constants
(open red circles) for a 40 mM solution of trityl (OX063) in a mixture of 60:30:10 vol.-%
d8-glycerol/D2O/H2O recorded at various microwave power levels. For details see text
and Table 2.
139
Although we believe the reduced build-up times are a result of an increase in
, it is requisite to show that this reduction is not caused by sample heating due to
microwave absorption in the sample rotor and/or the surrounding environment. High-
frequency structure simulation (HFSS) studies have shown that only a small amount of
the microwave power is actually dispersed inside the sample [40]; however, the sample
temperature might still be affected due to heating of the surrounding stator, and heat from
the components might lead indirectly to heat transfer to the sample. To determine if
sample heating by microwave absorption plays a significant role in the polarization
dynamics, we performed experiments at a magnetic field at which no DNP enhancement
is obtained and compare T1I with and without microwave irradiation. We chose a
magnetic field equivalent to a frequency offset of Ω0S 2π = −1014.3 MHz with respect
to the EPR transition. This offset ensures that no DNP enhancement is obtained.
Relaxation time measurements were performed using 8.90 W of microwave power and
compared to measurements without microwave power. The observed relaxation time
constants are given in Table 3. No significant difference can be observed when the probe
and sample are subject to microwave irradiation, leading to the conclusion that sample
heating is not a major issue under our experimental conditions. Thus, the observed
reduction of the polarization build-up time constants is safely ascribed to the polarization
transfer mechanism.
140
A deviation of the observed enhancements as function of incident microwave
power from linear behavior is expected, since significant polarization build-up in the 1H
bulk reduces the polarization gradient between electron spins and nuclear spins and
therefore also reduces the effective DNP rate. At this point we remind the reader of the
theoretical maximum enhancement factor of 660 for DNP of 1H. This behavior can be
better understood by further investigation of eqn. (40). In the limit of large KDNP, the
enhancement approaches PS
∞ PI ,eq . Assuming the electron spin preserves its thermal
equilibrium polarization due to fast relaxation, we achieve the maximum enhancement
given by γ S γ I . However, if the electron spin polarization is depleted from its thermal
equilibrium during DNP and has a lower steady-state polarization, the maximum
enhancement that can be obtained will be below that ratio.
With our present instrumentation, we have no knowledge of the electron spin
steady-state polarization during DNP or KDNP. Therefore, we cannot directly utilize (41)
to calculate KDNP from the measured value of ε∞ or vice versa. However, we can estimate
the steady-state electron polarization based on a model we have proposed recently [31].
This model describes the polarization transfer between electron spin and nuclear spins
TABLE 3. Comparison of nuclear spin relaxation time T1I at different field positions
with and without microwave irradiation.
(MHz) Pmw (W) T1 (s) T (K)
–212.426 0 26.2 82
–1014.287 0 26.8 84
–1014.287 8.90 27.1 89
141
that was observed for static (i.e. non-MAS) DNP experiments using the same polarizing
agent (i.e. trityl) inside a microwave cavity. In this model the nuclear spins are divided
into nearby and distant (i.e. bulk) spins, which are spatially separated by the spin-
diffusion barrier. During DNP, the electron spin transfers polarization to these nearby and
distant nuclei baths in parallel. Spins within the volume confined by the spin-diffusion
barrier experience fast, paramagnetically enhanced spin-lattice relaxation and are not able
to transfer the polarization to more distant nuclei before it relaxes to the lattice. DNP
transfer to more distant spins which are situated outside of the spin-diffusion barrier is
most likely slower due to reduced dipolar coupling (scaling with r-3); at the same time
paramagnetic relaxation is strongly attenuated since its effects scale as r-6. This allows the
polarization transferred to these distant spins to efficiently disperse the enhanced
polarization throughout the bulk. In other words, a significant amount of electron spin
polarization is “drained” from the electron spin by fast-relaxing nuclei in the close
proximity of the electron spin, whereas polarization transferred directly to spins more
distant to the electron leads to the observable DNP enhancement.
Another mechanism of electron spin depletion is off-resonant excitation of the
EPR transition which can be described by evoking one solution of the Bloch equations:
. (42)
is the resonance off-set between the excitation frequency and the electron spin
Larmor frequency and in our case is Ω0S 2π =ω0 I 2π = 212.428 MHz which is the
solid effect matching condition. Although we have experimentally determined relaxation
times T1S = 1.43 ms and T2S = 0.89 μs for trityl at 80 K [31], we do not know the
142
conversion factor, c, between the incident microwave power and the oscillating
microwave field strength amplitude B1:
c =
γ S B1S
2π Pmw
=ω1S
2π Pmw
, (43)
This prevents us from distinguishing between polarization depletion by off-resonant
excitation and by DNP transfer to fast-relaxing nuclei. Nanni et al. have estimated the
microwave field distribution inside a MAS stator used for 250 GHz DNP, and arrived at
an average conversion factor of 0.31 MHz W-0.5 for an experimental arrangement similar
to that used here [40]. Assuming the same conversion factor in our experiments leads us
to an average value of ω1S 2π = 1.15 MHz for the highest applied microwave power of
13.41 W. This field strength would lead to a depletion of ~4 % which can be considered
as negligible. Increasing the field strength to 2 or 3 MHz, however, would lead to 12 and
24 % depletion, respectively.
The remaining electron spin polarization which has been depleted by transfer to
nearby nuclei and off-resonant microwave irradiation − and is therefore available for
transfer to distant nuclei − can be approximated by knowledge of the longitudinal nuclear
spin relaxation time constant, DNP build-up time constant as well as the obtained
enhancement. Due to the fact that the build-up of the longitudinal nuclear spin
polarization is accelerated by the solid effect (as can be seen in Figure 5) an effective
DNP rate constant can be obtained from the observed build-up time constant TB and by
the nuclear spin-lattice relaxation time constant T1I. This effective DNP rate constant is
given by
143
kDNP
eff =1
TB
−1
T1I
. (44)
Assuming the polarization builds up equally fast on all bulk nuclear spins (i.e. 1H−1H
spin diffusion is fast compared to the DNP transfer rate) we can calculate the effective
electron spin polarization by
PS ,eqeff
PI ,eq
= 1kDNP
eff
ε∞
TB
− 1T1I
⎛
⎝⎜⎞
⎠⎟. (45)
This depleted steady-state polarization includes polarization loss by transfer to nearby
spins (which drain the polarization by fast paramagnetically enhanced longitudinal
relaxation) as well as saturation due to off-resonant excitation. The reader should note
that describes the effective electron spin polarization and replaces the thermal
equilibrium polarization. PS∞ , on the other hand, denotes the steady-state polarization
during the DNP of bulk nuclei; so and PS∞ are not generally equal. However, under
the assumption that DNP to distant nuclei does not significantly deplete the electron spin
polarization, we can use the condition PS
∞ = PS ,eqeff in eqn. (41) in order to calculate :
KDNPeff =
ε∞ −1( )PS ,eq
eff
PI ,eq
− ε∞⎛
⎝⎜
⎞
⎠⎟
(46)
Due to the low conversion factor between Pmw and ω1S as well the low effective ,
due to operation in high magnetic field, this assumption seems to be safe. However the
large ratio between nuclear spins and electron spins still might lead to a significant
depletion of electron spin polarization.
144
This set of equations now allow us to calculate the effective DNP rate constant,
the electron spin polarization available for DNP build-up on distant (bulk) nuclei as well
as validate the theoretically proposed linearity between as a measure of the solid
effect efficiency and the incident microwave power. A detailed analysis of the
experimental data is presented in Table 2 and depicted in Figure 6. We clearly see that
is linear with the microwave power, as expected; however, to calculate from
Figure 6. Calculation of Fit Parameters from Experiment
Top: calculated PS ,eq
eff PI ,eq according to eqn. (45) for various incident microwave power
levels. The dashed line represents the maximum value of 660. Bottom: according to
eqn. (46) as function of microwave power. For details see text and Table 2.
145
experiments, we assumed that PS
∞ = PS ,eqeff . If this assumption failed, then the calculated
would diverge from linearity. The validity of this assumption suggests that the
transfer of polarization from the electron to distant nuclei only has a minor effect on the
final electron polarization, PS∞ . However, Figure 6 shows that the effective electron spin
polarization, PS ,eq
eff PI ,eq , does change significantly during DNP. The effective electron
spin polarization undergoes a fast reduction at rather low microwave powers and a rather
small negative slope towards higher incident power. This might be explained by a fast
“saturation” (i.e. equilibration of electron and nuclear spin polarization) of the nearby
nuclei that cannot efficiently spread their polarization to the bulk due to hindered spin-
diffusion through the barrier.
4.4.3 Influence of spin-diffusion efficiency on DNP
In order to investigate the role of homonuclear spin-diffusion in SE DNP, we
prepared samples with a constant concentration of the polarizing agent trityl, but with
varying 1H concentrations. We modulated the 1H concentration by choosing the
appropriate ratio between D2O and H2O in the matrix or by substituting perdeuterated
glycerol (d8) with partially deuterated (d5) or fully protonated glycerol. Here we note that
a 2 ms CP contact time was used for all samples. We expect variations in CP efficiency
for different protonation levels, which are shown in Figure 7 (bottom) where we plot the
off-signal amplitude. However, the buildup times and enhancements for a given sample
do not depend on CP efficiency; these parameters only depend on measurement of
relative signal intensity within the set of experiments on one sample. We also note that
146
even in the worst case, the signal amplitude could only be increased by a maximum of
~25 % by shortening of the contact time.
In Figure 7 (top) the enhancements obtained with 40 mM trityl are shown in
comparison with a similar study using 10 mM Gd-DOTA as polarizing agent. Trityl
achieves the largest enhancement at minimum 1H concentration and the enhancement
drops monotonically if more protons are available. Gd-DOTA shows the opposite
behavior with increasing enhancements for higher 1H concentration, and reaches a
plateau value of ~12 above ~20 M 1H. Furthermore, in the case of trityl, the increase of
1H concentration has only a limited influence on the build-up time constants. The build-
up time constants are practically invariant at low 1H concentrations (with a constant value
of ~25 s) and only show a slight reduction of 30-40 % above 25 M 1H (see Figure 7,
middle). When using Gd-DOTA as a polarizing agent, the build-up kinetics show a very
rapid decrease of the build-up time constant if the 1H concentration is increased from the
minimum concentration. For a fully protonated matrix we observed a more than 4-fold
reduction in build-up time constants. The combination of higher enhancement and shorter
build-up time constant with higher 1H concentration renders the use of deuterated
solvents unnecessary in the case of Gd-DOTA.
147
Figure 7. Trityl and Gd-DOTA Enhancements at Various 1H Concentrations
Comparison of DNP parameters between the polarizing agents trityl (OX063) (green) and
Gd-DOTA (blue) for various 1H concentrations in the matrix. Top: enhancement factors,
obtained by comparing on- and off-signal amplitude after a polarization time of 1.26
times TB; middle: build-up time constants; bottom: non-DNP-enhanced off-signal
amplitudes after a recovery time of 1.26 times TB. Open circles represent samples
prepared with d8-glycerol, whereas open squares and diamond correspond to samples
prepared with d5- and h8-glycerol, respectively. The glycerol/water ratio was 60/40 (v/v)
in all cases, with 40 mM polarizing agent concentration for trityl and 10 mM in case of
Gd-DOTA. All data points were recorded under 8 W microwave power at a temperature
of ~86 K.
148
This overall behavior indicates that the solid effect is active in two different limits
when comparing the different polarizing agents. When using trityl, the kinetics during the
DNP process are dominated by a rate-limiting step which involves a relatively slow
initial polarization transfer from the electron spin to the nuclear spins; Gd-DOTA on the
other hand is limited by a relatively inefficient spin-diffusion whereas the initial
polarization step is not rate-limiting. We propose two causes for these differences: (i) The
polarization of the S = 1/2 electron spin in trityl relaxes much slower than in the case of
the S = 7/2 high-spin Gd(III). This might cause a significant depletion of the electron spin
polarization of trityl, leading to a smaller polarization gradient between electron spin and
nuclear spins and therefore lower enhancements if a greater number of nuclear spins are
to be polarized due to higher concentration of protons. Gd(III) might not be limited by
this effect because T1S is several orders of magnitude shorter. However, this explanation
is unlikely due to reasons discussed earlier in the preceding section (i.e. the transition
moment of the solid effect is too small to cause a significant saturation condition). This is
further supported by the fact that the enhancements shown in Figure 5 do not exhibit a
strong deviation from linearity, which would be the case if enhancements were limited by
saturation. (ii) The high-spin state of Gd-DOTA induces a transition moment that is four
times higher for all coherences involving transitions between the mS = –1/2 and mS = +1/2
states, including the solid effect transitions involved in DNP. This leads to a higher
efficiency of the initial polarization transfer. At the same time, the S = 7/2 spin induces a
much stronger paramagnetic relaxation of the surrounding nuclei which results in a larger
ratio of nearby to distant (i.e. bulk) nuclei. This renders a greater fraction of the protons
undetectable due to paramagnetic shifts, fast relaxation, and impeded spin-diffusion from
149
those spins to the bulk. This can be seen if one compares the off-signal amplitudes
between samples doped with trityl and those doped with Gd-DOTA. In Figure 7 (bottom)
we see that both 40 mM trityl and 10 mM Gd-DOTA have similar off-signal amplitudes,
although the concentration of trityl is 4-fold higher. This indicates that the volume of
exclusion due to paramagnetic effects is 4 times higher per electron spin in the case of
Gd-DOTA compared to trityl. The concentration dependence of the buildup dynamics
using Gd-DOTA versus trityl suggest that solid effect enhancements obtained with Gd-
DOTA are limited by spin-diffusion, whereas the initial polarization transfer rate limits
the solid effect enhancements for trityl; this may be due to a higher DNP transition
moment and stronger paramagnetic effects for Gd-DOTA as compared to trityl.
4.4.4 The solid effect at fields > 5 T
In light of the improvement in SE enhancements reported here and since higher
magnetic fields are generally of more interest to the NMR community, it is important to
consider the extent to which the solid effect can be used at >5 T. The experimental
findings in the preceding sections indicate that the SE is currently limited by the available
microwave field strength. Furthermore, the solid effect enhancements scale as ω0−2 ,[16-
18] due to the attenuation of state mixing at higher magnetic field. In eqn. (20) one can
see that this field dependence can be compensated if the microwave field strength scales
linearly with ω 0 . Concurrently, for a fixed conversion factor, the microwave field scales
as the square root of the power potentially leading to arcing and sample heating.
Enhancing the conversion factor between incident microwave power and field generated
by introduction of a resonant structure with a small MAS rotor might solve this problem
150
[41; 42]. This approach could be used with microwave frequencies in the range between
100-300 GHz, but for higher frequencies the dimensions of the resonator would drop
below 1 mm, at which point fabrication becomes a challenge. A Fabry-Perot resonator
design might be beneficial at frequencies >300 GHz, however, avoiding coupling
between the standing microwave field and the rf pickup device would be challenging.
A second possible problem associated with very high microwave fields might be
the off-resonant excitation of the single quantum EPR line. Earlier we have indicated that
saturation of the single quantum transition does not play a significant role for the power
levels used in this study and will not be a major issue even for microwave fields that are
~2-3 times larger. When transitioning to higher fields, the microwave field amplitude has
to be scaled linearly with the external magnetic field to maintain a constant transition
moment; on its own, the increase in microwave field will worsen off-resonant saturation.
However, we note that the frequency offset must also be scaled linearly with the external
magnetic field to meet the SE matching condition. Therefore, the increases in microwave
field amplitude and microwave frequency offset are proportional. By analyzing eqn. (42)
and assuming a constant ratio between nuclear Larmor frequency and microwave field
(i.e. ), as well as field independent relaxation time constants, we see that:
lim
ω0 I →∞PS ,eq
1+ω0 I2 T2S
2
1+ω0 I2 T2S
2 + b2ω0 I2 T1ST2S
⎛
⎝⎜
⎞
⎠⎟ =
PS ,eqT2S
T2S + b2T1S
. (47)
The polarization depletion by off-resonant irradiation converges to a constant
value if the frequency offset and the microwave field amplitude are scaled the same way
as the external magnetic field (in the case that the inhomogeneous linewidth is much
151
smaller than the nuclear Larmor frequency). Therefore we predict that off-resonant
saturation of the single quantum transition will not be of major concern at higher fields.
A final concern is the small but non-vanishing g-anisotropy of polarizing agents
such as trityl or BDPA. Whereas the g-anisotropy is barely visible at 5 T, it will lead to a
considerable broadening at higher magnetic fields and a reduction of the solid effect
efficiency. Recently we have introduced high-spin paramagnetic complexes based on
Gd(III) and Mn(II) [43]. Since these complexes show no spin-orbit coupling, they feature
a nearly isotropic electron Zeeman interaction. However, the central mS = –1/2 → +1/2
transition is broadened by a significant second-order zero-field splitting (i.e. electron
quadrupole interaction), which manifests itself in a 29 MHz full-width at half maximum
(FWHM) in the case of Gd-DOTA at 5 T, for example. 1H solid effect enhancements of
~13 have been reported with this polarizing agent under the same experimental
conditions used in this study. The line broadening second-order effect, however, is
expected to decrease linearly with ω0 [44]. This makes Gd-DOTA a potential candidate
as a polarizing agent for the solid effect at fields > 5 T, since the broadening disadvantage
of trityl with increasing field is substituted by a narrowing advantage of Gd-DOTA.
The combination of all these facts suggest that significant enhancements could be
reached with the solid effect even at magnetic fields > 5 T. However, this requires that
high microwave power is available at the respective frequencies, which may be
technically very challenging.
152
4.5 Conclusion
We have shown that significant enhancements of ~91 are observed with the solid
effect using trityl (OX063) as polarizing agent at a magnetic field of 5 T in MAS DNP.
The enhancement factors and polarization build-up dynamics have been recorded for
three different microwave power levels. The enhancements are currently limited by the
available microwave power and could likely be increased if higher power becomes
available. An analysis of the experimental data suggests good agreement with the
theoretical model proposed by Smith et al.,[31] which was developed using data recorded
under static (non-MAS) NMR conditions. The two data sets show no significant
mechanistic differences which leads us to conclude that both the initial DNP step of
polarization transfer from the electron spin to the nuclei as well as the further spreading
of polarization within the nuclear spin bath contribute similarly to the overall DNP
process.
We investigated the role of homonuclear spin-diffusion in solid effect DNP by
varying the 1H concentration of the sample using trityl and compared the results to those
obtained with the polarizing agent Gd-DOTA. Differences observed between the two
polarizing agents most probably arise because the solid effect is operating in two
different kinetic regimes: In the case of trityl the initial polarization transfer step between
the electron spin and the bulk nuclear spin bath is relatively slow and rate-limiting,
whereas the subsequent 1H–1H spin-diffusion is fast. For Gd-DOTA the opposite is the
case. While the initial polarization step is fast, the dispersion of polarization to the bulk
protons is rate-limiting. Possible reasons for this fundamental difference can be found in
drastically different electron spin relaxation properties of trityl and Gd-DOTA, a higher
153
ZQ and DQ transition moment due to the high-spin properties of Gd(III) as well as in
differences in paramagnetic relaxation of nuclear spins induced by the high-spin state.
We further discussed possible problems and advantages of the solid effect at
fields > 5 T. We predict that the solid effect will be a viable experiment at higher
magnetic fields, if sufficient microwave power becomes available at the required
frequencies.
4.6 Appendix
Derivation of DNP parameters for the zero quantum (ZQ) case
Here we derive the rate equations describing the excitation of the ZQ electron-
nuclear coherence. A similar treatment for the DQ case has been demonstrated in the
main text. We begin with the complete electron-nuclear spin Hamiltonian including the
microwave induced coupling terms in the eigenframe of the static Hamiltonian, in the
form given in eqn. (24):
. (A. 1)
We now select the ZQ subspace (line 2 in (24)) while neglecting the DQ subspace and the
SQ coherences for the same arguments made in the main text. The separated ZQ
subspace has the form:
154
. (A. 2)
In .
(A. 1) we have introduced as the offset between the
microwave frequency and the ZQ matching condition. Furthermore, we can drop the
secular part of the electron-nuclear coupling term, since within this subspace the
corresponding operator takes the form of an identity matrix and only leads to an offset of
the eigenstates in the ZQ subspace with respect to the DQ subspace. We are left with a
truncated ZQ Hamiltonian of the form
. (A. 3)
If we consider this Hamiltonian in the DQ subspace (as depicted in Figure 2), we
see that it resembles a simple one-spin Hamiltonian consisting of a Zeeman term and a
coupling term, driven by microwave irradiation. Accordingly, we can redefine a basis set
for this subspace, which resembles the basis set of a single-spin in Hilbert-space. This
basis set (incl. the respective raising and lowering operators) is given in Table I. We see
that if the ZQ matching condition is fulfilled, magnetization stored in
(corresponding to in the single spin case) is transformed into
(corresponding to ) under influence of the microwave term
155
(corresponding to ). magnetization will further evolve to
and subsequently to , and will finally return to
. If the microwave frequency is applied with an offset to the actual ZQ
matching condition, magnetization in will be in parallel evolving into
and then into , from where the microwave field will
drive it back to ; note that the return to results in a reduction of
the SE transfer efficiency. So we see that this spin system can be described by the
common Bloch equations in analogy to a single spin system.
To do so, we define the following set of expectation values that fulfill the usual
commutation relations:
. (A. 4)
where and are the polarization of the electrons and nuclear spins respectively, and
and are the transverse ZQ coherences. Now we can define a set of differential
equations which describe magnetization kinetics for the ZQ case:
156
dPS
dt= −ωDNP M y +
1T1S
PS ,eq − PS( )dPI
dt=ωDNP M y −
1T1I
PI ,eq − PI( )dMx
dt= −ΩDNP M y −
1T2,DNP
Mx
dM y
dt=ωDNP
PS + PI( )2
+ΩDNP Mx −1
T2,DNP
M y.
(A. 5)
In this case is the offset between the mw frequency and the ZQ transition
frequency.
Under the assumption of quasi-equilibrium (see main text) we find
. (A. 6)
and
, (A. 7)
which can be inserted into (A. 5), giving us effective terms describing the buildup and
decay of electron and nuclear polarization during microwave irradiation of the solid
effect transition:
. (A. 8)
Defining the rate constant
(A. 9)
157
yields
. (A. 10)
These rate equations only describe a pair of one electron and one nuclear spin. If one
electron spin polarizes several nuclear spins we have to generalize (A. 10) to:
. (A. 11)
If we now assume that spin-diffusion leads to a fast equilibration of all nuclei, we see that
. (A. 12)
Note that we have dropped all indices m because all nuclei are now indistinguishable.
During DNP, polarization will build up in the pool of nuclear spins and will
eventually reach a steady state polarization when
. (A. 13)
and describe the electron and nuclear polarization in the steady state, respectively.
We now introduce the DNP enhancement factor at infinite polarization time:
, (A. 14)
so we can now rearrang (A. 13) to obtain
158
. (A. 15)
This allows us to calculate our enhancement, if the DNP rate constant, nuclear
longitudinal relaxation time and the steady-state electron spin polarization are known:
(A. 16)
Here, KDNP = kDNPT1I has been introduced, which can be considered as an effective DNP
equilibrium constant between DNP buildup and polarization decay by longitudinal
nuclear spin relaxation. If the steady-state enhancement and the steady-state electron
polarization are known, can be calculated by:
. (A. 17)
4.7 Bibliography
[1] A.W. Overhauser, Physical Review 92 (1953) 411-415. [2] T.R. Carver, and C.P. Slichter, Physical Review 92 (1953) 212-213. [3] M. Abraham, R.W. Kedzie, and C.D. Jeffries, Physical Review 106 (1957) 165-166. [4] A. Abragam, and W.G. Proctor, Comptes Rendus Hebdomadaires Des Séances De
L'Academie Des Sciénces 246 (1958) 2253-2256. [5] C.D. Jeffries, Physical Review 117 (1960) 1056-1069. [6] A.V. Kessenikh, V.I. Lushchikov, A.A. Manenkov, and Y.V. Taran, Soviet Physics-
Solid State 5 (1963) 321-329. [7] C.F. Hwang, and D.A. Hill, Physical Review Letters 18 (1967) 110-&. [8] C.F. Hwang, and D.A. Hill, Physical Review Letters 19 (1967) 1011-&. [9] D.S. Wollan, Physical Review B 13 (1976) 3671-3685. [10] D.S. Wollan, Physical Review B 13 (1976) 3686-3696.
159
[11] M. Goldman, Spin Temperature and Nuclear Magnetic Resonance in Solids, Oxford University Press, London, 1970.
[12] A. Abragam, and M. Goldman, Reports on Progress in Physics 41 (1978) 395-467. [13] W.T. Wenckebach, Applied Magnetic Resonance 34 (2008) 227-235. [14] J.H. Ardenkjær-Larsen, I. Laursen, I. Leunbach, G. Ehnholm, L.G. Wistrand, J.S.
Petersson, and K. Golman, Journal of Magnetic Resonance 133 (1998) 1-12. [15] C.F. Koelsch, Journal of the American Chemical Society 79 (1957) 4439-4441. [16] T. Maly, G.T. Debelouchina, V.S. Bajaj, K.N. Hu, C.G. Joo, M.L. Mak-Jurkauskas,
J.R. Sirigiri, P.C.A. van der Wel, J. Herzfeld, R.J. Temkin, and R.G. Griffin, Journal of Chemical Physics 128 (2008) 19.
[17] A.B. Barnes, G. De Paepe, P.C.A. van der Wel, K.N. Hu, C.G. Joo, V.S. Bajaj, M.L. Mak-Jurkauskas, J.R. Sirigiri, J. Herzfeld, R.J. Temkin, and R.G. Griffin, Applied Magnetic Resonance 34 (2008) 237-263.
[18] R.G. Griffin, and T.F. Prisner, Physical Chemistry Chemical Physics 12 (2010) 5737-5740.
[19] K.N. Hu, V.S. Bajaj, M. Rosay, and R.G. Griffin, Journal of Chemical Physics 126 (2007) 7.
[20] K.N. Hu, H.H. Yu, T.M. Swager, and R.G. Griffin, Journal of the American Chemical Society 126 (2004) 10844-10845.
[21] C. Song, K.-N. Hu, C.-G. Joo, T.M. Swager, and R.G. Griffin, Journal of the American Chemical Society 128 (2006) 11385-11390.
[22] K.N. Hu, C. Song, H.H. Yu, T.M. Swager, and R.G. Griffin, Journal of Chemical Physics 128 (2008) 17.
[23] Y. Matsuki, T. Maly, O. Ouari, H. Karoui, F. Le Moigne, E. Rizzato, S. Lyubenova, J. Herzfeld, T. Prisner, P. Tordo, and R.G. Griffin, Angewandte Chemie-International Edition 48 (2009) 4996-5000.
[24] A.B. Barnes, B. Corzilius, M.L. Mak-Jurkauskas, L.B. Andreas, V.S. Bajaj, Y. Matsuki, M.L. Belenky, J. Lugtenburg, J.R. Sirigiri, R.J. Temkin, J. Herzfeld, and R.G. Griffin, Physical Chemistry Chemical Physics 12 (2010) 5861-5867.
[25] K.R. Thurber, W.M. Yau, and R. Tycko, Journal of Magnetic Resonance 204 (2010) 303-313.
[26] E.L. Dane, B. Corzilius, E. Rizzato, P. Stocker, T. Maly, A.A. Smith, R.G. Griffin, O. Ouari, P. Tordo, and T.M. Swager, The Journal of Organic Chemistry 77 (2012) 1789-1797.
[27] M.K. Kiesewetter, B. Corzilius, A.A. Smith, R.G. Griffin, and T.M. Swager, Journal of the American Chemical Society accepted (2012).
[28] Y. Hovav, A. Feintuch, and S. Vega, Journal of Magnetic Resonance 207 (2010) 176-189.
[29] A. Feintuch, D. Shimon, Y. Hovav, D. Banerjee, I. Kaminker, Y. Lipkin, K. Zibzener, B. Epel, S. Vega, and D. Goldfarb, Journal of Magnetic Resonance 209 (2011) 136-141.
[30] Y. Hovav, A. Feintuch, and S. Vega, Journal of Chemical Physics 134 (2011). [31] A.A. Smith, B. Corzilius, A.B. Barnes, T. Maly, and R.G. Griffin, Journal of
Chemical Physics 136 (2012) 015101. [32] A. Schweiger, and G. Jeschke, Principles of Pulse Electron Paramagnetic
Resonance, Oxford University Press, Oxford, 2001.
160
[33] K.N. Hu, G.T. Debelouchina, A.A. Smith, and R.G. Griffin, Journal of Chemical Physics 134 (2011) 19.
[34] F. Bloch, W.W. Hansen, and M. Packard, Physical Review 69 (1946) 680-680. [35] A.B. Barnes, M.L. Mak-Jurkauskas, Y. Matsuki, V.S. Bajaj, P.C.A. van der Wel, R.
DeRocher, J. Bryant, J.R. Sirigiri, R.J. Temkin, J. Lugtenburg, J. Herzfeld, and R.G. Griffin, Journal of Magnetic Resonance 198 (2009) 261-270.
[36] L.R. Becerra, G.J. Gerfen, R.J. Temkin, D.J. Singel, and R.G. Griffin, Physical Review Letters 71 (1993) 3561-3564.
[37] A.C. Torrezan, S.T. Han, I. Mastovsky, M.A. Shapiro, J.R. Sirigiri, R.J. Temkin, A.B. Barnes, and R.G. Griffin, Ieee Transactions on Plasma Science 38 (2010) 1150-1159.
[38] M. Borghini, W. de Boer, and K. Morimoto, Physics Letters A 48 (1974) 244-246. [39] T. Maly, L.B. Andreas, A.A. Smith, and R.G. Griffin, Physical Chemistry Chemical
Physics 12 (2010) 5872-5878. [40] E.A. Nanni, A.B. Barnes, Y. Matsuki, P.P. Woskov, B. Corzilius, R.G. Griffin, and
R.J. Temkin, Journal of Magnetic Resonance 210 (2011) 16-23. [41] A. Brinkmann, S.K. Vasa, H. Janssen, and A.P.M. Kentgens, Chemical Physics
Letters 485 (2010) 275-280. [42] V. Weis, M. Bennati, M. Rosay, J.A. Bryant, and R.G. Griffin, Journal of Magnetic
Resonance 140 (1999) 293-299. [43] B. Corzilius, A.A. Smith, A.B. Barnes, C. Luchinat, I. Bertini, and R.G. Griffin,
Journal of the American Chemical Society 133 (2011) 5648-5651. [44] A. Abragam, Principles of nuclear magnetic resonance, Oxford University Press,
New York, 1961.
161
Chapter 5
Observation of Strongly Forbidden Transitions via
Electron-Detected Solid Effect Dynamic Nuclear Polarization
Contributing: Björn Corzilius, Olesya Haze, Tim Swager
162
Abstract We present electron paramagnetic resonance (EPR) experiments for which solid
effect (SE) dynamic nuclear polarization (DNP) was observed indirectly via polarization
loss on the electron. Frequency profiles of the electron-detected SE obtained using trityl
radical showed intense saturation of the electron at the usual SE condition, which
involves a single electron and nucleus. However, higher order SE transitions involving
two or more nuclei were also observed with surprising intensity, although these
transitions did not lead to bulk nuclear polarization. Similar results were obtained for the
SA-BDPA radical, but significantly more structure was observed for the frequency
sweep. These results are discussed in light of recent theoretical predictions for SE. Also,
we note that the results are consistent with our recent model of the pathway of bulk
nuclear polarization in the SE.
163
5.1 Motivation Dynamic nuclear polarization (DNP) is a method of signal enhancement in nuclear
magnetic resonance (NMR) experiments which is achieved by the transfer of the large
electron Boltzmann polarization to nuclear polarization [1; 2]. The state of the art in DNP
has been improving in recent years with the introduction of a variety of new radicals [3-
10], and improved technique and instrumentation [11-15]. Because NMR is an inherently
insensitive technique, DNP allows a variety of otherwise prohibitively long experiments
to become feasible. There are a variety of recent applications of DNP including
biological studies [16-21], surface studies [22], and dynamics studies [23; 24] which
would not otherwise be possible.
Aside from pulsed-DNP, microwave-driven DNP mechanisms in the solid-state can
be categorized into three mechanisms, depending on the relative sizes of the homogenous
linewidth (δ), the inhomogeneous linewidth (Δ), and the nuclear Larmor frequency (ω0I):
The solid effect (SE) is a two-spin mechanism where microwave irradiation
simultaneously flips an electron and nucleus, and is dominant when ω0I> δ, Δ [25; 26].
The cross effect (CE) is a three-spin mechanism, for which microwave irradiation is used
to saturate one electron, and then a three-spin electron-electron-nuclear transition
polarizes the nucleus. CE is dominant when δ < ω0I < Δ [27-29]. Finally, thermal mixing
is a multi-electron process similar to CE which results in a nuclear spin flip and is
dominant when ω0I < δ, Δ [30-32].
Continued advancement of DNP requires a better understanding of the involved
mechanisms, and recently there has been a renewed effort to better characterize the SE
mechanism. In the case of an electron-nuclear pair, it is straightforward to obtain the
transition probability and matching condition of the SE, the latter given by Δω0S=±ω0I
(Δω0S is the offset of the microwave frequency from the electron Larmor frequency) [33;
34]. However, the many-spin mechanism is likely to be far more complicated, because of
higher order spin interactions that are accessible in strongly coupled systems, and
because of the complicated role that spin-diffusion plays in transporting polarization from
an electron to the bulk nuclei that are observed in an NMR experiment. Hovav et al. have
shown a variety of simulations of SE that both highlight the complicated nature of the
164
matching condition [35], and also discuss the role that higher order interactions play in
the polarization of bulk nuclei for larger spin systems [36]. We have also been working to
elucidate details of the SE mechanism for many spins via experimental methods. Recent
experiments were performed on static samples for which nuclear polarization buildup
rates and enhancements were measured. We were able to fit our results to a model in
which nuclei near an electron compete with bulk nuclei for polarization, and therefore
deplete the final bulk polarization [37]. More recently, we have shown our model is
consistent with results obtained for samples under magic angle spinning conditions [38].
In this paper, we show results for which we observe saturation of the electron
during SE-DNP at 5 T (140 GHz e-, 212 MHz 1H). Using trityl radical and a water-
soluble BDPA radical (SA-BDPA) [7], we sweep the saturating microwave field and
observe SE transitions (see Figure 1). Rather than only seeing slight saturation of the
electron at the SE matching condition, as one might expect from basic SE theory and
other recent results [39], we see a large saturation at the SE matching condition and
additionally see saturation at ωMW=ω0S±nω0I, where n=1,2,3,4. These results are
compared to the enhancement of bulk nuclear polarization, observed via NMR, where
enhancement is only seen for n=1. This result is consistent with our previously proposed
model [37]. We also see that the SE matching condition is heavily split by hyperfine
coupling of nearby nuclei when using SA-BDPA, which has many protons strongly
coupled to the electron. This is in contrast to trityl, which does not have many nearby
protons and therefore does not exhibit a splitting.
5.2 Theory In order to gain some insight into the SE mechanism, we start by considering the
Hamiltonian of a system of many nuclei in the rotating frame of the microwave field.
165
H = HZ + HIS + HII + HM
HZ = Δω0SSz − ω0I , j I jzj∑
HIS = AjSz I jz + BjSz I jx +C jSz I jy( )j∑
HII = I jDj,k Ikk> j∑
j∑
HM =ω1SSx.
(1)
HZ contains the Zeeman interactions, HIS is the electron-nuclear coupling, HII is the
nuclear-nuclear coupling, and H M is the microwave field. It is straightforward to
diagonalize HZ + HIS , by first rotating each nucleus about the z-axis by an angle χ j
( HZ is invariant to this rotation so we only show HIS ), which leads to
Uχ = exp[iχ j I jz]j∏
tanχ j =Cj Bj
UχHISUχ−1 = AjSz I jz + Bj
2 +Cj2( )1/2
Sz I jx⎛
⎝⎜
⎞
⎠⎟
j∑ .
(2)
We define Bj
* = Bj2 +C j
2( )1/2 for convenience, and complete the diagonalization of
HS + HI + HIS by repeating the steps given by Hu et al. for each nucleus, resulting in (3)
[34].
Uη = exp i(ηα , j −ηβ , j)Sz I jy + i2 (ηα , j +ηβ , j)I jy
⎡⎣⎢
⎤⎦⎥j
∏
tanηα , j =Bj
*
Aj −2ω0I , j, tanηβ , j =
Bj*
Aj +2ω0I , j
Uη HS + HI + HIS( )Uη−1 = Δω0SSz + −ω0I , j I jz + AjSz I jz( )
j∑
(3)
We have introduced additional terms to simplify the resulting Hamiltonian, which are
given by
166
ω0I , j = 12ω0I , j cosηα , j +cosηβ , j( )− 1
4 Aj cosηα , j −cosηβ , j( )− 1
4 Bj sinηα , j − sinηβ , j( )Aj = −ω0I , j cosηα , j −cosηβ , j( )+ 1
2 Aj cosηα , j +cosηβ , j( )+ 1
2 Bj sinηα , j + sinηβ , j( ).
(4)
Application of U χ and
Uη to HII and H M is straightforward, however we first
focus on the effects that the partial diagonalization has on H M . Defining
η j = (ηα , j −ηβ , j ) / 2 for convenience, we obtain
UηUχHmUχ−1Uη
−1 =
ω1S
Sx cosη jj∏ −2 SyI jy sinη j cosηl
l≠ j∏
j∑
+2 SxI jy Iky sinη j sinηk cosηl −…l≠ j,k∏
k> j∑
j∑
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
.
(5)
Examing (5), we see that the first term drives off resonant saturation of the electron. The
next term drives the usual SE transition, and all other terms drive higher order SE
transitions. Since the η j , ηk ,… are small, the probability of these transitions will be
increasingly weaker. The matching condition of a specific transition (excluding HII ), for
which the electron and nuclear spins j, k,… are flipped, and nuclear spins p, q,…
remaining fixed is given by
Sα /β I j,α /β Ik ,α /β…I p,α /β Iq,α /β… →
Sβ /α I j,β /α Ik ,β /α…I p,α /β Iq,α /β…
±Δω0S = ±ω0I , j ±ω0I ,k ±…( )+ 12 ±Ap ± Aq ±…( ),
(6)
where α / β indicates that a spin is in the spin-up or spin-down state, and the sign on a
term corresponding to the jth, kth, pth, or qth nucleus depends on its starting state. The
matching condition reduces to Δω0S = ±nω0 I when the electron-nuclear coupling of the
fixed spins is neglected, however if there is a strong electron-nuclear coupling to some of
the fixed spins, then this coupling cannot be ignored.
167
We do not explicitly calculate U χUηHIIUη
−1U χ−1 here, but note that the nuclear-
nuclear couplings in HII , when tilted, will lead to an additional coupling of the electron
to the nuclei, and will therefore contribute to the SE mechanism. We will see in our
experimental results, that U χUηH MUη
−1U χ−1 alone cannot account for the intensity of the
observed SE transitions, and therefore U χUηHIIUη
−1U χ−1 may play an important role.
5.3 Experimental We have performed electron-detected SE-DNP experiments on a 140 GHz pulsed-
EPR/DNP system that was recently described [40] using a fundamental mode TE011
coiled resonator [41]. In each experiment, the frequency-sweepable ELDOR channel of
the EPR bridge was used to apply continuous-wave microwave irradiation to the sample.
Figure 1: Pulse sequence and Polarizing Agents
(A) Pulse sequence for the electron-detected SE. (B) Trityl (OX063) and SA-BDPA
radicals used for SE.
168
After a period of irradiation and a short delay (10-20 μs to eliminate any electron
coherence), the on-resonant polarization was measured using a Hahn-echo, as shown in
Figure 1A [42]. Experiments were performed with samples using 40 mM radical
concentration, corresponding to typical concentrations used in DNP experiments. Two
different radicals were used: trityl, and SA-BDPA, which was recently shown to be an
effective radical for use in SE-DNP (Figure 1B) [7].
5.4 Results and Discussion Figure 2 shows the results of saturating the electrons of trityl radical at various
frequencies and for various lengths of time before measuring the on-resonant electron
polarization via a Hahn echo. As one should expect, there is severe saturation of the
electron when the swept frequency is the same as the observation frequency. This
saturation decreases as the swept frequency is moved away from the observation
frequency, but we again see significant saturation as the swept frequency approaches the
usual SE matching condition, Δω0S = ±ω0 I . As Δω0S is increased, we are surprised to
find that there are additional strong transitions at Δω0S = ±nω0 I for n=2,3 and 4.
169
Observation of transitions for n >1 is not entirely unexpected: Higher order terms
in (5) suggest that higher order transitions should exist. Additionally, the n=2 transition
was observed via NMR by de Boer, although the intensity was far lower than observed
here [43; 44]. In Figure 3, we show the same experiment (with a slightly different sample
Figure 2. Frequency sweep with variable irradiation time of the electron-detected
SE of 40 mM Trityl in 60:40 13C-glyerol:D2O at 80 K.
Polarization of the trityl radical was measured near the center of the trityl EPR spectrum
after irradiation at various frequencies. Polarization measurement was obtained using
Hahn echo centered at 140 GHz and a field of 4993.6 mT. Microwave field strength was
~2.5 MHz at Δω0S=0.
170
and lower microwave power), and instead of varying the length of the saturation time, we
fix the saturation time at 5 ms and vary the microwave power.
The results are very similar. Saturation is lower overall because of the lower
microwave power. Also, there is slightly more asymmetry in the microwave field strength
as the frequency is swept, causing the n=4 transition on the high frequency side to vanish.
One should note that as the microwave power is decreased, the overall shape of the
spectrum does not change significantly- only the degree of saturation changes. This
demonstrates that the observation of strong higher order transitions is not simply due to
high microwave field strengths, but rather is inherent to the sample.
171
We discuss briefly the possible causes of such intense higher order transitions.
The amplitudes of the transitions given in (5) are proportional to the product of the sinη j
Figure 3. Frequency sweep with variable power of the electron-detected solid effect
of 40 mM Trityl in 60:25:15 13C-glyerol:D2O:H2O at 80 K.
The power dependence of the electron-detected SE is shown, for 5 ms of microwave
irradiation. 50% power corresponds to ~1.8 MHz microwave field strength at
Δω0S=0.Polarization of the trityl radical was measured near the center of the trityl EPR
spectrum after irradiation at various frequencies. Polarization was measured using a Hahn
echo centered at 140 GHz and a field of 4993.6 mT.
172
of the involved spins (the cosη j are close enough to 1 that they can be neglected). For
the closest nucleus in trityl, the maximum value of Bj
* is ~1 MHz [45], and
ω0 I = 212 MHz , giving a maximum for 2sinη j = .001. The rate at which a particular
transition occurs if it is on resonant can be obtained by squaring of the amplitude.
Therefore, the ratio of the rate of n=2 transition for the closest nuclei to the n=2 transition
is at most |.01|2=1x10-4. We should note that there are many spins and considerably more
pathways for the n=2 transition than the n=1 transtion (any combination of two nuclear
spins). Therefore the saturation rate due to the n=2 transition would have roughly 2000
times more pathways for 2000 spins (there are ~1500-2000 protons per electron).
However even with 2000 spins, we fail to account for the intensity observed for n=2
(2000*1x10-4=2x10-1<<1). For the n>2 transition, the situation is worse, since the
probability is given by sinη j sinηk ...
2. Because we see this large intensity that cannot
easily be accounted for by the microwave field and electron-nuclear couplings alone, it
suggests that other processes are involved. A possible cause of the high degree of
electron saturation is the additional electron-nuclear coupling resulting from the term
U χUηHIIUη
−1U χ−1 . Hovav et al. have made similar arguments that processes involving the
nuclear-nuclear coupling contribute to polarization of bulk nuclei, and we note that the
unexpected intensity of the higher order transitions is consistent with their results [36]. If
this is the case, it is also a strong indication that the n=1 SE condition has major
contributions from high order coherences, brought about by strong nuclear-nuclear
couplings. One should also consider the possibility that other nearby radicals also play a
role.
173
In Figure 4A, we show a frequency profile for which we observe the 1H
polarization resulting from SE, rather than observing the electron polarization. An NMR
cross polarization experiment was used to transfer 1H polarization to 13C, after 10s of
microwave irradiation. As expected, we see DNP enhancements at the usual SE matching
condition, Δω0S = ±ω0 I . In light of our electron-detected SE results, one might expect to
also see enhancements at Δω0S = ±nω0 I (for n=1-4). However, this was not the case. In
Figure 4B, the microwave frequency was set to meet the condition Δω0S = ±2ω0 I and
also the microwave cavity was retuned to obtain maximum microwave field strength.
Whereas at the n=-1 condition, one sees an enhancement of 70, at the n=2 conditions,
there is not a significant enhancement.
Figure 4. NMR Detection of SE-DNP
(A) SE-DNP frequency profile observed after 10s microwave irradiation via an NMR
cross polarization from 1H to 13C. (B) NMR spectra at the SE-DNP conditions Δω0S=-ω0I
and Δω0S=±2ω0I, also observed via CP, where no significant enhancement can be seen for
Δω0S=±2ω0I. The microwave field strength is ~2.5 MHz.
174
The lack of bulk nuclear enhancement at the n=2 SE condition, despite a
significant polarization loss observed on the electron is consistent with our recent model
of electron to bulk polarization transfer [37]. In our model, we proposed that electron
polarization was transferred directly to both nearby and bulk nuclei. However, the
polarization transfer to nearby nuclei is not then transferred via spin-diffusion to the bulk,
but rather fast nuclear spin-lattice relaxation near the electron destroys this polarization.
As a result the nearby nuclei deplete the amount of polarization available for direct
electron-nuclear transfer to the bulk nuclei. The results shown here for the n=2 SE
condition suggest that polarization loss observed on the electron is primarily a result of
transfer of polarization to nearby nuclei, since the η j are largest for these nuclei, making
the transfer more likely. However, because these nuclei do not efficiently transfer
polarization to the bulk nuclei, a large loss of polarization is observed on the electron
without a significant gain of bulk nuclear polarization.
In Figure 5, we show the electron-detected SE experiment for the SA-BDPA
radical, with a fixed 10 μs irradiation time. Although we have performed a narrower
frequency sweep, we can still see the n=1 and n=2 transitions. In contrast to trityl, we see
a large number of individual peaks around each transition. Unlike trityl, for which there
are very few protons near the radical center, the SA-BDPA molecule has many nearby
protons with large hyperfine couplings [7]. Hovav et al. predicts the splitting of the DNP
matching condition due to the electron-nuclear couplings of nearby nuclei [35], as do we
in (6). When comparing the trityl and SA-BDPA frequency sweeps, we see that this
effect is clearly demonstrated here. Also, we note that one would not normally be able to
resolve this splitting when observing SE via NMR detection; the many different
orientations of the radical, each with a slightly different Larmor frequency, will broaden
the condition and cover the splitting. However, due to the orientation-selective nature of
the Hahn echo in EPR, we are able to observe the splitting.
175
5.5 Conclusions We have shown via electron-detected SE experiments that it is possible to observe
very intense saturation of the electron via the SE mechanism. This occurs both for the
expected n=1 transition, but also for n=2,3, and 4. The electron-nuclear dipole couplings
and microwave field cannot account for this saturation alone, giving merit to predictions
that the nuclear-nuclear dipole coupling may also play a major role in the SE mechanism.
We have also seen that electron saturation occurs for the n=2 condition, but bulk nuclear
polarization is notably absent, which is consistent with the polarization of nearby nuclei
that do not transmit polarization to the bulk, as predicted in our recent model of SE.
Finally, we have demonstrated that the SE matching condition is a complex function of
the couplings to many nearby nuclei, as recently predicted via theoretical methods.
5.6 Bibliography [1] T.R. Carver, and C.P. Slichter, Physical Review 92 (1953) 212. [2] A.W. Overhauser, Physical Review 92 (1953) 411.
Figure 5: Frequency sweep of the electron-detected solid effect of 40 mM SA-BDPA
in 60:40 13C-glyerol:D2O at 80 K.
Microwave irradiation is applied for 10 μs, followed by polarization detection at the
center of the SA-BDPA spectrum via a Hahn echo. Microwave field strength was ~2.5
MHz at Δω0S=0.
176
[3] C. Song, K.-N. Hu, C.-G. Joo, T.M. Swager, and R.G. Griffin, J. Am. Chem. Soc. 128 (2006) 11385-90.
[4] Y. Matsuki, T. Maly, O. Ouari, H. Karoui, F. Le Moigne, E. Rizzato, S. Lyubenova, J. Herzfeld, T.F. Prisner, P. Tordo, and R.G. Griffin, Angew. Chem. 48 (2009) 4996-5000.
[5] E.L. Dane, B. Corzilius, E. Rizzato, P. Stocker, T. Maly, A.A. Smith, R.G. Griffin, O. Ouari, P. Tordo, and T.M. Swager, J. Org. Chem. (2012).
[6] M. Kiesewetter, B. Corzilius, A.A. Smith, R.G. Griffin, and T.M. Swager, J. Am. Chem. Soc. (2012).
[7] O. Haze, B. Corzilius, A.A. Smith, R.G. Griffin, and T.M. Swager, J. Am. Chem. Soc. (2012).
[8] S. Macholl, H. Johannesson, and J.-H. Ardenkjaer-Larsen, Phys. Chem. Chem. Phys. 12 (2010) 5804-5817.
[9] B. Corzilius, A.A. Smith, A.B. Barnes, C. Luchinat, I. Bertini, and R.G. Griffin, J. Am. Chem. Soc. 133 (2011) 5648-5651.
[10] A. Zagdoun, G. Casano, O. Ouari, G. Lapadula, A.J. Rossini, M. Lelli, M. Baffert, D. Gajan, L. Veyre, W.E. Maas, M.M. Rosay, R.T. Weber, C. Thieuleux, C. Coperet, A. Lesage, P. Tordo, and L. Emsley, J. Am. Chem. Soc. (2012).
[11] V.S. Bajaj, M.K. Hornstein, K.E. Kreischer, J.R. Sirigiri, P.P. Woskov, M.L. Mak-Jurkauskas, J. Herzfeld, R.J. Temkin, and R.G. Griffin, J. Magn. Reson. 190 (2007) 86-114.
[12] A.B. Barnes, M.L. Mak-Jurkauskas, Y. Matsuki, V.S. Bajaj, P.C.A. van der Wel, R. DeRocher, J. Bryant, J.R. Sirigiri, R.J. Temkin, J. Lugtenburg, J. Herzfeld, and R.G. Griffin, Journal of Magnetic Resonance 198 (2009) 261-270.
[13] A.C. Torrezan, S.-T. Han, I. Mastovsky, M.A. Shapiro, J.R. Sirigiri, R.J. Temkin, A.B. Barnes, and R.G. Griffin, IEEE Transactions on Plasma Science (2010).
[14] J. Leggett, R. Hunter, J. Granwehr, R. Panek, A.J. Perez-Linde, A.J. Horsewill, J. McMaster, G. Smith, and W. Kockenberger, Phys. Chem. Chem. Phys. 12 (2010) 5883-5892.
[15] S. Jannin, A. Bornet, S. Colombo, and G. Bodenhausen, Chem. Phys. Lett. 517 (2011) 234-236.
[16] V.S. Bajaj, M.L. Mak-Jurkauskas, M. Belenky, J. Herzfeld, and R.G. Griffin, Proc. Natl. Acad. Sci. 106 (2009) 9244-9249.
[17] M. Rosay, J. Lansing, K. Haddad, W. Bachovchin, J. Herzfeld, R. Temkin, and R. Griffin, J. Am. Chem. Soc. 125 (2003) 13626-7.
[18] M.L. Mak-Jurkauskas, V.S. Bajaj, M.K. Hornstein, M. Belenky, R.G. Griffin, and J. Herzfeld, Proc. Natl. Acad. Sci. 105 (2008) 883-888.
[19] G.T. Debelouchina, M.J. Bayro, P.C.A. van der Wel, M.A. Caporini, A.B. Barnes, M. Rosay, W.E. Maas, and R.G. Griffin, Phys. Chem. Chem. Phys. 12 (2010) 5911-5919.
[20] Ü. Akbey, W.T. Franks, A. Linden, S. Lange, R.G. Griffin, B.-J.v. Rossum, and H. Oschkinat, Angew. Chem. Int. Ed. 49 (2010) 7803-7806.
[21] M.J. Bayro, G.T. Debelouchina, M.T. Eddy, N.R. Birkett, C.E. MacPhee, M. Rosay, W.E. Maas, C.M. Dobson, and R.G. Griffin, J. Am. Chem. Soc. 133 (2011) 13967-13974.
177
[22] A. Lesage, M. Lelli, D. Gajan, M.A. Caporini, V. Vitzthum, P. Mieville, J. Alauzun, A. Roussey, C. Thieuleux, A. Mehdi, G. Bodenhausen, C. Coperet, and L. Emsley, Journal of the American Chemical Society 132 (2010) 15459-15461.
[23] B.D. Armstrong, and S. Han, J. Am. Chem. Soc. 131 (2008) 4641-4647. [24] H. Zeng, Y. Lee, and C. Hilty, Analytical Chemistry 82 (2010) 8897-8902. [25] C.D. Jeffries, Physical Review Phys. Rev. PR 117 (1960) 1056. [26] C. Jeffries, D., Physical Review 106 (1957) 164. [27] A. Kessenikh, V., V. Lushchikov, I., A. Manenkov, A., and Y. Taran, V., Soviet
Physics - Solid State 5 (1963) 321-329. [28] A. Kessenikh, V., A. Manenkov, A., and G. Pyatnitskii, I., Soviet Physics - Solid
State 6 (1964) 641-643. [29] D. Wollan, S., Physical Review B: Condensed Matter 13 (1976) 3671. [30] M.J. Duijvestijn, A. Manenschijn, J. Smidt, and R.A. Wind, J. Magn. Reson. 64
(1985). [31] M. Duijvestijn, J., R. Wind, A., and J. Smidt, Physica B+C 138 (1986) 147-170. [32] M. Goldman, Spin temperature and nuclear magnetic resonance in solids, Clarendon
Press, Oxford, 1970. [33] G. Jeschke, and A. Schweiger, Mol. Phys. 88 (1996) 355-383. [34] K.-N. Hu, G.T. Debelouchina, A.A. Smith, and R.G. Griffin, J. Chem. Phys. 134
(2011). [35] Y. Hovav, A. Feintuch, and S. Vega, J. Magn. Reson. 207 (2010) 176-189. [36] Y. Hovav, A. Feintuch, and S. Vega, J. Chem. Phys. 134 (2011) 074509(1-20). [37] A.A. Smith, B. Corzilius, A.B. Barnes, T. Maly, and R.G. Griffin, J. Chem. Phys.
136 (2012) 015101-(1-16). [38] B. Corzilius, A.A. Smith, and R.G. Griffin, J. Chem. Phys. (2012). [39] V. Nagarajan, Y. Hovav, A. Feintuch, S. Vega, and D. Goldfarb, J. Chem. Phys. 132
(2010) 214504-1 to 13. [40] A.A. Smith, B. Corzilius, T. Maly, J. Bryant, R. DeRocher, P.P. Woskov, and R.G.
Griffin, J. Magn. Reson. (2012). [41] V. Weis, M. Bennati, M. Rosay, J.A. Bryant, and R.G. Griffin, J. Magn. Reson. 140
(1999) 293-299. [42] E.L. Hahn, Phys. Rev. 80 (1950) 580-594. [43] W. de Boer, Journal of Low Temperature Physics 22 (1976) 185-212. [44] A. Abragam, and M. Goldman, Reports on Progress in Physics 41 (1978) 395-467. [45] M. Bowman, C. Mailer, and H. Halpern, J Magn Reson 172 (2005) 254-67.
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Chapter 6
DNPsim: A Flexible Program for DNP Simulations
Contributing: Björn Corzilius
179
Abstract
A program designed for simulation of DNP experiments is presented. DNPsim is
designed to be highly flexible in the types and conditions of DNP experiments that can be
performed, while having a straightforward interface in MATLAB. A thorough discussion
of the theory underlying the program’s function is included; the generation of the
Hamiltonian, calculation of relaxation parameters, and propagation of the spin system are
addressed. Usage of the program is discussed, including how to define the spin system,
and explanations of various simulation options. Several examples of experiments are
given in order to demonstrate usage and the overall versatility. Finally, the role of high
order coherences in solid effect DNP is investigated for a system of six nuclei and one
electron. By providing researchers with a simple, but versatile program for performing
DNP simulations, we hope to encourage in depth investigation of DNP mechanisms using
a combination of theory, simulation, and experiment.
180
6.1 Motivation Dynamic nuclear polarization (DNP) is the enhancement of the signal in nuclear
magnetic resonance (NMR) experiments via a transfer of the high Boltzmann polarization
of electrons to nuclei [1; 2]. A variety of mechanisms exist for DNP experiments
performed in the solid state. Mechanisms that utilize continuous-wave (cw) microwave
irradiation include the solid effect (SE) [3-5], cross effect (CE) [6-9], and thermal mixing
(TM) [10-12]. Solid effect is a two-spin mechanism that is dominant when ω0 I > Δ,δ ,
ω0 I being the nuclear frequency, Δ being the electron inhomogeneous linewidth, and δ
being the electron homogeneous linewidth. Cross effect is a three-spin mechanism that is
dominant when Δ >ω0 I > δ , and thermal mixing is dominant when Δ,δ >ω0 I .
Additionally, pulsed-DNP methods have been proposed as means of circumventing an
unfavorable field dependence that exists for cw mechanisms. Several pulsed-DNP
experiments are the dressed-state solid effect (DSSE) [13], nuclear spin orientiation via
electron spin locking (NOVEL) [14], and nuclear rotating frame DNP (NRF-DNP) [15].
Although DNP is seeing considerably more use recently, there is still a number of
questions regarding the details of DNP mechanisms. In most DNP experiments, the
nuclei to be polarized greatly outnumber the electrons that are the source of polarization.
This has two important implications: (1) Relaxation is critical to DNP. Because the
electron spin-lattice relaxation rates ( 1/ T1S ) are several orders of magnitude faster than
the nuclear spin-lattice relaxation rates ( 1/ T1I ), a few electrons can polarize many nuclei.
(2) DNP mechanisms cannot be fully understood using only electron-nuclear or electron-
electron-nuclear spin systems, since the electron is coupled to many nuclei and transfers
polarization to these nuclei via multiple pathways. Therefore, to fully characterize a DNP
process, one must utilize a method of study that allows inclusion of relaxation and uses
many spins. Although creative experimentation can elucidate many details of the DNP
process, it can be very difficult to separate many effects in typical DNP samples. For
example, high-order coherences, which involve an electron and multiple nuclei, may play
a major role in DNP polarization transfer. However, direct observation of these
coherences has not been possible so far. For this reason, simulations that use multiple
181
spins and include appropriate relaxation treatment are useful in sorting through many
effects that underlie the DNP process.
Several papers have recently strived to address problems in DNP via simulation.
Hovav, Feintuch, and Vega have published two papers addressing the solid effect and one
addressing the cross effect [16; 17]. Additionally, Karabanov et al. has developed
methods to allow for simulations of large spin systems under solid effect conditions [18;
19]. The solid effect work by Hovav uses simulation methods for which the Hamiltonian
is diagonalized prior to the introduction of relaxation processes. As a result, many
relaxation rates are not considered in the simulation- an omission that is not thoroughly
addressed. The work of Karabanov et al. is more promising, since many spins (25) are
allowed for these simulations of 13C solid effect. It is not yet clear that the methods will
allow for systems of many strongly coupled spins (i.e. a concentrated 1H system).
However, the ability to handle many spins at once has the potential to greatly clarify
problems, such as the role of the spin-diffusion barrier in bulk polarization buildup.
The results from Hovav et al. and Karabanov et al. are useful in understanding
DNP processes. However, the ability to vary simulation parameters and adjust them to for
a variety of experiments and systems will lead to a far better understanding of DNP
processes in a wide community. For this reason, we want to provide the software to setup
and use a variety of different DNP simulations. Then, the user may perform
investigations into the roles of various experimental parameters and manipulate the
system as desired.
We begin with a thorough discussion of the theory used to develop our simulation
program in Section 6.2. This will include calculation of the Hamiltonian, a discussion of
the assumptions taken to derive relaxation parameters, and methods of system
propagation. In Section 6.3, the usage of the simulator is explained, including how to
specify the spin system and how to manipulate the available experimental and simulation
parameters. Several examples of both cw- and pulsed-DNP experiments are shown in
Section 6.4. Finally, the role of high-order coherences in SE is investigated in Section
6.5.
182
6.2 Theory The density matrix, σ (t) , describes the state of a spin system. In order to
understand the dynamics of electrons and nuclei under DNP conditions, we evaluate the
density matrix either in the absence of relaxation in Hilbert space, or with relaxation in
Liouville space. In Hilbert space, the evolution of the density operator is given by
ddtσ (t) = −i H (t),σ (t)⎡⎣ ⎤⎦ . (1)
σ (t) describes the spin system- it does not consider the states of quantum-mechanical
processes which are coupled to the spin-system, known as the bath. Likewise, H (t) only
describes interactions of the spin system. However, if we assume that the bath brings
about relaxation of the spin system, then (1) must be expanded to account for relaxation.
This can be done by including a relaxation superoperator, Γ , which allows one to
consider action of the bath on the spin system resulting in relaxation [20].
ddtσ (t) = −i H (t),σ (t)⎡⎣ ⎤⎦ − ˆ̂Γ{σ (t)−σ eq} . (2)
Note that ˆ̂Γ only accounts for action of the bath on the spin system, but does not
consider action of the spin system on the bath. Also, note that the function ˆ̂Γ is evaluated
for σ (t)−σ eq , which has the effect of pushing σ (t) towards thermal equilibrium, which
is given by σ eq .
In its current form, (2) can be difficult to evaluate because the function ˆ̂Γ cannot
usually be evaluated in the form of a commutator, and so there is no general solution to
(2). This requires one to evaluate the derivative at every time point and propagate it
stepwise. However, one may take the individual elements of the density matrix, σ (t) ,
and place them columnwise in a 1-dimensional vector, σσ (t) . If we define
H = H ⊗ 1 - 1⊗ H , where ⊗ is the direct product operator, 1 is the identity with the
same dimensions as H, and ~ indicates the complex conjugate, then we obtain
ddtσ (t) ={−iH (t)− Γ}(σ (t)−σ eq ) . (3)
183
In this case, ΓΓ is a matrix and so it is possible to evaluate (3) using linear methods.
In the remainder of the theory section, we will discuss three challenges that must
be addressed to solve equations (1) and (3). Section 6.2.1 will address the calculation of a
static Hamiltonian, and transformation of that Hamiltonian into the appropriate Liouville
space. Section 6.2.2 will discuss generation of the relaxation matrix. Finally, Section
6.2.3 will discuss propagation of the spin system.
6.2.1 Calculation of the Hamiltonian The general Hamiltonian for our spin system is given by
Hz = ω0S j
S jz + ω0 ImImz
m=1
NI
∑j=1
NS
∑ Electron and Nuclear Zeeman
HSS = S jd j ,k
SS Skk> j
NS
∑j=1
NS
∑ Electron-Electron Coupling
HII = Imdm,n
II Inn>m
NS
∑m=1
NI
∑ Nuclear-Nuclear Coupling
HIS = Aj ,mS jz Imz + Bj ,mS jz Imx +C j ,mS jz Imy
m=1
NI
∑j=1
NS
∑ Electron-Nuclear Coupling
HD = S j DjS j
j=1
NS
∑ Electron Zero-Field Coupling (4)
HQ = ImQmIm
m=1
NI
∑ Nuclear Quadrupole Coupling
H M = 2ω1S cos[ω MWt] S jx
j=1
NS
∑ Microwave Field
HRF = 2ω1Icos[ω RFt] Imx
m=1
NI
∑ Radio-Frequency Field
H0 = Hz + HSS + HII + HIS + HD + HQ Static Hamiltonian
H = H0 + H M + HRF Full Hamiltonian
Electrons are denoted as S j and nuclei are denoted as Im , where
S j = (S jx ,S jy ,S jz )
and
Im = (Imx , Imx , Imz ) . Zeeman interactions are given in Hz where ω0S j
and ω0 Im
are the
electronic and nuclear Zeeman frequencies, respectively. HSS contains the electron-
184
electron couplings, for which d j ,k
SS gives the coupling tensor. Similarly, dm,n
II is the
coupling tensor for nuclear-nuclear couplings, found in HII . HIS contains the electron
nuclear couplings, for which Aj ,m gives the secular coupling, and
Bj ,m and C j ,m give the
pseudo-secular couplings in the x- and y- directions, respectively. Electron zero-field
splittings are found in HD , given by the tensor Dj . Nuclear quadrupole couplings are
given in HQ , where the nuclear quadrupole tensor is Qm . Finally, microwave and radio
frequency (RF) fields are found in H M and HRF , where ω MW and ω RF give the
frequencies and ω1S and ω1I give the field strengths. In this form, the Hamiltonian is
time dependent if the microwave and/or RF fields are non-zero. To remove this
dependence, one performs a rotating frame transformation in the rotating frame of the
microwave field and/or the RF field. This is done by calculating
σ r (t) =
exp −i ω MW S jz + ω RF Imzm=1
NI
∑j=1
NS
∑⎛
⎝⎜⎞
⎠⎟⎡
⎣⎢⎢
⎤
⎦⎥⎥σ (t)exp i ω MW S jz + ω RF Imz
m=1
NI
∑j=1
NS
∑⎛
⎝⎜⎞
⎠⎟⎡
⎣⎢⎢
⎤
⎦⎥⎥
H r (t) =
exp −i ω MW S jz + ω RF Imzm=1
NI
∑j=1
NS
∑⎛
⎝⎜⎞
⎠⎟⎡
⎣⎢⎢
⎤
⎦⎥⎥
H (t)exp i ω MW S jz + ω RF Imzm=1
NI
∑j=1
NS
∑⎛
⎝⎜⎞
⎠⎟⎡
⎣⎢⎢
⎤
⎦⎥⎥
− ω MW S jzj=1
NS
∑ − ω RF Imzm=1
NI
∑
. (5)
In the case of an applied microwave field, the terms of the Hamiltonian are modified as
follows, where the microwave and radio frequencies are subtracted from the Zeeman
Hamiltonian:
185
Hzr = Δω0S j
S jzj=1
NS
∑ + ω0 ImImz
m=1
NI
∑
HSSr = 1
2 (d j ,kSS ,xx + d j ,k
SS ,yy )(S jxSkx + S jySky )+ d j ,kSS ,zzS jzSkz
k> j
NS
∑j=1
NS
∑
HISr = Aj ,m
zx S jz Imx + Aj ,mzy S jz Imy + Aj ,m
zz S jz Imzm=1
NI
∑j=1
NS
∑
HDr = 1
2 (Djxx + Dj
yy )(S jx2 + S jy
2 )+ DjzzS jz
2
j=1
NS
∑
H Mr =ω1S S jx
j=1
NS
∑
. (6)
In (6), Δω0S j
=ω0S j−ω MW , and the superscript r denotes the rotating frame. All terms are
now rotated at a frequency of ω MW about the S jz axes, and all rapidly oscillating terms
have been discarded. Similarly, if an RF field is applied, we must modify the
Hamiltonian as follows:
Hzr = ω0S j
S jzj=1
NS
∑ + Δω0 ImImz
m=1
NI
∑
HIIr = 1
2 (dm,nII ,xx + dm,n
II ,yy )(Imx Inx + Imy Iny )+ dm,nII ,zz Imz Inz
n>m
NI
∑m=1
NI
∑
HISr = Aj ,m
zz S jz Imzm=1
NI
∑j=1
NS
∑
HQr = 1
2 (Qmxx +Qm
yy )(Imx2 + Imy
2 )+Qmzz Imz
2
m=1
NI
∑
HRFr =ω1I Imx
m=1
NI
∑
. (7)
In (7), Δω0 Im
=ω0 Im−ω RF . As with the microwave field, all terms are now rotating at a
frequency of ω RF about the Imz axes, and oscillating terms have been dropped. Note that
HIS only contains the term Aj ,m
zz S jz Imz , because the static terms Aj ,m
xz S jx Imz and Aj ,m
yz S jy I jz
have been dropped. In this case, these pseudo-secular couplings are very small compared
to the electron-Zeeman field and therefore can be neglected. Note that it can help
accelerate computations if the nuclear-nuclear, electron-electron, quadrupole, and zero-
186
field couplings are truncated as in (6) and (7) even when there is not an applied field-
however caution should be taken with any truncation.
Once the rotating frame transformation has been made, and all oscillating terms
have been dropped, the terms in the Hamiltonian may be added together, resulting in
H0r = Hz
r + HSSr + HII
r + HISr + HD
r + HQr
H r = H0r + H M
r + HRFr
. (8)
At this point, the Hamiltonian, which is now static, may be used in Hilbert space, by
computing
ddtσ r (t) = −i H r ,σ r (t)⎡⎣ ⎤⎦ . (9)
One may obtain the Hamiltonian in Liouville space by computing H = H ⊗ 1− 1⊗ H .
In principle, H can be used directly in (3), but it can be beneficial to use a change of
basis set at this point. We use the basis set defined by the products of ( 1 , S jz ,
S j+ ,
S j− )
and ( 1 , Imz , Im+ , Im
− ) for all electron and nuclear spins. For S=1, the additional states (
S jz
2 − 23 1 ,
(S j
+ )2 , (S j
− )2 , S jzS j
+ + S j+S jz ,
S jzS j− + S j
−S jz ) are used. In this program, S>1 is
not allowed with relaxation. In order to make the transformation, a matrix, V, is
generated, which can be applied as ′H =V †H V , where ′H is in the new basis set.
Each column in V corresponds to one of the states in the new basis set. We obtain the
column by taking the desired state in Hilbert space (for example, S1+ I2
− ), and placing the
individual elements of the matrix columnwise into a vector. The vectors are then
normalized to one, so that V †V = 1 .
It may be useful to work with a truncated basis set, either to accelerate
calculations or to help understand the role of various states in a process. It is possible to
truncate the basis set by omitting columns from V that correspond to the states which are
to be truncated. When working with the full basis set, with N states, V is an N×N matrix.
However, if a truncated basis set is to be used, with only n states, then V is an N×n
matrix, and the resulting Hamiltonian superoperator, ′H , is an n×n matrix.
187
6.2.2 Relaxation 6.2.2.1 Construction of the Relaxation Matrix
Once H has been obtained, it is necessary to calculate the relaxation
superoperator, ΓΓ . Our goal is to be able to specify a few relaxation times, ideally taken
from experiment or other methods of computation, and use them to specify relaxation of
all states. Therefore, it is necessary to establish the relationship between the various
relaxation times under a set of assumptions. In order to account for relaxation, in addition
to the static Hamiltonian, H, we apply a stochastic field to the spin system, given by
H1(t) , where the stochastic field has a vanishing average in time. The evolution of a spin
system, σ (t) , is then given by
ddtσ (t) = H + H1(t),σ (t)⎡⎣ ⎤⎦ . (10)
In order to obtain the average behavior of the spin system we assume that there are many
spin systems, each given by their own σ (t) , and each evolving under H and H1(t) . Each
starts out in the same state, σ (0) , and evolves under the same static Hamiltonian, H.
However the stochastic Hamiltonian, H1(t) which leads to relaxation, is uncorrelated for
the individual spin systems. This approximates the behavior of a sample with many
similar spin systems, which are practically isolated from each other. Therefore, we are
interested in determining σ (t) , which is the average of the individual σ (t) .
We start by taking the explicit solution to (10), given by
σ (t) =σ (0)− i H + H1(t2 ),σ (t2 )⎡⎣ ⎤⎦dt2
0
t
∫ . (11)
This can then be inserted into (10), and the average over the σ (t) taken in order to obtain
ddtσ (t) = −i H + H1(t),σ (0)⎡⎣ ⎤⎦ − H + H1(t),[H + H1(t2 ),σ (t2 )]⎡⎣ ⎤⎦dt2
0
t
∫ . (12)
Immediately, we see that we can drop the term H1(t),σ (0)⎡⎣ ⎤⎦ since the average of H1(t)
is zero. At this point, it is necessary to further define the form of H1(t) . We let H1(t)
take the form of stochastic fields being applied in the x, y, and z directions to each
individual spin.
188
H1(t) = φλ ,p (t)Jλ ,p
p=1
N
∑λ=x ,y ,z∑ . (13)
Here, Jλ ,p is a spin operator, in the x, y, or z direction, operating on the pth spin (such that
J will be S or I depending on whether the pth spin is an electron or nucleus). φλ ,p (t)
specifies the strength and time dependence of the field being applied. Note that we do not
include modulation of couplings in our Hamiltonian leading to relaxation. Inclusion of
such terms could lead to transfer of populations between spins, which will not be
accounted for here. We further specify the form of φλ ,p (t) by stating that
lim′t →∞
1′tφλ ,p (t1)φχ ,q (t1 + t2 )dt1 =
1Tλ ,pq
δλ ,χδ (t2 )0
′t
∫ , (14)
This equation gives the correlation function for fields applied to spins p and q. If p=q
then this gives the strength and correlation time of a field applied to a single spin.
Otherwise, this indicates that fields applied to two different spins are correlated. In (14),
δλ ,χ is the Kronecker delta, which is 1 if λ = χ and 0 otherwise. δ (t2 ) is the Dirac delta;
therefore δ (t2 ) is infinite if t2 = 0 and is 0 otherwise, and its integral over all time is 1.
1/ Tλ ,pq gives the strength of the field being applied.
By giving the stochastic fields in H1(t) this form, we have made the correlation
time of the applied fields to be τ c = 0 . Because of this, the spectral density of the applied
fields takes on a uniform frequency distribution. By taking this assumption, we will be
able to specify the relaxation times of the various states without accounting for spectral
density of the applied field. This is useful if the relaxation times are known from another
source (i.e. computation or experiment). However, changes in the relaxation times due to
changes in the static Hamiltonian cannot be accounted for with this assumption. Note that
a similar assumption is taken by Cheng et al. [21] for treating noise in quantum
computations. Also, Goldman discusses the result for finite correlation times, and
therefore can account for the spectral density [22]. Note that usually,
1/ Tx ,pq = 1/ Ty ,pq = 1/ Tz ,pq , since the direction of the stochastic fields being applied to the
spins is arbitrary. However, because we have assumed τ c = 0 , we have to artificially
189
reintroduce the spectral density. Since stochastic fields in the x- and y- directions have to
overcome the large Zeeman field to induce relaxation, they require spectral density at
much higher frequencies than the field in the z-direction to be effective. To account for
this, we assume that 1/ Tx ,pq = 1/ Ty ,pq ≤1/ Tz ,pq . We may now evaluate (12), first by
expanding the commutator within the integral.
ddtσ (t) = −i H ,σ (0)⎡⎣ ⎤⎦
−H ,[H ,σ (t2 )⎡⎣ ⎤⎦ + H ,[H1(t2 ),σ (t2 )⎡⎣ ⎤⎦
+ H1(t),[H ,σ (t2 )]⎡⎣ ⎤⎦ + H1(t),[H1(t2 ),σ (t2 )]⎡⎣ ⎤⎦
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪dt2
0
t
∫. (15)
We note that because the correlation time is zero for the stochastic fields, infinitely many
applications of the stochastic field have occurred in any finite time period. The result is
that the individual σ (t) converge to a single value. Therefore, the commutators that have
exactly one instance of H1(t) average to zero, yielding
ddtσ (t) = −i H ,σ (0)⎡⎣ ⎤⎦ − H ,[H ,σ (t2 )⎡⎣ ⎤⎦ + H1(t),[H1(t2 ),σ (t2 )]⎡⎣ ⎤⎦{ }dt2
0
t
∫ . (16)
Combining the first two terms, we obtain
ddtσ (t) = −i H ,σ (t)⎡⎣ ⎤⎦ − H1(t),[H1(t2 ),σ (t2 )]⎡⎣ ⎤⎦dt2
0
t
∫ . (17)
As such, we have totally separated the action of the static Hamiltonian and relaxation due
to stochastic fields. Note that this was only possible because of the correlation time,
τ c = 0 ; otherwise, the relaxation rates depend on the relationship between the spectral
density of the stochastic fields and the static Hamiltonian, and therefore none of the terms
in the integral in (15) vanish. Since we do not know the value of H1(t) at any particular
point in time, it is necessary to take the time average of the integral over all time.
ddtσ (t) = −i H ,σ (t)⎡⎣ ⎤⎦ − lim
′t →∞
1′t 0
′t
∫ H1(t1),[H1(t1 + t2 ),σ (t2 )]⎡⎣ ⎤⎦dt20
t
∫ dt1 . (18)
190
There are several steps required to evaluate the double integral in (18) which are not
shown here, but discussed in detail in the Appendix. We give the result for the integral
from the Appendix.
− limt→∞
H1(t1),[H1(t1 + t2 ),σ (t2 )]⎡⎣ ⎤⎦dt2 dt10
t
∫0
′t
∫
= − 1Tλ ,pq
Jλ ,p ,[Jγ ,q ,σ (0)]⎡⎣ ⎤⎦q=1
N
∑p=1
N
∑λ=x ,y ,z∑
(19)
With this result, we must now evaluate the integral in (18) for the various states included
in the basis set of our simulation, given the form of the stochastic field found in (13) and
(14). For an example, we begin by taking a spin, S j , and evaluate the relaxation of the
state S j
+ . In this case, the field being applied is given by
H1(t) = φx , j (t)S jx +φy , j (t)S jy +φz , j (t)S jz . (20)
Taking the result in (19), we obtain
− limt→∞
1t
H1(t1),[H1(t1 + t2 ),S j+ ]⎡⎣ ⎤⎦
0
t1
∫ dt2 dt10
t
∫
= 1Tx , j
S jx[S jx ,S j+ ]⎡⎣ ⎤⎦ +
1Ty , j
S jy[S jy ,S j+ ]⎡⎣ ⎤⎦ +
1Tz , j
S jz[S jz ,S j+ ]⎡⎣ ⎤⎦
. (21)
If we evaluate the commutators, we obtain
− limt→∞
1t
H1(t1),[H1(t2 ),S j+ ]⎡⎣ ⎤⎦dt2 dt1
0
t1
∫0
t
∫
= −S j+ 1
2Tx , j
+ 12Ty , j
+ 1Tz , j
⎛
⎝⎜
⎞
⎠⎟
. (22)
Noting that 1/ Tx , j = 1/ Ty , j , and defining
1/ Tx , j +1/ Tz , j = 1/ T2 j this simplifies to
−S j+ 1
Tx , j
+ 1Ty , j
+ 1Tz , j
⎛
⎝⎜
⎞
⎠⎟ = −S j
+ 1Tx , j
+ 1Tz , j
⎛
⎝⎜
⎞
⎠⎟
= −S j+ 1
T2 j
. (23)
191
We apply this same treatment to all states, including those for spin 1, and give these in
Table I.
Table I: Relaxation Rates of 1-spin States
Spin State Relaxation rate Relaxation Rate
(in terms of T1, T2)
S j
+
1Tx , j
+ 1Tz , j
1T2 j
S j
−
1Tx , j
+ 1Tz , j
1T2 j
S jz
2Tx , j
1T1 j
S jz
2 − 23 1
6Tx , j
3T1 j
(S j
+ )2
2Tx , j
+ 4Tz , j
4T2 j
− 2T1 j
(S j
− )2
2Tx , j
+ 4Tz , j
4T2 j
− 2T1 j
S jzS j
+ + S j+S jz
5Tx , j
+ 1Tz , j
1T2 j
+ 2T1 j
S jzS j
− + S j−S jz
5Tx , j
+ 1Tz , j
1T2 j
+ 2T1 j
Therefore, if we specify the longitudinal and transverse relaxation times (T1 and
T2, respectively) for a specific spin, it is possible to calculate the relaxation rates of all
states for a spin-1 system under our assumptions. However, we need to consider a many
spin system for which it is necessary to obtain relaxation rates for all states, including
states involving more than one spin. Initially, to consider relaxation rates of states with
multiple spins, we will add an additional assumption, which will be removed again later.
192
We assume that fields acting on different spins are uncorrelated, as represented by a
modification to (14) to obtain
lim′t →∞
1′tφλ ,p (t1)φχ ,q (t1 + t2 )dt
0
′t
∫ = δλ ,χδ p,qδ (t2 ) . (24)
Under this assumption, it is possible to separate the relaxation effects acting on separate
spins for those states that involve multiple spins. For example, if we evaluate the integral
in (17) for S j
+Sk+ , we find that the relaxation rate of the state is the sum of the relaxation
rates of the single-spin states that are involved- here the relaxation rate is 1/ T2 j +1/ T2k .
We arrive at this result by the following steps.
H1, j (t) = φx , j (t)S jx +φy , j (t)S jy +φz , j (t)S jz
H1,k (t) = φx ,k (t)Skx +φy ,k (t)Sky +φz ,k (t)Skz
− lim′t →∞
1′t
H1, j (t1)+ H2,k (t1),[H1, j (t1 + t2 )+ H2,k (t1 + t2 ),S j+Sk
+ ]⎡⎣ ⎤⎦dt2 dt10
t
∫0
′t
∫
= −
1Tx , j
S jx ,[S jx ,S j+Sk
+ ]⎡⎣ ⎤⎦ +1
Ty , j
S jy ,[S jy ,S j+Sk
+ ]⎡⎣ ⎤⎦ +1
Tz , j
S jz ,[S jz ,S j+Sk
+ ]⎡⎣ ⎤⎦ +
1Tx ,k
Skx ,[Skx ,S j+Sk
+ ]⎡⎣ ⎤⎦ +1
Ty ,k
Sky ,[Sky ,S j+Sk
+ ]⎡⎣ ⎤⎦ +1
Tz ,k
Skz ,[Skz ,S j+Sk
+ ]⎡⎣ ⎤⎦
⎧
⎨⎪⎪
⎩⎪⎪
⎫
⎬⎪⎪
⎭⎪⎪
. (25)
Because the relaxing fields which act on S j and Sk are uncorrelated, we were able to
separate the two actions of the two fields. From here, the two integrals are evaluated as in
the single spin case, yielding
H1, j (t) = φx , j (t)S jx +φy , j (t)S jy +φz , j (t)S jz
H1,k (t) = φx ,k (t)Skx +φy ,k (t)Sky +φz ,k (t)Skz
− limt→∞
1t
H1, j (t1)+ H2,k (t1),[H1, j (t2 )+ H2,k (t2 ),S j+Sk
+ ]⎡⎣ ⎤⎦dt2 dt10
t1
∫0
t
∫
= − 1Tx , j
+ 1Tz , j
⎛
⎝⎜
⎞
⎠⎟ S j
+Sk+ − 1
Tx ,k
+ 1Tz ,k
⎛
⎝⎜
⎞
⎠⎟ S j
+Sk+
= − 1T2 j
+ 1T2k
⎛
⎝⎜
⎞
⎠⎟ S j
+Sk+
.
(26)
By default, our program assumes that uncorrelated fields are applied to the individual
spins, so relaxation of states with multiple spins will simply relax at a rate that is the sum
193
of the relaxation rates for the single-spin states. This is a good assumption for unlike
spins, such as an electron and a nucleus, since they will interact very differently with
their environment. However, two like spins that are near to each other may experience
correlated fields leading to their relaxation.
To understand the effects of having correlated fields applied to two spins
undergoing relaxation, we consider H1, j (t) and
H1,k (t) acting on spins S j and Sk . For
simplicity, we only apply a stochastic field in the z direction, although in the summary
table (Table II), we include fields in all directions.
H1, j (t) = φz (t)+ cχ z (t)( )S jz
H1,k (t) = ϕ z (t)+ (1− c)χ z (t)( )Skz
. (27)
In (27), we use different notation so that we can split φz , j (t) and
φz ,k into uncorrelated
and correlated parts. In this notation, φz (t) and ϕ z (t) are uncorrelated. However, both
spins have a correlated field acting on them, given by the χ z (t) , where c is a weighting
coefficient for the effect of the correlated field on each spin. The correlation times are
given by
lim′t →∞
φz (t1)φz (t1 + t2 )dt0
′t
∫ = 1Tz ,φ
δ (t2 )
lim′t →∞
ϕ z (t1)ϕ z (t1 + t2 )dt0
′t
∫ = 1Tz ,ϕ
δ (t2 )
lim′t →∞
χ z (t1)χ z (t1 + t2 )dt0
′t
∫ = 1Tz ,χ
δ (t2 )
lim′t →∞
φz (t1)ϕ z (t1 + t2 )dt0
′t
∫ = lim′t →∞
φz (t1)χ z (t1 + t2 )dt0
′t
∫ = lim′t →∞
ϕ z (t1)χ z (t1 + t2 )dt0
′t
∫ = 0
. (28)
For an example, we evaluate the integral in (17), applied to S j
+Sk+ .
194
− lim′t →∞
1′t
H1, j (t1)+ H1,k (t1),[H1, j (t1 + t2 )+ H1,k (t1 + t2 ),S j+Sk
+ ]⎡⎣ ⎤⎦0
t
∫ dt2 dt10
′t
∫
= − 1Tz ,φ
S jz ,[S jz ,S j+Sk
+ ]⎡⎣ ⎤⎦ −1
Tz ,ϕ
Skz ,[Skz ,S j+Sk
+ ]⎡⎣ ⎤⎦
− 1Tz ,χ
cS jz + (1− c)Skz ,[cS jz + (1− c)Skz ,S j+Sk
+ ]⎡⎣ ⎤⎦
= − 1Tz ,φ
+ 1Tz ,ϕ
+ 1Tz ,χ
⎛
⎝⎜
⎞
⎠⎟ S j
+Sk+
. (29)
In (29), the solution contains the correlation times of the fields resulting from φz (t) ,
ϕ z (t) , and χ z (t) . For reference, we give the solution to the integral applied to S j
+ and
Sk+ .
− limt→∞
1t
H1, j (t1),[H1, j (t2 ),S j+ ]⎡⎣ ⎤⎦
0
t1
∫ dt2 dt10
t
∫ = − 1Tz ,φ
+ c2
Tz ,χ
⎛
⎝⎜
⎞
⎠⎟ S j
+
− limt→∞
1t
H1,k (t1),[H1,k (t2 ),Sk+ ]⎡⎣ ⎤⎦
0
t1
∫ dt2 dt10
t
∫ = − 1Tz ,ϕ
+ (1− c)2
Tz ,χ
⎛
⎝⎜
⎞
⎠⎟ Sk
+
. (30)
As one can see, the double quantum coherence, S j
+Sk+ , no longer relaxes with the sum of
the relaxation rates of the single quantum coherences. If one simply sums up the
relaxation rates of the single quantum coherences, the result is too small, by a difference
of 2c / Tχz . Similarly, if one computes the relaxation rate of the zero-quantum coherence,
one obtains
− limt→∞
1t
H1, j (t1)+ H1,k (t1),[H1, j (t2 )+ H1,k (t2 ),S j+Sk
− ]⎡⎣ ⎤⎦0
t1
∫ dt2 dt10
t
∫
= − 1Tz ,φ
+ 1Tz ,ϕ
+ (1− 2c)2
Tz ,χ
⎛
⎝⎜
⎞
⎠⎟ S j
+Sk−
. (31)
In this case, if one sums up the relaxation rates of the single quantum coherences, the
result is too large, by a difference of (2c − 4c2 ) / Tz ,χ . Note that if the stochastic field
applied to each spin is identical, then 1/ Tz ,φ = 0 ,
1/ Tϕ ,z = 0 , and c = 0.5 . In this case, the
195
relaxation rate of the zero quantum coherence is zero; this is the origin of long-lived
coherences [23].
For spin states involving more than two spins, it becomes very difficult to account
for correlated fields that are acting on multiple spins with different strengths. For the
remainder of the discussion, it will be assumed that any correlated field acts on only two
spins. Also, correlated fields will only be considered for S=1/2.
In Table II, the relaxation rates are specified for states involving two spins. We
include fields being applied in the x and y directions, as well as the z direction, as given in
(32), where a is a weighting coefficient for the correlated field acting on each spin.
H1, j (t) =
φx (t)+ aχ x (t)( )S jx + φy (t)+ aχ y (t)( )S jy + φz (t)+ cχ z (t)( )S jz
H1,k (t) =
ϕx (t)+ (1− a)χ x (t)( )Skx + ϕ y (t)+ (1− a)χ y (t)( )Sky + ϕ z (t)+ (1− c)χ z (t)( )Skz
. (32)
Table II: Relaxation rates for two-spin states
Spin State Relaxation rate
(in terms of stochastic fields)
Relaxation
Rate (input)
S j
+ , S j−
1Tx ,φ
+ a2
Tx ,χ
+ 1Tz ,φ
+ c2
Tz ,χ
1T2 j
Sk+ , Sk
−
1Tx ,ϕ
+ (1− a)2
Tx ,χ
+ 1Tz ,ϕ
+ (1− c)2
Tz ,χ
1T2k
S jz
2Tx ,φ
+ 2a2
Tx ,χ
1T1 j
Skz
2Tx ,ϕ
+ 2(1− a)2
Tx ,χ
1T1k
S j
+Sk+ , S j
−Sk−
1Tx ,φ
+ 1Tx ,ϕ
+ 1− 2a + 2a2
Tx ,χ
+ 1Tz ,φ
+ 1Tz ,ϕ
+ 1Tz ,χ
1T2, jk
DQ
196
S j
+Sk− , S j
−Sk+
1Tx ,φ
+ 1Tx ,ϕ
+ (1− 2a)2
Tx ,χ
+ 1Tz ,φ
+ 1Tz ,ϕ
+ (1− 2c)2
Tz ,χ
1T2, jk
ZQ
S jzSkz
2Tx ,φ
+ 2Tx ,ϕ
+ 2(1− 2a)2
Tx ,χ
1T1, jk
LLS
S jzSk
+ , S jzSk−
2Tx ,φ
+ 1Tx ,ϕ
+ 1Tz ,ϕ
+ 4a2 − 3a +1Tx ,χ
+ (1− c)2
Tz ,χ
~ 1
T2k
S j
+Skz , S j−Skz
2Tx ,ϕ
+ 1Tx ,φ
+ 1Tz ,φ
+ 4a2 −5a + 2Tx ,χ
+ c2
Tz ,χ
~ 1
T2 j
S j+Sk
− + S j−Sk
+
→ 4S jzSkz
2a(1− a)Tx ,χ
1Tjk
trans
4S jzSkz →
S j+Sk
− + S j−Sk
+
4a(1− a)Tx ,χ
2Tjk
trans
S jzSk+ → S j
+Skz ,
S jzSk− → S j
−Skz
− a(1− a)
Tx ,χ
− 1
2Tjktrans
S j+Skz → S jzSk
+
S j−Skz → S jzSk
−
− a(1− a)
Tx ,χ
− 1
2Tjktrans
The correlated fields that are inducing relaxation in the two-spin states are also inducing
transfer between some of the states, which is specified by the rate 1/ Tjk
trans . Note that if
the fields applied to two spins are identical, then a = c = 0.5 and
1/ Tx ,φ = 1/ Ty ,φ = 1/ Tz ,φ = 1/ Tx ,ϕ = 1/ Ty ,ϕ = 1/ Tz ,ϕ = 0 . In this case, the
S j+Sk
− + S j−Sk
+ and
the S jzSkz states do not lose any population, although there is a transfer of population
between these states. This brings about long-lived singlet states [24].
One could consider applying correlated fields to all possible combinations of
spins. However, we limit the discussion here to correlated fields being applied to pairs of
spins, and will only allow for this in our simulations. Nevertheless, one must still apply
the relaxation to states involving more than two spins. For example, if we are interested
197
in the relaxation of the state S j
+Sk+Sl
− , then we must derive a relaxation rate from the input
rates, which are given in Table II. In order to do this, we start with the relaxation rate that
would be obtained if the fields applied to each spin were uncorrelated.
1T0
= 1T2 j
+ 1T2k
+ 1T2l
⎛
⎝⎜
⎞
⎠⎟ (33)
For the two-spin case, we note that because of the correlated fields, S j
+Sk+ relaxes faster
by a difference of 1/ T2, jk
DQ −1/ T2 j −1/ T2k , S j
+Sl− relaxes slower by a difference of
1/ T2, jl
ZQ −1/ T2 j −1/ T2l , and Sk+Sl
− relaxes slower by a difference of 1/ T2,kl
ZQ −1/ T2k −1/ T2l
. The same corrections can be made in the three-spin case, so that the relaxation rate of
S j
+Sk+Sl
− is given by
1T= 1
T2 j
+ 1T2k
+ 1T2l
⎛
⎝⎜
⎞
⎠⎟ +
1T2, jk
DQ − 1T2 j
− 1T2k
⎛
⎝⎜
⎞
⎠⎟ +
1T2, jl
ZQ − 1T2 j
− 1T2l
⎛
⎝⎜
⎞
⎠⎟
+ 1T2,kl
ZQ − 1T2k
− 1T2l
⎛
⎝⎜
⎞
⎠⎟
= 1T2, jk
DQ + 1T2, jl
ZQ + 1T2,kl
ZQ
⎛
⎝⎜
⎞
⎠⎟ −
1T2 j
+ 1T2k
+ 1T2l
⎛
⎝⎜
⎞
⎠⎟
. (34)
The fact that the same terms can be used to make the correction arises because there is
not a correlated field being applied to all three spins. One should also note that transfers
also occur for this case. The transfer rates are given as before. For example, there is a
transfer between S j
+Sk+Sl
− → 4S j+SkzSlz . The rate is the same as the rate for the transfer
between Sk+Sl
− → 4SkzSlz , given by 1/ Tkltrans .
At last, we know how to calculate the relaxation rates and transfer rates for a
many spin system. The relaxation rates that were calculated will appear as negative
values along the diagonal of the relaxation matrix, ΓΓ . The transfer rates that were
calculated appear as off diagonal elements that connect the involved states.
198
6.2.2.2 Recovery to Thermal Equilibrium
A critical part of DNP is the recovery of the spin system towards its thermal
equilibrium value. This both destroys nuclear polarization and generates additional
polarization on the electrons that is transferred to the nuclei via DNP. This recovery is
calculated using (3), where σσ eq represents the state of the system at equilibrium. If no
oscillating fields are being applied, then the equilibrium, σσ eq , is the thermal equilibrium,
given by
σσ eq
thermal = 1Z
exp −H0
kBT⎡
⎣⎢
⎤
⎦⎥ . (35)
In (35), H0 is the static Hamiltonian, Z normalizes σσ eq
thermal , kB is the Boltzmann
constant, and T is the temperature. This can be expanded as
σσ eqthermal = 1
Z1−
H0
kBT+ 1
2H0
kBT⎛
⎝⎜⎞
⎠⎟
2
− 16
H0
kBT⎛
⎝⎜⎞
⎠⎟
3
+…⎡
⎣⎢⎢
⎤
⎦⎥⎥
. (36)
Except at very low temperatures, this can be truncated at the second term; this is the high
temperature approximation. Note that the identity does not affect the evolution under the
density matrix formalism. Therefore, it can be dropped. For numerical evaluation, this is
important- if polarizations are small, the deviation of the density matrix from the identity
can be quite small, and small decimals can be lost if the identity is included. Therefore,
we typically use
σσ eq
thermal = − 1Z
H0
kBT, (37)
although the validity of the truncation is checked in our simulation before using this
approximation- otherwise the full exponential is calculated, with the identity subtracted
away for accuracy.
σσ eq
thermal should be used in place of σσ eq in (3) in the case that no oscillating fields
are being applied. Rewriting (3) for this case gives us
ddtσ (t) ={−iH 0 − Γ}(σ (t)−σ eq
thermal ) , (38)
199
where H 0 = H0 ⊗ E − E ⊗ H0 . However, in most cases we are interested in situations
where a microwave and/or RF field is being applied. In this case, we can no longer use
(3) as stated, because we do not know the value of σσ eq . In addition, we must make the
transformation into the rotating frame to generate a static Hamiltonian. To begin, we
rewrite (38) in the rotating frame.
ddtσ r (t) ={−iH 0
r − Γ}(σ r (t)−σ eqthermal ,r (t)) (39)
In (39), H 0r = H0
r ⊗ E − E ⊗ H0r and
σσ eq
thermal ,r is given by
σσ eqthermal ,r (t) =
exp −i ω MW S jzj=1
NS
∑ + ω RF Imzm=1
NI
∑⎛
⎝⎜⎞
⎠⎟t
⎡
⎣⎢⎢
⎤
⎦⎥⎥σ eq
thermal exp i ω MW S jzj=1
NS
∑ + ω RF Imzm=1
NI
∑⎛
⎝⎜⎞
⎠⎟t
⎡
⎣⎢⎢
⎤
⎦⎥⎥
(40)
Although time dependent terms arise in σσ eq
thermal ,r , these are dropped, since their fast
oscillation will make them negligible in calculations. The static part σσ eq
thermal ,r (t) is given
by
σσ eqthermal ,r = 1
Zexp − X
kBT⎡
⎣⎢
⎤
⎦⎥
X = H0r + ω MW S jz
j=1
NS
∑ + ω RF Imzm=1
NI
∑. (41)
As mentioned, we do not know what the value of σσ eq is for the full Hamiltonian,
including oscillating fields. However, we may assume that the recovery from relaxation
that occurs in the absence of oscillating fields is the same as when fields are applied.
Then, we obtain
ddtσ r (t) ={−iH r − Γ}σ r (t)+{iH 0
r + Γ}σ eqthermal ,r . (42)
Finally, this allows the calculation of the evolution of the density matrix under
irradiation. Note that the assumption that recovery from relaxation does not change when
oscillating fields are applied is not always valid. If the oscillating fields tilt the
eigenframe of the Hamiltonian significantly, then some of the relaxation rates may
change. For example, this arises when electrons or nuclei are spin-locked. In this case, the
200
transverse relaxation time often increases significantly, because the spin-locking field
overcomes relaxation processes that are being applied to the spin system in the z-
direction. However, accurate results can still be obtained by replacing the T2 with the
relaxation time under the spin-lock, T1ρ.
One should take careful note of the forms of the Hamiltonians appearing in (42).
σσ eq
thermal ,r is calculated from the static Hamiltonian in the lab frame( H0 ), and then is
subsequently brought into the rotating frame. H 0r is calculated from the static
Hamiltonian in the rotating frame ( H0r ). Finally, H r is calculated from the full
Hamiltonian, which includes microwave and RF fields, in the rotating frame ( H r ).
A last point that should be mentioned: we have discussed truncation of the basis
set for generation of the coherent Liouville matrix, H r . In the case of a truncated basis
set, the columns and rows in ΓΓ that correspond to the truncated states can simply be
omitted, and the lifetimes of these states do not need to be calculated.
6.2.3 Propagation of the Spin System Now that we have calculated all of the terms in (9) and (42), we must propagate the
system through time. This can be done with or without relaxation, either using (9) or (42)
respectively. Additionally, we must extract expectation values of particular states from
the total state of the spin system. We begin by discussing the propagation without
relaxation.
6.2.3.1 Propagation in Hilbert Space
We begin with the explicit solution to (9), where the Hamiltonian, H r , is time-
independent.
σ r (t) = exp −iH rt⎡⎣ ⎤⎦σ
r (0)exp iH rt⎡⎣ ⎤⎦ . (43)
It is straightforward to solve for σr (t) in this form. However, if it is necessary to
compute σr (t) for many time points, then this becomes computationally expensive, since
the exponentials are computed with a Taylor Series expansion at every time point. We
201
use two methods to avoid this problem: (1) Computation of a propagator if the time
points are equally spaced, or (2) diagonalization of the Hamiltonian, followed by
propagation in the diagonalized frame.
(1) Propagator Method
Computation of the propagator is done simply by computing
P = exp −iH rΔt⎡⎣ ⎤⎦ , (44)
where Δt = tn − tn−1 . Then, for the nth time point, tn , one may compute σr (tn ) from
σr (tn−1) .
σr (tn ) = Pσ r (tn−1)P (45)
Note that this method is only useful if the spacing between time points is uniform.
We usually are not interested in the full state of the spin system, but rather the
expectation values of a few states of the system. Once σr (tn ) is obtained, it is then
necessary to extract those expectation values. If we let X be the matrix representing the
state of interest, X † the complex transpose of X , and X the expectation value of X ,
then we may calculate
X =
tr X †σσ r (t)( )tr X † X( ) (46)
We use the notation tr x( ) , which is the sum of the diagonal elements of a matrix. For a
matrix of length N, the trace is given by
tr x( ) = xnn
n=1
N
∑ . (47)
Although relatively fast, this method still requires a matrix multiplication at every time
point, which may be avoided if one diagonalizes the Hamiltonian prior to the
propagation.
(2) Diagonalization Method
If the values of t, for which σr (t) are to be computed, are not uniformly spaced,
then it may be useful to diagonalize the Hamiltonian prior to propagation. We define U
and H r ,D such that
202
H r ,D =U −1HUH =UH r ,DU −1
. (48)
where H r ,D is a diagonal matrix. Then, we obtain
σ r (t) = exp −iUH r ,DU −1t⎡⎣ ⎤⎦σ
r (0)exp iUH rU −1t⎡⎣ ⎤⎦ . (49)
The U and U-1 matrices can be moved outside of the exponential. Multiplying both sides
of the equation in (49) on the left by U-1 and on the right by U yields
U −1σ r (t)U = exp −iH r ,Dt⎡⎣ ⎤⎦ U −1σ r (0)U( )exp iH r ,Dt⎡⎣ ⎤⎦ . (50)
Because the exponential is calculated for a diagonal matrix, it is not necessary to
calculate the matrix exponential, thus avoiding a Taylor series expansion on the full
matrix. Instead, one may do a scalar exponential for each of the elements on the diagonal
of the matrix, which is significantly less computationally demanding. One must only
transform the initial density matrix by calculating U−1σ r (0)U in order to obtain
U−1σ r (t)U .
Once we have obtained U−1σ r (t)U , one could transform back to the non-
diagonalized rotating frame. However, this adds additional computation at every time
point, and so it is faster to transform the detection matrix, to obtain U −1X †U . We then
calculate
tr U −1X †U U −1σσ r (t)U( )( )tr X † X( ) =
tr U −1X †σ r (t)U( )tr X † X( ) =
tr X †σ r (t)( )tr X † X( ) . (51)
Note that the trace is invariant to the transformation, which brings about the equality in
the latter part of (51).
6.2.3.2 Propagation in Liouville Space
In order to calculate the evolution of the spin system including relaxation, one
must solve for σσr (t) using (42). We note that the basis set used here includes products of
the states (1, S jz , S j
+ , S j−…) for electrons and
(1, I jz , I j
+ , I j−…) for nuclei. The products of
all possible states include the identity, 1 , in the spin-state vector, σσr (t) . We place this
203
state in the first element of the vector, and set its value to 1. By doing so, it is possible to
generate a matrix, L , such that
ddtσσ r (t) = Lσ r (t) . (52)
This matrix is given by
La,b = −iH r − ΓΓ⎡⎣ ⎤⎦a,b+
iH 0r + Γ( )σ eq
thermal ,r⎡⎣
⎤⎦b
a = 1
0 a ≠ 1
⎧⎨⎪
⎩⎪. (53)
Note that the first row and first column of −iH r − ΓΓ are all zero, since the identity
element does not evolve, and does not affect evolution. Therefore, (53) places the vector
iH 0
r + Γ( )σ eqthermal ,r , which brings about recovery to thermal equilibrium, into the first
column of L . This column is multiplied by the identity element in σσr (t) as shown in
(54).
ddt
1σ r (t)2
σ r (t)n
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
=
0 0 0iH 0
r + Γ( )σ eqthermal ,r⎡
⎣⎤⎦1
−iH r − ΓΓ⎡⎣ ⎤⎦2,2−iH r − ΓΓ⎡⎣ ⎤⎦n,2
iH 0r + Γ( )σ eq
thermal ,r⎡⎣
⎤⎦n
−iH r − ΓΓ⎡⎣ ⎤⎦2,n−iH r − ΓΓ⎡⎣ ⎤⎦n,n
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
1σ r (t)2
σ r (t)n
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
(54)
In (54), there are n possible states in the spin system. The first state is the identity, whose
magnitude is fixed at 1. The matrix multiplication causes the 1 in σσr (t) to be multiplied
by the vector iH 0
r + Γ( )σ eqthermal ,r , which appears in the first column of L. Since the first
row of L is entirely filled with zeros, the identity state will remain fixed at 1. Therefore,
we have consolidated all terms in (42) into a single matrix, L, for which the evolution of
σσr (t) is given by (52).
The explicit solution for (52) is given by
204
σσr (t) = exp Lt⎡⎣ ⎤⎦σ
r (0) . (55)
As was the case in Hilbert space, computing the exponential at many time points will be
very computation demanding, and so it is useful to consider other methods of
propagation. In fact, the situation is considerably worse in Liouville space, since the size
of L is given by 4N×4N where N is the number of S=1/2 spins, compared to 2N×2N for H in
Hilbert space. We discuss several methods reducing the computation difficulty here.
(1) Propagator Method
In this method, as in Hilbert space, we simply calculate a propagator for a specific
time step, Δt . In this case, one may calculate σσr (tn ) from σσ
r (tn−1) using P if
tn − tn−1 = Δt . If
P = exp LΔt⎡⎣ ⎤⎦ , (56)
then
σσr (tn ) = Pσ r (tn−1) . (57)
As was the case in Hilbert space, we want to extract the expectation value of a
state of interest, which is given by X . In this case X is no longer a 2-dimensional matrix,
but instead is a vector. The elements are the same as those in the density matrix X, but
spread columnwise into a column vector. We may then obtain X from (58).
X =
X †σσ eqr
X † X. (58)
Since X † is a row vector, X †σσ eq
r and X † X are scalars. As with the equilibrium position,
we may obtain X by calculating
X =X †
n σσ r (t)⎡⎣ ⎤⎦nn=1
N
∑
Xn† Xn
n=1
N
∑= X †σ r (t)
X † X. (59)
This method may be useful when a few time points which are equally spaced need to be
calculated. However, for many time points, or unequally spaced time points, the
following method is more efficient.
205
(2) Full Matrix Diagonlization
Ideally, one wants to avoid repeated calculation of matrix exponentials, since
these require a Taylor series expansion of a matrix, which is computationally demanding.
If U diagonalizes the matrix L such that
LD =U −1LUL =ULDU −1
, (60)
then it is possible to rewrite (55) as
σσ r (t) =U exp LDt⎡⎣ ⎤⎦ U −1σ r (0)( ) . (61)
If we multiply this by U −1 , we obtain
U −1σσ r (t) = exp LDt⎡⎣ ⎤⎦ U −1σ r (0)( ) . (62)
In this form, the exponential is on a diagonal matrix, so the scalar exponential of each
element can be computed instead of the full matrix exponential. The expectation value of
a state, X , can then be calculated as
X =
X †U( ) U −1σσ r (t)( )X † X
. (63)
Thus, we may calculate X †U( ) X † X( ) , and multiply by the result of (62) to obtain the
expectation value of the state.
(3) Block Diagonal Matrix Propagation
In some special cases, the matrix L may be block diagonal. This may occur in the
case of pure spin diffusion (no DNP), or with some truncation of the basis set. In this
case, L may be broken up into several matrices L{a} , where {a} indicates a set of states in
L that are block diagonal. Then, we may calculate σσ{a}
r (t) using a diagonalization step
with
σσ r
{a}(t) =U{a} exp LD{a}t⎡⎣ ⎤⎦ U −1
{a}σr{a}(0)( ) , (64)
where
L{a} =U{a}L{a}
D U{a}−1 . (65)
Then, the expectation value of a state, X , may be obtained with
206
X =X †U( ) U −1σσ r (t)( )
X † X
X = 1X † X
X{a}† U{a}( ) U{a}
−1σ{a}r (t)( )
{a}∑
. (66)
Because the matrices to be diagonalized are smaller, this method can be considerably
faster than full matrix diagonalized, but requires the special case of a block diagonal
matrix.
(4) Quasi-Equilibrium Propagation
In many cases in DNP, we may be interested in the long-term behavior of a
system. However, many states that are important to the DNP evolution will have lifetimes
that are significantly shorter than the time scale of the observation [25]. In this case, we
may rewrite the matrix L as
L =LLL LSL
LLS LSS
⎛
⎝⎜⎜
⎞
⎠⎟⎟
. (67)
In this case, we have specified states with short lifetimes with S and states with long
lifetimes with L (as compared to the observation time scale). Then, LLL connects the
long-lived states to other long-lived states. LSS connects the short-lived states to other
short-lived states. LLS and LSL connect the long-lived states to the short-lived states. We
can then rewrite (52) as
ddtσσ L
r (t)
0
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟=
LLL LSL
LLS LSS
⎛
⎝⎜⎜
⎞
⎠⎟⎟
σ Lr (t)
σ Sr (t)
⎛
⎝⎜⎜
⎞
⎠⎟⎟
. (68)
Note that we have replaced dσσ Sr (t) / dt with 0. We assume that the short-lived states,
given by σσ Sr (t) , see the long-lived states as being nearly static over the lifetime of the
short-lived states. Thus, σσ Sr (t) reaches an equilibrium value quickly compared to the
evolution of the long-lived states. We refer to this as quasi-equilibrium. The value of
σσ Lr (t) will change on the longer time scale, and σσ S
r (t) will adjust to this change.
However, the quasi-equilibrium state of σσ Sr (t) may be calculated in terms of σσ L
r (t) as a
207
normal equilibrium would be calculated: by setting the derivative of the state equal to
zero. The quasi-equilibrium solution for σσ Sr (t) is given by
σσ Sr (t) = −LSS
−1 LLSσ Lr (t) . (69)
This can be inserted into (68) to obtain
ddtσσ L
r (t) = LLL − LSLLSS−1 LLS( )σ L
r (t) . (70)
We may define a reduced matrix, Lred = LLL − LSLLSS−1 LLS , and then calculate
σσ Lr (t) = exp Lredt⎡⎣ ⎤⎦σ L
r (t) . (71)
This may be solved the same way we solved (61) for the full matrix diagonalization,
Ured
−1σσ Lr (t) = exp Lred
D t⎡⎣ ⎤⎦ Ured−1σ L
r (0)( ) , (72)
where
Lred =Ured LredD Ured
−1 . (73)
One then obtains the expectation values, X , by calculating
X =
X L†Ured( ) Ured
−1σσ Lr (t)( )
X † X. (74)
In (74), X L is the vector for the state X , but the short-lived states have been removed
from the vector. Elements of X corresponding to short-lived states should be zero;
otherwise, one detects short-lived states on a long time scale (our program will
automatically put any detected states into the long-lived state matrix).
Use of the quasi-equilibrium can significantly accelerate computations, since it avoids
diagonalizing a large L matrix, and instead only relies on computation of LSLLSS−1 LLS ,
which is a much faster calculation.
(5) Equilibrium Position
In some cases, we will only be interested in the equilibrium position of the spin
system under DNP conditions. If this is the case, then (42) can be solved directly by
setting the derivative to zero, resulting in
208
σ eq
r = iH r + Γ( )−1iH 0
r + Γ( )σ eqthermal ,r( ) . (75)
This is the simplest method, since no propagation is necessary.
Note that if a diagonlization method has been used to propagate the spin system,
then one eigenvalue of the diagonalized matrix, whether it is the full L or a reduced
matrix, is zero. This eigenvalue represents the steady-state value of the spin system. U
has a column and U −1 has a row that corresponds to this eigenvalue, which we denote by
U{0} and
U{0}
−1 , respectively. Then, we can obtain σσ eq
r by calculating
σσ eq
r =U{0}U{0}−1σ r (0) , (76)
where U{0}U{0}
−1 is a matrix.In either case, we may calculate the expectation value of the
states of interest, X , using (58) or (63).
We now know how to generate the Hamiltonian, how to generate the relaxation
matrix, and how to propagate the system and calculate the expectation values of interest.
At this point, we first discuss the capabilities and input methods of our simulation
program. Subsequently, we will show some examples, including a study of high order
coherence in the solid effect.
6.3 Simulator Usage DNPsim is designed for a combination of ease of use and high flexibility. This
includes options to sweep the magnetic field, microwave frequency, radio frequency, and
experiment length. Input describing the spin system may be in the form of tensors,
allowing for powder averaging, or coupling parameters may be input directly.
Additionally, it is possible to initiate and detect the system in an arbitrary state, making it
possible to investigate the roles of various states. One may use a variety of propagation
methods, with and without relaxation. Finally, basis set truncation can be used both to
accelerate simulations and investigate the roles of states in the basis set.
The DNPsim program is built in MATLAB, and takes advantage of its efficient
libraries for various linear algebra operations. DNPsim is called using two structure
209
variables, which we refer to as Sys and Par, and returns one structure variable, out. This
appears as follows in MATLAB. out = DNP_sim( Sys, Par );
Each of the structure variables contains fields giving some information either about the
experiment or the result. The Sys variable describes the spin-system itself, including
interactions between spins, interactions with the bath (relaxation), and interaction with
the external field. The Par variable describes the experimental conditions, including
strength of applied fields, parameters to be swept, starting states of the spin system and
states to be detected, and methods of simulation. The out variable contains all results of
the simulation. Note that all input and output units are in megahertz (MHz),
microseconds (μs), millitesla (mT), or Kelvin (K).
In the subsequent sections, we discuss how to setup the Sys and Par structure
variables, including a brief discussion of the output of the DNPsim. We also discuss the
basis set used in DNPsim, and conclude with a discussion of other programs included
with the DNPsim.
6.3.1 Spin System In order to specify the spin-system, we must give the interactions of each spin
with all applied fields and interactions with other spins. Additionally, in order to include
relaxation, the interactions of each spin with the bath must be described. Interactions of
the spins with applied fields and other spins can be input directly, as single parameters for
each interaction, or as tensors, which depend on orientation. These modes of input are
discussed in Section 6.3.1.1 and Section 6.3.1.2, respectively. The input of relaxation
parameters, to describe interactions with the bath, is discussed in Section 6.3.1.3. All
information on the spin-system is input into the simulation program using the structure
variable, Sys.
6.3.1.1 Direct Input
In the direct input mode, all interactions with the field and amongst the spins are
specified directly; in other words, they are not dependent on orientation. In this case,
210
there is no magnetic field specified because the Zeeman frequencies of the spins are
given directly to the program. In fact, DNPsim looks for specification of the external
magnetic field in the Par variable to determine whether direct input or orientation
dependent input is being used. If direct input is used, there are up to 13 fields in the Sys
structure that may be specified, excluding relaxation for which there are additional fields.
We consider a spin system with NS electrons and N I nuclei, where the electrons are
specified by S j and the nuclei by Im . In Table III, we give the 13 possible fields for the
Sys structure, a brief description of each field, the corresponding Hamiltonian, and the
appropriate size(s) for the given field. Note that some terms of the Hamiltonian may be
dropped during the rotating frame transformation, as discussed in the Theory section.
Table III: Direct Input Fields
Sys field Description Hamiltonian Field size
Sys.S Spin of each
electron – NS
Sys.I Spin of each
nucleus – N I
Sys.v0S Electron Zeeman
frequency 2π ν0S j
S jzj=1
NS
∑ NS
Sys.v0I Nuclear Zeeman
frequency 2π ν0 Im
I jzm=1
NI
∑ N I
Sys.A Secular electron-
nuclear coupling 2π Aj ,mS jz Imz
m=1
NI
∑j=1
NS
∑ N I × NS
Sys.B
Pseudo-secular
electron-nuclear
coupling (x-
direction) 2π Bj ,mS jz Imx
m=1
NI
∑j=1
NS
∑ N I × NS
Sys.C Pseudo-secular
electron-nuclear 2π C j ,mS jz Imy
m=1
NI
∑j=1
NS
∑ N I × NS
211
coupling (y-
direction)
Sys.d Electron-electron
dipolar coupling 2π d j ,k S jzSkz −
12
S jxSkx + S jySky( )⎡
⎣⎢
⎤
⎦⎥
k> j
NS
∑j=1
NS
∑ 12
NS NS −1( )
Sys.J Electron-electron
scalar coupling 2π J j ,k S jxSkx + S jySky + S jzSkz
⎡⎣ ⎤⎦k> j
NS
∑j=1
NS
∑ 12
NS NS −1( )
Sys.dnn Nuclear-nuclear
dipole coupling 2π d j ,k
nn Imz Inz −12
Imx Inx + Imy Iny( )⎡
⎣⎢
⎤
⎦⎥
n>m
NI
∑m=1
NI
∑ 12
N I N I −1( )
Sys.Jnn Nuclear-nuclear
scalar coupling 2π Jm,n
nn Imx Inx + Imy Iny + Imz Inz⎡⎣ ⎤⎦
n=1
NI
∑m=1
NI
∑ 12
N I N I −1( )
Sys.D Electron zero-
field coupling 2π 2
3D j S jz
2 − S jx2 + S jy
2( )⎡⎣
⎤⎦
j=1
NS
∑ NS
Sys.Q Nuclear
quadrupole
coupling 2π 2
3QM Imz
2 − Imx2 + Imy
2( )⎡⎣
⎤⎦
m=1
NI
∑ N I
Input for the d, j, dnn, and Jnn fields should have a length of 12 N N −1( ) , for the number
of electrons or nuclei ( N = NS , N I ). The order of coupling terms in each field then
corresponds to the (1,2), (1,3),…,(1,N), (2,3),…, (2,N),…(N-1,N) couplings. For
example, for 4 electrons, the couplings in the d field will be in the following order: (1,2),
(1,3), (1,4), (2,3), (2,4), (3,4).
If a high-spin electron or nucleus is specified, then it may be necessary to specify
either a zero-field or quadrupolar coupling, respectively. If the fields D or Q are used,
then they must have a length of NS or N I respectively, even if some of the electrons or
nuclei are not high spin. For S=1, and given value of D or Q will be disregarded.
One should also note that in direct input mode, none of the parameters in Table III
are required. However, if Sys.S or Sys.I are specified, and are non-empty, then Sys.v0S or
Sys.v0I must be specified with the appropriate size. All other parameters are optional, and
will be assumed to be zero if not specified.
212
6.3.1.2 Orientation-Dependent Input
In the orientation-dependent input mode, the various interaction parameters are
specified as either scalars or tensors, where the tensors may be rotated for a powder
average. Note that this mode of specifying Sys is designed to be compatible with the input
format of EasySpin, which is a popular software package for EPR simulations [26]. In
orientation-dependent mode, there are up to 14 parameters in the Sys structure that can be
specified. These correspond to the electron spin and specification of the nuclei in the spin
system (which do not an have orientation-dependent Zeeman interaction). Additionally,
there are 6 interactions that are orientation-dependent. Each of these are specified with a
field in the Sys variable, and an additional set of Euler angles which specify rotation of
the tensor from the principal axis frame of the tensor to the frame of the spin system, to
make 12 additional variables.
Table IV: Orientation Dependent Input Fields
Sys field Description Field size Sys.S Spin of all electrons NS
Sys.Nucs String specifying Nuclei (i.e.
‘1H,1H,13C’) Contains N I entries separated
by commas Sys.g Electron g-tensor
NS , NS × 3, 3NS × 3
Sys.gpa Euler angles for Sys.g NS × 3
Sys.A Electron-nuclear coupling tensor
N I × NS , N I × 2NS ,N I × 3NS , 3N I × 3NS
Sys.Apa Euler angles for Sys.A N I × 3NS
Sys.ee Electron-electron coupling
12 NS NS −1( ), 1
2 NS NS −1( )× 212 NS NS −1( )× 3,
3 12 NS NS −1( )× 3
Sys.eepa Euler angles for Sys.ee 12 NS NS −1( )× 3
213
Sys.nn Nuclear-nuclear coupling
12 N I N I −1( ), 1
2 N I N I −1( )× 212 N I N I −1( )× 3,
3 12 N I N I −1( )× 3
Sys.nnpa Euler angles for Sys.nn 12 N I (N I −1)× 3
Sys.D Electron zero-field coupling NS , NS × 2, NS × 3, 3NS × 3
Sys.Dpa Euler angles for Sys.D NS × 3
Sys.Q Nuclear quadrupole coupling N I , N I × 2, N I × 3, 3N I × 3
Sys.Qpa Euler angles for Sys.Q N I × 3
As can be seen in Table IV, the various parameters can be input in several
different formats. These formats are such that the tensor is either (1) specified with one
value (either a scalar or a dipolar coupling, depending on the term), (2) specified with
two values (e.g one value for scalar and one value for dipolar couplings) (3) specified
with 3 values for the tensor in its own principal axis frame, or (4) the full 3x3 tensor is
specified in the lab frame. Further details of the input formats are discussed subsequently.
In Table V, the tensor corresponding to a spin or spin-pair is given for the various input
formats. The j index corresponds to an electron, and the m index to a nucleus. For
electron-electron and nuclear-nuclear couplings, we use n for the nth coupling. This
corresponds to the same indexing system as described in Section 6.3.1.1, where the
ordering is (1,2), (1,3), etc. for the couplings. Also note that there is no entry for
Sys.Nucs. No anisotropy is taken into account for the nuclear Zeeman interaction, and so
no tensor is generated- only the isotropic value is calculated from the gyromagnetic ratio.
Table V: Generation of Principal Tensors from Input
Sys fields Field Size Principal Tensor
Sys.g NS
g0, j =
g(j) 0 0
0 g(j) 0
0 0 g(j)
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
214
NS × 3
g0, j =
g(j,1) 0 0
0 g(j,2) 0
0 0 g(j,3)
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
3NS × 3
g j =
g(3(j-1)+1,1) g(3(j-1)+1,2) g(3(j-1)+1,3)
g(3(j-1)+2,1) g(3(j-1)+2,2) g(3(j-1)+2,3)
g(3(j-1)+3,1) g(3(j-1)+3,2) g(3(j-1)+3,3)
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
Sys.A N I × NS
A0,( j ,m) =
A(m,j) 0 0
0 A(m,j) 0
0 0 A(m,j)
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
N I × 2NS
A0,( j ,m) =
A(m,2(j-1)+1) 0 0
0 A(m,2(j-1)+1) 0
0 0 A(m,2(j-1)+2)
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
N I × 3NS
A0,( j ,m) =
A(m,3(j-1)+1) 0 0
0 A(m,3(j-1)+2) 0
0 0 A(m,3(j-1)+3)
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
3N I × 3NS
A( j ,m) =
A(3(m-1)+1,3(j-1)+1) A(3(m-1)+1,3(j-1)+2) A(3(m-1)+1,3(j-1)+3)
A(3(m-1)+2,3(j-1)+1) A(3(m-1)+2,3(j-1)+2) A(3(m-1)+2,3(j-1)+3)
A(3(m-1)+3,3(j-1)+1) A(3(m-1)+3,3(j-1)+2) A(3(m-1)+3,3(j-1)+3)
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
Sys.d (Sys.dnn)
12 NS (NS −1)( 1
2 N I (N I −1))
d0,n =
−d(n) 0 0
0 −d(n) 0
0 0 2d(n)
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
12 NS (NS −1)× 2( 1
2 N I (N I −1)× 2)
d0,n =
−d(n,1) + d(n,2) 0 0
0 −d(n,1) + d(n,2) 0
0 0 2d(n,1) + d(n,2)
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
215
12 NS (NS −1)× 3( 1
2 N I (N I −1)× 3)
d0,n =
d(n,1) 0 0
0 d(n,2) 0
0 0 d(n,3)
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
3 12 NS (NS −1)× 3
(3 12 N I (N I −1)× 3)
dn =
d(3(n−1)+1,1) d(3(n−1)+1,2) d(3(n−1)+1,3)
d(3(n−1)+2,1) d(3(n−1)+2,2) d(3(n−1)+2,3)
d(3(n−1)+3,1) d(3(n−1)+3,2) d(3(n−1)+3,3)
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
Sys.D NS
D0, j =13
−D( j ) 0 0
0 −D( j ) 0
0 0 2D( j )
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
NS × 2
D0, j =
− 13 D( j ,1) + D( j ,2) 0 0
0 − 13 D( j ,1) − D( j ,2) 0
0 0 23 D( j ,1)
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
NS × 3
D0, j =
D( j ,1) 0 0
0 D( j ,2) 0
0 0 D( j ,3)
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
3NS × 3
Dj =
D(3( j−1)+1,1) D(3( j−1)+1,2) D(3( j−1)+1,3)
D(3( j−1)+2,1) D(3( j−1)+2,2) D(3( j−1)+2,3)
D(3( j−1)+3,1) D(3( j−1)+3,2) D(3( j−1)+3,3)
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
Sys.Q N I
Q0,m = 14Im(2Im −1)
−Q(m) 0 0
0 −Q(m) 0
0 0 2Q(m)
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
N I × 2
Q0,m =Q(m,1)
4Im(2Im −1)
−(1−Q(m,2) ) 0 0
0 −(1+Q(m,2) ) 0
0 0 2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
N I × 3
Q0,m =
Q(m,1) 0 0
0 Q(m,2) 0
0 0 Q(m,3)
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
216
3N I × 3
Qm =
Q(3(m−1)+1,1) Q(3(m−1)+1,2) Q(3(m−1)+1,3)
Q(3(m−1)+2,1) Q(3(m−1)+2,2) Q(3(m−1)+2,3)
Q(3(m−1)+3,1) Q(3(m−1)+3,2) Q(3(m−1)+3,3)
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
For all spin system interactions, one may specify three principal values to give the
value of the tensor in its principal axis frame directly. Additionally, it is possible to
specify the full tensor in the lab frame; however in this case, any Euler angle input for the
tensor will be ignored. In the case of the g-tensor and the electron-nuclear couplings, one
may specify only the scalar value of the interaction. In the case of the electron-electron
and nuclear-nuclear couplings, if only one value is specified, then this value will
correspond to a dipolar coupling (d), according to the point-dipole approximation. For
example, for an electron or nucleus, the Hamiltonian in the principal axis frame of the
dipole coupling is given by
HSS = d 3S jzSkz − S jSk( )HII = d 3I jmI jn − ImIn( ) , (77)
if only one value is specified. Note that this differs from EasySpin, where a single value
will be interpreted as a scalar coupling.
Additionally, the electron-nuclear couplings, the electron-electron couplings, the
nuclear-nuclear couplings, the electron zero-field couplings, and the nuclear quadrupole
couplings may be specified with two entries. In the case of electron-electron, and nuclear-
nuclear couplings, the first entry is assumed to be a dipole coupling (d) and the second
entry a scalar coupling (J), so that the coupling Hamiltonian in principal axis frame of the
tensor is given for an electron or nuclear pair by
HSS = d 3S jzSkz − S jSk( ) + JS jSk
HII = d 3Imz Inz − ImIn( ) + JImIn
. (78)
Again, this differs from EasySpin, where a two-entry format for electron-electron
couplings is not supported. For an electron-nuclear coupling, if only two values are
specified, then the coupling is assumed to be axial, and the input defines the
perpendicular ( A⊥ ) and parallel ( A ) components so that the Hamiltonian in the tensor’s
principal axis frame is
217
H IS = A⊥ S jx Imx + S jy Imy( ) + A S jz Imz . (79)
In the case of the electron zero-field coupling, the two terms are D and E such that the
zero-field Hamiltonian in its principal axis frame is
HD = D Sz
2 − S(S +1) / 3⎡⎣ ⎤⎦ + E Sx2 − Sy
2( ) . (80)
In the case of the nuclear quadrupole coupling, if two terms are specified for each
coupling, then the two terms are e2Qq / h and η , respectively such that the quadrupolar
Hamiltonian in its principal axis frame is
HQ = e2qQ / h
4I(2I −1)−(1−η)Ix
2 − (1+η)I y2 + 2Iz
2⎡⎣ ⎤⎦ . (81)
If the tensor is given by ΤP in its principal axis frame, it can be rotated into the
lab frame ( ΤL ), if its Euler angles angles are α , β , and γ , by the transformation
ΤL = Rz (γ )Ry (β )Rz (α )ΤP Rz (−α )Ry (−β )Rz (−γ ) . (82)
Note that the tensors are 3x3 matrices, and the rotation matrices, Rz (θ ) and Ry (θ ) , are
given by
Rz (θ ) =cosθ sinθ 0−sinθ cosθ 0
0 0 1
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
, Ry (θ ) =cosθ 0 −sinθ
0 1 0sinθ 0 cosθ
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
. (83)
As noted before, if the full 3x3 tensor is given, then any Euler angle input will be
ignored.
6.3.1.3 Relaxation Parameters
If the simulation being performed includes relaxation, then one must specify the
relaxation parameters. If one assumes that the fields acting on an individual spin is
uncorrelated from the fields acting on all other spins, then relaxation may be fully
specified by giving T1 and T2 for each spin. The inputs to specify T1 and T2 are given in
Table VI.
Table VI: Required Relaxation Parameters
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Sys Field States Field Size
Sys.T1e S jz 1, NS
Sys.T2e S j
+ , S j− 1, NS
Sys.T1n Imz 1, N I
Sys.T2n Im
+ , Im− 1, N I
It is possible to specify the same T1 and T2 for all electrons or all nuclei, by only
putting one value into the appropriate field. Otherwise, the length of the relaxation field
must be equal to the number of electron or nuclear spins. If we only specify the
parameters in Table VI, then the relaxation rates of states involving multiple spins will be
calculated by summing the relaxation rates of the single-spin states which are involved
(for example, S1+ I1
+ will relax at a rate of 1/ T2e,1 +1/ T2n,1 ). Additionally, if no additional
parameters are specified, then the relaxation rates for spin-1 will be calculated according
to Table I.
If additional relaxation parameters are specified, it is possible to change the
relationships between the various relaxation rates. Table VII gives additional fields that
can be used to specify relaxation rates of high spin states for spin-1.
Table VII: Spin-1 Relaxation Parameters
Sys Field States Size
Sys.QT1e S jz
2 − 23 1 1, NS
Sys.QT2e
S j+( )2
, S j−( )2
1, NS
Sys.QT1n Imz
2 − 23 1 1, N I
Sys.QT2n
Im+( )2
, Im−( )2
1, N I
Note that the additional spin-1 states, S jzS j
+ + S j+S jz and
S jzS j− + S j
−S jz will be relaxed with
the rate given in Table I.
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In addition to being able to specify the relaxation rates for states for spin-1, one
may also specify relaxation rates resulting from correlated fields being applied to pairs of
spins, as discussed in the Theory section.
Table VIII: Relaxation Rates from Correlated Fields
Sys Field States Size
Sys.TZQe S j
+Sk− , S j
−Sk+ 1, 1
2 NS (NS +1)
Sys.TDQe S j
+Sk+ , S j
−Sk− 1, 1
2 NS (NS +1)
Sys.TLLSe S jzSkz 1, 1
2 NS (NS +1)
Sys.Ttranse
4S jzSkz → S j+Sk
− + S j−Sk
+
S jzSk+ → S j
+Skz
S jzSk− → S j
−Skz
1, 12 NS (NS +1)
Sys.TZQn Im
+ In− , Im
− In+ 1, 1
2 N I (N I +1)
Sys.TDQn Im
+ In+ , Im
− In− 1, 1
2 N I (N I +1)
Sys.TLLSn Imz Inz 1, 1
2 N I (N I +1)
Sys.Ttransn
4Imz Inz → Im+ In
− + Im− In
+
Imz In+ → Im
+ Inz
Imz In− → Im
− Inz
1, 12 N I (N I +1)
The correlated relaxation rates are specified for spin pairs. Therefore, the size of the entry
must either be 12 N (N − I ) or must have size of 1. If the size is 1, then all correlated
relaxation rates for a spin type will be assumed to be the same. Otherwise, each pair is
specified using the same ordering as the electron-electron or nuclear-nuclear coupling.
Also note that the Sys.Transe and Sys.Transn fields specify the rate of relaxation of
S j
+Sk− + S j
−Sk+ → S jzSkz and Im
+ In− + Im
− In+ → Imz Inz ; the other transfer rates are scaled
accordingly, as given in Table II.
Note that it is possible to fully bypass the internal relaxation matrix setup used in
DNP_sim. In this case, one simply specifies the matrix Sys.Lrelax. This will override all
relaxation rates, and directly use Sys.Lrelax in the propagation of the spin system. Note
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that the system will still relax towards thermal equilibrium; the specification of the
thermal equilibrium position is discussed next.
The final set of entries into the Sys structure specifies the thermal equilibrium
position of the spin system. By default, the thermal equilibrium state of the spin system is
given by (37), and is normalized such that the thermal polarization of the first nucleus is
1, or if there are no nuclei, then it is normalized so that the polarization of the first
electron is one. It is recommended that this default mode is used, or that a temperature is
specified in the Par structure in order to obtain the absolute polarization
(Par.Temperature). However, it is possible to specify the equilibrium position directly,
by using Sys.sigmaeq. Note that Sys.sigmaeq should be a string, and will be evaluated
using the internal variables of DNPsim (see Implementation section for further
explanation of the internal variables). Finally, it is possible to specify Sys.PSeq and
Sys.PIeq, which will assign equilibrium polarization to the Zeeman states of the electrons
and nuclei. Note that recovery of non-polarization states to thermal equilibrium can play
an important role in the evolution of the spin-system, and so it is recommended that the
default equilibrium state or temperature specification be used in most cases.
6.3.2 Experimental Parameters
The Par structure specifies the experimental conditions for a simulation. The fields
in Par can be categorized as follows: sweep parameters, propagation parameters,
initiation and detection parameters, and general parameters. We begin by discussing the
sweep parameters.
6.3.2.1 Sweep Parameters
In DNPsim, it is possible to sweep the external magnetic field, the microwave
frequency, the RF frequency, the experimental time, and to perform a powder average.
Each of the parameters may also be fixed to one value, depending on the input. We begin
discussing the sweep of the external magnetic field.
Three parameters can be used in the Par structure to define the external magnetic
field. The first method of input uses just one of these parameters. Par.B0=5000;
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or Par.B0=linspace(4975,5025,51);
In the first case, a single value is used. Additionally, it is possible to list any set of values
to be swept over. These values may have any spacing.
The second entry may be replicated without use of the linspace command by specifying Par.B0Range=[4975 5005];
Par.B0Points=51;
Note that the external magnetic field must be input in millitesla (mT). Also, note that
usage of Par.B0, Par.B0Range, or Par.B0Points implies that Sys will be entered with
orientation dependent inputs.
As with the external magnetic field, the microwave frequency, the RF frequency,
and the experimental time may be swept. Microwave parameters are entered as Par.mwFreq=140e3;
or Par.mwRange=[139.5e3,140.5e3];
Par.mwPoints=101;
Similarly, the RF frequency can be entered as Par.rfFreq=[210 211 212 213 214];
or Par.rfRange=[210 214];
Par.rfPoints=51;
For both the microwave and RF frequency, the frequency is input in megahertz (MHz).
Inclusion of a microwave or RF frequency actually switches the simulation into the
rotating frame of the electrons or nuclei at the specified frequency. To actually apply the
oscillating field, one must also specify Par.mwStrength=3;
Par.rfStrength=.1;
which again are in MHz. Otherwise, the rotating frame transformation is taken, but there
is no field applied.
The sweep of experimental time is done similarly. One may specify
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Par.t=logspace(1,100,50);
where any arbitrary vector for Par.t may be used. One may also specify Par.maxt=1e3;
Par.tPoints=1000;
where the starting time is always 0, and the time vector is uniformly spaced.
Finally, one may “sweep” the orientations with a powder average. To specify a
powder average, one specifies Par.PowderPoints=40;
Par.Symmetry=’D2h’;
This performs a powder average using 40 uniformly distributed angles for θ, the angle
between the principal axis of the spin system and the magnetic field. Angles for φ are
determined so that the powder average is uniformly distributed over all possible
orientations. Par.Symmetry=’D2h’ causes the program to assume D2h symmetry for the
spin system. Possible entries for Par.Symmetry are ‘Ci’, ‘C2h’, and ‘D2h’. Additionally,
one may specify Par.Orientations=[0 0;pi/4 pi/4;pi/2 pi/2];
where each row specifies an orientation of the spin system.
Finally, note that for each of the sweep parameters, it is possible to over-specify
the parameters, for example by specifying Par.B0 and also specifying Par.B0Range and
Par.B0Points. In this case, Par.B0, Par.mwFreq, Par.rfFreq, Par.t, and Par.Orientations
will override other entries.
6.3.2.2 Propagation Parameters
DNPsim has a variety of options for preparing and propagating the spin system.
The first option determines if relaxation is considered. Par.Relaxation=’n’;
By default, relaxation is included. The above entry removes relaxation. Once it has been
determined whether to use relaxation or not, the method of propagation must be
determined. For simulations with and without relaxation, the options ‘Full’, ‘NoProp’,
and ‘NoDiag’ may be used. ‘Full’ diagonalizes and propagates the full Hamiltonian or
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Liouvillian, and is the default propagation method. ‘NoProp’ does not perform any
propagation, and ‘NoDiag’ performs the propagation without diagonalizing the
Hamiltonian or Liouvillian, which can be faster for only a few time points. These are
entered into the Par.PropMethod field as Par.PropMethod=’Full’;
Additional options are available if relaxation is used. The first option is used by inputting Par.PropMethod=’BlkDiag’;
This method is described in the Theory: Propagation section. DNPsim determines if the
matrix is block-diagonal, and if it is, propagates the block-diagonal parts separately.
Alternatively, it is possible to enforce a block-diagonal structure on the Liouvillian, even
if it is not actually block-diagonal. This is done by specifying Par.BlkDiag=BD;
Where BD is a numbered structure. Each element of the structure contains a list of the
rows and columns of a block-diagonal part of a matrix. For example, BD could have the
following form:
BD {1}=[1 2];
BD {2}=[3 4 5 6];
BD {3}=[7 8];
Thus, the Liouvillian is 8×8, and can be broken into two 2×2 matrices and one 4×4
matrix.
One may also use the quasi-equilibrium assumption to propagate the Liouvillian.
In this case, one specifies Par.PropMethod=’QuasiEq’;
By default, this propagation method will assume that all states with lifetimes, T, that are
shorter than 1/10 of the smallest time step ( T <1/ 10min(Δt) ), are in quasi-equilibrium.
This corresponds to the σσ Sr (t) in (68). It is also possible to specify which states are in
quasi-equilibrium. Par.LLstates=[1 2 5];
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This will set the states 1, 2, and 5 to be long-lived states, and all other states to be states
in quasi-equilibrium. The states corresponding to 1, 2, and 5 can be determined using the
State_List_Gen(Sys) function. Similarly, one may input Par.QEstates=[3 4 6 7 8 9 10 11 12 13 14 15 16];
which will set states 3, 4, and 6-16 to quasi-equilibrium, and the remaining states will be
long-lived states. Note that we have used linear indexing (a list of vector coordinates) in
our example, but it is also possible to use logical indexing (a list of 0s and 1s
corresponding to each vector element) for Par.LLstates and Par.QEstates.
Finally, if only interested in the equilibrium position of the system, then one may
input Par.PropMethod=’Eq’;
Usually, this input is not necessary because if there is no time vector specified, DNPsim
will use this option automatically. However, if the time vector is specified, this input will
prevent calculation of the propagation in time.
Truncation of the basis set may help accelerate calculations. Additionally, one
may truncate the basis set in order to investigate the role of particular states. Basis set
truncation can be used three ways. Par.TruncR=[2 4 10];
This method removes state 2, 4, and 10 from the basis set. Par.TruncA=[3 5:9 11:16];
This method removes all states except 1,3, 5-9, and 11-16. Note that state 1 will always
be included, since this state is used for relaxation towards equilibrium. As with the quasi-
equilibrium specification, one may obtain the numbering of the states using the
State_list_gen(Sys) function. Logical indexing may also be used for basis set truncation.
Finally, one may use the following input. Par.TruncO=3;
This method removes all states that involve more than 3 spins. For example, S1z I1+ I2
− I3z
would be removed in this case.
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6.3.2.3 Initiation and Detection Parameters
DNPsim allows the user to initiate the spin system in any arbitrary spin state, and
additionally one may detect any spin state. By default, the spin system will initiate at
thermal equilibrium. To start in a different state, one must specify Par.sigma0 or Par.v0. Par.sigma0=’Sx’;
This initiates the system with all electron spins oriented along the x-axis. One may use
any of the strings in Table IX.
Table IX: Initial State Strings
String Description Par.sigma0=’Sz’ Electron polarization on all electrons
Par.sigma0=’Iz’ Nuclear polarization on all nuclei
Par.sigma0=’S1z’ Electron polarization on 1st electron
Par.sigma0=’I1z’ Nuclear polarization on 1st nuclei Par.sigma0=’Sx’/’Sy’ Electron coherence on all electrons in x- or y-direction
Par.sigma0=’S1x’/’S1y’ Electron coherence on 1st electron in x- or y-direction
Par.sigma0=’Ix’/’Iy’ Nuclear coherence on all electrons in x- or y-direction
Par.sigma0=’I1x’/’I1y’ Nuclear coherence on 1st electron in x- or y-direction
Par.sigma0=’0’ All states unpopulated
In addition to using any of the states specified in Table IX, any other arbitrary state may
be used to initialize the simulation. In order to initialize in any arbitrary state, one creates
a string that calls the internal variable S. S is a numbered structure variable, which
contains Pauli matrices for every spin. The spins are numbered such that spins 1:NS are
the electrons, and NS+1:NS+NI are the nuclei. For each spin, fields exist for the Jx , J y ,
Jz , J + , and J − states, and for spin-1/2, fields also exist for the Jα and J β states. For
example, these fields are called for the first electron as given in Table X.
Table X: Spin states of first electron (S1)
State MATLAB call
S1x S{1}.x
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S1y S{1}.y
S1z S{1}.z
S1+ S{1}.p
S1− S{1}.m
S1α S{1}.alpha
S1β S{1}.beta
Using this formalism, one may initiate the spin system in any arbitrary spin state.
For example, suppose one wants to initiate a two-electron, three-nucleus system in the
zero-quantum substate of the first two nuclei. This substate is given by I1+ I2
− + I1− I2
+ .
Because there are two electrons, the first nucleus is spin #3, and the second nucleus is
spin #4. One may then input Par.sigma0=’S{3}.p*S{4}.m+S{3}.m*S{4}.p’;
Similarly, if one wanted to initiate in the double-quantum substate of the first electron
and last nucleus, one would need to use the substate S1+ I3
+ + S1− I3
− . The electron is spin #1,
and the nucleus is spin #5. Therefore, this can be input using Par.sigma0=’S{1}.p*S{5}.p+S{1}.m*S{5}.m’;
Therefore, using the states listed in Table IX, and the formalism shown above, it
is possible to access any initial spin state. Note that it is important to initialize the spin
system in a physically realistic state to obtain reliable results. For example, one should
not initialize the spin system in the S1+ state since this does not correspond to a real state,
although S1+ + S1
− would be acceptable, since this is twice the Sx state. All physically
realistic states can be written as linear combinations of products of the S jx ,
S jy , S jz and
Imx , Imy , Imz , using only real coefficients (
S1
+ = S1x + iS1y - the second coefficient is
imaginary).
Specification of the initial state may also be performed using Par.v0. When using
Par.sigma0, one specifies a string corresponding to the density matrix in Hilbert space
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describing the initial state. Par.v0 specifies the vector corresponding to the density
matrix. In this case, Par.v0 is a numerical column vector in Liouville space, using the
basis set given by products of the electron and nuclear states (i.e. 1 , S jz ,
S j+ ,
S j− and 1 ,
Imz , Im+ , Im
− ). On occasion, it may be useful to propagate a spin system under one set of
conditions, then change the conditions, but continue propagating. This may be done by
detecting the full state of the spin system, as discussed next. This is followed by plugging
the full state of the spin system into Par.v0.
One may also detect any arbitrary spin state. In this case, one also specifies a
string for detection. Any of the strings in Table X may be used for detection
Table XI: Detection Strings
String Detects Par.Detection=’Sz’ Polarization of each individual electron
Par.Detection=’Iz’ Polarization of each individual nucleus
Par.Detection=’Sp’; Sum of electron coherences, including real (Sx) and
imaginary (Sy) parts (transient EPR signal)
Par.Detection=’Ip’ Sum of nuclear coherences, including real (Ix) and
imaginary (Iy) parts (transient NMR signal)
Par.Detection=’Full’ Detection of the full state of the spin system, returned
in a column vector.
Additionally, any arbitrary spin state may be detected. These are specified the same way
that one specifies an arbitrary initial spin state. However, note that it is not necessary for
the detected state to be physically realistic; for example, S1+ is an acceptable state for
detection, and its real part will correspond to ⟨S1x ⟩ and its imaginary part will correspond
to ⟨S1y ⟩ .
It is possible to detect multiple states simultaneously. In order to detect multiple
states, one string is specified, where a comma separates each state to be detected. For
example, if one has a spin system with only two nuclei, and is interested in nuclear
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polarization, and nuclear zero-quantum coherence, then one would want to detect I1z , I2z
, and I1+ I2
− + I1− I2
+ . This would be specified using
Par.Detection=’Iz,S{1}.p*S{2}.m+S{1}.m*S{2}.p’;
Here, we use the shortcut string, ‘Iz’, which is given in Table XI to detect both I1z and
I2z , and use the full specification of the zero-quantum coherence. Note that if one is
unsure of the detected states of the output, this can be obtained using out.detected. out.detected
‘I1z,I2z,S{1}.p*S{2}.m+S{1}.m*S{2}.p’
This tells the user that the first, second, and third output are the polarization of the first
nucleus, the polarization of the second nucleus, and the zero-quantum coherence between
these two nuclei.
6.3.2.4 General Parameters
Several additional parameters are used to determine the behavior of the DNP
simulation. One may specify the temperature in order to obtain the absolute polarization
of the spin system. This will change the equilibrium position of the spin system,
according to (35). The temperature is specified as follows: Par.Temperature=80;
This sets the temperature to 80 K. Without a temperature entry, the simulation uses the
default equilibrium settings as described in Section 6.3.1.3, where the equilibrium is
normalized to the polarization of either the first nucleus or first electron, using the high
temperature approximation in (37).
It can be advantageous to partially truncate the Hamiltonian, by omitting
couplings that lead to double quantum nuclear-nuclear or electron-electron coherences-
i.e. one omits S j
+Sk+ and
S j−Sk
− for the electron and Im+ In
+ and Im− In
− . This make various
propagation operations faster, and the truncation can be done by specifying Par.HamTrunc=’y’;
Additionally, DNPsim provides warnings during usage if unexpected behavior
may occur. Also, a progress bar is provided for field, microwave frequency, and RF
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frequency sweeps, in addition to powder averages. Both of these can be suppressed using
the following entries. Par.SuppressWarnings=’y’;
Par.waitbar=’n’;
Finally, one may control the output variables of the DNPsim. The output is
specified in Par.output as a string, which lists all output variables, separated by commas.
Some possible output variables are listed in Table XII.
Table XII: Possible Output Variables
Par.output Description
=’M’
n dimensional matrix that gives the expectation values of the
detected states. The n dimensions are for (1) all detected states,
(2) time, (3) magnetic field, (4) RF frequency, and (5) microwave
frequency.
=’Meq’
n-1 dimensional matrix that gives the equilibrium expectation
values of the detected states. The n-1 dimensions are for (1) all
detected states, (2) field, (3) RF frequency, and (4) microwave
frequency =’B0’ Vector containing the magnetic field(s) used.
=’mwFreq’ Vector contain the microwave frequency(ies) used.
=’rfFreq’ Vector containing the RF frequency(ies) used.
=’t’ Vector contain the propagation time(s) used.
=’detected’ String listing the states that correspond to the first dimension of
the out.M and out.Meq fields
=’H0’ Static Hamiltonian matrix, without microwave or RF fields, in the
lab frame.
=’H’ Full Hamiltonian matrix, including microwave and RF fields, in
the rotating frame of both microwave and RF fields.
=’Lcoh’ Liouville matrix that only contains terms resulting from the
Hamiltonian.
230
=’Lrelax’ Liouville matrix that only contains terms resulting from
relaxation processes
=’L’ Full Liouville matrix including terms from Hamiltonian and
relaxation processes
=’Lreduc’ Liouville matrix calculated after using propagation with quasi-
equilibrium
Note that any internal variable may be retrieved from DNPsim, although only its
final value will be given. Also, multiple variables may be retrieved. For example, the
default setting for DNPsim is to return the magnetic field, the microwave and RF
frequencies, the times of propagation, the detected states, the expectation values of the
various detected states including time dependence, and the expectation value of the
detected states at equilibrium. The default output would be set as follows. Par.output=’B0,mwFreq,rfFreq,t,detected,M,Meq’;
Then, the output variable, out, will contain have the following form out=
B0: 4995
mwFreq: 140000
rfFreq: [1x101 double]
t: 10
detected: ‘S1z,I1z’
M: [2x101 double]
Meq: [2x101 double]
Note that if a variable is called that does not exist internally, it will be ignored.
6.3.3 Liouville Space Basis Set
DNPsim uses a basis set in Liouville space defined by direct products of 1 , S jz ,
S j
+ , S j
− , etc. for electrons and 1 , Imz , Im+ , Im
− , etc. for nuclei. This particular basis set
was chosen because it lends itself to basis set truncation, as demonstrated by Kuprov et
al. [27]. Some functions of DNPsim require specification of states in the basis set. For
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example, in order to truncate the Liouville space basis set, one has to specify which states
to remove. Therefore, we number the states in the basis set; we refer to that state number
here as sn. We describe how to determine the value of sn for a particular state here.
In our program, each of the 1 , S jz ,
S j+ ,
S j− , etc. corresponds to an integer, given
in Table XIII.
Table XIII: Spin-State Numbering
State Number
0
1
2
3
4
5
6
7
8
Internally, DNPsim generates a matrix called states (which is available for output as
specified in Section 6.3.2.4). states is a (NS + N I )× nstates matrix where NS and N I are
the number of electrons and nuclei respectively, and nstates is the total number of states in
Liouville space for the spin system ( 4NS+NI for all spin-1/2). Each column corresponds to
a state, and each row corresponds to a spin. Then the column for each state is a list of
integers: 0-3 for spin-1/2 and 0-8 for spin-1. The first NS rows correspond to the
electrons, and the next N I rows correspond to the nuclei. The product of the spin states
of the individual spins gives the total spin state of a particular column. For example, with
a one electron, one nucleus system, where both are spin-1/2, states is given by the
following matrix, where the corresponding states are listed below the columns, and the
state number is listed below the state:
1
Jz
J +
J −
Jz2 − 2
3 1
(J + )2
(J − )2
Jz J+ + J +Jz
Jz J− + J −Jz
232
states
0 0 0 0 1 1 1 1
0 1 2 3 0 1 2 3
1 Iz I+ I– Sz SzIz SzI+ SzI–
sn= 1 2 3 4 5 6 7 8
2 2 2 2 3 3 3 3
0 1 2 3 0 1 2 3
S+ S+Iz S+I+ S+I– S–
S–
Iz
S–
I+
S–
I–
sn= 9 10 11 12 13 14 15 16
It is then possible to specify particular states for truncation, quasi-equilibrium,
and block diagonalization by finding the states of interest in the state list, states. Note that
the function State_list_gen will also return states without running the full DNPsim
program. One may then uses states to help truncate the basis set. For example, consider a
system under solid effect DNP conditions with one electron and several spin-1/2 nuclei.
We expect that of the nuclear states, only zero-quantum states, and also single-quantum
states will be important to the process. Therefore, states such as Im+ In
+ , or Im+ In
− I p− Iq
−
should be truncated, since these are double-quantum states, whereas states such as Im+ In
−
or Im+ Inz should be kept since these are zero- and single-quantum states, respectively.
This could be done as follows. SL=State_list_gen(Sys);
Ip=sum(SL.states(2:end,:)==2,1);
Im=sum(SL.states(2:end,:)==3,1);
Truncate=abs(Ip-Im)>=2;
Par.TruncR=Truncate;
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The first line obtains the structure, SL, which contains the list of states, in
SL.states. The second line determines how many nuclear spins are in the J + state, and
the third line determines how many nuclear spins are in the J − state. Note that we index
the rows from 2:end to ignore the state of the electron, which is given in the first row. In
the fourth line, we determine where the absolute difference between total spins in J + and
J − states is at least 2. These states correspond to double-quantum or higher spin states on
the nuclei. In the last line, Par.TruncR is set to reject all states which are double-quantum
or higher. Note that in this case, we have used logical indexing to specify the states to be
truncated.
6.3.4 DNPsim Programs Several programs are included in the DNPsim package. Aside from DNP_sim, we
include GyroRatio, PowderOrientations, n_spin_system, Verify, and State_list_gen. Of
these, GyroRatio, PowderOrientations, n_spin_system, and Verify are called internally
from DNP_sim, and so must be on the MATLAB path when using DNP_sim. We provide
a brief description of each program.
6.3.4.1 GyroRatio
Gyroratio provides the spin and gyromagnetic ratio in MHz/T for a variety of
nuclear spins. It will also provide the gyromagnetic ratio for a free electron, but not the
spin. Gyroratio is called as follows: [gr I]=GyroRatio(‘1H’)
gr=
42.5760
I=
0.5000
The list of nuclei in GyroRatio is not comprehensive, although it is straightforward to add
additional nuclei inside the code of GyroRatio.
234
6.3.4.2 PowderOrientations
PowderOrientations generates a powder average that is uniformly distributed
across all possible orientations. One calls PowderOrientations with two arguments: the
first argument is the number of angles, θ , which will be returned in the powder average,
where θ is the angle between the z-axis of the lab frame and the z-axis of the principal
axes of the spin system. The second argument is a string specifying the symmetry of the
spin system. Possible inputs are ‘Ci’, ‘C2h’, and ‘D2h’. A powder average using ‘Ci’ will
have twice as many angles as ‘C2h’, and four times as many angles as ‘D2h’. This is
because higher symmetry makes some of the angles redundant. PowderOrientations is
called as follows. [theta phi]=PowderOrientations(20,’C2h’);
Note that because the powder average is uniformly distributed, it is not necessary to
include a weighting for the angles. PowderOrientations can easily be replaced with
another powder averaging program, although if a uniform distribution is not used some
internal changes to DNP_sim will be required.
6.3.4.3 n_spin_system
n_spin_system generates Pauli matrices for arbitrary spin systems. n_spin_system
supports multiple input and output formats, as well as generation of sparse matrices. The
simplest method of calling n_spin_system is done as follows, where we obtain Pauli
matrices for a one-spin system, with spin-1/2. S=n_spin_system;
The output, S, contains seven fields, corresponding to Sx , Sy , Sz , S + , S − , Sα , and S β .
S=
x: [2x2 double]
y: [2x2 double]
z: [2x2 double]
p: [2x2 double]
m: [2x2 double]
alpha: [2x2 double]
235
beta: [2x2 double]
It is also possible to call n_spin_system with a single argument, for example S=n_spin_system(1);
This will produce Pauli matrices for spin-1. If the spin is not 1/2, then the alpha and beta
fields will be omitted.
For more than one spin there are two output modes. We first discuss the multiple
output mode, where n_spin_system is called with at least two outputs. For example: [S1 S2 I1 I2 I3]=n_spin_system(1/2,1/2,1/2,1,1);
The number of outputs in the call for n_spin_system determines the number of spins. In
this case, there are five outputs and therefore five spins. Note that the number of inputs
matches the number of spins. This is not necessary, in general. If the number of inputs
exceeds the number of outputs, then the additional inputs are ignored. If the number of
outputs exceeds the number of inputs, then all additional spins are assumed to be spin-
1/2. [S1 S2 I1]=n_spin_system(1);
This call to n_spin_system will generate a three-spin system, where S1 is spin-1, but the
S2 and I2 are spin-1/2. Also, note that one may input a vector containing the spins of the
system, rather than listing them in the input. For example, one may input spin=[3/2 1 1/2];
[S1 S2 S3]=n_spin_system(spin);
This generates a three-spin system, with spin 3/2, 1, and 1/2.
For the second output mode, one only specifies one output, but multiple inputs. In
this case, n_spin_system generates a single structure that contains the Pauli matrices for
all specified spins. For example, S=n_spin_system(1/2,1/2,1/2);
generates a three-spin system, where all spins are spin-1/2. In this case, S has the
following form S=
[1x1 struct] [1x1 struct] [1x1 struct]
Then, the Pauli matrices for each individual spin are in the 3 structures. For example
236
S{1}.x
will call the Pauli matrix corresponding to the state S1x .
Finally, we note that n_spin_system can generate sparse matrices for the Pauli
matrices. Any input can be followed with the string ‘sparse’ to force n_spin_system to
generate sparse matrices. S=n_spin_system(1/2,1/2,1/2,’sparse’);
This will generate sparse Pauli matrices. If one creates a sufficiently large spin system,
n_spin_system will create sparse matrices automatically. If the total number of elements
in the output exceeds 1.5×106, then outputs will be sparse.
DNP_sim uses n_spin_system internally. In this case, it always uses the single
output mode, and always generates sparse matrices. The additional options are for ease of
creation of matrices for generating Hamiltonians and density matrices in MATLAB,
without using DNP_sim.
6.3.4.4 Verify
Verify takes two arguments, either the Sys or Par structure, and a string specifying
which structure has been input. Verify then returns a structure that contains logicals
corresponding to all possible fields of the Sys or Par structures. If the field is in use, the
logical is true, otherwise it is false. The functioning of DNP_sim requires periodic
knowledge of which inputs have been used, and Verify is used at the beginning of
DNP_sim to determine this. By storing the results in logical variables, rather than re-
testing at every occurrence saves significant computation time in certain situations.
6.3.4.5 State_list_gen
As discussed in Section 6.3.3, it is often necessary to specify particular states of
the spin system. These are specified by number, which can only be determined if there is
a list of the states. This list of states is generated internally in DNP_sim, and can be
returned in the output, if Par.output contains the string ‘states’. However, it is sometimes
convenient to have the list of states before running DNP_sim. This can be done with
State_list_gen. One may call State_list_gen as follows: SL=State_list_gen(Sys);
237
Sys is the same structure that one uses in DNP_sim. Actually, it is only necessary to
specify Sys.S and Sys.I or Sys.Nucs to use State_list_gen. The output, SL, is a structure
with three fields. SL.states is an (NS + N I )× nstates matrix, for which each column
corresponds to a state of the spin system, and each row corresponds to a spin, as
described in Section 6.3.3. SL.key is a list of the spins, where the order of the spins
corresponds to the rows of SL.states. For example, SL.key=
S1
I1
I2
for a system with one electron, and two nuclei. Therefore, the first row corresponds to the
first electron, the second row to the first nucleus, and the third row to the second nucleus.
Finally, SL.names contains fields for each spin, and gives the states corresponding to the
label numbers. For example, one may call SL.names.S1
0 = E
1 = S1z
2 = S1p
3 = S1m.
This tells us that state numbers 0,1,2, and 3 correspond to 1 , S1z , S1+ , and S1
− ,
respectively.
6.4 Examples Now that we have a basic understanding of the theory behind our methods of
simulation, and a working knowledge of the usage of the DNPsim, we show a variety of
examples of the DNPsim. For each example, we start with the MATLAB code to set up
the example, plot the result, and briefly discuss the example.
238
6.4.1 Solid Effect Field Profile
Sys.S=1/2; Par.B0Range=[4980 5007];
Sys.Nucs=’1H’; Par.B0Points=100;
Sys.g=[2.0027 2.0031 2.0034]; Par.PowderPoints=20;
Sys.A=[-1 -1 2]; Par.Symmetry=’D2h’;
Sys.T1e=1.4e3; Par.mwFreq=140e3;
Sys.T2e=.890; Par.mwStrength=3;
Sys.T1n=1e6;
Sys.T2n=1e3;
out=DNP_sim(Sys,Par);
plot(out.B0,out.Meq(2,:))
Figure 1: Solid Effect Field Profile.
A field profile of the solid effect is simulated, using one electron and one proton. The
microwave frequency is fixed at 140 GHz, and the magnetic field is swept.
239
We have simulated a solid effect field profile, using orientation-dependent input,
and a g-tensor. Note that we have not specified a time vector. Because we are only
interested in the field dependence of the enhancement, it is only necessary to calculate the
equilibrium position. Also, we used an electron-nuclear coupling that has the same
principal axis as the electron g-tensor. This gives the system D2h symmetry, giving ~4x
faster powder averaging than Ci symmetry.
6.4.2 Nuclear Orientation via Electron Spin-Locking Sys.S=1/2; Par.mwRange=[139.9e3 140.1e3];
Sys.I=1/2; Par.mwPoints=401;
Sys.v0S=140e3; Par.sigma0=’Sx’;
Sys.v0I=212; Par.mwFreq=140e3;
Sys.A=0.1 Par.mwStrength=212;
Sys.B=0.1 Par.maxt=10;
Sys.T1e=Inf; Par.tPoints=200;
Sys.T2e=40;
Sys.T1n=1e6;
Sys.T2n=1e3;
out=DNP_sim(Sys,Par)
MI=max(out.M(2,:,:));
plot(out.mwFreq,MI)
240
Nuclear spin-Orientation Via Electron spin-Locking (NOVEL) is a method of
electron-nuclear polarization transfer for which electron polarization is spin-locked, using
a microwave field strength close to that of the nuclear Larmor frequency [14]. In order to
simulate this effect, we initiate the system in the Sx state. Also note, that because this is a
spin-locked experiment, Sys.T2e is set to be significantly longer than if there were not a
spin-lock. A field is applied at several field strengths, and the microwave frequency is
swept. Because some oscillation of polarization between the electron and nucleus occurs,
we sweep the polarization time from 0-10 μs, and plot the maximum polarization
observed on the nucleus in Figure 2.
Figure 2: NOVEL with Various Powers.
NOVEL is performed with one electron, and one nucleus with a Larmor frequency of 212
MHz. The field strength of the electron is varied from 200 MHz up to 212 MHz, and the
frequency dependence of the matching condition is examined. When the field strength
matches the nuclear Larmor frequency, then the matching condition is on-resonant with
the electron. Otherwise, polarization transfer occurs when the microwave frequency is
off-resonant.
241
6.4.3 Cross Effect Sys.S=[1/2 1/2];
Sys.Nucs='1H,14N,14N';
Sys.g=[2.0097 2.0063 2.0022;2.0097 2.0063 2.0022];
Sys.A=[-.5 -.5 1 0 0 0;34.7640 18.4200 98.7008 0 0 0;
0 0 0 34.7640 18.4200 98.7008];
Sys.gpa=[0 0 0;90 90 90]*pi/180;
Sys.Apa=[0 0 0 0 0 0;0 0 0 0 0 0;0 0 0 90 90 90]*pi/180;
Sys.ee=31.4449;
Sys.eepa=[0 90 0]*pi/180;
Sys.Q=[0 3 3];
Sys.Qpa=[0 0 0;0 0 0;90 90 90]*180/pi;
Sys.T1e=.5e3;
Sys.T2e=.5;
Sys.T1n=[1e6 1e4 1e4];
Sys.T2n=1e3;
242
Par.mwRange=[139.3 140.7]*1e3;
Par.mwPoints=50;
Par.mwStrength=1;
Par.PowderPoints=30;
Par.Symmetry='D2h';
Par.B0=4985.6;
SL=State_list_gen(Sys);
Par.TruncR=(SL.states(4,:)==2|SL.states(4,:)==3|...
SL.states(5,:)==2|SL.states(5,:)==3)|...
(SL.states(1,:)==2&SL.states(2,:)==2)|...
(SL.states(1,:)==3&SL.states(2,:)==3);
out=DNP_sim_parallel(Sys,Par);
plot(out.mwFreq,out.Meq(3,:))
Figure 3: Cross Effect Field Profile.
Cross effect field profile for a TEMPO based biradical, using two electrons, two 14N
nuclei, and one 1H nucleus. Enhancements are on the 1H nucleus.
243
We generate a cross effect field profile under static DNP conditions. We use a
TEMPO biradical, with a 31.5 MHz electron-electron coupling. Note that we have given
the molecule D2h symmetry to speed up the powder average. Also, we have used basis set
truncation, and removed terms including electron-electron double quantum states, and
also coherences involving the nitrogens. Note that the field profile does not show a strong
effect from the 14N hyperfine coupling to the electrons. This is consistent with our
observations in Chapter 2, Figure 9, suggesting that the strong effect from the 14N
hyperfine coupling only occurs in MAS DNP experiments [28-30].
6.4.4 Dressed-State Solid Effect Sys.S=1/2; Par.mwFreq=140e3;
Sys.I=1/2; Par.mwStrength=3;
Sys.v0S=140e3; Par.sigma0=’Sx’;
Sys.v0I=212; Par.Detection=’Sp,Iz’
Sys.A=1; Par.rfRange=[207 217]
Sys.T1e=Inf; Par.rfPoints=200;
Sys.T2e=20; Par.t=linspace(0,200,101);
Sys.T1n=Inf;
Sys.T2n=1e2;
out=DNP_sim(Sys,Par)
subplot(2,1,1),mesh(out.rfFreq,out.t,2*real(squeeze(out.M(1,:,:))))
subplot(2,1,2),mesh(out.rfFreq,out.t,2*real(squeeze(out.M(2,:,:))))
244
We show a simulation of the dressed-state solid effect (DSSE), for which the
microwave field strength is matched to the offset of an RF field applied to the nucleus
[13]. Since we must spin-lock the electron, we initiate the simulation in the Sx state. We
Figure 4: Dressed-State Solid Effect.
Dressed-state solid effect simulation, showing the time evolution of the (A) spin-locked
electron coherence, and (B) the nuclear polarization for various RF offsets.
245
are interested in the evolution of the electron coherence, and the nuclear polarization and
so we detect S + and Iz . One sees in Figure 4A, the decay of the electron coherence, and
also the loss of coherence when the DSSE matching condition is satisfied. In Figure 4B,
the gain of polarization on the nucleus is observed.
6.4.5 Nuclear Rotating Frame DNP Sys.S=1/2; Par.mwFreq=140e3;
Sys.Nucs=’1H’; Par.mwStrength=.5;
Sys.g=[2.0027 2.0031 2.0034] Par.B0Range=[4990.6 4996.6];
Sys.A=[-1 -1 2]*.1 Par.B0Points=101;
Sys.T1e=1.4e3 Par.rfFreq=212.608;
Sys.T2e=.8; Par.rfStrength=.1;
Sys.T1n=1e7; Par.PowderPoints=20;
Sys.T1n=1e4; Par.Symmetry='D2h';
Par.sigma0='0';
Par.Detection='Ip';
Par.maxt=3*Sys.T2n;
Par.tPoints=201;
out=DNP_sim(Sys,Par)
subplot(2,1,1),plot(out.B0,out.Meq)
subplot(2,1,2),mesh(out.B0,out.t,real(squeeze(out.M(1,:,:))))
246
Nuclear rotating frame DNP transfers electron polarization to nuclear coherence
that is spin-locked with an RF field [15]. The matching condition is satisfied when the
field strength of the applied RF field matches the offset of the microwave field. Because
the EPR linewidth is considerably larger than the RF field, this means that transfers in
opposite directions are always competing, and greatly reducing the actual enhancement.
Figure 5: Nuclear Rotating Frame DNP
Nuclear rotating frame DNP simulation. (A) shows the equilibrium polarization, which is
found along the 1H spin locking field. In (B), we show the field dependent buildup of
polarization in the direction of the 1H spin lock.
247
Figure 5A shows the field profile for NRF-DNP, with the y-axis giving the absolute
enhancement, which is very small. However, the buildup of polarization, which can be
seen in Figure 5B for various fields, is much faster than for a typical NMR or DNP
experiment- requiring only milliseconds versus seconds for polarization buildup to occur.
In this simulation, we were interested in the buildup of nuclear coherence, so use the
option Par.Detection=’Ip’ in order to observe the nuclear coherence.
6.5 Investigating DNP via Simulation: The Solid Effect In the previous examples, we demonstrated simulations of basic continuous wave
and pulsed-DNP mechanisms. However, more in-depth studies are possible via
simulation. We examine the solid effect for a system of 6 nuclei and 1 electron, and
consider the role that high-order coherence is playing in this particular system. This
system models several protons on a glycerol molecule that is relatively close to a
polarizing agent. The electron T1 and T2 are taken from experimental values for trityl in a
60:40 glycerol:D2O sample at 80 K. Figure 6 shows the spatial configuration for the spin
system.
We want to examine the role of higher order coherences specifically in the DNP
processes (electron-nuclear transfer) and in the spin-diffusion processes (nuclear-nuclear
transfer). In order to do this, all states in the spin system, aside from polarization states,
will be assumed to be in quasi-equilibrium (a very similar assumption to that taken in the
Theory section of Chapter 3) [25]. Additionally, we control the spin-order of the
simulation. Spin-order is the number of spins involved in a state of the system. So, if we
truncate the spin order to 3, for example, any spin state involving more than three spins,
such as Sz I1+ I2
− I3z , would be omitted from the simulation [18]. For this simulation, we
vary the spin order from 2 to 7, which is the maximum possible order with seven spins.
The DNPsim input is as follows:
248
Sys.S=1/2;
Sys.I=1/2*ones(1,6);
Sys.v0I=212*ones(1,length(Sys.I));
Sys.v0S=140e3;
Sys.A=[-0.1092;-0.2532;-0.3556;-0.0448;0.0720;0.0449];
Sys.B=[-0.0204;-0.0812;-0.5029;-0.0634;-0.1620;-0.0610];
Sys.dnn=[0.00622;0.000778;0.000550;0.000389;0.0000687;0.00622; …
0.00311;0.000550;-0.000111;0.000550;-0.00311;-0.000389;0.00622;… 0.000550;-0.00311];
Sys.T1e=1.4e3;
Sys.T2e=.8;
Sys.T1n=1e5;
Sys.T2n=1e3;
Figure 6: Solid Effect Spatial Configuration
6 nuclei, corresponding to 1H found on a glycerol molecule, are placed near a radical
center, which is offset in the x, y, and z-directions by 1.5 Å, 1.1 Å, and 3.8 Å,
respectively.
249
Par.mwStrength=3.5;
Par.mwFreq=139.788e3;
Par.maxt=1e6;
Par.tPoints=1000;
Par.PropMethod='QuasiEq';
SL=State_list_gen(Sys);
Par.LLstates=sum(SL.states,1)==1;
Par.TruncO=2;
Par.output='B0,mwFreq,rfFreq,t,detected,M,Meq,Lreduc,EQstates,states';
out=DNP_sim(Sys,Par);
The input for the Sys structure is in the direct input format. The input for the Par structure
includes some more advanced commands. We use the ‘QuasiEq’ method of propagating
the spin system. This method calculates a reduced Liouville matrix according to (70). By
specifying Par.LLstates, we choose which states are not in quasi-equilibrium. In order to
select only the polarization states, we sum the integers in each column of the state list
(SL.states), and only take those states for which the sum is 1. We include the string
‘Lreduc’ in the Par.output field. This returns the reduced Liouville matrix using the
quasi-equilibrium assumption. Then, the elements of the reduced matrix are rate
constants of polarization transfer, so that one could write the evolution of the system as
follows:
250
ddt
PS = −k0PS − kDNPm PS − PI
m( )m=1
NI
∑ + 1T1S
PSeq − PS( )
ddt
PIm = kDNP
m Ps − PIm( ) + kSD
m,n PIn − PI
m( )n=1
NI
∑ + 1T1I
m PIeq − PI( )
(84)
Therefore, by extracting the elements of the matrix Lreduc, we are able to obtain the rate
constants in (84).
In addition to using the quasi-equilibrium propagation in order to obtain rate
constants, we also use basis set truncation to examine the role of high-order coherence.
We vary Par.TruncO from 2 to 7, in order to see the effects on the polarization buildup
on the 6 nuclei, and also the effect that is has on the rate constants. In Figure 7, we show
the polarization buildup on the 6 nuclei for truncation of the spin order from 2 to 7.
Figure 7: SE DNP for Spin-Orders of 2-7
Simulations of the DNP buildup for 6 nuclei, where the spin order is truncated to various
levels. One sees that the buildup behavior changes significantly as the spin order is
increased.
251
In Figure 7, we observe interesting behavior. When the spin system is truncated to
second order, there is very efficient transfer of polarization. The reason for this is that
both the DNP transfer rates and the spin-diffusion transfer rates are perfectly on
resonance. Normally, couplings to other nearby spins would offset the matching
conditions and slow down the various transfers. However, these couplings affect
intermediate coherences. For example, i I1
+ I2− − I1
− I2+( ) , which is involved in spin
diffusion, is transferred to Sz I1
+ I2− + I1
− I2+( ) by the electron-nuclear secular couplings
Sz I1z and Sz I2z . Because this is a state with spin order 3, it is not included in the spin
order 2 simulation. This faster transfer is seen between 2nd and 3rd order truncation, 4th
Figure 8: DNP Rate Constants for Spin Orders of 3, 5, 7
The DNP rate constants are extracted from Lreduc using DNPsim. We see significant
increases in the DNP rate constants for each nucleus as the spin order is increased. We
also show that the square of the pseudo-secular electron-nuclear coupling (B2, without
units) is nearly proportional to the rate constants for 3rd spin order.
252
and 5th order truncation, and 6th and 7th order truncation. Additionally, we see that
between 3rd, 5th , and 7th order truncation, there is a notable increase in the amount of
polarization transferred as the spin order is increased. This is strong evidence that the
higher order spin states make significant contributions to polarization transfer.
Although Figure 7 shows us the importance of high order spin states, it is not
clear whether those states are contributing primarily to the spin-diffusion transfer
efficiency (nuclear-nuclear) or the DNP transfer efficiency (electron-nuclear). In Figure
8, we plot the DNP rate constants, as given in (84), by extracting them from the matrix
Lreduc.
In Figure 8, the DNP rate constants ( kDNPm ) are plotted for each nucleus for spin
orders 3, 5, and 7. One can clearly see that the higher spin order contributes to the DNP
Figure 9: Spin-Diffusion Rate Constants for Spin Orders of 3, 5, 7
The spin-diffusion rate constants are extracted from Lreduc using DNPsim. Although
some of the rate constants increase for higher spin order, most do not see a significant
increase. The exception are the case where the spin-diffusion rate constant is already
exceptionally high. Therefore, it high order coherences play a less significant role in
enhancing spin diffusion.
253
rate constant, and thus increases the direct electron-nuclear transfer rate. Similarly, in
Figure 9 we plot the spin-diffusion rate constants ( kSDm,n ) for all nuclear spin pairs. In this
case, there is not a significant increase in the rate constant in most cases. The exception is
the transfer rate constant between nucleus 1 and 2, and less so the transfer rate between 2
and 5. However, these rates are already extremely high, and so the increase does not
affect the outcome of the polarization transfer much. Therefore, we can see that high-
order coherences play a large role in increasing the DNP transfer rate, but do not play as
significant a role in the spin-diffusion transfer rate. Note that this finding is in conflict
with the findings of Hovav et al., who refer to a “DNP assisted spin diffusion” in their
recent work [17]. Possible reasons for this discrepancy are a less rigorous treatment of
relaxation in the simulations of Hovav et al. Also, because we have used the quasi-
equilibrium of states to be able to obtain a reduced Liouville matrix, we have a much
clearer demarcation of the DNP and spin-diffusion processes.
6.6 Conclusions We have presented DNPsim, a program designed for the simulation of DNP
experiments. By providing a careful explanation of the method of simulation in Section
6.2, and instructions to use the MATLAB interface in Section 6.3, we hope to encourage
further studies of DNP that are supported by simulation. In Section 6.4, we have shown
examples of multiple types of experiments, including pulsed-DNP methods that transfer
spin-locked polarization to Zeeman polarization. Additionally, we have shown how the
quasi-equilibrium assumption can be used in order to separate DNP and spin-diffusion
rates, and how basis set truncation can elucidate the roles of different coherences in 6.5.
With the ability to set up a variety of spin-systems and manipulate the simulation method,
we hope that users are able to gain deeper insight into DNP mechanisms.
254
6.7 Appendix In order to assess the relaxation of a system, it is necessary to evaluate the integral
− lim
′t →∞
1′t
H1(t1),[H1(t1 + t2 ),σ (t2 )]⎡⎣ ⎤⎦dt2 dt10
t
∫0
′t
∫ , (85)
where H1(t) is given by
H1(t) = φλ ,p (t)Jλ ,p
p=1
N
∑λ=x ,y ,z∑ . (86)
In our case, the correlation functions of the φλ ,p (t) are given by
lim′t →∞
1′tφλ ,p (t1)φγ ,q (t1 + t2 )dt1 =
1Tλ ,pq
δλ ,γδ (t2 )0
′t
∫ . (87)
We can begin by inserting (86) into (1).
− lim
′t →∞
1′t
φλ ,p (t1)φγ ,q (t1 + t2 ) Jλ ,p ,[Jγ ,q ,σ (t2 )]⎡⎣ ⎤⎦dt2 dt10
t
∫0
′t
∫q=1
N
∑p=1
N
∑γ =x ,y ,z∑
λ=x ,yz∑ (88)
To solve this, we want to make the inner integral similar in form to the integral in (87).
We do this by changing the order of integration, and moving 1/ t into the outer integral.
− lim
′t →∞
1′tφλ ,p (t1)φγ ,q (t1 + t2 ) Jλ ,p ,[Jγ ,q ,σ (t2 )]⎡⎣ ⎤⎦dt1 dt2
0
′t
∫0
t
∫q=1
N
∑p=1
N
∑γ =x ,y ,z∑
λ=x ,y ,z∑ (89)
As mentioned in the main text, the σ (t) converge to a single value, due to the the fact
that τ c → 0 , and so the average over the spin systems only needs to be taken over the
first part of the integral. At this point, one can move the commutator to the outer integral,
and we have thus reproduced the form of our correlation function.
− lim′t →∞
Jλ ,p ,[Jγ ,q ,σ (t2 )]⎡⎣ ⎤⎦1′tφλ ,p (t1)φγ ,q (t1 + t2 )dt1 dt2
0
′t
∫0
t
∫q=1
N
∑p=1
N
∑γ =x ,y ,z∑
λ=x ,y ,z∑
= − lim′t →∞
Jλ ,p ,[Jγ ,q ,σ (t2 )]⎡⎣ ⎤⎦1
Tλ ,pq
δλ ,γδ (t2 )dt20
t
∫q=1
N
∑p=1
N
∑γ =x ,y ,z∑
λ=x ,y ,z∑
(90)
Because the function is non-zero only when t2 = 0 and λ = γ , we may simplify the
summation and drop the integral.
255
= − 1
Tλ ,pq
Jλ ,p ,[Jγ ,q ,σ (0)]⎡⎣ ⎤⎦q=1
N
∑p=1
N
∑λ=x ,y ,z∑ (91)
Therefore, it is possible to evaluate the relaxation rate simply by calculating the
commutators, and taking the strength of the stochastic fields, given by the 1/ Tλ ,pq .
6.8 Bibliography
[1] T.R. Carver, and C.P. Slichter, Physical Review 92 (1953) 212. [2] A.W. Overhauser, Physical Review 92 (1953) 411. [3] C. Jeffries, D., Physical Review 106 (1957) 164. [4] C.D. Jeffries, Physical Review Phys. Rev. PR 117 (1960) 1056. [5] A. Abragam, and W. Proctor, G., C. R. Acad. Sci. 246 (1958) 2253. [6] A. Kessenikh, V., V. Lushchikov, I., A. Manenkov, A., and Y. Taran, V., Soviet
Physics - Solid State 5 (1963) 321-329. [7] A. Kessenikh, V., A. Manenkov, A., and G. Pyatnitskii, I., Soviet Physics - Solid
State 6 (1964) 641-643. [8] C. Hwang, F., and D. Hill, A., Physical Review Letters 18 (1967) 110. [9] K.-N. Hu, G.T. Debelouchina, A.A. Smith, and R.G. Griffin, Journal of Chemical
Physics 134 (2011). [10] M. Goldman, Spin temperature and nuclear magnetic resonance in solids, Clarendon
Press, Oxford, 1970. [11] M. Duijvestijn, J., R. Wind, A., and J. Smidt, Physica B+C 138 (1986) 147-170. [12] M.J. Duijvestijn, A. Manenschijn, J. Smidt, and R.A. Wind, Journal of Magnetic
Resonance 64 (1985). [13] V. Weis, M. Bennati, M. Rosay, and R.G. Griffin, J. Chem. Phys. 113 (2000) 6795-
6802. [14] A. Henstra, P. Dirksen, J. Schmidt, and W.T. Wenckebach, J. Magn. Reson. 77
(1988) 389-393. [15] C. Farrar, D. Hall, G. Gerfen, M. Rosay, J. Ardenkjaer-Larsen, and R. Griffin, J
Magn Reson 144 (2000) 134-41. [16] Y. Hovav, A. Feintuch, and S. Vega, Journal of Magnetic Resonance 207 (2010)
176-189. [17] Y. Hovav, A. Feintuch, and S. Vega, Journal of Chemical Physics 134 (2011)
074509(1-20). [18] A. Karabanov, I. Kuprov, G.T.P. Charnock, A. van der Drift, L.J. Edwards, and W.
Köckenberger, Journal of Chemical Physics 135 (2011) 084106. [19] A. Karabanov, A. van der Drift, L.J. Edwards, I. Kuprov, and W. Köckenberger,
Phys. Chem. Chem. Phys. (2012). [20] R.R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of nuclear magnetic
resonance in one and two dimensions, Clarendon, Oxford, 1987. [21] Y.C. Cheng, and R.J. Silbey, Physical Review A 69 (2004) 052325-(1-15).
256
[22] M. Goldman, J. Magn. Reson. 149 (2001) 160-187. [23] P. Ahuja, R. Sarkar, P.R. Vasos, and G. Bodenhausen, ChemPhysChem 10 (2009)
2217-2220. [24] M.H. Levitt, Annu. Rev. Phys. Chem. 63 (2012) 5.1-5.17. [25] A.A. Smith, B. Corzilius, A.B. Barnes, T. Maly, and R.G. Griffin, Journal of
Chemical Physics 136 (2012) 015101-(1-16). [26] S. Stoll, and A. Schweiger, J Magn Reson 178 (2006) 42-55. [27] I. Kuprov, N. Wagner-Rundell, and P.J. Hore, Journal of Magnetic Resonance 189
(2007) 241-250. [28] C. Song, K.-N. Hu, C.-G. Joo, T.M. Swager, and R.G. Griffin, J. Am Chem. Soc 128
(2006) 11385-90. [29] Y. Matsuki, T. Maly, O. Ouari, H. Karoui, F. Le Moigne, E. Rizzato, S. Lyubenova,
J. Herzfeld, T.F. Prisner, P. Tordo, and R.G. Griffin, Angewandte Chemie 48 (2009) 4996-5000.
[30] M.K. Kiesewetter, B. Corzilius, A.A. Smith, R.G. Griffin, and T.M. Swager, J. Am. Chem. Soc. 134 (2012).
257
