Solid State Nuclear Magnetic Resonance 53 (2013) 20–26

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Solid State Nuclear Magnetic Resonance

0926-20http://d

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journal homepage: www.elsevier.com/locate/ssnmr

Complete description of the interactions of a quadrupolar nucleuswith a radiofrequency field. Implications for data fitting

T. Leigh Spencer, Gillian R. Goward, Alex D. Bain n

Department of Chemistry and Chemical Biology, McMaster University, Hamilton, ON, Canada L8S 4M1

a r t i c l e i n f o

Article history:Received 7 January 2013Received in revised form9 February 2013Available online 4 April 2013

Keywords:Radiofrequency fieldsNutationFlip anglesQuadrupolar nucleiLiouvilliansData fitting

40/$ - see front matter & 2013 Elsevier Inc. Ax.doi.org/10.1016/j.ssnmr.2013.03.002

esponding author. Fax: +1 905 522 2509.ail address: bain@mcmaster.ca (A.D. Bain).

a b s t r a c t

We present a theory, with experimental tests, that treats exactly the effect of radiofrequency (RF) fieldson quadrupolar nuclei, yet retains the symbolic expressions as much as possible. This provides amathematical model of these interactions that can be easily connected to state-of-the-art optimizationmethods, so that chemically-important parameters can be extracted from fits to experimental data.Nuclei with spins 41/2 typically experience a Zeeman interaction with the (possibly anisotropic) localstatic field, a quadrupole interaction and are manipulated with RF fields. Since RF fields are limited byhardware, they seldom dominate the other interactions of these nuclei and so the spectra show unusualdependence on the pulse width used. The theory is tested with 23Na NMR nutation spectra of a singlecrystal of sodium nitrate, in which the RF is comparable with the quadrupole coupling and is notnecessarily on resonance with any of the transitions. Both the intensity and phase of all three transitionsare followed as a function of flip angle. This provides a more rigorous trial than a powder sample wheremany of the details are averaged out. The formalism is based on a symbolic approach which encompassesall the published results, yet is easily implemented numerically, since no explicit spin operators or theircommutators are needed. The classic perturbation results are also easily derived. There are norestrictions or assumptions on the spin of the nucleus or the relative sizes of the interactions, so theresults are completely general, going beyond the standard first-order treatments in the literature.

& 2013 Elsevier Inc. All rights reserved.

1. Introduction

Radiofrequency (RF) irradiation is the main tool we use tomanipulate and observe spin dynamics in nuclear magneticresonance (NMR). Combined with free precession during delays,RF pulses are assembled into pulse sequences that can performwhat seems to be spectroscopic magic. For simple systems, such asthose with spins-1/2, the design and analysis of these pulsesequences is well understood, but for quadrupolar nuclei, thesituation is usually more complex. If the RF dominates all the otherinteractions of the nucleus, then a pulse is effectively instanta-neous and acts like a rotation of the frame of reference. Forquadrupolar nuclei, the interactions are large and the RF almostnever dominates. In the time domain, this means the spin systemis evolving under both the static Hamiltonian and the RF, so morethan one transition is being affected at the same time. Even thoughwe talk about a π pulse, this may not have the same effect asrotating the frame of reference through that angle. Finite pulseeffects are becoming more common in NMR and so a deep

ll rights reserved.

understanding of the effect of RF is essential. Such an importantaspect of NMR has been widely studied in the literature, of course,but often with some simplifying assumptions. This paper includesand extends existing work and presents a simple yet complete andexact approach to the effects of RF on a single static quadrupolarnucleus of arbitrary spin. No assumptions, such as first-ordercoupling or on-resonance irradiation are required. The theory issubjected to the most rigorous testing: nutation behavior of all the23Na transitions of a single crystal of sodium nitrate.

It is useful to divide the NMR of quadrupolar nuclei into twoclasses: first-order perturbed and second (or higher) order per-turbed. The first-order case is easy to deal with, but the spectracontain little information, since the central transitions all have thesame frequency regardless of orientation. Second-order perturbedsystems are more common (since quadrupole couplings tend to belarge and there are limits to magnetic field), and the powderpattern of the central transition shows a rich structure. Essentiallyall of the published analysis of RF effects is restricted to first-ordersystems. However, the theory is not restricted to this simple case,and it reveals a number of complications that arise in higher-ordersystems. These are observed experimentally in the often-unusualflip-angle dependence of second-order spectra. In this paper weestablish and test the method, and future work will give details ofthe effects of strong quadrupolar coupling.

T.L. Spencer et al. / Solid State Nuclear Magnetic Resonance 53 (2013) 20–26 21

We usually want to extract molecular parameters from experi-mental spectra. It is important to remember that the mathematicalmodel is a means to an end in this process. The parameters in themodel are usually varied systematically until a best fit to the datais obtained. This is an optimization problem [1] which can behandled in a number of ways. Roughly, such problems divide intocases where derivatives are available or where they are not.Derivative-based methods tend to be more efficient, but they canbe difficult to implement and numerically expensive. In this paper,we discuss both methods for the nutation experiment and showhow symbolic derivatives can be easily calculated within thisapproach. A numerical simulation will give numerical derivatives,which are typically less reliable. Furthermore, people typically fit asingle spectrum to a model based on the equilibrium state [2]. Ifwe can broaden this to a set of spectra as a function of a readily-controlled experimental parameter, such as flip angle, then we willget better data.

The spin dynamics of a quadrupolar nucleus subjected to theZeeman interaction (which includes chemical shielding aniso-tropy) and quadrupolar coupling is well-known [3–7], but the fullinteraction with a radiofrequency (RF) field, combined with theothers, is less so [8–38]. In the quest for simple analyticalformulae, many of these studies used simplifying, but welljustified assumptions, such as first-order coupling, on-resonanceirradiation or cylindrical tensors. Most of this published workassumes the quadrupole Hamiltonian is proportional to Iz

2, whichis equivalent to first-order perturbation. As well, it is often (withsome exceptions [28]) assumed that the RF is on-resonance withthe central transition—this would prevent us from analyzingpowders with a CSA contribution. There is also the question ofwhether the dependence on pulse width should be Fouriertransformed to give a nutation spectrum, or just left as a flip-angle dependence of the normal spectrum. Finally, most samplesare powders, so a powder average is then superimposed on thenutation behavior. This is appropriate for real samples, but it willaverage out many of the details. A single crystal study is a muchmore challenging experimental test of the theory.

In this paper, we present a unified theory which treats all ofthese interactions which follows on a preliminary communication[36]. The specific results can be derived by a number of othertechniques, including a very recent analysis using effective Hamil-tonians [37], so the stress here is on the method and its experi-mental test. Finally, we use optimization methods to fit the theoryto experiments and start a discussion of the calculation ofderivatives of the results with respect to the parameters—anecessary next step in fitting procedures.

The theory is exact and completely general, with no restrictionson the relative sizes of the RF, Zeeman and quadrupole interac-tions and it has proved useful in the case of third-order perturba-tions to the central transition [39]. Explicit formulae for the resultscan not be derived for the general case, but symbolic solutions areavailable for simple cases [40]. The goal is not to derive a series offormulae, rather it is to formulate it in terms of an eigenvalueproblem. There are explicit formulae for the matrix elements, sothat all that is needed is an eigenvalue (symbolic or numerical)routine, such as those available in higher-level programs such asMatlab (http://www.mathworks.com), Mathematica (http://www.wolfram.com) or Maple (http://www.maplesoft.com). If the pro-gram handles symbolic manipulation (e.g. Mathematica or Maple),then even the formulae can be easily generated by the program,for added reliability. In this way, there are no special cases—sincethe parameters are general, any case can be handled by the sameprogram. There is a very close analogy between this approach andthe way we deal with Hamiltonians: in both cases, there is a basisdenoted by spin quantum numbers and a set of formulae for thematrix elements. The original source of the formulae may be

daunting, but their use is very straightforward. This provides aneasy and accessible way to calculate the full spin dynamics for ageneral quadrupolar spin.

2. RF pulse effects

The effects of RF irradiation are well-known for spin-1/2 nuclei.If the RF dominates all interactions, the effects are equivalent tothe rotation of the frame of reference. Even if the RF is off-resonance, then it still acts as a rotation, albeit around a tiltedeffective axis [4,41–43]. For quadrupolar nuclei, the RF can affectthe central transition and the satellites simultaneously, so thesituation is more complicated. If the RF dominates the quadrupoleinteraction, then the behavior is like a spin-1/2. If the quadrupoledominates the RF, then the RF usually affects only a singletransition, which can be treated as a fictitious spin-1/2. The mostcomplicated case is a powder pattern for a system with a strongquadrupole. In this case, the central transition is shifted by thesecond-order quadrupole perturbation, which depends on orien-tation and can give substantial width to the central transitionlineshape. In practice, this is often the case encountered, and leadsto substantial flip-angle dependence of the powder lineshape.Within the powder, there will be orientations where the RFdominates and those where the quadrupole dominates and allthe in-between cases, so a general approach is essential. Describ-ing this is our ultimate goal.

Since NMR is such a simple quantum-mechanical system,these problems can already be treated fully with a number ofapproaches. They all must give the same answer, but variouspeople may find particular approaches more useful, so it isimportant to have alternatives. Numerical integration of theequation of motion of the density matrix is readily available withmodern computers and software, such as SIMPSON [44], SPINE-VOLUTION [45] or SpinDynamica (http://www.spindynamica.soton.ac.uk) to give an exact solution, but this may not give muchphysical insight and may be numerically inefficient because ofdifferent time scales. Also, the output may not be in a convenientformat for further processing. For smaller spin systems, particu-larly spin-3/2, reasonable approximations can be made and sym-bolic expressions can be derived. We always start with a symbolicstatement of the problem and end up with numerical values tocompare with experiments. The question arises as to when tomake this transition. If there is a full symbolic solution (e.g. fromperturbation theory), then there are no numerical issues. A fullynumerical solution is a good general way to proceed, but theremay be questions about step sizes and round-off errors. Thissituation is amplified if we try to calculate derivatives purelynumerically. If we start with a partly symbolic approach, then thismay focus the numerical issues in one step (perhaps an eigenvalueroutine) where the numerical issues are well-studied. There is no“best” way, since any numerical method has flaws. The approachwe present here tries to maintain as much of the symbolic solutionas possible, but also provides an easy route for exact numericalsimulation.

A challenging test for these theories is the spin nutationexperiment. For an isolated spin-1/2, the plot of signal againstpulse width will be a sine wave, and has often been used forcalibrations of the RF field. For a quadrupolar system, the plot is asum of sine waves whose frequencies and intensities depend onthe relative sizes of the quadrupole interaction and the RF field[8,11–13,16,26,28,29,31,36,37,46–49]. A related situation is thedirect creation of multiple-quantum coherence during a finitepulse [50–52]. In all these cases, the Zeeman, the quadrupoleand the RF interactions must be treated together, with noassumptions of their relative magnitudes.

T.L. Spencer et al. / Solid State Nuclear Magnetic Resonance 53 (2013) 20–2622

3. Theory

The theory in these descriptions usually involves the calcula-tion of the exponential of a matrix, either the Hamiltonian or theLiouvillian. This is the crux of the problem. These matrices can beeasily calculated in symbolic form, so the dependence on orienta-tion, RF field, quadrupole coupling, offset from resonance, etc. isclear. The exponential of a matrix is often calculated [53,54] bydiagonalizing the matrix and using the property that this matrix ofeigenvectors also diagonalizes the exponential. For simple cases, itmay be possible to calculate the eigenvalues and eigenvectorssymbolically, which leads to explicit expressions for the nutationbehavior and this has been the focus of much of the publishedwork. For general and practical cases, this may not be satisfactory,and numerical solutions are needed.

In this paper, we stress a method to achieve these goals. It doesnot require operators or their commutators [55], since theirproperties are “built into” the method. We use an approach thatmaintains the symbolic nature of the problem for as long aspossible. A set of explicit formulae give the matrix elements ofthe Liouvillian, which can then be diagonalized numerically for anexact solution. If the Hamiltonian can be diagonalized symbolicallyunder some assumptions, then so can the Liouvillian. This providesa simple, unified and systematic approach that is easily pro-grammed into high-level software. The approach clearly showsthe dependence on the initial state of the density matrix, theterms in the Liouvillian and the observable (central or satellitetransitions). It is the clear and straightforward nature of the theorythat gives a new view on the problem.

The density matrix, ρ, carries all the information about the spinsystem. We know its value at the start of the experiment, sincethat is usually the equilibrium state, then we can follow itsevolution in time with the Liouville-von Neumann equation

ddt

ρðtÞ ¼−i L ρðtÞ ð1Þ

We have written this equation in its superoperator [4,56] form,where L is the Liouvillian superoperator, the action of taking thecommutator with the Hamiltonian [4,55,57,58]. We use bold italicnotation to designate these operators in Liouville space, or super-operators in Hilbert space. Eq. (1) is perhaps more familiar in itsHamiltonian form in Hilbert space, but the two are completelyequivalent. In any case, we can propagate the density matrix,

ρðtÞ ¼ expð−i LtÞ ρð0Þ ð2Þ

and then extract the expectation value,⟨Q ⟩ of an observable bytaking the scalar product of the corresponding operator, Q, withthe density matrix

⟨Q⟩¼ ðρjQ Þ ð3Þ

The scalar product in Liouville space is just the standard dotproduct of two vectors and corresponds to the trace of the productof the two matrices in Hilbert space.

In practice, Eq. (2) is usually solved by diagonalizing theLiouvillian matrix, L, with a matrix, U containing the eigenvectorsas columns, to give a diagonal matrix, Λ, with the eigenvaluesdown the main diagonal.

U−1LU ¼Λ

L¼UΛU−1 ð4Þ

With this definition, Eq. (2) becomes

ρðtÞ ¼U expð−i ΛtÞU−1ρð0Þ ð5Þ

and the expectation value of the observable becomes

⟨Q⟩¼Q † U expð−i ΛtÞU−1ρð0Þ ¼ ðQ † UÞ expð−i ΛtÞ ðU−1ρð0ÞÞð6Þ

Provided we can calculate the Liouvillian, and identify theinitial state and the desired observable, we have a completesolution, since Eq. (6) is a simple set of matrix multiplications.The evolution is a sum of complex exponentials with coefficientsdetermined by the initial state and the final observable. The exactmatrix elements for the Liouvillian can be calculated from pub-lished [55] formulae, so all that is needed is an eigenvalue routine.This is very similar to the way we deal with a Hamiltonian in thecalculation of equilibrium spectra: a basis is defined by spinquantum numbers, matrix elements are calculated from formulaeinvolving the quantum numbers and the matrix is diagonalized. Insome cases, the eigenvalues and eigenvectors can be calculatedsymbolically, to give explicit equations for the observables. How-ever, the general case will require numerical diagonalization,which is readily available in high-level programs such as Matlab,Mathematica or Maple. This approach provides a simple numericaldescription of the exact spin dynamics, without any of thecomplications of operators or their commutators.

For an isolated quadrupolar nucleus in a static sample, there arethree important interactions. These are the Zeeman interaction,which includes both the main magnetic field and a possibly-anisotropic chemical shielding, the quadrupole interaction and theeffect of the radiofrequency (RF) field. Detailed discussion of theZeeman and quadrupole interactions have recently been published[40,55], and the RF interaction is straightforward. In a sphericaltensor basis, the RF terms in Liouville space are represented ascombinations of raising and lowering operators, just the same asin the more familiar Hilbert space.

4. Nutation experiments

The nutation experiment presents a rigorous test of the theory,since there are essentially no freely-adjustable parameters. Thephases and intensities of the central transition and the satellitesare governed by the quadrupole coupling, the offset of the RF fromthe central transition and the RF field strength. The quadrupolecoupling for a single crystal is obtained directly from the splittingbetween the satellites and the central transition. The offset fromresonance is defined by the spectrometer and the RF field can becalibrated from a liquid sample. The RF may be slightly different inthe single-crystal experiment because of differences in probetuning but its value can be adjusted. Looking at the satellites andthe central transition means that we are looking at differentobservables, with the same Liouvillian. Changing the offsetchanges the Liouvillian by adding a small Zeeman contribution.These experiments all start with a spin system in equilibrium, buta non-equilibrium set of z magnetizations [34,35] or spin-lockedmagnetization in the xy plane [59,60] can also be accommodated.If the theory fits the phase and intensity of three lines as a functionof pulse width over a wide range of nutation angles, then it can beconsidered a success.

5. Data fitting and derivatives

If we have a mathematical model (with variable parameters) ofour experiment, then we can search for the set of parameterswhich make the model fit the experimental data in the “best” way.This is a classic problem [1] with many solutions, and whichsolution is appropriate will depend on the particular problem.Much work has been done to improve the algorithms, and these

Fig. 1. Nutation spectrum of the central transition of 23Na in a single crystal ofNaNO3 with the RF on-resonance. The spectra were phased so that the shortestpulse width spectrum was pure absorption. The plot shows the signed intensity ofthe central transition as a function of pulse width plus the theoretical curve.The quadrupole coupling, e2qQ , used in the theory was measured from theseparation of the two satellites, and is 40+/0.5 kHz. The RF field was adjustedfrom that measured on a liquid sample and the value obtained from the fit is17.3 kHz. The only other variable parameters were the initial phase and intensity ofthe measured spectra. The calculated values match exactly the exact solutions forthe on-resonance case in references [46] and [32].

T.L. Spencer et al. / Solid State Nuclear Magnetic Resonance 53 (2013) 20–26 23

state-of-the-art methods are often available as open-sourcesoftware. These so-called “black-box” programs need not knowanything about the particular problem—they just need to calla program that takes the parameters as input and returns theobjective function (and perhaps its derivatives). The algorithm willthen generate a new set of parameters and the process willcontinue until convergence. The optimizer does not need tounderstand the problem, nor does the user need to understandthe subtleties of the best algorithms.

Simplex-type methods, such as that of Nelder and Mead [61],do not need derivatives and are widely used, perhaps because oftheir simplicity and surprising robustness. Improvements havebeen made to these derivative-free methods and good software isavailable. We have found NOMAD (http://www.gerad.ca/nomad/Project/Home.html) [62] to be extremely useful in several cases,including the present work.

For problems with many parameters, plus constraints,derivative-based methods are needed. For example, optimal con-trol methods (which use first derivatives) have been used inderiving high-performance shaped pulses [63–65]. Second deriva-tives make the methods more powerful, and typically, secondderivatives can be obtained fairly easily from first derivatives.Derivatives can be estimated numerically, but analytical deriva-tives are much preferred. We have recently applied this to pulsedesign [66], a problem with hundreds of parameters, using thevery powerful IPOPT black-box optimizer (https://projects.coin-or.org/Ipopt).

Derivatives can also provide useful information in designingthe experiment. The derivatives tell how sensitive a particular datapoint is to a given parameter, so we typically want to enhance thisif we can. Since the derivative is a function of the parameter itselffor a non-linear problem, we can optimize our range of para-meters. For instance, in the inversion-recovery experiment formeasuring T1, the derivatives show that the choice of variabledelays for recovery is not all that critical [67,68], as we alreadyknew from experience.

In the present case, the mathematical model involves theeigenvalues and eigenvectors of a matrix. Derivatives of thesewith respect to elements of the matrix are well-known [69], andMcClung and his group introduced them to the NMR world infitting slow-exchange data from selective-inversion experiments[70]. We prefer Wilkinson's formulae [69,71] to McClung's andimplemented them in a program called CIFIT. The mathematics inboth the slow-exchange situation and the present nutation experi-ment is identical, since both are described by sums of exponen-tials. The only difference is that the slow exchange case involvesreal exponents leading to decays, whereas the nutation involvesimaginary exponents, leading to oscillations.

This approach concentrates the numerical issues into theeigenvalue routine—up to that step, all the calculations are sym-bolic. The Maple program (and presumably many others) permitsnumerical work to be done with arbitrary precision through thedigits parameter, so quality control over the results is easilyachieved. In practice, NOMAD was sufficient for our experimentshere, since we have very high-quality data, but more challengingexperiments will require more sophisticated fitting.

6. Results and discussion

The figures demonstrate this success, so relatively little discus-sion is needed. The calculation of the nutation experiment isefficient, since all that is changed for each step in Eq. (6) is thevalue of t and so only those fifteen exponentials need be recalcu-lated. The model used the observed splitting of the satellites todefine a cylindrical quadrupole tensor oriented along the z axis

with the value e2qQ of 40.0 kHz. The true quadrupole coupling islarger, attenuated by the orientation, of course, but this approx-imation is fully justified here by the small size of the quadrupolecoupling. This is equivalent to using I

2z as the quadrupole Hamil-

tonian, as previous work has done [9,11,13,16]. The RF field and anarbitrary initial phase and scale factor for each line were used asfitting parameters—seven in total for each set of the on-resonanceand off-resonance data. Within this approximation, the transfor-mation to the rotating frame of reference essentially removes themain Zeeman interaction—a small contribution is left if the RF isnot on resonance. Fig. 1 shows the classic nutation experiment onthe central transition: a plot of the signed intensity as a function ofpulse width. If the RF is on-resonance there is a symbolic solution[11,46], which matches our calculations to within computernumerical accuracy. Strictly speaking, the RF was at the centre ofthe central transition, but this is not quite the same as trueresonance. If there are second-order effects, then the positive-going central transition is specifically mixed [40] with othertransitions and its frequency is shifted. The mixing depends onthe orientation of the quadrupole tensor. For most angles, theinteraction with the positive-going double-quantum transitionsdominate, so the central transition moves down in frequency [40],but there are angles where the interaction of the negative-goingsingle-quantum transitions takes over, moving the frequency up.This gives a rationalization of how the centre of gravity of thecentral transition is shifted. However, in this case, second-ordereffects are very small and these shifts are negligible. If the RF is off-resonance, then there is no symmetry and each line will behavedifferently. Figs. 2–4 show the real and imaginary magnetizationsof the central line and one of the satellites.

Note that these diagrams do not show fits obtained fromvarying chemical parameters—the important values used in calcu-lating the theoretical curves are measured independently. For eachline, an arbitrary initial phase and an intensity were varied. Thephase corrections were applied to the theoretical calculations,which were compared to the raw data in the figures. There werethree phase parameters (one for each line), but a plot of the phaseparameters against frequency was almost linear, so this corre-sponds closely to the usual practice of applying a zero- and first-order phase correction to the spectrum. Given the typical spectra,it would be difficult to phase correct them properly in the usualway. The data from all three lines were included in the fit with a

Fig. 2. Real (solid line) and imaginary (dotted) parts of the central transitionintensity with the RF off-resonance, as a function of pulse width. The intensities arereported relative to that of the spectrum obtained with the smallest flip angle. Notethat the spectra were not phase corrected and so a phase correction (obtained fromthe fit) was applied to the calculated results for comparison to the experimentaldata. For this line, the phase correction was 2.67 rad. The quadrupole coupling andthe RF field are the same as in the previous experiment and the RF was −15.6 kHzaway from the central transition frequency.

Fig. 3. Real (solid line) and imaginary (dotted) parts of the low-frequency satellitetransition intensity with the RF off-resonance, as a function of pulse width. Thesedata are from the same spectra as those in Fig. 2. Data from all three lines in a set ofexperiments were included in a global fit with a single value of the RF field. Notethat the spectra were not phase corrected and so a phase correction (obtained fromthe fit) was applied to the calculated results for comparison to the experimentaldata. For this line, the phase correction was 2.94 rad. The quadrupole coupling andthe RF field are the same as in the previous on-resonance experiment and the RFwas -15.6 kHz away from the central transition frequency.

Fig. 4. Real (solid line) and imaginary (dotted) parts of the high-frequency satellitetransition intensity with the RF off-resonance, as a function of pulse width. Thesedata are from the same spectra as those in Figs. 2 and 3. Data from all three lines ina set of experiments were included in a global fit with a single value of the RF field.Note that the spectra were not phase corrected and so a phase correction (obtainedfrom the fit) was applied to the calculated results for comparison to the experi-mental data. For this line, the phase correction was 2.467 rad. The quadrupolecoupling and the RF field are the same as in the previous on-resonance experimentand the RF was -15.6 kHz away from the central transition frequency.

T.L. Spencer et al. / Solid State Nuclear Magnetic Resonance 53 (2013) 20–2624

single value of the RF nutation frequency, so seven parameterswere used. The on-resonance and off-resonance data were fittedseparately, but the only difference was the offset from resonance.The best-fit values of γB1 were very similar for the two experi-ments: 17.2 and 17.3 kHz, close to, but not the same as an estimatefrom a liquid sample. Errors in parameters obtained from non-linear least squares fits are notoriously difficult to estimate [72],but we have found the profiling method [72,73] useful. Theagreement between the two values of the RF field is well withinthe experimental error. The apparent nutation is different for eachof these plots, but the dependence is always a sum of sinusoids ofthe same frequencies, so the differences are due to differentmixtures of them.

These results are for the special and challenging case of a singlecrystal. Since the treatment is exact and valid for all relative sizesof the Zeeman, quadrupole and RF interactions and any initialstate, it is simply a matter of embedding this in a powder averageto treat the usual powder sample. The nutation will be different foreach quadrupole coupling, i.e. each crystallite orientation, so thedetailed oscillations will be partially averaged in the final spec-trum. However, for strong quadrupole coupling, different orienta-tions will contribute to different parts of the powder lineshape ofthe central transition. This means that the lineshape will dependstrongly on the flip angle, but this dependence can easily besimulated in this approach. Future publications will explore thisaspect.

The theoretical approach here is just one of many possibilities.Those who are already familiar with the curious and fascinatingbehavior of quadrupolar nuclei may not need it, but those that aredaunted by operators and their commutators may find thisapproach more comfortable, since it does not require operators.The important point is that there is an easily-accessible exactdescription of the effects of RF on a static quadrupolar nucleus andcan be easily programmed. If a program is written, it can be readilyadapted to optimization methods to obtain parameters from fittedexperimental data. The derivatives here are derived from a singleeigenvalue problem. Similar methods could be applied to standardHilbert-space evolution of the density matrix, but since thisinvolves the product of matrices, the derivatives will be morecomplicated. The RF parameters are under our control, so we canexploit them to improve the acquisition and analysis of thesespectra.

7. Conclusions

The experimental results clearly show that this approachprovides an exact description of nutation spectra, with no restric-tions on the relative sizes of the quadrupole interaction, the RFoffsets or the strength of the RF field. No knowledge of the spinoperators or their commutators is required in the theory so thesetup is particularly simple. In a sense, this represents a stage inthe description of the spin dynamics of an isolated quadrupolar

T.L. Spencer et al. / Solid State Nuclear Magnetic Resonance 53 (2013) 20–26 25

nucleus. The nucleus is subject to the Zeeman, quadrupole andRF interactions, and this approach treats all of these exactly, whilemaintaining a clear symbolic approach as much as possible.This provides a rigorous mathematical model. Because the methodinvolves a single eigenvalue problem, the calculation of thederivatives with respect to parameters is straightforward. It canbe easily coded to give input and output designed to interface withstate-of-the-art optimization software. The prospect of a completesymbolic solution is dim, but the approach presented hereincludes all the symbolic perturbation results and also a systema-tic yet simple way to set up the numerical calculation.

8. Experimental

NaNO3 powder was purchased from Strem Chemicals, (499%).A saturated solution of NaNO3 was allowed to crystallize at roomtemperature, forming single crystals of NaNO3. One of the singlecrystals obtained was wrapped in Parafilm and packed into a7 mm MAS rotor. The crystal was rhombohedral in shape and hada mass of 12.870.2 mg.

Static solid state 23Na NMR was performed on a BrukerAvance I AV500 spectrometer using a standard 7 mm MASprobe. A reference solution of 1 M NaNO3 in water was used,and 23Na was referenced at 0 ppm. For this solution, at 1.00 dB a2π pulse required a 39.5 ms. For the solid NaNO3 crystal, thesample was oriented such that the quadrupolar coupling, or thedistance between the central transition and the satellite transi-tions, was about 20 kHz. For each experiment a spectral width of100 kHz was collected and a power level of 1.00 dB was used; arecycle delay of 40 s was allowed, and 16 transients werecollected. Two sets of experiments were performed, and foreach the pulse width was varied from 2 to 160 ms in incrementsof 2 μs to create the nutation experiment, with a total of 80points. For the first nutation experiment, the excitation fre-quency was at the central transition; and for the second, theexcitation frequency was 15.6 kHz off resonance, between thecentral transition and the low frequency satellite transition. Thetwo experiments were performed one after the other, so thequadrupole coupling and the RF field is the same in both.Spectral processing was performed in TopSpin 2.1 software. Nophase or baseline corrections were done in TopSpin, since theywere incorporated into the fitting procedure.

The phases and intensities of the lines were obtained byfitting the lines to a Lorentzian lineshape. Each line has a phase,an intensity, a position and a width. The position and width ofeach line was measured and fixed in the fitting procedure, butthe phase and intensity for the three lines, plus a three-parameter polynomial fit to the baseline, were varied. Thefitting of these nine parameters (phase and intensity of threelines, plus three parameters for the baseline) was done by theopen-source derivative-free optimizer, NOMAD [62]. The nuta-tion data were also fitted using NOMAD. In this case, there wasan arbitrary initial phase and an intensity for the nutation ofeach line, plus the value of the RF field, leading to seven variableparameters.

Acknowledgments

We would like to thank the Natural Sciences and EngineeringResearch Council of Canada (NSERC) for financial support, andDr. Bob Berno for technical assistance and helpful discussions andDr. Kris Harris for his input.

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