NMR: Relaxation Measurements. How to measure relaxation rates? T 1 : Longitudinal or spin-lattice relaxation. M z is restored, the system goes back to

NMR: Relaxation Measurements

How to measure relaxation rates?

T1: Longitudinal or spin-lattice relaxation. Mz is restored, the system goes back to equilibrium.

T2: Transverse or spin-spin relaxation. Transverse magnetization Mx,y vanishes, the observable signal disappears.

For measurements pulsed methods should be used

In principle we could calculate T2 according to

Dn1/2 = 1/pT2

from the width of the Lorentzian lineshape of the signals in our spectrum. But...

T2-Measurement

This value is strongly depends on the inhomogeneity of our B0-field.

We are rather interested in the 'pure' spin-spin relaxation component (which, in contrast to the B0-field, is a molecular property)!

Homogeneous and inhomogeneous linewidth• However, transverse relaxation can also proceed due to

statistical (and static) inhomogeneities in the precession frequency ω0. T

• Resulting rate of Free Induction Decay is denoted as

• The first contribution is the same for all molecules and thus defines the homogeneous linewidth.

• The last contribution defines the inhomogeneous linewidth. • It is quite common that

• There are methods of getting rid of the inhomogeneous linewidth!

w

DNMRspectrum

2

*2

11TT

2*

2 TT

Spin echo• Large inhomogeneous linewidth means very fast dephasing of the spin• However, dephased magnetization can be focused back by pulses• Let us consider pulse sequence π/2x - τ - π x

• Explanation: let us divide system into isochromates having the same frequency ω0. Their offsets are Δω=ω0–ω. At certain time they all have different phases

• But at t=2τ all have the same phase – there is an ‘echo’!

p/2 p echo

0 t 2tt

X´

Y´

t=0 j=0

X´

Y´

j=Dwt

X´

Y´

j=p–Dwt+Dw(t–t)

X´

Y´

t=2t j=p

Carr-Purcell method• Spin echo not only allows one to get rid of the inhomogeneous broadening but

also to measure T2. To do this, however, the pulse sequence should be modified because the repetition rate of the echo experiment is <1/T1 (quite low)

• Luckily, the whole echo decay can be measured while applying one pulse sequence (C-P). Let us apply the sequence π/2x - τ - πx – 2τ - πx – 2τ - …

• At t=2τ we will have the first echo (negative phase). Then spins start dephasing again, the next πx-pulse again focuses them (positive phase!). Thus, there are echoes at times 2τ, 4τ, 6τ, 8τ,… amplitudes decay with T2.

• Drawback: if the pulses are not set precisely, mistakes are accumulated with time. It is better to use CPMG sequence: π/2x - τ - πy - 2τ - πy - 2τ - …

Y´

X´

Y´

X´

Y´

X´

Y´

X´

Y´

X´ X´

Y´

X´

Y´

X´

Y´

T2-Measurement

Spin-echo sequence

t180o90o

tT2

I(t)=I(0)exp(-2t/T2)

T2-Measurement

Spin-echo sequence

t180o90o

t T2

Before 90o After 180o

Before acquisition

Before 180oAfter 90o

T2-Measurement

The spin-echo experiment:

Compensates for the component of T2 that origins from field inhomogeneity

The relaxation can be measured selectively Important dynamic properties of the molecule can

be extracted that way

T2-Measurement

The experiment is repeated a number of times with increasing delays t.

I (t)=I (0)exp(-2t/T2)

I (t)

t

I (0)

T2 is obtained from a plot of I(t) against t:

Inversion-recovery technique

• Determination of T1 is often quite important as well• Standard method is inversion-recovery• First we turn the spin(s) by pulse (usually π/2 or π) and then look how system

goes back to equilibrium (recovers Z-magnetization). If the pulse is a π-pulse magnetization will be inverted (maximal variation of magnetization) and then recovered

• Equation for Mz is as follows:

• The kinetic trace (t-dependence) gives T1-time• To detect magnetization at time t in NMR one more π/2-pulse is applied,

sequence is then πx - t (variable) - πx/2 - measurement• For broad lines spin echo is used for detection, the pulse sequence is then πx

- t (variable) - πx/2 - τ - πx - τ - measurement • Both sequences should be repeated many times at different delays t

100 /exp))0(()( TtMMMtM zz 1

–1

t

T1-Measurement

Inversion recorvery

t

180o 90o t = 0

t >> T1

t = ln(2)T1

Mz(t)=Mo[1-2exp(-t/T1)]

Inversion-Recovery


t

Inversion-Recovery


>> t T1 = t ln(2)T1

t

Mz

+1

-1

0

t = ln(2)T1

zero observable signal

Fast T1-Measurement

Inversion recorvery

t

180o 90o

For a quick estimation of T1: directly search for the time t, which results in zero intensity (tzero) and calculate T1 from this:

T1 = tzero/ln(2)

NMR: NMR spectrometer

Magnet (probe, sample)

Console (transmitter,receiver, interface)

Computer (pulse-programming, data processing)

Probe

What you see of it

Inside a Magnet

1 Bore tube2 Filling port (N2)3 Filling port (He)4 Outer housing5 Vacuum chambers/ radiation shields6 Nitrogen reservoir7 Vacuum valve8 Helium reservoir9 Magnet coil

Shimming coils (not shown here) are also very important:One should resolve tiny splittings!!! Homogeneity of the order of 10–

9 is necessary for NMR

What you (usually) don’t see of it

Tesla and MegaHertz

The strength of a magnetic field is meassured in Tesla (for strong fields) or Gauss (for weaker fields). 1 Tesla corresponds to 10000 Gauss. The earth magnetic field is about 0.5 Gauss.

The strength of an NMR magnet is usually given in terms of its 1H resonance frequency in MHz:

Tesla 2.3 8.4 11.7

14.1

16.5

17.6

21.1

MHz 100 360 500 600 700 750 900

Why go for stronger fields?

Another reason is resolution: It is always better to work with AX-systems and only zz-parts of the scalar couplings spectra are much simpler and better resolved

Signal-To-Noise Ratio S/N

S/N or the signal-to-noise ratio is a measure for the sensitivity of the NMR experiment:

S/N ~ n g5/2 B03/2 (NS)1/2

MHz 500 600 700 750 900

S/N 1.0 1.3 1.7 1.8 2.4

reso-lutio

n

1.0 1.2 1.4 1.5 1.8

Relative sensitivity and resolution of our spectrometer

Number of spins Number of scans

NMR probe

Locates the sample at homogeneous field;

RF curcuit and coil for irradiating the sample and detecting its subsequent response;

Additional functions (sample rotations, T stabilization, field gradients)

Transmitter, receiver, amplifiers Transmitter section: produces RF irradiation; consists of RF-

synthesizer, pulse gates and RF amplifierS(t)=Acos(ωt+φ(t))

φ(t) can be rapuidly switched

Receiver section: preamplifier, quadrature reciever (comparison of the signals with a reference wave to get rid of fast oscillations)

Mx(t)=M0cos(ω0t) M0cos(Ω0t) where Ω0=ω0–ωref going from 300 MHz to 1 MHzThe procedure does not distinguish positive and negative Ω0 receiver supplies two signals:

SA(t)=M0cos(Ω0t) and SB(t)=M0sin(Ω0t) full information is retained but phasing is necessary

Hardware (Summary)

Magnet (Dewar, coil, shims)

Probe

Transmitter, receiver, amplifiers

Acquisition computer (ADC)

Sensitivity

NMR: NOE

NMR: NOENOE=Nuclear Overhauser Effect

Overhauser effect (EPR and NMR meet): NMR enhancement after pumping EPR transitions; works on dipolar relaxation of electron and nucleus

NOE also works using dipolar relaxation of two nucleiApplications are quite different: not mainly enhancing

NMR signals but rather measuring distances between spins

NMR: NOE

RF

RF

RF

A B

regular spectrum

NOE, small molecule

NOE, large molecule

Nuclear Overhauser Effect

RF

RF

RF

A B

describes the change in intensity of a signal due to the NOE


eq

eq

M

)M(Mη

Energy Level Diagram

W1A

W1A

W1B

W1B

A0 = B0 = D

W0

W2

baab

bb

aa

With population differencesfor the A and B transitionsin the undisturbed system:

W0 and W2 involve simultan-eous transitions of spins Aand B.Spins relax together in this mechanism.When spin A is off-equilibrium spin B will feel it.Difference of W0 and W2 is important

A

A

A

A

W2 > W0

small molecules

W0 > W2

large molecules

W1A

W1A

W1B

W1B

A0 = B0 = D

A = 1.5 D

A = 0.5 D

W2

W0

A

A

A = A0 = DB = 0


Nuclear Overhauser EffectIn practice we find the NOE ranging from +0.5 for small up to -1.0 for large moleculesSign of NOE depends on whether W0 (minus) or W2 (plus) is dominating

0.5

0.0

-0.5

-1.0

0.01 0.1 1.0 10 100

w0tcfasttumbling

slowtumbling

tc = rotational correlation time (size of molecule)

r = distance between the two corresponding atoms

Distances from NOEs

tc

r6 ~

tc

r6

ref

=tc

ref. r6

ref

tc tcref

refr = rref 6

Distances from NOEs

Application for NOEs

• Information about short 1H-1H-distances in molecules (< 5Å)

• Translated into distance-constraints applied in Molecular Simulations

• Main source of structural information in NMR

• Information about short 1H-1H-distances in molecules (< 5Å)

• Translated into distance-constraints applied in Molecular Simulations

• Main source of structural information in NMR

• It will be explained how it works

Application for NOEs

NMR: 2D-NMR

NMR: 2D-NMRWhy is 1D (just NMR spectrum) not enough?

1-Dimensional NMR

1D FT-NMR(simplest case)

preparation - detection

S(t)FT

S(w)

A 1D-Spectrum of a Protein

For large proteins it is really hard to assign NMR signals and to obtain quantitative information from the spectra!Too many peaks Spectrum is a mess!

2-dimensional NMR

2D FT-NMR

Preparation - evolution - mixing - detection

t2 – direct domain; t1 – indirect domain

t1 tm t2

S(t1,t2)FT1, FT2

S(w1,w2)

A 2D-Spectrum of a Protein

A Signal of a 2D Spectrum

Contour plot of the same Signal

Compare: Topographical map (lines of equal height)

Now let us see how it worksHow to get to this second dimension???

t1

The size of the signal depends onthe evolution in t1: the signal is said to be 'modulated' with w1

For simplicity we look at a single frequency wwhich is the same in t1 and in t2 (no mixing)!

t2=0

FT (t2)

t2

The Second Time Domain

t2=0

FT (t2)

t2 t1 FT (t1)

w2

w1

The Second Time Domain

The SCOTCH Experiment

Spin COherence Transfer in (photo) CHemical reactions

Reaction A B with a proton at wA in A which resonates at wB in B. hn

t1 t2light

The corresponding pulse sequence

t1 t2light

The proton's magnetization is in t1

modulated with the frequency wA. After the light pulse, the same proton evolves with wB.

Subsequent FT of the both time domains results in a 2D spectrum with a peak at wA in F1 and wB in F2:


t1 t2light

The proton's magnetization is in t1

modulated with the frequency wA. After the light pulse, the same proton evolves with wB.

If A would not completely be converted to B by the light pulse, we would be able to observe a diagonal peak ofA as well:


Preparation - evolution - mixing - detection

t1 tm t2

In the mixing period the frequency modulationof one nucleus is transferred to another one!

General scheme of 2D NMR experiment

No mixing (tm = 0): Only diagonal peaks!Boring case

Mixing (tm > 0): We get cross correlated peaks (cross peaks)!Interesting case

The Mixingperiod

Some 2D NMR experiments: COSY and NOESY

SY = SpectroscopY (always in NMR)COSY = COrrelation SYNOESY = NOE SY

For both techniques there are also hetero-nuclear versions (for instance, proton-carbon, proton-nitrogen)

One can use other methods to obtain cross-peaks and acquire specific information on the spin system

One can go from 2D-NMR to 3D-NMR and even further

COSY experiment

COSY: J-coupling (through bond connectivities of neigh-boring atoms, max. ~3 bonds)

t1 t2

π/2 π/2

COSY experiment


t1 t2

How does it work?

Effect of the chemical shift:I1x I1xcos(ω1t)+I1ysin(ω1t)

Effect of J-coupling with spin 2:I1x I1xcos(J12t)+I1yI2zsin(J12t)

Why I1yI2z term?

y

x

β α

y

x

β α

x-component changes in the usual way; y-component is given by the population difference of the α- and β-states of spin 2, which is I2z

π/2 π/2

COSY experiment


t1 t2

How does it work?

I1z I1x I1y I1y I1x

x-magnetization stays on spin 1The efficiency of this pathway issin(ω1t1)cos(J12t1)sin(ω1t2)cos(J12t2)Diagonal peak will appear in the COSY-spectrum

π/2y t1 π/2y t2

π/2 π/2

COSY experiment


t1 t2

How does it work?

I1z I1x –2I1xI2z 2I1zI2x I2x

x-magnetization went from spin 1 to spin 2The efficiency of transfer issin(ω1t1)sin(J12t1) sin(ω2t2)sin(J12t2)Cross-peak will appear in the COSY-spectrumCross-peak is the direct evidence for J-coupling

π/2y J12 π/2y J12

Gain is two-fold:(1) Spectral resolution is increased because peaks become resolved in 2D;(2) Knowledge on additional coherence pathways can be obtained.

π/2 π/2

COSY experiment


t1 t2Result for more than 2 spins

When the spins are scalar coupled cross-peak will appearIn 2D peaks, which overlap in 1D-spectrum, become resolved

ω1→ω 2→

Ω1 Ω2 Ω3 Ω4

π/2 π/2

NOESY experiment

NOESY: dipolar couplings (through Space, NOEs give distances)

tm t2Cross-peaks come not from J but from NOE during the mixing period

t1

How does it work?

I1z –I1y –I1y –I1z –I2z I2y -I2x

x-magnetization went from spin 1 to spin 2The efficiency of transfer is different from the COSY casesin(ω1t1)sin(J12t1) sin(ω2t2)sin(J12t2)Cross-peak will appear in the NOESY-spectrumCross-peak gives information on NOE distance between the spins

π/2x t1 π/2x NOE π/2x t2

π/2 π/2 π/2

Some 2D NMR experiments: COSY and NOESY

SY = SpectroscopY (always in NMR)COSY = COrrelation SYNOESY = NOE SY

For both techniques there are also hetero-nuclear versions (for instance, proton-carbon, proton-nitrogen)

One can use other methods to obtain cross-peaks and acquire specific information on the spin system

One can go from 2D-NMR to 3D-NMR and even further

INEPT experiment not yet 2D, but often used in 2DNMR signal is proportional to the γ-ratio

4 times higher signals for protons than for 13C; even 10 higher than for 15N

Possible improvement is polarization transfer 1H→X-spin

NOE is not (always) the best solution: coherent mechanisms work better

http://www.chem.queensu.ca/facilities/nmr/nmr/webcourse/inept.htm

INEPT experiment not yet 2D, but often used in 2DNMR signal is proportional to the γ-ratio

4 times higher signals for protons than for 13C; even 10 higher than for 15N

Possible improvement is polarization transfer 1H→X-spin

NOE is not (always) the best solution: coherent mechanism and proper pulsing work better

INEPT=Insensitive Nuclei Enhanced by Polarization Transfer

http://www.chem.queensu.ca/facilities/nmr/nmr/webcourse/inept.htm

INEPT experiment: explanation

All spins are along x

τ

t2

τ

π/2 π π/2

π/2 π π/2

1H

X

INEPT: transferring polarization from proton to X-nucleus

y

x

β α


For τ=1/4J the angle between spins is 90-degree

τ

t2

τ

π/2 π π/2

π/2 π π/2

1H

X


y

x

β α


Components are flip by protons pulseTheir colors are exchanged by X-nucleus pulse

τ

t2

τ

π/2 π π/2

π/2 π π/2

1H

X


y

x

βα


Spins are along y for τ=1/4J

The last proton pulse results in one component positive and one negativeReminder: first both were positive

τ

t2

τ

π/2 π π/2

π/2 π π/2

1H

X


y

x

βα

INEPT experiment: explanationResulting populations

Now the final pulse for X-nucleus does the detectionGain is given by the ratio of gammasGain can be further increased when NMR of X is detected via protons

τ

t2

τ

π/2 π π/2

π/2 π π/2

1H

X


Pulsing really makes possible many nice tricks with the spins

The rest of 2D-NMR will be given by Prof. Robert Kaptein

The rest is:(i) Other methods;(ii) Their applications to proteins.

Documents

NMR: Relaxation Measurements. How to measure relaxation rates? T 1 : Longitudinal or spin-lattice relaxation. M z is restored, the system goes back to