1. NMR relaxation. BPP Theory
4th NMR Meets Biology Meeting
NMR relaxation. BPP Theory
Konstantin Ivanov (Novosibirsk, Russia)
Khajuraho, India, December 2018

2. Outline Phenomenological discussion, Bloch equations;
T1 and T2 relaxation;
Concepts from quantum mechanics;
Relaxation measurements;
Other relaxation times.

3. Macroscopic spin magnetization
Up to now we discussed a single spin ½, which is never the case in NMR
At thermal equilibrium we have almost the same amount of spins pointing up and down: the energy gap between the spin levels is much smaller than kT
We work with net magnetization of all spins; at equilibrium this is a vector parallel to B0 (longitudinal magnetization)
However, we do not measure the longitudinal component, but rotate M with RF pulses to obtain transverse magnetization and measured the signal from M┴ B0
Nuclear
paramagnetism:
the induced field is parallel to B0

4. Free precession We write down equations describing spin precession
Now we can describe simplest NMR experiments. Example:
Mz is flipped by 90 degrees by a resonant RF-pulse: φ=ω1τp=π/2
It starts rotating about the z-axis and decaying with T2
We detect My (or Mx) and collect the Free Induction Decay (FID) X Y Z
Never ending oscillations!
ω1τp t 1 -1
In the end, transverse magnetization must disappear
Free Induction Decay

5. T1 and T2 relaxation
Relaxation is a process, which brings a system to thermal equilibrium. Physical reason: fluctuating interaction of spins with molecular surrounding
For spins this means that Mz=M||=Meq and M┴=0
There are two processes, which are responsible for relaxation
Longitudinal, T1, relaxation: Meq is reached at t~T1:
Transverse, T2, relaxation: magnetization decays to zero at t~T1:
Generally, T1≠T2.
Taking all that into account we can write down equations taking into account precession and relaxation

6. Bloch equations and FID
We write down equations describing precession and add relaxation terms
Now we can describe simplest NMR experiments. Example:
Mz is flipped by 90 degrees by a resonant RF-pulse: φ=ω1τp=π/2
It starts rotating about the z-axis and decaying with T2
We detect My (or Mx) and collect the FID X Y Z
ω1τp t 1 -1
Oscillations decay
Mz recovers
Free Induction Decay

7. Origin of T1-relaxation
T1-relaxation: precession in the B0 field and a small fluctuating field Bf (t)
The precession cone is moving
Eventually, the spin can even flip
Spin flips up-to-down and down-to-up have slightly different probability (Bolztmann law!): Mz goes to Meq≠0
General expression for the transition rate B0 B0
B0+Bf (t)
Noise spectral density at the transition frequency
τc is the motional correlation time

8. Origin of T2-relaxation
T2-relaxation: kicks from the environment disturb the precession
Different spins precess differently and transverse net magnetization is gone
Generally the T2-rate
Dephasing
Two contributions:
Adiabatic and non-adiabatic (T1-related)

9. Origin of fluctuating interactions
Random events
Translational diffusion
Rotational diffusion
Vibrations
Conformational transitions
Chemical exchange
Example: molecular tumbling (reorientation)
Molecules are constantly moving!
Stokes’ law
(η is the viscosity, V is the volume)
Typical τc values:
1.7 ps for water
~100 ps for amino acids in aqueous solution

10. NMR relaxation mechanisms
Interaction terms in the spin Hamiltonian, which fluctuate upon molecular motion cause relaxation
In the case of molecular tumbling the interactions, which cause relaxation, are anisotropic interactions
In the NMR case, these are
Chemical shift anisotropy
Dipole-dipole interaction
Quadrupolar interaction
Upon vibrations, also scalar couplings can fluctuate
Relaxation can also be due to interactions with paramagnetic molecules added to the sample

11. Noise spectral density
Simple example: spins relaxed by fluctuating local fields, Bf (t)=0
However, the auto-correlation function is non-zero
Typical assumption
Lorentzian-like noise spectral density
time t
Exponential auto-correlation function
τc comes into play

12. From the Bloch to Redfield theory via BPP
Bloch theory: phenomenological equations with two relaxation times
Relaxation of multi-level systems is (generally) beyond the Bloch theory
BPP (Bloembergen, Purcell, Pound) theory:
Classical transition rate theory;
Relaxation times are derived from consideration of spin-lattice interactions, which fluctuate due to molecular motion;
Exponential auto-correlation function is used; τc is an input parameter.
Bloch-Wangness-Redfield theory:
Refinement of BPP: complex relaxation processes can be treated (relaxation of populations and coherences);
Semiclassical theory: classical molecular motions affect quantum spin properties;
New language: relaxation super-operator instead of the Hamiltonain operator

13. A bit of theory Classical transition rate theory:
Transition probability is
In addition to Pi we introduce ΔPi=Pi–Pi(eq) and write down the eqs.
We introduce:
Population difference, Pα–Pβ
Transition probability and relaxation rates β α
Noise spectral density Coupling matrix element
k takes account for relaxation to equilibrium populations at a fixed lattice temperature

14. A bit of theory At equilibrium, we obtain
The equations can be written as
Equation for longitudinal magnetization, Mz=Pα–Pβ, is as follows
The T1-relaxation rate is β α

15. Expressions for T1 and T2
Simple example: spins relaxed by fluctuating local fields, Bf (t)=0
However, the auto-correlation function is non-zero
General expressions
This dependence is explained by the J(ω) behavior

16. T1-measurement: inversion-recovery
Determination of T1 is often quite important as well
Standard method is inversion-recovery
First we turn the spin(s) by pulse (usually p/2 or p) and then look how system goes back to equilibrium (recovers Z-magnetization). If the pulse is a p-pulse magnetization will be inverted (maximal variation of m-n) and then recovered
Equation for Mz is as follows:
The kinetic trace (t-dependence) gives T1-time
To detect magnetization at time t one more p/2-pulse is applied, the sequence is then px - t (variable) - px/2 - measurement
The sequence should be repeated with different delays t t

17. Problem with T2 measurements: inhomogeneous linewidth
Problem: we need to discriminate two contributions to T2
Decay of the NMR signal also proceeds due to static inhomogeneities in the precession frequency w0. This can be due to external field gradients and local static interactions.
Resulting rate the signal decay
The first two contributions are the same for all the molecules and thus define the homogeneous linewidth.
The last contribution defines the inhomogeneous linewidth.
In solids usually w D
NMR
spectrum
Reason: δω·δt~2π

18. T2-measurement: spin echo
Large inhomogeneous linewidth means very fast dephasing of the spin
However, dephased magnetization can be focused back by pulses
Explanation: let us divide system into isochromates having the same frequency w0. Their offsets are Dw=w0–w. At certain time they all have different phases
But at t=2t all have the same phase: there is an ‘echo’!
The spin echo signal decays with T2 (real, not apparent T2)
p/2 p
echo t 2t
pulse sequence p/2x - t - px X Y
t=0
j=0
j=Dwt
j=p–Dwt+Dw(t–t)
t=2t
j=p

19. T2-measurement: spin echo
Basic spin echo sequence
Improvement: recording the entire relaxation time trace in a single experiment
Further improvement: CPMG (CP-Meiboom-Gill) sequence: change the phase of the 180-degree pulses. The sequence is 90x-180y-180y-180y-…
Question: how does it work and why do we need such a modification?
t/2
180o
90o T2
180o
90o
180o
180o
Carr-Purcell (CP) sequence

20. Other relaxation times
T1 in the rotation frame, T1ρ
The first pulse generates y-magnetization. The CW pulse applied along y “locks” magnetization: spins are parallel to the B1-field.
After turning off the CW-field we can detect transverse magnetization.
The duration of the spin-locking field is varied. Signal decays exponentially
For isotropic liquids T1ρ is the same as T2, unless there is chemical exchange
Comparison of T1, T2 and T1ρ: t
90x
CWy
Spin-locking
T1ρ is not the same as T2
Analysis of all three times provides important information on molecular mobility

21. Cross-relaxation
Cross-relaxation comes into play for two or more spins
Energy levels and relaxation transitions
single-quantum (W1I and W1S)
zero-quantum (W0) and double-quantum (W2)
The set of equations becomes
To solve these equations, it is better to introduce spin polarizations
W1S
W1I W0 W2
W1I
W1S

22. Cross-relaxation, Solomon equations
By introducing spin polarization
The set of equations becomes
Spin not only relax to equilibrium but also exchange polarization: cross-relaxation occurs
Cross-relaxation comes into play when (W2–W0)≠0
W1I
W1S W0 W2
ρI, ρS: auto-relaxation
σ: cross-relaxation

23. Nuclear Overhauser effect = NOE
Basic equations (Solomon equations)
Consequence is NOE: when one spin is off-equilibrium, the other spin feel that (through the transitions W0 and W2, which flip both spins)
If we excite one spin, polarization of the other spin will be altered, as described by η
Sign of the effect (increase or decrease of the line intensity) can be positive or negative)
NOE enables distance measurements: η~1/r6 for dipolar relaxation
W1I
W1S W0 W2 RF
S I

24. Overhauser effect: cross-relaxation
Relaxation processes in the system
w1 – nuclear T1-relaxation
w1' – electronic T1-relaxation
w0 – cross-relaxation, zero-quantum
w2 – cross-relaxation, double-quantum
In addition: EPR pumping
w1’ w1 w2 w0
Pumping EPR transitions equalizes populations for the two pairs of states
The fastest relaxation pathway, pure electronic relaxation, is blocked
Relaxation occurs via other channels: electron relaxes together with nuclei
Electron Boltzmann factor is transferred to the nuclei

25. Overhauser effect: a cartoon
Let us have only
w1' – electronic T1-relaxation
w2 – cross-relaxation, double-quantum
EPR pumping
Thermal equilibrium; pumping is off
w1’ w2
w1’

26. Overhauser effect: a cartoon
Let us have only
w1' – electronic T1-relaxation
w2 – cross-relaxation, double-quantum
EPR pumping
Pumping is on: pair-wise equal populations
w1’ w2
w1’

27. Overhauser effect: a cartoon
Let us have only
w1' – electronic T1-relaxation
w2 – cross-relaxation, double-quantum
EPR pumping
Cross-relaxation: For the DQ-transition the electronic Boltzmann factor comes into play
w1’ w2
w1’

28. Overhauser effect: a cartoon
Let us have only
w1' – electronic T1-relaxation
w2 – cross-relaxation, double-quantum
EPR pumping
Cross-relaxation: For the DQ-transition the electronic Boltzmann factor comes into play
w1’ w2
w1’
The nucleus has acquired polarization given by the electronic Boltzmann factor

29. Cross-correlated relaxation
Small modification to the previous scheme
Rates of single-quantum transitions are unequal
This can happen when there are two fluctuating interactions, which are correlated, e.g., CSA and DDI
The set of rate equations becomes
We add one more equation
W1S(2)
W1I(2) W0 W2
W1I(1)
W1S(1)
We introduce two spin order

30. Cross-correlated relaxation
Three kinds of spin order become mixed
Spectral manifestation: different lines of the NMR multiplet relax with different time constants (the IzSz terms are generated)
Applications: when two relaxation mechanisms are of the same size:
one of the rates becomes small. The same is true for transverse relaxation: one of the NMR lines is narrow. This is very important for NMR of large proteins.
This property is utilized in Transverse relaxation optimized spectroscopy, TROSY
W1I(2)
W1I(1)
W1S(2)
W1S(1) W0 W2

31. Summary Main concepts of relaxation are introduced;
T1 and T2 relaxation are discussed;
Basic relaxation measurements are discussed;
Some other types of relaxation are introduced.