NMR relaxation. BPP Theory 4th NMR Meets Biology Meeting NMR relaxation. BPP Theory Konstantin Ivanov (Novosibirsk, Russia) Khajuraho, India, 16-21 December 2018

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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’

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’

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’

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

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

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

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.