Introduction to Relaxation Theory James Keeler

EUROMAR Zürich, 24 Introduction to Relaxation Theory James Keeler University of Cambridge Department of Chemistry What is relaxation? Why might it be interesting? relaxation is the process which drives the spins to equilibrium (equilibrium z-magnetization, no xy-magnetization) a natural phenomenon, driven by molecular motion unusually slow in NMR useful probe of molecular motion the Nuclear Overhauser Effect (NOE) arises because of relaxation; estimation of distance Introduction and outline 2 Further reading Outline. How relaxation arises 2. Describing random motion the correlation time for more detail: James Keeler, Understanding NMR Spectroscopy, 2nd edit., Wiley 2 (Chapter 9) a PDF of this presentation is available to download at www-keeler.ch.cam.ac.uk 3. Motional regimes 4. Relaxation in terms of populations 5. Relaxation of a spin pair 6. Solomon equations and relaxation rate constants 7. Nuclear overhauser effect (NOE) 8. Transverse relaxation Introduction and outline 3 Introduction and outline 4

Behaviour of individual magnetic moments Local fields x z y bulk magnetization result of sum of magnetic moments of each spin each moment behaves as the overall magnetization i.e. precesses about z, rotated away from z by transverse fields oscillating at Larmor frequency pulse affects all the same way, but relaxation caused by local fields which are different for each spin B loc A z r B e.g. spin B generates magnetic field at A a local field only significant over a short distance local field varies in size and direction according to length and orientation of A B vector local field is random How relaxation arises 5 How relaxation arises 6 How random fields drive the system to equilibrium random fields fluctuating at close to the Larmor frequency should drive spins to equilibrium + (a) (b) (c) (d) Coming to equilibrium with the lattice random fields would appear to drive the z-magnetization to zero incorrect: equilibrium z-magnetization is finite z-component since surroundings are large and at equilibrium, greater chance of loosing energy to surroundings than gaining energy result is finite z-magnetization at equilibrium - time process is called longitudinal or spin lattice relaxation M z = 2 M z = 6 M z = 2 M z = How relaxation arises 7 How relaxation arises 8

Relaxation mechanisms Describing random motion the correlation time dipolar: local field goes as γ γ 2 /r 3 z B loc r B in solution, molecular collisions change orientation on about right timescale for relaxation chemical shift anisotropy (CSA): local field goes as B and typically depends on shift range B A B loc each collision only alters orientation by a small amount rotational diffusion correlation time, τ c, is average time it takes a molecule to move through radian correlation time describes the timescale of the random motion paramagnetic species (e.g. dissolved oxygen) How relaxation arises 9 Describing random motion the correlation time Rotational diffusion The correlation function representative molecules undergoing rotational diffusion (a) θ - molecule molecule 2 molecule 3 molecule 4 G(t, τ) = [ Bloc, (t)b loc, (t + τ) + B loc,2 (t)b loc,2 (t + τ) +... ] N = N B loc,i (t)b loc,i (t + τ) N i= = B loc (t)b loc (t + τ) (b) θ - time average of product of local field at time t with that at time (t + τ) (a) has longer correlation time than (b) usually only depends on τ: stationary random function Describing random motion the correlation time Describing random motion the correlation time 2

The correlation function Typical correlation function (a) τ = (b) τ < τ c (c) + τ >> τ c maximum at τ = G() = B loc (t)b loc (t) B loc () - = B 2 loc simplest form is an exponential + G(τ) = B 2 loc exp ( τ /τ c) B loc (τ) typical behaviour - B 2 loc τ c = τ min B loc () B loc (τ) + - G(τ) τ c = 2τ min τ c = 4τ min τ Describing random motion the correlation time 3 Describing random motion the correlation time 4 Reduced correlation function The spectral density need to know amount of motion at Larmor frequency time dependent part (max value ) g(τ) = exp ( τ /τ c ) Fourier transform of function of time, G(τ), gives function of frequency, J(ω) G(τ) Fourier transform J(ω) hence J(ω) gives amount of motion at frequency ω G(τ) = B 2 loc g(τ) recall that we need motion at the Larmor frequency to cause relaxation; g(τ) describes time dependence B 2 loc exp ( τ /τ c) FT B 2 2τ c loc + ω 2 τ 2 c J(ω) = B 2 2τ c loc + ω 2 τ 2 c Describing random motion the correlation time 5 Describing random motion the correlation time 6

Spectral density: interpretation Spectral density at Larmor frequency τ c = τ min J(ω ) J(ω) τ c = 2τ min τ c = 4τ min τ c = /ω τ c J(ω ) plotted against τ c ω maximum when τ c = /ω ; fastest relaxation with this value reduced spectral density, j(ω) area under curve is independent of τ c g(τ) FT j(ω) the shorter τ c, the higher the frequency present in motion exp ( τ /τ c ) FT 2τ c + ω 2 τ 2 c always has maximum value at zero frequency hence J(ω) = B 2 loc j(ω) Describing random motion the correlation time 8 Describing random motion the correlation time 7 Motional regimes Summary fast motion ω τ c << 2τ c j(ω ) = + ω 2 τ2 c fast motion: j(ω ) independent of frequency slow motion ω τ c >> slow motion: j(ω ) = 2 ω 2 τ c j(ω ) = 2τ c small molecules: τ c ps fast motion small protein: τ c ns slow motion or j(ω ) = j() ω 2 τ2 c rotational diffusion gives motion suitable for NMR relaxation rotational diffusion characterised by the correlation time τ c spectral density gives the frequency distribution of the motion e.g. J(ω) depends on τ c J(ω) = B 2 2τ c loc + ω 2 τ 2 c rate of longitudinal relaxation depends on spectral density at ω ; max. when ω τ c = Motional regimes 9 Motional regimes 2

Populations Rate equations useful to think in terms of populations of the spin states (energy levels) α ( spin up ) and β ( spin down ) n β z-magnetization due to a population difference between these two states M z = 2 γ(n α n β ) n α W α β W β α... just like chemical kinetics W are rate constants n α population of α state; n β population of β state Boltzmann distribution gives equilibrium magnetization omit constants M z = γ2 2 NB 4k B T M z = n α n β and M z = n α n β indicates equilibrium values rate from α to β = W α β n α rate from β to α = W β α n β rate of change of n α = +W β α n β } {{ } increase in n α rate of change of n β = +W α β n α } {{ } increase in n β W α β n α } {{ } decrease in n α W β α n β } {{ } decrease in n β Relaxation in terms of populations 2 Relaxation in terms of populations 22 Problem with the rate equations Modifying the rate equations n β W β α W α β n α at equilibrium, no change in population with time = W β α n β W α β n α = +W α β n α W β α n β instead of rate of change of n α = +W β α n β W α β n α rate of change of n β = +W α β n α W β α n β set W β α and W α β equal to W αβ and write hence rate of change of n α =W αβ (n β n β ) W αβ (n α n α) n α n β = W β α W α β rate of change of n β = W αβ (n β n β ) + W αβ (n α n α) rate depends on deviation from equilibrium population simple theory predicts W β α = W α β, hence n α = n β, which is wrong need more advanced theory, or... Relaxation in terms of populations 23 Relaxation in terms of populations 24

Relaxation in terms of populations Writing the magnetization in terms of the populations recall M z = n α n β, so rate of change of M z = rate of change of n α rate of change of n β using rate of change of n α =W αβ (n β n β ) W αβ (n α n α) rate of change of n β = W αβ (n β n β ) + W αβ (n α n α) gives rate of change of M z = 2W αβ (n β n β ) 2W αβ(n α n α) [ = 2W αβ (nα n β ) (n α n β )] = 2W αβ (M z Mz ) we have usually written or rate of change of M z = 2W αβ (M z M z ) dm z (t) = R z [ Mz (t) M z with R z = 2W αβ, longitudinal relaxation rate constant dm z (t) = T [ Mz (t) M z with T = /R z, time constant for longitudinal relaxation ] ] Relaxation in terms of populations 25 Relaxation in terms of populations 26 Longitudinal relaxation Longitudinal relaxation we have dm z (t) = R z [ Mz (t) M z implies that the rate of change of M z is proportional to the deviation of M z from the equilibrium value M z implies that M z tends to M z can integrate using M z = M z () at time t = to give ] M z (t)/m z (a)..5. -.5 -. different initial conditions time different rate constants (b) time M z (t) = [ M z () M z ] exp ( Rz t) + M z note M z always tends to equilibrium value Relaxation in terms of populations 27 Relaxation in terms of populations 28

Two spins: energy levels and transition rates Rate equations for the populations and z-magn. ββ 4 ββ 4 W (,β) W (2,β) W (,β) W (2,β) αβ 2 W βα 3 αβ 2 W βα 3 W (2,α) W 2 W (,α) W (2,α) W 2 W (,α) αα αα two spins, four energy levels dipolar interaction causes relaxation-induced transitions between any two levels rate constants W M, M gives change in M note W (,α) and W (,β) Relaxation of a spin pair 29 as before, look at gain and loss processes for level dn = W (2,α) (n n ) W(,α) (n n ) W 2(n n } {{ } ) loss from level + W (2,α) (n 2 n 2 } {{ } ) + W (,α) (n 3 n 3 ) + W 2 (n 4 n } {{ }} {{ 4 } ) gain from level 2 gain from level 3 gain from level 4 Relaxation of a spin pair 3 Rate equations for the populations and z-magn. define z-magn. for spin as population difference across the spin- transitions 3 and 2 4 similarly for spin 2 I z = (n n 3 ) + (n 2 n 4 ) I 2z = (n n 2 ) + (n 3 n 4 ) also need the difference in the population difference across the spin levels 2I z I 2z = (n n 3 ) (n 2 n 4 ) or 2I z I 2z = (n n 2 ) (n 3 n 4 ) and equilibrium values (2I z I 2z = at equil.) I z = n n 3 + n 2 n 4 I 2z = n n 2 + n 3 n 4 Relaxation of a spin pair 3 Rate equations for the populations and z-magn. after much algebra di z di 2z d 2I z I 2z = R () z (I z Iz ) σ 2(I 2z I2z ) () 2I z I 2z = σ 2 (I z Iz ) R(2) z (I 2z I2z ) (2) 2I z I 2z = () (I z Iz ) (2) (I 2z I2z ) R(,2) z 2I z I 2z rate constants in terms of the W R () z = W (,α) + W (,β) + W 2 + W R (2) z = W (2,α) + W (2,β) + W 2 + W σ 2 = W 2 W () = W (,α) W (,β) (2) = W (2,α) W (2,β) R (,2) z = W (,α) + W (,β) + W (2,α) + W (2,β) Relaxation of a spin pair 32

Rate equations for the populations and z-magn. Rate equations for the populations and z-magn. R z () I z σ 2 R z (2) I 2z R z () I z σ 2 R z (2) I 2z () (2) () (2) 2I z I 2z R z (,2) 2I z I 2z R z (,2) di z di 2z d 2I z I 2z = R () z (I z Iz ) σ 2(I 2z I2z ) () 2I z I 2z = σ 2 (I z Iz ) R(2) z (I 2z I2z ) (2) 2I z I 2z = () (I z Iz ) (2) (I 2z I2z ) R(,2) z 2I z I 2z Relaxation of a spin pair 33 pure dipolar relaxation: W (,α) = W (,β), W (2,α) = W (2,β) hence () = and (2) = 2I z I 2z not connected to I z or I 2z Relaxation of a spin pair 34 Solomon equations Solomon equations with di z di 2z d 2I z I 2z = R () z (I z I z ) σ 2(I 2z I 2z ) = σ 2 (I z I z ) R(2) z (I 2z I 2z ) = R (,2) z 2I z I 2z di z di 2z d 2I z I 2z = R () z (I z I z ) σ 2(I 2z I 2z ) = σ 2 (I z I z ) R(2) z (I 2z I 2z ) = R (,2) z 2I z I 2z R () z is rate constant for self relaxation of spin R () z = 2W () + W 2 + W R (2) z = 2W (2) + W 2 + W σ 2 = W 2 W R (,2) z = 2W () + 2W (2) likewise R (2) z for spin 2 σ 2 is cross-relaxation rate constant between spins and 2 cross relaxation connects the z-magnetizations of the two spins Solomon equations and relaxation rate constants 35 Solomon equations and relaxation rate constants 36

Relaxation rate constants Relaxation rate constants: dipolar dipolar only detailed theory shows rate constants W ij always given by expression of the form W ij = A ij Y 2 j(ω ij ) A ij is a quantum mechanical factor Y 2 relates to magnitude of local field (always squared) depends on e.g. distance between spins, γ j(ω ij ) reduced spectral density at ω ij, the transition frequency between the two levels W () = 3 4 b2 j(ω, ) W (2) = 3 4 b2 j(ω,2 ) W 2 = 3 b2 j(ω, + ω,2 ) W = 2 b2 j(ω, ω,2 ) magnitude factor hence b = µ γ γ 2 4πr 3 R () z = b 2 [ 3 2 j(ω,) + 3 j(ω, + ω,2 ) + 2 j(ω, ω,2 ) ] R (2) z = b 2 [ 3 2 j(ω,2) + 3 j(ω, + ω,2 ) + 2 j(ω, ω,2 ) ] σ 2 = b 2 [ 3 j(ω, + ω,2 ) 2 j(ω, ω,2 ) ] R (,2) z = b 2 [ 3 2 j(ω,) + 3 2 j(ω,2) ] Solomon equations and relaxation rate constants 37 Solomon equations and relaxation rate constants 38 Cross relaxation in the two motional regimes Cross relaxation in the two motional regimes homonuclear: ω, = ω,2 ω σ 2 = b 2 3 j(2ω 2 ) b 2 } {{ }} {{ j() } W 2 W fast motion j(ω) = 2τ c fast motion: σ 2 = b 2 3 j(2ω ) } {{ } W 2 2 b 2 } {{ j() } W = b 2 3 2τ c b 2 2 2τ c = 2 b2 τ c σ 2 = b 2 3 j(2ω 2 ) b 2 } {{ }} {{ j() } W 2 W slow motion j() = 2τ c, j(2ω ) negligible in comparison slow motion: σ 2 = b 2 3 j(2ω 2 ) b 2 } {{ }} {{ j() } W 2 W = b 2 2 2τ c = b2 τ c σ 2 negative in this limit σ 2 positive in this limit Solomon equations and relaxation rate constants 39 Solomon equations and relaxation rate constants 4

Cross relaxation as a function of τ c The nuclear overhauser effect (NOE) Solomon equation σ 2 τ c / ps 2 3 4 5 6 di z = R () z (I z I z ) σ 2(I 2z I 2z ) implies that if spin 2 not at equilibrium, spin will be affected computed for proton at 5 MHz cross over at 36 ps in this case ω τ c = 5 4 but only if cross-relaxation rate constant σ 2 cross relaxation is a feature of dipolar relaxation, hence detection of cross relaxation implies dipolar relaxation i.e. nearby spins origin of Nuclear Overhauser Effect Solomon equations and relaxation rate constants 4 Nuclear overhauser effect (NOE) 42 The transient NOE experiment Difference spectroscopy reveals the NOE (a) 8 spin 2 τ Ω Ω 2 (b) (a) (a) perturb spin 2 with a selective inversion pulse (b) wait time τ for cross relaxation to occur 9 pulse to give observable signal (c) = (a) - (b) (b) repeat without inversion pulse (reference spectrum) compute difference spectrum (a) (b) to reveal changes can analyze the experiment using the Solomon equations Nuclear overhauser effect (NOE) 43 difference reveals NOE Nuclear overhauser effect (NOE) 44

Transverse relaxation: non-secular contribution Transverse relaxation: secular contribution transverse relaxation is decay of xy-components of magnetization; determines rate of decay of FID and hence linewih transverse local fields, oscillating near to the Larmor frequency, cause longitudinal relaxation such fields can also affect the x- and y-components of individual magnetic moments, and therefore also cause transverse relaxation the z-component of the local field will cause a change in the (local) Larmor frequency individual magnetic moments will precess at slightly different Larmor frequencies and so get out of step with one another result is a decay in the transverse magnetization called the secular contribution to transverse relaxation called the non-secular contribution to transverse relaxation Transverse relaxation 45 Transverse relaxation 46 Transverse relaxation Chemical exchange F ax non-secular contribution to transverse relaxation: description similar to longitudinal relaxation; rate depends on j(ω ) secular contribution to transverse relaxation: rate depends on spectral density at zero frequency, j() chemical exchange is a useful analogy for the secular contribution F ax and F eq have different shifts: frequency difference between two resonances if rate constant for exchange is much less than the frequency difference two lines (slow exchange) F eq if rate constant for exchange is much greater than the frequency difference one line (fast exchange) Transverse relaxation 47 Transverse relaxation 48

Two-site chemical exchange Exchange processes from the point of view of single spins 5 k ex = s / 2 k ex = 5 s 5 5 k ex = s k ex = 25 s / 2 / 2 k ex = s k ex = 2 s consider behaviour of individual spins A k ex = s k ex = s B / 2 / 2 k ex = 5 s k ex = s either in environment A or B the spin jumps between them randomly shift difference 6 Hz initial broadening and then coalescence when exchange rate constant is 35 s the larger the exchange rate constant, the more frequent the jumps further increase in exchange rate constant results in a narrower line: exchange narrowing Transverse relaxation 49 Transverse relaxation 5 Simulation of two-site exchange Slow and fast exchange: interpretation (a) (a) (b) (c). time / s. spin spin 2 spin spin 2 (b) spin 3 spin 3 spin 4 spin 4 FID FID. time / s. time two cosine waves at. and.5 Hz; =.5 Hz A frequency B (a): observing for. s, can hardly see that the waves are at different frequencies slow intermediate fast Transverse relaxation 5 (b): observing for s, difference is clear phase difference over period τ is 2π τ; must be significant if frequencies are to be distinguished Transverse relaxation 52

Slow and fast exchange: interpretation The secular contribution to transverse relaxation (a). time / s. (b) (a) (b) (c) (d) (e) frequency. time / s. phase difference over period τ ex is 2π τ ex for significant phase difference τ ex >> (/ ) can only distinguish frequencies if τ ex >> (/ ); since τ ex = /k ex the condition for slow exchange is k ex << frequencies indistinguishable if τ ex << (/ ); the condition for fast exchange is k ex >> Transverse relaxation 53 continuous range of Larmor frequencies due to spread of local fields square profile in absence of motion: (a) molecular motion is fast compared to range of Larmor frequencies, so line is exchange narrowed: (e) Transverse relaxation 54 The secular contribution to transverse relaxation Secular and non-secular contribution to relaxation for A B, in fast exchange limit linewih π 2 AB 2k ex molecular motion: k ex /τ c ; AB W, wih of distribution of Larmor frequencies wih of narrowed line W 2 τ c W = khz, τ c = ps gives wih of Hz j() = 2τ c hence wih of narrowed line 2 W2 j() Summary non-secular: due to transverse local fields oscillating near to the Larmor frequency the same fluctuations cause longitudinal relaxation secular: due to a distribution of local fields along z giving a spread in Larmor frequencies molecular motion results in a linewih very much less than the spread of Larmor frequencies non-secular depends on j(ω ); secular depends on j() Transverse relaxation 55 Transverse relaxation 56

Relaxation by random fields Relaxation by random fields assume a randomly varying field with mean square value B 2 loc in all three directions R z = γ 2 B 2 loc j(ω ) R R xy transverse relaxation comparing R xy = 2 γ2 B 2 loc j() } {{ } secular + 2 γ2 B 2 loc j(ω ) } {{ } non-secular ω τ c = R z τ c R xy = 2 γ2 B 2 loc j() } {{ } secular + 2 R z }{{} non-secular i.e. non-secular part is precisely half of the overall longitudinal rate constant Transverse relaxation 57 Fast motion: R z = R xy Slow motion: R xy continues to increase due to secular term: J() τ c, but R z decreases as spectral density at Larmor frequency decreases Transverse relaxation 58 EUROMAR Zürich, 24 Introduction to Relaxation Theory The End Want more on the basic theory of NMR? Search for ANZMAG on YouTube The end 59