NMR Dynamics and Relaxation Günter Hempel MLU Halle, Institut für Physik, FG Festkörper-NMR 1
Introduction: Relaxation Two basic magnetic relaxation processes: Longitudinal relaxation: T 1 Relaxation Return to equilibrium Non-equilibrium states: Non-Boltzmannian level populations Transverse relaxation: T transverse phase correlations
Relaxation Thermal motion spin interactions Dipol-dipol interaction Chemical shift anisotropy Quadrupolar interaction Scalar coupling Spin-rotational interaction Nuclear spin system is forced to relax 3
Relaxation temperature dependence Two groups of relaxation times: ln T T 1, T 1ρ, T 1D, T SE, T eff T, T SE, T IS 1/T 4
Dipole-dipole interaction B 0 S 1 Detecting the resonance of the S spins in the neighbourhood of I we observe a freqency shift caused by the field of the I spin. I S B local = B 0 + B = B res = 0 + Local field > 0 for position 1 but < 0 for position. If I is antiparallel to B 0 then B and change sign. 5
Dipole-dipole interaction N S S N S N N S N S S N N S N S S N S N N S S N 6
Dipole-dipole interaction + thermal motion 7
Anisotropy of chemical shift x xx z B (1 ) B loc xx 0 z zz x B (1 ) B loc zz 0 The magnitude of the shielding of a nucleus depends on the spatial orientation of the electron cloud surrounding the nucleus. 8
Chemical Shift Anisotropy (CSA) B 0 z B loc x B 0X B 0Z B loc X, (1 XX) B 0X B loc Z, (1 ZZ) B 0Z H I σ B CSA 0 9
CSA + thermal motion 10
Quadrupolar interaction Nuclei with spin > 1/: Non-spherical charge distribution. electrical quadrupole moment eq + + + This interacts with the electric field gradient eq caused by the electron shell. I = 1/ I 1 H Qu µ 0 eqq 1 3IZ II 1 I I 4 4I I 1 Quadrupole coupling constant: C q = e qq/h 11
Quadrupolar interaction H NMR = H Z + H Q (+ H CS ) I=1 I=5/ Zeeman 1 st order H Q nd order H Q -h 0 m (hc q /40)(3cos -1) (9hC q /6400 0 ) H Z H Q -1 5(sin + sin 4 ) 0 3(sin 4 -sin ) +1 (sin 4 -sin ) observed splittings: 3 0 8 3cos 1 angle with B 0 -(sin 4 -sin ) -3(sin 4 -sin ) -5(sin + sin 4 ) 1
Scalar interaction H J I I J 1 Energy exchange between different nuclei Spin-rotation interaction H SR I C F Interaction of nuclear spin with the angular momentum of the molecule. 13
Effectiveness of the interactions I > ½: I = ½, I = ½, Quadrupolar interaction dominates (except on highly symmetric sites). small nuclei ( 1 H, 13 C, 9 Si), not too large fields: Dipolar interaction determines relaxation. large B 0 and/or larger nuclei: chemical-shift anisotropy can be the strongest mechanism. Spin-rotation interaction: Only detectable for small molecules and high temperatures Scalar interaction can be neglected as relaxation mechanism in almost all cases 14
Transitions between levels H H H Zeeman 't No fluctuation ω E / Medium fluctuation ω E / Fast fluctuation ω E / 15
Dephasing evolution evolution 16
Questions How we can obtain the relaxation time if the transition probability would be known? Master equations How to obtain the transition probability? time-dependent perturbation theory What is the frequency of a random process? description by correlation functions 17
Random functions Example: Two functions with values ±1 which are equally distributed, i.e. f = 0 Randomly changing sign. But they differ in the rate of change. f(t) 1 fs(t) 0-1 1-50 0 50 100 t f f (t) f(t) 0-1 -100-50 0 50 100 How to define a frequency of such a function? t 18
Correlation function Definition autocorrelation function: * K τ f t f t τ t Properties: τ 0 : K 0 f t (Mean square) τ : lim K τ f t (Square of the mean value) τ because: There is no correlation between f(t) and f(t+τ) hence the averaging of the product ia the product of the averages. 19
0
Numeric example: Resulting correlation function 1,0 0,5 K s () K() K f () 0,0-0,5-0 -10 0 10 0 1
Real systems Averaging over much more data points. Spectral density: () = K () τ J ω FT{ } Markov processes: 1,0 Exponential correlations τ c : correlation time () exp - τ K τ = K0 τc J() / x10-3 s 0,5 Example: τ c = 10-3 s Spectral density: K c J ω τ 0 1 ωτ c 0,0 0 4 6 log /( Hz)
Spectral densities Markov process: - -4-6 log J() -8-10 -1-14 c =10-3 s c =10-6 s c =10-9 s -16 0 4 6 8 10 log ( /) Other examples: See Noack, F., Nuclear Magnetic Relaxation Spectroscopy, in NMR Basic Principles and Progress Vol. 3, pp. 84, Springer 1971 3
Two-level system (One spin 1/) W AB W AB Level A: N A spins Level B: N B spins Magnetisation ΔN = N B N A Master equation: Solution: dn A dt dn B dt NW NW NW B BA A AB NW A AB B BA AB BA N N N N e 0 W W t ΔN 1 WAB T 1 W BA ΔN t 4
Four-level system (Two spins 1/) 1 3 W 13 W 1 Magnetisation N 3 N 1, because level does not contribute W1 dn 1 dt dn dt dn dt 3 Master equation: W N W N W N W N 1 1 13 1 1 13 3 4W N W N W N 1 1 1 1 3 W N W N W N W N 1 3 13 3 1 13 1 1 13 N N N N e Solution: 0 W W t 1 W W T 1 1 13 5
Time-dependent perturbation theory 1st order: Transition probability per time unit: 1 t W lim φ H ' t ' φ exp iω t' dt' ; ω t AB A B AB AB t 0 E A E B Transformation into a form where we can realize the correlation of H at different times: 1 t WAB FT KAB τ ; KAB τ H ' AB th ' AB * t τ 1 Particularly: Dipolar interaction within spin pairs (see above: W1 W13 ): T 1 1 J ω J ω T 1 1 0 13 0 ; J ω FT K τ AB AB 6
Dipolar Hamiltonian Two spins: H DD µ 4 r Ir Sr 0 IS 3 3 µ 4 r 0 3 A B C D E F r Dipolar alphabet A I S m 1 3cos z z 0 1 B 1 3cos IS IS 4 m 0 3 sin cos i C e ISz IzS m 1 3 sin cos i D e ISz IzS m 1 3 sin i E e I S m 4 3 sin i F e I S m 4 7
1 3 4 and two matrix elements: Basis set (order: I - S) Now we need H 1 and H 13. µ 0 3 µ 0 H' 1 C sincos e 3 3 4 r 4 4 r µ 0 3 µ 0 i H' 13 E sin e 3 3 4 r 4 4 r i Correlation functions: Time dependence is included in r,, and. We use the approach * t t F t F t F t exp t t µ 4 4 0 µ 0 4 4 1 6 13 6 1 3 1 3 H' t ; H' t 4 r 40 4 r 10 C Spectral density: 3 µ 0 4 4 1 C J1 0 6 10 4 r 1 C 0 C 1 C 6 µ 0 4 4 1 C J13 0 6 5 4 r 1 4 0 C exp exp it dt C 8
Result: BPP equations ln T 1 µ 0 C 4 C II ( 1) 6 T1 54 r 10C 140C T 1 1 1 µ 0 5C C II ( 1) 3 6 C T 54 r 1 0 C 14 0 C T 1 1 3 µ 0 C II ( 1) 6 T 54 r 1 4 1 1 C Bloembergen N, Purcell EM, Pound RV, Phys. Rev. 73 (1948) 679 Conditions: Isolated Pairs of equal spins; Sinle, isotropic motion (i.e. the correlation decays completely) τ C << T Asymptotic behaviour: High temperature ( 0 C << 1): 0 C = 0,615 1 C = 1/ T ln C µ 0 ( II1) 6 1 4 1 1 T T r Low temperature ( 0 C >> 1): 1 µ 0 4 1 II ( 1) 6 T 4 5 r C 1 0 C T : Third condition violated. 9
Summary and outlook Thermal motion Particular thermal motion spin interactions Creating transition probability reducing dephasing dephasing Correlation function What is the meaning of the correlation function? Possible temp. depend.: Arrhenius? WLF? Other? Shape of the correlation function? Order parameter? transition probability Relaxation time Nuclear spin system is forced to relax Properties of particular rel.times? Spin temperature Formalisms in intermediate range 30