Relaxation. Ravinder Reddy

Relaxation Ravinder Reddy

Relaxation What is nuclear spin relaxation? What causes it? Effect on spectral line width Field dependence Mechanisms

Thermal equilibrium ~10-6 spins leads to NMR signal!

T1 Spin-lattice relaxation time (T 1 ): Characteristic time required for relaxation of longitudinal magnetization (M z ) toward equilibrium value (M o ). dm z /dt = -(M z -M o )/T 1

T2 Spin-spin relaxation time (T 2 ): Characteristic time required for relaxation of transverse magnetization (M xy ) toward equilibrium value (M o ). dm xy /dt = -(M xy )/T 2

Thermal equilibrium? How is the thermal equilibrium established? ν B o = 0 B o = 1.5T

T1 Spin system Lattice β H W d W u ΔE ΔE α L probability per time of simultaneous transitions from β α and L H as W βl αh dn! dt = N " N L W "L#!H $ N! N H W!H #"L According to Quantum Mechanics W!L"#H = W #H "!L = W

T1 W u = N H W and W d = N L W As we can only measure the rate of change in the difference in populations of energy levels, let us introduce n = Nα N β N = N! + N " dn dt =!n(w d + W u ) + N(W d! W u )

T1 1 n o = N α 0 N β 0 n o (W d + W u ) + N(W d W u ) = 0 2 3 at thermal equilibrium dn/dt = 0 population difference is given by eqn 2 N α o and N β o are given by the Boltzman distribution (eqn 4) N! 0 N = exp(#$e /kt) = N H 0 " N L n o = N(W W ) d u (W d + W u ) From 3 5 From 1and 4 dn dt = (n n o )(W d + W u ) 4 6

T1 T1 definition dm z dt = (M z M 0 ) /T 1 dn dt = (n n o )(W d + W u ) T 1 = 1/(W d + W u ) This equation implies that T 1 depends on processes that increase the probability of transitions between α and β energy levels. What are those processes?--->

Rotational Motion

Frequency components In a sample there will be a distribution of Rotational freqs. Only frequencies at the Larmor Freq. Affect T1 ω o = γb o ====> Translational, vibrational motions?? NMR time scale ~1/ ω o

Temperature Increase in temp. increases the higher frequency components Reduces the number of molecular motions at o. Increases the T1 T1(280K)< T1(300K) Effect of Human body temp?

Effect of viscosity As the solution gets more viscous the number of molecules with high frequency components decrease. Viscosity of Tissues vary significantly. Biological tissues have different T1s. ν o

T2: Natural line width Δt ΔE ω ω o B o = 0 B o = 1.5T According to HUP (Heisennerg Unxertainity Principle) ΔEΔt h(cross) or Δω 1/ Δt Δt =T 1 ====> Δω 1/ 1

T 2 Process Fluctuating fields (Hz) which perturb the energy levels of the spin states cause transverse magnetization to dephase ΔE=γB o

T2 ΔE ν 1/2 = 1/πT 2 o ν 1/2 =FWHM

Observed line width inimum Spectral line width is limited by T1. This is known as natural line width. ν 1/2 1/ 1 Including effects of molecular motions ν 1/2 = 1/πT 2 Contributions to line width due to field inhomogeneities (ΔB o ) ν 1/2 (inhomo) = γδb o /2π Observed line = ν 1/2 = 1/πT * 2 1/T 2 * = 1/T 2 + γδb o /2

Relaxation Mechanisms H xy H z M xy H x and H y Must oscillate (fluctuate) at or close to ω o. NMR time scale (~1/ ω o ) Magnetic field that fluctuate more rapidly than ω o are time averaged to zero.

Relaxation Mechanisms Motion of nuclear magnetic moments generates a fluctuating magnetic fields H= ih x +jh y +kh z M= im x + jm y +km z (magnetization vector) Interaction between them (H x M)= i(h y M z -H z M y )+j(h z M x -H x M z ) +k(h x M y -H y M x ) H x,y ----> T1 and T2 relaxation H z ----> T2 relaxation -----> T1>T2

T2

SE 90 o 180 o Spin echo Sig = km o (1-exp(-TR/T1) exp(-te/t2) TE/2 TE/2

T1

IR 180 o 90 o TI FID

IR Sig = kmo [1-2exp(-TI/T 1 ]

Fluctuating fields These fluctuating fields have zero average: <B x (t)> = 0 Mean square fluctuating field <B x 2 (t)> 0 B x B o z M y x y

Fluctuating fields Since the square of the field is always +, the mean square field is not zero and is the same for all spins. The value of <B x2 (t)> is indicated by dashed line in the plots

How rapidly the fields fluctuate? Autocorrelation function of the field (convolution of a function with it self) defined as G( ) = <Bx(t) Bx(t+τ)> 0 It tells us how self similar a function is Fluctuating fields

Correlation time Let us examine the field at any one time point t with its value at t+τ. If τ is small compared to the timescale of the fluctuations, then the values of the field at the two time points tend to be similar. If τ is long, then the system loses its memory.

Autocorrelation function G( ) G(τ) is large for small values of τ, and tend to zero for large values of. One assumes an exponential form: G(t)= <B x2 > exp(- /τ c ) τ c is known as correlation time of the fluctuations. It indicates how long it takes before the random field changes sign.

Spectral density J( ) Spectral density is defined as the 2 FT of G(t): J( ) = 2 o G( ) exp{-i } For G(t)= <B x2 > exp(- /τ c ) The spectral density is given J(w) = 2 <B x2 > τ c /(1+ 2 τ c 2 ) If τ c is short then the SDF is broad and vice versa Normalized SDF, J(w o ) = τ c /(1+ 2 τ c 2 )

Spectral density As the solution gets more viscous the number of molecules with high frequency components decrease. Viscosity of Tissues vary significantly. Biological tissues have different T1s. J( ) o log( )

T1 1/T1 = 2W (transition probability per unit time between the states) W= (1/2) γ 2 <B x2 > J(w o ) For a model of fluctuating random field along x-axis 1/T 1 = γ 2 <B x2 > J(w o ) 1/T 1 = γ 2 [<B x2 >+ <B y2 >] J(w o, c )) 1/T 2 = γ 2 [<B z2 >] J(o, c ) + 1/(2T 1 ) In this equn. First term is due to static Hz contribution to T2 and the second term is due to finite life time of a spin in an excited energy state.

T1 In isotropic liquids motions do not have any preferential directions ----> B x2 = B y 2 = B z2 =B 2 1/T1 = γ 2 [<B x2 >+ <B y2 >] J(w o, c )) = 2 γ 2 [<B 2 >] J(w o, c )) 1/T2 = γ 2 [<B z2 >] J(o, c ) + 1/(2T1) = γ 2 [<H z2 >] [J(o, c ) + J(w o, c )]

T1 vs correlation times T1

T1 and T2 Variation of relaxation time of protons in water as a function of correlation time at a resonance frequency of 100 MHz (1/ o = 10-8 s) o c < 1, T1=T2 o c 1, T1>T2

Relaxation mechanism Combined T1 and T2 Dipole-dipole interactions Chemical shift anisotropy Spin rotation interaction Quadrupole interaction Pure T2 Processes Chemical shift (*) Spin-exchange Static field inhomogeneity(*) Sample inhomogeneity(*) diffusion