Relaxation Rates

The spin-lattice (or longitudinal) and spin-spin (or transverse) relaxation rates T-f1 and T2~2, respectively, are calculated taking into account the most important spin interaction. The direct dipolar coupling and the quadrupolar coupling are undoubtedly two of the most common and relevant in NMR relaxation. Considering the most common case where the direct dipolar coupling is between two spins (e.g., two 1H spins with I = 1/2) here labeled I and S, respectively, the Hamiltonian can be written as (Abragam, 1961; Kimmich, 1997)

with

and

в and p are the Euler angles between the interspin vector in the laboratory frame, and У(т), p) are spherical harmonics (Kimmich, 1997).

Considering the static Zeeman Hamiltonian for the two spins

the expected values of (Iz) and (Sz) can be calculates using Eq. 6.4 and the solution of Eq. 6.5. This calculus can be found in different text books (Abragam, 1961; Kimmich, 1997; Slichter, 1992) and the results is

where (Iz)0 and {Sz}0 are the equilibrium values, and

Using the Bloch/Wangsness/Redfield theory (Redfield, 1965) it is possible to obtain the expressions for the spin-lattice and spin- spin relaxation rates in terms of the spectral densities, Ja (Abragam, 1961; Kimmich, 1997; Kowalewski and Maler, 2006).

I

calculated from the autocorrelation functions, G(t),

For two identical spins I = S, we can expect (Iz) = (Sz), (Iz)0 = (Sz)0, and шI = rnS = ш0. Therefore, Eq. 6.13 and Eq. 6.14 merge and give

with 1/T1 = 1/Tl1 + 1/T1S and

Obviously, the number of identical spins in a sample per cubic centimeter is in the order of 1023 (i.e., of the order of the Avogadro's number), therefore for the calculus of the spin-lattice relaxation rate there are two cases to consider: (/) the motion of the identical spins are correlated or (ii) the motions are uncorrelated.

(1) If the spins are correlated then to a very good approximation a single exponential decay evolution is observed for the spin- lattice relaxation with a single relaxation rate where the spectral densities Eq. 6.19 are calculated using ensemble averages of the autocorrelation functions

(2) If the spin pairs are uncoupled then it is possible to write the relaxation rate taking into account all dipoles formed with a reference spin. The spin-lattice relaxation is still monoexponential but the relaxation rate is given by

The density matrix formalism can also be used to obtain the relaxation rates for the spin-spin (e.g., transverse) relaxation rate T and the spin-lattice relaxation rate in the rotating frame T-.

For the transverse relaxation due to the direct dipolar coupling of a pair of proton spins the relaxation rate is given by (Abragam, 1961; Kimmich, 1997)

It also possible to obtain the spin-lattice relaxation rate in the rotating frame, T—, when ш0 ^ ю1 ^ &>ioc (Abragam, 1961; Kimmich, 1997)

Spin-Lattice Relaxation in Aligned Systems

In the case of samples where it is possible to define an alignment direction of molecules (e.g., liquid crystals), it is necessary to consider the angle A between the alignment direction and the magnetic field. The spectral densities have to be calculated taking into account the existence of this angle between the two-axis frame of references (Ukleja etal., 1976)

where

Two limit cases are worth mentioning. The isotropic case where the spectral densities present the relative proportions

with c0 = 6, c1 = 1 e c2 = 4. In this case

and the spectral densities are obviously independent of A.

The second case is that of a sample with domains presenting an uniform distribution of orientation angles (e.g., like in a polycrystal). In this case it is necessary to average Eq. 6.28

Equation 6.27 becomes independent of A

 
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