Theory of Spin–Lattice Relaxation in Classical Liquids

Abstract

A set of coupled differential equations is obtained which represents an exact solution for the high-temperature spin autocorrelation function for spins in a liquid whose motion is governed by a classical isotropic rotational diffusion equation with a single rotational diffusion constant, D. If the diffusion is rapid, i.e., if D is large compared to the spin–lattice interaction, H1, then these equations can be solved by means of a perturbation expansion in (H1 / D). In this case, the dominant terms correspond to those in the well-known Redfield theory; in the absence of spin degeneracy the spectrum consists of Lorentzian lines whose widths T2− 1 are of the order of H12τ2 where τ2 = (6D)− 1, and whose frequencies are shifted by an amount of the order of ω0τ2T2− 1(1 + ω02τ22)− 1 from the Zeeman frequency, where ω0 is a characteristic spectral frequency difference. The present theory introduces a number of corrections: The linewidth should be corrected by terms of the order of ω0(τ2 / T2)3 / 2 (1 + ω02τ22)− 1 and T2− 1(τ2 / T2); the frequency shift should be corrected by terms of the order of T2− 1(τ2 / T2)1 / 2. Furthermore, a number of weak auxiliary Lorentzian lines at frequencies of the order of H1 from the Zeeman frequencies must be included; these lines have intensities which are of the order of (τ2 / T2) below that of the principal “Redfield lines” and their widths are τ2− 1. The superposition of these auxiliary lines on the Redfield lines gives rise to unsymmetrical, non-Lorentzian lines, but in the region (τ2 / T2) < 0.3, where this perturbation expansion is valid, the auxiliary lines contribute little to the central part of the composite lines, but they play a significant role in the wings. The coupled differential equations have been reformulated in order to treat the problem of slow diffusion, (D / H1) ≪ 1. In this case the spin Hamiltonian is diagonalized at each molecular orientation and the diffusion jumps between orientations are treated as a perturbation.