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, ${\mathfrak{H}}_1$ , then these equations can be solved by means of a perturbation expansion in $({\mathfrak{H}}_1\mathrm{\hspace{0.17em}/\hspace{0.17em}D)}$ . 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 $T_2^{\text{−\hspace{0.17em}1}}$ are of the order of ${\mathfrak{H}}_1^2{\tau }_2$ where ${\tau }_2{\text{\hspace{0.17em}=\hspace{0.17em}(6D)}}^{\text{−\hspace{0.17em}1}}$ , and whose frequencies are shifted by an amount of the order of ${\omega }_0{\tau }_2{T}_2^{\text{−\hspace{0.17em}1}}{\text{(1\hspace{0.17em}+\hspace{0.17em}ω}}_0^2{\tau }_2^2{)}^{\text{−\hspace{0.17em}1}}$ from the Zeeman frequency, where ${\omega }_0$ 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 ${\omega }_0{\mathrm{(\tau }}_2{\mathrm{\hspace{0.17em}/\hspace{0.17em}T}}_2{)}^{\text{3\hspace{0.17em}/\hspace{0.17em}2}}{\text{\hspace{0.17em}(1\hspace{0.17em}+\hspace{0.17em}ω}}_0^2{\tau }_2^2{)}^{\text{−\hspace{0.17em}1}}$ and $T_2^{\text{−\hspace{0.17em}1}}{\mathrm{(\tau }}_2{\mathrm{\hspace{0.17em}/\hspace{0.17em}T}}_2)$ ; the frequency shift should be corrected by terms of the order of $T_2^{\text{−\hspace{0.17em}1}}{\mathrm{(\tau }}_2{\mathrm{\hspace{0.17em}/\hspace{0.17em}T}}_2{)}^{\text{1\hspace{0.17em}/\hspace{0.17em}2}}$ . Furthermore, a number of weak auxiliary Lorentzian lines at frequencies of the order of ${\mathfrak{H}}_1$ from the Zeeman frequencies must be included; these lines have intensities which are of the order of ${\mathrm{(\tau }}_2{\mathrm{\hspace{0.17em}/\hspace{0.17em}T}}_2)$ below that of the principal “Redfield lines” and their widths are ${\tau }_2^{\text{−\hspace{0.17em}1}}$ . The superposition of these auxiliary lines on the Redfield lines gives rise to unsymmetrical, non‐Lorentzian lines, but in the region ${\mathrm{(\tau }}_2{\mathrm{\hspace{0.17em}/\hspace{0.17em}T}}_2\text{)\hspace{0.17em}<\hspace{0.17em}0.3}$ , 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, $\mathrm{(D\hspace{0.17em}/\hspace{0.17em}}{\mathfrak{H}}_1\text{)\hspace{0.17em}≪\hspace{0.17em}1}$ . In this case the spin Hamiltonian is diagonalized at each molecular orientation and the diffusion jumps between orientations are treated as a perturbation.