Quantum-Statistical Theory of Spin Echo In Liquid He³

Abstract

Quantum-statistical mechanics is applied to the spin echo method for liquid He³ in order, first, to derive quantum theoretically the expressions for the amplitudes, or the heights of peaks, of the echo signals and, secondly, to obtain the expression for the so-called “self-diffusion coefficient” measured by this method in terms of microscopic quantities of the system. Three cases are considered: the case of a 90°–180° pulse sequence, a 90°–180°–180° pulse sequence and the “B Method” by Carr and Purcell. It is found, among others, that the ratio of the amplitudes of two echoes in the second case is precisely expressed by the formula which has been obtained by several authors from classical or macroscopic standpoint, only with the spin diffusion coefficient D_s, of which the microscopic expression is given by Fukuda and Kubo, and also by Kadanoff and Martin, for the “self-diffusion coefficient”, under a usual experimental condition.