Generalized Theory of Relaxation

Abstract

This paper represents a further generalization, beyond that given recently, of an earlier theory by Wangsness and the author. In common with this theory, it is assumed that the spin system is weakly coupled to its molecular surroundings and that the latter can be considered as a heat reservoir which remains in thermal equilibrium at the absolute temperature $T$. The condition that the coupling is weak demands that its effects upon the spin system, calculated in first and second approximation and measured in a frequency scale, are small compared to the inverse of the correlation time ${\ensuremath{\tau}}_{c}$ as well as of the time $\ensuremath{\beta}=\frac{\ensuremath{\hbar}}{\mathrm{kT}}$. The principal progress over the earlier work consists in the fact that it imposes no additional conditions upon the energy of the spin system; in particular, it dispenses with the necessity that the dominant part of this energy is independent of the time. A linear differential equation of the first order for the distribution matrix is derived which is valid in this very general case and which contains the earlier results as special cases. The derivation of this Boltzmann equation is carried out in a form which is independent of any particular representation, used for the spin system, and which leads directly to a system of differential equations for the expectation values of spin functions. Beyond the earlier results, it is shown that the coupling with the molecular surroundings leads in second approximation not only to relaxation terms but also to a correction of the spin energy which has the nature of a "self-energy." As a considerably more restricted case, the situation is investigated where the dominant part of the spin energy varies little during a time of order ${\ensuremath{\tau}}_{c}$ and $\ensuremath{\beta}$. It is shown that, in this case, relaxation causes the distribution matrix to tend towards the form, corresponding to thermal equilibrium at the instantaneous value of the dominant part of the energy. The case where the frequencies of the spin system as well as their relative rate of variation are small compared to $\frac{1}{{\ensuremath{\tau}}_{c}}$ but arbitrary compared to $\frac{1}{\ensuremath{\beta}}$, is likewise discussed and shown to lead to a simple form of the Boltzmann equation. The general formalism is finally applied to a spin system with a single spin of value \textonehalf{}, exposed to a rotating field. One obtains in this case a phenomenological equation for the expectation value of the spin vector which is far more general than the one derived earlier; instead of requiring merely the knowledge of the longitudinal and the transverse relaxation time, relaxation is here characterized by five time constants which may all be different. The stationary solution is derived and applied to a number of familiar special cases.