The Dynamical Theory of Nuclear Induction

TLDR

The theory assumes independent nuclei, neglecting spin‑spin interactions, and draws an analogy to the classical Boltzmann equation to compute macroscopic spin averages over time. The study applies the derived result to the time evolution of macroscopic nuclear polarization, examining when it satisfies F.B.'s phenomenological differential equation. Using statistical methods, the authors derive a first‑order linear differential system for the distribution matrix that accounts for an arbitrary external field and molecular surroundings acting on a representative nucleus. The phenomenological equation does not describe line structures from neighboring spin interactions and is valid only for nuclei with spin greater than one when quadrupole relaxation is negligible; it requires isotropic surroundings and interaction frequencies much higher than the Larmor frequency, ensuring equal longitudinal and transverse relaxation times.

Abstract

Starting from the microscopic viewpoint, the dynamics of nuclear induction is derived by means of statistical methods. The only essential lack of generality lies in the assumption that the nuclei in the sample are independent of each other, so that the treatment does not account for features arising from spin-spin interaction. By considering the simultaneous action of an arbitrary external field and of the molecular surroundings upon a representative nucleus a system of linear differential equations of the first order is derived for the "distribution matrix." It is analogous to the classical Boltzmann equation for the distribution function and allows, upon integration, to determine the macroscopic average value of any spin function in its dependence upon time. This general result is particularly applied to the time dependence of the macroscopic nuclear polarization, and the conditions are investigated under which it satisfies the phenomenological differential equation originally proposed by one of the authors (F.B.). Besides the fact that this equation does not describe line structures caused by the interaction of neighboring spins its validity is found to be seriously restricted only for nuclei having a spin larger than unity and in cases where, in addition, quadrupole relaxation is essential. It demands in these cases that the molecular surroundings are isotropic, e.g., as in gaseous and liquid samples, and further, that their characteristic frequencies of interaction with the nuclei are large compared to the Larmor frequency so that there exists equality between the longitudinal and the transverse relaxation time.