A Mathematical Model for the Narrowing of Spectral Lines by Exchange or Motion

TLDR

The paper develops and partially justifies a simplified random frequency‑modulation model for exchange and motional narrowing. The model examines several cases, including dipolar‑broadened lines with exchange and diffusion‑induced narrowing in solids, applying Gaussian and Markovian assumptions. The model shows that extreme narrowing yields a resonance‑shaped core with steeper wings, that the half‑width scales with the mean square frequency breadth divided by the motion rate, and that it reproduces Van Vleck’s fourth‑moment results for dipolar exchange while confirming experimentally observed hyperfine line merging.

Abstract

In this paper a simplified mathematical model of exchange and motional narrowing, which we call the “random frequency-modulation model,” is developed, and, to a certain extent, justified. A number of cases are treated. The one general conclusion which is common to all is that in the case of extreme narrowing the central part of the line is resonance-shaped while the wings fall off more steeply than the resonance shape: the half-width is always of order of magnitude of the mean square frequency breadth divided by the rate of motion or exchange. In two cases the model makes some quantitative approach to realism: first, the case of a single dipolar-broadened line with exchange, where the “Gaussian” assumption can be made and the results have been fitted to Van Vleck's fourth moment calculation to give satisfactory numerical answers; and second, the case of narrowing by diffusion in solids, where the “Markoffian” assumption is valid. The problem of narrowing of hyperfine structure is considered, and on this model it is found that before merging the lines draw together, a result which is confirmed by experiments.