In regular crystals, the width of the absorption lines arising from the magnetic moment of the electron or nucleus is caused primarily by the interaction between the magnetic dipoles. It is prohibitively difficult to determine the precise shape of the absorption line theoretically, but the invariance of the diagonal sum in quantum mechanics permits the calculation of the second moment of the frequency deviation, and hence the r.m.s. line breadth. The latter agrees excellently with the observations of Pake and Purcell on the magnetic absorption of the F nucleus in Ca${\mathrm{F}}_2$, both in absolute magnitude, and in the dependence on the direction between the magnetic field and the principal cubic axes. The fourth moment was also computed to examine how good an approximation is the conventional assumption of a Gaussian shape. As long as no exchange is present (the nuclear case) the Gaussian model is moderately good. For the 100 direction in a cubic crystal, the theoretical ratio of root mean fourth to root mean square breadth is 1.25. Pake and Purcell's measurements yield 1.24. A Gaussian model would require 1.32. The theory is extended to include crystals with two kinds of spin moments (two types of nuclei, or simultaneous nuclear and electronic spin). Coupling between unlike moments is less effective (by a factor ⅔ in the r.m.s. width) than that between like in broadening the lines.

In the paramagnetic absorption caused by electronic spin, it is imperative to include the effect of exchange coupling. This interaction does not contribute to the second moment, but greatly increases the fourth. As a result, the lines are peaked much more sharply than one would compute from the second moment with the Gaussian model for line shape. This "exchange narrowing" explains why microwave paramagnetic absorption lines are much narrower than one first conjectures from the amount of dipolar coupling.

The theoretical calculations are given in Sections II-IV. The final sections V-VI give the comparison with the experiments of Pake and Purcell, and with the model of Bloembergen, Purcell, and Pound, for r-f absorption in liquids.

• Received 6 July 1948

DOI:https://doi.org/10.1103/PhysRev.74.1168