Paramagnetic Relaxation Times for Titanium and Chrome Alum

The relaxation times for spin-lattice coupling in titanium and chrome alum are computed on the basis of a specific model obtained by combining the thermodynamic theory of Casimir and du Pré with the writer's previous treatment of the normal modes of a cluster of the form X-6${\mathrm{H}}_2$O, where X contains an incomplete shell. The calculation includes both the first-order or direct processes important at low temperatures, and the second-order or "Raman" type of term predominant in the liquid-air region. There is no difficulty in understanding the observed absence of dispersion in titanium alum at liquid-air temperatures, but, barring crystal imperfections, it is hard to understand this absence at helium temperatures unless the nearest excited states are unreasonably deep. The agreement between the orders of magnitude of the calculated and experimental relaxation times is adequate in chromium both at high and low temperatures. The calculations predict, in agreement with experiments, that at liquid-air temperatures the relaxation time should increase when a constant field $H_0$ is applied and should be independent of the direction of $H_0$. The computed increase, however, is apparently not great enough. At helium temperatures, $\tau$ is theoretically not quite isotropic, and $\frac{d\tau }{d{H}_0}$ has the wrong sign, unless one abandons the usual formula ${\rho }_{\omega }\sim {\omega }^2$ for the density of lattice oscillators. The calculations on chromium should also apply qualitatively to iron alum, discussed at the very end.