Effect of the Spin-Phonon Interaction on the Thermal Conductivity

The effect of dilute concentration of the Kramers ion, ${\mathrm{Yb}}^{+++}$, and the non-Kramers ion, ${\mathrm{Ni}}^{++}$, on the thermal conductivity of Ca${\mathrm{F}}_2$ has been measured as a function of magnetic field at various temperatures in the range 0.3-1.3°K. The results have been compared with a theory due to Elliott and Parkinson based on the Jacobsen and Stevens dispersion relations. It has been found that it is possible to account for the effect of temperature on the change in conductivity with magnetic field in both cases, for fields such that $\frac{g\beta H}{\mathrm{kT}}<6\mathrm{}$. If the temperature dependence is factored out of the expression for the change in conductivity, the remaining terms predict that at low fields it should increase as $H^{\frac{7}{2}}$ for the Kramers ions, and as $H^{\frac{5}{2}}$ for the non-Kramers ions, unless a zero-field splitting is present which dominates, and then the change in conductivity should increase as $H^3$ increases. The Yb-doped crystal yielded an ${\mathrm{H}}^{\frac{7}{2}}$, and the Ni-doped yielded an $H^3$ dependence in agreement with the theory.