Contribution to the Kapitza conductance from spin-spin interactions

In this paper, we examine the theory of the rate at which spin-spin interactions produce energy transfer across an interface between a degenerate Fermi gas (liquid ${\mathrm{He}}^3$) and a system of electron spins. We consider first the case where the electron spins are conduction-electron spins in a metal, and examine the rate of energy transfer produced by spin-spin interactions of dipolar form, and also by a short-range exchange interaction. In the former case (dipolar coupling), the Kapitza conductance exhibits a $T^3\ln T$ dependence on temperature, but is very small in magnitude. In the second case (short-range interaction), the Kapitza conductance varies as $T^3$, with a magnitude that is also small. We conclude that neither dipolar interactions nor short-ranged interactions of the exchange type can produce a significant transfer of energy from the ${\mathrm{He}}^3$ spins to conduction-electron spins in metals. We also calculate the rate at which the ${\mathrm{He}}^3$ spins transfer energy to an array of paramagnetic local moments. We recover a result similar to that obtained earlier by Leggett and Vuorio. In contrast to Leggett and Vuorio, we use the full form of the dipolar interaction to couple the ${\mathrm{He}}^3$ spins to the local moments, and in all the calculations reported here, the effect of the surface on the single-particle wave functions is included by treating the surface as a barrier of infinite height.