Energy Gap and Thermal Conductivity of Pure and Impure Superconducting Tin

A theory is presented for the energy gap and the thermal conductivity in pure and impure tin. The central features are taken from the papers of Markowitz and Kadanoff and of Hohenberg, and include anisotropy in the pure material which is systematically eliminated by increased doping. First the usual theory of thermal conductivity is generalized to allow for gap anisotropy. Next an anisotropic gap function for pure tin is constructed using experimental results of other investigators. Synthesizing these two stages yields a prediction for the thermal conductivity of pure superconducting tin as a function of orientation and temperature. This prediction is in fair agreement with the data of Guénault. Turning to the impure metal, we find that the gap depends upon four quantities: energy $\omega$, angle $\Omega$, mean free path $l$, and temperature $T$. Simplifications are introduced whereby the complicated dependences are separated into simpler factors. Stress is laid upon the distinction between anisotropy in the gap edge and the effect of anisotropy on quantities such as the average gap and $T_c$. The gap edge is rendered isotropic by impurities well before the effect is felt by the other quantities. In fact it is the gap edge which is vital to thermal conductivity, since it is there that the carriers reside. Predictions for thermal conductivity in impure tin are compared with data of Pearson et al. and found to show good qualitative agreement. In particular, a reduction in the ratio of superconducting to normal thermal conductivity for certain sample directions is explained by the elimination of gap-edge anisotropy. However, this effect occurs at a much smaller impurity concentration than that predicted theoretically. This discrepancy and related matters are fully discussed.