Grüneisen Functions for Some Type-I Superconductors

The thermal expansion of some type-I superconductors (indium, lead, tantalum, and tin) is discussed thermodynamically in terms of Grüneisen functions, which describe the volume dependence of the entropy. The Grüneisen functions for the normal and superconducting states can be taken to differ only in their electronic components, ${\gamma }^{\mathrm{en}}$ and ${\gamma }^{\mathrm{es}}$. Values of ${\gamma }^{\mathrm{en}}$ obtained from thermal-expansion measurements in the normal state are shown to be more reliable than values obtained from critical-field measurements on superconductors, because the magnetic data are not accurate enough in the limiting low-temperature region determining ${\gamma }^{\mathrm{en}}$. The values of ${\gamma }^{\mathrm{es}}$ for the four metals all appear to be negative, unlike ${\gamma }^{\mathrm{en}}$. Theoretical values of ${\gamma }^{\mathrm{es}}$ calculated using the BCS theory for weak coupling and the similarity principle are consistent with experiment for all the metals except lead. The disagreement for lead is probably due to its particularly strong electron-phonon coupling. According to the theory, the values of ${\gamma }^{\mathrm{es}}$ depend through $\frac{d\ln T_c}{d\ln V}$ on the volume dependence of the electron-phonon interaction, as well as on properties of the normal state. The experimental evidence for significant changes in lattice properties is discussed; such a change appears to be established for indium but not for lead.