Thermodynamics of superconducting ${\mathrm{Nb}}_3$Al, ${\mathrm{Nb}}_3$Ge, ${\mathrm{Nb}}_3$Sn, and ${\mathrm{V}}_3$Ga

We have calculated the superconducting thermodynamic properties for several high-transition-temperature $A15\mathrm{}$ compounds: Nb-Al, Nb-Ge, Nb-Sn, and V-Ga. In our calculations we have used the tunneling electron-phonon—coupling spectra ${\alpha }^{2}F$ for all four systems considered, and in the case of Nb-Al and Nb-Ge we have also used ${\alpha }^{2}F=CG$, where $G$ is the measured generalized phonon density of states and $C$ is a constant. We find that all Nb-based $A15\mathrm{}$ compounds display similar thermodynamic properties, which do not depend explicitly on the band density of states: $\frac{{2\Delta }_0}{{\kappa }_B{T}_c}\cong 4.6$, $\frac{\Delta C}{\gamma T_c}\cong 2.5-2.6,\frac{-T_c{\left[\frac{{\mathrm{dH}}_c\mathrm{}\left(T\right)\mathrm{}}{\mathrm{dT}}\right]}_{T_c}}{H_c\mathrm{}\left(0\right)\mathrm{}\cong 2.1}$, $\gamma {\left[\frac{T_c}{H_c\mathrm{}\left(0\right)\mathrm{}}\right]}^2\cong 0.134$, and positive $D\left(t\right)\mathrm{}$'s with the maximum value around 0.02. For ${\mathrm{Nb}}_3$Sn we find good agreement between the calculated properties and the old specific-heat experimental results ($\gamma \cong 52$ mJ/mol ${\mathrm{K}}^2$). The same applies to ${\mathrm{V}}_3$Ga, where the theoretical results have been compared with the experiments of Junod et al. However, we do not find good agreement between calculated $\frac{\Delta C}{\gamma T_c}$, — $\frac{T_c{\left[\frac{{\mathrm{dH}}_c\mathrm{}\left(T\right)\mathrm{}}{\mathrm{dT}}\right]}_{T_c}}{H_c\mathrm{}\left(0\right)\mathrm{}}$, $\gamma {\left[\frac{T_c}{H_c\mathrm{}\left(0\right)\mathrm{}}\right]}^2$, and experimental values for ${\mathrm{Nb}}_3$Al and ${\mathrm{Nb}}_3$Ge, presumably due to broadened transitions. It is argued that the tunneling experiments underestimate the value of the gap which should be associated with the inverted ${\alpha }^{2}F$.