Influence of the phonon spectrum on the critical temperature of a superconductor at a type $2\frac12$ phase transition

Expressions for the critical temperature $T_c$ and the derivative $dT_c/d\varepsilon_{\mathrm F}'$ at the phase transition of the $2\frac12$ kind are obtained analytically on the basis of the phonon model in the weak coupling approximation ($\varepsilon_{\mathrm F}'=\varepsilon_{\mathrm F} -\varepsilon_{\mathrm K}$; $\varepsilon_{\mathrm F}$ is the Fermi energy, $\varepsilon_{\mathrm K}$ – critical energy at which the topology of the Fermi surface is changing). The expressions obtained are valid for arbitrary dispersion laws and provide the practical description of the quantities mentioned over all the range of the parameter $\varepsilon_{\mathrm F}'$. For the examples of the sound and Einstein dispersion laws it is shown that the phonon spectrum possesses only slight effect on the ratio $T_c/T_{c0}$ ($T_{c0}$ being the value of critical temperature in the absence of pressure and admixture) and the significant effect on the derivative $dT_c/d\varepsilon_{\mathrm F}'$.