Investigation of the electron-state densities in two-band super conductors with a nonmagnetic impurity

The electron-state densities $n_n(\omega)$ and also the related functions $\bar n_n(\omega)$, $m_n(\omega)$, and $\bar m_n(\omega)$ are calculated for a two-band superconductor with a nonmagnetic impurity in the limiting cases when the impurity has a low or a high concentration. The calculations are made for all frequencies in the range $\omega\geqslant\Omega_g$, where $\Omega_g$ is the single-particle spectrum energy gap of the two-band superconductor. It is shown that at low impurity concentrations $n_1$ and $n_2$ have sharp peaks near the corresponding energy gaps $\gamma_1$ and $\gamma_2$ of the pure two-band superconductor. Far from these values $\omega=\gamma_n$ a small impurity concentration hardly modifies the expressions for the densities given by the BCS-Bogolyubov theory for a pure two-band superconductor. If the nonmagnetic impurity has a high concentration, the functions $n_n$, $\bar n_n$, etc., of both bands differ little, and the state densities $n_n$ have a common maximum at the frequency $\Omega_m=\frac{\Gamma_1N_1+\Gamma_2N_2}{N_1+N_2}$. The energy gap $\Omega_g$ in this limiting case is somewhat less than $\Omega_m$. The densities $n_m(\omega)$ obtained in the different frequency ranges are fitted at the boundaries of these ranges.