Determination ofκ1(T)andκ2(T)for Type-II Superconductors with Arbitrary Impurity Concentration

Formulas are derived for the constants ${\kappa }_1$ and ${\kappa }_2$ of superconducting alloys for arbitrary temperatures and impurity concentrations. Beginning with Gorkov's equations in matrix form, we calculate the free-energy density up to fourth order in $\left|\Delta \right|$, using Abrikosov's assumption of a fluxoid lattice. The impurities are treated by the usual averaging technique, retaining only $s$ and $p$ scattering, and a special technique is developed to treat exactly the nonvanishing commutator of different components of the gauge-invariant derivative. The results contain the already known limiting cases. Intermediate values are obtained by performing machine calculations using the general formulas developed here. It is found that the values of ${\kappa }_1$ and ${\kappa }_2$ drop relatively strongly even for small impurity concentration, and also depend unexpectedly strongly on the ratio of $s$ and $p$ scattering. The recent result of Caroli, Cyrot, and de Gennes is confirmed according to which ${\kappa }_2$ is approximately equal to ${\kappa }_1$ in the dirty limit.