Spatial Variations of the Order Parameter in Superconductors Containing Magnetic Impurities

The problem of spatial variations of the order parameter in the vicinity of a paramagnetic impurity and near the transition temperature is reconsidered by means of a more rigorous formalism than in a discussion which appeared recently in the literature. General expressions are derived for the spatially inhomogeneous part of the order parameter and its effect on the critical temperature in the limit of very small concentrations of impurities. A detailed analysis shows that the dominant variations of the order-parameter range to a distance of the order of the coherence length ${\xi }_0\simeq \frac{v_F}{2\pi T_{c0\mathrm{}}}$ from a single impurity, but that this parameter also exhibits weak long-range variations in the region $\mathcal{r}\gg {\xi }_0$. For $\mathcal{r}<{\xi }_0$, both smooth spatial variations and fast oscillations show up in the expression of the superconducting order parameter. In the Born approximation the result for the transition temperature coincides with that of the Abrikosov-Gor'kov theory, where the average value of the order parameter is used. The next higher-order correction, due to spatial variations, leads to a small increase of the transition temperature, which is explicitly estimated.