Effect of the Mean Free Path on the Magnetic Behavior of Small Superconducting Particles

We have calculated the density of states of superconducting spheres much smaller in size than the coherence length ${\xi }_0$ and the penetration depth $\lambda$ for arbitrary mean free path $l$ and arbitrary fields at zero temperature. In agreement with experimental results, the theory predicts an appreciable deviation from results obtained in the limit $l\to 0$ even if the ratio $\frac{l}{{\xi }_0}$ is small, like $\frac{l}{{\xi }_0}\simeq 0.1$. We also discuss the dependence of the critical field on the mean free path and the boundary condition for arbitrary temperatures.