Fluctuating Spin $\mathit{g}$-Tensor in Small Metal Grains

In the presence of spin-orbit scattering, the splitting of an energy level ${\varepsilon }_{\mu }$ in a generic small metal grain due to the Zeeman coupling to a magnetic field $\overrightarrow{B}$ depends on the direction of $\overrightarrow{B}$, as a result of mesoscopic fluctuations. The anisotropy is described by the eigenvalues $g_j^2$ ( $j=1,2,3\mathrm{}$) of a tensor $G$, corresponding to the (squares of) $g$-factors along three principal axes. We consider the statistical distribution of $G$ and find that the anisotropy is enhanced by eigenvalue repulsion between the $g_j$.