Theory of a mixed-valent impurity

Abstract

We present a perturbative theory for the thermodynamic properties of a mixed-valent impurity in a metal. The impurity has two ionic configurations $f^{n-1\mathrm{}}$ and $f^n$ (nondegenerate and $n_{\lambda }$-fold degenerate, respectively) with energies ${\epsilon }_0$ and (${\tilde{\epsilon }}_f+\mu$), the difference (${\tilde{\epsilon }}_f-{\epsilon }_0$) being small. They mix via hybridization with conduction electrons (matrix element $V_{\mathrm{kf}}$). We show that for $D>({\tilde{\epsilon }}_f-{\epsilon }_0)\gtrsim -n_{\lambda }\Delta \ln \left(\frac{D}{n_{\lambda }\Delta }\right)$ a Brillouin-Wigner perturbation theory is convergent. Here $\Delta ={\left|V_{\mathrm{kf}}\right|}^2\rho \left(\mu \right)$ is the virtual level width and $2D$ is the conduction-electron bandwidth, $\rho \left(\mu \right)$ being the density of states at the Fermi level. The expansion parameter is the inverse of the orbital degeneracy $n_{\lambda }$. Since this is large (6 to 8), the expansion is quite convergent, and the lowest-order theory is accurate. This is checked by calculation of higher-order terms for various values of (${\tilde{\epsilon }}_f-{\epsilon }_0$). In the above range of (${\tilde{\epsilon }}_f-{\epsilon }_0$) the $f$-electron number is seen to change from ($n-1\mathrm{}$) to about $\mathrm{}(n-1\mathrm{})+0.80\mathrm{}$, so that there is a perturbative theory for a strongly-mixed-valent impurity. Hybridization stabilizes the singlet $f^{n-1\mathrm{}}$ relative to $f^n$, the maximum stabilization energy (level shift) being approximately $n_{\lambda }\Delta \ln \left(\frac{D}{n_{\lambda }\Delta }\right)$ for ${\epsilon }_0={\tilde{\epsilon }}_f$. This singlet ground state has been obtained variationally by Varma and Yafet, and from renormalization-group arguments by Haldane, and by Krishnamurthy, Wilkins, and Wilson; the Brillouin-Wigner perturbation theory has been used earlier by Bringer and Lustfeld. However, the recognition of ($\frac{1}{n_{\lambda }}$) as an expansion parameter and the consequent simplification of the theory are new. Physical properties such...