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-1and fⁿ (nondegenerate and n_λ-fold degenerate, respectively) with energies ε₀ and (ε^~_f+μ), the difference (ε^~_f-ε₀) being small. They mix via hybridization with conduction electrons (matrix element V_kf). We show that for D>(ε^~_f-ε₀)≳-n_λΔln(D/n_λΔ) a Brillouin-Wigner perturbation theory is convergent. Here Δ=|V_kf|²ρ(μ) is the virtual level width and 2D is the conduction-electron bandwidth, ρ(μ) being the density of states at the Fermi level. The expansion parameter is the inverse of the orbital degeneracy n_λ. 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 (ε^~_f-ε₀). In the above range of (ε^~_f-ε₀) the f-electron number is seen to change from (n-1) to about (n-1)+0.80, so that there is a perturbative theory for a strongly-mixed-valent impurity. Hybridization stabilizes the singlet f^n-1 relative to fⁿ, the maximum stabilization energy (level shift) being approximately n_λΔln(D/n_λΔ) for ε₀=e_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 (1/n_λ) as an expansion parameter and the consequent simplification of the theory are new. Physical properties such as valence, susceptibility, and specific heat are calculated as a function of (k_BT/Δ) for various values of (ε^~_f-ε₀). A simple way of including the effect of alloying pressure is described. Many characteristic properties of metallic dilute and concentrated mixed-valent systems, such as the temperature dependence of valence, the positive T² slope of the low-temperature susceptibility Χ(T), the broad maximum in it, the relation between Χ(0) and the Curie-Weiss temperature of high-temperature susceptibility, are qualitatively explained and quantitatively characterized for the first time. The results are directly applicable to dilute and nondilute alloys. They can also be applied to concentrated perfect lattice systems except at the lowest temperatures where relatively small intersite coupling leads to a uniform Fermi-liquid ground state. The Kondo limit, i.e., the nearly-fⁿ-valent singlet which occurs for (ε^~_f-ε₀)«-n_λΔ, is not described by the present theory.