Theory of a mixed-valent impurity

摘要: We present a perturbative theory for the thermodynamic properties of mixed-valent impurity in metal. The has two ionic configurations ${f}^{n\ensuremath{-}1}$ and ${f}^{n}$ (nondegenerate ${n}_{\ensuremath{\lambda}}$-fold degenerate, respectively) with energies ${\ensuremath{\epsilon}}_{0}$ (${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}+\ensuremath{\mu}$), difference \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}\ensuremath{-}{\ensuremath{\epsilon}}_{0}$) being small. They mix via hybridization conduction electrons (matrix element ${V}_{\mathrm{kf}}$). show that $Dg({\stackrel{\ifmmode \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}\ensuremath{-}{\ensuremath{\epsilon}}_{0})\ensuremath{\gtrsim}\ensuremath{-}{n}_{\ensuremath{\lambda}}\ensuremath{\Delta}\mathrm{ln}(\frac{D}{{n}_{\ensuremath{\lambda}}\ensuremath{\Delta}})$ Brillouin-Wigner perturbation is convergent. Here $\ensuremath{\Delta}={|{V}_{\mathrm{kf}}|}^{2}\ensuremath{\rho}(\ensuremath{\mu})$ virtual level width $2D$ conduction-electron bandwidth, $\ensuremath{\rho}(\ensuremath{\mu})$ density states at Fermi level. expansion parameter inverse orbital degeneracy ${n}_{\ensuremath{\lambda}}$. Since this large (6 to 8), quite convergent, lowest-order accurate. This checked by calculation higher-order terms various values \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}\ensuremath{-}{\ensuremath{\epsilon}}_{0}$). In above range $f$-electron number seen change from ($n\ensuremath{-}1$) about $(n\ensuremath{-}1)+0.80$, so there strongly-mixed-valent impurity. Hybridization stabilizes singlet relative ${f}^{n}$, maximum stabilization energy (level shift) approximately ${n}_{\ensuremath{\lambda}}\ensuremath{\Delta}\mathrm{ln}(\frac{D}{{n}_{\ensuremath{\lambda}}\ensuremath{\Delta}})$ ${\ensuremath{\epsilon}}_{0}={\stackrel{\ifmmode \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}$. ground state been obtained variationally Varma Yafet, renormalization-group arguments Haldane, Krishnamurthy, Wilkins, Wilson; used earlier Bringer Lustfeld. However, recognition ($\frac{1}{{n}_{\ensuremath{\lambda}}}$) as an consequent simplification are new. Physical such valence, susceptibility, specific heat calculated function ($\frac{{k}_{B}T}{\ensuremath{\Delta}}$) A simple way including effect alloying pressure described. Many characteristic metallic dilute concentrated systems, temperature dependence positive ${T}^{2}$ slope low-temperature susceptibility $\ensuremath{\chi}(T)$, broad it, relation between $\ensuremath{\chi}(0)$ Curie-Weiss high-temperature qualitatively explained quantitatively characterized first time. results directly applicable nondilute alloys. can also be applied perfect lattice systems except lowest temperatures where relatively small intersite coupling leads uniform Fermi-liquid state. Kondo limit, i.e., nearly-${f}^{n}$-valent which occurs $({\stackrel{\ifmmode \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}\ensuremath{-}{\ensuremath{\epsilon}}_{0})\ensuremath{\ll}\ensuremath{-}{n}_{\ensuremath{\lambda}}\ensuremath{\Delta}$, not described theory.