Electronic Properties of Alloys

The well-known criterion for characterizing approximately the electronic properties of concentrated alloys in pure-metal terms is that the mean free path of the electrons be many atomic spacings. For alloys whose solvent and solute have different valences, this condition can be expressed in terms of the energy shift $\Delta E\left(\mathbf{k}\right)$ of a given Bloch state as the impurities are adiabatically turned on. The criterion in terms of $\Delta E\left(\mathbf{k}\right)$ is that its real part should be much greater than its imaginary part, which is satisfied if that part of the $t$ matrix of the solutes which contributes to real scattering is small. The noble-metal alloys that obey the Hume-Rothery rules satisfy this weak-scattering criterion because the shielding cloud around dilute heterovalent impurities is more spread out than expected on the free-electron model, enhancing higher-angular-momentum phase shifts. Such a spreading of the shielding cloud is a consequence of band effects introduced by a large energy gap between the conduction band and the next unoccupied band. The small $t$ matrix or phase shifts cannot be calculated by perturbation theory because the change of wave function in the vicinity of the impurity is large; yet once they are known, their effects can be treated as small. It is shown for this type of alloy that a Bloch state with wave vector k in the pure metal will have exactly the same k in the alloy as the solutes are adiabatically turned on. The response of this alloy to electric and magnetic fields can be calculated from the same formulas as are used for pure metals, with the substitution of a phenomenological relaxation time appropriate for the alloy, and with energy levels of the alloy $E\left(\mathbf{k}\right)=E_0\left(\mathbf{k}\right)+\Delta E_r\left(\mathbf{k}\right)$ in place of the energy levels of the pure metal $E_0\left(\mathbf{k}\right)$, where $\Delta E_r\left(\mathbf{k}\right)$ is the real part of the energy shift $\Delta E\left(\mathbf{k}\right)$. In particular, de Haas-van Alphen measurements determine the shape of the Fermi surface of the alloy, which in general differs from that of the rigid-band model.