Electronic Structure of Transition Metals. II. Phonon Spectra

The use of the new transition-metal model potential (TMMP), developed in the preceding paper, in nearly-free-electron second-order perturbation theory of the electronic structure and properties, in particular the phonon spectra, of the transition metals is discussed. It is discussed. It is pointed out that, as a consequence of the strong nonlocality ($\mathcal{E}$ dependence) of the TMMP parameter, $A_2\left(\mathcal{E}\right)\propto {(\mathcal{E}-{\mathcal{E}}_d)}^{-1\mathrm{}}$ near the position ${\mathcal{E}}_d$ of the $d$ -band resonance, first-order (nearly-free-electron) perturbation theory involving the diagonal matrix element $〈\overrightarrow{\mathrm{k}}\mathrm{}\left|V\right|\mathrm{}\overrightarrow{\mathrm{k}}〉$ of the TMMP breaks down at ${\mathcal{E}}_{\overrightarrow{\mathrm{k}}}={\mathcal{E}}_d$ and hence describes $s-d$ hybridization, but second-order terms involving the off-diagonal matrix elements ${\left|〈\overrightarrow{\mathrm{k}}\mathrm{}\right|V|\mathrm{}\overrightarrow{\mathrm{k}}+\overrightarrow{\mathrm{q}}〉|}^2\equiv {\mathrm{}\left|V\right(q\left)\right|}^2$ are off resonance, i.e., ${\mathcal{E}}_{\overrightarrow{\mathrm{k}}+\overrightarrow{\mathrm{q}}}\ne {\mathcal{E}}_d$, and therefore small. Thus electron-phonon-interaction matrix elements, proportional to $V\left(q\right)\mathrm{}$ in the diffraction model, can be small in the transition metals. On the basis of this property of the TMMP perturbation method, systematic trends in the lattice dynamics of simple and transition metals of the $3d$, $4d$, and $5d$ series in the Periodic Table are compared in the jellium model, and first-principles calculation of the phonon spectra of ten transition metals—Cu, Ag, Ni, Pd, Fe, Cr, Mo, W, Nb, and Ta—are performed in local TMMP approximation. The theory provides a sound basis for quantitative understanding of the lattice dynamics of these metals, and agreement with the observed phonon spectra is quite good.