Lattice vibrations of the icosahedral solid α-boron

Abstract

Crystalline \ensuremath{\alpha}-boron consists of ${\mathrm{B}}_{12}$ icosahedra in a rhombohedral lattice of R3\ifmmode\bar\else\textasciimacron\fi{}m space-group symmetry. We here carry out a classical force-field analysis of the lattice vibrations. The q=0 Brillouin-zone vibrations are treated as those of a ${\mathit{D}}_{3\mathit{d}}$-point-group-symmetry ${\mathrm{B}}_{12}$ cluster perturbed by intericosahedral crystalline forces; valence-force constants are fitted to account for Raman and ir spectral data. Two-centered intericosahedral bonds are found to be twice as strong as intraicosahedral bonds, while three-centered crystalline bonds are almost as strong as those within a ${\mathrm{B}}_{12}$ unit. The highest-frequency Raman line arises from the breathing mode, strongly perturbed by the two-centered interactions. The lowest-observed-frequency Raman line is attributed to ${\mathrm{B}}_{12}$ libration. As crystal-force-field strengths are turned up, noncrossing of frequencies is encountered; we, therefore, correlate \ensuremath{\alpha}-crystal modes with ${\mathit{I}}_{\mathit{h}}$ regular-icosahedral and ${\mathit{D}}_{3\mathit{d}}$ ${\mathrm{B}}_{12}$-cluster modes through eigenvector expansions. Useful classical predictions are made of ir intensities for wave vector q=0 modes by considering adjacent bond stretching; a prediction of Raman intensities in terms of bond polarizabilities appears to be of very limited value. The phonon analysis is extended from the Brillouin-zone center to the edges by introducing phase-angle differences along two distinct (one ${\mathit{C}}_{3}$ and one ${\mathit{C}}_{2}$) rotational-symmetry axes. The acoustical-branch wave speeds are predicted to be 1.1\ifmmode\times\else\texttimes\fi{}${10}^{6}$ and 0.38\ifmmode\times\else\texttimes\fi{}${10}^{6}$ cm/sec for the c-direction longitudinal and transverse components, respectively. Finally, we consider possible origins of the 525-${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ Raman line with anomalous polarization and width; most likely this is a Raman electronic line associated with vacancies and ${\mathrm{B}}_{11}$ units.