Lattice vibrations of the icosahedral solid α-boron

Crystalline {α}-boron consists of B₁₂ icosahedra in a rhombohedral lattice of R3{}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 D3d-point-group-symmetry B₁₂ 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 B₁₂ 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 B₁₂ libration. As crystal-force-field strengths are turned up, noncrossing of frequencies is encountered; we, therefore, correlate {α}-crystal modes with Iₕ regular-icosahedral and D3d B₁₂-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 C₃ and one C₂) rotational-symmetry axes. The acoustical-branch wave speeds are predicted to be 1.1×{}10⁶ and 0.38×{}10⁶ cm/sec for the c-direction longitudinal and transverse components, respectively. Finally, we consider possible origins of the 525-cm^-1 Raman line with anomalous polarization and width; most likely this is a Raman electronic line associated with vacancies and B₁₁ units.