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angle angular frequency anharmonicity atoms binary axes Brillouin zone calculation character tables coefficients components contains corresponding critical points crystalline cubic crystal defined denoted dipole direction displacements effect eigenfunctions elastic waves electric electromagnetic waves electronic equation example expression factor group field formula Fourier matrix free molecule fundamental vibration given group 9 hamiltonian interaction ions irreducible representations Let us consider linear longitudinal matrix elements molecular molecule motion neutrons normal coordinates notation obtain optical orientation orthogonal oscillators permittivity phonon plane point group polarizability polarization potential energy primitive cell principal axis principal vibrations propagation quotient group radiation Raman scattering reciprocal lattice relation respectively rotation selection rules space group spectra spectrum subgroup symmetry operation tensor tion transformation transition translation transverse values vector q vibration modes wave functions wave vector zero