Dispersion relations of acoustic phonons in pyrope garnet; relationship between vibrational properties and elastic constants

Coherent inelastic neutron scattering measurements have been performed on a natural garnet pyrope from the Dora Maira Massif, Italian Western Alps, [(Mg_0.92Fe_0.05Ca_0.03)₃Al₂Si₃O₁₂, space group Ia3d, Z = 8, a = 11.459 Å] to determine the acoustic phonon-dispersion relations at room temperature. Phonon measurements have not previously been made on natural garnets, although the dispersion curves form a basis for developing and testing physically correct interatomic potentials for use in lattice dynamical models. Acoustic phonons propagating in the high-symmetry [100] and [110] directions were observed using the PRISMA spectrometer installed at the ISIS Neutron Spallation Source, U.K. The inelastic scans were performed around the (660), (080), (860), (10,6,0), and (0,10,0) Bragg points: Two transverse acoustic and one longitudinal acoustic phonon branches were measured. The observed acoustic phonon frequencies are in good agreement with those calculated on the basis of a semiempirical interatomic potential model. As a check, the elastic constants obtained from the slopes of the acoustic phonon branches are comparable with previous measurements obtained by Brillouin spectroscopy, although their large estimated errors reflect the complexity of the experimental measurements of phonon-dispersion curves on pyrope garnets.