Lattice Dynamics of Beryllium

Abstract

Measurements of the dispersion relations for waves propagating in the [0001] and [$01\overline{1}0$] directions in beryllium have been made using the MTR phased-chopper slow-neutron velocity selector. In the [$01\overline{1}0$] direction only one of the transverse modes was available for investigation, that having atom displacement normal to the basal plane. Two distinct frequencies were observed at the center of the Brillouin zone for the optical branches, and the corresponding branches are referred to as upper optical and lower optical. For both symmetry directions, the upper optical branch corresponds to the mode of vibration having the polarization vector parallel to the hexagonal axis. Pertinent frequencies in units of ${10}^{13}$ ${\sec }^{-1\mathrm{}}$ are: (at the center of the zone) upper optical, 1.99±0.07; lower optical, 1.33±0.04; (at the zone boundary in the [0001] direction) lower optical and transverse acoustical, 1.01±0.06; upper optical and longitudinal acoustical, 1.57±0.07; (at the zone boundary in the [$01\overline{1}0$] direction) upper optical, 1.69±0.09; lower optical, 1.63±0.08; longitudinal acoustical, 1.54±0.08; transverse acoustical, 1.21±0.05. The mode of vibration for a particular phonon was determined from the region of reciprocal space in which the transition was observed. The initial slopes of the acoustical branches agree well with elastic constant data, except for the longitudinal branch in the [0001] direction. Lattice dynamics models of Begbie and Born and of Slutsky and Garland, the latter extended to include interactions with fourth and fifth nearest neighbors, give limited agreement with the present data only when the force constants in the models are evaluated from the neutron-scattering data.