Elastic anisotropy and phonon focusing for tetragonal crystals: Application to γ-TiAl

Abstract

A theoretical formalism to investigate the elastic anisotropy and phonon focusing for tetragonal crystals is described in detail in this paper. This is applied for γ-TiAl and both of the numerical indicators and the three dimensional images of elastic anisotropy are derived. The percentage of bulk and shear anisotropies indicate γ-TiAl exhibits relatively high degree of anisotropy in shear but small anisotropy in compressibility. Anisotropy indexes A∗ and AU demonstrate γ-TiAl possesses high degree of anisotropy, and this feature is also reflected in the three dimensional Young’s modulus, shear modulus and Poisson’s ratio. Three slowness surfaces are constructed for γ-TiAl to investigate the phonon focusing patterns, and the group velocity surfaces are also obtained to gain more insight into the elastic anisotropy. Due to elastic anisotropy, both of the slowness surfaces and the group velocity surfaces are nonspherical. The anisotropy is highest for slow transverse mode, and then followed by fast transverse mode and longitudinal mode. Interestingly, only transverse phonons cause caustics for γ-TiAl. Furthermore, the phase velocity and group velocity are not collinear for transverse mode in some directions, resulting in cuspidal edges along these directions in the group velocity surfaces.