Elastic Constants of and Wave Propagation in Antimony and Bismuth

Ultrasonic wave velocities for 14 different modes were obtained on two differently oriented single-crystal antimony cubes from the time between successive unrectified radio-frequency pulse echoes. This redundant set of data was fitted by a least-squares technique to Voigt theory to yield the six room-temperature adiabatic elastic-stiffness constants. In units of ${10}^{10}$ dyn/${\mathrm{cm}}^2$, $c_{11}=99.4\mathrm{}\left(1\right)\mathrm{}$, $c_{33}=44.5\mathrm{}\left(9\right)\mathrm{}$, $c_{44}=39.5\mathrm{}\left(5\right)\mathrm{}$, $c_{66}=34.2\mathrm{}\left(3\right)\mathrm{}$, $c_{13}=26.4\mathrm{}\left(4\right)\mathrm{}$, and $c_{14}=+21.6\mathrm{}\left(4\right)\mathrm{}$, the positive sign for $c_{14}$ following from our choice of positive Cartesian axes. When similarly treated, Eckstein, Lawson, and Reneker's bismuth data yield in these same units: $c_{11}=63.22\mathrm{}$, $c_{33}=38.11\mathrm{}$, $c_{44}=11.30\mathrm{}$, $c_{66}=19.40\mathrm{}$, $c_{13}=24.40\mathrm{}\pm 0.09$, $c_{14}=+7.20\mathrm{}$. Also included are a visual method of fixing the laboratory coordinate system in antimony by means of an imperfect cleavage plane, a calculation of the pure-mode directions in the mirror plane, a simple formula for choosing the nonextraneous value of $c_{13}$ for trigonal crystals having six independent elastic constants without resorting to lattice-stability criteria, and a calculation of the deviation of elastic-wave particle displacement and energy-flux directions from the propagation direction. For waves propagating in the (0,1,1) and ($0,\overline{1},1\mathrm{}$) directions, the particle-displacement deviations for antimony and bismuth do not exceed 15° and 13°, respectively, and corresponding energy-flux deviations up to 45° and 27° are obtained.