X-Ray Determination of Phonon Dispersion in Vanadium

Phonon frequencies along principal symmetry directions were measured in vanadium by means of thermal diffuse scattering of x rays. A polishing and annealing procedure was adopted which eliminated extra scattering associated with impurities. The dispersion curves have initial slopes in agreement with the measured elastic constants. We found no evidence of any irregularity at low frequencies in any of the dispersion curves and in the calculated frequency spectrum. A low-frequency peak in the frequency spectrum has been reported in two inelastic neutron-scattering experiments and was attributed to a Kohn effect. A Born-von K\'arm\'an model with seven-neighbor forces gave an adequate fit to the dispersion curves within experimental uncertainty. The derived interatomic force constants are not consistent with an axially symmetric model. The calculated frequency spectrum is in generally good agreement with the inelastic neutron determinations, but indicates the presence of more detailed structure. The high-energy tail observed in many experiments is not confirmed, nor does the $\frac{g(\ensuremath{\nu})}{{\ensuremath{\nu}}^{2}}$ curve exhibit any hump in the low-frequency region.