Crystal Dynamics of Potassium. I. Pseudopotential Analysis of Phonon Dispersion Curves at 9°K

Abstract

The frequencies of normal modes of vibration of potassium at 9°K have been measured by inelastic-neutron-scattering techniques. Certain selected frequencies are (units ${10}^{12}$ cps): $H_{15}$ 2.21±0.02, $P_4$ ${1.78}_5$±0.02, ${N_1}^{\prime }$ 2.40±0.04, ${N_3}^{\prime }$ 1.50±${0.02}_5$, and ${N_4}^{\prime }$ 0.53±0.02. The results are very similar, apart from a scale factor 1.65, to those for sodium. Analysis of the results has been carried out in terms both of conventional Born-von Kármán models and of potential functions defined in reciprocal space. A fifth-neighbor, axially symmetric force model has been used to compute the frequency distribution function for the normal modes and the associated heat capacity. The reciprocal-space analysis was performed in two ways: (a) in terms of a total potential function, whose Fourier transform is the effective interatomic potential between "neutral pseudo-atoms" of potassium, and (b) in terms of the screened pseudopotential for the conduction-electron-ion interaction. Analysis (a) shows that a wide variety of interatomic potentials, both with and without long-range oscillatory character, can be found which give a satisfactory fit to the results. These potentials are compared with those obtained from an analysis of x-ray scattering data for liquid potassium. The pseudopotentials obtained from analysis (b) are in good agreement with that derived by Bortolani from the Heine-Abarenkov model. A reanalysis of the phonon dispersion curves for sodium leads to very similar conclusions, confirming earlier work by Cochran.