Neutron diffraction studies of anharmonic temperature factors in BaF₂

Abstract

Accurate integrated intensities for the Bragg reflexion of neutrons from BaF2 have been measured at various temperatures in the range 20–600°C and corrections for the effects of both thermal diffuse scattering and extinction have been applied. The experimental data show systematic deviations from the predictions of a model which assumes harmonic temperature factors for both types of atom, with observed intensities differing by up to ± 60% from the calculated `harmonic' values. Vibrational anharmonicity can be allowed for by using an effective one-particle potential of the form: Vj(r) = V0j + ½αj (x2 + y2 + z2) + βjxyz, where x, y and z are the coordinates defining the instantaneous displacement r of the nucleus of the jth atom and αj and βj are the coefficients of the quadratic and cubic terms respectively in the potential expansion. The site symmetry of the atoms allows the anharmonicity parameter βj to be non-zero only for the fluorine atoms, and introduction of this single parameter brings the observed and calculated structure factors into very good agreement (R ~ 1%) at all temperatures: the R value obtained for a harmonic model increases from 1.8% at room temperature to 8.9% at 600°C. The value obtained for βF is −3.06 × 10−12 erg.Å−3 at room temperature, falling to −2.52 × 10−12 erg.Å−3 at 600°C. The present measurements provide no evidence for the existence of any appreciable anisotropic quartic, or higher-order, contributions to the temperature factors. The ratio of nuclear scattering lengths, bBa/bF, was refined to 0.932 (± 0.004), which gives a value for the nuclear scattering length of barium of bBa = 0.522 (± 0.011) × 10−12 cm, assuming bF = 0.560 (± 0.010) × 10−12 cm.