A direct investigation of thermal vibrations of beryllium in real space through the maximum-entropy method applied to single-crystal neutron diffraction data

The thermal vibrations of beryllium metal were determined directly from the nuclear densities obtained by the maximum-entropy method (MEM) using neutron single-crystal data. A high-resolution nuclear density distribution of beryllium was obtained by applying the MEM to the 48 structure factors with sin θ/λ < 1.41 Å⁻¹ from a previous study [Larsen, Lehmann & Merisalo (1980). Acta Cryst. A36, 159–163], which showed small but significant cubic anharmonicity in beryllium by least-squares refinement of the structure factors. In the present study, quartic as well as cubic anharmonicities are clearly visible in the MEM nuclear density. In order to determine anharmonic thermal-vibration parameters, a three-dimensional function was fitted to the MEM nuclear density around the atom site. The one-particle potential was used to model the thermal vibrations up to quartic terms. The least-squares-fit values were γ = −0.306 eV Å⁻³ for the third- and α₄₀ = − 1.02, β₂₀ = 2.95 and γ₀₀ = − 3.28 eV Å⁻⁴ for the fourth-order anharmonic parameters. Thus, the atomic potential in the basal plane is hardened against the bipyramidal space around the tetrahedral holes of the hexagonal-close-packed structure. It is softened towards the center of the octahedral voids. Least-squares refinement of the MEM nuclear density gives a standard deviation of about 5 for the last digit of the anharmonic parameters. However, there is added uncertainty in the parameters because of the relationship of the reliability of the MEM density distribution to the standard deviations of the measured intensities. Judging from previous studies of the thermal parameters for beryllium based on least-squares refinement of observed structure factors, it is estimated that values determined here for the anharmonic parameters are reliable to the first digit after the decimal point.