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 fro m the nuclear densities obtained by the maximum-entropy method (MEM) u sing neutron single-crystal data. A high-resolution nuclear density di stribution of beryllium was obtained by applying the MEM to the 48 str ucture factors with sintheta/lambda < 1.41 angstrom-1 from a previous study [Larsen, Lehmann & Merisalo (1980). Acta Cryst. A36, 159-163], w hich showed small but significant cubic anharmonicity in beryllium by least-squares refinement of the structure factors. In the present stud y, quartic as well as cubic anharmonicities are clearly visible in the MEM nuclear density. In order to determine anharmonic thermal-vibrati on parameters, a three-dimensional function was fitted to the MEM nucl ear 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 gamma = -0.306 eV angstrom-3 for the third- and alpha 40 = -1.02, beta20 = 2.95 and gamma00 = -3.28 eV angstrom-4 for the fo urth-order anharmonic parameters. Thus, the atomic potential in the ba sal plane is hardened against the bipyramidal space around the tetrahe dral holes of the hexagonal-close-packed structure. It is softened tow ards the center of the octahedral voids. Least-squares refinement of t he MEM nuclear density gives a standard deviation of about 5 for the l ast digit of the anharmonic parameters. However, there is added uncert ainty in the parameters because of the relationship of the reliability of the MEM density distribution to the standard deviations of the mea sured intensities. Judging from previous studies of the thermal parame ters for beryllium based on least-squares refinement of observed struc ture factors, it is estimated that values determined here for the anha rmonic parameters are reliable to the first digit after the decimal po int.