An accurate determination of the thermal vibration of rutile from the nuclear density distribution of the maximum-entropy analysis

The maximum-entropy method (MEM) can provide a high-resolution nuclear density distribution purely from experimental neutron diffraction data. The distribution expresses thermal smearing, which is caused by all kinds of thermal-vibration modes both harmonic and anharmonic. If the effective one-particle potential (OPP) is assumed to describe thermal smearing of nuclei, the potential parameters can be determined by least-squares refinement of the nuclear density distribution. By this method, the OPP parameters of rutile (TiO₂) are directly determined from the nuclear density distribution originally derived by Sakata, Uno, Takata & Howard [J. Appl. Cryst. (1993), 26, 159-164]. In the rutile case, the x coordinate of the O atom located at (x, x, 0) has to be determined before the OPP parameters are analysed. The atomic position is defined as the position where the first-order moment of the nuclear density becomes zero. The obtained x coordinate is 0.30477, which shows excellent agreement with the previous study of Rietveld analysis by Howard, Sabine & Dickson [Acta Cryst. (1991), B47, 462-468], i.e. 0.30478 (6). The higher-order moments of nuclear density are calculated in order to build an adequate OPP model. Among these values, none of the sixth order is significant and hence the OPP model up to fourth-order anharmonicity is employed. The potential parameters are refined by least-squares analysis using the above OPP model. For the Ti atom, nine OPP parameters (three harmonic and six fourth-order anharmonic) are determined with reliability factor R = 0.73%. For the O atom, twelve OPP parameters (three harmonic, three third- and six fourth-order anharmonic) are determined with R = 3.83%. The nuclear density of the O atom shows substantial skewness in rutile owing to the third-order anharmonicities. It is shown that the present method is a very powerful technique to determine the precise values for both harmonic and anharmonic potential parameters based on the OPP model in comparison with conventional structure analysis.