スピネルMgAl<SUB>2</SUB>O<SUB>4</SUB>の高温構造解析における熱散漫散乱の補正と非調和熱振動の解析

Abstract

Thermal vibrations of atoms of a synthetic spinel MgA12O4 (a=8.0806Å, Z=8, Fd 3m) have been studied based on sets of X-ray single-crystal diffraction intensities collected at high temperatures up to 1700°C. The study was initiated by refining the temperature factor T (Q) of each atom in the form of the second-rankk tensor β, assuming the ellipsoidal model of harmonic vibration. Difference Fourier maps based on the structure factors obtained from the refinements at 1230°C (R=4.13 %), 1390°C (R=4. 32 %), and 1660°C (R=4. 31 %) revealed residual electron densities around the atoms at the tetrahedral site A (point symmetry 4 3 m), octahedral site B (3 m) and even around the oxygen atoms (3 m), the modes of occurrence suggesting anharmonic atomic thermal vibrations. In order to execute subsequent anharmonic refinement, the observed reflection intensities Iobs were then corrected for the intensities of thermal diffuse scattering ITDS, which is related to Ioba as Iobs=IBragg+ITDS-IBragg (1+α1), where IBragg is the true intensity of a Bragg reflection. The values of α1 were evaluated based on the model of one phonon and first order acoustic vibration and using the elastic constants cij. For anisotropic anharmonic refinement, we adopted a cumulant expansion of T (Q) in the form exp [∑n!inμpqrs……qpqqqrqs……](Johnson, 1969) and refined, in addition to βij, the coefficients γpqr and δpqrs of the third-and fourth-rank tensors, respectively. The result of the refinement includes : (1) The higher the heating temperature, the more intensified was the anharmonicity of the atomic thermal vibrations. (2) The oxygen u parameter was significantly shifted compared with the case in which anharmonic refinement was not made, the A-0 distances being contracted, while B-O lengthened. (3) The atom at the A site showed a large value of γ123, indicating that the thermal motion may well be represented by the shape of a tetrapod, while that at β showed a relatively large value of δ1122, thermal motion being represented by a set of eight lobes along the lines parallel to the cubic body diagonals. (4) The oxygen atom showed a conspicuous anisotropy particularly at 1660°C. In the difference Fourier map based on F_??_´s, obtained after the anharmonic refinement, the above-mentioned residual electron densities around the cation sites were almost cleared out.