E. Iguchi

Theoretical estimation of overlapping repulsive energies, polarizabilities, and lattice energy in WO3

Abstract Repulsive energies resulting from overlap of W 6+ W 6+ , W 6+ O 2− , and O 2− O 2− pairs were calculated within the framework of the free ion model constructed by Wedepohl. Polarizabilities of ions in the monoclinic structure of WO 3 were obtained following the theoretical treatments of Parker. Using these values, the lattice energy (cohesive energy) of the monoclinic structure of WO 3 , consisting of the long range coulombic interaction energy, the repulsive energy, van der Waals interaction energy, and the zero point energy, was evaluated to be −235.53 eV per WO 3 formula unit and compared with the experimental value determined by the Born-Haber cycle analysis, −(239.23 ∼ 243.44) eV per WO 3 formula unit, with a discrepancy of 3%. In comparison with other oxides, van der Waals interactions are found to contribute remarkably to the lattice energy in WO 3 . The lattice energy in the e-phase at low temperatures was also calculated, using the same parameters employed in the calculation of the lattice energy in the monoclinic structure. The lattice energy in the e-phase is −235.60 eV per WO 3 formula unit which is found to be in good agreement with the experimental value. In the e-phase, the contribution of van der Waals interactions plays also an important role in the lattice energy as well as the monoclinic structure.

Polaron interaction energies in reduced tungsten trioxide

Abstract Consideration of the properties of reduced tungsten trioxide suggest that the mobile charge carriers are polarons. As it is uncertain how the presence of polarons will influence the microstructures of the crystallographic shear ( CS ) planes present in reduced tungsten trioxide we have calculated both the polaron- CS plane and polaron-polaron interaction energy for a variety of circumstances. Three CS plane geometries were considered, {102}, {103}, and {001} 1 CS plane arrays, and the nominal compositions of the crystals ranged from WO 2.70 to WO 3.0 . The polarons were assumed to have radii from 0.6 to 1.0 nm and the polaron- CS plane electrostatic interaction was assumed to be screened. The results suggest that for the most part the total interaction energy is small and is unlikely to be of major importance in controlling the microstructures found in CS planes. However, at very high polaron densities the interaction energy could be appreciable and may have some influence on the existence range of CS phases.

Energy calculations for planar faults in reduced rutile (TiO2)

Calculations have been made of the energetics of reduced rutile (TiO2) crystals containing (a) an isolated {132} c. s. plane, (b) an ordered array of {132} c. s. planes corresponding to an oxide of composition Ti20O39, (c) a vacancy disc lying on {132} planes and (d) an ordered array of vacancy discs lying on {132} planes, which gives the crystal a composition of Ti20O39. The calculations were made by using the polarizable point ion shell model, and by taking the electronic polarizabilities of the ions involved into account. The results show that the c. s. planes are preferred to vacancy discs, and that this is largely due to the electronic polarization energy terms. The enthalpy of formation of an isolated {132} c. s. plane in rutile was calculated. A comparison with an estimate of the enthalpy of formation of the vacancy disc suggests that vacancy discs will be converted into c. s. planes in real crystals and will not exist as discrete defects. The change of the energy terms as the polarizabilities of the ions varied was also calculated to quantify the relation between preferred defect structure and dielectric constant. It was found that a change in dielectric constant did not have a significant effect upon the stability of vacancy discs, but did have a large effect on the energetics of c. s. planes, which suggests that c. s. planes are only favoured in crystals of high dielectric constant. It is shown that electronic polarizability is of importance in stabilizing {132} c. s. planes and that the polarizability of the O2 ─ ions, in particular, is of great significance.

The elastic strain energy of crystallographic shear planes in reduced tungsten trioxide

Abstract The elastic strain energy in reduced tungsten trioxide, which contains crystallographic shear ( CS ) planes, has been calculated using Fourier transform theory. This allows the effects of non nearest-neighbor CS planes to be evaluated, and also enables one to assess the relaxation energy of ions in the CS planes as well as the strain energy of the matrix between the CS planes. The results are presented for {10m} (2 ⩽ m ⩽ 7) and {001} CS plane types. They are compared with experimental data and also with the results of previous calculations using classical elasticity theory.

Strain energy between parallel {001} crystallographic shear planes in the tungsten trioxide structure

In order to clarify the appropriateness of the proposal of Iguchi and Tilley, in which the force between two parallel crystallographic shear (CS) planes at low separations is an attractive one and, at high separations, a repulsive one, the strain energy between {001} CS planes was calculated as a function of the separation. The strain energy was presumed to be due to the forces acting on cations in the CS planes and was calculated by using classical elasticity theory by way of the elastic Greens function. The result obtained is in agreement with the original proposal but the CS plane separation at which the force between them changes from an attractive to a repulsive one is much larger than was estimated previously. The strain-energy curve has, in addition, a plateau where the interaction of the paired CS planes is likely to be in the metastable state. The difference between other related results and ours is discussed in terms of the difference of the model employed.

The elastic strain energy and stability of some idealized lead-bismuth sulphides

Abstract The elastic strain energy of idealized structures related to the homologous series of chemically twinned phasesxPbS · Bi2S3, which includes the naturally occurring sulphides lillianite, Pb3Bi2S6, and heyrovskyite, Pb6Bi2S9, have been calculated using the Fourier transform method and an ionic model to represent the bonding. The elastic strain energy depends critically upon the distribution of the Pb and Bi atoms over the sites in the twin planes and the PbS-like matrix between the twin planes. The elastic strain energy terms so calculated do not explain why lillianite and heyrovskyite are the only stable compounds but do account for the fact that doping with other atoms can stabilize members of the homologous series not found in the pseudo-binary PbS Bi2S3 system.

Strain energy due to {001} crystallographic shear planes in materials possessing the ReO3 structure

Abstract The elastic strain energy in a structure of the ReO3 type containing ordered arrays of {001} CS planes has been calculated. The values obtained are for the elastic strain energy of the matrix between CS planes and also the relaxation energy of the ions in the CS planes themselves. Interactions from all the CS planes in the crystal are summed and not just those from nearest neighbors. The extent to which the CS planes can be considered to transmit the forces which strain the crystal is considered by including a variable parameter, α, in the calculations, which is related to the type of chemical bonding present in the CS planes. The results are compared with experimental observations in the WO3Nb2O5 and NbO2F systems. It is concluded that the value of α is high for WO3 doped with Nb2O5 and low for NbO2F in accord with the expectations of chemical bonding. The results also support the view that elastic strain energy is important in influencing the microstructures observed in crystals containing CS planes.

Elastic strain energy and microstructures of orthorhombic tin-tungsten bronzes

Abstract The elastic strain energy of the series of orthorhombic I bronzes which occur in the SnWO system has been calculated using the Fourier transform method. The results of these calculations are compared with microstructures and phase analysis carried out by transmission electron microscopy. The two are in good agreement, suggesting that elastic strain energy is important in materials containing planar faults other than crystallographic shear phases.

Electronic and ionic polarizabilities in some rutile type crystals

In several earlier papers (1-J), we suggested that certain transition metal oxides with a high dielectric constant as well as large refractive indices such as TiO&-utile), W03, and H-Nb205 use crystallographic shear (CS) planes to accommodate nonstoichiometry. In view of the correlation between a high dielectric constant and CS planes, the relaxation due to polarizabilities seems to be one of the factors which predisposes such a material to this type of stoichiometric change. To reveal this correlation in greater depth, we have calculated electronic and ionic polarizabilities of i-utile, because the most reliable experimental data are available for this crystal compared to W03 and H-Nb205 . We found that rutile has an extremely large value for the electronic polarizability of the anion (2, 3); this polarizability appears to be one of the most important factors in formation of CS planes. To compare the electronic polarizability of the anion in TiOz with that of other r-utile-type crystals, we have determined polarizabilities of Ti02 ,i SnOz , Pb02 , MgF2 , and ZnFz , the dielectric properties of which have been obtained qperimentally.