Dynamic deformation and the Debye–Waller factors for silicon-like crystals

Abstract

MITSUKO ONODA, MASANOBU SAEKI AND ISAO KAWADA 957 easier than that of the fully occupied Ti layer. It is probable that the stacking faults which occur at a low temperature such as 683 K are due only to the slide between the sandwiches, and the experimental data shown in Fig. 1 should be interpreted appropriately on the basis of the extended model.Experimental patterns which suggest the occurrence of stacking faults are often observed for the various temperatures and compositions in the Ti-S system. The method of analysis of the structure with stacking faults described above may be effectively used for considering the phase-relation problem in this system. The effect of dynamic deformation on the Debye-Waller factor of silicon and, to a slightly lesser extent, the other three elements, is investigated. The Debye-Waller factor for the shells only in the the shell models is calculated. The effect introduced by dynamic deformation whereby the Debye-Waller B value varies with scattering vector K is evaluated. Finally, the anisotropic Debye-Waller factor components for the bond charges are calculated for all four elements. It is found that the bond charges in the bond-charge model and the shells in the shell model vibrate substantially less than the main atomic cores. It is concluded that if the models are at all realistic then the effects of dynamic deformation on the Debye-Waller factors of these elements should be seriously considered.

A new theory for X-ray diffraction

“…to B $ 0.06 nm 2 , if the scattering follows kinematical theory, and B $ 0.08 nm 2 if it follows dynamical theory. This B factor is difficult to reconcile with the expected value of $0.02 nm 2 (Reid & Pirie, 1980). The experimentally measured parameter on bulk Si of B $0.046 nm 2 , by Aldred & Hart (1973), represents the maximum value, because their experimental conditions were highly biased towards the core value.…”

“…In strongly bonded Si the shell will be damped compared with the vibrations of the core, because it is linked via the bonding to other atoms (Reid & Pirie, 1980). The core Debye-Waller factor has been calculated and measured accurately, see for example the compilation and arguments in Reid & Pirie, and the measurements of Aldred & Hart (1973), who give a value close to the calculations at room temperature of B = 0.04613 nm 2 using high-order reflections; this value is therefore more closely related to the core.…”

“…The experimentally measured parameter on bulk Si of B $0.046 nm 2 , by Aldred & Hart (1973), represents the maximum value, because their experimental conditions were highly biased towards the core value. Reid & Pirie (1980) have critically reviewed the literature as well as having calculated this value for Si by numerous approaches, and concluded that B $0.02 nm 2 is the most likely value. The structure of Si is known and therefore there is little room for manoeuvre.…”

Modification of X-ray atomic scattering factors in electic fields – validation of the shell model

“…Non-dipole contributions to atomic scattering factors in H have been previously considered by Matthew & Gravano (1979). This suggests that the substantial body of work on deformation coupling on X-ray scattering factors and Debye-Waller factors by Melvin, Pirie & Smith (1968), Reid (1974a, b), Robertson & Reid (1979) and Reid & Pirie (1980) is well founded quantum mechanically. It gives support to their conclusion that the 'vibration' of the outer-electron charge distributions is much smaller than for the core electrons, and that this may lead to effective Debye-Waller factors B deviating by a few per cent from those of the ion cores for low-order reflections.…”

“…Melvin, Pirie & Smith (1968) and Reid (1974a, b) plausibly assign to the core and the shell quantum-mechanically calculated charge distributions characteristic of the inner and outer electrons in the ion and, using the core-shell displacements calculated from the lattice-dynamical shell-model calculations, estimate the resultant changes in diffuse X-ray intensity due to the modified scattering factors. Robertson & Reid (1979) have 0108-7673/84/060716-06501.50 made similar applications of the shell model to X-ray scattering from Si, while Reid & Pirie (1980) and Reid (1983) have used the model to estimate the effect of ionic deformation on Debye-Waller factors. Although March & Wilkins (1978) have developed a method of calculating elastic X-ray scattering from solids in terms of non-rigid pseudoatoms, this quantum-mechanical approach cannot be readily compared with semi-classical shell-model specifications of charge deformation.…”