Equations for diffuse scattering from materials with multiple sublattices

Abstract

General equations are presented for the diffuse scattering due to local ionic arrangements and displacements in systems with more than one ion per lattice point. Up to fourth-order terms are included. These are placed in a form easy to evaluate in specific cases; examples are given of a solid-solution oxide, and an oxide with vacant sites. Methods for employing these equations for the separation of the various contributions are discussed.

Short-range ordering of vacancies in TiO at 1323 K

“…2. The diffuse maximum at ],2,0, which we had observed at 1288 K and attributed to long-range vacancy-vacancy interactions out to the sixth neighbor shell The diffuse intensity I 0 (in absolute units) includes contributions from local order (IsRo) of the vacancies and from local static and thermal displacements of atoms from the average structure (Hayakawa & Cohen, 1975):…”

“…Accordingly we will make only qualitative comments on these terms, and there is no point in providing the detailed equations. (These can be readily obtained from the procedures described in Hayakawa & Cohen, 1975. ) After correcting the raw data from the two filters for surface roughness and parasitic scattering, these were placed on an absolute scale, calculated Compton scattering was subtracted (International Tables for X- ray Crystallography, 1962) and the ~i, ~,, t~, ~ were obtained by a least-squares fit to the total remaining intensity (Williams, 1972).…”

A study of the local atomic structure in Hg0.80Cd0.20Te using x-ray diffuse scattering

“…Quintana 16 has detailed the derivation of the general diffuse scattering equation for pseudobinary zincblende F43m structures following the method of Georgopoulos and Cohen 15 and using the multiple sublattice formalism presented by Hayakawa and Cohen. 17 He has shown that the Taylor expansion of the kinematic scattering equation used by Borie and Sparks 14 and Georgopoulos and Cohen 15 yields seventy independent terms in the general scattering equation for the pseudobinary zincblende structure. Cenedesee?…”

Equations for diffuse scattering from pseudobinary sphaleriteF\overline{4}3msystems

“…This method is not limited to binary systems. Hayakawa & Cohen (1975) have provided a formalism for extending the system to multiple-sublattice m3m systems under scatteringfactor assumptions similar to those used by Borie & Sparks (1971). However, for ternary or highercomponent systems or for systems with lower symmetry than m3m, the number of terms quickly increases and in some cases it is impractical to attempt to solve them with real data from a single measurement.…”

“…Equations relating the various terms at symmetry-related points in reciprocal space are included as well as the least-squares method required to solve the matrix equations from diffraction data taken in a volume of reciprocal space. Hayakawa & Cohen (1975) nor Georgopoulos & Cohen (1977) properly account for complex scattering factors introduced by the dispersion corrections [i.e.…”