A new method for calculation of crystal susceptibilities for X-ray diffraction at arbitrary wavelength

Abstract

A novel method for the calculation of the X-ray susceptibility of a crystal in a wide range of radiation wavelengths is described. An analytical interpolation of one-electron wave functions is built to approximate the solution to Hartree-Fock equations for all atoms and ions of the periodic system of elements with high accuracy. These functions allow the calculation of the atomic form factors in the entire range of a transmitted momentum as well as the description of their anisotropy taking into account external and intracrystalline fields. Also, an analytical approximation for the force matrix of an arbitrary crystal is obtained and the microscopic calculation of the Debye-Waller factor for crystals with a complicated unit cell is presented.

Many-Electron Atoms

“…One of the advantages of the OM approach is the smallness of the nonadditive contribution E 1 in comparison with the total energy, this is in contrast to the HF method [28]. The simple structure of the wave functions can be also used for accounting of a self-consistent relativistic contribution to the atomic Hamiltonian [14]. Direct physical interpretation of the OM wave functions can be used to find the change of the atomic form factors in the external field.…”

“…This section deals with the application of OM to above mentioned problems and construction of analytical solutions using the representation of oneelectron wave functions on the basis of effective charges model (ECM). This approach has been used in [14] to analytically derive the atomic scattering factors required for calculation of X-ray polarizabilities of crystals [24].…”

“…(8.133). The effective charges for all neutral atoms and some ions were listed in the paper [14], the atomic units are used in the calculations and the total Hartree-Fock energy for nonrelativistic atoms [26] is presented in the last column. The OM estimations for the total atomic energy are in a good agreement with the HF results.…”

“…The OM is also applicable in this case (Sect. 8.4), providing an effective charges model (ECM) [14,15] for basic set of one-electron wave functions. Such approach guarantees the accuracy of zeroth approximation for atomic characteristics, which is comparable to the results after HF method.…”

Calculation of the Debye–Waller factor of crystals using the n-dimensional Debye function involving binomial coefficients and incomplete gamma functions

“…Experimentally, the DWF can be measured using different techniques such as Xray diffraction [12][13][14][15][16][17][18], neutron diffraction [10], Mössbauer spectroscopy [19][20][21][22][23], and radioactive ion-beam technique [24]. The results obtained for the DWF are expressed, however, only at fixed temperatures, such as room temperature (293 K) [25,26], and hence these results are difficult to interpolate into different temperature regions and do not correspond completely to the experimental situation.…”