X-ray Form Factors Research Articles

Charge asphericity in vanadium metal

X-ray form factors have been calculated for metallic vanadium by the APW method. The charge asphericity is estimated using ratios of the integrated intensities of the X-ray scattering for reflection pairs having the same value of sin( theta / lambda ). The charge asphericity obtained from a state-dependent potential, which was made to reproduce the dimensions and shape of the Fermi hole pocket at N-points, is larger than that obtained from state-independent potentials. The ratios of the integrated intensities obtained by the state-dependent potential are in reasonable agreement with the experimental ones determined by Weiss and Demarco (1965) and Diana and Mazzone (1974) and are in excellent agreement with a recent experiment performed by Ohba et al. (1981).

Charge density of group IVA transition-metal dichalcogenides

A systematic study of electronic properties of the layered group IVA transition-metal dichalcogenides TiS2, TiSe2, ZrS2 and ZrSe2 has been performed using the fully self-consistent OPW method according to the local density formalism. The X-ray form factors and the charge density are presented. The anisotropy, the bonding properties and the layer-layer interaction are analysed and discussed in detail. Also a general conclusion on the electronic properties of the group IVA transition-metal dichalcogenides is given.

Form factors and total scattering intensities for the helium-like ions from explicitly correlated wavefunctions

Integral formulae necessary to compute X-ray form factors F( mu ) and total scattered X-ray intensities It( mu ) from explicitly correlated wavefunctions of the form psi (r1,r2,r12)=(4 pi )-1 Sigma k=1LCk(1+or-P12)exp(- alpha kr1- beta kr2- gamma kr12) are derived. They are then applied to such wavefunctions recently constructed by the authors for the ground states of the helium-like ions from H- through Mg10+. The results are presented in the form of convenient theoretically-based interpolation functions for F( mu ) and It( mu ). Numerous tests indicate that these interpolation functions are very accurate. Parameters that appear in the small and large mu expansions of F( mu ) and It( mu ) are tabulated along with some parameters that appear in the Bethe-Born asymptotic cross section for total inelastic scattering of fast charged particles. New internal consistency conditions for the exact and experimental F( mu ) and It( mu ) functions are derived.

On the use of generalized x-ray scattering factors for analysis of charge density from gas-phase electron diffraction intensities

Generalized x-ray scattering factors for H2 (1Σg+) and N2 (1Σg+) are reported for [2‖2] and [3‖3] expansions. Results for O2 (3Σg−) include [2‖2], [3‖3], and [4‖4] expansions. The multipole analysis of the molecular x-ray form factor of these diatomics is a guide to a rational choice for the pseudoatom multipoles and for the radial basis functions in the Kohl–Bartell model [J. Chem. Phys. 51, 2891 (1969)]. It appears that by using more multipoles, relatively simple radial functions for the pseudoatoms are needed for a Kohl–Bartell model in the analysis of electron diffraction intensities for charge density information.

X-ray form factors and the Compton profile in crystalline aluminum

X-ray form factors and the Compton profile in crystalline aluminum

Dependence of atomic Compton profiles upon the Xalphaexchange parameter

Non-relativistic atomic Compton profiles (in the impulse approximation) and X-ray form factors have been calculated for neon, argon and krypton in the Hartree-Fock approximation (HF) and in the Xalpha exchange approximation. A strong alpha dependence of the Compton profiles is found. Compton profiles and form factors in the Xalpha approximation match HF closely for argon and krypton when the virial theorem value of alpha is used. However for neon, the virial theorem Xalpha Compton profile at zero momentum transfer (2.766 au) is significantly higher than the HF value (2.726 au).

Metallic-Cu x-ray form factors

Two sets of experimental Cu x-ray form factors are compared for consistency in terms of a simple charge-distribution model. The results of Temkin, Henrich, and Raccah fit the model very well, while those of Hosoya and Yamagishi do not give a satisfactory fit.

Electronic charge density of aluminum

The valence electronic charge density is calculated for aluminum from wave-functions obtained via Ashcroft's pseudopotential. A contour plot of the charge density is presented in the (100) plane. The Fourier transform of the charge density is used to calculate the atomic form factors which are compared with experimental X-ray form factors. The accuracy of the wavefunctions are further tested by comparing calculations of the imaginary part of the frequency dependent dielectric function with and without the effect of core states.

The Z−1 Expansion of the Nuclear Magnetic Shielding Constant and X-Ray Form Factor for 2-, 3-, and 4-Electron Ions

The convergence of the Z−1 expansion of the nuclear magnetic shielding constant σ and the X-ray form factor F(μ) is studied within the Hartree–Fock (HF) approximation for the ground states of 2-, 3-, and 4-electron atoms and ions. The agreement of σ through first order with the total HF σ is excellent, whereas the agreement between first-order and total HF F(μ)'s is unsatisfactory.

A Consistency Test for X-Ray Form Factors

A Consistency Test for X-Ray Form Factors

On the relationship of the X-ray form factor to the 1-matrix in momentum space

The X-ray form factor F(μ) is shown to be proportional to the convolution of the momentum space charge density matrix.

Theoretical Form Factors of 3d Transition Metals

X-ray form factors of V, Cr, Fe, Ni and Cu, and magnetic form factors of Ni are calculated from the change density distributions obtained by the band calculation. The results depend much on the potential used, but not on the method, because Green function method gives almost the same results as APW method, if the potential is the same. Here, the form factors are evaluated from the wave functions obtained by the Green function method with the self-consistent potential. Then, the theoretical results are compared with those obtained by the experiments.

Application of the Method of Tight Binding to the Calculation of the Energy Band Structures of Diamond, Silicon, and Sodium Crystals

The method of tight binding is used to calculate the energy band structure of the diamond, silicon, and sodium crystals. The wave functions of the valence and conduction bands of diamond are expanded in terms of Bloch sums constructed from the $1s$, $2s$, and $2p$ Hartree-Fock atomic orbitals, and two different crystal potentials [the muffin-tin and the overlapping atomic potential (OAP)] were used. With the muffin-tin potential, the tight-binding and the augmented-plane-wave (APW) method yield nearly identical valence band structure, and their conduction bands show only minor differences. When the OAP is used, the results of the tight-binding scheme differ appreciably from those of Bassani and Yoshimine by the method of orthogonalized plane waves (OPW), the discrepancy being attrubuted to incomplete convergence of the latter calculations. The tight-binding structures of the valence band derived from the two different potentials agree quite well with each other, but considerable deviations are found in the conduction band. The x-ray form factors calculated by means of the tight-binding wave functions are in good agreement with experiment and represent a considerable improvement over a simple superposition of atomic charges. The convergence of the tight-binding method with respect to the higher atomic orbitals has been examined. It is found that addition of $3s$, $3p$, and $3d$ Bloch sums to the wave functions of diamond has only small effects on the energy of the valence and conduction bands. A similar tight-binding calculation has been performed for the band structure of silicon using OAP, and the results are in good agreement with those of a modified scheme of the method of OPW using as basis functions 609 OPW's as well as the Bloch sums of the core states. For the base of sodium, the method of tight binding gives conduction-band energies in good agreement with the APW-type calculations of Schlosser and Marcus. Generalization of the method of tight binding by using single-Gaussian Bloch sums is discussed, and the use of this scheme leads to substantial improvements for the case of diamond.

Self-Consistent Energy Bands and Related Properties of Boron Phosphide

A first-principles self-consistent orthogonalized-plane-wave energy-band calculation has been performed for cubic boron phosphide (BP) using a nonrelativistic formalism and Slater's free-electron exchange approximation. These are the first fully self-consistent energy-band solutions reported for BP. The imaginary part of the dielectric constant, spin-orbit splittings, effective masses, deformation energies, and the x-ray form factors (Fourier transforms of the electron charge density) have been calculated. The theoretical results are compared with the available experimental data.

Simple model potential approach to the X-ray form factor in aluminum

The conduction electron charge density in Al is computed to first order in terms of a simple model potential. Very good agreement is obtained with recent X-ray data.

Electronic Structure and Optical Spectrum of Boron Arsenide

A first-principles self-consistent orthogonalized-plane-wave (SCOPW) energy-band calculation has been performed for cubic BAs using a nonrelativistic formalism and Slater's free-electron exchange approximation. These are the first fully self-consistent energy-band solutions reported for BAs. The imaginary part of the dielectric constant, spin-orbit splittings, effective masses, deformation energies, and the x-ray form factors (Fourier transforms of the electron charge density) have been calculated. The theoretical results are compared with the available experimental data.

Energy-Band Structure of Aluminum Arsenide

A first-principles self-consistent orthogonalized-plane-wave energy-band calculation has been performed for cubic AlAs using a nonrelativistic formalism and Slater's free-electron-exchange approximation. These are the first fully convergent, fully self-consistent energy-band solutions reported for AlAs. The imaginary part of the dielectric constant, spin-orbit splittings, effective masses, and the x-ray form factors (Fourier transforms of the electron charge density) have been calculated. The theoretical results are compared with the available experimental data.

Electronic Band Structure and Related Properties of Cubic A1P

A first-principles self-consistent orthogonalized-plane-wave energy-band calculation has been carried out for cubic AlP using a nonrelativistic formalism and Slater's free-electron-exchange approximation. These are the first fully convergent, fully self-consistent energy-band solutions reported for AlP. The imaginary part of the dielectric constant, spin-orbit splitting, effective masses at $k=0$, and the x-ray form factors (Fourier transforms of the electron charge density) have been calculated. Since little is known experimentally about AlP, no comprehensive comparison can be made with experimental data.

Born Wave Calculation of Atom-Atom Inelastic Cross Sections: Description of Target Atoms by Elastic and Inelastic X-Ray Form Factors

Total electron-loss cross sections for H atoms in collision with He, Ne, Ar, Kr, C, N, and O over the range of incident energy 1 keV-100 MeV and total ($1s\ensuremath{-}2l$) excitation cross sections for H atoms in collision with He, Ne, Ar, and Kr over the range of incident energy 0.1 keV-10 MeV are calculated by using the first Born wave approximation and assuming closure. The relevant matrix elements for the target atoms are expressed in terms of elastic and inelastic x-ray form factors. The calculations agree, within experimental error, with high-energy measurements for He, N, and O targets. Good agreement between theoretical and experimental excitation cross sections is found at energies as low as 5 keV for He, Ne, and Kr targets when the Ne and Kr calculations are scaled with a velocity-dependent parameter calculated by matching theoretical and experimental ionization data. An explanation for the velocity-dependent scaling of the theoretical cross sections for many electron target atoms at energies up to 1 MeV is presented and discussed.

Soluble Model for Fibrous Structures with Steric Constraints

We consider a set of thin, long, flexible chains in two dimensions. The chains are in thermal equilibrium under a strong unidirectional stretching force. Configurations where different chains intersect each other are excluded. The chain configurations yn(x) are interpreted as the trajectories of an assembly of quantum mechanical particles, x playing the role of the time. The model is thus brought into correspondence with a one-dimensional gas of free fermions, and can be solved exactly. The system of chains has no long-range order, but a remarkably strong short-range order: the x-ray form factor, for a wave vector q normal to the stretching direction, shows a sharp peak at q = 2kF, where kF is the Fermi wave vector of the associated set of particles.