Analytical Expressions for Potentials of Neutral Thomas—Fermi—Dirac Atoms and for the Corresponding Atomic Scattering Factors for X Rays and Electrons

1963 ◽  
Vol 39 (9) ◽  
pp. 2200-2204 ◽  
Author(s):  
R. A. Bonham ◽  
T. G. Strand
Keyword(s):  
X Rays ◽  
Thomas Fermi ◽  
Atomic Scattering ◽  
Analytical Expressions

X-Ray Diffraction

2021 ◽  
pp. 408-480
Author(s):  
Kannan M. Krishnan
Keyword(s):  
Point Group ◽  
Polycrystalline Materials ◽  
X Rays ◽  
X Ray Diffraction ◽  
Symmetry Point Group ◽  
X Ray ◽  
Atomic Scattering ◽  
Scattering Factor ◽  
Diffraction Patterns ◽  
Lattice Planes

X-rays diffraction is fundamental to understanding the structure and crystallography of biological, geological, or technological materials. X-rays scatter predominantly by the electrons in solids, and have an elastic (coherent, Thompson) and an inelastic (incoherent, Compton) component. The atomic scattering factor is largest (= Z) for forward scattering, and decreases with increasing scattering angle and decreasing wavelength. The amplitude of the diffracted wave is the structure factor, F hkl, and its square gives the intensity. In practice, intensities are modified by temperature (Debye-Waller), absorption, Lorentz-polarization, and the multiplicity of the lattice planes involved in diffraction. Diffraction patterns reflect the symmetry (point group) of the crystal; however, they are centrosymmetric (Friedel law) even if the crystal is not. Systematic absences of reflections in diffraction result from glide planes and screw axes. In polycrystalline materials, the diffracted beam is affected by the lattice strain or grain size (Scherrer equation). Diffraction conditions (Bragg Law) for a given lattice spacing can be satisfied by varying θ or λ — for study of single crystals θ is fixed and λ is varied (Laue), or λ is fixed and θ varied to study powders (Debye-Scherrer), polycrystalline materials (diffractometry), and thin films (reflectivity). X-ray diffraction is widely applied.

X-ray Scattering Factors of Ions in Crystals

1979 ◽  
Vol 34 (12) ◽  
pp. 1471-1481 ◽  
Author(s):  
P. C. Schmidt ◽  
Alarich Weiss
Keyword(s):  
Gaseous Phase ◽  
Form Factors ◽  
Negative Ions ◽  
X Rays ◽  
Charged Sphere ◽  
Sphere Model ◽  
X Ray ◽  
Hartree Fock ◽  
X Ray Scattering ◽  
Atomic Scattering

AbstractThe atomic scattering factors for X - Rays are given for the ions Li⊕, Be2⊕, B3⊕, C4⊕, N5⊕, N3⊖, O2⊖, F⊖, Na⊕, Mg2⊕, Al3⊕, S2⊖, Cl⊖, K⊕, Ca2⊕, Sc3⊕, Ti4⊕, V5⊕, Ni, Cu⊕, Zn2⊕, Ga3⊕, Se2⊖, Br⊖, Rb⊕, Sr2⊕, Y3⊕, Pd, Ag⊕, Cd2⊕, I⊖, Cs⊕, and Ba2⊕ in the crystal. The crystal potential is simulated by a hollow charged sphere (Watson sphere model). The Hartree-Fock-Roothaan-method was used for the calculation. The crystal field affects most strongly the atomic form factors of the negative ions, especially the twofold and threefold ionized negative ions, which are unstable in the gaseous phase.

scholarly journals ROTATING FERMIONS IN TWO DIMENSIONS: A THOMAS–FERMI APPROACH

2001 ◽  
Vol 15 (19n20) ◽  
pp. 2799-2810
Author(s):  
SANKALPA GHOSH ◽  
M. V. N. MURTHY ◽  
SUBHASIS SINHA
Keyword(s):  
Ground State ◽  
Analytical Approach ◽  
State Energy ◽  
Critical Size ◽  
Energy Functional ◽  
Two Dimensions ◽  
Fermionic System ◽  
Thomas Fermi ◽  
Fermionic Systems ◽  
Analytical Expressions

Properties of confined mesoscopic systems have been extensively studied numerically over recent years. We discuss an analytical approach to the study of finite rotating fermionic systems in two dimension. We first construct the energy functional for a finite fermionic system within the Thomas–Fermi approximation in two dimensions. We show that for specific interactions the problem may be exactly solved. We derive analytical expressions for the density, the critical size as well as the ground state energy of such systems in a given angular momentum sector.

Bragg reflection and transmission of X-rays induced by the imaginary part of the atomic scattering factor

1995 ◽  
Vol 7 (42) ◽  
pp. 8089-8098 ◽  
Author(s):  
Xu Zhangcheng ◽  
Zhao Zongyan ◽  
Guo Changlin ◽  
Zhou Shengming ◽  
Tomoe Fukamachi ◽  
...  
Keyword(s):  
Imaginary Part ◽  
Bragg Reflection ◽  
X Rays ◽  
Reflection And Transmission ◽  
Atomic Scattering Factor ◽  
Atomic Scattering ◽  
Scattering Factor

THE THOMAS–FERMI–GOMBÁS ATOM MODELS IN WHICH THE ELECTRONS ARE GROUPED INTO n- AND nl-SHELLS AND A CALCULATION OF THE ATOMIC FORM FACTOR

2004 ◽  
Vol 18 (03) ◽  
pp. 409-419
Author(s):  
V. F. TARASOV
Keyword(s):  
Form Factor ◽  
Kinetic Energy ◽  
Total Energy ◽  
Hypergeometric Functions ◽  
Fermi Theory ◽  
Thomas Fermi ◽  
Atomic Form Factor ◽  
First Time ◽  
Analytical Expressions ◽  
Gradient Correction

This article, considers in detail P. Gombás's idea of grouping electrons into n- and nl-shells in the Thomas–Fermi theory of free atoms briefly, the TFG n- and TFG nl-models respectively). Using these models, exact analytical expressions for the total energy E and the atomic form factor F(κ) are obtained. All integrals of the TFG nl-model are computed by means of the hypergeometric functions 2F1(x), 3F2(x), F2(x,y) and FA(x1,…,x6) for the first time. In particular, Weizsäcker's gradient correction to the kinetic energy of the nl-th shell [Formula: see text] generates a new numerical triangle [Formula: see text] with coefficients bw=n+2l(n-l-1).

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