Calculation of incoherent X-ray scattering for argon by the statistical electron density distributions

1977 ◽  
Vol 42 (2) ◽  
pp. 137-141
Author(s):  
A. Dobay-Szegleth
Keyword(s):  
Electron Density ◽  
X Ray ◽  
Density Distributions ◽  
X Ray Scattering ◽  
Ray Scattering

scholarly journals Exploiting the full quantum crystallography

2018 ◽  
Vol 96 (7) ◽  
pp. 599-605 ◽  
Author(s):  
Lou Massa ◽  
Chérif F. Matta
Keyword(s):  
Quantum Mechanics ◽  
Electron Density ◽  
Mathematical Structure ◽  
Scattering Data ◽  
X Ray ◽  
X Ray Crystallography ◽  
X Ray Scattering ◽  
Fundamental Value ◽  
Quantum Crystallography ◽  
Ray Scattering

Quantum crystallography (QCr) is a branch of crystallography aimed at obtaining the complete quantum mechanics of a crystal given its X-ray scattering data. The fundamental value of obtaining an electron density matrix that is N-representable is that it ensures consistency with an underlying properly antisymmetrized wavefunction, a requirement of quantum mechanical validity. However, X-ray crystallography has progressed in an impressive way for decades based only upon the electron density obtained from the X-ray scattering data without the imposition of the mathematical structure of quantum mechanics. Therefore, one may perhaps ask regarding N-representability “why bother?” It is the purpose of this article to answer such a question by succinctly describing the advantage that is opened by QCr.

The Effect of Thermal Vibrations on the Intensities of the Diffracted Beams

1997 ◽  
Author(s):  
Philip Coppens
Keyword(s):  
Electron Density ◽  
Normal Modes ◽  
Unit Cell ◽  
Thermal Motion ◽  
X Ray ◽  
X Ray Scattering ◽  
Point Motion ◽  
Unit Cells ◽  
Thermal Vibrations ◽  
Ray Scattering

The atoms in a crystal are vibrating with amplitudes determined by the force constants of the crystal’s normal modes. This motion can never be frozen out because of the persistence of zero-point motion, and it has important consequences for the scattering intensities. Since X-ray scattering (and, to a lesser extent, neutron scattering) is a very fast process, taking place on a time scale of 10−18 s, the photon-matter interaction time is much shorter than the period of a lattice vibration, which is of the order Thus, the recorded X-ray scattering pattern is the sum over the scattering of a large number of 1/v, or ≈10−13s. instantaneous states of the crystal. To an extremely good approximation, the scattering averaged over the instantaneous distributions is equivalent to the scattering of the time-averaged distribution of the scattering matter (Stewart and Feil 1980). The structure factor expression for coherent elastic Bragg scattering of X-rays may therefore be written in terms 〈ρ(r)〉, of the thermally averaged electron density: . . . F(H)=∫unit cell〈ρ(r)〉 exp (2πi H ·r) dr (2.1) . . . The smearing of the electron density due to thermal vibrations reduces the intensity of the diffracted beams, except in the forward |S| = 0 direction, for which all electrons scatter in phase, independent of their distribution. The reduction of the intensity of the Bragg peaks can be understood in terms of the diffraction pattern of a more diffuse electron distribution being more compact, due to the inverse relation between crystal and scattering space, discussed in chapter 1. The reduction in intensity due to thermal motion is accompanied by an increase in the incoherent elastic scattering, ensuring conservation of energy. In this respect, thermal motion is much like disorder, with the Bragg intensities representing the average distribution, and the deviations from the average appearing as a continuous, though not uniform, background, generally referred to as thermal diffuse scattering or TDS. A crystal with n atoms per unit cell has 3nN degrees of freedom, N being the number of unit cells in the crystal.

scholarly journals X-ray scattering from unilamellar lipid vesicles

2005 ◽  
Vol 38 (1) ◽  
pp. 126-131 ◽  
Author(s):  
Michael R. Brzustowicz ◽  
Axel T. Brunger
Keyword(s):  
Form Factor ◽  
Lipid Bilayer ◽  
Electron Density ◽  
Lipid Vesicles ◽  
Lipid Vesicle ◽  
Asymmetric Structure ◽  
Bilayer Structure ◽  
X Ray ◽  
X Ray Scattering ◽  
Ray Scattering

An improved small-angle X-ray scattering (SAXS) method for determining asymmetric lipid bilayer structure in unilamellar vesicles is presented. From scattering theory, analytic expressions are derived for the bilayer form factor over flat and spherical geometries, assuming the lipid bilayer electron density to be composed of a series of Gaussian shells. This is in contrast to both classic diffraction and Guinier hard-shell SAXS methods which, respectively, are capable only of ascertaining symmetric bilayer structure and limited-resolution asymmetric structure. Using model fitting and direct calculation of the form factor, using only one equation, an asymmetric electron density profile of the lipid vesicle is obtained with high accuracy, as well as the average radius. The analysis suggests that the inner leaflet of a unilamellar lipid vesicle is `rougher' than the outer one.

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