Determination of the Radial Distribution Function and the Direct Correlation Function of Spheres by X‐Ray Small‐Angle Scattering

1971 ◽  
Vol 55 (10) ◽  
pp. 5095-5101 ◽  
Author(s):  
George W. Brady ◽  
C. C. Gravatt
Keyword(s):  
Distribution Function ◽  
Correlation Function ◽  
Radial Distribution Function ◽  
Radial Distribution ◽  
Small Angle ◽  
Small Angle Scattering ◽  
Direct Correlation Function ◽  
X Ray ◽  
Angle Scattering

Influence of neglected small-angle scattering in radial distribution function analysis

1971 ◽  
Vol 4 (4) ◽  
pp. 277-283 ◽  
Author(s):  
G. S. Cargill
Keyword(s):  
Distribution Function ◽  
Radial Distribution Function ◽  
Radial Distribution ◽  
Small Angle ◽  
Function Analysis ◽  
Small Angle Scattering ◽  
Density Fluctuations ◽  
Atomic Density ◽  
X Ray ◽  
Angle Scattering

Materials containing inhomogeneities (density-fluctuations) of much greater than atomic size produce scattering at very small angles, which may go unobserved in many X-ray, electron, and neutron scattering experiments. For liquids and for amorphous and polycrystalline solids composed of one atomic species, an approximate expression for the reduced radial distribution function obtained from intensity measurements which neglect the small-angle scattering is shown to be Gexp(r) = 4πr{ρ(r) − ρ0[1 + (\overline {\eta^2}η2(ω)/ρ0 2)γ(ω, r)]} where ρ(r) is the atomic distribution function, ρ0 is the average atomic density, \overline {\eta^2}(ω) is the average square of atomic density fluctuations, γ(ω,r) is the density fluctuation correlation function, and ω is a volume element larger than the average atomic volume but smaller than the scale of long-range density fluctuations. This expression is also valid for systems composed of more than one type of atom where ρ(r) is a weighted average of pair distribution functions and [\overline {\eta^2}(ω)/ρ0 2]γ(ω,r) for X-ray scattering describes electron density fluctuations The neglect of small-angle scattering gives rise to a G exp(r) which appears, from its slope at small r, to correspond to a material of greater average atomic density ρ0,exp than that of the sample being studied. These results are illustrated by application to fluid argon (ρ0,exp/ρ0 = 1.17 near the critical point), to amorphous silicon (ρ0,exp/ρ0 = 1.13), and to phase separated PbO–B2O3 glasses (ρ0,exp/ρ0 = 1.07 for 24 wt. % PbO).

scholarly journals Determination of G.P.(1) Zone Size in Al-Cu Alloys by the X-ray Small-Angle Scattering Method

1971 ◽  
Vol 35 (10) ◽  
pp. 1001-1007 ◽  
Author(s):  
Osamu Kawano ◽  
Tetsuya Hirouchi ◽  
Hiroyuki Yoshida ◽  
Yotaro Murakami
Keyword(s):  
Small Angle ◽  
Small Angle Scattering ◽  
Scattering Method ◽  
Zone Size ◽  
X Ray ◽  
Cu Alloys ◽  
Angle Scattering

Determination of the structure of tissue samples using X-ray small angle scattering

2000 ◽  
Vol 196 (12) ◽  
pp. 827-830 ◽  
Author(s):  
Martin Bradaczek ◽  
Hans Guski ◽  
Hans Bradaczek ◽  
Georgi G. Avtandilov
Keyword(s):  
Small Angle ◽  
Small Angle Scattering ◽  
Tissue Samples ◽  
X Ray ◽  
Angle Scattering

SQUARE-WELL FLUID AT LOW DENSITIES

1967 ◽  
Vol 45 (12) ◽  
pp. 3959-3978 ◽  
Author(s):  
J. A. Barker ◽  
D. Henderson
Keyword(s):  
Distribution Function ◽  
Correlation Function ◽  
Critical Temperature ◽  
Radial Distribution Function ◽  
Radial Distribution ◽  
Virial Coefficients ◽  
Hard Core ◽  
Direct Correlation Function ◽  
Virial Series ◽  
Square Well

Values for the radial distribution function and the direct correlation function at low densities and for the first five virial coefficients are obtained for a fluid of molecules interacting according to the square-well potential when the width of the attractive well is half the radius of the hard core. It is found that the higher-order coefficients are surprisingly large and, as a result, the virial series fails to converge even at temperatures and volumes significantly greater than the critical temperature and volume. Comparisons of these exact virial coefficients with those given by several approximate theories are made. Values are also given for the first five virial coefficients when the width of the attractive well is equal to the radius of the hard core.

Small-angle scattering analysis of the spherical half-shell

2007 ◽  
Vol 40 (2) ◽  
pp. 302-304 ◽  
Author(s):  
Wilfried Gille
Keyword(s):  
Correlation Function ◽  
Small Angle ◽  
Pair Correlation ◽  
Length Distribution ◽  
Small Angle Scattering ◽  
Scattering Intensity ◽  
Chord Length Distribution ◽  
Angle Scattering ◽  
Analytic Expressions

For a spherical half-shell (SHS) of diameter D, analytic expressions of the small-angle scattering correlation function \gamma_0(r), the chord length distribution (CLD) and the scattering intensity are analyzed. The spherically averaged pair correlation function p_0(r)\simeq r^2\gamma_0(r) of the SHS is identical to the cap part of the CLD of a solid hemisphere of the same diameter. The surprisingly simple analytic terms in principle allow the determination of the size distribution of an isotropic diluted SHS collection from its scattering intensity.

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