Chord length distribution of pentagonal and hexagonal rods: relation to small-angle scattering

2009 ◽  
Vol 42 (2) ◽  
pp. 326-328 ◽  
Author(s):  
Wilfried Gille ◽  
Narine G. Aharonyan ◽  
Hrachya S. Harutyunyan
Keyword(s):  
Small Angle ◽  
Length Distribution ◽  
Small Angle Scattering ◽  
Chord Length ◽  
Density Functions ◽  
Explicit Formulas ◽  
Chord Length Distribution ◽  
Regular Pentagon ◽  
Angle Scattering

Based on explicit formulas of chord length density functions (CLDs) for a regular pentagon and a hexagon, the CLDs of infinitely long regular homogeneous pentagonal/hexagonal cylinders are discussed. Characteristic properties of the small-angle scattering of these cylinders are studied.

Estimation of chord length distributions from small-angle scattering using indirect Fourier transformation

2003 ◽  
Vol 36 (5) ◽  
pp. 1190-1196 ◽  
Author(s):  
Steen Hansen
Keyword(s):  
Small Angle ◽  
Fourier Transformation ◽  
Length Distribution ◽  
Small Angle Scattering ◽  
Chord Length ◽  
Scattering Data ◽  
Additional Tool ◽  
Additional Information ◽  
Chord Length Distribution ◽  
Angle Scattering

It is shown that it is possible to estimate the chord length distribution from small-angle scattering data by indirect Fourier transformation. This is done for several examples of scatterers varying in structure from globular to elongated as well as scatterers consisting of separated parts. The presented examples suggest that the chord length distribution may give additional information about the scatterer. Therefore it may be relevant to consider estimation of the chord length distribution as an additional tool for analysis of small-angle scattering data.

The small-angle scattering correlation function of the cuboid

1999 ◽  
Vol 32 (6) ◽  
pp. 1100-1104 ◽  
Author(s):  
Wilfried Gille
Keyword(s):  
Correlation Function ◽  
Asymptotic Behaviour ◽  
Small Angle ◽  
Length Distribution ◽  
Chord Length ◽  
Scattering Intensity ◽  
Chord Length Distribution ◽  
Angle Scattering ◽  
Higher Derivatives ◽  
Derivatives Of

The analytical expression of the correlation function γ(r) for the cuboid with edgesa,bandcis established. The calculation is based on the chord-length distribution. Details of these structure functions at essentialrpositions are analysed, including higher derivatives of the correlation function at the maximum chord length. The result was checked on closer analysis of the corresponding scattering intensityI(h) and its asymptotic behaviourI∞(h).

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.

scholarly journals Chord-length distributions cannot generally be obtained from small-angle scattering

2020 ◽  
Vol 53 (1) ◽  
pp. 127-132
Author(s):  
Cedric J. Gommes ◽  
Yang Jiao ◽  
Anthony P. Roberts ◽  
Dominique Jeulin
Keyword(s):  
Porous Materials ◽  
Small Angle ◽  
Hollow Spheres ◽  
Boolean Model ◽  
Small Angle Scattering ◽  
Chord Length ◽  
Scattering Data ◽  
Angle Scattering ◽  
Dilute Suspension ◽  
Geometrically Complex

The methods used to extract chord-length distributions from small-angle scattering data assume a structure consisting of spatially uncorrelated and disconnected convex regions. These restrictive conditions are seldom met for a wide variety of materials such as porous materials and semicrystalline or phase-separated copolymers, the structures of which consist of co-continuous phases that interpenetrate each other in a geometrically complex way. The significant errors that would result from applying existing methods to such systems are discussed using three distinct models for which the chord-length distributions are known analytically. The models are a dilute suspension of hollow spheres, the Poisson mosaic and the Boolean model of spheres.

Characteristics of the SAS correlation function of long cylinders with oval right section

2010 ◽  
Vol 43 (2) ◽  
pp. 347-349 ◽  
Author(s):  
Wilfried Gille
Keyword(s):  
Correlation Function ◽  
Surface Area ◽  
Small Angle ◽  
Small Angle Scattering ◽  
Chord Length ◽  
Angle Scattering ◽  
The Mean ◽  
Two Parameters

An approximation for the small-angle scattering (SAS) correlation function (CF) β0(r) of a plane oval domainXis discussed. The approach is based on two parameters, the perimeteruand the surface areaSRS, ofX. The function β0(r) fixes the correlation function γ0(r) of the oval homogeneous rod with constant right sectionX. The mean chord lengthl1of such a rod is the root of the equation γ0(l1) = 1 − 8/(3π). For a dilute rod arrangement, the Porod lengthlpand γ′(l1), the value of the derivative of the sample CF atr=l1, are related by γ′(l1) = −4/(3πlp).

scholarly journals The chord length distribution function for regular polygons

2009 ◽  
Vol 41 (2) ◽  
pp. 358-366 ◽  
Author(s):  
H. S. Harutyunyan ◽  
V. K. Ohanyan
Keyword(s):  
Distribution Function ◽  
Regular Polygon ◽  
Length Distribution ◽  
Chord Length ◽  
Regular Hexagon ◽  
Regular Polygons ◽  
Chord Length Distribution ◽  
Regular Pentagon ◽  
Regular Triangle ◽  
Elementary Expression

In this paper we obtain an elementary expression for the chord length distribution function of a regular polygon. The formula is derived using δ-formalism in Pleijel identity. In the particular cases of a regular triangle, a square, a regular pentagon, and a regular hexagon, our formula coincides with the results of Sulanke (1961), Gille (1988), Aharonyan and Ohanyan (2005), and Harutyunyan (2007), respectively.

scholarly journals The chord length distribution function for regular polygons

2009 ◽  
Vol 41 (02) ◽  
pp. 358-366
Author(s):  
H. S. Harutyunyan ◽  
V. K. Ohanyan
Keyword(s):  
Distribution Function ◽  
Regular Polygon ◽  
Length Distribution ◽  
Chord Length ◽  
Regular Hexagon ◽  
Regular Polygons ◽  
Chord Length Distribution ◽  
Regular Pentagon ◽  
Regular Triangle ◽  
Elementary Expression

In this paper we obtain an elementary expression for the chord length distribution function of a regular polygon. The formula is derived using δ-formalism in Pleijel identity. In the particular cases of a regular triangle, a square, a regular pentagon, and a regular hexagon, our formula coincides with the results of Sulanke (1961), Gille (1988), Aharonyan and Ohanyan (2005), and Harutyunyan (2007), respectively.

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