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.

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.

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 Algebraic approximations of a polyhedron correlation function stemming from its chord-length distribution

2021 ◽  
Vol 77 (1) ◽  
pp. 75-80
Author(s):  
Salvino Ciccariello
Keyword(s):  
Fourier Transform ◽  
Correlation Function ◽  
Full Range ◽  
Length Distribution ◽  
Chord Length ◽  
Exact Form ◽  
Chord Length Distribution ◽  
Algebraic Approximation ◽  
Left And Right ◽  
The Common

An algebraic approximation, of order K, of a polyhedron correlation function (CF) can be obtained from γ′′(r), its chord-length distribution (CLD), considering first, within the subinterval [D i−1, D i ] of the full range of distances, a polynomial in the two variables (r − D i−1)1/2 and (D i − r)1/2 such that its expansions around r = D i−1 and r = D i simultaneously coincide with the left and right expansions of γ′′(r) around D i−1 and D i up to the terms O(r − D i−1) K/2 and O(D i − r) K/2, respectively. Then, for each i, one integrates twice the polynomial and determines the integration constants matching the resulting integrals at the common end-points. The 3D Fourier transform of the resulting algebraic CF approximation correctly reproduces, at large q's, the asymptotic behaviour of the exact form factor up to the term O[q −(K/2+4)]. For illustration, the procedure is applied to the cube, the tetrahedron and the octahedron.

scholarly journals Relating optical and microwave grain metrics of snow: The relevance of grain shape

2016 ◽  
Author(s):  
Quirine Krol ◽  
Henning Löwe
Keyword(s):  
Correlation Function ◽  
Correlation Length ◽  
Shape Parameter ◽  
Length Distribution ◽  
Empirical Finding ◽  
Geometrical Optics ◽  
Chord Length ◽  
Optical Measurements ◽  
Statistical Prediction ◽  
Chord Length Distribution

Abstract. While optical properties of snow are predominantly determined by the specific surface area (SSA), microwave measurements are often analyzed in terms of the exponential correlation length ξ. A statistical relation between both is commonly employed to facilitate forcing of microwave models by optical measurements. To improve the understanding of ξ and establish a link between optical and microwave grain metrics we analyzed the third order term in the expansion of the correlation function that can be regarded as a shape parameter related to mean and Gaussian curvature. We show that the statistical prediction of the correlation length via SSA is considerably improved by including the shape metric. In a second step we address the chord-length distribution as a key quantity for geometrical optics. We show that the second moment of the distribution can be accurately related to density, SSA and the shape parameter. This empirical finding is supported by a theoretical relation between the chord length distribution and the correlation function as suggested by small angle scattering methods. As a practical implication, we compute the optical shape factor $B$ from tomography data. Our results indicate a possibility of estimating ξ by a careful analysis of shape corrections in geometrical optics.

Contrast variation in small-angle scattering experiments on polydisperse and superparamagnetic systems: basic functions approach

2007 ◽  
Vol 40 (1) ◽  
pp. 56-70 ◽  
Author(s):  
Mikhail V. Avdeev
Keyword(s):  
Small Angle ◽  
Magnetic Particles ◽  
Scattering Length ◽  
Small Angle Scattering ◽  
Contrast Variation ◽  
Match Point ◽  
Scattering Intensity ◽  
Scattering Length Density ◽  
Angle Scattering ◽  
Basic Functions

The development of the basic functions approach [Stuhrmann (1995).Modern Aspects of Small-Angle Scattering, edited by H. Brumberger, pp. 221–254. Dordrecht: Kluwer Academic Publishers] for the contrast variation technique in small-angle scattering from systems of polydisperse and superparamagnetic non-interacting particles is presented. For polydisperse systems the modified contrast is introduced as the difference between the effective mean scattering length density (corresponding to the minimum of the scattering intensity as the function of the scattering length density of the solvent) and the density of the solvent. Then, the general expression for the scattering intensity is written in the classical way through the modified basic functions. It is shown that the shape scattering from the particle volume can be reliably obtained. Modifications of classical expressions describing changes in integral parameters of the scattering (intensity at zero angle, radius of gyration, Porod integral) with the contrast are analyzed. In comparison with the monodisperse case, the residual scattering in the minimum of intensity as a function of contrast (effective match point) in polydisperse systems makes it possible to treat the Guinier region of scattering curves around the effective match point quite precisely from the statistical viewpoint. However, limitations of such treatment exist, which are emphasized in the paper. In addition, the effect of magnetic scattering in small-angle neutron scattering from superparamagnetic nanoparticles is considered in the context of the basic functions approach. Conceptually, modifications of the integral parameters of the scattering in this case are similar to those obtained for polydisperse multicomponent particles. Various cases are considered, including monodisperse non-homogeneous and homogeneous magnetic particles, and polydisperse non-homogeneous and homogeneous magnetic particles. The developed approach is verified for two models representing the main types of magnetic fluids – systems of polydisperse superparamagnetic particles located in liquid carriers.

scholarly journals Effect of Pre-Shear on Agglomeration and Rheological Parameters of Cement Paste

Materials ◽  
2020 ◽  
Vol 13 (9) ◽  
pp. 2173
Author(s):  
Mareike Thiedeitz ◽  
Inka Dressler ◽  
Thomas Kränkel ◽  
Christoph Gehlen ◽  
Dirk Lowke
Keyword(s):  
Yield Stress ◽  
Mixing Time ◽  
Length Distribution ◽  
Chord Length ◽  
Rheological Parameters ◽  
Reflectance Measurement ◽  
Time Dynamic ◽  
Chord Length Distribution ◽  
In Situ Investigation

Cementitious pastes are multiphase suspensions that are rheologically characterized by viscosity and yield stress. They tend to flocculate during rest due to attractive interparticle forces, and desagglomerate when shear is induced. The shear history, e.g., mixing energy and time, determines the apparent state of flocculation and accordingly the particle size distribution of the cement in the suspension, which itself affects suspension’s plastic viscosity and yield stress. Thus, it is crucial to understand the effect of the mixing procedure of cementitious suspensions before starting rheological measurements. However, the measurement of the in-situ particle agglomeration status is difficult, due to rapidly changing particle network structuration. The focused beam reflectance measurement (FBRM) technique offers an opportunity for the in-situ investigation of the chord length distribution. This enables to detect the state of flocculation of the particles during shear. Cementitious pastes differing in their solid fraction and superplasticizer content were analyzed after various pre-shear histories, i.e., mixing times. Yield stress and viscosity were measured in a parallel-plate-rheometer and related to in-situ measurements of the chord length distribution with the FBRM-probe to characterize the agglomeration status. With increasing mixing time agglomerates were increasingly broken up in dependence of pre-shear: After 300 s of pre-shear the agglomerate sizes decreased by 10 µm to 15 µm compared to a 30 s pre-shear. At the same time dynamic yield stress and viscosity decreased up to 30% until a state of equilibrium was almost reached. The investigations show a correlation between mean chord length and the corresponding rheological parameters affected by the duration of pre-shear.

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