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.

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).

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.

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.

scholarly journals Contact and Chord Length Distribution Functions of the Poisson-Voronoi Tessellation in High Dimensions

2010 ◽  
Vol 42 (1) ◽  
pp. 48-68 ◽  
Author(s):  
L. Muche
Keyword(s):  
Voronoi Tessellation ◽  
Length Distribution ◽  
Distribution Functions ◽  
Chord Length ◽  
Dimensional Euclidean Space ◽  
Spherical Contact ◽  
Chord Length Distribution ◽  
Special Cases ◽  
Contact Distribution ◽  
Contact Distribution Function

In this paper we present formulae for contact distributions of a Voronoi tessellation generated by a homogeneous Poisson point process in the d-dimensional Euclidean space. Expressions are given for the probability density functions and moments of the linear and spherical contact distributions. They are double and simple integral formulae, which are tractable for numerical evaluation and for large d. The special cases d = 2 and d = 3 are investigated in detail, while, for d = 3, the moments of the spherical contact distribution function are expressed by standard functions. Also, the closely related chord length distribution functions are considered.

scholarly journals Volume degeneracy of the typical cell and the chord length distribution for Poisson-Voronoi tessellations in high dimensions

2008 ◽  
Vol 40 (04) ◽  
pp. 919-938 ◽  
Author(s):  
Kasra Alishahi ◽  
Mohsen Sharifitabar
Keyword(s):  
Space Dimension ◽  
Voronoi Tessellation ◽  
Length Distribution ◽  
Chord Length ◽  
Voronoi Tessellations ◽  
Constant Intensity ◽  
Chord Length Distribution ◽  
Typical Cell ◽  
Centered Ball ◽  
Contact Distribution

This paper is devoted to the study of some asymptotic behaviors of Poisson-Voronoi tessellation in the Euclidean space as the space dimension tends to ∞. We consider a family of homogeneous Poisson-Voronoi tessellations with constant intensity λ in Euclidean spaces of dimensions n = 1, 2, 3, …. First we use the Blaschke-Petkantschin formula to prove that the variance of the volume of the typical cell tends to 0 exponentially in dimension. It is also shown that the volume of intersection of the typical cell with the co-centered ball of volume u converges in distribution to the constant λ−1(1 − e−λu ). Next we consider the linear contact distribution function of the Poisson-Voronoi tessellation and compute the limit when the space dimension goes to ∞. As a by-product, the chord length distribution and the geometric covariogram of the typical cell are obtained in the limit.

scholarly journals The chord-length distribution of a polyhedron

2020 ◽  
Vol 76 (4) ◽  
pp. 474-488
Author(s):  
Salvino Ciccariello
Keyword(s):  
Distribution Function ◽  
Length Distribution ◽  
Chord Length ◽  
Square Root ◽  
Trigonometric Functions ◽  
Chord Length Distribution ◽  
Analytic Expressions ◽  
Analytic Expression ◽  
Positive Integers ◽  
Inverse Trigonometric Functions

The chord-length distribution function [γ′′(r)] of any bounded polyhedron has a closed analytic expression which changes in the different subdomains of the r range. In each of these, the γ′′(r) expression only involves, as transcendental contributions, inverse trigonometric functions of argument equal to R[r, Δ1], Δ1 being the square root of a second-degree r polynomial and R[x, y] a rational function. As r approaches δ, one of the two end points of an r subdomain, the derivative of γ′′(r) can only show singularities of the forms |r − δ|−n and |r − δ|−m+1/2, with n and m appropriate positive integers. Finally, the explicit analytic expressions of the primitives are also reported.

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