Procedures for reducing large datasets of crystal orientations using generalized spherical harmonics

2015 ◽  
Vol 88 ◽  
pp. 73-86 ◽  
Author(s):  
Marko Knezevic ◽  
Nicholas W. Landry
Keyword(s):  
Spherical Harmonics ◽  
Large Datasets ◽  
Crystal Orientations ◽  
Generalized Spherical Harmonics

Geographical interpretation of measurements of average phase velocities of surface waves over great circular and great semi-circular paths

1964 ◽  
Vol 54 (2) ◽  
pp. 571-610
Author(s):  
George E. Backus
Keyword(s):  
Spherical Harmonics ◽  
Rayleigh Waves ◽  
Seismic Station ◽  
Great Circle ◽  
Explicit Formulas ◽  
Phase Velocities ◽  
Rapid Accumulation ◽  
Generalized Spherical Harmonics ◽  
Geographical Position ◽  
The Difference

ABSTRACT If the averages of the reciprocal phase velocity c−1 of a given Rayleigh or Love mode over various great circular or great semicircular paths are known, information can be extracted about how c−1 varies with geographical position. Assuming that geometrical optics is applicable, it is shown that if c−1 is isotropic its great circular averages determine only the sum of the values of c−1 at antipodal points and not their difference. The great semicircular averages determine the difference as well. If c−1 is anisotropic through any cause other than the earth's rotation, even great semicircular averages do not determine c−1 completely. Rotation has negligible effect on Love waves, and if it is the only anisotropy present its effect on Rayleigh waves can be measured and removed by comparing the averages of c−1 for the two directions of travel around any great circle not intersecting the poles of rotation. Only great circular and great semicircular paths are considered because every earthquake produces two averages of c−1 over such paths for each seismic station. No other paths permit such rapid accumulation of data when the azimuthal variations of the earthquakes' radiation patterns are unknown. Expansion of the data in generalized spherical harmonics circumvents the fact that the explicit formulas for c−1 in terms of its great circular or great semicircular integrals require differentiation of the data. Formulas are given for calculating the generalized spherical harmonics numerically.

scholarly journals Addition Formula and Related Integral Equations for Heine—Stieltjes Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1231
Author(s):  
Hans Volkmer
Keyword(s):  
Integral Equations ◽  
Explicit Form ◽  
Spherical Harmonics ◽  
Reproducing Kernel ◽  
Addition Formula ◽  
Symmetric Products ◽  
Special Cases ◽  
Related Integral ◽  
Generalized Spherical Harmonics ◽  
The Relationship

It is shown that symmetric products of Heine–Stieltjes quasi-polynomials satisfy an addition formula. The formula follows from the relationship between Heine–Stieltjes quasi-polynomials and spaces of generalized spherical harmonics, and from the known explicit form of the reproducing kernel of these spaces. In special cases, the addition formula is written out explicitly and verified. As an application, integral equations for Heine–Stieltjes quasi-polynomials are found.

A New Method for Orientation Distribution Function Analysis

1985 ◽  
Vol 29 ◽  
pp. 443-449
Author(s):  
Munetsugu Matsuo ◽  
Koichi Kawasaki ◽  
Tetsuya Sugai
Keyword(s):  
Distribution Function ◽  
Spherical Harmonics ◽  
Orientation Distribution Function ◽  
Function Analysis ◽  
Orientation Distribution ◽  
Symmetry Operation ◽  
Composite Function ◽  
Odd Order ◽  
Pole Figures ◽  
Generalized Spherical Harmonics

AbstractAs a means for quantitative texture analysis, the crystallite orientation distribution function analysis has an important drawback: to bring ghosts as a consequence of the presence of a non-trivial kernel which consists of the spherical harmonics of odd order terms. In the spherical hamonic analysis, ghosts occur in the particular orientations by symmetry operation from the real orientation in accordance with the symmetry of the harmonics of even orders. For recovery of the odd order harmonics, the 9th-order generalized spherical harmonics are linearly combined and added to the orientation distribution function reconstructed from pole figures to a composite function. The coefficients of the linear combination are optimized to minimize the sum of negative values in the composite function. Reproducibility was simulated by using artificial pole figures of single or multiple component textures. Elimination of the ghosts is accompanied by increase in the height of real peak in the composite function of a single preferred orientation. Relative fractions of both major and minor textural components are reproduced with satisfactory fidelity In the simulation for analysis of multi-component textures.

Crystallite Orientation Analysis for Rolled Cubic Materials

1967 ◽  
Vol 11 ◽  
pp. 454-472 ◽  
Author(s):  
Peter R. Morris ◽  
Alan J. Heckler
Keyword(s):  
Spherical Harmonics ◽  
Jacobi Polynomials ◽  
Numerical Technique ◽  
Orientation Distribution ◽  
Angular Width ◽  
Crystallite Orientation ◽  
Cubic Materials ◽  
Generalized Spherical Harmonics ◽  
Data Points ◽  
Physical Symmetry

AbstractRoe's method for deriving the crystallite orientation distribution in a series of generalized spherical harmonics is applied to the analysis of texture in rolled cubic materials. The augmented Jacobi polynomials, which are the basis of the generalized spherical harmonics, have been derived for cubic crystallographic symmetry and orthotopic physical symmetry through the sixteenth order. Truncation of the series expansions at the sixteenth order should permit treatment of textures having a maximum of 17 times random and a minimum angular width at half maximum of 34°. A numerical technique has been developed which permits approximate evaluation of the integral equations from a finite array of data points. The method is illustrated for commercial steels and is used to elucidate the primary recrystalization texture of a decarburized Fe-3%Si alloy.

Spherical harmonics analysis based on the Reuss model in elastic macro strain and stress determination by powder diffraction

2017 ◽  
Vol 50 (6) ◽  
pp. 1735-1743 ◽  
Author(s):  
Nicolae C. Popa
Keyword(s):  
Stress Analysis ◽  
Stress Tensor ◽  
Rietveld Refinement ◽  
Powder Diffraction ◽  
Spherical Harmonics ◽  
Ground State ◽  
Stress Determination ◽  
New Approach ◽  
Generalized Spherical Harmonics ◽  
Strain Stress

In this paper a new approach to macro strain/stress analysis by generalized spherical harmonics is presented. It consists of expanding the stress tensor weighted by texture in a series of generalized spherical harmonics with the ground state of expansion specific to the classical Reuss model of an isotropic polycrystal. Like previously reported models having a ground state of hydrostatic type [Popa & Balzar (2001).J Appl Cryst.34, 187–195] and of Voigt type [Popaet al.(2014).J Appl Cryst.34, 154–159], the actual model is appropriate for use with Rietveld refinement.

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