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.

Elastic macro strain and stress determination by powder diffraction: spherical harmonics analysis starting from the Voigt model

2013 ◽  
Vol 47 (1) ◽  
pp. 154-159 ◽  
Author(s):  
Nicolae C. Popa ◽  
Davor Balzar ◽  
Sven C. Vogel
Keyword(s):  
Stress State ◽  
Spherical Harmonics ◽  
Ground State ◽  
Strain Tensor ◽  
New Approach ◽  
Generalized Spherical Harmonics ◽  
Strain Stress ◽  
Voigt Model ◽  
Spherical Harmonics Expansion

A new approach for the determination of the elastic macro strain and stress in textured polycrystals by diffraction is presented. It consists of expanding the strain tensor weighted by texture in a series of generalized spherical harmonics where the ground state is defined by the strain/stress state in an isotropic sample in the Voigt model. In contrast to similar expansions already reported by other authors, this new approach provides expressions valid for any sample and crystal symmetries and can easily be implemented in whole powder pattern fitting, including Rietveld refinement. An earlier article [Popa & Balzar (2001).J. Appl. Cryst.34, 187–195] reported a similar model, but with a spherical harmonics expansion around the hydrostatic strain/stress state of the isotropic polycrystal. The availability of several different models is beneficial in order to allow one to select the representation in which the ground state is the closest to the actual stress state in the sample.

scholarly journals Elastic macro strain and stress determination by powder diffraction: spherical harmonics analysis starting from the Voigt model. Corrigenda

2015 ◽  
Vol 48 (1) ◽  
pp. 311-311 ◽  
Author(s):  
N. C. Popa ◽  
D. Balzar ◽  
S. C. Vogel
Keyword(s):  
Powder Diffraction ◽  
Spherical Harmonics ◽  
Stress Determination ◽  
Voigt Model

Corrections to the paper by Popa, Balzar & Vogel [J. Appl. Cryst.(2014),47, 154–159] are provided.

Quantitative powder diffraction phase analysis with a combination of the Rietveld method and the addition method

2002 ◽  
Vol 17 (4) ◽  
pp. 287-289 ◽  
Author(s):  
T. Balić-Žunić
Keyword(s):  
Rietveld Refinement ◽  
Phase Analysis ◽  
Amorphous Phase ◽  
Powder Diffraction ◽  
Rietveld Method ◽  
Original Sample ◽  
Addition Method ◽  
Original Mixture ◽  
The Absolute ◽  
New Variables

The Rietveld method can be combined with the addition method to determine the absolute quantities of the phases treated by Rietveld refinement plus the quantity of phase(s) not treated by it (amorphous or unobserved). If q is the added proportion of a defined phase already present in the sample, and a1 and a2 its relative proportions as determined by Rietveld refinement prior and after the addition, the proportion of the amorphous (untreated) phase(s) in the original sample is calculated as xo=[a2−(1−q)a1−q]/(1−q)(a2−a1). The absolute quantities of the phases treated by Rietveld refinement are then determined by a correction for the content of the amorphous phase(s), or they can be calculated directly from specific equations. The advantage of the method is that no new variables are introduced in the refinement when the added standard already is a part of the original mixture.

Maximum substitution of magnesium for calcium sites in Mg-β-TCP structure determined by X-ray powder diffraction with the Rietveld refinement

2009 ◽  
Vol 118 (2-3) ◽  
pp. 337-340 ◽  
Author(s):  
J.C. Araújo ◽  
M.S. Sader ◽  
E.L. Moreira ◽  
V.C.A. Moraes ◽  
R.Z. LeGeros ◽  
...  
Keyword(s):  
Rietveld Refinement ◽  
Powder Diffraction ◽  
X Ray

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.

Rietveld refinement of the solid-solution series: (Cu,Mg)SO4·H2O

1995 ◽  
Vol 10 (3) ◽  
pp. 189-194 ◽  
Author(s):  
C. L. Lengauer ◽  
G. Giester
Keyword(s):  
Solid Solution ◽  
Rietveld Refinement ◽  
Powder Diffraction ◽  
Rietveld Method ◽  
Structure Refinement ◽  
Solid Solution Series ◽  
Cation Site ◽  
Tg Analysis ◽  
X Ray ◽  
Solution Series

The kieserite-type solid-solution series of synthetic (Cu,Mg)SO4·H2O was investigated by TG-analysis and X-ray powder diffraction using the Rietveld method. Representatives with Cu≥20 mol% are triclinic distorted () analogous to the poitevinite (Cu,Fe)SO4·H2O compounds. Cation site ordering with preference of Cu for the more distorted M1 site was additionally proven by the structure refinement.

Sign in / Sign up

Export Citation Format

Share Document