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Whole powder pattern modelling: microstructure determination from powder diffraction data
International Tables for Crystallography (2019). Vol. H, ch. 3.6, pp. 288-303 [ doi:10.1107/97809553602060000951 ]
... the profile, which in turn can be represented as (Scardi & Leoni, 2002 ) where includes all constant or known functions of (e.g. ... dislocations and faults (Cheary & Coelho, 1992 ; van Berkum, 1994 ; Scardi & Leoni, 2002 ). Taking these into account, where TIP(L) and ... for several cases of practical interest (see, for example, Scardi & Leoni, 2002 , 2004 , 2005 ; Leoni & Scardi, 2004 ; Leineweber & Mittemeijer, 2004 ; ...
Histogram (arbitrary) distributions
International Tables for Crystallography (2019). Vol. H, Section 5.1.10, pp. 533-534 [ doi:10.1107/97809553602060000966 ]
... 5.1.70) is still valid in this case. As shown by Leoni & Scardi (2004 ), the analytical distribution can be replaced by a ... profiles corresponding to each size component. References David, W. I., Leoni, M. & Scardi, P. (2010). Domain size analysis in the ...
Analytical distributions
International Tables for Crystallography (2019). Vol. H, Section 5.1.9, p. 533 [ doi:10.1107/97809553602060000966 ]
... coefficients and for the profiles in some particular cases (Scardi & Leoni, 2001 ; Scardi et al., 2006 ). For instance, the Fourier ... defined as with moments can easily be obtained as (Scardi & Leoni, 2001 ) whereas the corresponding profile can just be calculated via ... moments The Fourier coefficients can easily be calculated as (Scardi & Leoni, 2001 ) An approximation of the corresponding line profile was ...
Extension of the Fourier approach to a size distribution
International Tables for Crystallography (2019). Vol. H, Section 5.1.8, pp. 531-533 [ doi:10.1107/97809553602060000966 ]
Extension of the Fourier approach to a size distribution 5.1.8. Extension of the Fourier approach to a size distribution To what extent can the Scherrer formula be employed to the study of a real powder where a distribution of sizes is present? In 1955, Stokes wrote `It must be observed that ...
Meaning of the size Fourier coefficients: mean size and column-length distribution
International Tables for Crystallography (2019). Vol. H, Section 5.1.7, p. 531 [ doi:10.1107/97809553602060000966 ]
... we can therefore build a column-length distribution (CLD) p(M), such that gives the fraction of columns with a length between M and M + dM. For instance, for a sphere of diameter D. ...
Calculation of the intensity profile for simple shapes
International Tables for Crystallography (2019). Vol. H, Section 5.1.6, pp. 530-531 [ doi:10.1107/97809553602060000966 ]
Calculation of the intensity profile for simple shapes 5.1.6. Calculation of the intensity profile for simple shapes According to equation (5.1.22) , the knowledge of the common volume function suffices to calculate the corresponding diffraction profile in reciprocal space. For a single sphere, the intensity profile assumes a particularly simple mathematical form ...
Common volume function for simple shapes
International Tables for Crystallography (2019). Vol. H, Section 5.1.5, pp. 528-530 [ doi:10.1107/97809553602060000966 ]
... unit-cell volume: and, through Cholesky decomposition, , the orthogonalization matrix M. This will give us the general values for the A ... al. (1983 ) (hexagonal prisms), Grebille & Bérar (1985 ) (convex polyhedra), Scardi & Leoni, 2001 (sphere, simple and general polyhedra, distributions of spheres and ... of simpler convex polyhedra [cf. Grebille & Bérar (1985 ) and Scardi & Leoni (2001 )], leading to the general result that the common ...
The peak profile and the common volume function (`ghost')
International Tables for Crystallography (2019). Vol. H, Section 5.1.4, pp. 526-528 [ doi:10.1107/97809553602060000966 ]
... is the basis of the whole powder pattern modelling (Scardi & Leoni, 2002 ; Chapter 3.6 ), the second one is the basis of ... of spherical particles. Phys. Rev. B, 73, 1-7.GoogleScholar Paterson, M. S. (1952). The effects of lattice imperfections on X ... line broadening. J. Appl. Phys. 23, 499-500.GoogleScholar Scardi, P. & Leoni, M. (2002). Whole powder pattern modelling. Acta Cryst. ...
Fourier methods
International Tables for Crystallography (2019). Vol. H, Section 5.1.3, p. 526 [ doi:10.1107/97809553602060000966 ]
Fourier methods 5.1.3. Fourier methods In the Scherrer formula, and in general in the methods based on the integral breadth, just part of the information contained in the pattern is employed in the analysis. The information on the microstructure is coded into the whole shape of the peak, so the breadth ...
The Scherrer formula and integral-breadth methods
International Tables for Crystallography (2019). Vol. H, Section 5.1.2, p. 524 [ doi:10.1107/97809553602060000966 ]
... of crystallite size. J. Appl. Cryst. 11, 102-113.GoogleScholar Laue, M. von (1926). Lorentz-Faktor und Intensitätsverteilung in Debye ...
International Tables for Crystallography (2019). Vol. H, ch. 3.6, pp. 288-303 [ doi:10.1107/97809553602060000951 ]
... the profile, which in turn can be represented as (Scardi & Leoni, 2002 ) where includes all constant or known functions of (e.g. ... dislocations and faults (Cheary & Coelho, 1992 ; van Berkum, 1994 ; Scardi & Leoni, 2002 ). Taking these into account, where TIP(L) and ... for several cases of practical interest (see, for example, Scardi & Leoni, 2002 , 2004 , 2005 ; Leoni & Scardi, 2004 ; Leineweber & Mittemeijer, 2004 ; ...
Histogram (arbitrary) distributions
International Tables for Crystallography (2019). Vol. H, Section 5.1.10, pp. 533-534 [ doi:10.1107/97809553602060000966 ]
... 5.1.70) is still valid in this case. As shown by Leoni & Scardi (2004 ), the analytical distribution can be replaced by a ... profiles corresponding to each size component. References David, W. I., Leoni, M. & Scardi, P. (2010). Domain size analysis in the ...
Analytical distributions
International Tables for Crystallography (2019). Vol. H, Section 5.1.9, p. 533 [ doi:10.1107/97809553602060000966 ]
... coefficients and for the profiles in some particular cases (Scardi & Leoni, 2001 ; Scardi et al., 2006 ). For instance, the Fourier ... defined as with moments can easily be obtained as (Scardi & Leoni, 2001 ) whereas the corresponding profile can just be calculated via ... moments The Fourier coefficients can easily be calculated as (Scardi & Leoni, 2001 ) An approximation of the corresponding line profile was ...
Extension of the Fourier approach to a size distribution
International Tables for Crystallography (2019). Vol. H, Section 5.1.8, pp. 531-533 [ doi:10.1107/97809553602060000966 ]
Extension of the Fourier approach to a size distribution 5.1.8. Extension of the Fourier approach to a size distribution To what extent can the Scherrer formula be employed to the study of a real powder where a distribution of sizes is present? In 1955, Stokes wrote `It must be observed that ...
Meaning of the size Fourier coefficients: mean size and column-length distribution
International Tables for Crystallography (2019). Vol. H, Section 5.1.7, p. 531 [ doi:10.1107/97809553602060000966 ]
... we can therefore build a column-length distribution (CLD) p(M), such that gives the fraction of columns with a length between M and M + dM. For instance, for a sphere of diameter D. ...
Calculation of the intensity profile for simple shapes
International Tables for Crystallography (2019). Vol. H, Section 5.1.6, pp. 530-531 [ doi:10.1107/97809553602060000966 ]
Calculation of the intensity profile for simple shapes 5.1.6. Calculation of the intensity profile for simple shapes According to equation (5.1.22) , the knowledge of the common volume function suffices to calculate the corresponding diffraction profile in reciprocal space. For a single sphere, the intensity profile assumes a particularly simple mathematical form ...
Common volume function for simple shapes
International Tables for Crystallography (2019). Vol. H, Section 5.1.5, pp. 528-530 [ doi:10.1107/97809553602060000966 ]
... unit-cell volume: and, through Cholesky decomposition, , the orthogonalization matrix M. This will give us the general values for the A ... al. (1983 ) (hexagonal prisms), Grebille & Bérar (1985 ) (convex polyhedra), Scardi & Leoni, 2001 (sphere, simple and general polyhedra, distributions of spheres and ... of simpler convex polyhedra [cf. Grebille & Bérar (1985 ) and Scardi & Leoni (2001 )], leading to the general result that the common ...
The peak profile and the common volume function (`ghost')
International Tables for Crystallography (2019). Vol. H, Section 5.1.4, pp. 526-528 [ doi:10.1107/97809553602060000966 ]
... is the basis of the whole powder pattern modelling (Scardi & Leoni, 2002 ; Chapter 3.6 ), the second one is the basis of ... of spherical particles. Phys. Rev. B, 73, 1-7.GoogleScholar Paterson, M. S. (1952). The effects of lattice imperfections on X ... line broadening. J. Appl. Phys. 23, 499-500.GoogleScholar Scardi, P. & Leoni, M. (2002). Whole powder pattern modelling. Acta Cryst. ...
Fourier methods
International Tables for Crystallography (2019). Vol. H, Section 5.1.3, p. 526 [ doi:10.1107/97809553602060000966 ]
Fourier methods 5.1.3. Fourier methods In the Scherrer formula, and in general in the methods based on the integral breadth, just part of the information contained in the pattern is employed in the analysis. The information on the microstructure is coded into the whole shape of the peak, so the breadth ...
The Scherrer formula and integral-breadth methods
International Tables for Crystallography (2019). Vol. H, Section 5.1.2, p. 524 [ doi:10.1107/97809553602060000966 ]
... of crystallite size. J. Appl. Cryst. 11, 102-113.GoogleScholar Laue, M. von (1926). Lorentz-Faktor und Intensitätsverteilung in Debye ...
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