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Use of an ellipsoid model for the determination of average crystallite shape and size in polycrystalline samples

Published online by Cambridge University Press:  10 January 2013

Tonči Balić Žunić  and
Jesper Dohrup
Tonči Balić Žunić
Affiliation:
Geological Institute, Øster Voldgade 10, DK-1350, Copenhagen K, Denmark
Jesper Dohrup
Affiliation:
Haldor Topsøe Research Laboratories, Nymøllevej 55, DK-2800 Lyngby, Denmark
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Abstract

A mathematical model for interpreting the anisotropical broadening of the powder diffraction lines by an average crystallite in the form of a triaxial ellipsoid is developed. The model covers satisfactorily a broad range of averaged crystallite shapes in polycrystalline samples of all crystal symmetries and provides simple formulas for use in powder pattern fitting routines. When ra, rb, rc are the principal ellipsoid radii, and ca, cb, cc direction cosines of diffraction vector related to the principal axes of ellipsoid, the average dimension of crystallites along the diffraction vector (Dhkl) is: Dhkl=K/ca2/ra2+cb2/rb2+cc2/rc2. The coefficient K has the value 3/2 if Dhkl is the volume average dimension of crystallites along the diffraction vector, or 4/3 in the case of the surface average dimension. The appropriate expression for use in whole pattern fitting routines is: b11h2+b22k2+b33l2+2b12hk+2b13hl+2b23kl=K2/Lhkl2dhkl2, where bij are the elements of a second-rank symmetric tensor. Finding eigenvalues and vectors of tensor b gives dimensions and orientations of the principal ellipsoid radii in reciprocal lattice values.

Type
Research Article
Information
Powder Diffraction , Volume 14 , Issue 3 , September 1999 , pp. 203 - 207
Copyright
Copyright © Cambridge University Press 1999

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References

Bertaut, E. F. (1950). “Raies de Debye-Scherrer et Répartition des Dimensions des Domaines de Bragg dans les Poudres Polycristallines,” Acta Crystallogr. 3, 14–18.CrossRefGoogle Scholar
Caglioti, G., Paoletti, A., and Ricci, F. P. (1958). “Choice of collimators for a crystal spectrometer for neutron diffraction,” Nucl. Instrum. 3, 223–228.CrossRefGoogle Scholar
Giacovazzo, C. (1992). Fundamentals of Crystallography (International Union of Crystallography, Oxford University Press, New York).Google Scholar
Grebille, D., and Bérar, J.-F. (1985). “Calculation of diffraction line profiles in the case of a major size effect: Application to boehmite AlOOH,” J. Appl. Crystallogr. 18, 301–307.CrossRefGoogle Scholar
Langford, J. I., and Louër, D. (1982). “Diffraction line profiles and Scherrer constants for materials with cylindrical crystallites,” J. Appl. Crystallogr. 15, 20–26.CrossRefGoogle Scholar
Langford, J. I., and Wilson, A. J. C. (1978). “Scherrer after sixty years: A survey and some new results in the determination of crystallite size,” J. Appl. Crystallogr. 11, 102–113.CrossRefGoogle Scholar
Le Bail, A., and Jouanneaux, A. (1997). “A qualitative account for anisotropic broadening in whole-powder-diffraction-pattern fitting by second-rank tensors,” J. Appl. Crystallogr. 30, 265–271.CrossRefGoogle Scholar
Lutterotti, L., and Scardi, P. (1990). “Simultaneous structure and size–strain refinement by the Rietveld method,” J. Appl. Crystallogr. 23, 246–252.CrossRefGoogle Scholar
Nandi, R. K., Kuo, H. K., Schlosberg, W., Wissler, G., Cohen, J. B., and Crist, B. Jr. (1984). “Single-peak methods for Fourier analysis of peak shapes,” J. Appl. Crystallogr. 17, 22–26.CrossRefGoogle Scholar
Scherrer, P. (1918). “Bestimmung der Grösse und der inneren Struktur von Kolloidteilchen mittels Röntgenstrahlen,” Nachr. Ges. Wiss. Göttingen 26 Juli, 98–100.Google Scholar
Smith, W. L. (1972). “Crystallite sizes and surface areas of catalysts,” J. Appl. Crystallogr. 5, 127–130.CrossRefGoogle Scholar
Solovyov, L. A. (1998). “Allowance for anisotropic line broadening in the crystal structure solution of [Pd(NH 3)4][Pd(C 2O 4)2],”Mater. Sci. Forum 278–281, 885–890.CrossRefGoogle Scholar
Toraya, B. H. (1989). “The determination of direction-dependent crystallite size and strain by X-ray whole-powder-pattern fitting,” Powder Diffr. 4, 130–136.CrossRefGoogle Scholar
Vargas, R., Louër, D., and Langford, J. I. (1983). “Diffraction line profiles and Scherrer constants for materials with hexagonal crystallites,” J. Appl. Crystallogr. 16, 512–518.CrossRefGoogle Scholar
Warren. (1969). X-Ray diffraction (Addison-Wesley, Reading, MA).Google Scholar
Wilson, A. J. C. (1962). X-Ray Optics, 2nd ed. (Methuen, London).Google Scholar