Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-15T07:35:29.313Z Has data issue: false hasContentIssue false
  • English
  • Français

Comparison methods of variance and line profile analysis for the evaluation of microstructures of materials

Published online by Cambridge University Press:  29 February 2012

V. Soleimanian and
S. R. Aghdaee
V. Soleimanian
Affiliation:
Department of Physics, Iran University of Science and Technology, Narmak, 16844 Teheran, Iran
S. R. Aghdaee*
Affiliation:
Department of Physics, Iran University of Science and Technology, Narmak, 16844 Teheran, Iran
*
a)Author to whom correspondence should be addressed. Electronic mail: aghdaee@iust.ac.ir
Get access

Abstract

A comparison of different methods of X-ray diffraction analysis for the determination of crystallite size and microstrain; namely, line profile analysis, Rietveld refinement, and three approaches based on the variance method, is presented. The analyses have been applied to data collected on a ceria sample prepared by the IUCr Commission on Powder Diffraction. In the variance method, split Pearson VII, the Voigt function, and its approximation pseudo-Voigt function were fitted to X-ray diffraction line profiles. Based on the fitting results, the variances of line profiles were calculated and then the crystallite size and root mean square strain were obtained from variance coefficients. A SS plot of Langford as well as a Fourier analysis and Rietveld refinement have been carried out. The average crystallite size and microstrain were determined. The values of area-weighted domain size determined from the variance method are in agreement with those obtained from line profile analysis within a single (largest) standard uncertainty, and the volume-weighted domain sizes derived from the SS plot, Fourier size distribution, and Rietveld refinement agree within a single standard uncertainty. The results of rms strain calculated from variance and Pearson VII shape function and those from Rietveld refinements fall within a single esd. However, the variance method in conjunction with pseudo-Voigt and Voigt functions produce rms strains substantially larger than those determined from line profile analysis and Rietveld refinements.

Keywords

line broadeningX-ray diffraction (XRD)microstructure

Information

Type
Technical Articles
Information
Powder Diffraction , Volume 23 , Issue 1 , March 2008, pp. 41 - 51
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Balzar, D., Audebrand, N., Daymond, M. R., Fitch, A., Hewat, A., Langford, J. I., Le Bail, A., Louër, D., Masson, O., McCowan, C. N., Popa, N. C., Stephens, P. W., and Toby, B. H. (2004). “Size-Strain Line-Broadening Analysis of the Ceria Round-Robin Sample,” J. Appl. Crystallogr. JACGAR 10.1107/S0021889804022551 37, 911–924.Google Scholar
Caglioti, G., Paoletti, A., and Ricci, F. P. (1958). “Choice of Collimators for Crystal Spectrometers for Neutron Diffraction,” Nucl. Instrum. NUINAO 10.1016/0369-643X(58)90029-X 3, 223–228.Google Scholar
Dong, Y. H. and Scardi, P. (2000) “MarqX: A New Program for Whole-Powder-Pattern Fitting,” J. Appl. Crystallogr. JACGAR 10.1107/S002188989901434X 33, 184–189.Google Scholar
Halder, N. C. and Wagner, C. N. J. (1966). “Separation of Particle Size and Lattice Strain in Integral Breadth Measurements,” Acta Crystallogr. ACCRA9 10.1107/S0365110X66000628 20, 312–313.CrossRefGoogle Scholar
Hall, M. M. Jr., Veeraraghavan, V. G., Rubin, H., and Winchell, P. G. (1977) “The Approximation of Symmetric X-Ray Peaks by Pearson Type VII Distributions,” J. Appl. Crystallogr. JACGAR 10.1107/S0021889877012849 10, 66–68.CrossRefGoogle Scholar
de Keijser, Th., Mittemeijer, E. J., and Rozendaal, H. C. F. (1983). “The Determination of Crystallite-Size and Lattice-Strain Parameters in Conjunction with the Profile-Refinement Method for the Determination of Crystal Structures,” J. Appl. Crystallogr. JACGAR 10.1107/S0021889883010493 16, 309–316.CrossRefGoogle Scholar
Klug, H. P. and Alexander, L. E. (1974). X-Ray Diffraction Procedures for Polycrystalline and Amorphous Materials, 2nd ed. (Wiley, New York).Google Scholar
Langford, J. I. (1968a). “The Variance and Other Measures of Line Broadening in Powder Diffractometry. I. Practical Considerations,” J. Appl. Crystallogr. JACGAR 10.1107/S002188986800498X 1, 48–59.CrossRefGoogle Scholar
Langford, J. I. (1968b). “The Variance and Other Measures of Line Broadening in Powder Diffractometry. II. Determination of Particle Size,” J. Appl. Crystallogr. JACGAR 1, 131–138.CrossRefGoogle Scholar
Langford, J. I. (1978). “A Rapid Method for Analysing the Breadths of Diffraction and Spectral Lines Using the Voigt Function,” J. Appl. Crystallogr. JACGAR 10.1107/S0021889878012601 11, 10–14.CrossRefGoogle Scholar
Langford, J. I. (1982). “The Variance as a Measure of Line Broadening: Corrections for Truncation, Curvature and Instrumental Effects,” J. Appl. Crystallogr. JACGAR 10.1107/S0021889882012047 15, 315–322.CrossRefGoogle Scholar
Langford, J. I. (1992) “The Use of the Voigt Function in Determining Microstructural Properties from Diffraction Data by Means of Pattern Decomposition,” in Proceedings of the International Conference: Accuracy in Powder Diffraction II (NIST Special Publication 846), edited by Prince, E. and Stalick, J. K. (US. Government Printing Office,Washington, D.C.), pp. 110–126.Google Scholar
Langford, J. I., Louër, D., Sonneveld, E. J., and Visser, J. W. (1986). “Applications of Total Pattern Fitting to a Study of Crystallite Size and Strain in Zinc Oxide Powder,” Powder Diffr. PODIE2 1, 211–221.CrossRefGoogle Scholar
Michette, A. G. and Pfauntsch, S. J. (2000). “Laser Plasma X-Ray Line Spectra Fitted Using the Pearson VII Function,” J. Phys. D JPAPBE 10.1088/0022-3727/33/10/308 33, 1186–1190.CrossRefGoogle Scholar
Mitra, G. B. and Mukherjee, P. S. (1981). “Application of the Method of Moments to X-Ray Diffraction Line Profiles from Paracrystals: Native Cellulose Fibres,” J. Appl. Crystallogr. JACGAR 14, 421–431.Google Scholar
Naidu, S. V. N. and Houska, C. R. (1982). “Profile Separation in Complex Powder Patterns,” J. Appl. Crystallogr. JACGAR 15, 190–198.CrossRefGoogle Scholar
Popa, N. C. (1998). “The (hkl) Dependence of Diffraction-Line Broadening Caused by Strain and Size for All Laue Groups in Rietveld Refinement,” J. Appl. Crystallogr. JACGAR 10.1107/S0021889897009795 31, 176–180.CrossRefGoogle Scholar
Sánchez-Bajo, F. and Cumbrera, F. L. (1997). “The Use of the Pseudo-Voigt Function in the Variance Method of X-Ray Line-Broadening Analysis,” J. Appl. Crystallogr. JACGAR 10.1107/S0021889896015464 30, 427–430.Google Scholar
Sánchez-Bajo, F., Oritz, A. L. F., and Cumbrera, F. L. (2006). “Analytical Formulation of the Variance Method of Line-Broadening Analysis for Voigtian X-Ray Diffraction Peaks,” J. Appl. Crystallogr. JACGAR 39, 598–600.CrossRefGoogle Scholar
Schoening, F. R. L. (1965). “Strain and Particle Size Values From X-Ray Line Breadths,” Acta Crystallogr. ACCRA9 10.1107/S0365110X65002335 18, 975–976.CrossRefGoogle Scholar
Solovyov, L. A. (2004) “Full-Profile Refinement by Derivative Difference Minimization,” J. Appl. Crystallogr. JACGAR 37, 743–749.CrossRefGoogle Scholar
Stokes, A. R. (1948). “A Numerical Fourier-Analysis Method for the Correction of Widths and Shapes of Lines on X-Ray Powder Photographs,” Proc. Phys. Soc. London PPSOAU 10.1088/0959-5309/61/4/311 61, 382–391.Google Scholar
Suortti, P., Ahtee, M., and Unonius, L. (1979). “Voigt Function Fit of X-Ray and Neutron Powder Diffraction Profiles,” J. Appl. Crystallogr. JACGAR 12, 365–369.CrossRefGoogle Scholar
Thompson, P., Cox, D. E., and Hastings, J. B. (1987). “Rietveld Refinement of Debye–Scherrer Synchrotron X-Ray Data from Al2O3,” J. Appl. Crystallogr. JACGAR 10.1107/S0021889887087090 20, 79–83.CrossRefGoogle Scholar
Toraya, H., Yoshimura, M., and Somiya, S. (1983). “A Computer Program for the Deconvolution of X-Ray Diffraction Profiles with the Composite of Pearson Type VII Functions,” J. Appl. Crystallogr. JACGAR 16, 653–657.CrossRefGoogle Scholar
Warren, B. E. and Averbach, B. L. (1950). “The Effect of Cold-Work Distortion on X-Ray Patterns,” J. Appl. Phys. JAPIAU 10.1063/1.1699713 21, 595–599.Google Scholar
Wilson, A. J. C. (1962a). X-Ray Optics (Methuen, London).Google Scholar
Wilson, A. J. C. (1962b). “On Variance as a Measure of Line Broadening in Diffractometry General Theory and Small Particle Size,” Proc. Phys. Soc. London PPSOAU 10.1088/0370-1328/80/1/333 80, 286–294.CrossRefGoogle Scholar
Young, R. A. and Wiles, D. B. (1982). “Profile Shape Functions in Rietveld Refinements,” J. Appl. Crystallogr. JACGAR 10.1107/S002188988201231X 15, 430–438.CrossRefGoogle Scholar