Hostname: page-component-cb9f654ff-pvkqz Total loading time: 0 Render date: 2025-08-23T09:53:43.448Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of the Broadening of Powder Pattern Peaks using Variance, Integral Breadth, and Fourier Coefficients of the Line Profile

Published online by Cambridge University Press:  06 March 2019

N. C. Halder  and
C. N. J. Wagner
N. C. Halder
Affiliation:
Yale University, New Haven, Connecticut
C. N. J. Wagner
Affiliation:
Yale University, New Haven, Connecticut
Get access

Abstract

The broadening of powder pattern peaks has been studied by three methods—Fourier analysis, integral breadth measurements, and variance of the line profiles. The results obtained from the variances are compared with those obtained from the integral breadths and Fourier coefficients.

Tungsten filings were prepared at room temperature and their powder pattern peaks were recorded with a Norelco diffractometer using filtered Cu K α radiation. The variances, integral breadths, and Fourier coefficients were calculated with the IBM 7094 computer. The results indicate that the variance is very sensitive to the range of integration s 2 − s 1 = (2θ2 − 2θ1) cos θ0/λ. An error of ± 10% in this range due to the difficulty in choosing the correct background changes the values of the variance significantly and the integral breadth to a lesser extent. However, the same error does not affect the values of the Fourier coefficients.

Comparing the particle sizes and strains obtained by the three methods, it was found that the strains agreed remarkably well. The particle size calculated from the variance was smaller (DeW = 150Å) than that evaluated from the initial slope of the Fourier coefficients (De – 210Å) and from the integral breadths 2De ≃ D 1 = 430Å.

Information

Type
Research Article
Information
Advances in X-Ray Analysis , Volume 9: Fourteenth Annual Conference on Applications of X-ray Analysis, August 25-27, 1965 , 1965 , pp. 91 - 102
Copyright
Copyright © International Centre for Diffraction Data 1965

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

1. von Laue, M., “The Lorentz Factor and the Intensity Distribution in Debye-Scherrer Ring,” Z.Krist. 64: 115, 1926.Google Scholar
2. Wagner, C. N. J. and Aqua, E. N., “ Analysis of the Broadening of Powder Pattern Peaks from Cold-Worked Face-Centered and Body-Centered Cubic Metals,” in: Mueller, W. M., Mallett, G. R., and Fay, M. J. (eds.) Advances in X-Ray Analysis, Vol. 7, Plenum Press, New York, 1963, pp. 46–65.Google Scholar
3. Warren, B. E. and Averbach, B. L., “The Effect of Cold-Work Distortion on X-Ray Patterns,” J. Appl. Phys. 21: 595, 1950.Google Scholar
4. Wilson, A. J. C., “Variance as a Measure of Line Broadening,” Nature 193: 568–569, 1962.Google Scholar
5. Wilson, A. J. C., “On Variance as a Measure of Line Broadening in Diffractometry, General Theory and Small Particle Size,” Proc. Phys. Soc. (London) 80: 286, 1962,Google Scholar
6. Langford, J. I. and Wilson, A. J. C., Crystallography and Crystal Perfection, Academic Press, London, 1963, p. 207.Google Scholar
7. Haider, N. C. and Mitra, G. B., “Particle Size and Strain in Deformed Tungsten,” Proc. Phys. Soc. (London) 82: 557–560, 1963.Google Scholar
8. Aqua, E. N., “The Separation of Particle Size and Strain by the Method of Variance,” Acta Cryst. (in press).Google Scholar
9. Warren, B. E., “X-Ray Studies of Deformed Metals,” Progr. Metal Phys. 8: 147–202, 1959.Google Scholar
10. Wagner, C. N. J., “Analysis of the Broadening and Changes in Postiion of X-Ray Powder Pattern Peaks,” in: J. B. Cohen and J. E. Hilliard (eds,), Local Atomic Arrangements Studied by X-Ray Diffraction, Gordon and Breach, New York, 1966, Chapter 6.Google Scholar
11. Jones, F. W., “Measurement of Particle Size by the X-Ray Method,” Proc. Roy. Soc. (London) Ser. A. 166: 16, 1938.Google Scholar
12. Stokes, A. R., “ A Numerical Fourier-Analysis Method for the Correction of Widths and Shapes of Lines on X-Ray Powder Photographs,” Proc. Phys. Soc, (London) B 61: 382, 1948,Google Scholar
13. Vasil'ev, D. M. and Smimoy, B. I., “Certain X-Ray Diffraction Methods of Investigating Cold-Worked Metals,” Soviet Phys.-Usp. 4: 226–259, 1961.Google Scholar
14. Haider, N. C., “A Study on the X-Ray Line Profile,” Physica 30: 1044–1050, 1964.Google Scholar
15. Ruland, W., “The Integral Width of the Convolution of a Gaussian and/or Cauchy Distribution,” Acta Cryst. 18: 581, 1965.Google Scholar
16. Hall, W. H., “X-Ray Line Broadening in Metals,” Proc. Phys. Soc. (London) 62: 741–743, 1949.Google Scholar
17. Kurdyumov, G. V. and Lysak, L. I., J. Tech. Phys. U.S.S.R. 17: 993, 1947.Google Scholar
18. Schoening, F. R. L., “Strain and Particle Size Values from X-Ray Line Breadths,” Acta Cryst. 18: 975–976, 1965.Google Scholar
19. Haider, N. C. and Wagner, C. N. J., “Separation of Particle Size and Lattice Strain in Integral Breadth Measurements,” Acta. Cryst, 20: 312, 1966.Google Scholar
20. Pitts, E. and Willets, F. W., “Determination of Crystallite Size from Diffraction Profiles by using Standard Deviation as a Measure of Breadth,” Acta Cryst. 14: 1302–1303, 1961.Google Scholar
21. Willets, F. W., “An Analysis of X-Ray Diffraction Line Profiles Using Standard Deviation as a Measure of Breadth,” Brit. J. Appt. Phys. 16: 323, 1965.Google Scholar
22. Sato, Shinichiv, “Lattice Defects in Marten site by X-Ray Diffraction,” Japan J. Appl. Phys. 1: 210–217, 1962.Google Scholar
23. Wilson, A. J. C., Mathematical Theory of X-Ray Powder Diffractometry, Centrex Publishing Company, Eindhoven, 1963, p. 66.Google Scholar
24. Mitra, G. B. and Haider, N. C., “Stacking Fault Probabilities in Hexagonal Cobalt,” Acta Cryst. 17: 817–822, 1964.Google Scholar
25. Wagner, C. N. J. and Aqua, E. N., “X-Ray Diffraction Study of Imperfections in Rhenium,” J. Less-Common Metals, 8: 51–62, 1965.Google Scholar
26. Bertaut, E. F., “Debye-Scherrer Lines and Distribution of the Dimensions of the Bragg Domains in Polycrystalline Powders,” Acta Cryst. 3: 14–18, 1950.Google Scholar