Back in 1918 Scherrer [1] realized that small crystallites cause broadening of diffraction lines. However, more than a quarter of century was needed before the more complex theory of line broadening, which includes the strain as another source of broadening, was formulated by Stokes and Wilson [2]. Shortly thereafter, a new impulse was given to the theory, pushing aside the simple integral-breadth methods: Stokes [3] adapted the Fourier-deconvolution method to obtain the pure physically broadened line profile from the observed profile and, through developments of Bertaut [4] and Warren and Averbach [5], the more detailed and accurate analysis of complete line-profile shape became available. However, because of serious drawbacks in a deconvolution process in cases of line overlapping or weak structural broadening, its application was limited to the materials with the highest crystallographic symmetry. Moreover, after the development of the Rietveld [6] and similar full-pattern decomposition procedures [7], the integral-breadth methods experienced a new impetus, especially after Langford [8] introduced a Voigt function in the field of x-ray powder diffraction.
The complete process of line-broadening analysis will be discussed, beginning with the experimental procedure. Two approaches for a removal of instrumental broadening are currently in use: Fourier (or similar) deconvolution method or convolution of assumed physically broadened line-profile function with the predetermined instrumental profile, followed by the least-squares adjustment to the observed profile. After a removal of instrumental contribution, the physically broadened line profile is to be analyzed. The main emphasis will be given to the widely used methods of separation of size and strain broadening: the Warren-Averbach approximation [9] and the integral-breadth methods. The integral-breadth methods will be collated and their reliability discussed. It will be shown that systematic differences exist for both domain size and strain among different methods. The close attention will be given to an assumed Voigt-function profile shape for both size-broadened and strain broadened profiles [10], because it provides a bridge between Fourier Warren-% Averbach and integral-breadth methods [11]. It will be proven that both size and strain parameters, yielded by two different approaches, are related and compatible under the assumption of Gaussian distribution of mean-square strain. Some characteristic results from the literature will illustrate these findings.
1. P. Scherrer: Nachr. Gött. 2 (1918) 98.
2. A. R. Stokes and A. J. C. Wilson: Proc. Phys. Soc. London
56 (1944) 174.
3. A. R. Stokes: Proc. Phys. Soc. London 61 (1948) 382.
4. E. F. Bertaut: Acta Cryst. 3 (1950) 14.
5. B. E. Warren and B. L. Averbach: J. Appl. Phys. 21 (1950)
595.
6. H. M. Rietveld: Acta Cryst. 22 (1967) 151.
7. G. S. Pawley: J. Appl. Cryst. 14 (1981) 357.
8. J. I. Langford: J. Appl. Cryst. 11 (1978) 10.
9. B. E. Warren and B. L. Averbach: J. Appl. Phys. 23 (1952)
497.
10. J. I. Langford: Accuracy in Powder Diffraction, NBS
Special Publication No. 567, edited by S. Block and C. R.
Hubbard, Washington, D. C., National Bureau of Standards,
1980, p. 255.
11. D. Balzar and H. Ledbetter: J. Appl. Cryst. 26 (1993) 97.