X-ray diffraction broadening analysis

Stanko Popović, Željko Skoko

Abstract


The microstructure is very important in research aimed to the development of new materials. The microstructural parameters, crystallite size, crystallite size distribution, crystallite strain, dislocation density and stacking fault probability, play a major role in physical and chemical properties of the material. These parameters can be determined by a proper analysis of X-ray diffraction line profile broadening. The observed XRD line profile of the studied sample, h(ε), is the convolution of the instrumental profile, g(ε), inherent in diffraction, and pure diffraction profile, f(ε), caused by small crystallite (coherent domain) sizes, by faultings in the sequence of the crystal lattice planes, and by the strains in the crystallites. That is, f(ε) is the convolution of the crystallite size/faulting profile, p(ε), and the strain profile, s(ε). The derivation of f(ε) can be performed from the measured h(ε) and g(ε) by the Fourier transform method, usually referred to as the Stokes method. That method does not require assumptions in the mathematical description of h(ε) and g(ε). The analysis of f(ε) can be done by the Warren-Averbach method, which is applied to the Fourier coefficients obtained by the deconvolution. On the other hand, simplified methods (which may bypass the deconvolution) based on integral widths may be used, especially in studies where a good relative accuracy suffices. In order to obtain the relation among integral widths of f(ε), p(ε) and s(ε), one assumes bell-shaped functions for p(ε) and s(ε). These functions are routinely used in the profile fitting of the XRD pattern and in the Rietveld refinement of the crystal structure. The derived crystallite size and strain parameters depend on the assumptions for the profiles p(ε) and s(ε). Integral width methods overestimate both strain and crystallite size parameters in comparison to the Warren-Averbach-Stokes method. Also, the crystallite size parameter is more dependent on the accuracy, with which the profile tails are measured and how they are truncated, than it is the strain parameter. The integral width also depends on the background level error of the pure diffraction profile. The steps and precautions, which are necessary in order to minimize the errors, are suggested through simple examples. The values of the crystallite size and strain parameters, obtained from integral widths derived by the Stokes deconvolution, are compared with those which followed from the Warren-Averbach treatment of broadening. Recent approaches in derivation of microstructure are also mentioned in short.

Keywords


X-ray diffraction broadening; crystallite size and strain; deconvolution; integral width; Warren-Averbach method

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References


R. Delhez, Th. de Keijser, E. J. Mittemeijer, Accuracy of crystallite size and strain

values from X-ray diffraction line profiles using Fourier series; in: Accuracy in Powder

Diffraction, NBS Special Publication No. 567, S. Block, C. R. Hubbard (Eds),

Washington DC, National Bureau of Standards, 1980, pp. 213–253.

H. P. Klug, L. E. Alexander, X-ray Diffraction Procedures, New York, John Wiley, 1974.

J. G. M. van Berkum, G. J. M. Sprong, Th. de Keijser, R. Delhez, E. J. Sonneveld, The

optimum standard specimen for X-ray diffraction line-profile analysis, Powder

Diffraction 10, 129–139 (1995).

D. Balzar, N. Audebrand, M. Daymond, A. Fitch, A. Hewat, J. I. Langford, A. Le Bail,

D. Louër, O. Masson, C. N. McCowan, N. C. Popa, P. W. Stephens, B. Toby, Size-strain

line-broadening analysis of the ceria round-robin sample, J. Appl. Cryst. 37, 911–924

(2004).

B. E. Warren, X-ray Diffraction, Reading, MS, Addison-Wesley, 1969 / Dover, NY,

Mineola, 1990.

A. R. Stokes, A numerical Fourier-analysis method for the correction of widths and

shapes of lines on X-ray powder photographs, Proc. Phys. Soc. London, 61, 382–391

(1948).

B. E. Warren, B. L. Averbach, The effect of cold-work distortion on X-ray patterns,

J. Appl. Phys. 21, 595–599 (1950).

J. I. Langford, A rapid method for analysing the breadths of diffraction and spectral lines

using the Voigt function, J. Appl. Cryst. 11, 10–14 (1978).

P. Thompson, D. E. Cox, J. B. Hastings, Rietveld refinement of Debye-Scherrer

synchrotron X-ray data from Al2O3, J. Appl. Cryst. 20, 79–83 (1987).

J. I. Langford, Diffraction line broadening analysis; in: Accuracy in Powder Diffraction,

NBS Special Publication No. 567, S. Block, C. R. Hubbard (Eds), Washington DC,

National Bureau of Standards, 1980, pp. 255–269.

J. I. Langford, Use of pattern decomposition or simulation to study microstructure:

theoretical considerations; in: IUCr Monographs on Crystallography, Defect and

Microstructure Analysis by Diffraction, R. L. Snyder, J. Fiala, H. J. Bunge, (Eds),

Oxford, Oxford University Press, 1999, pp. 59–81.

W. Ruland, The separation of line broadening effects by means of line-width relations,

J. Appl. Cryst. 1, 90–101 (1968).

D. Balzar, S. Popović, Reliability of the simplified integral-breadth methods in

diffraction line-broadening analysis, J. Appl. Cryst. 29, 16–23 (1996).

F. R. L. Schoening, Strain and particle size values from X-ray line breadths, Acta Cryst.

, 975–976 (1965).

N. C. Halder, C. N. J. Wagner, Separation of particle size and lattice strain in integral

breadth measurements, Acta Cryst. 20, 312–313 (1966).

S. Popović, Application of bell-shaped functions in X-ray diffraction broadening

analysis, Croat. Chem. Acta 57, 749–755 ( 1984).

S. Popović, Ž. Skoko, G. Štefanić, Factors affecting diffraction broadening analysis,

Z. Kristallogr. Proc. 1, 55–62 (2011).

Ž. Skoko, S. Popović, G. Štefanić, Dependence of microstucture on errors in X-ray

diffraction profile measurement, J. Mater. Sci. Eng. A3, 690–697 (2013).

S. Popović, B. Čelustka, D. Bidjin, X-ray diffraction measurement of lattice parameters

of In2Se3, Phys. Stat. Sol. A 6, 301– 304 (1971).

S. Popović, A. Tonejc, B. Gržeta, B. Čelustka, R. Trojko, Revised and new crystal data

for indium selenides, J. Appl.Cryst. 12, 416–420 (1979).

S. Popović, Analysis of X-ray diffraction line broadening, Izvj. Jugoslav.centr. krist.,

Yugoslav (Croatian) Academy of Sciences and Arts 12, 47–80 (1977).

C. L. Cronan, F. J. Micale, M. Topić, H. Leidheiser Jr., A. C. Zettlemoyer, S. Popović,

Surface properties of Ni(OH)2 and NiO, J. Colloid Interface Sci. 55, 546–557 (1976).

Ž. Skoko, J. Popović, K. Dekanić, V. Kolbas, S. Popović, XBroad–program for

extracting basic microstructure information from XRD pattern in few clicks, J. Appl.

Cryst. 45, 594–597 (2012).

R. A. Young, The Rietveld Method. Oxford, Oxford University Press, 1993.

J. I. Langford, D. Louër, Powder diffraction, Rep. Prog. Phys. 59, 131–234 (1996).

P. Scardi, A new whole-powder pattern-fitting approach; in: IUCr Monographs on Crystallography, Defect and Microstructure Analysis by Diffraction, R. L. Snyder, J. Fiala, H. J. Bunge (Eds), Oxford, Oxford University Press, 1999, pp. 570–596.

P. Scardy, M. Leoni, Y. H. Dong, Whole powder pattern modelling, IUCr Commission on Powder Diffraction, Newsletter, 24, 23–24 (2000).




DOI: http://dx.doi.org/10.20450/mjcce.2015.642

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