Tech Note 1014

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Crystallite size, size distribution and strain



The broadening of diffraction lines occurs for two principal reasons: instrumental effects and physical origins [for the most current review articles on this subject, consult the recent monographs edited by Snyder et al. (1999) and Mittemeijer & Scardi (2004)]. The latter can be roughly divided into diffraction-order-independent (size) and diffraction-order dependent (strain) broadening in reciprocal space. Because many common crystalline defects cause line broadening to behave in a similar way, it is often difficult to discern the type of defect dominating in a particular sample. Therefore, it would be desirable to have standard samples with different types of defects to help to characterize unequivocally the
particular sample under the investigation.
Another point for consideration is the analysis of line
broadening for the purpose of extracting information about
crystallite size and structure imperfections. Quantification of
line-broadening effects is not trivial and there are different
and sometimes conflicting methods. Roughly, they can be
divided into two types: phenomenological ‘top–bottom’
approaches, such as integral-breadth methods (summarized by
Klug & Alexander, 1974; see also Langford, 1992) and Fourier
methods (Bertaut, 1949; Warren & Averbach, 1952). Both
approaches estimate physical quantities (coherently
diffracting domain size and lattice distortion/strain averaged
over a particular distance in the direction of the diffraction
vector) from diffraction line broadening. Only after the
analysis, is an attempt made to connect the thus-obtained
parameters with actual (i) defects and strains in the sample,
based on the behavior of certain parameters and a rather loose
association with underlying physical effects (see, for instance,
Warren, 1959), or (ii) crystallite size and shape in strain-free
samples (see, for example, Loue¨ r et al., 1983). Conversely,
there are physically based ‘bottom–top’ approaches that
attempt to model the influence of simplified dislocation
configurations (Krivoglaz, 1996; Unga´ r, 1999) or similar
defects (van Berkum, 1994), or crystallite size distributions
(Langford et al., 2000) on diffraction lines. Conditionally, we
can call these two approaches a posteriori and a priori,
respectively, according to when the correspondence of domain
size and strain parameters with the underlying microstructure
is made. Lately, there have been significant efforts (Unga´ r et
al., 2001, and references therein) to bridge these sometimes
diverging approaches. Even among the a posteriori approaches,
there are a variety of methods that yield conflicting
results for identically defined physical quantities. Simplified
integral-breadth methods that assume either a Gaussian or
Lorentzian function for a size- and/or strain-broadened profile
were shown to yield systematically different results (Balzar &
Popovic´, 1996). Nowadays, it is widely accepted that a ‘double-
Voigt’ approach, that is, a Voigt-function approximation for
both size-broadened and strain-broadened profiles (Langford,
1980, 1992; Balzar, 1992) is a better model than the simplified
integral-breadth methods. This model also agrees with the
Warren–Averbach (1952) analysis on the assumption of a
Gaussian distribution of strains (Balzar & Ledbetter, 1993;

Theoretical consideration


Integral Breadth Method


The simplified multiple line integral breadth method generally uses either of the following equations for size strain separation

,                                                       (1)

 ,                                             (2)

.                                                 (3)

These three equations are denoted as Cauchy–Cauchy (or often called the Williamson–Hall plot), Cauchy–Gaussian (or intermediate parabolic) and Gaussian–Gaussian approximations where β is the integral breadth, D denotes the volume weighted domain size and ε the upper limit of microstrain. The domain size and strain are generally expressed as,





where s2 = 2s1, x = β2/β1, c1 = 1/β1, c2 = β1/(2s1), s = 2sinθ/λ and s2 = 2s1. Here, the subscripts refer to first and second order of a particular reflection.


Why important

Mineral particle size distributions may yield geological information about a mineral's provenance, degree of metamorphism, degree of weathering, etc. We currently are using this program for research applications in the earth sciences. However, this program also would be useful to many types of manufacturers who use or synthesize clay-size (i.e. very fine grained) crystalline materials, because a material's particle size and structural strain may strongly influence its physical and chemical properties (e.g. its rheology, surface area, cation exchange capacity, solubility, reflectivity, etc.).

Clay-size crystals generally are too fine to be measured by light microscopy (~2 to 100 nm in thickness). Laser scattering methods give only average particle sizes, and particle size cannot be measured in a particular crystallographic direction. Also, the particles measured by laser techniques may be composed of several different phases, and some particles may be agglomerations of individual crystals. Individual particle dimensions may be measured by electron and scanning force microscopy, but it often takes several days of intensive effort to measure a few hundred particles per sample, which may yield an accurate mean size for a sample, but is often too few measurements to determine an accurate size distribution. Furthermore, such instrumentation is usually not available outside a research setting.

We are working with to test and improve our size determination for nano- and microspheres.

Measurement of size distributions by X-ray diffraction (XRD) solves these shortcomings. An X-ray scan of a sample is automated, taking a few minutes to a few hours. The resulting XRD peaks average diffraction effects from billions of individual clay-size particles. The size that is measured by XRD may (see below) be related to the "fundamental" particle size of a mineral, i.e. to the size of the individual crystalline domains, rather than to the size of particles formed by the agglomeration of crystals. Furthermore, one can determine the size of an individual phase within a mixture, and the dimension of particles in a particular crystallographic direction. Crystallite shape can be determined by measuring crystallite size in several different crystallographic directions.

The XRD method is based on the regular broadening of XRD peaks as a function of decreasing crystallite size. This broadening is a fundamental property of XRD, described by well-established theory.

An example

Below we give the crystallite size distribution studies nanosize titanium dioxides. These materials and analytical results are proprietary to Corpuscular, Inc. and reproduced with permission.




1. H. P. Klug and L. E. Alexander, X-ray Diffraction Procedures, 2nd edition (John Wiley,      New York, 1974).

2. D. Balzar and H. Ledbetter, J. Appl. Cryst. 26 (1993) 97-103.

3. D. Balzar, J. Res. Natl. Inst. Stand. Technol. 98 (1993) 321-353.

4. D. Balzar, J. Appl. Cryst. 25 (1992) 559-570.

5. D. Balzar and H. Ledbetter, J. Mater. Sci. Lett. 11 (1992) 1419-1420.

6. D. Balzar, H. Ledbetter, and A. Roshko, Pow. Diffr. 8 (1993) 2-6.

7. J. I. Langford, Accuracy in Powder Diffraction II, NIST Special Publication 846 (U.S.      Government Printing Office, Washington, D.C., 1992) p. 110-126.

8. Th. H. de Keijser, J. I. Langford, E. J. Mittemeijer, and A. B. P. Vogels, J. Appl. Cryst.       15 (1982) 308-314.

9. Th. H. de Keijser, E. J. Mittemeijer, and H. C. F. Rozendaal, J. Appl. Cryst. 16 (1983)       309-316.

10. B. E. Warren, X-ray Diffraction (Addison Wesley, Reading, MA, 1969).


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