**
Introduction**
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,
_{} (4)
and
_{} (5)
where *s*_{2}
= 2*s*_{1}, *x *= *β*_{2}/*β*_{1},
*c*_{1} = 1/*β*_{1}, *c*_{2} = *β*_{1}/(2*s*_{1}),
*s *= 2sin*θ*/*λ *and *s*_{2} = 2*s*_{1}.
Here, the subscripts refer to first and second order of a particular
reflection.
³
**
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
www.microspheres-nanospheres.com 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.
**
References**
1. H. P. Klug and
L. E. Alexander, *X-ray Diffraction Procedures*, 2^{nd}
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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