Cubic spinel sample :

Ungár, T., Leoni, M. & Scardi, P. (1998). J. Appl. Cryst. 32, 290-295.
 
 
 
 
 
 
 
 
 

Anisotropic line broadening of X-ray diffraction profiles due to line and plane lattice defects was also said to be Fourier modelled (Scardi & Leoni, 1999). Applications to face-centred cubic structure materials provided detailed information on the defect structure : dislocation density and cut-off radius, stacking- and twin-fault probabilities were refined together with the structural parameters.
 

Cu-tablet.  Mixed model, dislocations + stacking and twin faults :

Scardi, P. & Leoni, M. (1999). J. Appl. Cryst. 32, 671-682.
 
 
 

A simple procedure for the experimental determination of the average contrast factor of dislocations has been established. The character of the dislocations can be determine in terms of a simple parameter q which can be used in Rietveld structure refinement procedures.
Ungár, T., Dragomir, I., Révéz, Á & Borbély, A. (1999). J. Appl. Cryst. 32, 992-1002.
 
 
 
 
 
 

Size-effect only is much easier to consider than both size-microstrain. How being sure that only a size effect occurs ? This could be shown by transmission electron micrograph of the powder. Then, the size distribution of single-crystal nanoparticles can be estimated by different approaches. One approach consists in Monte Carlo fitting of wide-angle X-ray scattering peak shape (Di Nunzio & Martelli, 1999). Another method applies maximum entropy for determining the column-length distributions from size-broadened diffraction profiles (Armstrong & Kalceff, 1999) as well as for removing instrument broadening.
Di Nunzio, P. E. & Martelli, S. (1999). J. Appl. Cryst. 32, 546-548.
Armstrong, N. & Kalceff, W. (1999). J. Appl. Cryst. 32, 600-613.
 
 
 
 
 

Dislocations are at the hearth of microstrain effects.

Many studies are undertaken in order to describe dislocations scattering effects.

If dislocations are described by statistical effects in powder diffraction, such an approach may prove soon to be oversimplified. More accurate description of dislocations may be required in order to access to more accuracy in the diffraction pattern simulation.
 
 
 

Synchrotron topography technique, simulation of micropipe-related superscrew dislocations in silicon carbide crystals, allowed to build a model capable of revealing the detailed diffraction behavior of the highly distorted region around the dislocation core (Huang et al., 1999). Considering that the distorted regions consist of small misoriented crystallites which diffract X-rays kinematically according to their local lattice orientation, they have developed a simplified numerical model for simulating the direct images of superscrew dislocations in synchrotron topographs.
Huang, X. R., Dudley, M., Vetter, W. M., Huang, W., Si, W. & Carter Jr., C. H. (1999). J. Appl. Cryst. 32, 516-524.
 
 
 
 
 
 
 

The theory of dislocation-induced X-ray or neutron diffraction line broadening has been adapted for Rietveld refinement by fitting a Voigt function to each peak (Wu, Gray & Kisi, 1998). Information on both the type of slip system and the density of dislocations in the crystallites may then be found by evaluating the shape parameter and the index-dependent breadth of the Voigt function. Precise description of the dislocation model seems to be possible by using this method. Applications (Wu, Kisi & Gray, 1998) were on deuterium-cycled LaNi₅ and b-PdD_0.66 neutron powder diffraction data.
Wu, E., Gray, E. Mac A. & Kisi, E. H. (1998). J. Appl. Cryst. 31, 356-362.
Wu, E., Kisi, E. H. & Gray, E. Mac A. (1998). J. Appl. Cryst. 31, 363-368.
 
 
 
 
 
 

X-ray scattering by crystals with local lattice rotation fields was examined by Barabash and Klimanek (1999). The peculiarities of the intensity distribution is dependent on the dislocation arrangements.
Barabash, R. I. & Klimanek P. (1999). J. Appl. Cryst. 32, 1050-1059.
 
 
 
 
 
 
 
 

What to do when a material is highly disordered and a space group cannot be defined so that the Rietveld method cannot be utilized directly ? An answer was given by Schilling and Dahn (1988) who fitted complex patterns of disordered manganese dioxides. In their paper, the g-MnO₂ structure is defined as an intergrowth of ramsdellitic and pyrolusitic domains. They proposed an analytic expression for the scattered intensity calculated from a stochastic stacking of four different types of layers.

Schilling, O. & Dahn, J. R. (1998). J. Appl. Cryst. 31, 396-406.
 
 
 
 
 

Convoluting or deconvoluting is a choice to be made at the beginning of a study of microstructures. Either the instrumental contribution is convoluted with the sample effect in order to regenerate the raw powder pattern, or a deconvoluted could be performed for extracting the sample contribution. The latter option is still not in use due to non preservation of intensity positivity and presence of spurious oscillations in the deconvoluted profile. One can note also that the separation of the Ka1 component from Ka2 is not at all recommended for analogous reasons (sometimes improperly described as a deconvolution instead of decomposition or desummation).
 

Armstrong, N. & Kalceff, W. (1998). J. Appl. Cryst. 31, 453-460.
 
 
 
 
 
 
 
 
 

However, deconvolution has to be done and the Fourier coefficients of the individual profile are needed if one wants to realize a Warren and Averbach (1950) analysis.
 
 
 
 
 
 
 

When possible, this is the best to do as was done in the study of dislocations and grain size in electrodeposited nanocrystalline Ni (Ungár, Révéz & Borbély, 1998), applying too the Williamson and Hall (1953) plot. The Scherrer particle size continues to be estimated with assumption of spherical particles (Klug & Alexander, 1974), for instance in the investigation of germanium nanoclusters (Bläsing et al., 1998). A recent tool (AXES software) for estimation of crystallite size and shape by Williamson-Hall analysis was proposed by Mändar et al. (1999).
Ungár, T., Révéz, Á & Borbély, A. (1998). J. Appl. Cryst. 31, 554-558.
Bläsing, J. Kohlert, P., Zacharias, M. & Veit, P. (1998). J. Appl. Cryst. 31, 589-593.
Mändar, H., Felsche, J., Mikli, V. & Vajakas, T. (1999). J. Appl. Cryst. 32, 345-350. AXES is available electronically at http://www.ccp14.ac.uk/.
 
 
 

Recent applications include the microstructural investigation of plastically deformed Pb_(1-x)Sn_x alloys through a profile fitting approach within the framework of the Warren-Averbach and the Williamson-Hall methods.
Chatterjee, P. & Sen Gupta, S. P. (1999). J. Appl. Cryst. 32, 1060-1068.
 
 
 
 
 

When even more disorder is detected, using variants of the Rietveld method has proved to be useful. Application to the characterization of disordered and small crystals  as found in semicrystalline polymers was recently published (Dupont, Jonas & Legras, 1997). The possible tilt angle of chain axes versus the large faces of the lamellar crystals were introduced in the model of PEEK [poly(ether-ether-ketone)] (Dupont et al. 1999). However, models with or without chain tilt give similar goodness-of-fit parameters, indicating that results coming from other techniques than X-ray diffractometry are required in order to characterize the dimensions and shape of crystals in isotropic polymer samples.

Dupont, O., Jonas, A. M. & Legras, R. (1997). J. Appl. Cryst. 30, 921-931.
Dupont, O., Ivanov, D. A., Jonas, A. M. & Legras, R. (1999). J. Appl. Cryst. 32, 497-504.
 
 

Studying 10 and 8.4 Å hydrates of kaolinite proves to be still difficult, if one looks at the best fits (not Rietveld) obtained by Jemai (1999).

Kaolinite
 
 

Hydrate 8.4 A
 
 
 

Hydrate 10 A
 

Jemai, S., Ben Haj Amara, A., Ben Brahim, J. & Plançon, A. (1999). J. Appl. Cryst. 32, 968-976.
 
 
 

The effect of sample transparency in powder diffractometry is generally treated in the Rietveld method by a simple peak displacing law. Ida and Kimura (1999a) treat this effect for Bragg-Brentano geometry as a convolution with an asymmetric aberration function. Also, they show that the flat-specimen effect on the peak profile can quantitatively be treated as a convolution with an asymmetric window function (Ida & Kimura, 1999b).
Ida, T. & Kimura, K. (1999a). J. Appl. Cryst. 32, 982-991.
Ida, T. & Kimura, K. (1999b). J. Appl. Cryst. 32, 634-640.
 
 
 

Additional references