Stacking Disorder and its Analysis by X-ray Diffraction

Ernesto Estevez-Rams, Cristy Azanza-Ricardo, Beatriz Aragon-Fernandez*, Arbelio Penton-Madrigal

Instituto de Materiales y Reactivos. Universidad de la Habana. San Lazaro y L. CP 10400. C. Habana. Cuba

*Centro de Investigaciones Metalurgicas, Calle 51 esq a 240 . La Lisa. C. Habana. Cuba

The study of planar faulting by X-ray diffraction has a long history dating from the first works in diffractions. (Early contributions have been reviewed in [1], while further developments can be found in [2]).

Two types of methods have been so far used, (1) those which derive a set of useful parameters related to some assumed faulting model and (2) those based on computer simulations and Monte Carlo procedures which match simulated patterns with the experimental ones.

There a several limitations to both approaches. In the first type, Warren's use of difference equations [1, 3] is the most used: peak shift, broadening and asymmetry are related to the probabilities of deformation and twin faulting. As pointed out recently [4], the original approach can lead to wrong predictions of the faulting effects on diffraction profiles. Another drawback is the lack of generality, the methods need of specific derivations for different structures and faulting types.

Concerning Monte Carlo schemes, these are trial and error procedures matching simulated patterns with experimental ones. Simulation procedures are better suited than Warren's approach for investigating complex faulted structures. Yet, simulations strongly rely on the ability of the researchers to propose faulting models suitable to the investigated problem, and they make little use of the observed experimental data until the last stages of analysis, where comparison is performed.

A recent approach by the authors derives quantitative information about the stacking order in layer crystals directly from diffraction data, without assuming any prior model for the stacking disorder [5, 6]. The original formulation was valid for powder diffraction data with one type of layer in the stacking sequence and a integer number of times a constant displacement vector between the different possible layer positions. The solution of the diffraction equations allowed, in a general framework, to derive features of powder diffraction patterns of faulted layer crystals and better understand the effect of faulting in the diffraction pattern [6, 7]. The relation between the symmetric and asymmetric component of broaden peaks were explored for the general close packed case [7]. This allowed avoiding the arbitrariness in the modeling of peak profiles affected by faulting.

In this presentation we will review the latest work done by the authors in the analysis of diffraction pattern of samples with planar disorder. Some theoretical developments will be reviewed together with applications to real data will be shown.

If we lift the restriction on the same type of layer for the diffraction equations the expression for the interference functions (definitions and more detailed mathematical background can be found in [4, 5] ) can written as:

(1)

Expression (1) reduces to the ones already derived in [4] for structures with all layers having the same structure factor:

(2)

In the case of a constant structure factor per layer, the interference function (experimental observable) can be related to the probability correlation function which describes the stacking ordering of the layers [5, 6]. The problem of the diffraction pattern of planar disordered structures then reduces to the extraction from the available data of the probability correlation function.

It has been shown in [7] that (2) forces a relation between the symmetric and asymmetric component of the peak broadening. This biunivocal relation implies that once a symmetrical component is chosen the asymmetrical component is completely determined, which reduces, at least in this sense, the arbitrariness of the asymmetry modeling. The relations can also be used to address the implications of an assumed peak profile model in the underlying planar disorder.

The following table shows the corresponding expression of the asymmetric component for common used symmetrical profiles:

Profile	Symmetric Term	Asymmetric Term
Gaussian
Lorentzian
PseudoVoigt
Pearson VII

Acknowledgement

EER will like to thank the Alexander von Humboldt Foundation for the sustained support. This work was partially done under an Alma Mater grant.

[1] Welberry, T. R., 1985, Rep. Prog. Phys., 48, 1543.

[2] Ustinov, A. I., 1994, Defect and Microstructure Analysis by Diffraction, edited by R. L. Snyder, J. Fiala and H. J. Bunge (Oxford: Oxford Science Publications), chapter 15, pp. 264–316.

[3] Warren, B. E., 1990, X-ray Diffraction (New york: Dover Publications).

[4] Velterop, L., Delhez, R., Keijser, Th. H., Mittemeijer, E. J., and Reefman, D., 2000, J. Appl. Crystallogr., 33, 296.

[5] Estevez-Rams, E., Martinez, J., Penton-Madrigal, A., and Lora-Serrano, R., 2001, Phys. Rev. B, 63, 54 109.

[6] E. Estevez-Rams,1, B. Aragon-Fernandez, H. Fuess, and A. Penton-Madrigal, 2003, Phys. Rev. B, 68, 064111.

[7] E. Estevez-Rams, M. Leoni, P. Scardi, B. Aragon-Fernandez, Phil. Mag. , 2003, 83, 4045.