Modelling effects of small crystallite size and lattice defects on powder diffraction lines

 

Z. Matěj, R. Kužel

 

Department of Condensed Matter Physics, Faculty of Mathematics and Physics,

Charles University in Prague, Ke Karlovu 5, 121 16 Praha 2, Czech Republic

matej@karlov.mff.cuni.cz

 

Introduction

X-ray powder diffraction (XRD) techniques are very appropriate for characterisation of various materials as non-destructive and containing rich information on phase composition, lattice parameters, crystal structure and other aspects of material structure on nanometre and submicrometre scale, which can be related to application properties and hence to be also of technological interest. Presence of lattice defects, their type and concentration as well as crystallite size can be determined from width and shape of diffraction lines. The aim of this contribution is to give a short overview of classical approaches of a line profile analysis (LPA) complemented with methods based on simulations and fitting of whole diffraction patterns advanced during the last decade. The theoretical summary is supplemented with some practical examples.  

Direct and whole profile modelling techniques

Already at the birth of the XRD analysis Scherrer (1918) [1] utilized a simple relation between diffraction peak width and particle size. Variations of the lattice parameter between individual crystallites or lattice deformations around defects were then included in the Williamson-Hall plot method [1], which links information from multiple hkl reflections to separate the size and strain related broadening effects. Fourier transformation of intensity data and again the analysis of reflections of different order are the basis of the Warren-Averbach method [1]. This technique can reveal finer details in the real space without preliminary assumptions about the microstructure model.

Figure 1. Simulated diffraction profile for a fcc copper polycrystalline specimen with intrinsic stacking faults. 311 reflection, stacking fault probability a = 0.05. Thick black line depicts the whole diffraction profile. Colour lines show its subcomponents. The stacking fault effect was convoluted with size broadening to avoid the delta peak from unaffected components. Crystallites size was set to D ~ 200 nm. Similar figure can be seen in Balogh et al. [4]. (q = 4p sin Q / l)

 

Generally any LPA method introduces some approximations and indeed some of them require ill-posed steps, such us deconvolution. A common attribute of all the methods mentioned is that reflections in the powder pattern are analysed separately, which becomes problematic due to strong peak overlap especially in the case of nanocrystalline or low symmetry materials. It can be solved by introduction of the Rietveld method. All peaks in the diffraction pattern are fitted simultaneously on a basis of some model, which parameters are refined. Most of models used also by direct methods can be generalised and adapted also for the Rietveld technique. E.g. anisotropy (hkl dependence) of peak broadening can be accounted for any type of crystal symmetry [2,3]. For computational simplicity analytical profile function, e.g. pseudo-Voigt function, are utilized to describe peak profiles in the classical Rietveld programs. Nevertheless in many cases, see  Fig. 1 showing peak broadening in fcc metal sample induced by stacking fault defects [4], peak shape can be quite complex. Hence it was proposed by Scardi&Leoni (2000) [5] and Ribárik&Ungár [6] to simulate shape of diffraction profiles from a microstructural model suitable for a particular material and problem under investigation. This technique of the whole powder pattern modelling (WPPM) makes it possible to determine e.g. stacking fault probabilities or dislocation densities in severely deformed (SPD) metals [6].

At first the WPPM method was applied to some model cubic [5-6] or hexagonal materials. However, as in the last decade mainly nano-materials stay in the centre of interest, validity of an additional approximation present also in WPPM was discussed. This important assumption is known in Warren [1] as a “powder diffraction theorem” or in Beyerlein (2011) [7] as a “tangent plane approximation”. The problem can be intrinsically solved if the powder pattern is calculated from the atomistic model using a well known Debye formula. This technique can be used for calculation of scattering from various nano-objects as nanotubues [8] etc., but it is hardly scalable to larger 3D objects because of its computational complexity.

Lattice defects

In WPPM the microstructural models are usually build in the real space, where Fourier coefficients are derived and the profile is calculated by means of Fourier transformation. In this contribution a formalism introduced by Wilkens [9] and Krivoglaz&Ryaboshapka [10] for dislocation induced broadening will be briefly presented. Its application will be demonstrated on SPD copper samples and nanocrystalline metal samples, where also stacking fault defects (Fig. 1) play an important role.

Figure 2. Williamson-Hall plot of a TiO2 sample prepared by hydrolysis of n-butoxide [11] at temperature 300 °C (circles) and at 450 °C (triangles). The integral breadths of anatase reflection with instrumental broadening already deconvoluted are plotted. It has to be also considered that there is a very strong peak overlap - especially for the sample prepared at 300 °C (see Fig. 4).

Figure 3. Williamson-Hall plot of a TiO2 sample prepared by hydrolysis of isopropoxide [11] at temperature 400 °C. A significant microstrain broadening is evident. 

Nanostructures and size effects

Determination of crystallite size distribution will be illustrated on model TiO2 samples prepared by different chemical routes. Nevertheless in these samples the broadening effect is dominating (Fig. 2), it should be borne in mind that also presence of microstrain can not be always neglected (Fig. 3). A pattern fit of a model samples is depicted in Fig. 4. Beside the approximations mentioned above also an effect of crystallite shape will be discussed (Fig. 5).

Other effects

When compared to the direct methods analysing reflections separately, the whole powder pattern modelling techniques have also a significant drawback. There is complex information encoded by nature in the experimental pattern. The direct methods can simply separate individual effects, by neglecting their mutual influence in the analysis, e.g. it is assumed that reflection position is not related to its width and shape. Contrary the whole pattern fitting procedure has to account for several effects mixed together to reach good pattern fit and reliable results. Background scattering, residual stresses and texture have to be also included.

Computer programs

There are many computer programs suitable for modelling powder diffraction profiles. Some of them (PM2k [12], Maud [13] and MStruct [14]) will be briefly introduced.

Figure 4. Pattern fit of a TiO2 sample prepared by hydrolysis of n-butoxide at temperature 300 °C [11]. Small magenta ticks at the bottom mark the anatase reflections, whereas the cyan ticks above indicate reflections from the minor phase (here brookite).

 

 

Figure 5. Simulated Williamson-Hall plots for anatase crystallites of bipyramidal shape and different areas ratio of {101} and {001} facets. It is visible that some particular configurations (depicted by squares) are hardly distinguishable from the isotropic one for the spherical crystallites (line).

References

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3.     P. W. Stephens, J. Appl. Cryst., 32, (1999), 281.

4.     L. Balogh, G. Ribárik, T. Ungár, J. Appl. Phys., 100, (2006), 023512.

5.     P. Scardi, M. Leoni, Acta Cryst. A, 58, (2002), 190.

6.     G. Ribárik, T. Ungár, J. Gubicza, J. Appl. Cryst., 34, (2001), 669.

7.     K. R. Beyerlein, R. L. Snyder, P. Scardi, J. Appl. Cryst., 44, (2011), 945.

8.     T. Brunátová, S. Daniš, R. Kužel, D. Králová, Z. Kristallogr. Proc., 1, (2011), 229.

9.     M. Wilkens, in Fundamental aspects of dislocation theory, 317, (1970), 1191.

10.          M. A. Krivoglaz, X-Ray and Neutron Diffraction in Nonideal Crystals. Springer. 1996.

11.  L. Matějová, Z. Matěj, O. Šolcová, Micro. Mesoporo. Materials, 154, (2012), 187.

12.  M. Leoni, T. Confente, P. Scardi, Z. Kristallogr. Suppl., 23, (2006), 249.

13.  L. Lutterotti, M. Bortolotti, IUCr: Comput. Commission Newsletter, 1, (2003), 43, www.ing.unitn.it/~maud.

14.  Z. Matěj, Materials Struct. Chem. Biol. Phys. Tech., 17-2a, (2010), k99, xray.cz/mstruct.