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15 nm if adjacent crystallites are strongly preferentially oriented. The most important criterion for the maximum distance of the par- tially coherent crystallites is the coherence length of X-rays [15] that is, according to the Heisenberg'

s uncertainty prin- ciple, related to the spectral quality of the radiation. More details regarding the coherence length of X-rays can be found in [16]. The maximum disorientation of the partially coherent neighbouring crystallites depends on their size, but it is usually below 2° [13, 17, 18]. Most frequently, the phenomenon of the partial coherence of crystallites is ob- served in nanocrystalline thin films [14, 17-20]. However, the partial coherence of crystallites was also reported for powders with a strong local preferred orientation of crys- tallites [21, 22]. The physical background of the partial co- herence of crystallites was described in [14] and is summarised in the next Section. Some applications of the phenomenon of the partial coherence of crystallites for ? Krystalografická spoleènost Materials Structure, vol. 14, no.

2 (2007)

67 microstructure studies on nanocrystalline materials are il- lustrated by experimental examples in the experimental sections. Phenomenon of the partial coherence of crystallites The phenomenon of the partial coherence of crystallites can be explained using the microstructure model, which as- sumes that the material under study consists of nearly de- fect-free crystallites, which have slightly different macroscopic orientations (Fig. 1). Such a microstructure model is well applicable for impacted powders with a high local preferred orientation of crystallites, as well as for compact samples, which microstructure can be described with the aid of the Mughrabi composite model of plastic deformation [23]. According to the Mughrabi model, re- gions with very low dislocation density are separated by re- gions with a high dislocation density. The latter are called dislocation walls. If the defect density inside the disloca- tion walls is very high, it can be assumed that the disloca- tion walls do not contribute to the Bragg peaks diffracted by the defect-free crystallites. Thus, the contribution of the defect-free crystallites to the diffraction pattern can be sep- arated from the contribution of the dislocation walls. A modification of the Mughrabi model can be applied for de- scription of microstructure in nanocrystalline materials and nanocomposites that consist of nearly defect-free nano- crystallites and of strongly distorted regions between them. Each individual crystallite can be described by a single reciprocal lattice. According to the kinematical diffraction theory [24], the size of the reciprocal lattice points is in- versely proportional to the size of the (nearly defect-free) crystallite that is known as the size effect in the XRD line profile analysis. The reciprocal lattice points from nano- crystallites are extremely broadened. The mutual disorientations of individual crystallites cause rotation of the reciprocal lattices around their join origin (Fig. 2). If the neighbouring crystallites have a strong local preferred ori- entation, then their reciprocal lattices are only slightly mu- tually disoriented that leads to a partial overlap of the reciprocal lattice points near the origin of the reciprocal space, as it is shown in Fig.

2 for two slightly disoriented crystallites with the face-centred cubic (fcc) crystal struc- ture. The degree of the overlap of the reciprocal lattice points depends obviously on their size, on the mutual dis- orientation of the reciprocal lattices and on the distance of the respective reciprocal lattice points from the origin of the reciprocal space. In the kinematical diffraction theory, the intensity of the X-ray radiation scattered on an ensem- ble of the scattering centres is proportional to the modulus of the sum of the amplitudes scattered by individual scatter- ing centres, i.e. to the modulus of the sum of the structure factors of the scattering centres, taking the respective phase shift into account: I F q iq R F q iq R F n n n n n n ? * = = - * * ? ( )exp( ) ( )exp( ) ( * r r r r r r r