Test of applicability of some powder diffraction tools to nanocrystals

Z. Kaszkur

Institute of Physical Chemistry PAS, Warszawa, 01-224 Poland

Most of the diffraction structure analysis methods developed for polycrystals meet their application limit when crystal size decreases below few nanometers. Measurable effects of nanocrystallinity on the analysis can be noticed already for 10 nm crystallites. The Bragg law itself ceases to apply strictly [1] what appears to be a direct consequence of short atom rows and thus of short, truncated Fourier series in the peak harmonic representation. The surface relaxation effect adds only a minor term to the lattice constant calculated directly from a single peak position. With advent of nanotechnologies and rising interest in experimental analysis of nano-sized structures it is increasingly important to test application limits of the available structural methods.

The structural methods affected include the full profile analysis, methods of separation of size and strain (the Williamson-Hall plot as well as the Warren-Averbach method), methods of a lattice constant determination, quantitative analysis (linearity of the peak intensity – number of atoms dependence) etc.

The tests were performed on the model nanocrystals having distribution of sizes following the log-normal distribution of a crystallite volume. The maximum was centred ~5nm and the model crystallites were cubooctahedral, closed shell fcc structures having from 561 to 24739 atoms. The structure was chosen to be that of palladium metal and the interatomic potentials used for its relaxation followed the Sutton-Chen N-body scheme [2]. Both non-relaxed and energy relaxed models were used to estimate the effects of relaxation.

The diffraction patterns for the model were calculated followig the Deby'e formula. The patterns were analysed using PEAKFIT program [3] via decomposing the profiles onto constituent Pearson VII peaks. Due to lack of unequivocal background definition the peak fit parameters may be obtained with some error – this howeveraffects the results only for strongly overlapping peaks. Such peaks were excluded from the data presented in figs. 1,2. Fig.1 shows the apparent lattice constant as obtained from the consecutive peaks position using the Bragg law. It is remarkable that the peaks with the odd Miller indices give the lattice constant value systematically greater than that produced by the even indices. The Williamson-Hall plot (figure 2) displays crystallite size close to the real one and evidently not vanishing stress parameter for the relaxed system pattern.

Fig. 1

Fig. 2

The Warren-Averbach analysis for the same pattern (002 peak family) enables reconstruction of the original column-length distribution and confirms the presence of a not vanishing stress distribution for the relaxed model. For the models of bimodal log-normal distribution the same analysis fails however in reconstructing the column length distribution in both: maxima positions and their amplitude ratio [4].

The discussed effects are not negligible in a full profile analysis of nanocrystals and are more significant the grater stress is induced to the nanocrystal structure.

Acknowledgement

This study was supported by the State Committee for Scientific Research (KBN) grant no. 4T09A 180 24.

[1] Z. Kaszkur, J. Appl. Crystallogr., 33 (2000) 87.

[2] A.P.Sutton and J.Chen, Philos.Mag.Lett., 61, 139 (1990).

[3] Jandel Scientific GmbH , PEAKFIT (1990), v.3.11, D-40699 Erkrath, Germany.

[4] Z.Kaszkur, B.Mierzwa, to be published.