Evaluation of size and strain parameters from X-ray peaks by the momentum method

A. Borbély¹, Á. Révész¹, I. Groma¹

¹Eötvös University, Department of General Physics, H-1518, POB. 32, Budapest, Hungary.

Determination of meaningful and reliable size and strain parameters from X-ray peaks is a challenging task for nowadays evaluation techniques. In this respect microstructurally based models are welcome since they predict the shape of X-ray peaks, which can be directly compared with experiment. This is especially true for nanomaterials when due to the small particle size a nearly Lorentzian peak shape is expected. If the particles contain lattice defects, then the resulting profile is given the convolution of the transform of the size profile and the transform of the profile characteristic for the relevant lattice defect. It is known that in case of dislocations (the most frequently encountered lattice defect) the tail of the profile varies as q^-3, where q is the deviation from the reciprocal lattice point. According to the general theory of dislocation induced X-ray peak broadening [1,2] only this asymptotic behaviour can be anticipated, the shape of the whole profile being unknown. Exception from this is the special case of restrictedly random distribution of dislocations, a model developed by Wilkens, who has calculated the entire peak shape [3]. It is however, questionable if this special dislocation distribution is valid in any practical situation. If not, it is safer to consider only the asymptotic behaviour of the X-ray peaks. This doesn't mean however, that the Wilkens model and its incorporation in multi-profile fitting programs [4], to replace less physically justified peak-functions, is not applicable. We only want to stress that in such cases a microstructural justification of the selected evaluation method should be given. If the selected method cannot be justified, then only the general model is reliable.

Since a general peak-function applicable to each investigated case has not been found yet, we will discuss the asymptotic method. The kinematic theory of X-ray scattering predicts at large q values a q^-2 and q^-3 dependence of the scattered intensity, for the cases of small crstallite size [5] and dislocation [2] produced broadening, respectively. Commonly the measurements contain statistical errors, which may be reduced if an integral evaluation method is selected. Extending the variance method of Wilson [5], the authors have proposed recently a momentum method for the evaluation of the average particle size and dislocation density [6], when both sources of broadening are present. According to the q dependencies mentioned above the different order moments of the scattered intensity have typical behaviours. For example the fourth order moment divided by q² is constant when broadening in produced by dislocations and shows linear q dependence for particle type broadening. The great advantage of the momentum method is that one can readily see the type of broadening present in the experiment and verify if the assumptions of small particle size or presence of dislocations applies. The method is exemplified on measurements done on ball-milled and heat-treated aluminium powder samples. An error analysis of the evaluated parameters is presented also.

Acknowledgement

The authors acknowledge the financial support of the Hungarian Research Found OTKA under the contracts no. T034999 and T043519.

[1] Groma, I., Ungár, T., Wilkens, M. J. Appl. Cryst. 21 (1988) 47.

[2] Groma, I. Phys. Rev. B 57 (1998) 7535.

[3] Wilkens, M. Fundamental Aspects of Dislocation Theory, National Bureau of Standards (US) Special Publication No. 317, Vol. II, edited by J. A. Simmons, R. de Wit & R. Bullough, (1970) pp. 1195-1221. Washington, DC: N.B.S.

[4] Ungár, T., Gubicza, J., Ribarik, G. and A. Borbély, J. Appl. Cryst., 34 (2001) 298.

[5] A. J.C. Wilson, Proc. Phys. Soc. 80 (1962) 286; ibid. 81 (1963) 41.

[6] Borbély, A. és Groma, I. Applied Physics Letters, 79 (2001) 1772.