International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 3.2, pp. 260-261
Section 3.2.3.6. Extinction^{a}Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794–3800, USA |
The analysis of diffracted intensities in this chapter has been premised on the assumption that the diffracted beam produced by each crystallite is much weaker than the incident beam, known as the kinematic approximation. This is a consequence of the Born approximation, whereby the diffracted intensity from one grain is proportional to the number of atoms within it. However, as the size of the crystallite grows, so will the intensity of the diffracted wave within it. The diffracted radiation will be re-diffracted back into the incident beam, leading to a measured integrated intensity less than the kinematic value, by a ratio y(G) = I_{obs}(G)/I_{kinematic}(G), which depends on the particular Bragg reflection G. This is the phenomenon of primary extinction, which can be understood within the framework of the dynamical theory of diffraction. The following discussion ignores absorption.
The relevant physical parameter is the extinction length Λ. For X-rays,where V is the unit-cell volume, r_{e} = 2.82 × 10^{−15} m, A_{G} is the structure factor, and P is the polarization factor: 1 or cos^{2} 2θ for S or P polarization, respectively. For neutrons,As an example, the 111 reflection of Si has an extinction length of 7.2 µm for 1.54 Å X-rays and 50 µm for 1.59 Å neutrons. Intensities from the kinematic theory are correct in the limit that the coherent grain size is much less than Λ. Note that extinction is most significant for the strongest reflections in a powder-diffraction pattern.
There are no exact calculations of extinction available for sample geometries applicable to powder samples, but Thorkildsen and Larsen have obtained a rigorous series solution for spherical particles of radius R in powers of Λ/R (Thorkildsen & Larsen, 1998). To lowest order,This first term gives y accurate to 3% for R/Λ ≤ 0.4. Fig. 3.2.6 shows their calculated result up to fifth order.
References
Thorkildsen, G. & Larsen, H. B. (1998). Primary extinction in cylinders and spheres. Acta Cryst. A54, 172–185.Google Scholar