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International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 2.6, p. 104
Section 2.6.1.7.3. Calculation of scattering intensities
O. Glattera
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The scattering curves can be calculated semi-analytically for simple triaxial bodies and for models composed of some of these bodies. The scattering amplitude for regular bodies like ellipsoids, parallelepipeds, and cylinders can be calculated analytically for any orientation. The spatial averaging has to be performed numerically. Such calculations have been performed for a large number of different models by Porod (1948![[link]](/graphics/greenarr.gif)
), Mittelbach & Porod (1961a,b, 1962), and by Mittelbach (1964). More complicated structures can be described by models composed of several such triaxial bodies, but the computing time necessary for such calculations can be hours on a mainframe computer.
Models composed only of spherical subunits can be evaluated with the Debye formula (Debye, 1915): ![[I(h)=i_{\rm el} (h) \sum^N_{i=1} \sum^N_{k=1} \rho_iV_i\rho_kV_k\Phi_i(h)\Phi_k(h) {\textstyle \sin (hd_{ik})\over \textstyle hd_{ik}},\eqno (2.6.1.67)]](/teximages/cbch2o6/cbch2o6fd95.svg)
where the spatial average is carried out analytically. Another possibility would be to use spherical harmonics as discussed in the previous section but the problem is how to find the expansion coefficients for a certain given geometrical structure.
References
Debye, P. (1915). Zerstreuung von Röntgenstrahlen. Ann. Phys. (Leipzig), 46, 809–823.Google Scholar
Mittelbach, P. (1964). Zur Röntgenkleinwinkelstreuung verdünnter kolloider Systeme. VIII. Acta Phys. Austriaca, 19, 53–102.Google Scholar
Mittelbach, P. & Porod, G. (1961a). Zur Röntgenkleinwinkelstreuung verdünnter kolloider Systeme. Die Berechnung der Streukurven von Parallelepipeden. Acta Phys. Austriaca, 14, 185–211.Google Scholar
Mittelbach, P. & Porod, G. (1961b). Zur Röntgenkleinwinkelstreuung verdünnter kolloider Systeme. VI. Acta Phys. Austriaca, 14, 405.Google Scholar
Mittelbach, P. & Porod, G. (1962). Zur Röntgenkleinwinkelstreuung verdünnter kolloider Systeme. VII. Die Berechnung der Streukurven von dreiachsigen Ellipsoiden. Acta Phys. Austriaca, 15, 122–147.Google Scholar
Porod, G. (1948). Die Abhängigkeit der Röntgen-Kleinwinkelstreuung von Form und Grösse der kolloiden Teilchen in verdünnten Systemen. IV. Acta Phys. Austriaca, 2, 255–292.Google Scholar
