Absorbing crystals

Authier, A.

doi:10.1107/97809553602060000569

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 5.1, pp. 547-548 | 1 | 2 |

Section 5.1.7.2.2. Absorbing crystals

A. Authier^a ^*

^a Laboratoire de Minéralogie-Cristallographie, Université P. et M. Curie, 4 Place Jussieu, F-75252 Paris CEDEX 05, France
Correspondence e-mail: authier@lmcp.jussieu.fr

5.1.7.2.2. Absorbing crystals

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Reflected intensity. The intensity of the reflected wave for a thin absorbing crystal is $[\eqalign{I_{h} &= |\gamma | \left|{D_{h}^{(a)} \over D_{o}^{(a)}}\right|^{2}\cr &= \left|{F_{h} \over F_{\bar{h}}}\right| {\cosh 2b - \cos 2a \over L\cosh 2b + (L^{2} - 1)^{1/2}\sinh 2b - \cos (2a + 2\Psi')},} \eqno(5.1.7.12)]$ where $[\eqalign{2a &= [\pi t/\Lambda_{B}] \rho \cos (\beta + \omega),\cr 2b &= [\pi t/\Lambda_{B}] \rho \sin (\beta + \omega)}.]$ L, ρ and ψ′ are defined in (5.1.7.5), β is defined in (5.1.3.7) and ω is the phase angle of $[(\eta^{2} - 1)^{1/2}]$ .

Comparison with geometrical theory. When $[t / \Lambda_{B}]$ decreases towards zero, expression (5.1.7.12) tends towards $[[\sin (\pi t \eta / \Lambda_{B}) / \eta]^{2}]$ ; using (5.1.3.5) and (5.1.3.8), it can be shown that expression (5.1.7.12) can be written, in the non-absorbing symmetric case, as $[I_{h} = {R^{2} \lambda^{2} C^{2} |F_{h}|^{2} t^{2} \over V^{2} \sin^{2} \theta} \left\{{\sin [2\pi k \cos (\theta) t \Delta \theta] \over [2\pi k \cos (\theta) t \Delta \theta]}\right\}^{2}, \eqno(5.1.7.13)]$ where d is the lattice spacing and Δθ is the difference between the angle of incidence and the middle of the reflection domain. This expression is the classical expression given by geometrical theory [see, for instance, James (1950)].

References

James, R. W. (1950). The optical principles of the diffraction of X-rays. London: G. Bell and Sons Ltd.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 5.1, pp. 547-548