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

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

Section 5.1.7.2.2. Absorbing crystals

A. Authiera*

aLaboratoire 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

| top | pdf |

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)[link], β is defined in (5.1.3.7)[link] 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)[link] tends towards [[\sin (\pi t \eta / \Lambda_{B}) / \eta]^{2}]; using (5.1.3.5)[link] and (5.1.3.8)[link], it can be shown that expression (5.1.7.12)[link] 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)[link]].

References

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








































to end of page
to top of page