Integrated intensity

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. 544-545 | 1 | 2 |

Section 5.1.6.5. Integrated intensity

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

5.1.6.5. Integrated intensity

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5.1.6.5.1. Non-absorbing crystals

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The integrated intensity is the ratio of the total energy recorded in the counter when the crystal is rocked to the intensity of the incident beam. It is proportional to the area under the line profile: $[I_{hi} = {\textstyle\int\limits_{-\infty}^{+\infty}} I_{h} \hbox{ d} (\Delta \theta). \eqno(5.1.6.8)]$

The integration was performed by von Laue (1960). Using (5.1.3.5), (5.1.6.6) and (5.1.6.7) gives $[I_{hi} = A {\textstyle\int\limits_{0}^{2\pi t\Lambda_{L}^{-1}}} J_{0} (z) \hbox{ d}z,]$ where $[J_{0}(z)]$ is the zeroth-order Bessel function and $[A = {R\lambda^{2} |C F_{h}| (\gamma)^{1/2} \over 2V \sin 2\theta}.]$ Fig. 5.1.6.7 shows the variations of the integrated intensity with $[t/\Lambda_{L}]$ .

Figure 5.1.6.7| top | pdf |

Variations with crystal thickness of the integrated intensity in the transmission case (no absorption) (arbitrary units). The expression for A is given in the text.

5.1.6.5.2. Absorbing crystals

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The integration was performed for absorbing crystals by Kato (1955). The integrated intensity in this case is given by $[\eqalign{I_{hi} &= A |F_{h}/F_{\bar{h}}| \exp \left[-1/2 \mu_{o} t (\gamma_{o}^{-1} + \gamma_{h}^{-1})\right]\cr &\quad \times \left[\textstyle\int\limits_{0}^{2\pi t\Lambda_{L}^{-1}} J_{0}(z) \hbox{ d}z - 1 + I_{0} (\zeta)\right],}]$ where $[\zeta = \mu_{o} t \left\{\left[\left|C\right|^{2} |F_{ih}/F_{io}|^{2} \cos^{2} \varphi + (\gamma_{h} - \gamma_{o})/(4\gamma_{o} \gamma_{h})\right] / (\gamma_{o} \gamma_{h})\right\}^{1/2}]$ and $[I_{0} (\zeta)]$ is a modified Bessel function of zeroth order.

References

Kato, N. (1955). Integrated intensities of the diffracted and transmitted X-rays due to ideally perfect crystal. J. Phys. Soc. Jpn, 10, 46–55.Google Scholar

Laue, M. von (1960). Röntgenstrahl-Interferenzen. Frankfurt am Main: Akademische Verlagsgesellschaft.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 5.1, pp. 544-545