Special cases

Maslen, E. N.

doi:10.1107/97809553602060000602

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 6.3, p. 600

Section 6.3.3.1. Special cases

E. N. Maslen^a ^‡

^a Crystallography Centre, The University of Western Australia, Nedlands, Western Australia 6009, Australia

6.3.3.1. Special cases

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For special cases, the integral can be solved analytically, and in some of these the expression reduces to closed form. These are listed in Table 6.3.3.1.

Table 6.3.3.1| top | pdf |
Transmission coefficients

(1) Reflection from a crystal slab with negligible transmission; the crystal planes are inclined at an angle $[\varphi]$ to the extended face, and the normal in the plane of the incident and diffracted beams

$[A={\sin(\theta-\varphi)\over \mu\{\sin(\theta-\varphi)+\sin(\theta+\varphi)\}}]$ (1a) $[\varphi]$ = 0 $[A=1/2\mu]$

(2) Reflection from a crystal slab of thickness t, with planes parallel to the extended face

$[A=\{1-\exp\,(-2\mu t{\;\rm cosec}\;\theta)\}/2\mu]$

(3) Transmission through a crystal slab of thickness t; the crystal planes are at $[\pi/2-\varphi]$ to the surface, with the normal in the plane of the incident and reflected beams

$[A={{\exp\{-\mu t \sec(\theta+\varphi)\}-\exp\{-\mu t \sec (\theta-\varphi) \}}\over{\displaystyle \mu\biggl [ 1-{{\sec(\theta+\varphi)}\over{ \sec(\theta-\varphi )}} \biggr] }}]$ (3a) $[\varphi=0]$ $[A=t\sec\theta\exp(-\mu t\sec\theta)]$

(4) Transmission through a sphere of radius R (i.e. for a uniform X-ray beam and $[\theta=0^\circ]$ )

$[A={3\over 2(\mu R)^3}[1/2-e^{-2\mu R}\{1/2+\mu R+ (\mu R)^2\}]]$

(5) Reflection from a sphere of radius R (i.e. for a uniform X-ray beam, and $[\theta=90^\circ]$ )

$[A={3\over 4\mu R}\left\{1/2 - {1\over16(\mu R)^2}\;[1-(1+4\mu R)\,e^{-4\mu R}]\right\}]$

References

International Tables for Crystallography (2006). Vol. C. ch. 6.3, p. 600