Table 5.2.4.1

Parrish, W.; Wilson, A. J. C.; Langford, J. I.

doi:10.1107/97809553602060000596

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. 5.2, p. 494

Table 5.2.4.1

W. Parrish,^a ^‡ A. J. C. Wilson^b ^‡ and J. I. Langford^c

^a IBM Almaden Research Center, San Jose, CA, USA,^bSt John's College, Cambridge CB2 1TP, England, and ^cSchool of Physics & Astronomy, University of Birmingham, Birmingham B15 2TT, England

Table 5.2.4.1 | top | pdf |
Centroid displacement 〈Δθ/θ〉 and variance W of certain aberrations of an angle-dispersive diffractometer ; for references see Wilson (1963, 1965c, 1974) and Gillham (1971)

For the Seemann–Bohlin arrangement, S and R are given by equations (5.2.4.1) and (5.2.4.2); for the symmetrical arrangement, they are equal to R₀. Other notation is explained at the end of the table.

Aberration	$[\langle \Delta(2\theta)\rangle]$	W
Zero-angle calibration	Constant	0
Specimen displacement	$[-s\{R^{-1}\cos(2\theta-\varphi)+S^{-1}\cos \varphi\}]$	0
Specimen transparency
Thick specimen	$[-\sin2\varphi/\mu(R+S)]$	$[\sin^22\varphi/\mu^2(R+S)^2]$
Thin specimen	See Wilson (1974, p. 547)
2:1 mis-setting	Zero if centroid of illuminated area is centred	$[\beta^2 A^2[R^{-1}\cos(2\theta- \varphi)+S^{-1}\cos \varphi]^2/3]$
Inclination of plane of specimen to axis of rotation	Zero if centroid of illuminated area on equator of specimen	$[\gamma^2h^2[R^{-1}\cos(2\theta- \varphi)+S^{-1}\cos \varphi]^2/3]$ for uniform illumination
Flat specimen	$[-A^2\sin2\theta/3 \,RS]$	$[4A^4\sin^22\theta/45\,R^2S^2]$
Focal-line width	Small	$[\sim f^2_1/12S^2]$
Receiving-slit width	Small	$[\sim r^2_1/12R^2]$
Interaction terms	Small if adjustment reasonably good	See Wilson (1963, 1974)
Axial divergence No Soller slits, source, specimen and receiver equal	$[-h^2[(S^{-2}+R^{-2})\cot2\theta+(RS)^{-1}\,{\rm cosec}\, 2\theta]/3]$	$[\eqalign{h^4[\{&7S^{-4}+2(RS)^{-2}+7R^{-4}\}\cot^22\theta \cr &+14(RS)^{-1}(S^{-2}+R^{-2})\cot 2\theta\,{\rm cosec}\, 2\theta \cr &+19(RS)^{-2}\,{\rm cosec}\,^2\,2\theta]/45}]$
Narrow Soller slits
One set in incident beam	$[-[\Delta^2/12+h^2/3R^2]\cot 2\theta]$	$[\eqalign{ 7[&\Delta^4/720+h^4/45R^2]\cot^2 2\theta \cr &+h^2\,{\rm cosec}^2\,2\theta/9R^2}]$
One set in diffracted beam	Replace R by S in the above
Two sets	$[-(\Delta^2\cot2\theta)/6]$	$[\Delta^4(10+17\cot^2\,2\theta)/360]$
Wide Soller slits	Complex. See Pike (1957), Langford & Wilson (1962), Wilson (1963, 1974), and Gillham (1971)
Refraction	$[\sim -2\delta\tan\theta]$	$[\sim\delta^2[-6\ln(\Delta/2)+25]/4\mu p]$
Physical aberrations	See Wilson (1963, 1965c, 1970a, 1974) and Gillham & King (1972)

Notation: 2A = illuminated length of specimen; β = angle of equatorial mis-setting of specimen; γ = angle of inclination of plane of specimen to axis of rotation; Δ = angular aperture of Soller slits; μ = linear absorption coefficient of specimen; r₁ = width of receiving slit (varies with θ in some designs of diffractometer); s = specimen-surface displacement; f₁ = projected width of focal line; h = half height of focal line, specimen, and receiving slit, taken as equal; 1 − δ = index of refraction; p = effective particle size.