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

International Tables for Crystallography (2006). Vol. C, ch. 5.2, p. 495

Section 5.2.4. Angle-dispersive diffractometer methods: conventional sources

W. Parrish,a A. J. C. Wilsonb and J. I. Langfordc

aIBM 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

5.2.4. Angle-dispersive diffractometer methods: conventional sources

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The symmetrical Bragg–Brentano (Parrish) and the Seemann–Bohlin angle-dispersive diffractometers are fully described in Chapter 2.3[link] . The centroid and peak displacements and the variances of the aberrations of the symmetrical diffractometer have been collected by Wilson (1961[link], 1963[link], 1965c[link], 1970a[link], 1974[link]). For the Seemann–Bohlin type, they are collected in Table 5.2.4.1[link], mainly from Wilson (1974[link]). They are expressed in inverse powers of the source–specimen distance S and the specimen–detector distance R, and tend to be larger for the Seemann–Bohlin arrangement than for the symmetrical arrangement. For the latter, S and R are constant and equal to the radius, say [R_0], of the diffractometer, whereas, for the former, [S=2R_0\sin\varphi \eqno (5.2.4.1)]and [R=2R_0\sin(2\theta-\varphi), \eqno (5.2.4.2)]where [\varphi] is the constant angle that the incident X-rays make with the specimen surface. In the Seemann–Bohlin case, S will be constant at a value depending on the choice of angle [\varphi], but usually less than [R_0], and R will vary with [2\theta], approaching zero as θ approaches [\varphi/2]. There will thus be a range of [2\theta] for which the Seemann–Bohlin aberrations containing R become very large. Mack & Parrish (1967[link]) have confirmed experimentally the expected differences in favour of the symmetrical arrangement for general use, even though the effective equatorial divergence (`flat-specimen error') can be greatly reduced by curving the specimen appropriately in the Seemann–Bohlin arrangement. The aberrations for the symmetrical arrangement are found by putting [R=S=R_0], [\varphi=\theta] in the expression in Table 5.2.4.1[link]; they are given explicitly by Wilson (1963[link], 1965c[link], 1970a[link]).

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[link], 1965c[link], 1974[link]) and Gillham (1971[link])

For the Seemann–Bohlin arrangement, S and R are given by equations (5.2.4.1[link]) and (5.2.4.2[link]); 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[link], 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[link], 1974[link])
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[link]), Langford & Wilson (1962[link]), Wilson (1963[link], 1974[link]), and Gillham (1971[link])
Refraction [\sim -2\delta\tan\theta] [\sim\delta^2[-6\ln(\Delta/2)+25]/4\mu p]
Physical aberrations See Wilson (1963[link], 1965c[link], 1970a[link], 1974[link]) and Gillham & King (1972[link])

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.

References

Mack, M. & Parrish, W. (1967). Seemann–Bohlin X-ray diffractometry. II. Comparison of aberrations and intensity with conventional diffractometer. Acta Cryst. 23, 693–700.
Wilson, A. J. C. (1961). A note on peak displacements in X-ray diffractometry. Proc. Phys. Soc. London, 78, 249–255.
Wilson, A. J. C. (1963). Mathematical theory of X-ray powder diffractometry. Eindhoven: Centrex.
Wilson, A. J. C. (1965c). Röntgenstrahlpulverdiffractometrie. Mathematische Theorie. Eindhoven: Centrex.
Wilson, A. J. C. (1970a). Elements of X-ray crystallography. Reading, MA: Addison-Wesley.
Wilson, A. J. C. (1974). Powder diffractometry. X-ray diffraction, by L. V. Azaroff, R. Kaplow, N. Kato, R. Weiss, A. J. C. Wilson & R. A. Young, Chap. 6. New York: McGraw-Hill.








































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