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 . The centroid and peak displacements and the variances of the aberrations of the symmetrical diffractometer have been collected by Wilson (1961, 1963, 1965c, 1970a, 1974). For the Seemann–Bohlin type, they are collected in Table 5.2.4.1, mainly from Wilson (1974). 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 , of the diffractometer, whereas, for the former, and where 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 , but usually less than , and R will vary with , approaching zero as θ approaches . There will thus be a range of for which the Seemann–Bohlin aberrations containing R become very large. Mack & Parrish (1967) 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 , in the expression in Table 5.2.4.1; they are given explicitly by Wilson (1963, 1965c, 1970a).

 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 W Zero-angle calibration Constant 0 Specimen displacement 0 Specimen transparency Thick specimen Thin specimen See Wilson (1974, p. 547) 2:1 mis-setting Zero if centroid of illuminated area is centred Inclination of plane of specimen to axis of rotation Zero if centroid of illuminated area on equator of specimen for uniform illumination Flat specimen Focal-line width Small Receiving-slit width Small Interaction terms Small if adjustment reasonably good See Wilson (1963, 1974) Axial divergence No Soller slits, source, specimen and receiver equal Narrow Soller slits One set in incident beam One set in diffracted beam Replace R by S in the above Two sets Wide Soller slits Complex. See Pike (1957), Langford & Wilson (1962), Wilson (1963, 1974), and Gillham (1971) Refraction 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.

### 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.