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, pp. 493-494

Section 5.2.3.2. Extrapolation, graphical and analytical

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

5.2.3.2. Extrapolation, graphical and analytical

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Equation (5.2.1.4)[link] indicates that for a given error in θ the fractional error in the spacing d approaches zero as θ approaches 90°. The errors in θ – expressed as [\Delta(2\theta)=KF(2\theta)] in (5.2.3.2)[link] – arising from any specified aberration may increase as θ increases, but ordinarily this increase is insufficient to outweigh the effect of the [\cot\theta] factor. In the simple cubic case, one can write [a_{\rm true} = [(h^2+k^2+l^2){}^{1/2}\lambda/2\sin\theta]+KF(\theta), \eqno (5.2.3.3)]where K is a proportionality factor and [F(\theta)] represents the angular variation of the systematic errors in the lattice parameter. The functions F in (5.2.3.2)[link] and (5.2.3.3)[link] are not exactly the same; they are transformed into one another by the use of (5.2.1.4)[link]. Functions suitable for different experimental arrangements are quoted in the following sections; see, for example, equation (5.2.8.1)[link] for the Debye–Scherrer camera and Tables 5.2.4.1[link] and 5.2.7.1[link] for diffractometers. Simple graphical extrapolation is quick and easy for cubic substances, and by the use of successive approximations it can be applied to hexagonal (Wilson & Lipson, 1941[link]), tetragonal, and even orthorhombic materials. It is, however, very cumbersome for non-cubic substances, and impracticable if the symmetry is less than orthorhombic.

Analytic extrapolation seems to have been first used by Cohen (1936a[link],b[link]). It is now usual even in the cubic case: programs are often included in the software accompanying powder diffractometers, and many others are available separately. Some programs that are frequently referred to are described by Appleman & Evans (1973[link]), Mighell, Hubbard & Stalick (1981[link]), and Ferguson, Rogerson, Wolstenholme, Hughes & Huyton (1987[link]); for a comparison, see Kelly (1988[link]). If the precision warrants it, the single function [KF(\theta)] may be replaced by a sum of functions [K_iF_i(\theta)], one for each of the larger aberrations listed in Tables 5.2.4.1[link], 5.2.7.1[link], and 5.2.8.1[link]. Two – the zero error and a function corresponding to specimen-surface displacement and transparency – must be used routinely; one or two more may be added if the precision warrants it.

References

Appleman, D. E. & Evans, H. T. (1973). Indexing and least-squares refinement of powder diffraction data. US Department of Commerce, National Technical Information Service, 5286 Port Royal Rd, Springfield, VA 22151, USA.Google Scholar
Cohen, M. U. (1936a). Elimination of systematic errors in powder photographs. Z. Kristallogr. 94, 288–298.Google Scholar
Cohen, M. U. (1936b). Calculation of precise lattice constants for X-ray powder photographs. Z. Kristallogr. 94, 306–310.Google Scholar
Ferguson, I. F., Rogerson, A. H., Wolstenholme, J. F. R., Hughes, T. E. & Huyton, A. (1987). FIRESTAR-2. A computer program for the evaluation of X-ray powder measurements and the derivation of crystal lattice parameters. United Kingdom Atomic Energy Authority, Northern Division Report ND-R-909(S). London: HMSO, February 1987.Google Scholar
Gillham, C. J. (1971). Centroid shifts due to axial divergence and other geometrical factors in Seemann–Bohlin diffractometry. J. Appl. Cryst. 4, 498–506.Google Scholar
Gillham, C. J. & King, H. W. (1972). Measurements of centroid and peak shifts due to dispersion and the Lorentz factor at very high Bragg angles. J. Appl. Cryst. 5, 23–27.Google Scholar
Kelly, E. H. (1988). A summary of a `round-robin' exercise comparing the output of computer programs for lattice-parameter refinement and calculations. British Crystallographic Association.Google Scholar
Langford, J. I. & Wilson, A. J. C. (1962). Counter diffractometer: the effect of specimen transparency on the intensity, position and breadth of X-ray powder diffraction lines. J. Sci. Instrum. 39, 581–585.Google Scholar
Mighell, A. D., Hubbard, C. R. & Stalick, J. K. (1981). NBS*EXAIDS83. A Fortran program for crystallographic data evaluation. Natl Bur. Stand. (US) Tech. Note, No. 1141, April 1981.Google Scholar
Pike, E. R. (1957). Counter diffractometer – the effects of vertical divergence on the displacement and breadth of powder diffraction lines. J. Sci. Instrum. 34, 355–361.Google Scholar
Wilson, A. J. C. (1963). Mathematical theory of X-ray powder diffractometry. Eindhoven: Centrex.Google Scholar
Wilson, A. J. C. (1965c). Röntgenstrahlpulverdiffractometrie. Mathematische Theorie. Eindhoven: Centrex.Google Scholar
Wilson, A. J. C. (1970a). Elements of X-ray crystallography. Reading, MA: Addison-Wesley.Google Scholar
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.Google Scholar
Wilson, A. J. C. & Lipson, H. (1941). The calibration of Debye–Scherrer X-ray powder cameras. Proc. Phys. Soc. London, 53, 245–250.Google Scholar








































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