Tables for
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 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 Extrapolation, graphical and analytical

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Equation ([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 ([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 (]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 ([link] and ([link] are not exactly the same; they are transformed into one another by the use of ([link]. Functions suitable for different experimental arrangements are quoted in the following sections; see, for example, equation ([link] for the Debye–Scherrer camera and Tables[link] and[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[link],[link], and[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.


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