International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 190-191   | 1 | 2 |

Section 2.1.2.1. Mathematical background

U. Shmuelia* and A. J. C. Wilsonb

aSchool of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and bSt John's College, Cambridge, England
Correspondence e-mail:  ushmueli@post.tau.ac.il

2.1.2.1. Mathematical background

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The process may be illustrated by evaluating, or attempting to evaluate, the average intensity of reflection by the three processes. The intensity of reflection is given by multiplying equation (2.1.1.1)[link] by its complex conjugate: [\eqalignno{I &= FF^{*} &(2.1.2.1) \cr &= \textstyle\sum\limits_{j, \, k}f_{j}\;f_{k}^{*}\exp\{2\pi i[h(x_{j}-x_{k})+\ldots]\} &(2.1.2.2) \cr &= \Sigma + \textstyle\sum\limits_{j \neq k}f_{j}\;f_{k}^{*}\exp\{2\pi i[h(x_{j}-x_{k})+\ldots]\}, \qquad &(2.1.2.3)}%2.1.2.3] where [\Sigma = \textstyle\sum\limits_{j}f_{j}\;f_{j}^{*} \eqno(2.1.2.4)] is the sum of the squares of the moduli of the atomic scattering factors. Wilson (1942[link]) argued, without detailed calculation, that the average value of the exponential term would be zero and hence that [\langle I \rangle = \Sigma. \eqno(2.1.2.5)] Averaging equation (2.1.2.3)[link] for [hkl] fixed, [xyz] ranging uniformly over the unit cell – the first process described above – gives this result identically, without complication or approximation. Ordinarily the second process cannot be carried out. We can, however, postulate a special case in which it is possible. We take a homoatomic structure and before averaging we correct the f's for temperature effects and the fall-off with [\sin \theta], so that [ff^{*}] is the same for all the atoms and is independent of [hkl]. If the range of [hkl] over which the expression for I has to be averaged is taken as a parallelepiped in reciprocal space with h ranging from [-H] to [+H], k from [-K] to [+K], l from [-L] to [+L], equation (2.1.2.2)[link] can be factorized into the product of the sums of three geometrical progressions. Algebraic manipulation then easily leads to [\eqalignno{\langle I \rangle & = ff^{*} \sum _{j}\sum _{k}{{\sin \pi N_{H}(x_{j}-x_{k})}\over{N_{H}\sin \pi (x_{j}-x_{k})}} \cr &\quad\times {{\sin \pi N_{K}(y_{j}-y_{k})}\over{N_{K}\sin \pi (y_{j}-y_{k})}} {{\sin \pi N_{L}(z_{j}-z_{k})}\over{N_{L}\sin \pi (z_{j}-z_{k})}}, \cr & &(2.1.2.6)}] where [N_{H} = 2H+1], [N_{K} = 2K+1] and [N_{L} = 2L+1]. The terms with [j = k] give [\Sigma], but the remaining terms are not zero. Because of the periodic nature of the trigonometric terms, the effective coordinate differences are never greater than 0.5 and in a structure of any complexity there will be many much less than 0.5. For [HKL = 000], in fact, [\langle I \rangle] becomes the square of the modulus of the sum of the atomic scattering factors, [\langle I \rangle = \Phi\Phi^{*}, \eqno(2.1.2.7)] where [\Phi = \textstyle\sum\limits_{j = 1}^{N}f_{j}, \eqno(2.1.2.8)] and not the sum of the squares of their moduli; for larger [HKL], [\langle I \rangle] rapidly decreases to [\Sigma] and then oscillates about that value. Wilson (1949[link], especially Section 2.1.1) suggested that the regions of averaging should be chosen so that at least one index of every reflection is [\geq 2] if [\langle I \rangle] is to be identified with [\Sigma], and this has proven to be a useful rule-of-thumb.

The third process of averaging replaces the sum over integral values of the indices by an integration over continuous values, the appropriate values of the limits in this example being [-(H+1/2)] to [+(H+1/2)]. The effect is to replace the sines in the denominators, but not in the numerators, of equation (2.1.2.6)[link] by their arguments, and this is equivalent to the approximation [\sin x \simeq x] in the denominators only. This is a good approximation for atoms close together in the structure and thus giving the largest terms in the sums in equation (2.1.2.6)[link], and gives the correct sign and order of magnitude even for x having its maximum value of [\pi/2].

References

First citationWilson, A. J. C. (1942). Determination of absolute from relative intensity data. Nature (London), 150, 151–152.Google Scholar
First citationWilson, A. J. C. (1949). The probability distribution of X-ray intensities. Acta Cryst. 2, 318–320.Google Scholar








































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