International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 190-191
Section 2.1.2.1. Mathematical background^{a}School of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and ^{b}St John's College, Cambridge, England |
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) by its complex conjugate: where is the sum of the squares of the moduli of the atomic scattering factors. Wilson (1942) argued, without detailed calculation, that the average value of the exponential term would be zero and hence that Averaging equation (2.1.2.3) for fixed, 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 , so that is the same for all the atoms and is independent of . If the range of over which the expression for I has to be averaged is taken as a parallelepiped in reciprocal space with h ranging from to , k from to , l from to , equation (2.1.2.2) can be factorized into the product of the sums of three geometrical progressions. Algebraic manipulation then easily leads to where , and . The terms with give , 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 , in fact, becomes the square of the modulus of the sum of the atomic scattering factors, where and not the sum of the squares of their moduli; for larger , rapidly decreases to and then oscillates about that value. Wilson (1949, especially Section 2.1.1) suggested that the regions of averaging should be chosen so that at least one index of every reflection is if is to be identified with , 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 to . The effect is to replace the sines in the denominators, but not in the numerators, of equation (2.1.2.6) by their arguments, and this is equivalent to the approximation 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), and gives the correct sign and order of magnitude even for x having its maximum value of .
References
Wilson, A. J. C. (1942). Determination of absolute from relative intensity data. Nature (London), 150, 151–152.Google ScholarWilson, A. J. C. (1949). The probability distribution of X-ray intensities. Acta Cryst. 2, 318–320.Google Scholar