International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 9798

Cruickshank consolidated and extended Cochran's derivations in a series of classic papers (Cruickshank, 1949b, 1950, 1952, 1956). He was able to show that all the coefficients involved in the righthand side and normal matrix of the leastsquares method could be calculated by means of suitable differential Fourier syntheses even when the atoms overlap. This remarkable achievement lay essentially dormant until its independent rediscovery by Agarwal in 1978 (Section 1.3.4.4.7.6).
To ensure rigorous equivalence between the summations over (in the expressions of leastsquares righthand side and normal matrix elements) and genuine Fourier summations, multiplicitycorrected weights were introduced by:where Gh denotes the orbit of h and its isotropy subgroup (Section 1.3.4.2.2.5). Similarly, derivatives with respect to parameters of symmetryunique atoms were expressed, via the chain rule, as sums over the orbits of these atoms.
Let be the label of a parameter belonging to atoms with label j. Then Cruickshank showed that the pth element of the righthand side of the normal equations can be obtained as , where is a differential synthesis of the formwith a polynomial in (h, k, l) depending on the type of parameter p. The correspondence between parameter type and the associated polynomial extends Booth's original range of differential syntheses, and is recapitulated in the following table.Unlike Cochran's original heuristic argument, this result does not depend on the atoms being resolved.
Cruickshank (1952) also considered the elements of the normal matrix, of the formassociated with positional parameters. The block for parameters and may be writtenwhich, using the identitybecomes(Friedel's symmetry makes redundant on the last line). Cruickshank argued that the first term would give a good approximation to the diagonal blocks of the normal matrix and to those offdiagonal blocks for which and are close. On this basis he was able to justify the `nshift rule' of Shoemaker et al. (1950). Cruickshank gave this derivation in a general space group, but using a very terse notation which somewhat obscures it. Using the symmetrized trigonometric structurefactor kernel of Section 1.3.4.2.2.9 and its multiplication formula, the above expression is seen to involve the values of a Fourier synthesis at points of the form .
Cruickshank (1956) showed that this analysis could also be applied to the refinement of temperature factors.
These two results made it possible to obtain all coefficients involved in the normal equations by looking up the values of certain differential Fourier syntheses at or at . At the time this did not confer any superiority over the standard form of the leastsquares procedure, because the accurate computation of Fourier syntheses was an expensive operation. The modified Fourier method was used by Truter (1954) and by Ahmed & Cruickshank (1953a), and was incorporated into the program system described by Cruickshank et al. (1961). A more recent comparison with the leastsquares method was made by Dietrich (1972).
There persisted, however, some confusion about the nature of the relationship between Fourier and leastsquares methods, caused by the extra factors which make it necessary to compute a differential synthesis for each type of atom. This led Cruickshank to conclude that `in spite of their remarkable similarities the leastsquares and modifiedFourier methods are fundamentally distinct'.
References
Ahmed, F. R. & Cruickshank, D. W. J. (1953a). A refinement of the crystal structure analysis of oxalic acid dihydrate. Acta Cryst. 6, 385–392.Cruickshank, D. W. J. (1949b). The accuracy of atomic coordinates derived by leastsquares or Fourier methods. Acta Cryst. 2, 154–157.
Cruickshank, D. W. J. (1950). The convergence of the leastsquares and Fourier refinement methods. Acta Cryst. 3, 10–13.
Cruickshank, D. W. J. (1952). On the relations between Fourier and leastsquares methods of structure determination. Acta Cryst. 5, 511–518.
Cruickshank, D. W. J. (1956). The determination of the anisotropic thermal motion of atoms in crystals. Acta Cryst. 9, 747–753.
Cruickshank, D. W. J., Pilling, D. E., Bujosa, A., Lovell, F. M. & Truter, M. R. (1961). Crystallographic calculations on the Ferranti Pegasus and mark I computers. In Computing Methods and the Phase Problem in XRay Crystal Analysis, edited by R. Pepinsky, J. M. Robertson & J. C. Speakman, pp. 32–78. Oxford: Pergamon Press.
Dietrich, H. (1972). A reconsideration of Fourier methods for the refinement of crystal structures. Acta Cryst. B28, 2807–2814.
Shoemaker, D. P., Donohue, J., Schomaker, V. & Corey, R. B. (1950). The crystal structure of L_{S}threonine. J. Am. Chem. Soc. 72, 2328–2349.
Truter, M. R. (1954). Refinement of a noncentrosymmetrical structure: sodium nitrite. Acta Cryst. 7, 73–77.