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. 1.3, pp. 8889

It was the use of Fourier syntheses in the completion of trial structures which provided the incentive to find methods for computing 2D and 3D syntheses efficiently, and led to the Beevers–Lipson strips. The limited accuracy of the latter caused the estimated positions of atoms (identified as peaks in the maps) to be somewhat in error. Methods were therefore sought to improve the accuracy with which the coordinates of the electrondensity maxima could be determined. The naive method of peakshape analysis from densities recalculated on a grid using highaccuracy trigonometric tables entailed 27 summations per atom.
Booth (1946a) suggested examining the rapidly varying derivatives of the electron density rather than its slowly varying values. If then the gradient vector of at can be calculated by means of three Fourier summations from the vector of Fourier coefficients Similarly, the Hessian matrix of at can be calculated by six Fourier summations from the unique elements of the symmetric matrix of Fourier coefficients:
The scalar maps giving the components of the gradient and Hessian matrix of will be called differential syntheses of 1st order and 2nd order respectively. If is approximately but not exactly a maximum of , then the Newton–Raphson estimate of the true maximum is given by: This calculation requires only nine accurate Fourier summations (instead of 27), and this number is further reduced to four if the peak is assumed to be spherically symmetrical.
The resulting positions are affected by seriestermination errors in the differential syntheses. Booth (1945c, 1946c) proposed a `backshift correction' to eliminate them, and extended this treatment to the acentric case (Booth, 1946b). He cautioned against the use of an artificial temperature factor to fight seriestermination errors (Brill et al., 1939), as this could be shown to introduce coordinate errors by causing overlap between atoms (Booth, 1946c, 1947a,b).
Cruickshank was able to derive estimates for the standard uncertainties of the atomic coordinates obtained in this way (Cox & Cruickshank, 1948; Cruickshank, 1949a,b) and to show that they agreed with those provided by the leastsquares method.
The calculation of differential Fourier syntheses was incorporated into the crystallographic programs of Ahmed & Cruickshank (1953b) and of Sparks et al. (1956).
References
Ahmed, F. R. & Cruickshank, D. W. J. (1953b). Crystallographic calculations on the Manchester University electronic digital computer (Mark II). Acta Cryst. 6, 765–769.Booth, A. D. (1945c). Accuracy of atomic coordinates derived from Fourier synthesis. Nature (London), 156, 51–52.
Booth, A. D. (1946a). A differential Fourier method for refining atomic parameters in crystal structure analysis. Trans. Faraday Soc. 42, 444–448.
Booth, A. D. (1946b). The simultaneous differential refinement of coordinates and phase angles in Xray Fourier synthesis. Trans. Faraday Soc. 42, 617–619.
Booth, A. D. (1946c). The accuracy of atomic coordinates derived from Fourier series in Xray structure analysis. Proc. R. Soc. London Ser. A, 188, 77–92.
Booth, A. D. (1947a). The accuracy of atomic coordinates derived from Fourier series in Xray structure analysis. III. Proc. R. Soc. London Ser. A, 190, 482–489.
Booth, A. D. (1947b). The accuracy of atomic coordinates derived from Fourier series in Xray structure analysis. IV. The twodimensional projection of oxalic acid. Proc. R. Soc. London Ser. A, 190, 490–496.
Brill, R., Grimm, H., Hermann, C. & Peters, C. (1939). Anwendung der röntgenographischen Fourieranalyse auf Fragen den chemischen Bindung. Ann. Phys. (Leipzig), 34, 393–445.
Cox, E. G. & Cruickshank, D. W. J. (1948). The accuracy of electrondensity maps in Xray structure analysis. Acta Cryst. 1, 92–93.
Cruickshank, D. W. J. (1949a). The accuracy of electrondensity maps in Xray analysis with special reference to dibenzyl. Acta Cryst. 2, 65–82.
Cruickshank, D. W. J. (1949b). The accuracy of atomic coordinates derived by leastsquares or Fourier methods. Acta Cryst. 2, 154–157.
Sparks, R. A., Prosen, R. J., Kruse, F. H. & Trueblood, K. N. (1956). Crystallographic calculations on the highspeed digital computer SWAC. Acta Cryst. 9, 350–358.