An electron density map obtained through SIR phasing is the superposition of the true electron density and theSingle Anomalous Difference (SAD) phasing:inverseof the true electron density convoluted with the Fourier transform of exp(2 i phi_sub), where phi_sub are the phases of the heavy atom substructure. The Fourier coefficients of this map are given by (Ramachandran & Srinivasan, 1970, Fourier Methods in Crystallography, p. 123):F + conjugate(F) exp(2 i phi_sub)

An electron density map obtained through SAD phasing is the superposition of the true electron density and theIn general the second terms are expected only to contribute to the background in the SIR or SAD maps. However, the case when the substructure has a centrosymmetric configuration in a noncentrosymmetric crystal is special. If the centre of inversion of the substructure is (without loss of generality) placed at the origin of the unit cell, all phases phi_sub are either 0 or 180 degrees, and exp(2 i phi_sub) = 1 for either value. Therefore the equations above reduce to F + conjugate(F) and F - conjugate(F), respectively. The SIR or SAD map will therefore be the superposition of the true electron density with its exact inverse or negative inverse, respectively, and the interpretation of the map will therefore be significantly more difficult.negative inverseof the true electron density convoluted with the Fourier transform of exp(2 i phi_sub) (p. 190 in the same book):F - conjugate(F) exp(2 i phi_sub)

Often it is not immediately obvious that a substructure is centrosymmetric. This server provides a very simple method for testing for this condition:

- Given the crystallographic data of the substructure the phases phi_sub are computed.
- The coefficients exp(2 i phi_sub) are Fourier transformed.
- The resulting map is searched for peaks.
- The peaks are sorted by height and plotted.

To see examples, try space group P1 with one or two atoms in any position, or space group P6 with one atom at x1,y1,0 and optionally another atom at x2,y2,1/2. To see a good counter example, try four or more randomly placed atoms in space group P31.

This server can also be used to show that some maps will be more difficult to interpret than others even if the substructure is not centrosymmetric. For example, compare the plots for a randomly placed atom in space group P3 and space group P31.

The general rule is:

- The pessimistic viewpoint: A SIR or SAD map is more difficult to interpret if the distribution in the plot is sharp.
- The optimistic viewpoint: A SIR or SAD map is easier to interpret if the distribution in the plot is flat.

* The name of this server is derived from the term *double-phasing*
used by Ramachandran & Srinivasan.

** The Fourier transform of a constant function (e.g. exp(2 i phi_sub) = 1) is a delta function: it is different from zero only at the origin, and exactly zero everywhere else. If the centre of inversion of a substructure is shifted away from the origin the expression exp(2 i phi_sub) is no longer a constant, but the Fourier transform is still a delta function which is different from zero only at the location (2 xc, 2 yc, 2 zc), where (xc, yc, zc) are the coordinates of the centre of inversion.

[Index of services]