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Fourier transform

The equation of the structure factor:

is equivalent to a Fourier transform, an integral with very convenient properties. Outside the sample, ρ(r) is zero, and the integral above can be extended over all space without changing its value. The physical meaning of the equation above is that the structure factor is a Fourier transform of the object.
Because F(S) is the Fourier transform of ρ(r), a second Fourier integral must exist that relates these two quantities, and this is the inverse Fourier transform:


The integral is over all reciprocal space, and V is a constant that contains (2π)3 and other constants that compensate for difference in unit cell volume of sample space r and reciprocal space S.

The last equation means: if we have measured or calculated values of F(S) extending over all reciprocal space, we can compute the electron density distribution of the object.

The Fourier transformation allows a complete and reversible transformation between direct domain representation and reciprocal domain representation of the same physical reality. Complex structure factors representing reciprocal space can be converted into electron density representing direct space and back into complex structure factors. The continuous Fourier integrals are replaced by a discrete summation over all structure factor indices h, k, l or all density grid points x,y,z in the back-transform, respectively.

References and Links

- Kevin Cowtan's Book of Fourier
Biomolecular Crystallography By Bernhard Rupp
Techniques for the study of biological structure and function By Charles R. Cantor, Paul Reinhard Schimmel