In order to produce a centrosymmetric electron
density distribution, only cosine terms are required in the Fourier series:
because the cosine function is centrosymmetric; sine is not. Thus, the electron
density equation for a centrosymmetric unit cell reduces to:
in which a calculation of the Fourier electron density map requires that the signs of the
individual structure factors be determined. Even with these simplifications possible for
centrosymmetric cells, the solution of a structure is a formidable task. If each
structure factor can be either + or -, and if there are 1000 reflections in a data set (a
rather small structure), there would be approximately 10300 possible solutions
to the structure (10300 electron density maps to inspect). In order to have any
realistic hope of solving a structure, there must be some way of “getting at” the phases of
the structure factors.
