With this simplification, even a small structure (one with, e.g., 1000 symmetry-unique reflections;
perhaps 20 atoms in the asymmetric unit) would have 21000 possible solutions. If a subset
of reflections is used (e.g., the 100 strongest reflections) there are still 2100 possible
solutions; there would still be about 1030 electron density maps to inspect for the
correct solution. Clearly even a small structure like this cannot be solved by simple trial-and-error
methods in which all possible phase combinations are attempted. And the situation for non-centrosymmetric
structures is even worse - all phase angles (from 0° to 360°) are possible.
Recall from the earlier discussion of
the phase problem that for centrosymmetric structures, determining the phases of the structure
factors amounts to deciding whether the Fhkl's are + or - :
To gain some insight into Direct Methods of solving X-ray structures, it is useful to picture the
contributions of those cosine terms in the above equation to the electron density wave; thus, in one
as the h index increases, the contribution of that term to the electron density wave has more nodes.