Fourier Synthesis and the Phase Problem
A diffraction pattern gives |F|, and the electron density is the Fourier transform of the structure factors — so with amplitudes and phases the structure follows by summation. The phases are not measured. This sums a real structure both ways, so you can see what the missing half was carrying.
- You supply
- One of the named structures, and how far the series should run. Everything else — the coefficients, their signs, the electron count — is computed from the atoms.
- Reading it
- Summing over h alone gives the density projected down b and c, not a section — atoms sharing an x land on top of one another. And a truncated series rings: it merges neighbours the resolution cannot separate, and can dip below zero between the atoms.
Worked examples: zirconia — 7 of 12 signs negative · aragonite, alternating signs · quartz at 6 terms — the silicons merge · quartz at 30 terms — and separate · caesium chloride — where the demonstration is empty
Input
Gaps in the row of bars are systematic absences: a centred lattice or a glide plane makes whole classes of h00 vanish, so the series has fewer terms than its length suggests and the projection repeats more often than the cell does.
The half a measurement throws away
A diffraction pattern gives you |F|, the amplitude of each reflection. The electron density is the Fourier transform of the structure factors — so with the amplitudes and the phases, the structure follows by summation, and there is nothing left to solve. The phases are not measured. Recovering them is the central problem of the subject.
Choose a structure and this sums its F(h00) into the electron density projected along a, then does it again with every phase set to zero — which is what you would have if you used the measurement alone.
For some of the named structures the second sum is identical to the first. That is a result rather than an omission, and the page says which ones and why.
Try it
zirconia — 7 of 12 signs negative · aragonite, alternating signs · quartz at 6 terms — the silicons merge · quartz at 30 terms — and separate · caesium chloride — where the demonstration is empty