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 projected electron density
baddeleyite, monoclinic zirconia, P21/c — 12 non-zero coefficients out of 12, carrying detail to 0.42 Å. The second curve is the same sum with the phases discarded.
Summing over h alone gives the density projected down b and c onto the a axis — not a section through the cell. So a peak sits at an atom’s x whatever its y and z are, and two atoms sharing an x project on top of one another: in rock salt the sodium and the chlorine both land at 0 and at ½, and the projection cannot tell them apart. A true section along x would need every hkl.
The phase problem
7 of the 12 coefficients are negative — a phase of 180°. A diffractometer measures the height of each bar and nothing else, so those signs are exactly what a measurement does not give you. Take every coefficient positive and the sum collapses: at x = 0 every cosine is 1, so every term reaches its maximum together and the map has one large peak at the origin. That peak is a property of having thrown the phases away, not of this crystal — it appears at the origin for any structure treated this way.
Each bar is one coefficient, drawn with its sign. A measurement gives the height of every bar and not which side of the line it is on.
What was summed
| Cell edge a | 5.1505 Å |
|---|---|
| Atoms in the cell | 12 |
| Distinct projected sites | 6 |
| F(000) | 223.87 electrons |
| Mean of the curve | 223.87 — which is F(000) again, and it never entered the series |
| Coefficients used | 12 of 12 |
| Negative coefficients | 7 |
| Highest value, phases discarded | 1,292.1 at x = 0 |
| Correlation of the two curves | -0.1450 |
The series stops, and a stopped series rings. Between the atoms the sum overshoots and undershoots, leaving small maxima where the crystal has nothing. Here it stays above zero, which is not a general property: a few light atoms on a large F(000) baseline do not ripple that far, while rock salt at the same term count goes visibly negative. Atoms closer together in x than the series can resolve merge into one maximum: quartz’s two silicons, 0.056 apart, come out as a single peak at ½ until the series is long enough to separate them.
Where the atoms are
| x | Atoms there | Nearest maximum |
|---|---|---|
| 0.0703 | O | 0.0075 away |
| 0.2758 | Zr | 0.0008 away |
| 0.4423 | O | 0.0133 away |
| 0.5577 | O | 0.0133 away |
| 0.7242 | Zr | 0.0008 away |
| 0.9297 | O | 0.0075 away |
The distance is to the nearest maximum in the curve above, and it is an annotation rather than a test: not every atom has a peak of its own once the projection has merged it with a neighbour, and not every maximum is an atom.
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