Fourier Synthesis and the Phase Problem
https://xraytools.com/fourier
A diffraction pattern gives intensities, and correcting those gives |F|2, so amplitudes. The phases are never recorded at all, and the page opens by saying what that costs. This sums a real structure both ways, so you can see what the missing half was carrying.
- Before this
- A structure factor has an amplitude and a phase, and only the first survives a measurement. If that is not yet concrete, build one first.
- 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
Earlier on the path: Structure Factor Calculator Next on the path: The Patterson Function On From intensities to a structure, step 2 of 6
Notation here: F, |F| — what each one means here
See also: Structure Factor Calculator · The Patterson Function · Difference Map
What each input changes
- Terms
- How many reflections the sum runs over — the resolution of the map. Too few and two atoms merge into one peak, which is a limit of the DATA and not of the method.
Teaching with this page
- Objective
- After this page a learner can state the phase problem precisely and show what a correct set of amplitudes with wrong phases produces.
- Start from
- this worked example
- Ask first
- A synthesis from six terms shows a maximum where the structure has no atom. Is the structure wrong?
- Watch for
- “Yes — density appears where the electrons are”
- Then
- The Patterson Function
Check yourself: A synthesis from six terms shows a maximum where the structure has no atom. Is the structure wrong?
No — a series cut short has ripples of its own Yes — density appears where the electrons are
The density is a sum over all reflections and any measurement supplies a finite number of them. Stopping at six is multiplying the true transform by a box, and a box in one space is a ripple in the other — so features appear beside real atoms and between them. Adding terms shrinks them and no number of terms removes them. Add some here and watch it happen.
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 experiment records intensities — counts under each reflection. Correcting those for geometry, polarisation and absorption gives |F|2, and its square root is |F|, the amplitude of that reflection; the corrections page is that step on its own. 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. No step of that chain records a phase. 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
Where this comes from
- The determination of parameters in crystal structures by means of Fourier series
W. L. Bragg, Proc. R. Soc. A 1929, 123, 537–559 · doi:10.1098/rspa.1929.0083
Summing the series back into a picture of the density, which is what this page does. It is also where the phase problem stops being an abstraction: the amplitudes are measured and the signs are chosen.