Fourier Synthesis and the Phase Problem

The projected electron density

low albite, NaAlSi₃O₈, C1 — 6 non-zero coefficients out of 12, carrying detail to 0.61 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 amplitudes this page sums are not what a detector records. On a single crystal the chain runs: an integrated count over the time a reflection passes through the diffracting condition, divided by the Lorentz factor for how long that took, divided by the polarisation factor for the beam, corrected for absorption through the crystal and for extinction where a reflection is strong, and scaled — which gives I ∝ |F|², so |F| is a square root away and the sign is already gone. The corrections page works that chain through in detail for a powder diffractometer; the factors have the same names in both geometries and different forms, because the Lorentz factor is a statement about how a reflection sweeps through the Ewald sphere and a spinning crystal does not sweep the way a powder ring does. Take the shape of the chain from there and not the formulae.

The phase problem

A structure factor is a wave, and a wave has a size and a starting point. |F| is the size — how strongly that set of planes scatters. The phase is the starting point: where the crests of that wave sit relative to the origin of the cell. Adding the waves up puts density where crests from many reflections coincide, so the phases are what decide where the atoms are, and the amplitudes only how much scattering there is to place. That is why moving the origin changes every phase and no amplitude: shift the cell by t and F(h) is multiplied by exp(2πi h·t), which turns each wave without resizing it. An individual phase is therefore partly a statement about a choice of origin, which is exactly why direct methods work with combinations — like the triplet on the sign-relation page — that survive that choice.

1 of the 6 coefficients is 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.

Every coefficient in this sum is +|F| or −|F|, so “the phase problem” here is a choice between two signs rather than an angle anywhere on a circle. That is a property of these structures, not of the transform. F(h00) comes out real, and the projection centrosymmetric, exactly when the space group has an operation sending x to −x; every structure offered here has one, including the five with no centre of symmetry of their own — quartz, berlinite, cristobalite, zinc blende and urea. A structure in P1 has none, and its projection is not centrosymmetric: one carbon at 0.1, 0.2, 0.3 gives F(100) a phase of 36° on the structure-factor page. Restore the other two indices and the phases go back to being continuous even here: a general hkl of a non-centrosymmetric structure has a phase that is not 0 or 180°, and recovering it is a harder problem than choosing a sign.

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

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, 0.0606 apart, come out as a single peak at ½ until the series is long enough to separate them.

Where the atoms are

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.