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.
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.
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 ownYes — 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 projected electron density
Point at a tick along the foot to see which atoms project onto it. Click it to keep it; click it again, click empty space, or press Escape to let go.
α-quartz, SiO2,
P3221 —
30 non-zero
coefficients out of
30, carrying detail to
0.14 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(2πih·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.
7 of the 30 coefficients are negative — a phase of 180°. A corrected, scaled measurement gives 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.
Point at any bar for its value, its phase and what the side of the line means. Click it to keep it; click it again, click empty space, or press Escape to let go.
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
4.9134 Å
Atoms in the cell
9
Distinct projected sites
9
F(000)
89.99 electrons
Mean of the curve
89.99 — which is
F(000), the constant term of the series: every cosine averages to zero over the
cell
Coefficients used
30 of 30
Negative coefficients
7
Highest value, phases discarded
330.3 electrons per unit x, at
x = 0
Correlation of the two curves
0.3537
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
x
Atoms there
Nearest maximum
0.0000
Si
0.0000 away
0.1466
O
0.0022 away
0.2669
O
0.0002 away
0.4135
O
0.0115 away
0.4697
Si
0.0003 away
0.5303
Si
0.0003 away
0.5865
O
0.0115 away
0.7331
O
0.0002 away
0.8534
O
0.0022 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.
The determination of parameters in crystal structures by means of Fourier series W. L. Bragg, Proc. R. Soc. A1929, 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.