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.
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 phase problem
7 of the 30 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.
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) again, and it never entered the series
Coefficients used
30 of 30
Negative coefficients
7
Highest value, phases discarded
330.3 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, 0.056 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
0.0000
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.