Structure Factor Calculator

Why a reflection Bragg's law allows can still be strong, weak or absent. Give a cell, a space group and the atoms of the asymmetric unit; the page expands them by the symmetry, sums the scattered waves, and shows what each atom contributed.

You supply: A unit cell, a space group and one atom per line of the asymmetric unit. Coordinates may be written as fractions. Occupancy and an isotropic B are optional.
Reading it: |F| is what a measurement gives and the phase is what it loses — recovering it is the central problem of structure solution. Scattering factors here are those of free, spherical atoms, with no anomalous dispersion, so Friedel's law holds exactly.

Worked examples: NaCl (200) · Cu (111) · α-Fe (110) · CsCl (110) · ZnS (111)

Unit cell

Å

Space group

Number or symbol

Conditions belong to a setting, not to a space group number. Pnma, Pbnm and Pmcn are one space group with its axes labelled three ways, and the three tables differ: the mirror that empties hk0 in one of them empties 0kl in another. Every setting in the International Tables is here with its own operations, so P2₁/n and P2₁/a answer for themselves.

Atoms in the asymmetric unit

Label, x, y, z, and optionally occupancy and B

One atom per line, for example Na 0 0 0 or O 1/3 2/3 1/4 1 0.8. Coordinates may be fractions, which matters: a site at exactly ⅓ is not a site at 0.333333. Ions are written as the Tables label them — Na1+, O2-. Only the asymmetric unit is needed; the space group supplies the rest.

Reflection

F is summed over every atom in the unit cell -- the asymmetric unit expanded by the space group. |F| is what a measurement gives; the phase is what it loses, and recovering it is the central problem of structure solution.

Structure factor

\|F\|	85.3863 electrons
Phase	0.0°
F as a complex number	85.3863 + 0i
sin θ/λ	0.177305 Å⁻¹, so d = 2.82 Å
Atoms in the cell	8, holding 111.972 electrons — which is F(000)

What each atom contributed

Atom	Sites	f₀	exp(−B s²)	Contribution to F
Na	4	8.64872	1	34.5949 + 0i
Cl	4	12.6978	1	50.7914 + 0i

The site count is the Wyckoff multiplicity: how many copies of that atom the space group puts in the cell. An atom on a symmetry element has fewer than a general position, because the operations that fix it carry it onto itself.

The sum, drawn

Each atom’s contribution laid head to tail, and F as the resultant from the origin; a dot marks the end of each step. The dashed circle is F(000), the electron count of the whole cell — the largest |F| could ever be, and the same circle for every reflection of this structure, so two reflections can be compared by eye. An arrow that falls short of it does so for one of two reasons: the atoms are partly cancelling, or the scattering factors have simply fallen away with angle. An extinct reflection is a walk that closes back on where it started. This setting has a centre of symmetry at the origin, so F is real and every phase is 0° or 180° — a sign rather than an angle, which is why centrosymmetric structures were the first to be solved.

Anomalous dispersion is not included: close to an absorption edge the scattering factor gains f' and if'', and the imaginary part is what makes a Friedel pair unequal. Everything below therefore obeys Friedel's law exactly.