Structure Factor Calculator

https://xraytools.com/structurefactor

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.

Before this: The asymmetric unit is the part of the cell the symmetry has not yet copied; the page expands it for you, and the sum runs over every atom that results. The space group is what does the expanding.
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: A diffraction experiment records intensities; |F| is derived from them once the corrections are applied, and the phase is what no measurement records — the results panel says what that costs. Corrections is the step in between. Scattering factors here are those of free, spherical atoms, with no anomalous dispersion, so Friedel's law holds exactly.

Worked examples: cubic – NaCl, Fm3m (200) · cubic – Cu, Fm3m (111) · cubic – α-Fe, Im3m (110) · cubic – CsCl, Pm3m (110) · cubic – ZnS, F43m (111) · hexagonal – quartz, P3₂21 (101) · tetragonal – cristobalite, P4₁2₁2 (101) · hexagonal – berlinite, P3₁21 (101) · tetragonal – rutile, P4₂/mnm (110) · orthorhombic – aragonite, Pmcn (111) · monoclinic – ZrO₂, P2₁/c (111) · triclinic – albite, C1 (111) · tetragonal – urea, P42₁m (110)

Earlier on the path: Space Group from Absences Next on the path: Fourier Synthesis and the Phase Problem On From planes to a powder pattern, step 7 of 7 · From intensities to a structure, step 1 of 6

Notation here: s · F, |F| · Z, V · f₀, f′, f″ — what each one means here

Terms here: asymmetric unit · general position · multiplicity · setting · zone

What each input changes

Atoms in the asymmetric unit: Which atoms are summed, after the space group has copied them. Moving one atom changes every reflection, because every term of the sum carries its position.
Space group: How many copies each atom gets and where they sit. It is what makes some reflections cancel exactly.

Teaching with this page

Objective: After this page a learner can account for a reflection being strong, weak or exactly absent by the terms of the sum rather than by a rule.
Start from: this worked example
Ask first: A detector records 4,000 counts for one reflection. Which quantity is that, before any correction?
Watch for: “|F|, which is what diffraction measures”
Then: Fourier Synthesis and the Phase Problem

Check yourself: A detector records 4,000 counts for one reflection. Which quantity is that, before any correction?

An intensity, from which |F| is eventually derived |F|, which is what diffraction measures

The chain is counts → corrected intensity → |F|² → |F|, and every arrow is work: background, scale, Lorentz-polarisation, absorption. The corrections page is that chain. Calling the measurement |F| is expert shorthand and it hides the step where most of the difficulty is.

Unit cell

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 glide that empties hk0 in one of them empties 0kl in another — a glide, because it is the fractional translation that makes a whole zone cancel, and a pure mirror carries none and empties nothing. 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. A diffraction experiment measures intensities; |F| is what survives the reduction of those counts, and the phase is what no measurement records. 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

What each atom contributes to F. CSV
Atom	Sites	f₀	exp(−B s²)	Contribution to F
Na	4	8.6487	1.0000	34.5949 + 0i
Cl	4	12.6978	1.0000	50.7914 + 0i

The site count is the site 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. (This is the count a Wyckoff letter carries in the International Tables; the site computes it and does not otherwise use that vocabulary.)

How this is calculated

$F (h k l) = \sum_{j} f_{j} (s) \exp (- B_{j} s^{2}) \sum_{r} \exp (2 π i (h x_{r} + k y_{r} + l z_{r}))$

j runs over the atoms you listed and r over every position the space group generates from each of them, so the inner sum is where the symmetry enters. A reflection is systematically absent when those terms cancel for every atom at once, whatever the atoms are — which is why an absence is a fact about the space group rather than about the structure.

$s = \frac{\sin (θ)}{λ} = \frac{1}{2 d} = \frac{1}{2 \times 2.82 Å} = 0.177305 Å^{- 1}$

$Re F_{Na} = 8.64872 \times (\cos (0 °) + \cos (0 °) + \cos (0 °) + \cos (0 °)) = 34.5949$

$Re F_{Cl} = 12.6978 \times (\cos (0 °) + \cos (0 °) + \cos (0 °) + \cos (0 °)) = 50.7914$

This setting has an inversion centre at the origin, so every position is matched by one at minus its coordinates and the sines cancel in pairs. F is therefore real, and its phase can only be 0° or 180°.

$Re F = 34.5949 + 50.7914 = 85.3863$

$| F | = \sqrt{{(Re F)}^{2} + {(Im F)}^{2}}$

$| F | = \sqrt{85.3863 \times 85.3863 + 0 \times 0} = 85.3863 electrons$

Nearest neighbours

Nearest neighbours of each atom, out to 3.6 Å. CSV
atom	neighbour	distance (Å)	how many
Na	Cl	2.8200	6
Cl	Na	2.8200	6

These are distances, not bonds. The cell says where the atoms are; whether two of them are bonded is a chemical judgement this page does not make, and a short contact between ions of the same charge is a repulsion rather than a bond. Distances are computed from the atomic coordinates through every symmetry operation of the space group and across cell boundaries, so the neighbour counts are the full coordination and not only what lies inside one cell.

The sum, drawn

All 8 terms: the sum so far is 85.3863 + 0i, so |F| = 85.3863 electrons.

⇤ start ← back add one → all ⇥

One arrow per atom in the cell, 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.

What the phase is a statement about. Each reflection is one wave running through the cell, with wavelength d(hkl) and its crests on the hkl planes. |F| says how strong that wave is; the phase says where it sits — how far its crest is from the cell origin, as a fraction of d written as an angle, so 180° is half a spacing. That is why the origin has to be fixed before a phase means anything, and why moving the origin changes every phase on this page while leaving every |F| exactly where it was: the waves have not moved, the point they are measured from has. A structure is those waves added up, which is what makes the missing phases the whole problem — the amplitudes say how much of each wave, and nothing in a measured intensity says where any of them sits.

Every term of the sum

Each row is one atom in the cell — the asymmetric unit expanded by the space group — contributing occ · f₀ · exp(−B s²) turned through the phase 2π(hx + ky + lz). The running total is the structure factor of everything above that line, which is what the arrows beside it are drawing.

#	Atom	x	y	z	Weight	This term	Running \|F\|
1	Na	0.0000	0.0000	0.0000	8.6487	8.64872 + 0i	8.6487
2	Na	0.0000	0.5000	0.5000	8.6487	8.64872 + 0i	17.2974
3	Na	0.5000	0.0000	0.5000	8.6487	8.64872 + 0i	25.9462
4	Na	0.5000	0.5000	0.0000	8.6487	8.64872 + 0i	34.5949
5	Cl	0.5000	0.5000	0.5000	12.6978	12.6978 + 0i	47.2927
6	Cl	0.5000	0.0000	0.0000	12.6978	12.6978 + 0i	59.9906
7	Cl	0.0000	0.5000	0.0000	12.6978	12.6978 + 0i	72.6884
8	Cl	0.0000	0.0000	0.5000	12.6978	12.6978 + 0i	85.3863

The weight is what the atom scatters at this angle before the phase is applied: occupancy times f₀(s) times the temperature factor. Two atoms of the same element at the same occupancy and the same B therefore have the same weight and differ only in their phase — which is why a systematic absence is a property of where the atoms are rather than of what they are. All three factors are per atom here, so a list that gives one site a partial occupancy or its own B breaks that equality deliberately; the absences of the space group survive it, because they cancel term against term whatever the weights are.

Where this comes from

X-ray scattering factors computed from numerical Hartree-Fock wave functions
D. T. Cromer and J. B. Mann, Acta Cryst. A 1968, 24, 321–324 · doi:10.1107/S0567739468000550
The four-Gaussian fit to f₀ that every term in the sum above is evaluated from. The site ships International Tables Volume C, Table 6.1.1.4, which prints coefficients of this form.
Interferenz von Röntgenstrahlen und Wärmebewegung
P. Debye, Ann. Phys. 1913, 348, 49–92 · doi:10.1002/andp.19133480105
Where the temperature factor comes from, and why thermal motion costs the high-angle reflections first — the optional B beside each atom above is this factor’s. In German.