xraytools.

Interatomic Distances and Angles

This sheet: https://xraytools.com/geometry?cellsa=2&cellsc=2&structure=berlinite

Page: https://xraytools.com/geometry

No diffraction data

The first thing anybody asks of a solved structure: how far apart are the atoms, and at what angles. Give a cell, a space group and the asymmetric unit — the page expands them by the symmetry, finds every neighbour inside a radius you choose, and names each one by the operation that produced it.

Before this
The atoms you supply are the asymmetric unit; the neighbours a distance is measured to are mostly symmetry copies the page generates. A distance to an atom you never typed is the normal case, not an error.
You supply
A unit cell, a space group and one atom per line of the asymmetric unit. Any constant or coordinate may carry its standard uncertainty in brackets, as a CIF writes it, and those are what the uncertainties below are propagated from.
Reading it
Uncertainties are propagated as if the parameters were uncorrelated, because a CIF does not carry the refinement’s variance–covariance matrix. That is not what a refinement program prints: for two atoms of one rigid group the correlation is usually positive, which makes the figure here larger. It is not an upper limit either.

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

Earlier on the path: CIF Parser Next on the path: Displacement Parameters and NPD Atoms On How to read a published structure, step 2 of 4

Notation here: U, B — what each one means here

Terms here: asymmetric unit · setting · zone

See also: CIF Parser · Displacement Parameters and NPD Atoms

What each input changes
How far to look
How far out neighbours are looked for. It decides which contacts are listed and nothing about their values; a bond does not become a bond by raising it.
Atoms in the asymmetric unit
The asymmetric unit. Most of the neighbours in the answer are symmetry copies the page generates, so a distance to an atom you never typed is the normal case.
Teaching with this page
Objective
After this page a learner can decide whether two reported distances are significantly different.
Start from
this worked example
Ask first
Two bonds are reported as 1.943(4) Å and 1.947(4) Å. Are they different?
Watch for
“No — the difference is inside the uncertainty, so they are equal”
Then
Displacement Parameters and NPD Atoms
Check yourself: Two bonds are reported as 1.943(4) Å and 1.947(4) Å. Are they different?

Not resolved, which is not the same as being equal No — the difference is inside the uncertainty, so they are equal

The difference is 0.004 Å against a combined uncertainty near 0.006 — well under the 3 s.u. that is conventionally taken as the threshold for calling a difference real — so the data cannot separate them. That is a statement about the data and not about the bonds: failing to resolve a difference is not evidence that there is none, and better data might resolve it. The combined figure is approximate as well — this page propagates as if the refined parameters were uncorrelated, and a CIF carries no variance–covariance matrix to do better.

Unit cell
Å
Å
Å
°
°
°

A constant may carry its standard uncertainty in brackets, exactly as a CIF writes it — 4.9137(2). That is what the uncertainties below are built from.

Space group

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

One atom per line, for example Si 0.4697(1) 0 1/6. This is the same notation the structure factor page reads, so a list works on either — and a coordinate may carry its uncertainty, which only this page uses. A coordinate written as a fraction is one the symmetry fixes, and carries no uncertainty by definition.

How far to look
Å

Every neighbour inside this radius is listed. The angle table grows as the square of the neighbour count, so a small increase here is a large one there.

These are distances and angles, not bonds. Every neighbour inside the radius is listed, whether or not anything holds the two atoms together, and a short contact between ions of the same charge is a repulsion rather than a bond. Contacts are found through every symmetry operation of the space group and across cell boundaries, so the list is the full environment and not only what lies inside one cell. Which of these contacts are bonds is asked separately below, by sorting them on how far each one exceeds the two covalent radii and cutting at the largest gap in that order — and it is that answer, not this table, that the picture draws.

The cell, filled

Al — O2, 1.761 ÅO2 at 0.4164, 0.257, −0.1186, outside the cell, drawn for a bondbP — O1, 1.512 ÅO1 at −0.1235, 0.5843, 0.0657, outside the cell, drawn for a bondAl — O1, 1.739 ÅO1 at 0.5843, 0.8765, −0.0657, outside the cell, drawn for a bondP — O2, 1.507 ÅO2 at −0.257, 0.1594, 0.2147, outside the cell, drawn for a bondP — O1, 1.512 ÅP — O2, 1.507 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP at 0, 0.4669, 0.1667Al — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 0.5339, 0.5339, 0Al — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 0.257, 0.4164, 0.1186Al — O1, 1.739 ÅO1 at 0.1235, −0.2922, 0.2677, outside the cell, drawn for a bondAl — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 0.1235, 0.7078, 0.2677Al — O2, 1.761 ÅO2 at 1.4164, 0.257, −0.1186, outside the cell, drawn for a bondAl — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 0.8765, 0.5843, 0.0657aAl — O1, 1.739 ÅO1 at 1.5843, 0.8765, −0.0657, outside the cell, drawn for a bondAl — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 0.743, 0.1594, 0.2147Al — O2, 1.761 ÅO2 at 0.743, 1.1594, 0.2147, outside the cell, drawn for a bondAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 0.4661, 0, 0.3333P — O1, 1.512 ÅP — O2, 1.507 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP at 1, 0.4669, 0.1667Al — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 1.5339, 0.5339, 0Al — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 0.4661, 1, 0.3333Al — O2, 1.761 ÅO2 at −0.1594, 0.5836, 0.5481, outside the cell, drawn for a bondP — O2, 1.507 ÅAl — O2, 1.761 ÅO2 at 1.257, 0.4164, 0.1186Al — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 0.4157, 0.2922, 0.399Al — O1, 1.739 ÅO1 at 0.4157, 1.2922, 0.399, outside the cell, drawn for a bondAl — O1, 1.739 ÅO1 at 1.1235, −0.2922, 0.2677, outside the cell, drawn for a bondP — O1, 1.512 ÅAl — O1, 1.739 ÅO1 at 1.1235, 0.7078, 0.2677Al — O2, 1.761 ÅO2 at 0.5836, −0.1594, 0.4519, outside the cell, drawn for a bondAl — O1, 1.739 ÅO1 at −0.2922, 0.1235, 0.7323, outside the cell, drawn for a bondAl — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 0.5836, 0.8406, 0.4519Al — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 1.8765, 0.5843, 0.0657Al — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 0, 0.4661, 0.6667P — O1, 1.512 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP — O2, 1.507 ÅP at 0.5331, 0.5331, ½Al — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 0.2922, 0.4157, 0.601Al — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 1.743, 0.1594, 0.2147Al — O2, 1.761 ÅO2 at 1.743, 1.1594, 0.2147, outside the cell, drawn for a bondAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 1.4661, 0, 0.3333P — O1, 1.512 ÅP — O2, 1.507 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP at 2, 0.4669, 0.1667Al — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 1.4661, 1, 0.3333Al — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 0.8406, 0.5836, 0.5481P — O2, 1.507 ÅO2 at 0.1594, −0.257, 0.7853, outside the cell, drawn for a bondP — O2, 1.507 ÅO2 at 2.257, 0.4164, 0.1186, outside the cell, drawn for a bondAl — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 0.1594, 0.743, 0.7853Al — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 1.4157, 0.2922, 0.399Al — O1, 1.739 ÅO1 at 1.4157, 1.2922, 0.399, outside the cell, drawn for a bondP — O1, 1.512 ÅO1 at 2.1235, 0.7078, 0.2677, outside the cell, drawn for a bondAl — O2, 1.761 ÅO2 at 1.5836, −0.1594, 0.4519, outside the cell, drawn for a bondAl — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 0.7078, 0.1235, 0.7323Al — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 1.5836, 0.8406, 0.4519P — O1, 1.512 ÅO1 at 0.7078, 1.1235, 0.7323, outside the cell, drawn for a bondP — O1, 1.512 ÅP — O2, 1.507 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP at 0.4669, 0, 0.8333Al — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 1, 0.4661, 0.6667P — O1, 1.512 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP — O2, 1.507 ÅP at 1.5331, 0.5331, ½P — O1, 1.512 ÅP — O2, 1.507 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP at 0.4669, 1, 0.8333Al — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 1.2922, 0.4157, 0.601Al — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 0.4164, 0.257, 0.8814P — O2, 1.507 ÅO2 at 0.4164, 1.257, 0.8814, outside the cell, drawn for a bondP — O1, 1.512 ÅO1 at −0.1235, 0.5843, 1.0657, outside the cell, drawn for a bondP — O1, 1.512 ÅO1 at 0.5843, −0.1235, 0.9343, outside the cell, drawn for a bondAl — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 0.5843, 0.8765, 0.9343P — O2, 1.507 ÅO2 at −0.257, 0.1594, 1.2147, outside the cell, drawn for a bondAl — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 1.8406, 0.5836, 0.5481P — O2, 1.507 ÅO2 at 1.1594, −0.257, 0.7853, outside the cell, drawn for a bondcAl — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 1.1594, 0.743, 0.7853P — O1, 1.512 ÅP — O2, 1.507 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP at 0, 0.4669, 1.1667Al — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 0.5339, 0.5339, 1Al — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 0.257, 0.4164, 1.1186Al — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 1.7078, 0.1235, 0.7323P — O1, 1.512 ÅO1 at 1.7078, 1.1235, 0.7323, outside the cell, drawn for a bondP — O1, 1.512 ÅP — O2, 1.507 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP at 1.4669, 0, 0.8333Al — O1, 1.739 ÅO1 at 0.1235, −0.2922, 1.2677, outside the cell, drawn for a bondAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 2, 0.4661, 0.6667P — O1, 1.512 ÅP — O2, 1.507 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP at 1.4669, 1, 0.8333Al — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 0.1235, 0.7078, 1.2677Al — O1, 1.739 ÅO1 at 2.2922, 0.4157, 0.601, outside the cell, drawn for a bondAl — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 1.4164, 0.257, 0.8814P — O2, 1.507 ÅO2 at 1.4164, 1.257, 0.8814, outside the cell, drawn for a bondAl — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 0.8765, 0.5843, 1.0657P — O1, 1.512 ÅO1 at 1.5843, −0.1235, 0.9343, outside the cell, drawn for a bondAl — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 1.5843, 0.8765, 0.9343Al — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 0.743, 0.1594, 1.2147Al — O2, 1.761 ÅO2 at 0.743, 1.1594, 1.2147, outside the cell, drawn for a bondAl — O2, 1.761 ÅO2 at 2.1594, 0.743, 0.7853, outside the cell, drawn for a bondAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 0.4661, 0, 1.3333P — O1, 1.512 ÅP — O2, 1.507 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP at 1, 0.4669, 1.1667Al — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 1.5339, 0.5339, 1Al — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 0.4661, 1, 1.3333Al — O2, 1.761 ÅO2 at −0.1594, 0.5836, 1.5481, outside the cell, drawn for a bondP — O2, 1.507 ÅAl — O2, 1.761 ÅO2 at 1.257, 0.4164, 1.1186Al — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 0.4157, 0.2922, 1.399Al — O1, 1.739 ÅO1 at 0.4157, 1.2922, 1.399, outside the cell, drawn for a bondAl — O1, 1.739 ÅO1 at 1.1235, −0.2922, 1.2677, outside the cell, drawn for a bondP — O1, 1.512 ÅAl — O1, 1.739 ÅO1 at 1.1235, 0.7078, 1.2677Al — O2, 1.761 ÅO2 at 0.5836, −0.1594, 1.4519, outside the cell, drawn for a bondAl — O1, 1.739 ÅO1 at −0.2922, 0.1235, 1.7323, outside the cell, drawn for a bondAl — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 0.5836, 0.8406, 1.4519Al — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 1.8765, 0.5843, 1.0657Al — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 0, 0.4661, 1.6667P — O1, 1.512 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP — O2, 1.507 ÅP at 0.5331, 0.5331, 1½Al — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 0.2922, 0.4157, 1.601Al — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 1.743, 0.1594, 1.2147Al — O2, 1.761 ÅO2 at 1.743, 1.1594, 1.2147, outside the cell, drawn for a bondAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 1.4661, 0, 1.3333P — O1, 1.512 ÅP — O2, 1.507 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP at 2, 0.4669, 1.1667Al — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 1.4661, 1, 1.3333Al — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 0.8406, 0.5836, 1.5481P — O2, 1.507 ÅO2 at 0.1594, −0.257, 1.7853, outside the cell, drawn for a bondP — O2, 1.507 ÅO2 at 2.257, 0.4164, 1.1186, outside the cell, drawn for a bondAl — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 0.1594, 0.743, 1.7853Al — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 1.4157, 0.2922, 1.399Al — O1, 1.739 ÅO1 at 1.4157, 1.2922, 1.399, outside the cell, drawn for a bondP — O1, 1.512 ÅO1 at 2.1235, 0.7078, 1.2677, outside the cell, drawn for a bondAl — O2, 1.761 ÅO2 at 1.5836, −0.1594, 1.4519, outside the cell, drawn for a bondAl — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 0.7078, 0.1235, 1.7323Al — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 1.5836, 0.8406, 1.4519P — O1, 1.512 ÅO1 at 0.7078, 1.1235, 1.7323, outside the cell, drawn for a bondP — O1, 1.512 ÅP — O2, 1.507 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP at 0.4669, 0, 1.8333Al — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 1, 0.4661, 1.6667P — O1, 1.512 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP — O2, 1.507 ÅP at 1.5331, 0.5331, 1½P — O1, 1.512 ÅP — O2, 1.507 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP at 0.4669, 1, 1.8333Al — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 1.2922, 0.4157, 1.601Al — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 0.4164, 0.257, 1.8814P — O2, 1.507 ÅO2 at 0.4164, 1.257, 1.8814, outside the cell, drawn for a bondP — O1, 1.512 ÅO1 at 0.5843, −0.1235, 1.9343, outside the cell, drawn for a bondAl — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 0.5843, 0.8765, 1.9343Al — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 1.8406, 0.5836, 1.5481P — O2, 1.507 ÅO2 at 1.1594, −0.257, 1.7853, outside the cell, drawn for a bondAl — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 1.1594, 0.743, 1.7853Al — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 0.5339, 0.5339, 2Al — O2, 1.761 ÅO2 at 0.257, 0.4164, 2.1186, outside the cell, drawn for a bondAl — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 1.7078, 0.1235, 1.7323P — O1, 1.512 ÅO1 at 1.7078, 1.1235, 1.7323, outside the cell, drawn for a bondP — O1, 1.512 ÅP — O2, 1.507 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP at 1.4669, 0, 1.8333Al — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 2, 0.4661, 1.6667P — O1, 1.512 ÅP — O2, 1.507 ÅP — O1, 1.512 ÅP — O2, 1.507 ÅP at 1.4669, 1, 1.8333Al — O1, 1.739 ÅO1 at 2.2922, 0.4157, 1.601, outside the cell, drawn for a bondAl — O2, 1.761 ÅP — O2, 1.507 ÅO2 at 1.4164, 0.257, 1.8814P — O2, 1.507 ÅO2 at 1.4164, 1.257, 1.8814, outside the cell, drawn for a bondAl — O1, 1.739 ÅO1 at 0.8765, 0.5843, 2.0657, outside the cell, drawn for a bondP — O1, 1.512 ÅO1 at 1.5843, −0.1235, 1.9343, outside the cell, drawn for a bondAl — O1, 1.739 ÅP — O1, 1.512 ÅO1 at 1.5843, 0.8765, 1.9343Al — O2, 1.761 ÅO2 at 2.1594, 0.743, 1.7853, outside the cell, drawn for a bondAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl — O1, 1.739 ÅAl — O2, 1.761 ÅAl at 1.5339, 0.5339, 2Al — O2, 1.761 ÅO2 at 1.257, 0.4164, 2.1186, outside the cell, drawn for a bondAl — O1, 1.739 ÅO1 at 1.8765, 0.5843, 2.0657, outside the cell, drawn for a bond
a−2+b1+c−2+Atomsball & stickvan der Waalsspace-fillingShowasymmetric unitunit cellBondsshownhiddendrag to rotate · scroll to zoomLook along

P Z 15 · rcov 1.07 Å Al Z 13 · rcov 1.21 Å O Z 8 · rcov 0.66 Å

Hover an atom to name it. Click one to pin it and mark every copy of that site.

Every atom of the cell is drawn, including the symmetry-equivalent ones. An atom lying on a face, an edge or a corner belongs to each cell it touches and is drawn in all of them, so counting the spheres overcounts the contents of one cell. Colours are the standard CPK ones, and spheres are drawn at half the covalent radius (Cordero et al., 2008), reduced further only if that would make two of them run into each other — at full size two bonded atoms touch by definition. These are not ionic radii: in a salt the cation is drawn larger than the anion, which is the opposite of the ionic picture. A stick is drawn where both atoms' own bond ladders put the other below their cut, so every bond here is one the table above lists — and where the two ladders disagree the table is the fuller answer, not this picture. A sphere outside the cell is drawn only because a bond from inside reaches it, and its own bonds are not completed in turn.

The symmetry the atoms obey

The atoms obey exactly the 6 operations of the space group given, and no others that this cell would permit.

Distances

Interatomic distances, in Å. CSV
atomneighbourdistancesymmetry of the neighbour
AlO11.7390x,y,z
O11.7390x-y,-y,-z+2/3
O21.7610-y+1,x-y,z-2/3
O21.7610-x+1,-x+y,-z+4/3
P3.0836-y,x-y-1,z-2/3
P3.0836-x+y+1,-x+1,z-1/3
P3.0867-y+1,x-y,z-2/3
P3.0867-x+y+1,-x,z-1/3
PO21.5065x,y,z
O21.5065x-y,-y,-z+5/3
O11.5120-y+1,x-y,z+1/3
O11.5120-x+1,-x+y,-z+4/3
Al3.0836-y+1,x-y,z+1/3
Al3.0836-x+y+1,-x,z+2/3
Al3.0867-y,x-y-1,z+1/3
Al3.0867-x+y+1,-x+1,z+2/3
O1P1.5120-x+y+1,-x+1,z-1/3
Al1.7390x,y,z
O12.4511y,x,-z+1
O22.4742-x+1,-x+y+1,-z+4/3
O22.4745-x+y+1,-x+1,z-1/3
O22.8033-x+1,-x+y,-z+4/3
O22.8598-y+1,x-y,z-2/3
O12.8852x-y,-y,-z+2/3
O2P1.5065x,y,z
Al1.7610-x+y+1,-x+1,z+2/3
O22.4388x-y,-y,-z+5/3
O12.4742-x+1,-x+y,-z+4/3
O12.4745-y+1,x-y,z+1/3
O12.8033-x+1,-x+y+1,-z+4/3
O12.8598-x+y+1,-x+1,z+2/3
O22.9335y,x,-z+2

Which of these are bonds

Contacts ranked by slack against the covalent radii, in Å. The rule marks the largest gap. CSV
atomneighbourdistanceradius sumslackgap
Al2 × O11.73901.8700−0.1310—
2 × O21.76101.8700−0.10900.0220
2 × P3.08362.28000.80360.9126
2 × P3.08672.28000.80670.0032
Al: 4 neighbours below the cut — 2 × O1, 2 × O2. The gap is 0.9126 Å, 41.48 times the next largest.
P2 × O21.50651.7300−0.2235—
2 × O11.51201.7300−0.21800.0054
2 × Al3.08362.28000.80361.0216
2 × Al3.08672.28000.80670.0032
P: 4 neighbours below the cut — 2 × O2, 2 × O1. The gap is 1.0216 Å, 188.28 times the next largest.
O1P1.51201.7300−0.2180—
Al1.73901.8700−0.13100.0871
O12.45111.32001.13111.2621
O22.47421.32001.15420.0230
O22.47451.32001.15450.0003
O1: 2 neighbours below the cut — P, Al. The gap is 1.2621 Å, 3.84 times the next largest; 3 further contacts inside this radius are not shown.
O2P1.50651.7300−0.2235—
Al1.76101.8700−0.10900.1145
O22.43881.32001.11881.2278
O12.47421.32001.15420.0354
O12.47451.32001.15450.0003
O2: 2 neighbours below the cut — P, Al. The gap is 1.2278 Å, 3.73 times the next largest; 3 further contacts inside this radius are not shown.

The cut is where the largest gap falls, not a criterion anyone chose. Slack is the contact's length less the sum of the two covalent radii, so a bond has little of it and a passing neighbour has a great deal; sorting by slack puts the coordination shell at the top whatever elements it is made of. Nothing here decides that a contact is a bond — it shows you where the evidence changes, and you decide.

A covalent radius is the wrong radius for a large ion, and that is where this fails. Where a published coordination number exists to compare against, the cut reproduces it for 26 of 35 atoms at a 5 Å radius, and 5 of the 9 misses have a soft cation — rock salt's sodium, caesium chloride's caesium, rutile's titanium, aragonite's calcium, baddeleyite's zirconium. The radius decides as much as the chemistry: the same 35 atoms give 23 right at 3.2 Å and 25 at 4 Å. A wrong cut usually has a gap barely bigger than the next one, so the ratio beside each cut is worth reading — but it is an association and not a test, and albite's sodium breaks it at 3.2 Å with a gap 3.4 times the next and the wrong answer.

Angles

Angles at each atom, in degrees. CSV
atbetweenangle
AlO1 and O1112.101
O1 and O2109.587
O1 and O2106.435
O1 and P126.739
O1 and P17.174
O1 and P92.717
O1 and P119.336
O1 and O2106.435
O1 and O2109.587
O1 and P17.174
O1 and P126.739
O1 and P119.336
O1 and P92.717
O2 and O2112.792
O2 and P91.572
O2 and P109.062
O2 and P17.656
O2 and P114.860
O2 and P109.062
O2 and P91.572
O2 and P114.860
O2 and P17.656
P and P142.693
P and P106.450
P and P91.407
P and P91.407
P and P106.450
P and P122.489
PO2 and O2108.078
O2 and O1110.124
O2 and O1110.105
O2 and Al90.956
O2 and Al123.240
O2 and Al123.989
O2 and Al20.765
O2 and O1110.105
O2 and O1110.124
O2 and Al123.240
O2 and Al90.956
O2 and Al20.765
O2 and Al123.989
O1 and O1108.306
O1 and Al19.854
O1 and Al112.166
O1 and Al90.805
O1 and Al111.158
O1 and Al112.166
O1 and Al19.854
O1 and Al111.158
O1 and Al90.805
Al and Al122.508
Al and Al106.450
Al and Al91.418
Al and Al91.418
Al and Al106.450
Al and Al142.656
O1P and Al142.973
P and O135.847
P and O234.876
P and O234.866
P and O2109.025
P and O2131.161
P and O1162.882
Al and O1137.448
Al and O2153.859
Al and O2108.887
Al and O237.052
Al and O235.461
Al and O133.950
O1 and O260.315
O1 and O260.304
O1 and O2102.796
O1 and O2158.972
O1 and O1129.537
O2 and O259.053
O2 and O2138.864
O2 and O2119.884
O2 and O1159.923
O2 and O279.871
O2 and O2100.717
O2 and O1139.966
O2 and O262.386
O2 and O160.344
O2 and O158.410
O2P and Al141.579
P and O235.961
P and O135.019
P and O135.010
P and O1155.469
P and O1107.650
P and O2135.692
Al and O2157.712
Al and O1108.204
Al and O1133.135
Al and O136.513
Al and O134.953
Al and O233.604
O2 and O160.480
O2 and O160.468
O2 and O1160.475
O2 and O1138.270
O2 and O2126.286
O1 and O159.381
O1 and O1138.864
O1 and O177.919
O1 and O2103.531
O1 and O1122.254
O1 and O1102.229
O1 and O2157.773
O1 and O161.246
O1 and O259.752
O1 and O257.862

Every pair of neighbours inside the radius is here, which is more than the bond angles: two atoms that are not bonded to each other still subtend an angle at the middle one, and a small value usually means the two legs are of very different length. The distance table above is what says which neighbours are close enough to be bonds.

Torsions

Torsion angles about each bond, in degrees. CSV
aboutchaintorsionsectorchains
O1-PAl–O1–P–O1+−104.707−ac3
Al–O1–P–O2+15.746sp3
Al–O1–P–O2+134.828+ac3
O2-PAl–O2–P–O1+−96.768−ac3
Al–O2–P–O1+22.596sp3
Al–O2–P–O2+142.920+ac3
Al-O1O1–Al–O1–P+150.757ap3
Al-O2O1–Al–O2–P+17.813sp3
O1–Al–O2–P+139.233+ac3
Al-O1O2–Al–O1–P+−91.298−ac3
O2–Al–O1–P+30.954+sc3
Al-O2O2–Al–O2–P+−100.563−ac3

A torsion angle is measured looking along the middle bond: it is the angle from the first atom to the fourth, projected onto the plane across that line. It is the first quantity on this page whose sign carries information no distance or angle does — mirror a crystal and every distance and every angle is unchanged, while every torsion changes sign. Quartz is the example: its two enantiomorphs give an identical set of bond lengths and the exactly opposite set of torsions. Two values are their own opposite and so carry no sign here: 0, where the chain is eclipsed, and 180, where it is anti. Each of those is superimposable on its own mirror image, so there is no hand to report.

The sign follows the convention of Klyne and Prelog, which IUPAC adopted as the standard for describing conformation: look from the first atom along the middle bond towards the fourth, and the torsion is positive when the near bond has to turn clockwise, through less than 180°, to eclipse the far one. Which end you look from does not change the answer — reading the chain backwards gives the same number, sign included, which is why a chain and its reverse are one row above. The form of it a reader can check against a picture rather than against algebra: a right-handed helix has positive torsions.

What this page computes, with b1 = B−A, b2 = C−B and b3 = D−C:

τ = atan2( |b2| b1·(b2×b3), (b1×b2)·(b2×b3) )

Two arguments rather than one, which is the whole point: an arc cosine of the angle between the two planes gives the size and throws the sign away, and a sign applied afterwards is a second convention to get wrong. Here it comes out of the arithmetic.

The sector column names the range, in IUPAC's terms:

  • sp synperiplanar, 0–30°
  • ±sc synclinal, 30–90°
  • ±ac anticlinal, 90–150°
  • ap antiperiplanar, 150–180°

IUPAC gives those ranges with shared endpoints — 0 to ±30 synperiplanar, 30 to 90 synclinal — so a torsion of exactly 30° is in two of them and the recommendation does not settle which. This page gives a boundary to the sector nearer zero; that is a choice, not a standard. The sign goes on sc and ac because +sc and −sc are two different sectors, while sp is one sector straddling zero and ap one straddling 180° — a sign on those would only repeat the number beside it.

Each of the three bonds in a chain comes from the ladder above, so every reservation there applies here three times over. Where a cut runs through a large soft cation the chains built on it are numerous and the least trustworthy on the page: read the ladder before the table.

Hydrogen bonds

This structure has no hydrogen in it, so it has no hydrogen bonds. Every other worked example below is an inorganic solid without hydrogen; load urea to see this table with something in it.

A hydrogen bond D–H···A is a hydrogen held between the atom it is covalently bonded to and a second one it is not. This table takes every hydrogen, finds its covalent bond from the ladder above, and keeps the contacts with the hydrogen genuinely between the two — which is the D–H···A angle exceeding 90°, and needs no cutoff to say. What survives is ranked by the same ladder, so the cut here and the cut above are one rule. One thing here is a convention rather than a measurement, and it is the elements: the donor must be N, O, F or S and the acceptor one of those or a halogen, because a hydrogen bond needs a polarised bond at one end and a lone pair at the other and neither is visible in a list of coordinates. That excludes C–H···O, which is a real if weaker interaction — the contact is still in the distance table above, it is only the name that is withheld. Urea is the example the page ships: each of its four N–H donates to a carbonyl oxygen, and each oxygen accepts four.

Every row rests on the ladder twice — once for the D–H bond it starts from and once for the cut among what is left — so each reservation there applies here twice. And the shortest contact of a hydrogen is often not its hydrogen bond: in urea the two closest neighbours of H1 are the other hydrogen on the same nitrogen and the carbon two bonds away, both of which sit beside it rather than in front of it.

Packing and voids

Empty space against probe radius, cell volume 231.6 Å3. CSV
probe radius / Åvoid fractionvoid volume / Å3
0.00.130230.1
0.20.03488.1
0.40.00090.2
0.60.00000.0
0.80.00000.0
1.20.00000.0

Packing fraction 0.8698 — that fraction of the cell is inside an atom.

The packing fraction is the row at a probe radius of zero: how much of the cell lies inside an atom. The rows below it ask a different question — how much space is left for a sphere of that radius to sit in without overlapping anything, which is what a crystallographer means by a void. A water molecule is usually given 1.2 Å.

Measured by sampling 110,592 points on a grid of 483, offset from the cell origin by an irrational fraction. The offset is not cosmetic: a grid that lines up with the lattice samples whole planes of points onto sphere boundaries, and its error then depends on the arithmetic relationship between the grid and the cell rather than on the resolution — refining it does not help. Against the four lattices whose packing fraction is an exact constant, this grid is right to about 5×10−4, so the fourth decimal above is the last one worth reading.

Radii are van der Waals radii from Alvarez (2013), the revision of Bondi's set derived from the Cambridge Structural Database. That choice is a convention and it is the only one on this table — the volumes themselves are geometry. A van der Waals radius describes how close a non-bonded neighbour comes, so in a metal or an ionic solid, where every contact is a bond, the spheres overlap and the packing fraction is 1 by construction. The number means what it says for molecular crystals.

About the uncertainties

No uncertainties are quoted, because none were given. Nothing in the cell or the coordinates above carries a bracket, so there is nothing to propagate. An absent uncertainty is not a zero one.

Write a constant as 4.9137(2) or a coordinate as 0.4697(1) and every distance and angle below will carry its own, split into the part that comes from the cell and the part that comes from the coordinates.

Contacts were searched out to 3.2 Å. A neighbour is named by the operation that produces it from the atom in the list above, written out in full rather than as a numbered code.

Where this comes from