Interatomic Distances and Angles

The cell, filled

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

Which of these are bonds

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

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

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

Packing fraction 0.9042 — 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.