Reduced Cell and Bravais Lattice
A lattice has infinitely many cells, so two descriptions of one crystal can look nothing alike. This reduces any cell to the Niggli reduced cell — the one the lattice itself picks — with the integer matrix that gets you there and back, and then finds the lattice’s own point group to name its Bravais type. A primitive cell with three 60° angles turns out to be face-centred cubic.
- You supply
- Six cell constants, and nothing else — no space group, no atoms. Any cell of the lattice gives the same answer, which is the whole point. The tolerance decides how close two measured constants have to be before the search treats them as equal.
- Reading it
- Everything here describes the lattice, not the crystal. A metrically cubic lattice does not make the structure cubic — the atoms may have far less symmetry, and a monoclinic structure on a metrically cubic lattice is neither impossible nor rare. Finding the true space group needs the atoms as well. The symmetry answer also depends on the tolerance, and that is not a defect: at 0.001 a cell with β = 90.02° is orthorhombic and at 0.000001 it is monoclinic, and which is right depends on how well β is known.
Worked examples: Cu, the primitive cell – secretly cubic F · α-Fe, the primitive cell – secretly cubic I · a cell with β = 90.02° – orthorhombic, or not · quartz – already reduced, and unchanged
A lattice has infinitely many cells: any three non-coplanar lattice vectors describe the same lattice with six different numbers. The reduced cell is the one chosen by a rule that depends only on the lattice, so two descriptions of one lattice reduce to the same six numbers — which is what makes it the thing to compare, and what makes a database search for "have I made this before?" possible at all.
The lattice
The point group of this lattice metric has 48 operations, so the lattice is cubic — holohedry m3m. Its Bravais type is cF: cubic, face-centred on all three faces.
The seven holohedries have seven different orders — 2, 4, 8, 12, 16, 24 and 48 — so counting the operations names the family outright, with no table to look anything up in.
Everything here is a statement about the lattice, which is not the same as a statement about the crystal. A lattice whose metric is cubic does not make the structure cubic: the atoms may sit at positions with far less symmetry, and a metrically cubic monoclinic structure is neither impossible nor rare. Finding the true space group needs the atoms as well — this page is what that search starts from, not what it ends with.
The cells
| a / Å | b / Å | c / Å | α / ° | β / ° | γ / ° | V / Å3 | |
|---|---|---|---|---|---|---|---|
| as typed | 2.5561 | 2.5561 | 2.5561 | 60.000 | 60.000 | 60.000 | 11.809 |
| reduced | 2.5561 | 2.5561 | 2.5561 | 60.000 | 60.000 | 60.000 | 11.809 |
| conventional | 3.6149 | 3.6149 | 3.6149 | 90.000 | 90.000 | 90.000 | 47.237 |
- as typed — the cell you gave.
- reduced — the Niggli reduced cell, type I (all three angles acute, which the lattice fixes rather than the reduction choosing) — the cell you typed was already this one.
- conventional — cF: face-centred on all three faces, holding 4 lattice points and therefore 4 reduced cells. This is the cell a paper would quote.
A cell that comes back unchanged is the case worth seeing: a page that only ever shows the cell moving never demonstrates that it can leave one alone.
How to get there
Your cell → the reduced cell
...and back again
The reduced cell → the conventional cell
This one has determinant 4 rather than 1, which is not a different kind of thing — it is the statement that the conventional cell holds 4 lattice points, which is what the centring letter says in words.
The matrix has the new basis vectors as its rows, written in terms of the old ones: the first row is the new a as a combination of the old a, b and c. The column convention is equally common and gives the transpose of every matrix below, which no length or angle can tell you apart — so it is worth checking against before comparing with another program. Every entry is a whole number and the determinant is 1, so the transformation can be applied and undone by hand.
About the tolerance
At this tolerance, and for a cell of about 2.56 Å, two edges count as equal when they differ by less than about 0.0013 Å, and an angle counts as a right angle when it is within about 0.057° of one.
A lattice is more symmetric at a looser tolerance and less at a tighter one, and neither answer is the true one on its own — what settles it is how well the constants are actually known. A cell refined to ±0.001 Å has no business being asked about agreement at 0.0001.