xraytools.

Reduced Cell and Bravais Lattice

https://xraytools.com/reduce

Powder or single crystal

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.

Before this
A lattice and a cell are different things: one lattice has infinitely many cells, and reduction is the rule that picks one of them. The cell arithmetic is assumed.
You supply
Six cell constants and nothing else, for everything above the last panel — 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. Naming a space group as well is optional and adds one question: what twinning this particular cell would permit.
Reading it
Everything here describes the lattice, not the crystal, and the panel says plainly what that leaves undone. 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

Notation here: a*, b*, c* — what each one means here

Terms here: centring · point group · holohedry · setting

See also: Line Broadening · Powder Indexing · Reciprocal Cell Calculator · Space Group Reflection Conditions

What each input changes
How close counts as equal
How close two cell edges have to be before they count as equal. It decides which Bravais lattice comes out, so a borderline cell can change type on this number alone.
Teaching with this page
Objective
After this page a learner can explain why one lattice has many cells and what a reduction rule is choosing between.
Start from
this worked example
Ask first
A lattice point sits at the corner of every cell you draw. What is at that point?
Watch for
“An atom, since that is what repeats”
Check yourself: A lattice point sits at the corner of every cell you draw. What is at that point?

Nothing need be — it marks a repeat, not a position An atom, since that is what repeats

A lattice is a set of points expressing the repeat; the motif is what sits at each one, and it can be one atom, twenty, or none centred there at all. This is why one lattice has infinitely many cells and why reduction has to pick one: the choice is about the repeat, and the atoms come along with it.

The cell you have
Å
Å
Å
°
°
°

Any cell of the lattice will do, and that is the point: the answer does not depend on which one you type.

How close counts as equal

Measured constants are never exactly equal, so the symmetry search needs to be told how close is close enough. Sweep this and watch the answer change — that is not a defect, it is the question.

The space group, if you know it

Optional. Name the group the structure was refined in and the page also answers what twinning this cell would permit — both the laws that hold exactly and the ones it comes close enough to reach.

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

The cell as typed, reduced, and in its conventional setting. CSV
a / Å b / Å c / Å α / ° β / ° γ / ° V / Å3
as typed2.55612.55612.556160.00060.00060.00011.809
reduced2.55612.55612.556160.00060.00060.00011.809
conventional3.61493.61493.614990.00090.00090.00047.237
  • 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

(100010001)

...and back again

(100010001)

The reduced cell → the conventional cell

(1−1−11−11−1−11)

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, so the transformation can be applied and undone by hand. Its determinant is the volume ratio: 1 between two cells of the same volume, and the number of lattice points in the conventional cell when a primitive cell is turned into a centred one — 2 for I and C, 4 for F.

Its number in the tables

This reduced cell is lattice character 1 of the standard tabulation — a type I cell, listed there as cF. That is the same lattice as the cF the operation count gave at the top of this page — two routes, one answer, and neither used the other.

The other rows this cell also fits

Rows of the tabulation whose conditions this cell meets at ε = 0.001. CSV
Character Type Bravais Needs ε
1 this one I cF exact
2 I hR exact
9 I hR exact
10 I mS exact
19 I oI exact
20 I mS exact
27 I mS exact
31 I aP exact

A cell of high symmetry meets the conditions of every row that asks for less — a primitive cubic one fits seventeen of the forty-four — so the tabulation on its own does not say which is the character. What picks the row above out of that list is the Bravais type this page derived by counting operations, which needed no table at all.

The lattice character is the reduced cell's row number in the standard tabulation of International Tables A, Table 3.1.3.1 — 44 rows, each a set of equalities among the six numbers above, each carrying a Bravais type and a matrix to the conventional cell. It is what an indexing program means by "character 25", which is the reason it is printed here. This page does not use it: measured constants satisfy an exact equality never, so every row has to be met with a tolerance, and at a loose one several rows fit while at a tight one only the general triclinic row is left, which is a name for the absence of any symmetry rather than an answer. The Bravais type above was found the other way — by counting the symmetry operations of the lattice itself, which needs no table and degrades gently. The number is the name; the operation count is the answer.

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.

How much would have to bend

Every lattice symmetry below is one this cell has at some tolerance, and the figure beside it is the smallest tolerance that reaches it — computed exactly for each operation rather than found by trying values. Read the first row as what the cell is, and any row after it as how far its numbers would have to bend to be something more symmetric. A higher symmetry appearing just above your tolerance is the warning worth having: it means the answer above depends on where you drew the line, which is how a structure ends up refined in the wrong space group.

Lattice Holohedry Operations Needs ε i.e. edges within and angles within
cubic reached m3m 48 exact — —

Rows marked reached are the ones your tolerance of 0.001 already accepts; the last of them is the answer at the top of this page.

A small threshold is not evidence that the higher symmetry is real. It says the six numbers are close to it and nothing more — the atoms decide, and they are not on this page. Nor is a large one evidence against: a cell measured at low resolution can sit further from its own true symmetry than a good measurement of a genuinely distorted one.

Twinning on part of the lattice

A reticular twin shares only part of its lattice. The twin operation maps a SUBLATTICE onto itself rather than the whole lattice, so one reflection in n is common to both orientations and the rest are separate — n is the twin index. That is the case the two panels above cannot see: there n is 1 and every reflection coincides.

96 operations turned up across the 178 sublattices searched, and they describe 10 twin laws: all 10 hold exactly, with nothing bending.

The index is exact and is the whole of the diagnosis: at index 3 one reflection in three is shared, so two thirds of the pattern belongs to one orientation or the other. The threshold is how far this cell has to bend before the sublattice accepts the operation — zero where it holds outright, and the same quantity the ladder above this panel is measured in.

Twin laws by reticular merohedry CSV
Twin index Reflections shared Threshold ε Twin operation Equivalent operations
3 1 in 3 exact two-fold rotation about [120] — equivalently the reflection in the plane perpendicular to it 12
3 1 in 3 exact two-fold rotation about [102] — equivalently the reflection in the plane perpendicular to it 12
3 1 in 3 exact two-fold rotation about [012] — equivalently the reflection in the plane perpendicular to it 12
3 1 in 3 exact two-fold rotation about [111] — equivalently the reflection in the plane perpendicular to it 12
5 1 in 5 exact two-fold rotation about [113] — equivalently the reflection in the plane perpendicular to it 8
5 1 in 5 exact two-fold rotation about [131] — equivalently the reflection in the plane perpendicular to it 8
5 1 in 5 exact two-fold rotation about [112] — equivalently the reflection in the plane perpendicular to it 8
5 1 in 5 exact two-fold rotation about [121] — equivalently the reflection in the plane perpendicular to it 8
5 1 in 5 exact reflection in (431) — equivalently the two-fold about its normal 8
5 1 in 5 exact reflection in (413) — equivalently the two-fold about its normal 8

Every sublattice of index up to 6 is searched, so the answer is complete to that index; nothing about the direction of the axis or the indices of the plane is bounded. Beyond it a twin shares less than one reflection in six, which the measured intensities separate without help. Directions are given in the cell you typed — give a primitive cell and the axes are indexed on that cell, not on the conventional one.

As above, every law here is an opportunity rather than an observation: it says the lattice would let the crystal twin this way, not that it has.

Where this comes from