The centring survives the limit

Every list of the fourteen Bravais lattices is arranged by crystal system, from triclinic at one end to cubic at the other, and the arrangement suggests a story about deformation. A triclinic cell whose angles happen to be right angles is orthorhombic; an orthorhombic cell with two equal edges is tetragonal; a tetragonal one with all three equal is cubic. Read that way the systems form a chain, and a crystal whose cell drifts towards higher symmetry climbs it.

The chain is accurate about the systems and silent about centring, and the centring is where the story goes wrong. A body-centred tetragonal lattice whose axial ratio runs to one becomes body-centred cubic; run the same ratio to the square root of two and it becomes face-centred cubic. A primitive tetragonal lattice whose ratio runs to one becomes primitive cubic, and no deformation of a primitive tetragonal lattice ever arrives at either of the centred cubic ones. The tetragonal step above the cube is not one step. It is two, and which of them leads where is fixed.

This essay computes the whole relation: for each of the fourteen lattices, the types whose lattices come arbitrarily close to it. The computation needs nothing beyond what seventy-three, without a search already built, and its answer is a diagram of nineteen edges that no chain of systems can draw.

What it means for one lattice to be a limit of another type #

The convention first, because the result depends on it. Two lattices are close when they have bases whose Gram matrices, the tables of dot products between basis vectors, are close entry by entry. A lattice of type $A$ is a limit of type $B$ when lattices of type $B$ come as close to it as anybody likes. The lattices are taken up to rotation and scale, and only lattices proper are allowed: a sequence of cells that flattens to a plane has no limit among lattices, and is not counted.

Every type is trivially a limit of itself. What is interesting is the set of other types that reach a given one. A limit can only gain symmetry, never lose it, because a symmetry that holds for every lattice in a convergent sequence holds for their limit as well. So the relation runs upward, from lower symmetry to higher, and the question is which upward steps exist.

The obvious guess is the chain: a type is a limit of every type whose point group is a subgroup of its own. The guess is wrong, and the reason it is wrong is the content of this essay. The point group is a group of rotations and reflections of space. A lattice’s group is a group of integer matrices on a basis of the lattice, and two lattices with the same point group can carry it as integer matrices in different ways. The way they carry it is a discrete datum, and a discrete datum cannot change continuously.

The rule, and why it is finite #

The rule that settles every case is short. A lattice $L$ is a limit of type $B$ exactly when some subgroup of $L$’s own group has a Bravais group of type $B$.

The Bravais group of a finite group of integer matrices is the object seventy-three, without a search turned on. Take the quadratic forms the group fixes, which make a linear space; take a generic form in that space; ask which integer matrices preserve it. The answer contains the group, depends only on it, and is the group of one of the fourteen lattice types: the most general lattice the group can act on.

One direction of the rule follows directly from that. If a subgroup $S$ of $L$’s group has Bravais group of type $B$, then $L$’s own metric is one of the forms $S$ fixes, so it lies in $S$’s space of invariant forms. The generic forms of that space are dense in it, and every generic form is a lattice of type $B$. Lattices of type $B$ therefore come as close to $L$ as anybody likes.

The other direction is where integers matter. Suppose lattices of type $B$ converge on $L$. Choose bases converging to a basis of $L$; the integer matrices preserving a metric near $L$’s have bounded entries, so there are finitely many candidates, and any one that preserves infinitely many of the approaching metrics preserves $L$’s metric too. Passing to a subsequence, the approaching lattices all have the same group $H$, written on converging bases, and $H$ sits inside $L$’s group. Since $H$ is the full group of lattices of type $B$, it is its own Bravais group. That is the subgroup the rule asks for.

So the relation is computed from finite data. The fourteen lattice groups have between them five hundred and ten subgroups, thirty-two from fourteen matrices found all of them, and each needs one Bravais group and one name.

Naming a type without deciding conjugacy #

The name is the step with a trap in it. A Bravais group comes out of the computation as a set of integer matrices on some basis, and deciding which of the fourteen it is means deciding whether it is conjugate, by an integer change of basis, to one of fourteen representatives. That is the search the previous essay was built to avoid, and it is avoidable again.

The order of the group separates the systems: two elements for triclinic, four for monoclinic, eight for orthorhombic, sixteen for tetragonal, twelve for rhombohedral, twenty-four for hexagonal, forty-eight for cubic. It does not separate the centrings, since the four orthorhombic types all have order eight and the three cubic types all have forty-eight. What separates them is arithmetic the matrices carry with them. Collect the lattice vectors that lie on the group’s rotation axes, and ask what index the sublattice they span has in the whole lattice. Do the same with the vectors lying in its mirror planes.

For the orthorhombic types the axes’ vectors span the whole lattice when it is primitive, half of it when it is C-centred or body-centred, and a quarter when it is face-centred; the mirrors’ vectors span the whole lattice except when it is body-centred, where they span half. That gives four different pairs of indices for four types. For the two tetragonal types the two-fold axes alone decide it, index one or two. For the three cubic types the four-fold axes do, with index one, two and four.

Every one of those indices is a conjugation invariant. A change of basis carries an axis to an axis and a mirror to a mirror, and a sublattice of index two to a sublattice of index two. So the indices name a type without any conjugacy being decided, and they are checked rather than trusted: each of the fourteen representatives is named as itself, and each centred type’s group, conjugated by a randomly chosen integer change of basis, is still named correctly.

The plane first #

The plane has five lattice types, and its version of the relation fits on one line.

The square is a limit of both rectangular types. A rectangle whose sides become equal is a square, and so is a rhombus whose angle becomes a right angle, since a rhombus is a centred rectangle drawn on its short vectors. The hexagonal lattice is a limit of rhombi too, as a rhombus whose angle becomes sixty degrees. It is not a limit of rectangles.

The reason is visible in the hexagonal net itself. It has six mirror lines, and the lattice vectors along any two perpendicular ones span a sublattice of index two: with unit spacing, the shortest vector along one mirror has length one, the shortest along the perpendicular mirror has length $\sqrt{3}$, and the rectangle they make has area $\sqrt{3}$ against a cell area of $\sqrt{3}/2$. Every rectangle drawn on a hexagonal net is a centred one. A sequence of primitive rectangular lattices converging on the hexagonal lattice would need a primitive rectangle in the limit, and there is none.

That is the whole mechanism, in its smallest case. Both rectangular types carry the same point group, $mm$. They carry it as different integer matrices, and the difference is an index, one or two. A limit keeps it.

The fourteen #

In space the same computation runs over five hundred and ten subgroups.

The table can be read row by row. The triclinic lattice’s group has two subgroups, the identity and the inversion, and both have the triclinic Bravais group, so the triclinic lattice is a limit of nothing but itself. Primitive monoclinic’s five subgroups give two triclinic and three monoclinic primitive. Primitive cubic’s ninety-eight give two triclinic, nine primitive monoclinic, eighteen C-centred monoclinic, five primitive orthorhombic, fifteen C-centred orthorhombic, twenty-four primitive tetragonal, twenty rhombohedral, and five cubic: eight types in all.

The three cubic rows are the clearest place to see the centring. Each has twenty-four subgroups with a tetragonal Bravais group, which are the tetragonal point groups placed along each of the three axes. In primitive cubic all twenty-four are primitive tetragonal. In body-centred and face-centred cubic all twenty-four are body-centred tetragonal. There is no mixture in any row. The same subgroups, defined by the same rotations about the same axes, sit in the three cubic lattices as integer matrices of two different kinds.

The index explains it in a sentence. In a body-centred cubic lattice, the lattice vectors along the three cube axes and along the face diagonals span only the cube corners, half the lattice; the body centres lie on none of those axes. In a face-centred cubic lattice the face diagonals carry the face centres, but the cube axes and the face diagonals together still miss half the points, in the arrangement a body-centred tetragonal cell, turned forty-five degrees, describes. In primitive cubic they span everything. A tetragonal lattice keeps whatever index its axes have as it is deformed, so a primitive tetragonal lattice cannot arrive at a lattice whose tetragonal axes span half of it.

The same reading accounts for the other surprises in the table. Rhombohedral is a limit of triclinic and of C-centred monoclinic and of nothing else: none of its subgroups has an orthorhombic Bravais group, because its two-fold axes are three, in one plane, at sixty degrees to each other, and no two are perpendicular. The hexagonal lattice is a limit of C-centred orthorhombic and not of primitive orthorhombic, which is the plane’s fact about rectangles stacked into space. And primitive monoclinic, which a reader might expect to sit beneath every lattice with a two-fold axis, sits beneath only six of the fourteen: itself, primitive orthorhombic, C-centred orthorhombic, primitive tetragonal, hexagonal and primitive cubic. Eleven of the fourteen are limits of C-centred monoclinic.

Nineteen edges #

Drawing only the steps with nothing between them turns the table into a diagram.

There are forty-five ordered pairs of distinct types in the relation and nineteen covering edges, and the relation passes the two checks any closure relation must: it is transitive, so a limit of limits of a type is a limit of that type, and no two different types are limits of each other.

Several things in the diagram are invisible on a list of systems. The cubic lattices are reached by two roads each, and the roads differ. Primitive cubic sits above primitive tetragonal and above rhombohedral. Body-centred and face-centred cubic both sit above body-centred tetragonal and above rhombohedral, and their sets of predecessors are identical, all seven of them. Rhombohedral reaches all three cubic lattices, because a rhombohedral cell whose angle passes through sixty, ninety and $109.47$ degrees passes through face-centred, primitive and body-centred cubic in turn.

The hexagonal lattice has a single way in, from C-centred orthorhombic. It is not a limit of rhombohedral lattices and rhombohedral lattices are not limits of it: the two types share a crystal family and nothing else, as a lattice described on somebody else’s axes found when the rhombohedral centring of a hexagonal cell turned out to be a different lattice rather than a smaller symmetry of the same one.

Primitive tetragonal is a limit of C-centred orthorhombic, through a square net drawn as a centred rectangle, while body-centred tetragonal is a limit only of the body-centred and face-centred orthorhombic types. The two tetragonal types share no predecessor above the monoclinic.

Two paths a metallurgist knows #

The diagram’s statements about cubic lattices are not abstractions. Two of its edges have been followed by real metals for a century.

In 1924 Edgar Bain proposed that the face-centred cubic structure of austenite could turn into the body-centred structure of martensite by a homogeneous strain: take a body-centred tetragonal cell inside the face-centred cubic lattice, whose axial ratio is $\sqrt{2}$, and compress it along its four-fold axis until the ratio is one. The left half of the figure is that path, with the group order computed at every step rather than taken on trust. It sits at sixteen all the way, and rises to forty-eight exactly at the two ends.

The diagram adds something to Bain’s observation that the observation itself does not say. Any continuous path between the two centred cubic lattices that keeps a four-fold axis the whole way must run through body-centred tetragonal lattices, since those are the only tetragonal lattices of which either centred cubic lattice is a limit. A path through primitive tetragonal lattices would have to change an integer on the way. The alternative road the diagram allows is rhombohedral, the right half of the figure, which is the trigonal deformation path discussed alongside Bain’s: a rhombohedral cell sheared along its three-fold axis passes from face-centred cubic at sixty degrees through primitive cubic at ninety to body-centred cubic at $109.47$, keeping twelve symmetries in between.

What the paths do not say is which of them a real transformation takes, or whether it takes either. A martensitic transformation involves shuffles of atoms within the cell, interfaces between transformed and untransformed regions, and an invariant plane, none of which a lattice of points knows about. The diagram constrains the homogeneous part of the strain and nothing else. It says which symmetry a path between two cubic structures must pass through if it keeps a given axis, and it is silent on energies.

A symmetry nothing in the computation knew about #

A computation like this is easy to get subtly wrong. A Bravais group named with the wrong centring would add or remove an edge, and the diagram would still be transitive and still look plausible. So it is checked against a fact that has nothing to do with how it was computed.

Every lattice has a reciprocal lattice, whose Gram matrix is the inverse of the original’s on the dual basis, and whose group is the original group transposed and inverted. Taking reciprocals is continuous, so a limit of reciprocals is the reciprocal of the limit, and the map sending each type to the type of its reciprocal must carry the relation onto itself.

The reciprocal of a body-centred lattice is face-centred and the other way round, in both the orthorhombic and the cubic systems; every other type is its own reciprocal’s type. The computation finds exactly that, by naming the group of each inverted Gram matrix, and then checks the relation pair by pair. It survives. That is why the two centred cubic lattices have identical predecessors: the reciprocal exchanges them and fixes everything below them, so whatever reaches one must reach the other. A naming error that confused body-centred with face-centred orthorhombic in one row and not in its dual would have broken the symmetry, and nothing in the table would have hinted at it otherwise.

What the picture cannot show #

The diagram is a relation between types, and a type is a set of lattices with a continuum of parameters. An edge says that some sequence of lattices of the lower type converges to each lattice of the higher one. It does not say how many such sequences there are, how many parameters they have, or how close a given measured cell is to any of them. A cubic cell measured with a small error is close to many lower types at once, and a cubic cell has fourteen chances to fail is about what happens when each of those proximities is tested statistically. That essay’s walk ran over the strata a primitive cubic cell sits on and said that a centred cubic cell would have other strata above it. The diagram here says which: body-centred tetragonal where the primitive cell had primitive tetragonal, and the body-centred and face-centred orthorhombic types where it had the primitive and C-centred ones.

Nor does the diagram distinguish ways of approaching. A tetragonal lattice can approach a cubic one along any of three axes, and the twenty-four tetragonal subgroups of each cubic group are those three directions times eight tetragonal point groups each. The relation records only that one of them exists.

Who found it and what is computed here #

The arrangement of the fourteen types by specialisation is a classical part of lattice geometry. Louis Michel and his collaborators studied the space of lattices as a stratified set, each stratum the lattices of one Bravais type, and drew the incidence of those strata; the treatment in Engel, Michel and Senechal’s monograph on lattice geometry is the standard modern one. Diagrams of the same shape appear in discussions of going up costs the cell a parameter and in the literature on lattice-distortive transformations, where the Bain and trigonal paths are the usual examples.

What this essay adds is a derivation from integer matrices alone. The fourteen groups come from Gram matrices with generic parameters, as in twenty-five cells, and fourteen lattices; their five hundred and ten subgroups come from closing small generating sets; each subgroup’s Bravais group comes from a generic invariant form; and each Bravais group is named by the indices its axes and mirrors span. No table of the relation was consulted, and the reciprocal lattice is the independent check.

The enumeration that produced each count in this essay is that pipeline. The forty-five pairs and nineteen edges are counted on its output, and the subgroup tallies in the table are its intermediate results.

Five tests and one refusal #

The first test conjugates six centred types’ groups by random integer changes of basis and requires their names to survive, which is what makes the naming a function of the type rather than of the basis it happened to be written on. The second requires transitivity and antisymmetry. The third and fourth state the cubic results directly, with the counts that produce them. The fifth is the reciprocal lattice.

The refusal is the claim this essay began by distrusting. Asked at the level of point groups, the question has the wrong answer: primitive cubic contains $4/mmm$, body-centred tetragonal lattices have point group $4/mmm$, and the order of a tetragonal Bravais group is sixteen whichever centring it has. A test that named Bravais groups by order alone would let primitive cubic be reached from body-centred tetragonal and would add an edge that does not exist. All twenty-four tetragonal subgroups of primitive cubic’s group have index one along their axes, and none of them is body-centred.

Still open: four dimensions, where the diagram has sixty-four nodes #

The method uses nothing specific to three dimensions. A four-dimensional lattice has one of sixty-four Bravais types, and the limit relation among them is decided by the same rule: subgroups of each type’s group, their Bravais groups, and a name for each. The two obstacles are size and naming. The four-dimensional lattice groups are much larger, the largest of order 1,152, and their subgroups number in the thousands. And the indices that named the fourteen here may not separate the sixty-four, since four dimensions has lattice groups whose rotations fix no line at all, as the twelves found.

What would be interesting there is whether the reciprocal lattice is still the only symmetry of the diagram. In three dimensions it exchanges two pairs. In four dimensions the self-dual types are more varied and some lattices, the $D_4$ root lattice among them, are similar to their own reciprocal by a rotation and a change of scale that no symmetry of the lattice supplies, which could give the diagram a symmetry that no three-dimensional type has.