Symmetry of crystals

Symmetry of crystals

Here we are going to summarize the symmetry of crystals, including the translational symmetry, rotational symmetry, the combination of them, and a lot of terminologies.

Translational symmetry:

Translational symmetry is the most fundamental symmetry of a crystal, or the definition of a crystal. In principle, a crystal may not have rotational symmetry, but will always have translational symmetry.

Translational symmetry is described by the primitive lattice, or the basis a1,a2,a3. However, they are not unique. One has may ways to choose a1,a2,a3, because by definition, as long as all the points in the lattice can be reproduced by n1a1+n2a2+n3a3, you have the basis. However, there are some rules for chosen primitive axis to unify the usage of them. (e.g. a1=i, a2=j, and a3=k).

Note that we want to avoid the term sc, bcc, fcc and etc. here because they contain information more than just translational symmetry. Let us emphasize that translational symmetry means only primitive axis. To exagerate, for every point on the lattice, we can put a potato. It will still be a lattice as long as the potato have the same orientations.

Primitive cell and unit cell

Primitive cells are the cells that has the smallest repeating
volume. The axes are defined by a1,a2 and a3. As mentioned above, the choice of primitive cells is infinite. Most of them do not reflect the symmetry of the lattice. For example, for fcc a1=(i+j)/2,  a2=(k+j)/2, a1=(i+k)/2. This will generate a primitive cell that has no obvious symmetry.

It is always nice to choose a primitive cell to reflect the symmetry of the lattice. So called W-S cell is to do that. W-S cell has all the symmetry of the lattice.

Sometimes, for simplicity, people also choose unit cell that's larger than the primitive cell, just to retain the symmetry. For example, choose the cube for bcc.

Rotational symmetry:

If we talk about rotational symmetry alone, there are infinite number of rotational symmetry, for example, Cn, where n is an integer. However, since we are talking about crystal, we should focus on the rotational symmetries that are only allowed in crystals. (Note that, here, introduction of rotational symmetry alreay implies the coupling between translation and rotational symmetry.) It is well know only n=1,2,3,4,6 is valid for Cn in a crystal.

We have to keep in mind that this is only the rotational symmetry of the lattice (not the crystal). Only when the crystal have simple lattice, we have have the same rotational symmetry for the crystal too.

Lattice system and Bravais lattice

Continuing the discussion of rotational symmetry, we said n can only be 1,2,3,4,6. This combined with the translational symmetry, will only result 7 lattice systems and 14 Bravais lattices. Just to list them, they are

Triclinic
Monoclinic (s, bc 2)
Orthorhombic (s,bc,bc1,fc 4)
Tetragonal (s, bc)
Cubic (sc, bcc, fcc 3)
Trigonal
Hexagonal

Symmetry group (macroscopic symmetry)

So far, we just talked about the microscopic view of a crystal. Next, we need to talk about macroscopic symmetry, e.g. the symmetry of the physical properties such as dielectric constant.

This has to do with the complexity of the lattice point.

Let's define:

simple lattice: lattice points are spheres (isotropic)
complex lattice: lattice points have structures (e.g. a potato)

To emphasize again, when we talk about the rotational lattice, we did not care about the structure of every lattice point. Here we have to because we care about the rotational symmetry now.

Obviously, for the same lattice, the macroscopic symmetry is very different, depending on if you put a sphere on lattice points or if you put a potato. For example, for sc lattice, simple lattice gives cubic macroscopic symmetry (O_h), complex lattice (potato) gives C1 macroscopic symmetry.

A more academic example is the comparison between diamond and ZnS. They both have fcc lattice. But diamond has O_h macroscopic symmetry and ZnS has T_d.

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Space group

Another effect of coupling the translational and rotational symmetry is the space group. When the unit cell has several atom of the same element, there can be screw axis, glide plane which are combination of point group operator and translations.

For example, for BiFeO3. If we don not consider the glide plane, the point group symmetry is C₃. With glide plane, the point group will be C_3v. the space group is R_3c ={{C₃|R}{C_3v|τ+R}}.

Reciprocal space and Brillouvin zone.

The reciprocal lattice can be found by the definition:

b_i=2πa_jXa_k/V, where V is the unit cell volume.

The Brillouvin zone is chosen to be the W-S primitive cell of the reciprocal space.

The symmetry of wave functions