What Are the Key Symmetries and Degeneracies in Lattice Structures?

[Niles]

Hi all

What symmetries are there usually in a lattice? Let us say for example that I look at a lattice having the form (each "x" is an atom)

x x x x x x x x x
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x


Of course there is translational symmetry (assuming homogeneous system). But what other symmetries are usually responsible for degeneracies?

 

[kanato]

If this is a 2D square lattice, you have inversion, three reflection symmetries, and a four fold rotation symmetry. Say if each atom has p orbitals on it, then px and py will be equivalent, because you have at least one symmetry operation that maps px onto py. In this case it's the 4 fold rotation, or the reflection through the [110] axis. This will make px and py degenerate.

 

[Niles]

The three reflections are longitudinal, horizontal and diagonal, right?
The four rotations are longitudinal, horizontal, diagonal and what is the fourth?

I can't see what you mean by inversion - isn't that included in the reflections?

 

[kanato]

A 4 fold rotation axis means you have an axis where you can rotate the system in steps of 360/4 degrees. Translational symmetry is compatible only with 2, 3, 4, and 6 fold rotation axes.

You are correct about the three reflection planes. And inversion in this case is the composite of the horizontal and vertical reflections. But they are distinct symmetry operations, and not every crystal which has inversion will have the other reflections.

 

[Niles]

You are correct about the three reflection planes.

By the way, aren't there four? Two diagonals, one vertical and one horizontal.

 

[kanato]

Yes, that's right, there should be four.