Crystal Symmetry

A primitive cubic unit cell with 8 corner atoms. (Created using CrystalMaker software.)

In undergraduate mechanical engineering program, students start connecting macroscopic concepts - such as temperature, pressure, stress - they learn from solid mechanics, fluid mechanics and thermodynamics with microscopic objects - bonding energy, crystallography, dislocations - when they study materials science.

One challenging topic in materials science is crystallography. It has to do with how we characterize and classify arrangements of atoms in three-dimensional space. It is mathematically abstract. We classify different atoms by paying attention to the symmetries of these arrangements.

Symmetry is not a thing, it is a process done to these atomic arrangement. The process changes an arrangement of these atoms to another arrangement without changing how these atoms relate to each other in space.

To study symmetry of atoms in a solid, we imagine the atoms to be arranged in a regular three-dimensional pattern. Because these atoms are placed periodically, we can come up with a building block, a unit volume that can be cloned as many times as required to create a solid of any size we want. This building block is called a unit cell. We call a solid that has a periodic arrangement of atoms a crystal.

One periodic way to arrange atoms in a crystal is to place them at 8 corners of a cubic unit cell. Each corner atom contributes 1/8 of an atom, so one cubic unit cell is occupied by 1 atom. We need to imagine we have other unit cells surrounding the first unit cell forming a crystal filled with atoms. For illustration, however, I just show one unit cell with 8 corner atoms (shown above). I am going to use this cubic unit cell to explain the crystal symmetry idea.

A cubic unit cell viewed squarely onto one of its 6 faces.

The picture above shows the cubic unit cell viewed squarely onto one of its 6 faces. One process that can change the arrangement shown is to rotate it by 90 degree about an axis at the center of the face. We get a new arrangement but this arrangement is identical to the before-rotation arrangement. We thus say that the cubic unit cell obeys the 90-degree-rotation symmetry. We don't call the 180-degree rotation a symmetry of the cubic unit cell since it can be reproduced by two 90-degree rotations. We don't call the 360-degree-rotation a symmetry of the cubic unit cell since it is trivial.

The 6 faces of the cubic unit cell can be divided into 3 groups, and each group has 2 faces. We can perform the 90-degree operation to each face group. So there are total 3 90-degree rotation symmetries.

Another symmetry that the cubic unit cell has is a mirror symmetry. The square face above has four mirror planes. One mirror plane exchanges the 2 left corner atoms with the 2 right corner atoms. Another mirror plane exchanges the 2 top corner atoms with the 2 bottom corner atoms. The other two mirror planes exchange the corner atoms face diagonally. Since there are those 3 groups of faces, there are thus a total of 12 mirror symmetries.

A cubic unit cell viewed along the body diagonal direction.

The cubic unit cell also has rotation and mirror symmetries operating on an axis running body diagonally as shown above. It has a 120-degree rotation symmetry. There are 3 body diagonal axes in the cubic unit cell, so there are 3 120-degree rotation symmetries.

There are also 3 mirror symmetries for each body diagonal axis, so there are 9 mirror symmetries.

A cubic unit cell viewed from the face diagonal direction.

Along the face diagonal direction, the cubic unit cell looks like above. We can rotate this picture 180-degree and the arrangement of atoms does not change. There are 2 axes of this 180-degree rotation for each face group. There are thus 6 180-degree rotation along its face diagonal axes.

There is 1 inversion symmetry about the center point of the cubic unit cell. This inversion symmetry exchanges one corner atom with another one body diagonally away from each other.

The total symmetry operations the cubic unit cell has is therefore

a) 3 90-degree rotation on its faces
b) 12 mirror symmetry operations on its faces
c) 3 120-degree rotation along its body diagonal axes.
d) 6 180-degree rotation along its face diagonal axes.
e) 1 inversion symmetry about the cubic center.

I so far identified 25 symmetry operations for the cubic unit cell. Are there others you can find?