Lattice and Cubic Unit Cells | Symmetry of Crystals

Symmetry of Crystals
A symmetry element is a geometrical object such as a point, a line, or a plane that some symmetry operation(s) can be carried out with respect to.

A symmetry operation is an action, such as a rotation or a reflection of an object, that after it is completed, the product of this operation and the original object are indistinguishable.

The following is a description of some common symmetry operations.

Symmetry Elements and Operations:
Label Symmetry Element Symmetry Operation Description
E Identity Zero Nothing changes.
i Center of symmetry or inverted center Inversion Projects the object through the center (inverts about the center)
Cn n-fold improper axis of rotation Rotation Rotates (360/n)° in the clock-wise or anticlockwise direction about the axis
σ Mirror plane Reflection Reflects across a plane
Sn n-fold improper axis of rotation with a plane of reflection Rotation followed by a reflection Rotates (360/n)° in the clockwise or anticlockwise direction about the axis followed by a reflection across a plane perpendicular to the rotation axis

A tetrahedral structure has the following symmetries:

Identity (E): Each structure is equivalent to itself, so it always has an identity.

Rotation Symmetry (Cn): A rotation about the axis by (360/n)°, where n is an integer. If we say that a molecule has symmetry Cn about a given axis, we mean that n equal rotations of (360/n)° each about that axis will get us back to where we started, and that each of the n rotations will leave the molecule in a position identical to its starting position.

[ Download full size video (avi/4.03MB) ] C2 is a rotation on an axis (shown in purple) by 180°. A tetrahedral structure, which is represented by the four gold atoms extending out from the blue central atom with tetrahedral bonds, has three C2 axes perpendicular to each other (3C2).
C3 is a rotation by 120°. A tetrahedral structure has four 3-fold rotation axes (C3) along each of the four tetrahedral bonds, and these four axes are also 3-fold rotation axes in the opposite direction (C3-1, the inverse of C31). A tetrahedral has 8C3 symmetry operations, a clockwise and a counter-clockwise along each of the four tetrahedral bonds. [ Download full size video (avi/3.83MB) ]
[ Download full size video (avi/3.81MB) ] σ is the mirror (reflection) plane (see animation, the dark grey plane is the mirror plane). A tetrahedral structure has six mirror reflection planes (6σ), each passing the center atom and two of the other four atoms (take two from four, and there are six different combinations).
The symmetry of Sn is composed of two symmetry operations or we say it is the product of two symmetry operations:
  • Rotations by (360/n)° radians about an axis, and
  • Reflection through a plane perpendicular to the axis.
S4 is a rotation by 90° followed by a mirror reflection (see animation). A tetrahedral structure has three 4-fold rotation axes followed by a reflection plane (S4), and these 4-fold rotation axes are the same as the three 2-fold rotation axes discussed above. Each of the three axes is also a 4-fold rotation axes in the opposite direction followed by a reflection plane (S4-1, the inverse of S4). So the total number of S4 operation is 6. If you operate S4 twice, it will be the same as a C2 operation. 		[ Download full size video (avi/6.94MB) ]

In summary, a tetrahedral Structure has the following 24 symmetry operations: 1 E, 3C2, 8C3 (= 4C3 + 4C2-1), 6σ, and 6 S4 (= 3S4+ 3S4-1).