2.3: Compatibility of Rotations and Lattices

Although it may seem intuitive that any rotational symmetry of a lattice must occur at every lattice point, we can prove this conclusion using the accompanying sketch. Let x be the general location perpendicular to a rotation axis by angle \(\alpha\) and let there be a lattice point at r from x. The notion of rotational symmetry means that there will be another lattice point at \(C_{\alpha}\boldsymbol{r}\).

Figure 2.18: Rotation of vector r by C subscript alpha yielding new lattice point.

Now, these two lattice points must be related by a lattice vector \(\boldsymbol{T}_{mnp}\), so that the following relationship emerges:

\[C_{\alpha}\boldsymbol{r} = \boldsymbol{r} + \boldsymbol{T}_{mnp} \nonumber \]

If \(\textbf{r}\) is a lattice vector, then \(C_{\alpha}\boldsymbol{r}\) must also be a lattice vector, and this rotational symmetry exists at every lattice point.

What rotations \(\alpha\) are compatible with translational periodicity?

Figure 2.19: Geometrical derivation of allowed rotations on a lattice. Read accompanying text.

To answer this question, we apply a geometrical argument. Two adjacent lattice points separated by vector T are part of a 1-d sublattice of points along this line. Now, at the lattice point designated “1”, consider a counterclockwise rotation by angle α to create the new lattice point 1′ off this line. If this rotation is a symmetry operation of the lattice, there must also be the corresponding inverse or clockwise rotation. By applying the clockwise rotation at the neighboring lattice point 2, the new lattice point 2′ is generated, which creates a segment 1′⋅⋅⋅2′ that is parallel to the original line. For this new segment to be part of a 2-d lattice, the vector connecting points 1′ and 2′ must be an integer multiple of T. Using the geometry of the construction:

\[1′⋅⋅⋅2′ = \(N\boldsymbol{T} = \boldsymbol{T} + 2\boldsymbol{T}\ \cos\alpha = (1 + 2\cos\alpha)\boldsymbol{T} \nonumber \]

so that \(\left( 1 + 2\cos\alpha \right) =\) integer.

This outcome restricts the allowed values of α to 180°, 120°, 90°, 60°, and 0°. Therefore, the allowed rotational symmetries of any lattice include 2-fold C₂, 3-fold C₃, 4-fold C₄, and 6-fold C₆ rotations and no others. The absence of rotational symmetry (C₁) is always allowed.

In 3-d, these rotational symmetries can be combined with translational periodicity to create seven crystal systems that give seven distinct unit cell shapes:

Figure \(\PageIndex{1}\): Unit cell (parallelepiped) showing a-, b-, and c-directions. (1) α: between b and c, (2) β: between a and c, and (3) γ: between a and b

Table: seven crystal systems that give seven distinct unit cell shapes

Crystal System	Lattice Symmetry		Unit Cell Shape
Triclinic	\mathcal{C}_{i} – no C_{n} axes (C_{1} only)		a ≠ b ≠ c; α ≠ β ≠ γ
Monoclinic	\mathcal{C}_{2v} – one C_{2} axis (|| b)		a ≠ b ≠ c; α = γ = 90°, β ≠ 90°
Orthorhombic	\mathcal{D}_{2h} – three ⊥ C_{2} axes		a ≠ b ≠ c; α = β = γ = 90°
Tetragonal	\mathcal{D}_{4h} – one C_{4} axis (|| c)		a = b ≠ c; α = β = γ = 90°
Cubic	\mathcal{O}_{h} – four C_{3} axes		a = b = c; α = β = γ = 90°
Trigonal	\mathcal{D}_{3d} – one C_{3} axis	(|| [111]) (|| c)	a = b = c; α = β = γ a = b ≠ c; α = β = 90°, γ = 120°
Hexagonal	\mathcal{D}_{6h} – one C_{6} axis (|| c)		a = b ≠ c; α = β = 90°, γ = 120°

The rotational symmetry of a crystal system restricts the relative lengths and angles of the unit cell parameters. In the table, the symbol “≠” means that there is no restriction for the two quantities to be equal, i.e., they may be equal, but not for any symmetry constraints. Important features of the unit cell shapes include:

• the trigonal system offers two distinct shapes, one of which is identical to the shape of the hexagonal system, so that these two lattices are the same;
• the cubic and one of the trigonal unit cells are closely related to each other because the three cell lengths are equal and the three angles are also equal, but they are exactly 90° for the cubic cell; and
• the orthorhombic, tetragonal, and cubic unit cell shapes have mutually perpendicular lattice vectors, and increasing restrictions on their relative lengths.