2.6: Bravais Lattices (2-d)

The five 2-d crystal systems and the point groups of their corresponding lattices identify the four general shapes of unit cells that define real space primitive lattices, which are designated “p”. Now, are there any other distinct shapes for primitive cells that are consistent with the rotational symmetry for each of these crystal systems? There are different geometrical and arithmetical strategies to answer this question. We will use a geometrical procedure called lattice centering and examine each crystal system separately and systematically. The general idea is to add new points throughout the lattice at equivalent sites in every primitive cell that have the same point symmetry as the lattice. These sites for each crystal system were identified in the previous slide. Then, the important question posed by lattice centering is, does the new arrangement of points generated by the centering action create a new type of lattice (unit cell) in the same crystal system?

Figure 2.31: Oblique lattice with one shaded unit cell and the 2-fold rotation axes noted.

OBLIQUE LATTICES

The general primitive cell is a parallelepiped with two unrelated sides (a ≠ b) and an unrestricted angle (γ ≠ 90°). The lattice symmetry is \(2 = \mathcal{C}_{2}\). Sites at the edge and cell centers also have symmetry \(2\), so we examine both of these lattice centering possibilities.

Edge-Centering

Adding points to the centers of every horizontal edge does not create a new type of lattice, but just changes the length of one lattice constant. The new unit cell remains a parallelepiped with two unrelated sides and an unrestricted angle: a′ ≠ b′; γ′ ≠ 90°.

Figure 2.32: Oblique cell resulting from edge-centering.

Cell-Centering

Adding points to the centers of every cell also does not create a new type of lattice. Again, the unit cell is a parallelepiped with two unrelated sides and an unrestricted angle: a″ ≠ b″; γ″ ≠ 90°. Therefore, the 2-d oblique system has just a primitive (p) lattice.

Figure 2.33: Oblique cell resulting from cell-centering.

To determine whether the oblique system allows other lattices by applying group theory, identify the 2×2 matrices for each member of the lattice point symmetry group using the primitive cell vectors as the basis. Therefore,

\[2 = \mathcal{C}_{2} = \left\{ 1 = E = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix},\ 2 = C_{2} = \begin{pmatrix} - 1 & 0 \\ 0 & - 1 \\ \end{pmatrix} \right\}}\nonumber \]

These diagonal matrices are the same for each lattice centering possibility, so group theory also indicates just one type of 2-d oblique lattice.

RECTANGULAR LATTICES

The general primitive cell is a rectangle with two unrelated but perpendicular sides (a ≠ b; γ = 90°). The lattice point symmetry is \(2mm = \mathcal{C}_{2v}\). The sites at the edge and cell centers also have \(2mm\) symmetry, so, we examine lattice centering at these locations.

Figure 2.34: Rectangular lattice with one shaded unit cell and the 2-fold rotation axes and mirrors noted.

Edge-Centering

Adding points to the centers of every horizontal edge does not create a new type of lattice, but just changes the length of one lattice constant. The new unit cell is still a rectangle with two unrelated and perpendicular sides a′ ≠ b′; γ′ = 90°.

Figure 2.35: Rectangular cell resulting from edge-centering.

Cell-Centering

Adding points to the centers of every cell creates a rhombic primitive cell with two equal cell edges related to each other by mirror planes. The angle between cell edges is unrestricted. This primitive cell is distinctly different from the primitive rectangular cell and represents a new type of lattice: a″ = b″; γ″ ≠ 90°.

Figure 2.36: Rectangular cell resulting from edge-centering.

If the new primitive lattice vectors are combined, a″ + b″ = a = and –a″ + b″ = b, then a rectangular cell emerges (a ≠ b; γ = 90°) with two lattice points, one at the corners and one in the middle of the cell, and twice the area of the primitive cell. This new lattice is called centered rectangular and is designated “c”.

As a result, the 2-d rectangular system allows primitive (p) and centered (c) lattices.

Using group theory, the 2×2 matrices for each member of the lattice point symmetry group for the two different primitive cells are:

• p-lattice: \[2mm = \left\{ 1 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix},\ 2 = \begin{pmatrix} - 1 & 0 \\ 0 & - 1 \\ \end{pmatrix},\ m_{\boldsymbol{a}} = \begin{pmatrix} - 1 & 0 \\ 0 & 1 \\ \end{pmatrix},m_{\boldsymbol{b}} = \begin{pmatrix} 1 & 0 \\ 0 & - 1 \\ \end{pmatrix} \right\} \nonumber \]

• c-lattice: \[2mm = \left\{ 1 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix},\ 2 = \begin{pmatrix} - 1 & 0 \\ 0 & - 1 \\ \end{pmatrix},\ m_{\boldsymbol{a}} = \begin{pmatrix} 0 & - 1 \\ - 1 & 0 \\ \end{pmatrix},m_{\boldsymbol{b}} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix} \right\} \nonumber \]

The identity and inversion (2-fold) operations are the same diagonal matrices in both sets, but the matrices for the reflections are different. Since these matrices describe the same operations, they must be related by similarity transformations, i.e., for some matrix \(X\), \(Xm_{\boldsymbol{b}}(c)X^{- 1} = m_{\boldsymbol{b}}(p)\) or, written in another way: \(Xm_{\boldsymbol{b}}(c) = m_{\boldsymbol{b}}(p)X\). If these two sets of operations describe the same type of lattice, then matrix \(X\) must have rational matrix elements and must be orthogonal (\(X^{T} = X^{- 1}\) with det \(X\) = 1). So, let \(X\) be a general 2×2 orthogonal matrix, which can be expressed as \(\small{\begin{pmatrix} p & q \\ - q & p \\ \end{pmatrix}}\) with det X = p² + q² = 1. Then we must find solutions to the following equation:

\[Xm_{\boldsymbol{b}}(c) = \begin{pmatrix} p & q \\ - q & p \\ \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & - 1 \\ \end{pmatrix}\begin{pmatrix} p & q \\ - q & p \\ \end{pmatrix} = m_{\boldsymbol{b}}(p)X \nonumber \]

or

\[\begin{pmatrix} q & p \\ p & - q \\ \end{pmatrix} = \begin{pmatrix} p & q \\ q & - p \\ \end{pmatrix} \nonumber \]

The only possible solution is p = q = \(\frac{1}{\sqrt{2}}\), which is irrational. Therefore, there is no rational, orthogonal matrix \(X\) that relates the two sets by a similarity transformation, so these two lattices must be distinct in the rectangular crystal class.

TETRAGONAL LATTICES

The general primitive cell is a square with two equal, perpendicular sides (a = b; γ = 90°). The lattice point symmetry is \(4mm\). Only sites at the cell centers have the same \(4mm\) point symmetry. Edge-centering destroys tetragonal symmetry and is not a valid centering option.

Figure 2.37: Tetragonal lattice with one shaded unit cell and the 4-fold rotation axes noted.

Cell Centering

Adding points to the centers of every cell generates a lattice with a cell that is also a square, with two equal and perpendicular sides, so this centering option does not generate a new type of lattice. The 2-d tetragonal system has just a primitive (p) lattice.

Figure 2.38: Tetragonal cell resulting from cell-centering.

Using group theory, the 2×2 matrices for each member of the lattice point symmetry is:

\[\small{4mm = \begin{Bmatrix} 1 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}, & 4 = \begin{pmatrix} 0 & - 1 \\ 1 & 0 \\ \end{pmatrix}, & 4^{2} = 2 = \begin{pmatrix} - 1 & 0 \\ 0 & - 1 \\ \end{pmatrix}, & 4^{3} = \begin{pmatrix} 0 & 1 \\ - 1 & 0 \\ \end{pmatrix}, \\ m_{\boldsymbol{a}} = \begin{pmatrix} - 1 & 0 \\ 0 & 1 \\ \end{pmatrix}, & m_{\boldsymbol{b}} = \begin{pmatrix} 1 & 0 \\ 0 & - 1 \\ \end{pmatrix}, & m_{\boldsymbol{a} + \boldsymbol{b}} = \begin{pmatrix} 0 & - 1 \\ - 1 & 0 \\ \end{pmatrix}, & m_{\boldsymbol{a} - \boldsymbol{b}} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix} \\ \end{Bmatrix}}. \nonumber \]

After cell-centering, the same set of matrices occurs, which confirms that the tetragonal 2-d system has just the primitive lattice.

TRIGONAL LATTICES

The general primitive cell is a rhombus with two equal sides at an interior angle of 120° (a = b; γ = 120°). The point symmetry at lattice points is \(3m\). Two interior sites have the same \(3m\) point symmetry.

Figure 2.39: Trigonal lattice with one shaded unit cell and the 3-fold rotation axes noted.

So, edge-centering will destroy trigonal symmetry and is not a valid centering option. If we add points at one set of interior sites (below left), the resulting network is not a lattice. If we add points at both sets (below right), then the resulting primitive cell is a rhombus (a″ = b″; γ″ = 120°) and it is not a new lattice.

Figure 2.40: Trigonal lattice with 3-fold rotation axes noted and one set of interior points included - this is not a lattice.

Figure 2.41: Trigonal lattice with 3-fold rotation axes noted and both sets of interior points included - this is a smaller trigonal lattice.

Therefore, the 2-d trigonal system has just a primitive (p) lattice.

HEXAGONAL LATTICES

The general primitive cell is a rhombus with two equal sides at an angle of 120° (a = b; γ = 120°). The lattice point symmetry is \(6mm\). No other sites in the primitive cell have \(6mm\) point symmetry, so there are no centered hexagonal lattices.

Therefore, the 2-d hexagonal system has just a primitive (p) lattice.

Figure 2.42: Hexagonal lattice with one shaded unit cell and the 6-fold rotation axes noted.

In summary, there are five distinct 2-d Bravais lattices: (1) primitive oblique; (2) primitive rectangular; (3) centered rectangular; (4) primitive tetragonal; and (5) primitive trigonal and hexagonal (same lattice due to inversion). Primitive 2-d lattices are designated p; the centered 2-d lattice is designated c.