1.2: The Lattice as a Group

Each description of a lattice is a set that forms a mathematical group. Using the vector description for 2-d space \(\left\{{\boldsymbol T}_{n_1n_2} \right\}\), the members of the lattice set combine under vector addition. This set obeys the four characteristics of every group:

1. the set is closed: \[{\boldsymbol T}_{m_1m_2} + {\boldsymbol T}_{n_1n_2} = {\boldsymbol T}_{m_1 + n_1m_2 + n_2},  \nonumber \] which is a member of the same set because addition of integers gives integers;
2. vector addition is associative: \[\left({\boldsymbol T}_{m_1m_2} + {\boldsymbol T}_{n_1n_2} \right) + {\boldsymbol T}_{p_1p_2} = {\boldsymbol T}_{m_1m_2} + \left({\boldsymbol T}_{n_1n_2} + {\boldsymbol T}_{p_1p_2} \right)  \nonumber \]
3. the identity exists \({\boldsymbol T}_{00}\): \[{\boldsymbol T}_{n_1n_2} + {\boldsymbol T}_{00} = {\boldsymbol T}_{n_1n_2} = {\boldsymbol T}_{00} + {\boldsymbol T}_{n_1n_2} \nonumber \]
4. an inverse exists for every member \(T_{n_1 n_2}\): \[\boldsymbol T_{- n_1- n_2} = - \boldsymbol T_{n_1n_2} \nonumber \] and \[{\boldsymbol T}_{n_1n_2} + {\boldsymbol T}_{{- n}_1{- n}_2} = {\boldsymbol T}_{{- n}_1{- n}_2} + {\boldsymbol T}_{n_1n_2} = {\boldsymbol T}_{00}.  \nonumber \]

Since vector addition is also commutative, i.e., \({\boldsymbol T}_{m_1m_2} + {\boldsymbol T}_{n_1n_2} = {\boldsymbol T}_{n_1n_2} + {\boldsymbol T}_{m_1m_2}\), the set of lattice translations is an abelian group. As a result, each member \(T_{n_1n_2}\) belongs to its own class, which has important implications for features of the electronic states of crystals.

Identifying Lattices and Unit Cells in Periodic Structures

The Dutch graphic artist M.C. Escher, who lived 1898-1972, constructed diagrams that include periodic wallpaper patterns. These diagrams exhibit 2-d periodicity and were inspired by his visit to El Alhambra, a Spanish palace and fortress complex that contains numerous tilings. Here are two examples:

Figure 1.12 Source: https://mcescher.com/gallery/symmetry/ ( The M.C. Escher Company B.V.)

To identify the lattice and unit cells for each 2d pattern, follow these steps:

1. Place a first lattice point at any site in the pattern. Although this is an arbitrary decision, we are probably drawn to positions of highest rotational symmetry in the diagram. For example, the left figure has points with 3-fold symmetry.
2. Examine the pattern for all positions that have identical environments as the location of the “first lattice point”. These environments must also be identically oriented in the plane. Place points at these positions in the diagram. You have now successfully drawn the lattice.
3. From one lattice point, draw connections to the nearest points in two different directions, which identify two sides of the unit cell. Complete the cycle of connections by drawing two parallel connections, and you have a unit cell for the lattice.

Figure 1.13

Using these two patterns, one can find different locations for sets of lattice points in a repeating pattern. Nevertheless, the shapes and sizes of the smallest unit cells are the same, regardless of where the lattice points are placed in a pattern.

Example \(\PageIndex{1}\)

Four networks of atoms (points) and bonds (lines) are shown. Which ones are lattices; which ones are not?

Figure 1.14: Four networks of atoms.  A and B: two-dimensional nets; C and D: three-dimensional nets in perspective.

Solution

Synonyms of “lattice” include “frame”, “framework”, “grid”, “net”, “network”, and “trellis”, all of which are less strict than any mathematical definition, like the one described above. To establish whether or not a network of atoms is a lattice, then its surroundings must be completely identical in shape and orientation in space.

A: The 2-d triangular net. Every point is surrounded by 6 neighboring atoms in a regular hexagon. The Schläfli symbol of this 2-d network is 3⁶, which means that every point is surrounded by 6 triangles (“3”). The primitive unit cell is a rhombus. YES, the 2-d triangular net is a lattice.

B: The 2-d Kagomé net. Every point is surrounded by 4 atoms, but they occur in three different orientations. Therefore, not all points in the network are translationally equivalent. The Schläfli symbol is 3636. The unit cell that describes the translational periodicity of this structure contains 3 points. NO, the 2-d Kagomé net is not a lattice.

Representations of 3-d structures are more challenging because they must be projections in 2-d. Learning how to visualize 3-d structures is an important skill for chemists that requires imagination and practice.

Figure 1.15: Body-centered cubic structure of atoms. Unit cell highlighted in transparent red.

C: The 3-d body-centered cubic (BCC) network. Every point is surrounded by a cube with 8 neighbors. YES, the 3-d BCC net is a lattice.

Figure 1.16: Diamond structure with unit cell.

D: The 3-d diamond network. Every point is surrounded by 4 others in a local tetrahedral arrangement, but with two different orientations. Therefore, like the 2-d Kagomé net, not all points in the diamond net are translationally equivalent. A unit cell that represents 3-d translational periodicity, i.e., one that connects a set of points with identically oriented environments, is a face-centered cubic (FCC) lattice. NO, the diamond net is not a lattice.

(NOTE: given the broader definition of “lattice” mentioned above, you may find the term “diamond lattice” in the literature.)

Example \(\PageIndex{1}\)

1. For each structure illustrated, draw a lattice and one unit cell. How many atoms fall within one unit cell?
2. For your choice of unit cell in structure (I), evaluate the lattice constants and determine the fractional coordinates of the atoms in one unit cell using the interatomic distances shown.

Figure 1.17: Two-dimensional crystal structures.  Left: (I) one atom type forming squares and octagons.  Distance in every atom pair is 2.000 angstroms.  Right: (II) same pattern as (I) but with two atom types forming squares of all black and all white atoms alternating in the plane.  Octagons alternate two black atoms and two white atoms. 

Solution

There are multiple satisfactory choices for assigning the location of lattice points in each structure, but there are symmetry-based guidelines to select the best choice for the lattice and its unit cells. Lattice points should have the highest-order rotational symmetry or should coincide with inversion centers of the structure. For many cases, the locations with highest order point symmetry include inversion centers, but this is not always the case. When these two specifications yield different positions in the crystal, then the lattice will have two settings.

(a) Using these guidelines, the sites of highest rotational symmetry coincide with inversion centers in both structures I and II. Nevertheless, there are two possible solutions for each structure:

Figures 18 and 19: Structure (I) with 4 atoms per unit cell

There is just one kind of atom in structure (I) and four-fold symmetry occurs at the centers of the squares and octagons, characteristics that give two acceptable solutions for setting the lattice. In both cases, the unit cells are identical in size and shape. When the lattice points are placed at the centers of octagons, it is straightforward to count 4 atoms in one unit cell. For the other choice, the atoms fall on unit cell edges, which are shared between two adjacent cells. Since there are 8 atoms on the edges, then there are 8/2 or 4 atoms per unit cell.

Figures 20 and 21: Structure (II) with 8 atoms per unit cell

There are two kinds of atoms forming two different squares in structure (II). As a result, four-fold symmetry occurs only at the centers of the squares; the centers of octagons have two-fold symmetry. Again, there are two acceptable solutions, with lattice points located at the centers of one type of square. In each case, there are 8 atoms per unit cell: 4 open circles and 4 closed circles.

(b) For structure (I), all interatomic distances are 2.000 Å and the interior angles of the four-membered rings are 90°. Then, apply straightforward geometry as follows:

Figures 21: Structure (I) unit cell with lattice points at centers of octagons: Four atoms labeled (1)-(4) starting at top and going clockwise.

lattice constants:

\[a_1=a_2=(1.000 \AA)+\sqrt{2}(2.000 \AA)+(1.000 \AA)=4.82 \nonumber \]

\[\alpha_3=90^{\circ} \nonumber \]

fractional coordinates:

• Atom 1: \(\left( \frac{1.000\mathring{\mathrm{A}}}{4.828\mathring{\mathrm{A}}},\frac{1}{2} \right) = \left( 0.2071,\frac{1}{2} \right)\)
• Atom 2: \(\left( \frac{1}{2},\frac{3.828\mathring{\mathrm{A}}}{4.828\mathring{\mathrm{A}}} \right) = \left( \frac{1}{2},0.7929 \right)\)
• Atom 3: \(\left( \frac{3.828\mathring{\mathrm{A}}}{4.828\mathring{\mathrm{A}}},\frac{1}{2} \right) = \left( 0.7929,\frac{1}{2} \right)\)
• Atom 4: \(\left( \frac{1}{2},\frac{1.000\mathring{\mathrm{A}}}{4.828\mathring{\mathrm{A}}} \right) = \left( \frac{1}{2},0.2071 \right)\)

Figures 22: Structure (I) unit cell with lattice points at centers of squares: Four atoms labeled (1)-(4) correspond to same numbers in Fig. 1.22.

lattice constants:

\[a_1=a_2=\frac{(2.000 \AA)}{\sqrt{2}}+(2.000 \AA)+\frac{(2.000 \AA)}{\sqrt{2}}=4.828 \AA \text {; } \nonumber \]

\[\alpha_3=90^{\circ} \nonumber \]

fractional coordinates:

• Atom 1: \(\left( \frac{3.414\mathring{\mathrm{A}}}{4.828\mathring{\mathrm{A}}},0 \right) = (0.7071,0)\)
• Atom 2: \(\left( 0,\frac{1.414\mathring{\mathrm{A}}}{4.828\mathring{\mathrm{A}}} \right) = (0,0.2929)\)
• Atom 3: \(\left( \frac{1.414\mathring{\mathrm{A}}}{4.828\mathring{\mathrm{A}}},0 \right) = (0.2929,0)\)
• Atom 4: \(\left( 0,\frac{3.414\mathring{\mathrm{A}}}{4.828\mathring{\mathrm{A}}} \right) = (0,0.7071)\)

As expected, geometry gives the identical sizes and shapes for the two unit cells, but the fractional coordinates of the atoms depend on their exact locations in each cell. Nevertheless, the numbers assigned in each diagram are for the same atoms in the structure.