09 | The unit cell

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A unit cell defines the smallest enclosed space from which an entire periodic structure can be generated by symmetry operations of pure translation. It is equivalent to the wavelength in wave mechanics or the monomer in polymer chemistry and be defined in 1D as a line, 2D as a parallelogram or in 3D as a parallelepiped, Fig. 11.

Unit cells are useful substructures to identify for a number of reasons. In computations, the electronic structure and an infinitely large periodic system can be easily calculated by only considering the few atoms whose positions are within the unit cell. Similarly, the X-ray diffraction pattern of a material can be modeled by knowing only the chemical composition of the crystal. From this information along the atomic positions within the unit cell can be accurately determined to within a fraction of an Ångstrom.

Fig. 11 Unit cells in the 1D polyacetylene, the 2D graphene, and 3D wurtzite crystals.

Identifying unit cells

A unit cell can be identified in a known structure by first identifying a set of lattice points that are commensurate with the the crystal structure. Lattice points are a set of points in an infinite periodic structures that are in an environment that is completely indistinguishable from the location for any other lattice point. For a given crystal, the exact location of a lattice point is arbitrary. However once we pick the location of one lattice points the location of every other lattice point is also determined. For example in the case of graphene in Fig. 11, the vertices of the unit cell describe a subset of 4 lattice points in the infinite lattice and are located at the exact center of each hexagon. However an equally valid set of lattice points, and likewise a unit cell, exists where the lattice points overlay 1/2 of the carbon atoms, Fig. 12.

Fig. 12 (Click image) Unit cell and lattice point choices for graphene. Lattice points are shown as orange circles and the unit cell is drawn in black.

As noted above, a unit cell is an enclosed space form which a periodic structure can be generated by only applying symmetry operations of pure translation. The conventional standard unit cells for any 3D crystal can be described as a parallelepiped in 3D or a parallelogram in 2D. However there are valid unit cells of different shapes. For example. In the case of graphene, we could describe a hexagonal unit cell that when translated along any of this principal axes will also generate the entire crystal structure. This construction is called the Wigner-Seitz cell and is often used in electronic structure calculations. For now, and throughout our discussion of X-ray diffraction and the description the atomic structure of materials we will only deal with conventional standard unit cells (parallelograms and parallelepipeds).

Unit cell axes

In order to describe the positions or atoms in a crystal and the geometry of their structures we need to define a coordinate system (aka crystal system). However, because unit cells are not always squares or cubes we cannot easily describe their periodic structures in a familiar Cartesian system—that is, a set of mutually orthogonal unit vectors. Instead, is it makes more sense to have the axes of our coordinate system have point along the edges of the the unit cell and have their lengths scaled in proportion to the length of each side of the unit cell. In the case of 2D graphene it is clear that doing so results in a non-orthogonal coordinate system. In order to distinguish between the crystal coordinate system and the canonical Cartesian system the coordinate axes are always referred to as \(a\), \(b\), and \(c\).

Fig. 13 The unit cell coordinate systems for the unit cell of a wurtzite (ZnS) crystal shows as the basis vectors \(|\vec{a}|, |\vec{b}|, |\vec{c}|\).

The relationship between crystal coordinates and Cartesian coordinates can be describe using as described in §1.1. The basis vectors for a unit cell are a set of vectors connecting lattice points as to form a right handed coordinate system. In the case of graphene, the basis vectors in units of Ångstroms are

\[\begin{split} \vec{a} = \begin{bmatrix} 2.46 \\ 0 \end{bmatrix}\; \vec{b}=\begin{bmatrix} 1.23 \\ 2.13 \end{bmatrix}. \end{split}\]

However, it is most common to report the geometry of a unit cell as the unit cell parameters. For a 3D crystal these are \(|\vec{a}|, |\vec{b}|, |\vec{c}|, \alpha, \beta, \gamma\). By convention:

• \(\alpha\) is the angle between \(|\vec{b}|\) and \(|\vec{c}|\)

• \(\beta\) is the angle between \(|\vec{a}|\) and \(|\vec{c}|\)

• \(\gamma\) is the angle between \(|\vec{a}|\) and \(|\vec{b}|\),.

In two dimensions we only need to define three parameters to define the unit cell: \(|\vec{a}|, |\vec{b}|\) and \(\gamma\). The unit cell parameters for graphene are \(|\vec{a}|=|\vec{b}|=2.46\,Å,\,\gamma=120°\). Basis vectors can be converted to unit cell parameters using vector dot products. For example,

\[ \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos{\gamma} \]

This conversion can be accomplished succinctly by forming the metric tensor, \(G\), of the basis vectors. In 3D we can form the basis vectors into a 3x3 matrix, \(A\), as

\[\begin{split} A = \begin{bmatrix} a_x & b_x & c_x \\ a_y & b_y & c_y \\ a_z & b_z & c_z \\ \end{bmatrix}, \end{split}\]

and the metric tensor as

\[\begin{split} G = A^TA = \begin{bmatrix} a \cdot a & a \cdot b & a \cdot c \\ b \cdot a & b \cdot b & a \cdot c \\ c \cdot a & c \cdot b & c \cdot c \\ \end{bmatrix} = \begin{bmatrix} a^2 & ab\cos{\gamma} & ac\cos{\beta} \\ ab\cos{\gamma} & b^2 & bc\cos{\alpha} \\ ac\cos{\beta} & bc\cos{\alpha} & c^2 \\ \end{bmatrix}. \end{split}\]

Unit cell volume

The area of a 2D unit cell defined by two basis vectors is their cross product,

\[ Area = \vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin{\gamma}. \]

The volume or a 3D unit cell can be calculated in the general case as the vector triple product:

\[ V=c\cdot a\times b, \]

or

\[ V^2 = \det{G}, \]

or from the lattice parameters as

(1)\[ V = abc\sqrt{1-\cos^2{\alpha}-\cos^2{\beta}-\cos^2{\gamma}+2\cos{\alpha}\cos{\beta}\cos{\gamma}}. \]

Formula unit, assymetric unit, and \(Z\)

In molecules the stoichiometry must contain only discrete values. In materials we can imagine an analogous formula unit. The composition of the unit cell is an integer multiple of formula unit. In turn the composition of a perfect crystal is an integer multiple of the formula unit and thus the unit cell as well. In the 2D structure of graphene, it is obvious from the conventional standard unit cell setting #1 in Fig. 12 that there are 2 carbon atoms in the unit cell. The formula unit is \(C\) and the unit cell composition is \(C_2\).

There is another important fundamental unit of a periodic structure called the asymmetric unit. While the unit cell defines a substructure from which a 3D periodic structure can be generated using only symmetry operations of pure translation. From the asymmetric unit is a smaller substructure from which the entire unit cell can be generated using every possible symmetry operation. In the case of graphene the composition of the asymmetric unit is \(C\). Importantly this implies that by only determining the position of a single carbon atom and a set of unit cell parameters the entire structure of graphene can be generated by symmetry.

Note that it is not always true that the composition of the asymmetric unit is equivalent to the formula unit nor does its composition need to contain integer multiples of the elements. Provided the composition of the unit cell is stoichiometric, the composition of the asymmetric unit can contain non-integer values. Nonetheless the composition of the asymmetric unit and the composition of the unit cell are always related by an integer multiple value \(Z\). That is the \(Z\) is the number of times the asymmetric unit appears in the unit cell. In graphene \(Z=2\).

Crystallographic density

By calculating the volume of a unit cell form the unit cell parameters (or basis vectors) and calculating the mass from the chemical composition of the unit cell.

(2)\[ \rho_{crystal} = \frac{mass_{unit\,cell}}{V_{unit\,cell}} =\frac{Z\,mass_{assym}}{V_{unit\,cell}} \]

In the case of the wurzite polymorph of zinc sulfide shown in Fig. 13 the unit cell parameter are:

\(a = b = 3.823\,Å\)
\(c= 6.261\, Å\)
\(\alpha = \beta = 90°\)
\(\gamma= 120°\)

From (1) the unit cell volume is 79.231 ų. The formula weight or ZnS is 97.44 g/mol and from Fig. 13 2 Zn and 2 S atoms are positioned inside the unit cell. Densities are conventionally reported in g/cm³ and from (2) the crystallographic density of the mineral wurtzite is: 4.08 g/cm³.

Points inside the unit cell

Previously in §1.2 we saw the structure of a molecule can be described by simply listing the positions of atoms along with their xyz coordinates in cartesian space. The positions of atoms in crystals can be described similarly but in the crystal coordinate system rather than a cartesian coordinate system.

In the cartesian system of an .xyz files the coordinates of each atom have units of Angstroms. In contrast, crystal coordinates are unitless and represent multiples of the unit cells three basis vectors such that a position \(X\) can be defined as

\[ X=x\vec{a}+y\vec{b}+z\vec{c}. \]

All atoms in a crystal must reside within the unit cell. The fractional coordinates (\(x,y,z\)) are therefore restricted to the values \(0 \leq x < 1\). We can describe a position out side of these bounds but by translational symmetry in the periodic lattice it would be indistinguishable from one within these bounds. By adding or subtracting 1 repeatedly until the crystal coordinate is between 0 and 1 we can always locate the equivalent position within the first unit cell located at the origin, Fig. 14.

Fig. 14 Some fractional coordinates \((x,y,z)\) in a face-centered cubic unit cell. Note for coordinates along any axes \(1 \equiv 0\).

Note that apart from bounding the values of the fractional coordinates (\(x,y,z\)) to be between 0 and 1 the mechanics by which we defined a position in space using a position vector is not different from that described for basis vectors in §1.1.