Chemistry · Ch 1 — The Solid State

Primitive and Centred Unit Cells

1.4.1

Primitive and Centred Unit Cells

Unit cells fall into two broad categories — primitive and centred.

(a) Primitive unit cells

When the constituent particles are present only at the corners of the unit cell, it is a primitive (simple) unit cell.

(b) Centred unit cells

When a unit cell has particles at positions in addition to the corners, it is a centred unit cell. There are three kinds:

Body-centred: one extra particle at the body centre, besides those at the corners.
Face-centred: one extra particle at the centre of each face, besides those at the corners.
End-centred: one extra particle at the centre of two opposite faces, besides those at the corners.

The seven crystal systems and Bravais lattices

Studying a wide variety of crystals shows that all of them conform to one of seven basic shapes, called the seven crystal systems (shown in Fig. 1.11). Which system a crystal belongs to is decided by the angles between its faces and the number of axes needed to describe its shape.

The French mathematician Bravais showed that only 14 distinct three-dimensional lattices are possible — the 14 Bravais lattices. Table 1.3 lists the seven crystal systems together with the centred variations each one can form.

Crystal system	Possible variations	Edge lengths	Axial angles	Examples
Cubic	Primitive, Body-centred, Face-centred	$a=b=c$	$\alpha=\beta=\gamma=90^\circ$	NaCl, zinc blende, Cu
Tetragonal	Primitive, Body-centred	$a=b\neq c$	$\alpha=\beta=\gamma=90^\circ$	White tin, SnO₂, TiO₂, CaSO₄
Orthorhombic	Primitive, Body-, Face-, End-centred	$a\neq b\neq c$	$\alpha=\beta=\gamma=90^\circ$	Rhombic sulphur, KNO₃, BaSO₄
Hexagonal	Primitive	$a=b\neq c$	$\alpha=\beta=90^\circ,\ \gamma=120^\circ$	Graphite, ZnO, CdS

Table 1.3Seven Primitive Unit Cells and their Possible Variations as Centred Unit Cells

Crystal system	Possible variations	Axial distances or edge lengths	Axial angles	Examples
Cubic	Primitive, Body-centred, Face-centred	a = b = c	alpha = beta = gamma = 90 deg	NaCl, Zinc blende, Cu
Tetragonal	Primitive, Body-centred	a = b != c	alpha = beta = gamma = 90 deg	White tin, SnO2, TiO2, CaSO4
Orthorhombic	Primitive, Body-centred, Face-centred, End-centred	a != b != c	alpha = beta = gamma = 90 deg	Rhombic sulphur, KNO3, BaSO4
Hexagonal	Primitive	a = b != c	alpha = beta = 90 deg, gamma = 120 deg	Graphite, ZnO, CdS
Rhombohedral or Trigonal	Primitive	a = b = c	alpha = beta = gamma != 90 deg	Calcite (CaCO3), HgS (cinnabar)

Figure 1.10Illustration of parameters of a unit cell

What this figure shows. A single 3D parallelepiped unit cell with the three edge vectors labelled a, b, c and the three interedge angles labelled alpha (between b and c), beta (between a and c), gamma (between a and b), showing the six unit-cell parameters. …

Figure 1.11Seven crystal systems

What this figure shows. A chart of seven 3D parallelepiped shapes, one per crystal system: cubic, tetragonal, orthorhombic, hexagonal, rhombohedral/trigonal, monoclinic and triclinic, each drawn to show its characteristic edge lengths and axial a …

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