Chemistry · Ch 6 — Solid State
Primitive and Non-Primitive Unit Cell
Primitive and Non-Primitive Unit Cell
Unit cells come in two basic kinds. A primitive unit cell contains only one type of lattice point -- lattice points sitting solely at the 8 corners of the cell, with nothing extra inside or on a face. A non-primitive unit cell, by contrast, carries additional lattice points beyond the corners, either centred on one or more faces or somewhere within the body of the cell itself.
There are exactly seven primitive crystal systems -- cubic, tetragonal, orthorhombic, hexagonal, monoclinic, triclinic and rhombohedral -- distinguished purely by how their three edge lengths and three inter-edge angles relate to one another (for instance, cubic has all three edges equal and all three angles at 90 degrees, while triclinic has all three edges unequal and none of the three angles equal to 90 degrees or to each other). Once the different possible centrings (body-centred, face-centred, base-centred) of these seven systems are also counted, Bravais showed that only 14 distinct lattices -- the 14 Bravais lattices -- are geometrically possible in total (Table 6.2): 3 for cubic (primitive, body-centred, face-centred), 2 for tetragonal, 1 for hexagonal, 4 for orthorhombic, 2 for monoclinic, 1 for trigonal/rhombohedral and 1 for triclinic. …
What this figure shows. A four-panel schematic on a dotted 2D lattice background. Top row, labelled 'Primitive': two panels each showing a single parallelogram-shaped cell (pink, then a differently-shaped pink cell) built from only one type of lattice point, at the corners. Bottom row, labelled 'Non-Primitive': two panels (green shapes) each showing a cell that encloses an additional lattice point beyond its corners -- illustrating that a non-primitive cell carries extra lattice points either on a face or within the cell, unli …
What this figure shows. Seven small 3D unit-cell diagrams (green blocks with pink sphere vertices), each labelled with its axial relations: Cubic (a=b=c, alpha=beta=gamma=90 deg); Rhombohedral (a=b=c, alpha=beta=gamma not-equal 90 deg); Hexagonal (a=b not-equal c, alpha=beta=90 deg, gamma=120 deg); Tetragonal (a=b not-equal c, alpha=beta=gamma=90 deg); Orthorhombic (a not-equal b not-equal c, alpha=beta=gamma=90 deg); Monoclinic (a not-equal b not-equal c, alpha=gamma=90 deg, beta not-equal 90 deg); and Triclinic (a not-equal b not-equal c, alpha not-equal beta not-equal gamma not-equal 90 deg). A small axis-convention diagram alongside defines a, b, c as the three crystallographic …
| Crystal system | Primitive | Body-centred | Face-centred | Base-centred |
|---|---|---|---|---|
| Cubic | yes | yes | yes | -- |
| Tetragonal | yes | yes | -- | -- |
| Hexagonal | yes | -- | -- | -- |
| Orthorhombic | yes | yes | yes | yes |
| Monoclinic | yes | -- | -- | yes |
| Trigonal (Rhombohedral) | yes | -- | -- | -- |
| Triclinic | yes | -- | -- | -- |