Unit cells, crystal systems and Bravais lattices

A crystal is one box copied in every direction

Intuition

Wallpaper is one printed block repeated up, down and sideways. A crystal is the same trick in three dimensions. Describe the block and the copy rule, and you have described the whole crystal.

Point symmetry leaves at least one point unchanged. Translational symmetry moves everything. The feature that defines a lattice is translation: shifting the whole structure by one unit cell edge leaves it looking the same. If an atom is at $(x, y, z)$ then there is one at $(1+x, y, z)$ and at $(x, 3+y, z)$.

A unit cell is the smallest unit that, repeated along the unit cell vectors, produces the whole crystal. In a 2D pattern it is the smallest part of the drawing that generates the pattern when repeated. Many different choices exist, all of the same size. Lecture 4 says you need three things to describe a material: the size and shape of the unit cell, the symmetry elements inside it, and where the atoms sit. It says these are found by diffraction, most commonly with X-rays but also with neutrons and electrons (Lecture 6).

The cell is described by three vectors and three angles

The vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ are the unit cell vectors, with lengths $a = |\mathbf{a}|$ and so on. The angles between them are $\alpha$, $\beta$ and $\gamma$. A right-handed coordinate system is used. Any position inside the cell is $\mathbf{r} = x\mathbf{a} + y\mathbf{b} + z\mathbf{c}$, where $x, y, z$ are fractional coordinates with $0 \le x, y, z \le 1$.

By definition a crystal has translational symmetry. Any position in the crystal is $\mathbf{r} = (u+x)\mathbf{a} + (v+y)\mathbf{b} + (w+z)\mathbf{c}$. The integers $u, v, w$ say which cell you are in. The fractions $x, y, z$ say where you are inside it.

Try it: worked example

A cell has an atom at fractional coordinates (0.25, 0.5, 0.75). What is the position in the cell with u = 2, v = 0, w = 1?

Show answer and steps

Answerr = 2.25 a + 0.5 b + 1.75 c.

1. Use $\mathbf{r} = (u+x)\mathbf{a} + (v+y)\mathbf{b} + (w+z)\mathbf{c}$.
2. $u + x = 2 + 0.25 = 2.25$.
3. $v + y = 0 + 0.5 = 0.5$.
4. $w + z = 1 + 0.75 = 1.75$.
5. So $\mathbf{r} = 2.25\,\mathbf{a} + 0.5\,\mathbf{b} + 1.75\,\mathbf{c}$.

Seven cell shapes cover every crystal

Intuition

A box can be a perfect cube, a stretched cube, a squashed one or a fully tilted one. Crystallographers found there are seven kinds of box, each with its own pattern of equal edges and right angles.

Exam detail

The seven crystal systems from the slide. Triclinic: $a \ne b \ne c$, $\alpha \ne \beta \ne \gamma \ne 90^\circ$, no essential symmetry. Monoclinic: $a \ne b \ne c$, $\alpha = \gamma = 90^\circ$, $\beta \ne 90^\circ$, one twofold axis and/or a mirror plane. Orthorhombic: $a \ne b \ne c$, all angles $90^\circ$, three perpendicular twofold axes and/or mirror planes. Tetragonal: $a = b \ne c$, all angles $90^\circ$, one fourfold axis. Trigonal: $a = b = c$, $\alpha = \beta = \gamma \ne 90^\circ$, one threefold axis. Hexagonal: $a = b \ne c$, $\alpha = \beta = 90^\circ$, $\gamma = 120^\circ$, one sixfold axis. Cubic: $a = b = c$, all angles $90^\circ$, four threefold axes tetrahedrally arranged.

The cell is defined by: triclinic $a, b, c, \alpha, \beta, \gamma$; monoclinic $a, b, c, \beta$; orthorhombic $a, b, c$; tetragonal $a, c$; trigonal $a$; hexagonal $a, c$; cubic $a$.

Adding centring gives the 14 Bravais lattices

Each crystal system can be primitive (P), base-centred (C), body-centred (I) or face-centred (F). The lecture’s example is orthorhombic, which takes all four. Not all combinations are unique, so the total is the 14 Bravais lattices.

Pitfall

Do not multiply 7 by 4 and call it 28. The lecture says not all combinations are unique, and the total is 14.

Optional aside

The last slides of Lecture 4 list the minimum symmetry required for each system in Hermann–Mauguin notation and mark it beyond scope. The symbol table on the final slide was not transcribed. Both are outside the exam.