Crystal symmetry: glides, screws and space groups

Point groups plus translation make space groups

Intuition

A molecule has symmetry about a fixed point. A crystal has that, plus the habit of repeating. Combine the two and you get the full set of ways a crystal can be symmetric. Those combinations are space groups.

Atoms in a crystal obey translational symmetry, and often more: rotation, mirror planes, inversion, and combinations of rotation or mirror planes with translation. The lecture calls this combination the grand idea. Point groups plus translational symmetry give space groups, where “space” means we now fill space with a symmetric object instead of growing an object around a point.

Not every point group fits every Bravais lattice. A triclinic cell cannot have rotation axes or mirror planes, only inversion centres. A periodic crystal cannot have 5-fold, 7-fold, 8-fold (and similar) rotation. The result is 32 point groups plus 14 Bravais lattices giving 230 space groups.

Crystals use the same symmetry, with crystal symbols

The crystal symbol for rotation is $n$, so a twofold axis is $2$ and a threefold axis is $3$ (like $C_2$ and $C_3$). The symbol for a mirror plane is $m$ (like $\sigma$). Inversion is $\bar{1}$ (like $i$). A rotary inversion axis is $\bar{n}$. The lecture’s examples: a twofold axis in CsC$_2$N$_5$, a threefold axis in cristobalite (SiO$_2$), a mirror plane in BaSb$_2$O$_6$, inversion centres in aspirin and $\bar{3}$ axes in a metal-organic framework.

Exam detail

Test 1 asks for definitions of the symmetry operations, including those that carry a translation. Know glide planes and screw axes cold, with their symbols and handedness behaviour.

A glide plane is a mirror plus a slide

Intuition

Imagine footprints in sand. Each left foot is the mirror image of the right, and each is shifted forward by a step. Flip then shift gives the same pattern. That is a glide.

A glide plane is a translation followed by a reflection in a plane whose normal is parallel to a rotational symmetry element. The translation is usually half the unit cell edge or diagonal involved. The symbols are $a$, $b$ and $c$ for a glide of $\tfrac12$ along that edge. The diagonal glide $n$ is $\tfrac12$ along the diagonal. The $d$ glide is $\tfrac14$ along a face or diagonal. In aspirin the lecture shows a reflection perpendicular to $b$ with translation of $c/2$. Glide planes change handedness, because they contain a reflection.

A screw axis is a turn plus a slide

Intuition

A spiral staircase: you turn a bit and rise a bit, again and again. The pattern copies itself up the stairs.

A screw axis describes helical symmetry. The symbol $m_n$ means translate $n/m$ of the unit cell and then rotate about the axis by $360^\circ/m$.

Symbol	Translate	Rotate
$2_1$	1/2 of the cell	180°
$3_1$	1/3 of the cell	120°
$4_2$	2/4 of the cell	90°
$6_3$	3/6 of the cell	60°

By the crystallographic convention, rotations are clockwise with respect to translation in the away direction (right-hand rule, thumb is translation and fingers are rotation). This matters only for helical axes such as $3_1$ and $3_2$, which are chiral. Screw axes do not change handedness, because they contain a rotation. The lecture shows $2_1$ axes in aspirin and $6_5$ axes in a metal-organic framework.

Compare

Glide plane

Reflection plus translation. Changes handedness.

Screw axis

Rotation plus translation. Keeps handedness.

To describe a crystal packing fully you need the cell lengths $a, b, c$, the angles $\alpha, \beta, \gamma$, the Bravais lattice (P, C, I or F), the extra symmetry (rotation, screw, mirror, inversion, glide) and the unit cell contents.

Pitfall

Both glides and screws mix a move with a translation, but only the glide has a reflection. That is why one changes handedness and the other does not.

Optional aside

The slides name aspirin as P2$_1$/c and caffeine hydrate as P2$_1$/a. They are given as forward references to Lecture 5, where the symbols are read.