Past exam of the materials science course of the University of Cambridge / 2022 / Paper 1 / 1

A periodic crystal cannot have a fivefold crystallographic rotation axis. The crystallographic restriction theorem permits rotation orders $1,2,3,4,6$ for a periodic Bravais lattice. A $72^{\circ }$ rotation cannot preserve such a lattice. This restriction concerns exact symmetry of the infinite crystal structure; an isolated finite cluster with fivefold symmetry is a different object.

A motif can have fivefold symmetry. For example, repeat an identically oriented regular pentagonal molecular motif at every point of a square translation lattice. Each isolated motif has fivefold rotational symmetry, while the crystal structure as a whole does not: rotating about one motif fails to map the other motif centres onto lattice points. The symmetry of an atomic basis of a crystal need not be a symmetry of its Bravais lattice.

Reduce the supplied fractional coordinates modulo one. The rutile structure has Ti at $(0,0)$ with height $z=0$ and at $(1/2,1/2)$ with height $z=1/2$. The oxygen projections are $(0.30,0.30),(0.70,0.70)$ at $z=0$, and $(0.80,0.20),(0.20,0.80)$ at $z=1/2$. Include equivalent corner sites to show the boundary of a complete unit cell; the cell edges are $a$ along $[100]$ and $[010]$.

Each Ti has coordination number six, with oxygen neighbours forming a distorted octahedron. Each oxygen has coordination number three, with titanium neighbours forming a planar triangle. For oxygen at $(u,u,0)$, the nearest Ti positions can be taken as $(0,0,0)$ and $(1/2,1/2,\pm 1/2)$; all three lie in the plane $x=y$. The triangle is not generally equilateral. Equal Ti–O bond lengths in the following calculation do not make every angle ideal.

In the tetragonal crystal system, physical squared distances use $a^2(\Delta x^2+\Delta y^2)+c^2\Delta z^2$. Two Ti–O distances from the corner Ti have length $\sqrt{2}ua$; the other four have length $\sqrt{2a^2(1/2-u{)}^2+c^2/4}$. Equating both to $r$ givesThus, within the equal-bond model,The rutile structure remains tetragonal and its octahedral angles remain distorted; these numbers are not a claim about measured ambient-pressure lattice parameters.

The Weiss zone law gives $u+w=0$ and $u-v=0$, so a common direction is $[11\bar{1}]$. Its physical vector is $(a,a,-c)$, while $[100]$ is parallel to $(a,0,0)$. Their dot product therefore givesTaking a Euclidean angle directly between the index triples would incorrectly treat the tetragonal axes as equal in length. Reversing the chosen common direction gives the supplementary oriented angle.