Crystal Structure and Crystallography

Miller indices of planes and directions

Miller indices are a three-number notation that labels crystal planes and directions relative to the unit-cell axes \mathbf{a}, \mathbf{b} and \mathbf{c}. A plane is written (hkl) and a direction is written [uvw], and the two bracket types must not be confused. To find the indices of a plane, take its intercepts on the three axes in lattice units from a chosen origin, take the reciprocals of the intercepts, and clear any fractions by multiplying through by the smallest integer that makes all three values integers. A plane parallel to an axis has an intercept at infinity, so its index on that axis is 0. A negative intercept gives a bar over the index, as in (\bar{1}10). A plane that passes through the origin cannot be read directly, so the origin must be moved to another corner or the parallel plane must be used instead. For example, a plane with intercepts 1, 1 and \tfrac{1}{2} on \mathbf{a}, \mathbf{b} and \mathbf{c} has reciprocals 1, 1 and 2, so it is (112).

In a figure, a shaded region represents a plane and an arrow represents a direction, so the first step in any matching question is to classify each figure before reading its indices.

• (hkl) — one specific plane or a set of parallel planes.
• [uvw] — one specific direction.
• {hkl} — the family of all symmetry-equivalent planes.
• <uvw> — the family of all symmetry-equivalent directions.

To obtain the indices of a direction, express the vector from the origin to a point on the line in lattice units and reduce it to the smallest integers. For instance, the vector (\tfrac{1}{2}, 1, 1) becomes [121] after multiplication by 2. In cubic crystals only, the normal to plane (hkl) is parallel to direction [hkl]. If two non-parallel vectors lie in a plane, their cross product gives the normal, and the components of the normal reduced to integers give the plane indices. The indices (43\bar{2}) and (\bar{4}\bar{3}2) denote the same family of parallel planes, because reversing the sign of all indices reverses the normal without changing the plane.

Planar atomic density

Planar atomic density is the number of atoms whose centres lie in a crystal plane divided by the area of that plane, so that \rho_p = N/A. Linear density is the analogous quantity for a line of atoms. To count atoms in a plane section of a cell, each atom is weighted by the fraction of it that lies inside the section. An atom at the centre of the cell lies wholly inside a plane that contains it and contributes 1. A corner atom of a rectangular section is shared by four cells in the plane and contributes \tfrac{1}{4} to the cell. A face-centre atom in the plane contributes \tfrac{1}{2}.

For example, in a simple cubic cell the (100) face contains four corner atoms, giving 4\times\tfrac{1}{4}=1 atom in an area a^2, so its planar density is 1/a^2. In a body-centred cubic crystal, the (110) plane is a centred rectangle with sides a and \sqrt{2}a, and the planar density of this plane is \sqrt{2}/a^2. This is why the (110) plane is the most densely packed plane of the BCC structure.

Planar density also governs surface energy in the simple broken-bond model, in which surface energy is proportional to the number of nearest-neighbour bonds cut per unit area. In BCC, each atom has eight nearest neighbours. A \{100\} surface has four bonds per atom crossing to the next plane, while a \{110\} surface has two bonds per atom crossing, so the ratio of surface energies is S_{100}/S_{110} = (4/a^2)/(2\sqrt{2}/a^2) = \sqrt{2} \approx 1.4.

Bravais lattice classification

A Bravais lattice is a distinct arrangement of lattice points that is the same when viewed from any lattice point. There are 14 Bravais lattices in three dimensions, which are grouped into the 7 crystal systems according to their symmetry and lattice parameters. The centring types are P (primitive), I (body-centred), F (face-centred), C (base-centred) and R (rhombohedral). The 14 lattices are: cubic P, I and F; tetragonal P and I; orthorhombic P, I, F and C; hexagonal P; rhombohedral R; monoclinic P and C; and triclinic P.

A centring type is accepted only if the resulting cell cannot be described by a smaller cell with a higher symmetry. A base-centred cubic cell does not appear in the list because its primitive cell has two equal perpendicular edges of length a/\sqrt{2} and a third edge of length a, which is a simple tetragonal lattice. Thus the base-centred cubic cell is the same as the simple tetragonal cell. Lattice points are also not generated by a base centring that would require a threefold axis, and a cubic lattice must have threefold axes along the body diagonals. Adding base-centre points destroys these axes, so the lattice cannot be cubic.