Crystallography

Crystal and crystallography

A crystal is a solid in which atoms, or groups of atoms, are arranged in a regular periodic three dimensional pattern such that every atom or group has an exactly identical surrounding. Crystallography is the study of the geometric form and the other physical properties of crystalline solids, worked out using X-rays, electron beams and neutron beams.

Space lattice

A space lattice is an array of points arranged in a regular periodic three dimensional pattern such that each point has an exactly identical surrounding. Those points are the lattice points or lattice sites, and a plane containing them is a lattice plane. No atoms are involved yet. The lattice is a mathematical object.

The position of any point in the lattice is written using three vectors a, b and c, called the basis vectors or primitive vectors. In two dimensions every point is reached by translating a and b:

OR = r = n₁a + n₂b

In three dimensions the third vector joins the list:

OR = r = n₁a + n₂b + n₃c

Basis and crystal structure

Place the same type of atom, or the same group of atoms, at or near every lattice point and you have a crystal structure. The atom or group associated with each point is the basis. The basis must be identical in composition, arrangement and orientation, otherwise the surroundings of each point would not appear the same.

Space lattice + Basis → Crystal structure

A single atom basis gives a monoatomic crystal, which covers all the metals such as aluminium, copper, sodium and iron. A two atom basis gives a diatomic crystal such as NaCl, KCl or CsCl. A three atom basis gives a triatomic crystal such as CaF₂. Sodium chloride and diamond can share a lattice and still be completely different substances, because the basis differs.

Unit cell and primitive cell

The unit cell is the smallest block of a space lattice that reproduces the complete lattice by repeated displacement. A unit cell can be chosen in a number of ways. A cell carrying lattice points only at its corners and nowhere else is a primitive cell. A cell carrying additional points alongside the corner points is a non-primitive cell. In two dimensions a unit cell is a parallelogram, and in three dimensions it is a parallelepiped.

The lengths a, b and c of the primitive vectors are the edges of the unit cell, and they are also called the lattice constants. The angle α lies between b and c, the angle β between c and a, and the angle γ between a and b. Those three are the interaxial angles. Together the three lattice constants and the three interaxial angles are the six lattice parameters, and they are all a unit cell needs.

Seven crystal systems

The six lattice parameters can take only seven distinct sets of values, so the crystal lattices fall into seven systems. They run from cubic, where all three edges are equal and all three angles are right angles, down to triclinic, where nothing is equal to anything.

If atoms sit only at the corners of those seven cells there would be seven crystal structures, all of them primitive, and each cell would be a simple cell. Alongside the seven primitive cells there are seven more non-primitive cells, of three kinds: body centred, face centred and base centred. So there are four types of unit cell in all.

Bravais lattices

Impose the condition that every lattice point must have an exactly identical environment and only fourteen space lattices survive. Those fourteen are the Bravais lattices. Seven systems with four centrings each would suggest twenty eight, but most of those combinations are not new. A face centred tetragonal cell, to take the usual example, redraws as a smaller body centred tetragonal cell describing an identical set of points. Strike out every duplicate and fourteen genuinely distinct lattices remain.

Characteristics of the cubic cell

Once the cell type is known, seven more quantities follow from geometry alone. The cubic cell is the one worth working through in full, because it is the one the chart tabulates for simple cubic, body centred cubic and face centred cubic.

The volume of the cell is a³. The effective number of atoms per cell, written Z, comes from how each atom is shared. A corner atom is shared between the eight cells meeting there and contributes one eighth. A face atom is shared between two cells and contributes a half. A body centred atom belongs to its own cell and counts as one. That gives Z = 1 for simple cubic, Z = 2 for body centred cubic and Z = 4 for face centred cubic.

The coordination number is the number of nearest neighbours simultaneously in contact with one atom. It is 6 for simple cubic, 8 for body centred cubic and 12 for face centred cubic. The atomic packing fraction (A.P.F.) is the share of the cell that the atoms actually fill:

A.P.F. = Z · v / V

Working that through gives π/6 or 52 % for simple cubic, √3π/8 or 68 % for body centred cubic and √2π/6 or 74 % for face centred cubic. Whatever the atoms do not fill is void space, which is 1 minus the packing fraction, so 48 %, 32 % and 26 % respectively. That 74 % figure is the densest packing possible for identical spheres in three dimensions, a claim Kepler made in 1611 and nobody proved until 1998.

Knowing Z lets you predict a bulk property from cell geometry alone. The mass of one atom is M/N_A, so the density of the crystal is:

ρ = Z · M / (a³ N_A)

Crystal symmetry

A symmetry operation is one that, performed on a crystal, leaves a configuration identical to the one it started from. A crystal possesses the symmetry element corresponding to that operation. There are three kinds of element to count in the cubic system.

A centre of symmetry is a point such that any straight line drawn through it meets the crystal surfaces at equal distances on both sides and joins identical points. It is also called the centre of inversion. For a cube it is the point where the body diagonals cross, and there is exactly one.

An axis of symmetry is a line about which a rotation through a definite angle brings the crystal back to a configuration identical to the original. Take the normal MN through the midpoints of a pair of opposite faces. In one complete turn about MN the cube comes back onto itself four times, once every 90°, so MN is a four-fold or tetrad axis. There are three of them. An axis CD through diagonally opposite corners repeats three times per turn, once every 120°, making it a three-fold or triad axis, and there are four. An axis KL through the midpoints of opposite edges repeats twice per turn, once every 180°, making it a two-fold or diad axis, and there are six. Thirteen axes in all.

A plane of symmetry is a mirror plane. It divides the crystal into two parts such that each is the exact mirror image of the other. The plane PQRS through the middle of the cube and parallel to one pair of faces is one, and there are three planes parallel to the faces. The diagonal plane KLMN through a pair of opposite parallel edges is another, and there are six of those. Nine planes in all.

Thirteen axes, nine planes and one centre make twenty-three symmetry elements. That is the largest collection any crystal system has, and it is the reason cubic crystals are usually the first ones anybody studies.