3.1 The Unit Cell

We have seen that the requirement for electronic neutrality in crystals, together with coordination, provides important constraints on the arrangements of atoms within crystals. One important consequence of these constraints is that structures within individual crystals are repeated on a regular basis. For example imagine that we were sufficiently small (smaller by a factor of ~10 -11 !) that we were able to travel along the tetrahedral chain in a single chain inosilicates such as a pyroxene. As we traveled along we would pass by an identical structural and chemical environment every two tetrahedrons. This similarity would extend to the position and coordination of the adjacent inter-chain cations as well as the distribution and orientation of the surrounding tetrahedra (one interesting consequence would be that without some independent frame of reference we could not possibly tell that we were making any headway as we traveled along the chain as we would constantly be returning to an identical and therefore indistinguishable environment). The scale of this repeat distance is astonishingly small; along the length of the tetrahedral chains in pyroxenes it is ~ 5.3 A (that is 5.3 * 1-10 metres).

The recognition that identical structures are repeated within crystals allows us to identify a fundamental 3 dimensional repeat unit called the unit-cell. The unit cell is a parallelepiped (you do know what a parallelepiped is, don't you?) with edges defined by three non-coplanar directions (these directions are defined as x, y and z). Importantly there is no requirement that these directions, which define the edges of the unit cell, are orthogonal (although they sometimes are); they simply define the edges of the (unit-cell) parallelepiped which has the following properties:

(1) an infinite crystal can be made up simply by repetition of the parallelepiped along the x, y and z directions.

(2) the axes and the dimensions of the edges of the parallelepiped are chosen so that the volume of the parallelepiped is the smallest possible unit which on repetition is capable of producing the structure of the infinite crystal.

The lengths of the unit cell parallel to x, y, and z are respectively denoted a, b and c, and the interaxial angles a = y^z, b = z^x and g = x^y.