Page 38 - Instant notes

Physical Chemistry 24

basic building block is referred to as the unit cell, and is the smallest unit which contains
all the components of the whole assembly. It may be used to construct the entire array by
repetition of simple translation operations parallel to any of its axes. By convention, the
three unit cell edge lengths are denoted by the letters a, b and c. Where a, b and c are
identical, all three edges are denoted a, and where two are identical, these are denoted a,
with the third denoted by c. The angles between the axes are likewise denoted α, β and γ.
Because there are an infinite number of possible unit cells for any given array, several
principles govern unit cell selection:
(i) the edges of the unit cell should be chosen so as to be parallel with symmetry axes or
perpendicular to symmetry planes, so as to best illustrate the symmetry of the crystal;
(ii) the unit cells should contain the minimum volume possible. The unit cell lengths
should be as short as possible and the angles between the edges should be as close to
90° as possible;
(iii) where angles deviate from 90°, they should be chosen so as to be all greater than 90°,
or all smaller than 90°. It is preferable to have all angles greater than 90°;
(iv) the origin of the unit cell should be a geometrically unique point, with centres of
symmetry being given the highest priority.
Fig. 2 illustrates some of these points for a two-dimensional lattice. Some possible unit
cells for the rhombohedral array of points (Fig. 2a) are shown in Fig. 2b, and whilst
repetition of any one of these unit cells will generate the entire lattice, only one of them
complies with (i), (ii) and (iii) above. Fig. 2c and Fig. 2d illustrate principle (iv) above.
Taking the most appropriate unit cell for this array its position is selected so as to place
its origin at a geometrically unique point (i.e. on a lattice element). Therefore, whilst the
unit cell shown in Fig. 2c is permissible, the unit cell shown in Fig. 2d is preferred for its
compliance with principle (iv).
It is worth noting that these principles are only guidelines, and other considerations
may occasionally mean that it is beneficial to disregard one or more of them. One might,
for example, select a unit cell so as to better illustrate a property of the crystal, or so as to
allow easier analysis through diffraction methods (see Topic A6).
Unit cells themselves have a limited number of symmetry properties, since they must
be capable of packing together to completely fill an area or volume of space. For a two-
dimensional array, this restriction means that the possible unit cells must possess 1-, 2-,
3-, 4- or 6-fold rotational symmetry, with no other possible symmetries. Five- or seven-
fold unit cell symmetries, for example, would require tiling pentagons or heptagons on a
flat surface, an operation which is easily demonstrated to be impossible. Two-
dimensional packing is therefore limited to five basic types of unit cell.
For three-dimensional crystalline arrays, the same fundamental arguments apply, and
fourteen basic unit cells exist which may be packed together to completely fill a space.
These are referred to as the Bravais lattices. There are seven primitive unit cells
(denoted P), with motifs placed only at the vertices. Two base centered unit cells (C)
may be formed from these primitive unit cells by the addition of a motif to the center of
two opposing unit cell faces. Two face centered unit cells (F) result from adding a motif
to all six face centers. Three body centered unit cells (B) are generated by placing a
motif at the center of the unit cell.