envisaged. Hence, cubic crystals can have angles between faces that do not
equal 90 8 but also, for instance, 135 8. The angles are measured inside the
crystal; they are rarely larger than 180 8.
To characterize the various faces, several systems have been devised.
The most common one is theMiller index(hkl); the values ofh,k, andlare
established as follows. The coordinate system used is directly derived from
the unit cell; this implies that the angles between axes can be different from
908. The plane considered cuts off from the axes lengths ofA,B, andCtimes
the corresponding lengths of the unit cell (a,b,andc); these numbers can be
1 ; 2 ; 3 ;...;?. The Miller index of the face is now given by (N/AN/BN/C),
whereNis chosen such thath, k,andlare the smallest possible natural
numbers, including zero (N=?). This is illustrated in Figure 15.4 for two
dimensions. Miller indices of crystal faces are, for instance, (100), (110) or
(021); numbers larger than 3 rarely occur. It should further be noted that
different faces can have the same index, because the axes have a sign and a
face may cut through an axis on the positive side (‘‘above’’ the origin) or on
the negative side; cf. lines (21) andð 21 Þin the figure (if need be, the negative
FIGURE15.4 Illustration of various faces that can occur for a simple crystal
lattice, and of the determination of the Miller indices (the numbers between
brackets). Two-dimensional case.