illustrated in Figure 15.3, which gives theprimitive cells, i.e., only the shape;
the number of Bravais lattices in each system is indicated. The hexagonal
system differs from the others in that the unit cell can be considered trigonal
(if one allowsa¼b6¼cin the trigonal system); however, three of these cells
can be stacked to form a hexagonal prism, as illustrated.
As shown, the unit cells are characterized by three axes of lengtha,b,
andc; and by three angles between the axes,a,b, andg. The dimensions
(lengths and angles) of the unit cell can generally be determined by x-ray
diffraction. In principle, the orientation of the molecules in the cell can also
be established.
A unit cell may contain one or more molecules, for instance, a sugar
and a water molecule. In such a case, the pair of molecules is considered to
be the building entity. If the cell contains two identical molecules, these do
not have identical orientation: otherwise, the unit cell could also be defined
as containing one molecule, and be half as large.
Anisotropy. Since the distances between atoms or molecules, as
well as the bond strengths between them, varies with direction (with respect
to the axes of the unit cell), physical properties of a crystal may vary with
direction. This is called anisotropy (cf. Section 9.1). For instance, the elastic
modulus or the breaking strength of a crystal may depend on the direction
of the force. Anisotropy tends to be more pronounced for crystal lattices in
which the unit cell is more asymmetric.
A well known example is optical anisotropy, especiallybirefringence.
The latter implies that the refractive index of the material depends on
direction. When a narrow light beam passes through the crystal, it will be split
into two beams, or even into three. It also means that a linearly polarized beam
of light will attain elliptical polarization; such a material can be seen in a
polarized light microscope if the specimen is between crossed polarizers. These
phenomena do not occur if the light beam is parallel to the optical axis. As
given in Figure 15.3, some crystal systems have one optical axis, others are
biaxial; cubic crystals are isotropic and hence do not cause birefringence.
Crystal Morphology. Since a crystal can be seen as a stack of unit
cells, the simplest shape that a crystal can have directly follows from the unit
cell: theanglesbetween the faces are the same. Since the stacking need not
amount to the same number of units cells in each direction, the relative sizes
of the various faces van vary. Moreover, other (plane) faces can form, as
illustrated in Figure 15.4 for the two-dimensional case. The lines denoted
(10) and (01) are parallel to the axesaandb, respectively, but other lines can
be drawn through corner points of unit cells. The same applies to the three-
dimensional case, where plane surfaces rather than straight lines are