value is indicated by a dash above the number). If a crystal has a face (110),
it likely also has the facesð 110 Þ,ð 110 Þ,andð 110 Þ, for symmetry reasons.
Finally, the sense (and in some cases also the name) of the axes can be taken
arbitrarily, but various workers use the same convention for a given crystal.
Crystals show enormous variation inexternal shapeorhabit, although
all these shapes arise from ordered stacking of unit cells. This is illustrated in
Figure 15.5, which speaks for itself. However, for a given unit cell, the angles
between faces that can exist are fixed. This thus allows determination of the
crystal system and of the dimensions of the unit cell from a crystal (if it is
‘‘perfect’’ and not too small). Figure 15.5 shows that considerable variation
in shape can be encountered. Even far more intricate shapes are observed in
ice crystals present in snow (which are formed by desublimation of water
from the air). The variation in shape is caused by variation in the growth
rate of the various faces of a crystal, which rates often depend on the
composition of the solution; see Section 15.2.
In a perfect crystal, the edges (where two faces meet), and especially
the corners, have a veryhigh curvature. According to the Kelvin equation
(10.19), this means that at the edge the solubility of the crystalline material is
greatly enhanced. It is often observed by microscopy that corners and edges
are not sharp but rounded. Small ice crystals (some micrometers in
diameter) in foods such as ice cream are often seen to be almost spherical. In
large crystals, curved faces are rare but not impossible. Some needlelike
crystals have a slight twist, for instance.
Polymorphism. Polymorphic literally means multiform, but the
term does not refer to variation in external shape. It indicates that crystals of
the same molecules have different unit cells, be it of the same or of a
different crystal system. The phenomenon is quite common. There are two
types of polymorphism.Enantiotropic polymorphseach are stable within a
certain range of temperature and pressure. Consequently, a phase diagram
of the various polymorphs can be made. The prime example is ice (Section
15.3.1). Ifmonotropic polymorphsexist, all but one of these are unstable.
There is no phase diagram and, given time, only the most stable form will
remain. The prime examples are compounds with long paraffinic chains,
including most lipids (especially acylglycerols), where three main
polymorphs exist (a,b^0 , andb).
Strictly speaking, a material that can crystallize with and without
solvent molecules, such as CaCO 3 and CaCO 3 ?6H 2 O, does not show
polymorphism: the composition of both types of crystals is not the same.
Neither is it polymorphism when thea- and theb-anomer of a reducing
sugar, such as glucose, can crystallize, since these are different molecules.