Engineering Chemistry S Datta

482 ENGINEERING CHEMISTRY
Fundamental laws of crystallography
Geometrical crystallography is concerned with the outward spatial arrangement of crystal
planes and the geometrical shapes of the crystals and thus crystallography is dependent upon
the three following fundamental laws.
• Law of constancy of interfacial angle
• Law of rational indices, and
• Law of symmetry.
Law of constancy of interfacial angle or Steno’s law
The law states that the angles between the corresponding faces on various
crystals of the same substance are constant. The crystals of substances are bounded by
plane surfaces which are called faces. These faces always intersect at an angle, called interfacial
angle. Interfacial angle has a characteristic value for a given crystalline solid.
It is often seen that the crystal faces are unequally
developed, leading to various shapes of the crystals. But
the angle of intersection of the two corresponding faces will
be same for any crystal of the same substance. In Fig. 22.1,
two crystals are represented two-dimensionally. The shapes
are different but having the same interfacial angle.
For example, NaCl crystallises as cubes from aqueous
solution and as octahedral from urea solution but interfacial
Fig. 22.1 Interfacial angle.
angles of all crystals of NaCl are found to be 90°.
Law of rational indices
In 1784 Haüy proposed the law of rational
indices or rational intercepts. This can be
understood in the following way. For describing the
geometry of a crystal usually three non-co-planar
co-ordinate axes are selected arbitrarily. These are
crystallographic axes. According to this law, the
ratio between intercepts on crystallographic
axes for the different faces of a crystal can
always be represented by rational numbers.
In other words, all faces cut a given axis at
distances from the origin, which bear a simple ratio
to one another.
To illustrate, let us consider a plane ABC in
the crystal as shown below:
This plane has intercepts OA, OB and OC along X, Y and Z axes at distances 2a, 3b and
4c respectively, where OL = a, OM = b and ON = c are the unit distances chosen along the three
co-ordinates. These intercepts are in the ratio of 2a : 3b : 4c, where 2, 3 and 4 are simple
integral whole numbers and the standard intercepts are a, b and c. The ratio of the intercepts
in terms of the standard is 2 : 3 : 4. These ratios characterise and represent any plane of the
crystal. The coefficients are known as Weiss indices of the plane given. If any plane is parallel
to one axis, then it will cut it at infinity. In such cases the use of Weiss indices are rather
awkward and have consequently been replaced by Miller indices. The Miller indices of a
plane are obtained by taking the reciprocals of the coefficients of a, b, and c i.e., Weiss indices
X
A
L
Z
a
C
N
c b
O
Fig. 22.2 Intercepts of crystallographic
planes.
M
B
Y