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The Crystal Lattice System
tion. From observations and measurements we find that it is the regular
crystalline structure that leads to certain special properties and behavior
of the associated materials. In this section we develop the basic ideas that
enable us to describe crystal structure analytically, so as to exploit the
symmetry properties of the crystal in a systematic way.

2.2.1 Symmetries of Crystals

Consider a regular rectangular arrangement of points on the plane. The
points could represent the positions of the atoms that make up a hypo-
thetical two-dimensional crystal lattice. At first we assume that the atoms
are equally spaced in each of the two perpendicular directions, say by a
b
a
pitch of and . More general arrangements of lattice points are the
rule.

Translational Consider a vector that lies parallel to the horizontal lattice direction
a
Invariance and with magnitude equal to the pitch . Similarly, consider vector in
b
a
the other lattice direction with magnitude equal to the pitch . Then,
b
starting at point p , we can reach any other lattice point q with
i j
q = α a + β b , where α and β are integers. Having reached another
j j j j j
interior point q , the vicinity is the same as for point p , and hence we
j i
say that the lattice is invariant to translations of the form α a + β b , see
j j
the example in Figure 2.10.
q = 2a + 5b
j
Figure 2.10. In this 2-dimensional
infinitely-extending regular lattice q
j
the 2 lattice vectors are neither
perpendicular nor of equal length.
Given a starting point, all lattice
points can be reached through b p i
b
q = α a + β b . The vicinities of a
j j j
p and q are similar. a
i i

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