Page 44 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Crystal Structure
Rotational
Symmetry If we consider the vicinity of an interior point p i in our lattice, and let us
assume that we have very many points in the lattice, we see that by rotat-
ing the lattice in the plane about point p by an angle of 180° , the vicin-
i
ity of the point p remains unchanged. We say that the lattice is invariant
i
to rotations of 180° . Clearly, setting a = b and a ⊥ b makes the lattice
invariant to rotations of 90° as well. Rotational symmetry in lattices are
due to rotations that are multiples of either 60° , 90° or 180° , see
Figure 2.11. If the underlying lattice has a rotational symmetry, we will
expect the crystal’s material properties to have the same symmetries.

180° 90° 60°

Figure 2.11. Illustrations of lattice symmetries w.r.t. rotations.

Bravais With the basic idea of a lattice established, we now use the concept of a
Lattice Bravais lattice to model the symmetry properties of a crystal’s structure.
A Bravais lattice is an infinitely extending regular three-dimensional
array of points that can be constructed with the parametrized vector
q = α a + β b + γ c (2.3)
j j j j
where , and are the non-coplanar lattice vectors and α , β and
c
ab
j j
γ are arbitrary (positive and negative) integers. Note that we do not
j
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