Direct Energy, 2018a

38 2.3 Piezoelectric Devices
Figure 2.7: Shapes used to illustrate symmetry elements.
As an example ofidentifying symmetry elements, consider the 2D shapes
in Fig. 2.7. The T-shaped gure has one symmetry element, a mirror plane
symmetry. The shape looks the same ifit is reected over the mirror plane
shown in the gure by a dotted line. The Q-shape has no symmetry elements.
The hexagon has multiple symmetry elements. It contains 2-fold
rotation because it looks the same when rotated by 180 ◦ . It also has 3-fold
and 6-fold rotation symmetries because it looks the same when rotated by
60 ◦ and 30 ◦ respectively. It also has multiple mirror planes shown by dotted
lines in the gure. In this example, symmetry elements of2D shapes
are identied, but material scientists are interested in identifying symmetries
of3D crystal structures to gain insights in the properties ofmaterials.
Materials are classied into categories called crystal point groups based on
the symmetries oftheir crystal structures.
We generalize about crystalline materials based on whether or not their
crystal structure possesses inversion symmetry. What is the inversion operation?
In 2D, inversion is the same as a rotation by 180 ◦ . In 3D, a
shape or crystal structure contains inversion symmetry ifit is identical
when rotated by 180 ◦ and inverted through the origin [24, p. 269]. More
specically, draw a vector −→ V from the center of the shape to any point
on the surface. If the shape has inversion symmetry, then for any such
vector −→ V , the point a distance − −→ V from the origin is also on the surface
ofthe shape. The example on the left ofFig. 2.8 has inversion symmetry
because for any such vector −→ V from the center of the shape to a point on
the surface, there is a point on the surface a vector − −→ V away from the
origin too. The example on the right does not contain inversion symmetry
as illustrated by the vector −→ V shown by the arrow.
Ifa crystal structure has inversion symmetry, we say the crystal has
a center of symmetry otherwise we say it is noncentrosymmetric. Crystal
structures are classied into classes called crystal point groups, and twenty-