Direct Energy, 2018a

40 2.3 Piezoelectric Devices
c
a
β
α
γ
b
Figure 2.9: Labels on a primitive cell of a lattice.
2.9, the angles of a primitive cell of a lattice are labeled α, β, and γ, and
the lengths of the sides are labeled a, b, and c. Crystal point groups can
be classiedbasedon the angles andlengths of the primitive cell of the
lattice which belongs to that group. The literature contains multiple subtly
dierent ways of dening crystal systems [30]. The information in the
thirdcolumn follows reference [28]. The fourth column gives relationships
between the angles of the primitive cell. The fth column gives relationships
between the lengths of sides of the primitive cell. Combinations of
angles andlengths are not unique to a specic row. For example, classes
C 2 and C 1h both have α =90 ◦ , β ≠90 ◦ , γ =90 ◦ , and a ≠ b ≠ c. However,
crystal structures belonging to these crystal point groups contain dierent
symmetry elements. For more details on specically which symmetry elements
are containedin which crystal point group, see [24] [26] [27] [28].
The sixth column lists whether or not the crystal point group has inversion
symmetry. Crystal structures with no inversion symmetry or center of
symmetry, called noncentrosymmetric, are both piezoelectric andPockels
electro-optic. The last column lists whether or not crystalline materials
whose crystal structure belongs to the various crystal point groups can be
pyroelectric.
It is possible to start with a crystal structure of a material, derive
the symmetry elements it contains, derive whether or not the material is
piezoelectric, andderive whether or not the material is pyroelectric. Furthermore,
it is possible to derive along which axes piezoelectricity or pyroelectricity
can occur in the material. However, this derivation is beyond
the scope of this book. For further details, see [6] [27] [28] [31].
To predict whether or not a dielectric crystalline material is piezoelectric,
identify its lattice and crystal basis to identify its crystal structure.
Identify the symmetries of the crystal structure to classify its crystal structure
into a particular crystal point group. If that crystal structure contains
inversion symmetry, the material can be piezoelectric. We often do not