Direct Energy, 2018a

2 CAPACITORS AND PIEZOELECTRIC DEVICES 35
Simple cubic:
Body centered cubic:
Face centered cubic:
Asymmetric triclinic:
Figure 2.6: Illustrationof some Bravais lattices.
where l is the length of the primitive cell. Carbon, Si, Ge, and Sn all crystallize
inthis diamond structure with cell lengths of l =0.356, 0.543, 0.565,
and 0.646 nm respectively [25].
While there are only 14 possible three dimensional lattices, there are
signicantly more possible crystal structures because the crystal structure
also incorporates the basis. It is not possible to list all possible crystal
structures. Instead, they are classied based on the symmetries they contain.
Possible symmetry operations are 2-fold, 3-fold, 4-fold, and 6-fold
rotations, horizontal and vertical mirror planes, and inversion. Crystal
structures are grouped based on the symmetry elements they contain into
classes called crystal point groups. There are 32 possible crystal point
groups, and they are listed in the Table 2.2.
Some authors classify crystal structures into crystal space groups instead
of crystal point groups [6] [26]. While there are 32 crystal point groups,
there are 230 crystal space groups. Crystal space groups are based on
symmetry transformations which can incorporate not only rotations and
mirror planes but also combination of translations along with rotations
and mirror planes. Crystal space groups will not be discussed further in
this text.