Chemistry Reference
In-Depth Information
1.2.1 Crystal Lattice and Symmetry
A crystal structure can be simply expressed by a point lattice as shown in Figure 1.1(a).
The point lattice represents the three-dimensional arrangement of the atoms in the
crystal structure. It can be imagined as being comprised of three sets of planes, each set
containing parallel crystal planes with equal interplane distance. Each intersection of
three planes is called lattice point and represents the location of the center of an atom,
ion, or molecule in the crystal. A point lattice can be minimally represented by a unit
cell, highlighted in bold in the bottom left corner. A complete point lattice can be
formed by the translation of the unit cell in three-dimensional space. This feature is
also referred to as translation symmetry. The shape and size of a unit cell can be defined
by three vectors a, b,andc all starting from any single lattice point as shown in
Figure 1.1(b). The three vectors are called the crystallographic axes of the cell. As each
vector can be defined by its length and direction, a unit cell can also be defined by the
three lengths of thevectors (a, b,andc) aswell as the angles between them(a, b, andg).
The six parameters (a, b, c, a, b,andg) are referred to as the lattice constants or lattice
parameters of the unit cell.
One important feature of crystals is their symmetry. In addition to the translation
symmetry in point lattices, there are also four basic point symmetries: reflection,
rotation, inversion, and rotation-inversion. Figure 1.2 shows all four basic point
symmetries on a cubic unit cell. The reflection plane is like a mirror. The reflection
plane divides the crystal into two sides. Each side of the crystal matches the mirrored
position of the other side. The cubic structure has several reflection planes. The
rotation axes include two-, three-, four-, and sixfold axes. A rotation of a crystal about
an n-fold axis by 360
/n will bring it into self-coincidence. A cubic unit cell has
several two-, three- and fourfold axes. The inversion center is like a pinhole camera;
the crystal will maintain self-coincidence if every point of the crystal inverted through
the inversion center. Any straight line passing through the inversion center intersects
with the same lattice point at the same distance at both sides of the inversion center.
A cubic unit cell has an inversion center in its body center. The rotation-inversion
center can be considered as a combined symmetry of rotation and inversion.
c
c
a
b
b
b
g
a
a
(
a
)
(
b
)
FIGURE 1.1 A point lattice (a) and its unit cell (b).
