Unit Cell in 3D

1.1 Periodicity: Crystal Structures
Most of solid materials possess crystalline structure that means spatial periodicity or translation symmetry. All the lattice can be obtained by repetition of a building block called
basis. We assume that there are 3 non-coplanar vectors a1, a2, and a3 that leave all the
properties of the crystal unchanged after the shift as a whole by any of those vectors. As
a result, any lattice point R0 could be obtained from another point R as
R0 = R + m1a1 + m2a2 + m3a3 (1.1)
where mi are integers. Such a lattice of building blocks is called the Bravais lattice. The
crystal structure could be understood by the combination of the propertied of the building
block (basis) and of the Bravais lattice. Note that
• There is no unique way to choose ai. We choose a1 as shortest period of the lattice,
a2 as the shortest period not parallel to a1, a3 as the shortest period not coplanar to
a1 and a2.
• Vectors ai chosen in such a way are called primitive.
• The volume cell enclosed by the primitive vectors is called the primitive unit cell.
• The volume of the primitive cell is V0 The natural way to describe a crystal structure is a set of point group operations which
involve operations applied around a point of the lattice. We shall see that symmetry provide important restrictions upon vibration and electron properties (in particular, spectrum
degeneracy). Usually are discussed:
Rotation, Cn: Rotation by an angle 2?/n about the specified axis. There are restrictions
for n. Indeed, if a is the lattice constant, the quantity b = a + 2a cos ? (see Fig. 1.1)
Consequently, cos ? = i/2 where i is integer.
Inversion, I: Transformation r ? ?r, fixed point is selected as origin (lack of inversion
symmetry may lead to piezoelectricity);
Reflection, ?: Reflection across a plane;
Improper Rotation, Sn: Rotation Cn, followed by reflection in the plane normal to the
rotation axis.There are 14 types of lattices in 3 dimensions. Several primitive cells is shown in Fig. 1.4.
The types of lattices differ by the relations between the lengths ai and the angles ?i.
Cubic and Hexagonal Lattices. Some primitive lattices are shown in Fig. 1.5. a,
b, end c show cubic lattices. a is the simple cubic lattice (1 atom per primitive cell),
b is the body centered cubic lattice (1/8 × 8 + 1 = 2 atoms), c is face-centered lattice
(1/8 × 8 + 1/2 × 6 = 4 atoms). The part c of the Fig. 1.5 shows hexagonal cell.
We shall see that discrimination between simple and complex lattices is important, say,
in analysis of lattice vibrations.
The Wigner-Seitz cell
As we have mentioned, the procedure of choose of the elementary cell is not unique and
sometimes an arbitrary cell does not reflect the symmetry of the lattice (see, e. g., Fig. 1.6,
and 1.7 where specific choices for cubic lattices are shown). There is a very convenient procedure to choose the cell which reflects the symmetry of the lattice. The procedure is
as follows:
1. Draw lines connecting a given lattice point to all neighboring points.
2. Draw bisecting lines (or planes) to the previous lines.
The procedure is outlined in Fig. 1.8. For complex lattices such a procedure should be
done for one of simple sublattices. We shall come back to this procedure later analyzing
electron band structure.
The crystal periodicity leads to many important consequences. Namely, all the properties,
say electrostatic potential V , are periodic
It implies the Fourier transform. Usually the oblique co-ordinate system is introduced, the
axes being directed along ai. If we denote co-ordinates as ?s having periods as we get
V (r) =
? X
k1,k2,k3=??
Vk1,k2,k3 exp "2?i X s ka s? ss# . (1.4)
Then we can return to Cartesian co-ordinates by the transform
?i = X
k
?ikxk (1.5)
Finally we get
V (r) = X
b
Vbeibr . (1.6)
From the condition of periodicity (1.3) we get
V (r + an) = X
b
Vbeibreiban . (1.7)
We see that eiban should be equal to 1, that could be met at
ba1 = 2?g1, ba2 = 2?g2, ba3 = 2?g3 (1.8)
where gi are integers. It could be shown (see Problem 1.4) that
b
g ? b = g1b1 + g2b2 + g3b3 (1.9)
where
b1 = 2?[a2a3]
V0 , b2 = 2?[V a0 3a1], b3 = 2?[V a0 1a2] . (1.10)
It is easy to show that scalar products
aibk = 2??i,k . (1.11)
Vectors bk are called the basic vectors of the reciprocal lattice. Consequently, one can construct reciprocal lattice using those vectors, the elementary cell volume being (b1[b2, b3]) =
(2?)3/V0.
Reciprocal Lattices for Cubic Lattices. Simple cubic lattice (sc) has simple cubic
reciprocal lattice with the vectors’ lengths bi = 2?/ai. Now we demonstrate the general
procedure using as examples body centered (bcc) and face centered (fcc) cubic lattices.
First we write lattice vectors for bcc as
a1 =
a 2
(y + z ? x) ,
a2 =
a 2
(z + x ? y) ,
a1 =
a 2
(x + y ? z)