Chapter 2
Unit Cell Construction
2.1 Construction of Wigner–Seitz Unit Cells
A Wigner–Seitz unit cell is an alternative way of selecting a primitive unit cell of
area ð~ a  ~ b ¼ ab sin cÞ
or volume (~ a: ~ b Â~ c ¼ abc 1 À cos
2
a À cos
2
b À cos
2
c þ 2 cos a cos b cos c
ð
Þ
1=2 )
equal to other conventional unit cell in a given (2-D or 3-D) lattice. This unit cell is
constructed around a lattice point according to the following procedure:
1. Select a reference point in a given lattice and draw lines to connect this point
with all other nearest lattice points.
2. At mid points of each line segment between the reference and neighboring
lattice points, draw a line (in 2-D) and a plane (in 3-D) as perpendicular bisector.
3. Smallest area (in 2-D) and volume (in 3-D) enclosed around the reference lattice
point gives the required Wigner–Seitz primitive unit cell.
The characteristic feature of a Wigner–Seitz unit cell is that it retains the symmetry of the original lattice.
Solved Examples
Example 1 Draw Wigner–Seitz unit cell for each of the five plane lattices.
Solution: Consider each lattice one by one. In each case, take the central atom as
the reference atom and proceed according to the above said procedure. Mark the
midpoint of each line. Draw new lines through their midpoints. Join the new lines
with each other to get the required Wigner–Seitz unit cells for five lattices as shown
in Fig. 2.1 in terms of three-step process.
Example 2 Construct a Wigner–Seitz unit cell for a simple cubic lattice.
Solution: Ideally eight simple cubes are needed to be drawn side by side to get the
central atom and all other six neighboring atoms as shown in Fig. 2.2a. However, to
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
M. A. Wahab, Numerical Problems in Crystallography,
https://doi.org/10.1007/978-981-15-9754-1_2
41
Unit Cell Construction
2.1 Construction of Wigner–Seitz Unit Cells
A Wigner–Seitz unit cell is an alternative way of selecting a primitive unit cell of
area ð~ a  ~ b ¼ ab sin cÞ
or volume (~ a: ~ b Â~ c ¼ abc 1 À cos
2
a À cos
2
b À cos
2
c þ 2 cos a cos b cos c
ð
Þ
1=2 )
equal to other conventional unit cell in a given (2-D or 3-D) lattice. This unit cell is
constructed around a lattice point according to the following procedure:
1. Select a reference point in a given lattice and draw lines to connect this point
with all other nearest lattice points.
2. At mid points of each line segment between the reference and neighboring
lattice points, draw a line (in 2-D) and a plane (in 3-D) as perpendicular bisector.
3. Smallest area (in 2-D) and volume (in 3-D) enclosed around the reference lattice
point gives the required Wigner–Seitz primitive unit cell.
The characteristic feature of a Wigner–Seitz unit cell is that it retains the symmetry of the original lattice.
Solved Examples
Example 1 Draw Wigner–Seitz unit cell for each of the five plane lattices.
Solution: Consider each lattice one by one. In each case, take the central atom as
the reference atom and proceed according to the above said procedure. Mark the
midpoint of each line. Draw new lines through their midpoints. Join the new lines
with each other to get the required Wigner–Seitz unit cells for five lattices as shown
in Fig. 2.1 in terms of three-step process.
Example 2 Construct a Wigner–Seitz unit cell for a simple cubic lattice.
Solution: Ideally eight simple cubes are needed to be drawn side by side to get the
central atom and all other six neighboring atoms as shown in Fig. 2.2a. However, to
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
M. A. Wahab, Numerical Problems in Crystallography,
https://doi.org/10.1007/978-981-15-9754-1_2
41