these midpoints. Volume enclosed by the intersection of these planes (the resulting
shape is a rhombic dodecahedron) is the required Wigner–Seitz unit cell. The
three-step process for its construction is shown in Fig. 2.5.
Example 6 Show that the ratio of the lengths of the diagonals of each parallelogram face of the Wigner–Seitz unit cell for fcc lattice is
ffiffi ffi
2
p : 1.
Proof: Consider the Wigner–Seitz unit cell constructed for fcc in Fig. 2.5c. Further,
consider triangular region lying over a parallelogram face, shown separately
(Fig. 2.6) whose base is one of the diagonals of the parallelogram. Other dimensions are known through its construction. To determine the diagonal, we can write
x
2
¼
a
2
2 À
ffiffi ffi
2
p
a
4
2
¼
a
4
2 À
a
8
2
or x ¼
a
2
ffiffi ffi
2
p
and 2x ¼ d 1 say
ð Þ ¼
a
ffiffi ffi
2
p
Further, the side of the small cube is a/2. Therefore, its body diagonal is
ffiffi ffi
3
p
a=2
ð Þ. However, the side of the parallelogram is half the body diagonal of the
cube, which is
ffiffi ffi
3
p
a=4
ð Þ as shown in Fig. 2.6. To determine the other diagonal say
d 2
ð Þ we can write
d 2
2
2
¼
ffiffi ffi
3
p
a
4
2
À
a
2
ffiffi ffi
2
p
2
Fig. 2.5 Three-step process of Wigner–Seitz unit cell in fcc
2.1 Construction of Wigner–Seitz Unit Cells
45