¼
3a
16
2
À
a
8
2 ¼
a
4
2
or d 2 ¼
a
2
Therefore, the ratio of
d 1
d 2
¼
a/
ffiffi ffi
2
p
a/2
¼
ffiffi ffi
2
p
1
Example 7 Construct a Wigner–Seitz unit cell for a simple hexagonal lattice.
Solution: Draw two simple hexagonal unit cells one above the other as done in fcc.
The atom lying at the center of the middle hexagonal plane is taken as the reference
atom. Connect this atom with all first nearest neighbor atoms. Mark the midpoints
on these lines. Draw planes through these midpoints. Volume enclosed by the
intersection of these planes (the resulting shape is a simple hexagon) is the required
Wigner–Seitz unit cell. The construction of W–S cell is shown in Fig. 2.7.
Example 8 Construct a Wigner–Seitz unit cell for a trigonal lattice.
Solution: Draw two trigonal unit cells one above the other. The atom lying at the
center of the middle hexagonal plane is taken as the reference atom. Connect this
atom with all first nearest neighbor atoms. Mark the midpoints on these lines. Draw
planes through these midpoints. Volume enclosed by the intersection of these
planes (the resulting polyhedral shape) is the required Wigner–Seitz unit cell. The
construction of W–S cell is shown in Fig. 2.8.
Example 9 Construct a Wigner–Seitz unit cell for CsCl lattice.
Solution: Draw 27 CsCl unit cells and consider the Cs
+ ion at the body-centered
position of the central CsCl unit cell as the reference ion. Connect this with six
other neighboring Cs
+ ions. Mark midpoints on these lines. Draw planes through
Fig. 2.6 Showing top view
of the parallelogram in
Fig. 2.5c
46
2 Unit Cell Construction