these midpoints. Volume enclosed by the intersection of these planes (the resulting
polyhedral shape for either ionic species) is the required Wigner–Seitz unit cell
shown in Fig. 2.9.
Example 10 Construct a Wigner–Seitz unit cell for NaCl lattice.
Solution: The Na
+ ion lying at the center is taken as the reference ion. Connect this
atom with all first nearest neighbor Na
+ ions. Mark the midpoints on these lines.
Draw planes through these midpoints. Volume enclosed by the intersection of these
planes (the resulting rhombic dodecahedral shape) is the required Wigner–Seitz unit
cell shown in Fig. 2.10.
2.2 Construction of Reciprocal Lattice
(a) For a given direct (primitive) lattice (irrespective of its dimension) there exists a
corresponding reciprocal lattice. It is interesting to note that both direct and
reciprocal lattices exhibit the same symmetries and hence can be represented by
the same type of coordinate axes. For a given direct lattice, the corresponding
reciprocal lattice can be obtained according to the following procedure.
Fig. 2.9 Three-step process
of Wigner–Seitz unit cell in
CsCl
48
2 Unit Cell Construction