Solved Examples
Example 1 Determine the radius of the Fermi circle for a monovalent metal. Show
that this circle lies well within the first B-Z obtained from a square lattice of side a.
Solution: Given: 2-D square lattice of side a, monovalent metal, k F = ?
We know that the reciprocal lattice of a square lattice of side a is a square lattice of
side 2p/a. Thus the area of the first B-Z is given by
A ¼
4p
2
a 2
Further, for a monovalent metal, the area occupied by an electron is half the area
of the first B-Z (allowing two possible spin states for each electron). Therefore,
pk
2
F ¼
1
2
Â
4p
2
a 2
or k F ¼
ffiffiffi
2
p
r p
a
¼ 0:798
p
a
where p/a is the distance of the zone boundary from the center of the zone. Also, the
value of k F lies between
0\k F \
p
a
This indicates that the Fermi circle lies well within the first B-Z, as shown in
Fig. 2.29.
Example 2 Determine the radius of the Fermi circle for a divalent metal. Show
that this circle does not remain confined to the first B-Z obtained from a square
lattice of side a.
Solution: Given: 2-D square lattice of side a, divalent metal, k F = ?
We know that for a square lattice of side a, the corresponding area of the first
B-Z is given by
A ¼
4p
2
a 2
Further, for a divalent metal, the area occupied by two electrons is equal to the
area of the first B-Z (allowing two possible spin states for each electron). Therefore,
pk
2
F ¼
2
2
Â
4p
2
a 2
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2 Unit Cell Construction