Example 6 Calculate the volume atomic density in (atom/ m
3 ) of fcc nickel whose
atomic mass is 58.71 kg/k mol and density 8.94 Â 10
3 kg/m
3 .
Solution: Given: Crystal structure is fcc, so that n = 4, atomic mass = 63.54 kg/k
mol, density = 8.94 Â 10
3 kg/m
3 , q v ¼ ?
We know that the volume atomic density in a crystal is given by
q v ¼
qN
M
¼
8:94 Â 10
3
 6:023  10
26
58:71
¼ 9:17 Â 10
28 atoms=m
3
Example 7 Calculate the volume atomic density in (atom/ m
3 ) of fcc copper whose
lattice parameter is 3.61 Å and atomic mass 63.54 kg/k mol.
Solution:
Given:
Crystal
structure
is
fcc,
so
that
n = 4,
a = 3.61 Å = 3:61 Â 10
À10 m, atomic mass = 63.54 kg/k mol, q v ¼ ?
We know that the volume atomic density in a crystal is given by
q v ¼
qN
M
¼
n
a 3 ¼
1
3:61 Â 10 À10
ð
Þ
3
¼ 8:50 Â 10
28 atoms=m
3
Example 8 Calculate the atomic density in (100), (110) and (111) planes of a
simple cubic structure whose lattice parameter is 2.5 Å.
Solution: Given: Crystal planes are: (100), (110) and (111). Crystal structure is
simple cubic, so that n = 1, a = 2.5 Å = 2:5 Â 10
À10 m, q p ¼ ?
We know that the atomic density in a crystal plane is given by
q p ¼
nd
V
Further, for simple cubic structure
d 100 ¼ a ; d 110 ¼
a
ffiffi ffi
2
p and d 111 ¼
a
ffiffi ffi
3
p ; V ¼ a
3
Therefore,
q ð100Þ ¼
1 Â d 100
a 3
¼
a
a 3 ¼
1
a 2 ¼
1
2:5 Â 10 À10
ð
Þ
2
¼ 1:60 Â 10
19 atoms=m
2
Similarly,
q ð110Þ ¼
1 Â d 110
a 3
¼
a
ffiffi ffi
2
p
a 3
¼
1
ffiffi ffi
2
p
a 2
¼
1
ffiffi ffi
2
p
2:5 Â 10 À10
ð
Þ
2
¼ 1:13 Â 10
19 atoms=m
2
q ð111Þ ¼
1 Â d 111
a 3
¼
a
ffiffi ffi
3
p
a 3
¼
1
ffiffi ffi
3
p
a 2
¼
1
ffiffi ffi
3
p
2:5 Â 10 À10
ð
Þ
2
¼ 9:24 Â 10
18 atoms=m
2
118
3 Unit Cell Calculations