We know that the density of crystal material is given by
q ¼
Mass
Volume
¼
M Â n
N Â a 3
) a
3
¼
M Â n
N Â q
or
a ¼
M Â n
N Â q
1=3
¼
2 Â 55:85
6:023 Â 10 26 Â 7860
1=3
¼ 2:86 Â 10
À10 m ¼ 2:86 ˚
A
Further, in a bcc structure, we have
4r ¼
ffiffi ffi
3
p
a
where r is the radius of the atom. Therefore,
r ¼
ffiffi ffi
3
p  2:86
4
¼ 1:24 ˚
A
Example 6 Zinc has hcp structure. The height of its unit cell is 4.94 Å, nearest
neighbor distance 2.7 Å and the atomic weight of zinc 65.38, respectively.
Calculate the density of zinc.
Solution: Given: a ¼ b ¼ 2:7 ˚
A ¼ 2:7 Â 10
À10 m, c ¼ 4:94 ˚
A ¼ 4:94 Â 10
À10 m,
at. wt. of Zn = 65.37, structure is hcp, so that n = 6, q ¼ ?
We know that the volume of hcp cell is given by
V ¼
3
ffiffi ffi
3
p
a
2 c
2
¼
3 Â 1:732 Â 2:7 Â 10
À10
ð
Þ
2 Â4:94 Â 10
À10
2
¼ 93:56 Â 10
À30 m
3
Now, the density of the crystal material is obtained by
368
9 Determination of Crystal Structure Parameters