cos a ¼
0
1 þ 4 þ 0
ð
Þ
1=2 0 þ 0 þ 1
ð
Þ
1=2
¼
0
5
ð Þ
1=2 1
ð Þ
1=2
¼ 0
So that, a = 90°
and the angle between the edges with the end coordinates: 3, 1, 0 and 0, 0, 1
cos b ¼
0
9 þ 1 þ 0
ð
Þ
1=2 0 þ 0 þ 1
ð
Þ
1=2
¼
0
10
ð Þ
1=2 1
ð Þ
1=2
¼ 0
So that, b = 90°
(iii) Volume of the new unit cell with a = 16.16 Å, b = 13 Å, and c = 7 Å;
a = b = 90°, c = 45° is:
V n ¼ 16:16 Â 13 Â 7 1 À cos
2 45
À
Á 1=2 ¼ 1470:56 Â sin 45 ¼ 1039:84 ˚
A
3
(iv) Ratio of the two volumes is
V n
V o
¼
1040
210
¼ 4:95 ffi 5
) There are five lattice points in the new unit cell.
3.5 Atomic Density in Crystals
Sometimes, it is important to know the atomic density along a particular direction or
in a particular plane of the given crystal lattice. Similarly, sometimes it is also
important to know the atomic density in crystal to determine other related quantities. They are defined as follows:
(i) Linear Atomic Density
Linear atomic density q l is defined as the number of atomic diameter
intersected by the selected line in the given direction as
q l ¼
Number of atoms in the given direction ðlineÞ
Length of the given line
¼
n
L
ð3:17Þ
(ii) Planar Atomic Density
Planar atomic density q p is defined as the equivalent number of atoms whose
centers are intersected by the selected area in the given plane as
114
3 Unit Cell Calculations