1. There are not more than 4 ways of arranging
spheres in any shape of unit cell
These are Primitive, Body Centered, Face
Centered & End Centered

2. a = 2r
1
2
4
3
5
6
7
8
Unit Cell shape view
Unit Cell arrangement view

3. Layer arrangement view

5. Volume occupied by a sphere in the unit cell V
8
1

Total volume occupied by all the spheres in the unit
cell
V
V 1
8
8
1




6. Packing Fraction
Fraction of the Unit cell’s volume occupied by the
spheres
cell
unit
of
volume
cell
unit
the
inside
spheres
the
by
occupied
Volume

3
3
3
4
a
r


 3
3
2
3
4
r
r


52
.
0


7. Coordination number
6

8. Unit Cell shape view
Unit Cell arrangement view

9. Layer arrangement view

10. a
r 3
4 
2
3
2
a
r  a

a > 2r

12. Packing Fraction
3
3
3
4
3
4
2








r
r

68
.
0

Volume occupied by a corner sphere in the unit cell V
8
1

Volume occupied by the central sphere in the unit cell V
1

Total Volume occupied by the spheres in the unit cell V
2

Packing Fraction

13. Coordination number
8

14. Unit Cell shape view
Unit Cell arrangement view

15. a
a
r 2
4 
2
2
a
r  a

r
2


17. Packing Fraction
3
3
2
4
3
4
4








r
r

74
.
0

Volume occupied by a corner sphere in the unit cell V
8
1

Volume occupied by a face centered sphere in the unit
cell
V
2
1

Total Volume occupied by the spheres in the unit cell
4
2
1
6
8
1
8 




Packing Fraction
Highest Packing Fraction of all shapes and of
all arrangements

18. Coordination number
x
y
z
y-z plane
x-z plane x-y plane

19. Coordination number
a/2
a/2
2
a

21. Minimum rc/ra for various coordination numbers
2r
a
B
O
A
3
2
30
sec 


 
a
a
c
r
r
r
AB
OA
155
.
0
1
3
2



a
c
r
r
Coordination number - 3

22. 4
3
4
4
4
2
2
2
a
a
a
a
r
r
AB a
c 























a
r
a 4
2 
Coordination number - 4
z-
axis
A
B
(0,0,0)
(a/4, a/4,a/4)








2
4
4
3 a
a
c
r
r
r
a
a
c r
r
r
2
3


2
3
1 

a
c
r
r
225
.
0
1
2
3



a
c
r
r

23. Coordination number - 4 (square planar) or 6 (octahedron)
B
A 2
a
r
r
AB a
c 


a
r
a 2

a
a
a
c r
r
r
r 2
2
2




414
.
0
1
2 



a
c
r
r

24. Coordination number - 8 (cube)
a
c r
r
a


2
3
a
r
a 2

  a
c
a r
r
r 

 2
2
3
732
.
0
1
3 



a
c
r
r

25. Final Radius Ratios
Radius Ratio, rc/ra
Co-ordination No.
<0.155 2
[0.155, 0.225) 2 or 3
[0.225, 0.414) 2 or 3 or 4 Td
[0.414, 0.732) 2 or 3 or 4 Td, 4 sq. pl or 6 Oh
[0.732, 0.99) 2 or 3 or 4 Td, 4 sq. pl or 6 Oh
or 8