This document discusses the four types of unit cell arrangements in crystals - primitive, body centered, face centered, and end centered. It also examines coordination numbers, packing fractions, radius ratios, and the relationships between unit cell shape and sphere arrangement. Key details covered include that primitive unit cells have a coordination number of 6, face centered unit cells have the highest packing fraction of 74%, and different radius ratios correspond to specific coordination numbers.

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