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

3. Layer arrangement view

4. Primitive Cubic

5. Primitive Cubic Volume occupied by a sphere in the unit cell
Total volume occupied by all the spheres in the unit cell
Primitive Cubic

6. Primitive Cubic Packing Fraction
Fraction of the Unit cell’s volume occupied by the spheres
Primitive Cubic

7. Coordination number 6
Primitive Cubic

8. Body Centered Cubic
Unit Cell shape view
Unit Cell arrangement view

9. Layer arrangement view

10. a
> 2r
Body Centered Cubic

11. Body Centered Cubic

12. Body Centered Cubic Packing Fraction
Volume occupied by a corner sphere in the unit cell
Volume occupied by the central sphere in the unit cell
Total Volume occupied by the spheres in the unit cell
Packing Fraction
Body Centered Cubic

13. Coordination number 8
Body Centered Cubic

14. Face Centered Cubic
Unit Cell shape view
Unit Cell arrangement view

15. a
Face Centered Cubic

16. Face Centered Cubic

17. Highest Packing Fraction of all shapes and of all arrangements
Volume occupied by a corner sphere in the unit cell
Volume occupied by a face centered sphere in the unit cell
Total Volume occupied by the spheres in the unit cell
Packing Fraction
Highest Packing Fraction of all shapes and of all arrangements
Face Centered Cubic

18. Face Centered Cubic Coordination number y-z plane x-y plane x-z plane

19. Coordination number
a/2
a/2
Face Centered Cubic

20. End Centered

21. Radius Ratios Minimum rc/ra for various coordination numbers

22. Radius Ratios Coordination number - 4 (0,0,0) A B (a/4, a/4,a/4)
z-axis
Radius Ratios

23. Coordination number - 4 (square planar) or 6 (octahedron)
Radius Ratios

24. Coordination number - 8 (cube)
Radius Ratios

25. Radius Ratios Final Radius Ratios Co-ordination No.
Radius Ratio, rc/ra
Co-ordination No. 2
<0.155
[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
Radius Ratios