1. Paul Sundaram University of Puerto Rico at Mayaguez
Cubic systems
Paul Sundaram
University of Puerto Rico at Mayaguez

2. Review Seven crystal systems Fourteen Bravais lattices
4/15/2017
Review
Seven crystal systems
Fourteen Bravais lattices
Cubic and Hexagonal systems: 90% of all metals have a cubic or hexagonal structure

3. Cubic system characteristics
Unit cell a=b=c, a= b = g =90˚
Face diagonal and body diagonal
Number of atoms per unit cell
Coordination number:number of nearest neighbor atoms
Close-packed structures
Atomic Packing Factor (APF)
APF=(vol.of atoms in unit cell)/(vol. of unit cell)
Atom positions, crystallographic directions and crystallographic planes (Miller indices)
Planar atomic density & linear atomic density

4. Some concepts Number of atoms per unit cell
Corner atom = 1/8 per unit cell
Body centered atom = 1
Face centered atom = 1/2
Body diagonal=
Face diagonal=

5. Simple cubic(P)

6. Simple cubic

7. Simple cubic

8. Body centered cubic(I)

9. Real picture

10. Body centered cubic

11. Body centered cubic

12. Face centered cubic(F)

13. Real picture

14. Face centered cubic

15. Face centered cubic
*Highest packing possible in real structures

16. Questions

17. Atomic Positions Z (1/2,1/2,1) (0,1,1) (0,0,1) (1/2,1/2,1/2)
(1/2,0,1/2) Y
(0,0,0) X

18. Crystallographic directions
Concept of a vector & components R
R cos(90-f) f
R cos(f)

19. Examples Components X:a cos 90=0 Y:a cos 90=0 Z:a cos 0=a
Miller index:[001]
Examples
Components
X:a cos 90=0
Y:a cos 0=a
Z:a cos 90=0
Miller index:[010]
Components
X:a cos 0=a
Y:a cos 90=0
Z:a cos 90=0
Miller index:[100]
Components
X:a cos 90=0
Y:a cos 0=a
Z:a cos 90=0
Miller index:[010]
Family
<100> <010> <001>

20. Examples Components X: 0 Y: a Z: a Miller index:[011] Components X: a

21. Examples Components X: 0 Y: -a Z: -a Miller index:[0 1 1] Components
Family
<110> <011> <101>

22. Examples Components X: -a Y: -a Z: -a Miller index:[111] Components
Family
<111>

23. Crystallographic planes
1.Intersections with X,Y,Z axes
  1
2. Take the inverse
1/ 1/ 1/1
Miller index(0 0 1) Z
How to determine indices of plane
1.Intersections with X,Y,Z axes
1  
2. Take the inverse
1/1 1/ 1/
Miller index(1 0 0) Y X
1.Intersections with X,Y,Z axes
 1 
2. Take the inverse
1/ 1/1 1/
Miller index(0 1 0)
Family {100}

24. Example 1 1  1/1 1/1 1/  Miller index(1 1 0) Z
How to determine indices of plane
1.Intersections with X,Y,Z axes
1 1 
2. Take the inverse
1/1 1/1 1/ 
Miller index(1 1 0) Y X
Family {110}

25. Example 1 1 1 1/1 1/1 1/1 Miller index(1 1 1) Z
How to determine indices of plane
1.Intersections with X,Y,Z axes
1 1 1
2. Take the inverse
1/1 1/1 1/1
Miller index(1 1 1) Y X
Family {111}

26. Examples Components X: 1/2 Y: 1/2 Z: 1 [1/2 1/2 1] [112] Components
[-1 1 1/2]
[2 2 1]
Components
X: -1
Y: -1/2
Z: 1/2
[-1 -1/2 1/2]
[2 1 1]

27. Examples Intersections -1,-1,1/2 Inverse Intersections -1 -1 2
(1 1 2)
Intersections
1/2,1,1/2
Inverse
2 1 2
(212)
Intersections
1/6,-1/2,1/3
Inverse
(6 2 3)
Intersections
-1/2,1/2,1
Inverse
-2 2 1
(2 2 1)