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Cubic systems Paul Sundaram University of Puerto Rico at Mayaguez.

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Presentation on theme: "Cubic systems Paul Sundaram University of Puerto Rico at Mayaguez."— Presentation transcript:

1 Cubic systems Paul Sundaram University of Puerto Rico at Mayaguez

2 Review n Seven crystal systems n Fourteen Bravais lattices n Cubic and Hexagonal systems: 90% of all metals have a cubic or hexagonal structure

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

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

5 Simple cubic(P)

6 Simple cubic

7

8 Body centered cubic(I)

9 Real picture

10 Body centered cubic

11

12 Face centered cubic(F)

13 Real picture

14 Face centered cubic

15 *Highest packing possible in real structures

16 Questions

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

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

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

20 Examples Components X: a Y: a Z: 0 Miller index:[110] Components X: 0 Y: a Z: a Miller index:[011] Components X: a Y: 0 Z: 1 Miller index:[101]

21 Examples Components X: -a Y: -a Z: 0 Miller index:[1 1 0] Components X: 0 Y: -a Z: -a Miller index:[0 1 1] Components X: -a Y: 0 Z: -a Miller index:[1 0 1] Family

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

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

24 Example X Y 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) Family {110}

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

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

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


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