Paul Sundaram University of Puerto Rico at Mayaguez

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Presentation transcript:

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

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

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

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=

Simple cubic(P)

Simple cubic

Simple cubic

Body centered cubic(I)

Real picture

Body centered cubic

Body centered cubic

Face centered cubic(F)

Real picture

Face centered cubic

Face centered cubic *Highest packing possible in real structures

Questions

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

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

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>

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

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

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

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}

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}

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}

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]

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 (6 2 3) Intersections -1/2,1/2,1 Inverse -2 2 1 (2 2 1)