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Cubic systems Paul Sundaram University of Puerto Rico at Mayaguez

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

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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

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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=

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Simple cubic(P)

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Simple cubic

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Body centered cubic(I)

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Real picture

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Body centered cubic

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Face centered cubic(F)

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Real picture

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Face centered cubic

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*Highest packing possible in real structures

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Questions

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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)

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

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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

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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]

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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

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

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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)

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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}

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Example X Y 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) Family {111}

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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]

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

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