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**Paul Sundaram University of Puerto Rico at Mayaguez**

Cubic systems Paul Sundaram University of Puerto Rico at Mayaguez

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**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

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**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

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**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=

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Simple cubic(P)

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Simple cubic

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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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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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Face centered cubic *Highest packing possible in real structures

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Questions

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**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

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**Crystallographic directions**

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

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**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>

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

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

Family <110> <011> <101>

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

Family <111>

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**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}

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**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}

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**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}

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**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]

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**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)

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