Cubic systemsPaul SundaramUniversity of Puerto Rico at Mayaguez

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, = = =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

Face diagonal=

Body diagonal= 3a

2a

Simple cubic(P)

Number of atomsper unit cell 1/8 X 8 = 1

Coordinationnumber

Atomic packingfactor

Simple cubic

Number of atomsper unit cell 1/8 X 8 = 1

Coordinationnumber 6

Atomic packingfactor

Simple cubic

Number of atomsper unit cell 1/8 X 8 = 1

Coordinationnumber 6

Atomic packingfactor

0.52

Body centered cubic(I)Number of atoms

per unit cell 1/8 X 8 + 1 = 2

Coordinationnumber

Atomic packingfactor

Real picture

Body centered cubicNumber of atoms

per unit cell 1/8 X 8 + 1 = 2

Coordinationnumber 8

Atomic packingfactor

Body centered cubicNumber of atoms

per unit cell 1/8 X 8 + 1 = 2

Coordinationnumber 8

Atomic packingfactor

0.68

Face centered cubic(F)

Number ofatoms per unit

cell1/8 X 8 + 1/2 X 6 = 4

Coordinationnumber

Atomic packingfactor

Real picture

Face centered cubic

Number ofatoms per unit

cell1/8 X 8 + 1/2 X 6 = 4

Coordinationnumber 12

Atomic packingfactor

Face centered cubic

Number ofatoms per unit

cell1/8 X 8 + 1/2 X 6 = 4

Coordinationnumber 12

Atomic packingfactor

0.74*

*Highest packing possible in real structures

Questions

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)

Crystallographic directions

R

R cos()

R cos(90-)

Concept of a vector & components

Examples

ComponentsX:a cos 0=aY:a cos 90=0Z:a cos 90=0Miller index:[100]

ComponentsX:a cos 90=0Y:a cos 90=0Z:a cos 0=aMiller index:[001]

ComponentsX:a cos 90=0Y:a cos 0=aZ:a cos 90=0Miller index:[010]

ComponentsX:a cos 90=0Y:a cos 0=aZ:a cos 90=0Miller index:[010]

Family<100> <010> <001>

Examples

ComponentsX: aY: aZ: 0Miller index:[110]

ComponentsX: 0Y: a Z: aMiller index:[011]

ComponentsX: aY: 0Z: 1Miller index:[101]

Examples

ComponentsX: -aY: -aZ: 0Miller index:[1 1 0]

ComponentsX: 0Y: -a Z: -aMiller index:[0 1 1]

ComponentsX: -aY: 0Z: -aMiller index:[1 0 1]

Family<110> <011> <101>

Examples

ComponentsX: aY: aZ: aMiller index:[111]

ComponentsX: -aY: -aZ: -aMiller index:[111]

Family<111>

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 12. Take the inverse 1/ 1/ 1/1 Miller index(0 0 1)

Example

X

Y

ZHow 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}

Example

X

Y

ZHow to determine indices of plane 1.Intersections with X,Y,Z axes

1 1 12. Take the inverse

1/1 1/1 1/1

Miller index(1 1 1)Family {111}

Examples

ComponentsX: -1Y: 1Z: 1/2[-1 1 1/2][2 2 1]

ComponentsX: 1/2Y: 1/2Z: 1[1/2 1/2 1][112]

ComponentsX: -1Y: -1/2Z: 1/2[-1 -1/2 1/2][2 1 1]

Examples Intersections1/2,1,1/2Inverse2 1 2(212)

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

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

Intersections1/6,-1/2,1/3Inverse6 -2 3(6 2 3)