Cubic systems Paul Sundaram University of Puerto Rico at Mayaguez
Feb 13, 2016
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)