face centered cubic, fcc

Atoms are arranged in a periodic pattern in a crystal.

The atomic arrangement affects the macroscopic properties of a material.

Crystals are relatively easy to model.

Many important materials (silicon, steel) are crystals

Institute of Solid State Physics

Crystal Structure Technische Universität Graz

body centered cubic, bcc

simple cubic

Crystals

unit cell

Bravais lattice Crystal

=

1 1 2 2 3 3r n a n a n a

1 2 3lattice vectors , ,a a a

a1

a3

a2

•Primitive Vectors:

a1  =  ½ a Y + ½ a Za2  =  ½ a X + ½ a Za3  =  ½ a X + ½ a Y

•Basis Vectors:

B1  =     0     (Na)   

B2  =  ½ a1 + ½ a2 + ½ a3  =  ½ aX + ½ aY + ½ aZ  (Cl)

Example NaCl

http://cst-www.nrl.navy.mil/lattice/struk/b1.html

14 Bravais lattices

http://en.wikipedia.org/wiki/Bravais_lattice

Points of a Bravais lattice do not necessarily represent atoms.

Unit Cell

Choice of unit cell is not unique

1 2 3a a a

volume of a unit cell =

diamond

a1

a3

a2

Wigner-Seitz Cells

bcc fcc

Rhombic dodecahedron

http://britneyspears.ac/physics/crystals/wcrystals.htmhttp://en.wikipedia.org/wiki/Rhombic_dodecahedronhttp://en.wikipedia.org/wiki/Truncated_octahedron

Truncated octahedron

Coordination number

Number of atoms touching one atom in a crystal

Diamond 4Graphite 3bcc 8fcc 12hcp 12sc 6

atomic packing density

HCP FCC

close packing density = 0.74random close pack = 0.64simple cubic = 0.52diamond = 0.34

From: Hall, Solid State Physics

Fcc

conventional unit cell showing close packed plane

Primitive unit cell Wigner-Seitz cell

Crystal planes and directions: Miller indices

bcc Wigner Seitz cell

KOH rapidly etches the Si <100> planes

[ ] specific direction< > family of equivalent directions( ) specific plane{ } family of equivalent planes

Cementite - Fe3C

Unit cell

cell 5.09000 6.74800 4.52300 90.000 90.000 90.000 natom 3Fe1 26 0.18600 0.06300 0.32800 Fe2 26 0.03600 0.25000 0.85200 C 6 0.89000 0.25000 0.45000 rgnr 62Cohenite (Cementite) Fe3 C

Asymmetric unit

Generated by PowderCell

GroupsCrystals can have symmetries: translation, rotation, reflection, inversion,...

1 0 00 cos sin0 sin cos

x xy yz z

Symmetries can be represented by matrices.All such matrices that bring the crystal into itself form the group of the crystal.

AB G for A, B G32 point groups (one point remains fixed during

transformation)230 space groups

http://www.pdb.org/robohelp/data_download/biological_unit/asymmetric_unit.htm

Asymmetric Unit

http://it.iucr.org/A/

simple cubic

http://cst-www.nrl.navy.mil/lattice/

Po

Number: 221

Primitive Vectors:

a1  =  a Xa2  =  a Ya3  =  a Z

•Basis Vector:

B1 = 0  

fcc

http://cst-www.nrl.navy.mil/lattice/

Al, Cu, Ni, Sr, Rh, Pd, Ag, Ce, Tb, Ir, Pt, Au, Pb, Th

Primitive Vectors:

a1 = ½ a Y + ½ a Za2 = ½ a X + ½ a Za3 = ½ a X + ½ a Y

Basis Vector:

B1 = 0  

Number 225

hcp

http://cst-www.nrl.navy.mil/lattice/

Mg, Be, Sc, Ti, Co, Zn, Y, Zr, Tc, Ru, Cd, Gd, Tb, Dy, Ho, Er, Tm, Lu, Hf, Re, Os, Tl

bcc

http://cst-www.nrl.navy.mil/lattice/

W Na K V CrFe Rb Nb Mo Cs Ba EuTa Primitive Vectors:

Basis Vector:

B1 = 0

a1  =  - ½ a X + ½ a Y + ½ a Z

a2  =  + ½ a X - ½ a Y + ½ a Z

a3  =  + ½ a X + ½ a Y - ½ a Z

NaCl

http://cst-www.nrl.navy.mil/lattice/

CsCl

http://cst-www.nrl.navy.mil/lattice/

perovskite

http://cst-www.nrl.navy.mil/lattice/

ybco

http://cst-www.nrl.navy.mil/lattice/

graphite

http://cst-www.nrl.navy.mil/lattice/

diamond

http://cst-www.nrl.navy.mil/lattice/

CSiGe

•Primitive Vectors:

•Basis Vectors:

Number: 227

a1  =  ½ a Y + ½ a Za2  =  ½ a X + ½ a Za3  =  ½ a X + ½ a Y

B1  =  - 1/8 a1 - 1/8 a2 - 1/8 a3  =  - 1/8 a X - 1/8 a Y - 1/8 aZ

B2  =  + 1/8 a1 + 1/8 a2 + 1/8 a3  =  + 1/8 a X + 1/8 a Y + 1/8 aZ

http://cst-www.nrl.navy.mil/lattice/

zincblende

ZnSGaAsInP

wurtzite

http://cst-www.nrl.navy.mil/lattice/

ZnOCdSCdSeGaNAlN

Quartz

http://cst-www.nrl.navy.mil/lattice/

body centered cubic, bcc

simple cubic face centered cubic, fcc