Institute of Solid State Physics
Technische Universität Graz
Crystal Structure
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
simple cubic
body centered
cubic, bcc
face centered
cubic, fcc
Crystals
=
unit cell
Bravais lattice
Crystal
lattice vectors a1 , a2 , a3
a3
r  n1a1  n2a2  n3a3
a1
a2
Example NaCl
•Primitive Vectors:
a1
=
½aY+½aZ
a2
=
½aX+½aZ
a3
=
½aX+½aY
•Basis Vectors:
B1
=
B2
=
0
½ a1 + ½ a2 + ½ a3
=
(Na)
½ aX + ½ aY + ½ aZ
http://cst-www.nrl.navy.mil/lattice/struk/b1.html
(Cl)
14 Bravais lattices
Points of a Bravais lattice do not
necessarily represent atoms.
http://en.wikipedia.org/wiki/Bravais_lattice
Unit Cell
a3
Choice of unit cell is not unique
a1
a2
volume of a unit cell =
a1  a2  a3
diamond
Wigner-Seitz Cells
bcc
Truncated octahedron
fcc
Rhombic dodecahedron
http://britneyspears.ac/physics/crystals/wcrystals.htm
http://en.wikipedia.org/wiki/Rhombic_dodecahedron
http://en.wikipedia.org/wiki/Truncated_octahedron
Coordination number
Number of atoms touching one atom in a crystal
Diamond 4
Graphite 3
bcc 8
fcc 12
hcp 12
sc 6
atomic packing density
HCP
FCC
close packing density = 0.74
random close pack = 0.64
simple cubic = 0.52
diamond = 0.34
Fcc
conventional unit cell
showing close packed
plane
Primitive unit cell
Wigner-Seitz cell
From: Hall, Solid State Physics
Crystal planes and directions: Miller indices
[ ] specific direction
< > family of equivalent directions
( ) specific plane
{ } family of equivalent planes
bcc Wigner Seitz cell
KOH rapidly etches the Si <100> planes
Cementite - Fe3C
Unit cell
cell 5.09000 6.74800 4.52300 90.000 90.000 90.000
natom 3
Fe1 26 0.18600 0.06300 0.32800
Fe2 26 0.03600 0.25000 0.85200
C 6 0.89000 0.25000 0.45000
rgnr 62
Cohenite (Cementite) Fe3 C
Asymmetric unit
Generated by PowderCell
Groups
Crystals can have symmetries: translation,
rotation, reflection, inversion,...
0
0  x 
 x   1
  
 
 y    0 cos  sin   y 
 z    0  sin  cos   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  G
32 point groups (one point remains fixed during transformation)
230 space groups
Asymmetric Unit
http://www.pdb.org/robohelp/data_download/biological_unit/asymmetric_unit.htm
http://it.iucr.org/A/
simple
cubic
Po
Number: 221
Primitive Vectors:
a1
=
aX
a2
=
aY
a3
=
aZ
•Basis Vector:
B1 = 0
http://cst-www.nrl.navy.mil/lattice/
fcc
Al, Cu,
Ni, Sr,
Rh, Pd,
Ag, Ce,
Tb, Ir,
Pt, Au,
Pb, Th
Number 225
Primitive Vectors:
a1 = ½ a Y + ½ a Z
a2 = ½ a X + ½ a Z
a3 = ½ a X + ½ a Y
Basis Vector:
B1 = 0
http://cst-www.nrl.navy.mil/lattice/
hcp
Mg, Be,
Sc, Ti,
Co, Zn,
Y, Zr, Tc,
Ru, Cd,
Gd, Tb,
Dy, Ho,
Er, Tm,
Lu, Hf,
Re, Os,
Tl
http://cst-www.nrl.navy.mil/lattice/
bcc
W
Na
K
V
Cr
Fe
Rb
Nb
Mo
Cs
Ba
Eu
Ta
Primitive Vectors:
a1
=
-½aX+½aY+½aZ
a2
=
+½aX-½aY+½aZ
a3
=
+½aX+½aY-½aZ
Basis Vector:
B1 = 0
http://cst-www.nrl.navy.mil/lattice/
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
C
Si
Ge
Number: 227
•Primitive Vectors:
a1
=
½aY+½aZ
a2
=
½aX+½aZ
a3
=
½aX+½aY
•Basis Vectors:
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
ZnS
GaAs
InP
http://cst-www.nrl.navy.mil/lattice/
wurtzite
ZnO
CdS
CdSe
GaN
AlN
http://cst-www.nrl.navy.mil/lattice/
Quartz
http://cst-www.nrl.navy.mil/lattice/
simple cubic
body centered
cubic, bcc
face centered
cubic, fcc