Atomic arrangement
• Short range order
(amorphous materials)
(c) 2003 Brooks/Cole
Publishing / Thomson
Learning™
Atomic arrangement
• Long range order
• Positions of the atoms
can be described by
simple translation
Crystals
Translation
Simple cubic lattice
• a=b=c
• ===90
• Po
Tetragonal lattice
• a=bc
• ===90
• In, Sn (if T13 )
Orthorombic lattice
• a  bc
• ===90
• Ga, U
Rombohedral lattice
• a=b=c
•   90,   90,
  90
• Hg, Bi, As
• sometimes called
„trigonal”
Hexagonal lattice
• a=b  c
•  = =90, =120
• Cd, Mg, Zn, graphite
Monoclinic lattice
• a  bc
•   90,   90,
 = 90
• sulfur
Triclinic lattice
• a  bc
•   90,   90,
  90
• Se, Te
The 14 Bravais-lattice
Primitive vs.non-primitive lattice
As primitive:
rombohedral
As non-primitive: face
centered cubic (FCC)
Miller-indices
• Points
• Directions
• Planes
For cubic lattices only
[hkl]  (hkl) !!!
Angle between two planes in
cubic systems
Angle between two planes:
since
(h1k1l1)[h1k1l1] és (h2k2l2)[h2k2l2],
and    
r1  r2  r1  r2 cos  ,
 
r1  r2
h1  h2  k1  k2  l1  l2
cos     
r1  r2
h12  k12  l12  h22  k22  l22
Interception line of two planes
in cubic systems
The interception line of two planes is the
vector product of the normals of the planes:

i

j

k
 
r1  r2  h1
k1



l1  i (k1l2  l1k 2 )  j (h1l2  l1h2 )  k (h1k 2  k1h2 )
h2
k2
l2
Distances between lattice planes,
only for cubic!
d
a
h k l
2
2
2
TKK (110)
a
a 2
d110 

2
2
Crystallographic properties
coordination number
number of atoms in the unit cell
atomic diameter vs. lattice constant
atomic packaging factor (APF)
largest free space (diameter, place)
closest packed direction, plane
planar density (PD)
linear density (LD)
Coordination number
(c) 2003 Brooks/Cole Publishing / Thomson
Learning™
Primitive cubic
Body centered cubic (BCC)
Number of atoms in the unit cell
Atomic diameter vs. lattice constant
(c) 2003 Brooks/Cole Publishing / Thomson Learning™
SC = Simple Cubic
BCC = Body Centered Cubic
FCC = Face Centered Cubic
Atomic packaging factor (APF)
APF= total volume of atoms / volume of cell
a = lattice constant = edge of the unit cell
d = atomic diameter
E.g. BCC → 2 atoms in a cell
d 
2
6

TT 
a3
3
  a 3 
 

2


3a 3
3

a 3 3  3


 0.68
3
8a
8
Largest free space
(c) 2003 Brooks/Cole Publishing / Thomson Learning™
Primitive cubic
Lattice
Metals
Coord. No.
PC
Po
6
Atomic
diameter
a
Number of
atoms
1
APF
0,52
Largest
free space
0,73 a
in the centre
Closest
packed
{100}
<100>
Body centered cubic (BCC)
Lattice
Metals
Coord. No.
Atomic
diameter
BCC
Na, K, Cr,
Mo, W, Ti,
Fe
8
3
a
2
Number of
atoms
2
Low formability, oxidative,
low conductivity, brittle-ductile transition
APF
0,68
Largest
free space
0,252 a
½¼0
Closest
packed
{
1
1
0
}
<
1
1
1
>
Face centered cubic (FCC)
Lattice
Metals
Coord. No.
Atomic
diameter
FKK
Cu, Au, Ag,
Pb, Ni, Pt,
Fe
12
2
a
2
Number of
atoms
4
APF
0,74
Maximális!
Largest
free space
0,293 a
½00
½
½
½
Good formability, chemical stability, good conductivity
Closest
packed
{
1
1
1
}
<
1
1
0
>
Diamond lattice
Lattice
Metals
Diamond
C, Si, Ge,
Sn
Coord.
No.
4
Atomic
diameter
3
a
4
Number of
atoms
8
APF
Closest packed
0,34
{111}
<110>
Do not touch each other!
Hexagonal close packed (HCP)
Lattice
Metals
Cord. nr.
Atomic
diameter
Nr. of
atoms
APF
Largest
free space
HCP
Be, Mg, Zn,
Cd, Ti
12
c/a=1,63
6
0,74
Max!
0,235 a
Closest
packed
plane and
direction
{0001}
<1120>
Comparioson of FCC and HCP
(hexagonal close packed) lattice
ABCABC
ABABAB
Coordination number
Coordination number
Coordination number