Metals
Ionic Compounds
anion
cation
Radius Ratio Rules
CN (cation)
2
Geometry
linear
min rc/RA (f)
3
trigonal planar
0.155
4
tetrahedral
sites occur within
0.225
close-packed arrays
6
octahedral
0.414
8
cubic
12
cubo-octahedral
none
common in ionic
compounds
0.732
1
if rc is smaller than fRA, then the space is too big and the structure is unstable
Summary: Sites in HCP & CCP
2 tetrahedral sites / close-packed atom
1 octahedral site / close-packed atom
sites are located between layers: number of sites/atom same for ABAB & ABCABC
Common Ionic Structure Types
• Rock salt (NaCl) sometimes also ‘Halite’
– Derive from cubic-close packed array of Cl-
• Zinc blende (ZnS)
– Derive from cubic-close packed array of S=
• Fluorite (CaF2)
– Derive from cubic-close packed array of Ca2+
• Cesium chloride (CsCl)
– Not derived from a close-packed array
• Complex oxides
– Multiple cations
Example: CaF2 (Fluorite)
•
F-
~ 1.3 Å;
Ca2+
~ 1.0 Å;
– rc/RA = 0.77
• Ca2+ is big enough for CN = 8
CN
f
4
0.225
6
0.414
8
0.732
– But there are no 8-fold sites in close-packed arrays
• Consider structure as CCP cations
– F- in tetrahedral sites
– RA / rc> 1  fluorine could have higher CN than 4
• Ca:F = 1:2  all tetrahedral sites filled
• Places Ca2+ in site of CN = 8
• Why CCP not HCP? - same reason as NaCl
Fluorite
Ca2+
FCN(F-) = 4
CN(Ca2+) = 8 [target]
CsCl
• Cl- ~ 1.8 Å; Cs+ ~ 1.7 Å;
– rc/RA = 0.94
• Cs+ is big enough for CN = 8
– But there are no 8-fold sites in close-packed arrays
• CsCl unrelated to close-packed structures
– Simple cubic array of anions
– Cs+ in cuboctahedral sites
– RA / rc> 1  chlorine ideally also has large CN
• Ca:Cl = 1:1  all sites filled
Cesium Chloride
Cl-
1 Cs+/unit cell
1 Cl-/unit cell
CN(Cs) = 8
Cs+
Why do ionic solids stay bonded?
electrostaic
• Pair: attraction only
E pair

 Z1Z 2 e
2
4  o r
• Solid: repulsion between like charges
• Net effect? Compute sum for overall all possible pairs
Madelung Energy
electrostatic
E solid cluster

1


2
i
Can show
electrostatic
E solid
 N0
j
 ZiZ je
4  o rij
 ( Z e )
4  o r
2
2
Sum over a cluster
beyond which energy
is unchanged
For simple structures
Single rij
|Z1| = |Z2|
 = Madelung constant
Structures of Complex Oxides
• Multiple cations
– Perovskite
• Capacitors
• Related to high Tc
superconductors
– Spinel
• Magnetic properties
• Covalency
– Zinc blende
• Semiconductors
– Diamond
• Semiconductors
– Silicates
• Minerals
Perovskite
– Perovskite: ABO3 [B  boron]
• A2+B4+O3
A3+B3+O3
A1+B5+O3
• CaTiO3
LaAlO3
KNbO3
above/below
• Occurs when RA ~ RO and RA > RB
• Coordination numbers
A
– CN(B) = 6; CN(A) = 12
– CN(O) = 2B + 4A
• CN’s make sense? e.g. SrTiO3
– RTi = 0.61 Å
– RSr = 1.44 Å
– RO = 1.36 Å
O
B
RTi/RO = 0.45
RSr/RO = 1.06
http://abulafia.mt.ic.ac.uk/shannon/ptable.php
Tolerance factor
close-packed directions
A
B
Covalent Compounds
sp3
s2p1 s2p2 s2p3 s2p4
s2
Covalent Structures
Recall: zinc blende  both species tetrahedral
ZnS: +2 -2
or sp3
GaAs: +3 -3
single element: C or Si or Sn
diamond
Structural Characteristics
• Metals
– Close-packed structures (CN = 12)
– Slightly less close-packed (CN = 8)
• Ionic structures
– Close-packed with constraints
– CN = 4 to 8, sometimes 12
• Covalent structures
– Not close-packed, bonding is directional
• Any can be strongly or weakly bonded (Tm)
Diamond vs. CCP
8 atoms/cell, CN = 4
4 atoms/cell, CN = 12
½ tetrahedral
sites filled
2a  4R
3a  8 R
3
3
 8 
3
3
V a 
R

98.5
R

3


3
V / ato m  1 2 .3 R
3
 4 
3
3
3
V a 
R

22.6
R

2


V / ato m  5 .7 R
3
Computing density
• Establish unit cell contents
• Compute unit cell mass
• Compute unit cell volume
– Unit cell constant, a, given (or a and c, etc.)
– Or estimate based on atomic/ionic radii
• Compute mass/volume, g/cc
• Example: NaCl
–
–
–
–
Contents = 4 Na + 4 Cl
Mass = 4(atom mass Na + atomic mass Cl)/No
Vol = a3
Avogardo’s #
g / m ol
3
Units = cm 3  # / m o l  g / cm
Cl
Na
Single Crystal vs. Polycrystalline
Rb3H(SO4)2
Diamond
Quartz (SiO2)
Ba(Zr,Y)O3-d
Periodicity extends uninterrupted
throughout entirety of the sample
External habit often reflects
internal symmetry
Regions of uninterrupted periodicity
amalgamated into a larger compact
= grains
delineated by grain boundaries
Isotropic vs. Anisotropic
graphite*
polycrystalline averaging
* http://www.electronics-cooling.com/assets/images/2001_August_techbrief_f1.jpg
diamond