Types of Primary Chemical Bonds
Isotropic, filled outer shells
• Metallic
– Electropositive: give up electrons
• Ionic
– Electronegative/Electropositive
• Colavent
– Electronegative: want electrons
+
+
e-
e+
+
+
+
+
+
+
+
-
+
-
+
-
+
-
+
e-
– Shared electrons along
bond direction
Close-packed
structures
Review: Common Metal Structures
hcp
ABABAB
ccp (fcc)
ABCABC
bcc
not close-packed
Features
• Filled outer shells  spherical atom cores, isotropic bonding
• Maximize number of bonds  high coordination number
• High density
Metals
• single element, fairly electropositive
• elements similar in electronegativity
Ionic Compounds
• elements differing
in electronegativity
cation
anion
Ionic Bonding & Structures
• Isotropic bonding
• Maximize packing density
• Maximize # of bonds, subject to constraints
– Like atoms should not touch
– Maintain stoichiometry
– Alternate anions and cations
Ionic Bonding & Structures
Isotropic bonding; alternate anions and cations
–
–
–
+
–
–
–
–
–
–
–
–
–
–
–
+
–
–
+
–
–
Just barely stable
 Radius Ratio “Rules”

Cubic Coordination: CN = 8
2 RA  a
2( RA  rc )  3a
2(rc + RA)
a
 rc  RA  
RA
3
rc
 3  1  0.732
RA
2RA
Cuboctahedral: CN = 12
2RA
rc + RA
rc + RA = 2RA
rc = RA  rc/RA = 1
Radius Ratio Rules
CN (cation)
2
Geometry
min rc/RA
none
(linear)
3
0.155
(trigonal planar)
4
0.225
(tetrahedral)
CN
6
Geometry
min rc/RA
0.414
(octahedral)
8
0.732
(cubic)
12
1
(cuboctahedral)
Ionic Bonding & Structures
• Isotropic bonding
• Maximize # of bonds, subject to constraints
– Like atoms should not touch
•
•
•
•
‘Radius Ratio Rules’ – rather, guidelines
Develop assuming rc < RA
But inverse considerations also apply
n-fold coordinated atom must be at least some size
– Maintain stoichiometry
• Simple AaBb compound: CN(A) = (b/a)*CN(B)
– Alternate anions and cations
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
Local Coordination  Structures
• Build up ionic structures from closepacked metallic structures
• Given range of ionic radii: CN = 4, 6, 8
tetrahedral
octahedral
occur in closepacked structures
HCP: tetrahedral sites
4 sites/unit cell
2 sites/close-packed atom
HCP: octahedral sites
2 sites/unit cell
1 site/close-packed atom
Sites in cubic close-packed
8 tetrahedral sites/unit cell
2 tetrahedral sites/close-packed atom
4 octahedral sites/unit cell
1 octahedral site/close-packed atom
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: NaCl (rock salt)
• Cl- ~ 1.81 Å; Na+ ~ 0.98 Å; rc/RA = 0.54
• Na+ is big enough for CN = 6
– also big enough for CN = 4,
but adopts highest CN possible
• Cl- in cubic close-packed array
• Na+ in octahedral sites
• Na:Cl = 1:1  all sites filled
CN
f
4
0.225
6
0.414
8
0.732
Rock Salt Structure
Cl
Na
ccp array with sites shown
CN(Cl-) also = 6
RA/rc > 1  Cl- certainly large enough for 6-fold coordination
Lattice Constant Evaluation
rock salt
ccp metal
a
a
R
R
4R = 2 a
a = 2(RA + rc) > ( 4/2)RA
Example: ZnS
• S2- ~ 1.84 Å; Zn2+ ~ 0.60 – 0.57 Å;
– rc/RA = 0.326 – 0.408
•
•
•
•
•
CN
Zn2+ is big enough for CN = 4
4
6
S2- in close-packed array
8
2+
Zn in tetrahedral sites
Zn:S = 1:1  ½ tetrahedral sites filled
Which close-packed arrangement?
– Either! “Polytypism”
– CCP: Zinc blende or Sphaelerite structure
– HCP: Wurtzite structure
f
0.225
0.414
0.732
ZnS: Zinc Blende
 CCP
anions as CP atoms
fill 4/8 tetr sites
S2z=0
y
z=½
z=1
x
x
x
x
x
y
z=½
x
ZnS: Zinc Blende
S2-
Zn2+
CN(S2-) also = 4
RA/rc > 1  S2- certainly large enough for 4-fold coordination
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?
• Pair: attraction only
E electrostaic
pair
 Z1Z 2e 2

4 o r
• Solid: repulsion between like charges
• Net effect? Compute sum for overall all possible pairs
Madelung Energy
electrostatic
Esolid
cluster
 Zi Z j e2
1
 
2 i j 4 o rij
Sum over a cluster
beyond which energy
is unchanged
 ( Ze)
 N0
4 o r
For simple structures
Single rij
|Z1| = |Z2|
 = Madelung constant
Can show
electrostatic
Esolid
2
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