Crystalline State
Basic Structural Chemistry
Structure Types
Lattice Energy
Pauling Rules
Degree of Crystallinity
Crystalline – 3D long range order
Single-crystalline
Polycrystalline - many crystallites of different sizes and
orientations (random, oriented)
Paracrystalline - short and medium range order,
lacking long range order
Amorphous – no order, random
 Single Crystalline
 Polycrystalline
 Semicrystalline
 Amorphous
Degree of Crystallinity
Grain boundaries
A crystalline solid: HRTEM
image of strontium titanate.
Brighter atoms are Sr and
darker are Ti.
Degree of Crystallinity
A TEM image of amorphous
interlayer at the Ti/(001)Si
interface in an as-deposited
sample.
The building blocks of these two are
identical, but different crystal faces
are developed
Crystal Structure
Cleaving a crystal of rocksalt
Conchoidal fracture in chalcedony
X-ray structure analysis with single crystals
Single crystal X-ray diffraction
structure analysis
a four circle X-ray diffractometer
CAD4 (Kappa Axis Diffractometer)
IPDS (Imaging Plate Diffraction System)
Crystals
• Crystal consist of a periodic arrangement of structural motifs =
building blocks
• Building block is called a basis: an atom, a molecule, or a group
of atoms or molecules
• Such a periodic arrangement must have translational symmetry
such that if you move a building block by a distance:
then it falls on another identical building block with the same
orientation.
• If we remove the building blocks and replace them with points,
then we have a point lattice or Bravais lattice.
vectors.are,,andintegers,areand,,where 321
321
cbannn
cnbnanT 
Planar Lattice 2D
Five Planar Lattices
Unit Cell: An „imaginary“ parallel sided region of a structure from which
the entire crystal can be constructed by purely translational
displacements
Content of a unit cell represents its chemical composition
Space Lattice: A pattern that is formed by the lattice points that have
identical environment.
Coordination Number (CN): Number of direct neighbours of a given
atom (first coordination sphere)
Crystal = Periodic Arrays of Atoms
Lattice point
(Atom, molecule, group of molecules,…)
Translation Vectors
Primitive Cell:
• Smallest building block for
the crystal lattice.
• Repetition of the primitive cell
gives a crystal lattice
a
c
ba, b , c
Seven Crystal Systems
All angles 90
Fourteen Bravais Lattices
Seven Crystal Systems
+ Centering
Lattices and Space Groups
Bravais Lattice
(Lattice point = Basis of
Spherical Symmetry)
Crystal Structure
(Structural motif = Basis of
Arbitrary Symmetry)
Number of
point groups:
7
(7 crystal systems)
32
(32 crystallographic point
groups)
Number of
space groups:
14
(14 Bravais lattices)
230
(230 space groups)
Add one atom at the
center of the cube
Body-Centered Cubic (BCC)
a
c
b
a = b = c
a  b  c
Simple Cubic (SC)
Add one atom
at the center of
each face
Face-Centered Cubic (FCC)
Conventional Cell = Primitive Cell
Conventional Unit Cell  Primitive Cell
Primitive Cell
a
a
a
Body-Centered
Cubic (I)
Unit Cell Primitive Cell
A primitive cell of the lattice = volume of space translated through all
the vectors in a lattice that just fills all of space without overlapping or
leaving voids.
A primitive cell contains just one Bravais lattice point.
The primitive cell is the smallest cell that can be translated throughout
space to completely recreate the entire lattice.
There is not one unique shape of a primitive cell, many possible shapes.
The primitive cell for the simple cubic lattice is equal to the simple cubic
unit cell (they are identical).
(magenta)
Primitive Cell
A primitive cell of the lattice may be constructed in 2 ways:
 The primitive cell may have the lattice point confined at its
CENTER = the WIGNER-SEITZ cell
 The primitive cell may be formed by constructing lines
BETWEEN lattice points, the lattice points lie at the
VERTICES of the cell
Primitive Cell
Primitive Cell of BCC
•Rhombohedron primitive cell
0.53a
109o28’
The primitive cell is smaller
or equal in size to the unit
cell.
The unit cells possesses the
highest symmetry present in
the lattice (for example
Cubic).
Nonprimitive Unit Cell vs. Primitive Cell
a
a
a
Face-Centered
Cubic (F)
a
Rotated 90º
Primitive Cell
Unit Cell
The primitive cell is smaller or equal in size to the unit cell.
The unit cells possesses the highest symmetry present in the
lattice (for example Cubic).
Nonprimitive Unit Cell vs. Primitive Cell
The primitive cell is smaller or equal in size to the unit cell.
The unit cells possesses the highest symmetry present in the
lattice (for example Cubic).
1) Find the intercepts on the axes in terms of the lattice constants a, b,
c. The axes may be those of a primitive or nonprimitive unit cell.
2) Take the reciprocals of these numbers and then reduce to three
integers having the same ratio, usually the smallest three integers.
The result enclosed in parenthesis (hkl), is called the index of the
plane.
Index System for Crystal Planes
(Miller Indices)
Miller Indices
Miller Indices
Crystals and Crystal Bonding
• metallic (Cu, Fe, Au, Ba, alloys )
metallic bonding
• ionic (NaCl, CsCl, CaF2, ... )
ionic bonds, cations and anions, electrostatic
interactions
• covalent (diamond, graphite, SiO2, AlN,... )
atoms, covalent bonding
• molecular (Ar, C60, HF, H2O, organics, proteins )
molecules, van der Waals and hydrogen bonding
Three Cubic Cells
SC or Primitive (P) BCC (I) FCC (F)
Three Cubic Cells
a
a
a
d
D
a = edge
d = face diagonl
(d2 = a2 + a2 = 2a2)
D = body diagonal
(D2 = d2 + a2 = 2a2 + a2 = 3a2)
a2 d a3 D
Cube
CN 6
Simple Cubic SC = Polonium
Space filling 52%
Z = 1
Space filling 68%
CN 8
BCC = W, Tungsten
a
d
D
r
Fe, Cr, V, Li-Cs, Ba
Z = 2
BCC
Space filling 74%
CN 12
FCC = Copper, Cu = CCP
d
r
Z = 4
Close Packing in Plane 2D
B and C holes cannot be occupied at the same time
Close Packing in Space 3D
Cubic
CCP
Hexagonal
HCP
hexagonal
cubic
cubichexagonal
hexagonal cubic
cubic
hexagonal
Mg, Be, Zn, Ni, Li, Be, Os, He,
Sc, Ti, Co, Y, Ru
Cu, Ca, Sr, Ag, Au, Ni, Rh, solid
Ne-Xe, F2, C60, opal (300 nm)
Structures with Larger Motifs
Structures with Larger Motifs
TEM images of
superlattices composed of
11.3 nm Ni nanoparticles
Structures with Larger Motifs
Coordination Polyhedrons
Coordination Polyhedrons
Space Filling
a = lattice
parameter
Atom Radius,
r
Number of
Atoms (lattice
points), Z
Space filling
SC a/2 1 52%
BCC 3a/4 2 68%
FCC 2a/4 4 74%
Diamond 3a/8 8 34%
CCP = FCC
(ABC)
Close packed layers of CCP are oriented perpendicularly to the
body diagonal of the cubic cell of FCC
CCP FCC
Two Types of Voids (Holes)
Tetrahedral Holes TN
cp atoms in lattice cell
N Octahedral Holes
2N Tetrahedral Holes
Tetrahedral Holes T+ Octahedral Holes
Two Types of Voids (Holes)
Two Types of Voids (Holes)
Tetrahedral HolesOctahedral Holes
Z = 4
number of atoms in the
cell (N)
N = 8
number of tetrahedral
holes (2N)
Tetrahedral Holes (2N)
Octahedral Holes (N)
Z = 4
number of atoms in
the cell (N)
N = 4
number of octahedral
holes (N)
Two Types of Voids (Holes)
N cp atoms in lattice cell
N Octahedral Holes
2N Tetrahedral Holes
Tetrahedral Holes (2N)
Characteristic Structures of Solids = Structure
Types
Rock salt NaCl LiCl, KBr, AgCl, MgO, TiO, FeO, SnAs, UC, TiN, ...
Fluorite CaF2 BaCl2, K2O, PbO2 ...
Lithium bismutide Li3Bi
Sphalerite (zinc blende) ZnS CuCl, HgS, GaAs ...
Nickel arsenide NiAs FeS, PtSn, CoS ...
Wurtzite ZnS ZnO, MnS, SiC
Rhenium diboride ReB2
ICSD
3555 NaCl
3438 MgAl2O4
2628 GdFeO3
Structure Types Derived from CCP = FCC
Structure Types Derived from CCP = FCC
Structure Types Derived from CCP = FCC
Anions/cell (= 4) Oct. (Max 4) Tet. (Max 8) Stoichiometry Compound
4 100% = 4 0 M4X4 = MX NaCl
(6:6 coord.)
4 0 100% = 8 M8X4 = M2X Li2O
(4:8 coord.)
4 0 50% = 4 M4X4 = MX ZnS, sfalerite
(4:4 coord.)
4 50% = 2 0 M2X4 = MX2 CdCl2
4 100% = 4 100% = 8 M12X4 = M3X Li3Bi
4 50% = 2 12.5% = 1 M3X4 MgAl2O4,
spinel
o/t fcc(ccp) hcp
all oct. NaCl NiAs
all tetr. CaF2 ReB2
o/t (all) Li3Bi (Na3As) (!) problem
½ t sphalerite (ZnS) wurtzite (ZnS)
½ o CdCl2 CdI2
Comparison between structures
with filled octahedral and tetrahedral holes
Fluorite CaF2 and antifluorite Li2O
Fluorite structure = a face-centered cubic array (FCC)
of cations = cubic close packing (CCP) of cations with
all tetrahedral holes filled by anions = a simple cubic
(SC) array of anions.
Antifluorite structure = a face-centred cubic (FCC)
array of anions = cubic close packing (CCP) of anions,
with cations in all of the tetrahedral holes (the reverse
of the fluorite structure).
K2[PtCl6], Cs2[SiF6], [Fe(NH3)6][TaF6]2
Fluorite (CaF2, antifluorite Li2O)
F / Li
Fluorite structures (CaF2, antifluorite Li2O)
Oxides: Na2O, K2O, UO2,
ZrO2, ThO2
alkali metal sulfides,
selenides and tellurides
K2[PtCl6], (NH4)2[PtCl6],
Cs2[SiF6],
[Fe(NH3)6][TaF6]2.
CaF2, SrF2, SrCl2, BaF2, BaCl2, CdF2, HgF2, EuF2, -PbF2, PbO2
Li2O, Li2S, Li2Se, Li2Te, Na2O, Na2S, Na2Se, Na2Te, K2O, K2S
Fluorite structures (CaF2, antifluorite Li2O)
Sphalerite (zincblende, ZnS)
Cubic close packing of anions
with 1/2 tetrahedral holes
filled by cations
Sphalerite (zincblende, ZnS)
Sphalerite (zincblende, ZnS)
13-15 compounds: BP, BAs, AlP, AlAs, GaAs, GaP, GaSb, AlSb, InP, InAs, InSb
12-16 compounds: BeS, BeSe, BeTe, -MnS (red), -MnSe, -CdS, CdSe, CdTe, HgS,
HgSe, HgTe, ZnSe, ZnTe
Halogenides: AgI, CuF, CuCl, CuBr, CuI, NH4F
Borides: PB, AsB
Carbides: -SiC
Nitrides: BN
Diamond
6,16Å
2,50 Å
4,10Å
cubic hexagonal
SiO2 cristobalite SiO2 tridymite
ice
Diamond
Cubic Diamond
Diamond Structure
C, Si, Ge, -Sn
• Add 4 atoms to a FCC
• Tetrahedral bond arrangement
• Each atom has 4 nearest neighbors and 12 next nearest neighbors
Elements of the 14th Group
Cuprite Cu2O Cubic Lattice
Wurzite, ZnS
Hexagonal close packing of anions
with 1/2 tetrahedral holes filled by
cations
Wurzite, ZnS
ZnO, ZnS, ZnSe, ZnTe, BeO, CdS, CdSe, MnS, AgI, AlN
Semiconductors of 13-15 and 12-16 type
Rock Salt, NaCl
Cubic close packing of anions with all
octahedral holes filled by cations
Rock Salt, NaCl
Anion and cation sublattices
Rock Salt, NaCl
Rock salt structures (NaCl)
Hydrides: LiH, NaH, KH,
NH4BH4 – H2 storage material
Borides: ZrB, HfB
Carbides: TiC, ZrC, VC, UC
Nitrides: ScN, TiN, UN, CrN, VN, ZrN
Oxides: MgO, CaO, SrO, BaO, TiO, VO, MnO, FeO,
CoO, NiO
Chalcogenides: MgS, CaS, SrS, BaS, -MnS, MgSe,
CaSe, SrSe, BaSe, CaTe
Halides: LiF, LiCl, LiBr, LiI, NaF, NaBr, NaI, KF,
KCl, KBr, KI, RbF, RbCl, RbBr, AgCl, AgF, AgBr
Intermetallics: SnAs
Other
FeS2 (pyrite), CaC2, NaO2
Rock salt structures (NaCl)
FeS2 (pyrite), CaC2, NaO2
SiO2 (pyrite - high pressure polymorph,
Uranus and Neptune core)
NiAs - type
Hexagonal close packing of
anions with all octahedral holes
filled by cations
NiS, NiAs, NiSb, NiSe, NiSn, NiTe, FeS,
FeSe, FeTe, FeSb, PtSn, CoS, CoSe,
CoTe, CoSb, CrSe, CrTe, CoSb,
PtB (anti-NiAs structure)
NiAs - type
Hexagonal close packing of anions with all octahedral holes filled by cations
ReB2 - type
Hexagonal close packing of
anions with all tetrahedral holes
filled by cations
[Cr(NH3)6]Cl3, K3[Fe(CN)6]
bcc
Li3Bi - type (anti BiF3)
Li3Bi - type (anti BiF3)
Fe3Al
[Cr(NH3)6]Cl3
K3[Fe(CN)6]
Cubic close packing of anions
with all tetrahedral and
octahedral holes filled by
cations
Li3Bi - type (anti BiF3)
M3C60
Cubic close packing of C60
3
anions with all tetrahedral
and octahedral holes filled by
cations
CsCl Primitive cubic packing of
anions with all cubic holes
filled by cations
Primitive cubic packing of
CsCl8 cubes sharing all faces
CsCl
CsCl is not BCC
CsBr, CsI, CsCN, NH4Cl, NH4Br, TlCl, TlBr, TlI, CuZn, CuPd, LiHg
NaTl
Niggli – 230 space groups – restrictions on arrangement of atoms:
There ar only 4 possible AB cubic structures: NaCl, ZnS-sfalerite, CsCl, and NaTl
Both sublattices form independent diamond
structures.
The atoms sit on the sites of a bcc lattice with
abcc = ½ a.
SC of ReO6 octahedra
ReO3
NaCl structure with 3/4 of cations removed and 1/4 of
anions removed
UO3, MoF3, NbF3, TaF3, Cu3N
Perovskite, CaTiO3
Two equvivalent views of the unit cell of perovskite
Ti
CaO
Ti
O
Ca
Cubic "close packing" of Ca and O with 1/4 octahedral holes filled by Ti cations
Perovskite, CaTiO3
Two equvivalent views of the unit cell of perovskite
Cubic "close packing" of Ca and O with 1/4 octahedral holes filled by Ti cations
TiO6 – octahedra
CaO12 – cuboctahedra
(Ca2+ and O2- form a cubic close packing)
preferred structure of piezoelectric,
ferroelectric and superconducting
materials
Perovskite structure CaTiO3
Goldschmidt’s tolerance factor
Similarity to CsCl
Perovskite, CaTiO3
Cubic "close packing" of A and X with 1/4 octahedral holes filled by B cations
Perovskite, CaTiO3
MgSiO3, CaSiO3
KNbO3, KTaO3, KIO3,
NaNbO3, NaWO3, LaCoO3,
LaCrO3, LaFeO3, LaGaO3,
LaVO3, SrTiO3, SrZrO3,
SrFeO3
ThTaN3, BaTaO2N
Perovskite, BaTiO3
Perovskite - ferroelectric BaTiO3
Perovskite - ferroelectric BaTiO3
Perovskite structure of YBCO
Perovskite structure of CH3NH3PbI3
Rutile, TiO2
CN – stoichiometry Rule
AxBy
CN(A) / CN(B) = y / x
Distorted hexagonal close packing of anions with 1/2
octahedral holes filled by cations (giving a tetragonal
lattice)
Rutile, TiO2
GeO2, CrO2, IrO2, MoO2, NbO2, -MnO2, OsO2, VO2
(>340K), RuO2, CoF2, FeF2, MgF2, MnF2
TiO6 – octahedra
OTi3 – trigonal planar
(alternative to CaF2 for highly
charged smaller cations)
The rutile structure: TiO2
Three polymorphs of TiO2
anatase (a), rutile (b) and brookite (c)
fcc array of O2- ions, A2+
occupies 1/8 of the tetrahedral
and B3+ 1/2 of the octahedral
holes
 normal spinel:
AB2O4
 inverse spinel:
B[AB]O4
Fe3O4 = Fe3+[Fe2+Fe3+]O4
 basis structure for several
magnetic materials
The spinel structure: MgAl2O4
Spinel
AB2X4 Spinel normal: Cubic close packing of anions with 1/2 octahedral
holes filled by B cations and 1/8 tetrahedral holes by A cations
MgAl2O4, CoAl2O4, MgTi2O4, Fe2GeO4, NiAl2O4, MnCr2O4
AB2X4 Spinel inverse: As for spinel but A cations and 1/2 of B cations
interchanged
MgFe2O4, NiFe2O4, MgIn2O4, MgIn2S4, Mg2TiO4, Zn2TiO4, Zn2SnO4,
FeCo2O4.
δ = the inversion parameter
(AδB1-δ)A[A1-δB1+δ]BO4
Values from δ = 1 (normal) to δ = 0 (inverse)
May depend on synthesis conditions
Corundum
Al2O3 consists of hcp O2− ions
Al3+ fill ……. of all octahedral holes
The Al centres are surrounded by ……. oxides
Oxide ligands are ……………coordinated by Al
Corundum
AlO6 octahedral units are linked in both facesharing
and edge-sharing orientations as parallel
and perpendicular to the c-axis, respectively. The
relative orientation of the metal centres causes a
pseudo Peierls distortion, resulting in
neighbouring metal centres that are rotated at an
angle of 64.3° away from each other. Elongation in
pairs of the surrounding oxide ligands results in a
pentagonal bi-pyramidal geometry belonging to
the space group R-3c. The material is largely ionic
in nature with a wide band gap of 9.25 eV.
Garnets
Naturally occuring garnets A3B2Si3O12 = A3B2(SiO4)3
A3 = divalent cation (Mg, Fe, Mn or Ca) dodecahedral
B2 = trivalent (Al, Fe3+, Ti, or Cr) octahedral
Si3 = tetravalent, tetrahedral
Since Ca is much larger in radius than the other divalent cations, there
are two series of garnets: one with calcium and one without:
pyralspite contain Al (pyrope, almandine, spessartine)
ugrandite contain Ca (uvarovite, grossular, andradite)
Synthetic garnets A3B5O12
A3 = trivalent cations, large size (Y, La,…)
B5 = trivalent (Al, Fe3+, Ti, or Cr) 2B octahedral, 3B tetrahedral
Y3Al5O12
Y3Fe5O12
Garnets
Garnet Y3Al5O12
Y3 = red - dodecahedral
trivalent cations, large size
Al5 = blue
2 octahedral
3 tetrahedral
O12
Fullerides
M1C60 all the octahedral (O) sites (dark blue) are occupied (NaCl)
M2C60 all the tetrahedral (T) sites (light blue) are occupied (CaF2)
M3C60 both the O and the T sites are occupied (BiF3)
M4C60 rearranged to a body-centered tetragonal (bct) cell and both the O and
the T sites of the bct lattice are occupied
M6C60 a bcc lattice and all its T sites are occupied
Fullerides
BCC unit cell of Rb6C60 and Cs6C60
Layered Structures
CdI2 Hexagonal close packing of anions with 1/2 octahedral holes filled
by cations
CoI2, FeI2, MgI2, MnI2, PbI2, ThI2, TiI2, TmI2, VI2, YbI2, ZnI2, VBr2,
TiBr2, MnBr2, FeBr2, CoBr2, TiCl2, TiS2., TaS2.
CdCl2 Cubic close packing of anions with 1/2 octahedral holes filled by
cations
CdCl2, CdBr2, CoCl2, FeCl2, MgCl2, MnCl2, NiCl2, NiI2, ZnBr2, ZnI2,
Cs2O* (anti-CdCl2 structure)
CdI2 Hexagonal Close Packing
CdCl2 Cubic Close Packing
CdCl2 Cubic close packing
Bonding models for covalent and ionic
compounds
Organic vs inorganic bonding
G. N. Lewis 1923
Electron pair sharing
Orbital overlap
Chemical bond
Number of bonds = atomic valence
Born, Lande, Magelung, Meyer
1918
Electrostatic attraction (Coulomb)
Repulsion
1911
The lattice enthalpy change, L, is the standard molar enthalpy change
for the process:
M+
(gas) + X
(gas)  MX(solid)
The formation of a solid from ions in the gas phase is always exothermic
Lattice enthalpies are usually negative
The most stable crystal structure of a given compound is the one with the
highest (most negative) lattice enthalpy.
(entropy considerations neglected)
H L
0
Lattice Enthalpy, L
(L)
Lattice Enthalpy, L, kJ/mol
∆Hform
o = - 411 kJ mol1
∆Hsubl
o = 108 kJ mol1
½ D= 122 kJ mol1
EA = - 355 kJ mol1
IE = 502 kJ mol1
L = ?Na(s) + 1/2 Cl2 (g)
Na(g) + 1/2 Cl2 (g)
Na(g) + Cl (g)
Na+
(g) + Cl (g)
Na+
(g) + Cl-
(g)
NaCl (s)
0 = ∆Hform
o + ∆Hsubl
o + 1/2 D + IE + EA+ L
0 = 411 + 108 +122 + 502 + (-355) + L
L =  788 kJ mol1
Born-Haber cycle
all enthalpies: kJ mol-1 for normal conditions  standard enthalpies
Lattice Enthalpy
L = Ecoul + Erep
One ion pair
Ecoul = (1/40) zA zB / d
(calculated exactly)
Erep = B / dn
(modelled empirically)
n = Born exponent
(experimental measurement of
compressibilty)
B = a constant
Lattice Enthalpy
1 mol of ions
Ecoul = NA A (e2 / 4  0) (zA zB / d)
A = Madelung constant - a single ion interacts with all other ions
Erep = NA B / dn
L = Ecoul + Erep
Find minimum dL/d(d) = 0
nA
BA
A
d
B
N
d
eZZ
ANL 
0
2
4
d
ezz
ANE ACoul
0
2
4


Coulombic contributions to lattice enthalpies
Calculation of Lattice Enthalpies
Coulomb potential of
an ion pair
ECoul : Coulomb potential (electrostatic potential)
A : Madelung constant (depends on structure type)
NA: Avogadro constant
z : charge number
e : elementary charge
o: dielectric constant (vacuum permittivity)
d : shortest distance between cation and anion
Madelung Constant
Ecoul = (e2 / 4  0)*(zA zB / d)*[+2(1/1)  2(1/2) + 2(1/3)  2(1/4) + ....]
Ecoul = (e2 / 4  0)*(zA zB / d)*(2 ln 2)
Count all interactions in the crystal lattice
Madelung constant A = 1.3863…
for an infinite linear chain of ions
= sum of convergent series
The simplest example : 1D lattice
Madelung Constant for NaCl
Ecoul = (e2 / 4  0) * (zA zB / d) * [6(1/1) - 12(1/2) + 8(1/3) - 6(1/4) + 24(1/5) ....]
Ecoul = (e2 / 4  0) * (zA zB / d) * A
convergent series
3D ionic solids:
Coulomb attraction and
repulsion
a single ion interacts with all
other ions
74756.1....
5
24
2
6
3
8
2
12
6 A
Madelung Constants for other Structural Types
Structural Type A
NaCl 1.74756
CsCl 1.76267
CaF2 2.519
ZnS Sfalerite 1.63805
ZnS Wurtzite 1.64132
Linear Lattice 1.38629
Ion Pair ?
Born repulsion Erep
Because the electron density of atoms decreases
exponentially towards zero at large distances
from the nucleus the Born repulsion shows the
same behavior
approximation:
d
E nrep
B

B and n are constants for a given atom type; n can be
derived from compressibility measurements (~8)
r
r0
Repulsion arising from
overlap of electron clouds
Total lattice enthalpy from Coulomb interaction
and Born repulsion
 EE repCoulL
 min
0
)
1
1(
4 0
2
0
n
N
d
ezz
A AL
 

(set first derivative of the sum to zero)
Measured (calculated) lattice enthalpies (kJ mol-1):
NaCl: –772 (–757)
CsCl: –652 (–623)
(measured from Born Haber cycle)
The Kapustinskii equation
Kapustinskii found that if the Madelung constant for a given structure is
divided by the number of ions in one formula unit () the resulting values
are almost constant:
Structure Madelung constant (A) A/ Coordination
CsCl 1.763 0.88 8:8
NaCl 1.748 0.87 6:6
CaF2 2.519 0.84 8:4
-Al2O3 4.172 0.83 6:4
 general lattice energy equation that can be applied to
any crystal regardless of the crystal structure










 rr
G
rr
ZZ
vKL BA
1
K, G = constants
structure M CN stoichm M / v
CsCl 1.763 (8,8) AB 0.882
NaCl 1.748 (6,6) AB 0.874
ZnS sfalerite 1.638 (4,4) AB 0.819
ZnS wurtzite 1.641 (4,4) AB 0.821
CaF2 fluorite 2.519 (8,4) AB2 0.840
TiO2 rutile 2.408 (6,3) AB2 0.803
CdI2 2.355 (6,3) AB2 0.785
Al2O3 4.172 (6,4) A2B3 0.834
v = the number of ions in one formula unit
Kapustinski
Most important advantage of the Kapustinski equation
 it is possible to apply the equation for lattice calculations of crystals with
polyatomic ions (e.g. KNO3, (NH4)2SO4 ...).
 a set of „thermochemical radii“ was derived for further calculations of lattice
enthalpies
Lattice Enthalpy







nd
eZZ
ANL BA
A
1
1
4 0
2

El. config. n Example
He-He 5 LiH
Ne-Ne 7 NaF, MgO
Ar-Ar 9 KCl, CaS, CuCl, Zn2+,
Ga3+
Kr-Kr 10 RbBr, AgBr, Cd2+, In3+
Xe-Xe 12 CsI, Au+, Tl3+
Born–Mayer
d* = 0.345 Å
Born–Lande







d
d
d
eZZ
ANL BA
A
*
0
2
1
4
For compounds of mixed ion types, use
the average value (e.g., for NaCl, n = 8).
Lattice Enthalpy of NaCl
Born–Lande calculation L =  765 kJ mol1
Only ionic contribution
Experimental Born–Haber cycle L =  788 kJ mol1
Lattice Enthalpy consists of ionic and covalent contribution
Applications of lattice enthalpy calculations:
 thermal stabilities of ionic solids
 stabilities of oxidation states of cations
 Solubility of salts in water
 calculations of electron affinity data
 lattice enthalpies and stabilities of „non existent“
compounds
Five principles which could be used to determine the
structures of complex ionic/covalent crystals
Pauling’s Rule no. 1 Coordination Polyhedra
A coordinated polyhedron of anions is formed about each cation.
Cation-Anion distance is determined by sums of ionic radii.
Cation coordination environment is determined by radius ratio.
Pauling’s Rules
Coordination Polyhedra
Different Types of Radii
P – Pauling radius
G – Goldschmidt radius
S – Shannon radius
Variation of the electron density
along the Li – F axis in LiF
Variation of ionic radii with coordination number
The radius of one ion was fixed to a
reasonable value
(r(O2-) = 140 pm) (Linus Pauling)
That value is then used to compile a
set of self consistent values for all
other ions.
Variation of atomic radii
through the Periodic table
R.D. Shannon and C.T. Prewitt, Acta Cryst. B25, 925-945 (1969)
R.D. Shannon, Acta Cryst. A32, 751-767 (1976)
Ionic Radii
As the coordination number (CN) increases, the Ionic Radius increases
Sr 2+
CN Radius, Å
6 1.32
8 1.40
9 1.45
10 1.50
12 1.58
As the oxidation state increases, cations get smaller
(6-fold coordination, in Å)
Mn2+ 0.810
Mn3+ 0.785
Mn4+ 0.670
Ti2+ 1.000
Ti3+ 0.810
Ti4+ 0.745
Ionic Radii
The radius increases down a group in the periodic table.
The exception - 4d/5d series in the transition metals - the lanthanide contraction
(6-fold coordination, in Å)
Al3+ 0.675
Ga3+ 0.760
In3+ 0.940
Tl3+ 1.025
Ti4+ 0.745
Zr4+ 0.86
Hf4+ 0.85
Right to left across the periodic table the radius decreases.
(6 coordinate radii, in Å)
La3+ 1.172
Nd3+ 1.123
Gd3+ 1.078
Lu 3+ 1.001
1. Ionic radii increase down a group.
(Lanthanide contraction restricts the increase of heavy ions)
2. Radii of equal charge ions decrease across a period
3. Ionic radii increase with increasing coordination number
the higher the CN the bigger the ion
4. The ionic radius of a given atom decreases with increasing
charge (r(Fe2+) > r(Fe3+))
5. Cations are usually the smaller ions in a cation/anion
combination (exceptions: r(Cs+) > r(F-))
6. Frequently used for rationalization of structures:
„radius ratio“ r(cation)/r(anion) (< 1)
General trends for ionic radii
Cation/anion Radius Ratio
CN r/R
12 – hcp/ccp 1.00 (substitution)
8 – cubic 0.732 – 1.00
6 – octahedral 0.414 – 0.732
4 – tetrahedral 0.225 – 0.414
optimal radius
ratio for
given CN
ions are in touch
Structure Map
Structural map as function of
radius ratios for AB
compounds.
Structural map as function of
radius ratios for A2BO4
compounds.
Dependence of the structure type on parameters, such as ionic radii, ionicity,
electronegativity etc.
Structure Map
Dependence of the structure type (coordination number) on the
electronegativity difference and the average principal quantum
number (size and polarizability)
AB compounds
Pauling’s Rule no. 2 Bond Strength
The strength of an electrostatic bond = valence / CN
The bond valence sum of each ion equals its oxidation state.
The valence of an ion (Vi, equal to the oxidation state of the ion) is equal to a sum of
the valences of its bonds (sij).
In a stable ionic structure the charge on an ion is balanced by the sum of electrostatic
bond strengths (sij) to the ions in its coordination polyhedron.
TiO2 (Rutile) Ti - oxidation state of +4, coordinated to 6 oxygens.
VTi = +4 = 6 (sij) sij = +2/3
The bond valence of oxygen, coordinated by 3 Ti atoms
Vo = 3 (sij) = 3 (-2/3) = 2
Each bond has a valence of sij with respect to the cation
and sij with respect to the anion.
Pauling’s Rules
Bond Strength
Brown, Shannon, Donnay, Allmann:
Correlation of the valence of a bond sij with the (experimental) bond distance dij.
Rij = standard single bond lenght - determined empirically from (many) structures
where bond distances and ideal valences are accurately known.
Tables of Rij values for given bonding pairs (i.e. Nb-O, Cr-N, Mg-F, etc.) have
been calculated, just as tables of ionic radii are available.
A constant b = 0.37
b
dR
s
ijij
ij

 exp
Bond Strength
Correlation of the valence of a bond sij with the (experimental) bond distance dij.
Use of the bond valence concept
A) To check experimentally determined structures for correctness, or bonding
instabilities
B) To predict new structures
C) To locate light atoms such as hydrogen or Li ion, which are hard to find
experimentally
D) To determine ordering of ions which are hard to differentiate experimentally, such
as Al3+ and Si4+, or O2- and F-
b
dR
s
ijij
ij

 exp
CN
z
sv i
iji 
Pauling’s Rule no. 3 Polyhedral Linking
The presence of shared edges, and particularly shared faces decreases the
stability of a structure. This is particularly true for cations with large
valences and small coordination number.
Avoid shared polyhedral edges and/or faces.
Pauling’s Rules
Polyhedral Linking
Polyhedral Linking
The Coulombic interactions - maximize the cation-anion interactions
(attractive), and minimize the anion-anion and cation-cation interactions
(repulsive).
The cation-anion interactions are maximized by increasing the coordination
number and decreasing the cation-anion distance. If ions too close - electronelectron
repulsions.
The cation-cation distances as a function of the cation-anion distance (M-X)
Polyhedron/Sharing
Corner Edge Face
2 Tetrahedra 2 M-X 1.16 MX 0.67 MX
2 Octahedra 2 M-X 1.41 MX 1.16 MX
The cation-cation distance decreases, (the Coulomb repulsion increases) as
the
•degree of sharing increases (corner < edge < face)
•CN decreases (cubic < octahedral < tetrahedral)
•cation oxidation state increases (this leads to a stronger Coulomb repulsion)
Pauling’s Rule no. 4 Cation Evasion
In a crystal containing different cations those with large valence and
small coord. number tend not to share polyhedral elements (anions).
Pauling’s Rules
Perovskite, CaTiO3
CaII 12-coordinate CaO12 cuboctahedra share FACES
TiIV 6-coordinate TiO6 octahedra share only VERTICES
Pauling’s Rule no. 5 Environmental Homogeneity
the rule of parsimony
The number of chemically different coordination environments for a given
ion tends to be small.
Once the optimal chemical environment for an ion is found, if possible all
ions of that type should have the same environment.
Pauling’s Rules
High Pressure Transformations
•high pressure phases
•higher density
•higher coodination number
•higher symmetry
•transition to from nonmetal to metal
•band mixing
•longer bonds
Pressure/Coordination Number Rule: increasing pressure – higher CN
Pressure/Distance Paradox: increasing pressure – longer bonds