Basic Structural Chemistry
Crystalline state
Structure types
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
•The building blocks of these two
are identical, but different crystal
faces are developed
Crystal Structure
• Cleaving a crystal of rocksalt
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
Contents of unit cell represents 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
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)
Seven Crystal Systems
Fourteen Bravais Lattices
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 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).
Primitive Cell of BCC
•Primitive Translation Vectors:
•Rhombohedron primitive cell
0.5√3a
109o28’
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).
Primitive Cell of FCC
•Angle between a1, a2, a3: 60o
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)
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
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
Coordination Polyhedrons
Coordination Polyhedrons
Space Filling
74%4√2a/4FCC
34%8√3a/8Diamond
68%2√3a/4BCC
52%1a/2SC
Space fillingNumber of
Atoms (lattice
points), Z
Atom Radius,
r
a = lattice
parameter
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)
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)
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
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
0.225 – 0.4144 – tetrahedral
0.414 – 0.7326 – octahedral
0.732 – 1.008 – cubic
1.00 (substitution)12 – hcp/ccp
r/RCN
optimal radius
ratio for
given CN
ions are in touch
Structure Map
Dependence of the structure type (coordination number) on the
electronegativity difference and the average principal quantum number
(size and polarizability)
AB compounds
The lattice enthalpy change is the standard molar enthalpy change for
the following process:
M+
(gas) + X(gas)
→ MX(solid)
Because the formation of a solid from a „gas of ions“ is always exothermic
lattice enthalpies (defined in this way) are usually negative.
If entropy considerations are neglected the most stable crystal structure of a
given compound is the one with the highest lattice enthalpy.
H LΔ
0
H LΔ
0
Lattice Enthalpy
A Born-Haber cycle for KCl
(all enthalpies: kJ mol-1 for normal
conditions → standard enthalpies)
standard enthalpies of
- formation: 438
- sublimation: +89 (K)
- ionization: + 425 (K)
- atomization: +244 (Cl2)
- electron affinity: -355 (Cl)
- lattice enthalpy: x
Lattice enthalpies can be determined by a
thermodynamic cycle → Born-Haber cycle
∆Hsluč
o = - 411 kJ mol−1
∆Hsubl
o = 108 kJ mol−1
½ D= 121 kJ mol−1
EA = - 354 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 = −∆Hsluč
o + ∆Hsubl
o + 1/2 D + IE + EA + L
0 = 411 + 108 +121 + 502 + (-354) + 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 = (1/4πε0) zA zB / d
Erep = B / dn
n = Born exponent
(experimental measurement of
compressibilty)
Lattice Enthalpy
1 mol of ions
Ecoul = NA (e2 / 4 π ε0) (zA zB / d) A
Erep = NA B / dn
L = Ecoul + Erep
Find minimum dL/d(d) = 0
nA
BA
A
d
B
N
d
eZZ
ANL +=
0
2
4πε
N
r
ezz
AV
AB
AB
0
2
4πε
−+−=
Coulombic contributions to lattice enthalpies
Calculation of lattice enthalpies
Coulomb potential
of an ion pair
VAB: Coulomb potential (electrostatic potential)
A: Madelung constant (depends on structure type)
N: Avogadro constant
z: charge number
e: elementary charge
εo: dielectric constant (vacuum permittivity)
rAB: shortest distance between cation and anion
Madelung Constant
Ecoul = (e2 / 4 π ε0)*(zA zB / d)*[+2(1/1) - 2(1/2) + 2(1/3) - 2(1/4) + ....]
Ecoul = (e2 / 4 π ε0)*(zA zB / d)*(2 ln 2)
Count all interactions in the crystal lattice
Madelung constant A
(for linear chain of ions)
= sum of convergent series
Calculation of the Madelung constant
Na
Cl
...
5
24
2
6
3
8
2
12
6 +−+−=A
3D ionic solids:
Coulomb attraction and
repulsion
Madelung constants:
CsCl: 1.763
NaCl: 1.748
ZnS: 1.641 (wurtzite)
ZnS: 1.638 (sphalerite)
ion pair: 1.0000 (!)
= 1.748... (NaCl)
(infinite summation)
Madelung constant for NaCl
Ecoul = (e2 / 4 π ε0) * (zA zB / d) * [6(1/1) - 12(1/√2) + 8(1/√3) - 6(1/√4) + 24(1/√5) ....]
Ecoul = (e2 / 4 π ε0) * (zA zB / d) * A
convergent series
Madelung Constants for other Structural Types
1.64132ZnS Wurtzite
1.63805ZnS Sfalerite
2.519CaF2
1.76267CsCl
1.74756NaCl
AStructural Type
Born repulsion VBorn
Because the electron density of atoms decreases
exponentially towards zero at large distances
from the nucleus the Born repulsion shows the
same behavior
approximation:
r
V nBorn
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
).(0
VVMin BornABL +=ΔΗ
)
1
1(
4 00
2
0
n
N
r
ezz
A
L
−−=ΔΗ −+
πε
(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 (ν) the resulting values
are almost constant:
6:40.834.172α-Al2O3
8:40.842.519CaF2
6:60.871.748NaCl
8:80.881.763CsCl
CoordinationA/νMadel. const.(A)Structure
→ general lattice energy equation that can be applied to
any crystal regardless of the crystal structure
−+
−+
⋅
⋅⋅
−=ΔΗ
rr
zz
L
ν10
5
0 079.1
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
MNL BA
A
1
1
4 0
2
πε
nEl. config.
10Kr
12Xe
9Ar
7Ne
5He
Born – Mayer
d* = 0.345 Å
Born – Lande
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=
d
d
d
eZZ
MNL BA
A
*
0
2
1
4πε
Lattice Enthalpy
Kapustinski
M/v je přibližně konstantní pro všechny typy struktur
v = počet iontů ve vzorcové jednotce
M je nahrazena 0.87 v, není nutno znát strukturu
⎟
⎠
⎞
⎜
⎝
⎛
−=
dd
ZZ
vL BA 345,0
11210
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
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
Cation/Anion Radius Ratio
0.225 – 0.4144 – tetrahedral
0.414 – 0.7326 – octahedral
0.732 – 1.008 – cubic
1.00 (substitution)12 – hcp/ccp
r/RCN
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
Lu3+ 1.001
Pauling’s Rule no. 2 Bond Strength
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
Correlation of the valence of a bond sij with the bond distance dij.
b = 0.37, Rij is determined empirically from 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.
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
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
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)
1.16 MX1.41 MX2 M-X2 Octahedra
0.67 MX1.16 MX2 M-X2 Tetrahedra
FaceEdgeCorner
Polyhedron/Sharing
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 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
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
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, β-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, β-MnS (red), β-MnSe, β-CdS, CdSe, CdTe, HgS,
HgSe, HgTe, ZnSe, ZnTe
Halogenides: AgI, CuF, CuCl, CuBr, CuI, NH4F
Borides: PB, AsB
Carbides: β-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, α-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
Diamond Lattice (111) Hard Sphere Model
Diamond Lattice (111) Hard Sphere Model
Face Centered Cubic Lattice (111) Hard Sphere Model
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, α-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
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
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
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
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
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, β-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
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.
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
Pyrope Mg3Al2(SiO4)3
Almandine Fe3Al2(SiO4)3
Spessartine Mn3Al2(SiO4)3
Uvarovite Ca3Cr2(SiO4)3
Grossular Ca3Al2(SiO4)3
Andradite Ca3Fe2(SiO4)3
Garnets
Garnet Y3Al5O12
Y3 = red - dodecahedral
trivalent cations, large size
Al5 = blue
2 octahedral
3 tetrahedral
O12
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
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
X-ray structure analysis with single crystals
Principle of a four circle X-ray diffractometer
for single crystal structure analysis
CAD4 (Kappa Axis Diffractometer)
IPDS (Imaging Plate Diffraction System)