Crystalline State
Basic Structural Chemistry
Structure Types
1
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, distortions, lacking long range order
Amorphous – no order, random
2
• Single Crystalline
• Polycrystalline
• Nanocrystalline
• Amorphous
Degree of Crystallinity
Grain boundaries
3
Metallic glass
A crystalline solid: HRTEM
image of strontium titanate
SrTiO3
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
4
Single-Crystal X-ray Diffraction
Structure Analysis
a four circle X-ray diffractometer 5
Single-Crystal X-ray Diffraction
Structure Analysis
6
Crystals
• Crystal = 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
• 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 
7
Crystal Structure
(POINT) LATTICE
the geometrical pattern repeating
periodically in space (2D or 3D)
formed by points that have
identical environment
representing the locations of
basis or motifs
MOTIF (BASIS)
the repeating unit of a pattern
(an atom, a group of atoms, a
molecule etc. ) inside the unit
cell
UNIT CELL
the smallest repetitive volume of
the crystal, which when stacked
together with replication
reproduces the whole crystal
8
Crystal Structure = Lattice + Motifs
An „imaginary“ parallel sided region (parallelepiped) of a structure from
which the entire crystal can be constructed by purely translational
displacements
Contains one unit of the translationally repeating pattern
Content of a unit cell represents its chemical composition
The unit cells that are commonly formed by joining neighbouring lattice
points by straight lines, are called primitive unit cells
Unit Cell
9
Primitive
unit cell
Unit cell
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
10
Five Planar Lattices
11
graphene
Lattice points of
spherical symmetry
a
e
b,c
d
Ten Planar Point Groups
12
Symmetry
preservation of form and
configuration across a
point, a line, or a plane symmetry
elements
Symmetry Element
a geometric entity (line,
point, plane) about which
a symmetry operation
takes place
Symmetry Operation
a permutation of atoms
such that an object
(molecule or crystal) is
transformed into a state
indistinguishable from
the starting state
Point Group
the collection
of symmetry
elements of an
isolated shape,
does not
consider
translation
Lattice points
occupied by
motifs of
nonspherical
symmetry
17 Plane Space Groups - Wallpaper
13
A space group = a complete set of all symmetry elements and translations
Seven Crystal Systems in 3D
14
Lattice points
of
spherical
symmetry
Fourteen Bravais Lattices in 3D
7 Crystal Systems
+ Centering
= 14 Bravais Lattices
15
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 16
Centering
3D 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)
17
Primitive Cell
a
a
a
Body-Centered
Cubic (I)
Unit Cell Primitive Cell
• The smallest cell that can be translated throughout space to
completely recreate the entire lattice
• Volume of space translated through all the vectors in a lattice
that just fills all of space without overlapping or leaving voids
• Contains just one Bravais lattice point (Z = 1)
• There is not one unique 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)
18
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
19
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) 20
1) Coordinates within a unit cell
2) Express the coordinates u, v, w as fractions of unit cell
vectors (lattice parameters) a, b, and c: (h, k, l)
3) Do not clear fractions
4) Entire lattice can be referenced by one unit cell
Index System for Points
21
Central point coordinates?
Index System for Directions
(Miller Indices)
1) Determine coordinates of two points in
direction of interest (simplified – origin):
u1 v1 w1 and u2 v2 w2
2) Subtract coordinates of the second point
from those of the first point:
u’ = u1 - u2, v’ = v1 - v2, w’ = w1 - w2
3) Clear fractions from the differences to give
indices in lowest integer values
4) Write indices in [ ] brackets - [uvw]
5) Negative = a bar over the integer
A = [100]
B = [111]
C = [1¯2¯2]
22
Index System for Directions
(Miller Indices)
In the cubic system directions having the same
indices regardless of order or sign are equivalent
For cubic crystals, the directions are all
equivalent by symmetry:
[1 0 0], [ 1¯ 0 0], [0 1 0], [0 1¯ 0], [0 0 1], [0 0 1¯ ]
Families of crystallographic directions
e.g. <1 0 0>
Angled brackets denote a family of
crystallographic directions
23
1. If the plane passes through the origin, select an equivalent plane or
move the origin
2. 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
3. Take the reciprocals of these numbers and then reduce to three
integers having the same ratio, usually the smallest three integers
4. (1/∞ = 0)
5. The result enclosed in parenthesis (hkl), is called the index of the
plane
Index System for Crystal Planes
(Miller Indices)
24
Index System for Crystal Planes
(Miller Indices)
25
Index System for Crystal Planes
(Miller Indices)
Cubic system - planes having the same indices regardless of order or
sign are equivalent - braces {hkl}
(111), (11¯1), (111¯) ….
belong to {111} family
(100), (1¯00), (010), and (001) …..
belong to {100} family
26
Index System for Crystal Planes
(Miller Indices)
The Miller indices (hkl) is the same vector as the normal to the plane [hkl]
27
Index System for Crystal Planes
(Miller Indices)
28
2 nm
Atomic planes influence
• Optical properties
• Reactivity
• Surface tension
• Dislocations
Quasiperiodic Crystals
29
2 nm
Quasiperiodic crystal = a structure that is ordered but not periodic
continuously fills all available space, but it lacks translational symmetry
Penrose - a plane filled in a nonperiodic
fashion using two different types of tiles
Five-fold symmetry
Only 2, 3, 4, 6fold
symmetry
allowed to fill 2D
plane
completely
Crystals and Crystal Bonding
• metallic (Cu, Fe, Au, Ba, alloys )
metallic bonding, electron delocalization
• ionic (NaCl, CsCl, CaF2, ... )
ionic bonds, cations and anions, electrostatic
interactions, ions pack into extremely regular crystalline
structures, in an arrangement that minimizes the lattice
energy (maximizing attractions and minimizing
repulsions). The lattice energy is the summation of the
interaction of all sites with all other sites.
• covalent network solid (diamond, graphite, SiO2, AlN,... )
atoms, covalent bonding, a chemical compound (or
element) in which the atoms are bonded by covalent
bonds in a continuous network extending throughout the
material, there are no individual molecules, the entire
crystal or amorphous solid may be considered a
macromolecule
• molecular (Ar, C60, HF, H2O, organics, proteins )
molecules, van der Waals and hydrogen bonding
30
Covalent Network Solids
31
Three Cubic Cells
SC or Primitive (P) BCC (I) FCC (F)
32
33
SC or Primitive (P) BCC (I) FCC (F)
Coordination
number
Z = number of
lattice points
per unit cell
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
34
CN 6
Simple Cubic SC = Polonium
Space filling 52%
Z = 1
35
Space filling 68%
CN 8
BCC = W, Tungsten
a
d
D
r
-Fe, Cr, V, Li-Cs, Ba
Z = 2
36
BCC
37
Zeolite
Octasilicate = motif
Space filling 74%
CN 12
FCC = Copper, Cu = CCP
d
r
Z = 4
38
Close Packing in Plane 2D
39
Close Packing
The second layer - B and C holes cannot be occupied at the same time
40
Close Packing in Space 3D
41
The third layer decides:
Hexagonal
HCP
Cubic
CCP
42
Hexagonal
HCP
Cubic
CCP
Mg, Be, Zn, Ni, Li, Os, Sc, Ti,
Co, Y, Ru, solid He
Cu, γ-Fe (austenite), Ca, Sr, Ag, Au, Ni,
Rh, solid Ne-Xe, F2, C60, opal (300 nm)
43
Hexagonal
HCP
Cubic
CCP
= FCC
CCP = FCC
(ABC)
Close packed layers of CCP are oriented perpendicularly to the
body diagonal of the cubic cell of FCC
CCP FCC
44
Structures with Larger Motifs
45
C60 - FCC = CCP
SEM - Opal – 300 nm SiO2 FCC = CCP
Structures with Larger Motifs
TEM images of
superlattices composed of
11.3 nm Ni nanoparticles
46
Structures with Larger Motifs
47
Coordination Polyhedrons
48
Which is HCP and which is CCP?
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%
49
50
Two Types of Voids (Holes)
in Close-Packed Structures (CCP and HCP)
51
Tetrahedral Holes TN
cp atoms in a lattice
cell
N Octahedral Holes
2N Tetrahedral Holes
Tetrahedral Holes T+ Octahedral Holes
52
Two Types of Voids (Holes)
Tetrahedral Holes (T)Octahedral Holes (O)
53
Z = 4
number of atoms in the CCP
cell (N)
T = 8 number of tetrahedral
holes (2N)
Z = 4
number of atoms in the CCP
cell (N)
O = 4 number of octahedral
holes (N)
Two Types of Voids (Holes)
HCP
N hcp atoms in a lattice cell
N Octahedral Holes (OC)
2N Tetrahedral Holes (TE)
54
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: Fe3Al, M3C60
Sphalerite (zinc blende) ZnS: CuCl, HgS, GaAs ...
Nickel arsenide NiAs: FeS, PtSn, CoS ...
Wurtzite ZnS: ZnO, MnS, SiC
ICSD
3555 NaCl
3438 MgAl2O4
2628 GdFeO3
55
Structure Types Derived from CCP = FCC
56
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)
57
Fluorite CaF2 and Antifluorite Li2O
F / Li
58
Ca / O
Fluorite CaF2 and 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 59
Fluorite CaF2 and Antifluorite Li2O
60
Pyrochlores = Disordered Fluorite
61
CaF2 (Ca4F8) Pyrochlore A2B2O7
(Na,Ca)2Nb2O6(OH,F)
Y2Ti2O7, La2Zr2O7
Sphalerite (zincblende, ZnS)
Cubic close packing of
anions with 1/2 tetrahedral
holes filled by cations
62
Sphalerite (zincblende, ZnS)
13-15 compounds: BP, BAs, AlP, AlAs, GaAs, GaP, GaSb, AlSb, InP, InAs, InSb
12-16 compounds: BeS, BeSe, BeTe, b-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
63
Cubic Diamond
64
6,16Å
2,50 Å
4,10Å
Sphalerite
Cubic = chairs only
Wurzite
Hexagonal = chairs + boats
SiO2 cristobalite
Replace C-C with Si-O-Si
Lonsdaleite
SiO2 tridymite
Ice-hexagonal
Replace C-C with O-HꞏꞏꞏꞏO
Diamond
65
Diamond Structure
66
Elements of Group 14: C, Si, Ge, grey-Sn
• Add 4 atoms to FCC/CCP to ½ of tetrahedral holes
• Tetrahedral bond arrangement of all atoms
• Each atom has 4 nearest neighbors and 12 next nearest neighbors
Cuprite Cu2O - Cubic Diamond Lattices
67
Two interpenetrating diamond lattices
Replace C-C with O-Cu-O
Wurzite, ZnS
Hexagonal close packing of
anions with 1/2 tetrahedral
holes filled by cations
68
ZnO, ZnS, ZnSe,
ZnTe, BeO, CdS,
CdSe, MnS, AgI,
AlN, GaN
Lonsdaleite
Zincite, ZnO
69
Semiconductors of 13-15 and 12-16 type
70
Rock Salt, NaCl
Cubic close packing of anions with all
octahedral holes filled by cations 71
Anion and cation sublattices = same FCC/CCP
Rock salt (NaCl) = Anti-rock salt (ClNa)
Rock Salt, NaCl
72
Rock Salt Structures (NaCl)
Hydrides: LiH, NaH, KH,
NH4BH4 – H2 storage material Pd(H)
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, a-MnS,
MgSe, CaSe, SrSe, BaSe, CaTe
Halides: LiF, LiCl, LiBr, LiI, NaF, NaBr, NaI, KF,
KCl, KBr, KI, RbF, RbCl, RbBr, RbI, CsF, AgCl,
AgF, AgBr
Intermetallics: SnAs
Other
FeS2 (pyrite), CaC2, NaO2
73
Rock Salt Structures (NaCl)
74
Rock Salt Structures (NaCl)
FeS2 (pyrite), CaC2, NaO2
SiO2 pyrite - high pressure polymorph,
in Uranus and Neptune core
75
Nickel Arsenide, NiAs
Hexagonal close packing of
anions (As) with all
octahedral holes filled by
cations (Ni)
NiS, NiAs, NiSb, NiSe, NiSn, NiTe,
FeS, FeSe, FeTe, FeSb, PtSn, CoS,
CoSe, CoTe, CoSb, CrSe, CrTe,
CoSb,
PtB (anti-NiAs structure)
76
unit cell
Ni
As
ReB2 - type
Hexagonal close packing of
anions with all tetrahedral
holes filled by cations
77
Li3Bi (anti BiF3)
Fe3Al
[Cr(NH3)6]Cl3
K3[Fe(CN)6]
M3C60
Cubic close packing of
anions with all tetrahedral
and octahedral holes filled
by cations
78
CsCl
Primitive cubic packing of
anions with all cubic holes
filled by cations
79
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
80
NaTl
Niggli – 230 space groups – restrictions on arrangement of atoms:
There are only 4 possible AB cubic structures:
NaCl, ZnS-sphalerite, CsCl, and NaTl
Both sublattices form independent diamond structures.
The atoms sit on the sites of a bcc lattice with abcc = ½ a
81
SC of ReO6 octahedra
ReO3
NaCl structure with 3/4 of cations removed and
1/4 of anions removed
Cubic-WO3, UO3, MoF3, NbF3, TaF3, AlF3, Cu3N 82
What type of unit cell?
sc, bcc, fcc
1839 G. Rose named mineral after
C. A. Perovski
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 83
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
84
Perovskite, ABX3
MgSiO3, CaSiO3
KNbO3, KTaO3, KIO3,
NaNbO3, NaWO3,
LaCoO3, LaCrO3,
LaFeO3, LaGaO3, LaVO3,
SrTiO3, SrZrO3, SrFeO3
ThTaN3, BaTaO2N
85
0.8 < t < 0.9 orthorhombic/monoclinic
0.9 < t < 0.97 cubic
0.97 < t < 1.02 tetragonal
Goldschmidt’s tolerance factor
Perovskite - Ferroelectric BaTiO3
86
Perovskite, BaTiO3
87
Tc = critical temperature
Perovskite - Ferroelectric BaTiO3
88
Perovskite Structure of YBCO
89
Perovskite Structure of CH3NH3PbI3
90
Three Polymorphs of TiO2
anatase (a), rutile (b) and brookite (c)
91
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)
92
Rutile, TiO2
GeO2, CrO2, IrO2, MoO2, WO2, NbO2, -MnO2, OsO2, VO2
(>340 K), RuO2, CoF2, FeF2, MgF2, MnF2
93
TiO6 – octahedra
OTi3 – trigonal planar
(alternative to CaF2 for highly
charged smaller cations)
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
94
Magnetite (Fe3O4) and Maghemite (γ-Fe2O3)
95
Cubic inverse spinel
O2- atoms are arranged in closepacked
FCC lattice
Fe2+ occupy ½ of OCT sites
Fe3+ are split evenly across the
remaining OCT and TET sites
Fully oxidized form of magnetite
Inverse spinel with cation
deficiency
One of every six octahedral sites
in magnetite is vacant
Stoichiometry Fetet(Fe5/3□1/3)octO4
Spinels, AB2X4
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
96
Corundum, Al2O3
Al2O3 lattice consists of HCP array of O2−
ions
Al3+ ions fill ……. of all octahedral holes
The Al centres are surrounded by …….
oxides
Oxides are coordinated by …… Al3+ ions
97
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
98
Synthetic Garnets A3B5O12
YAG
Garnet Y3Al5O12
Y3 = red - dodecahedral
trivalent cations, large
size
Al5 = blue
2 octahedral
3 tetrahedral
O12
99
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
100
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
Mg(OH)2 - brucite
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)
101
CdI2 Hexagonal Close Packing
102
HCP of anions with 1/2 octahedral
holes filled by cations
Fully occupied and completely
empty planes alternate
CdCl2 Cubic Close Packing
103
CCP of anions with 1/2 octahedral holes filled by
cations, fully occupied and completely empty planes
alternate
Vocabulary of terms
104
Parallelepiped = rovnoběžnostěn