Unit cell
Kap. 5
Crystallography and
crystal structures
z
y
x
y
z
x
Unitcell
Transtaltion along x, y, z
Counting of atoms in 2D
Atoms in a corner = ¼
Atoms on an edge = 1/2
Atoms inside the cell = 1
V = a ⋅ (b × c )
Counting of atoms in 3D
A corner-atom is shared between 8 cells ⇒
1/ atoms pr. cell
8
An edge-atom is shared between mellom 4
cells ⇒ 1/4 atom pr cell
A surface-atom is shared between 2 cells ⇒
1/ atom pr cell
2
A atom inside one cell ⇒ 1 atom pr cell
Positioning of atoms
(0.5,0.5,0.5)
(2.5,0.5,0.5)
(1.5,0.5,0.5)
Can add and subtract whole numbers at will.
Crystal plane and crystal directions
A plane
(h k l)
A set of equivalent planes
{h k l}
A direction
[h k l]
A set of equivalent directions
<h k l>
The equivalent planes and directions are a
result of the systems symmetry
e.g.
fcc
<111>
[111] [111] [111] [111]
[111] [111] [111] [111]
Miller indices, 2D
P
I
(111)
(100)
(100)
(200)
(200)
(200)
Crystal plane and crystal directions
Directions
[112]
[101]
A plane
(h k l)
A set of equivalent planes
{h k l}
A direction
[h k l]
A set of equivalent directions
<h k l>
(1,1,2)
(2,0,2)
(½, ½,1)
(1,0,1)
(0,0,0)
[½ 0 ½] = [101] = [202] = n[101]
The equivalent planes and directions are a
result of the systems symmetry
e.g.
fcc
<111>
[111] [111] [111] [111]
Parallel directions have same index
[111] [111] [111] [111]
Atoms as spheres:
- ions
- metal atoms
- molecules
Ionic bonding
Metal bonding
Kulepakking
Van der Waals bonding
Covalent bonding
Closest (densest) packing of spheres:
Spherepacking
74% of the volume is filled by the spheres
The entities have to be:
•Spherical
•Of same type (size)
•Non-compressible
•Non-repulsive / contractive
Ideal sphere packing model
Any observed deviation from the ideal model will be explained by that
the requirements are not fully met.
26% voids / vacant space
The voids/holes will have different appearance:
•Octahedral shape
•Tetrahedral shape
•(Trigonal prismatic holes)
•(Trigonal bipyramidale holes)
The voids/holes may be filled with atoms
•of the same type as the packing spheres
•of different type
Density of packing
Coordination
number (CN)
Name
Density
6
8
8
10
12
Simple cubic
Simple hexagonal
Body-centred cubic
Body-centred tetragonal
Closest packing
0.5236
0.6046
0.6802
0.6981
0.7405
Dense sphere packing
A
AB
ABA
hcp
ABC
ccp
Octahedra holes
CN = 6
Tetraeder hole +
Tetraeder hole -
Octaeder hole
Tetrahedra holes
CN = 4
Type of hole
Cuboctahedron
Cube
Trigonal prismatic
Octahedral
Tetragonal
Triangle
Hexagonal packing (AA..)
Trigonal prismatic holes
CN = 6
A
Hexagonal closepacked (AB..)
Trigonal bipyramidal
CN = 5
B
A
Number
N
N
2N
N
2N
N
Max. radius
1
0.732
0.528
0.414
0.225
0.155
(111)
Diagonal = 4 r,
Volume of cube = (2√2 r)3
Volume of 4 spheres = 4*π*4/3 r3
Density = 16π/3 / (2√2)3 = 0.7405
60°
120°
fcc, Cubic F-centered
lattice
bcc,
cucic, I-centered
(0,0,0) + (1/2,1/2,1/2)
CN = 8
Structure = lattice +
basis (motif)
F-centered lattice with
metal in (0,0,0)
CsCl-type structure, CN = 8
M in (0,0,0)
X in (1/2,1/2,1/2)
Not I-centered, -> P
NaCl-type structure =
cubic + basis
F-centered lattice:
Na in (0,0,0)
Cl in (1/2,0,0)
hcp (hexagonal close packed)
ccp (cubic close packed)
Z=2
Z=4
2 atoms in unitcell:
4 atoms in unitcell:
(2/3, 1 /3, 1 /2)
(0, 0, 0) (0, 1 /2, 1 /2)
(0, 0, 0),
(1 /2, 0, 1 /2) (1 /2, 1 /2, 0)
bcc
Structure (types) derived from dense closepacking of spheres
Principe:
A
Closepacked layers of different types of spheres
B
Filling of holes with smaller spheres
(octahedra-, tetrahedra-, trig. bipyramidal.- holes)
C
Combinations of
Z=2
2 atoms in unitcell:
(0, 0, 0) (1 /2, 1 /2, 1 /2)
A
and
B
A
TiAl3
WAl5 = WAl3 + Al2
B
C
Filling of holes (interstitial possitions)
ABn
MmX
X = Packing sphere
Filling degree
All octaederholes
All tetraederholes
½ teraederholes
½ octaederholes
1/ octaeerholes
3
Mixed spheres in dense packed layers
+ filling of interstitial holes
A, B
X
ABn
MmX
Spherepacking
ccp
hcp
AB
AB2
AB3
AB
AB2
MX
M1/2X
M1/3X
MX
NaCl
M2X
CaF2
ZnS(bl.)
CdCl2
CrCl3
cations
anion
A and X of similar size
B is so small that it fits into octaeder holes
NiAs
ZnS(wu.)
CdI2[Cd(OH)2]
BiI3, β-ZrCl3
AX3 densepacked layers
Those octahedra holes with 6 neighbours of X type is filled with B
ABX3
perovskite type structure
Perovskite
Perovskite
A
A
O
Perovskite
Perovskite
A
A
O
O
B
Perovskite
Perovskite
A
A
B
B
O
O
Perovskite
Perovskite
A
B
O
ABO3
Space group: Pm-3m (No. 221)
High symmetry
Large Mandelung constant
Large crystal energy
Enables costly electron configurations
Interesting properties
A
B
O
Space filling of polyhedra
Perovskite
Property
Compound examples
Insulator
HighHigh-K dielectric
Semiconductivity
Half metallicity
LaGaO3, LaAlO3, LaCrO3, LaFeO3
BaTiO3, Ba2EuZrO5.5, CaCu3Ti4O12
LaMnO3, PbCrO3, RTiO3 (R = La...Tm)
LaBaMn2O5.5, YBaMn2O5.5
Sr2FeMoO6, Ba2FeMoO6, Ca2FeMoO6,
Ca2FeReO6
Metallic conductivity LaNiO3
Superconductivity YBa2Cu3O7, HgBa2CuO4, La1.5Nd0.5CaBa2Cu5Oz,
Bi2Sr2Ca2Cu3O10–
10–d, HgBa2Ca2Cu3O8+d
Colossal
magnetoresistance A0.3La0.7MnO3 (A = Ca, Sr,
Sr, Pr, Pb)
Pb)
Multi ferroics
BiMnO3, BiFeO3,
Ferroelacticity
LaCoO3
Ferromagnetic
SrRuO3, LaMnO3.15, La1–xCaxMnO3, Sr1–xLaMnO3
Anti ferro
BiMnO3, LaFeO3, LaMnO3
Piezoelectricity
PbZr0.47Ti0.53O3
Spin glass
CaRuO3
Multi valence
materials
Ca3Co2O6, Sr4Fe4O11, YBaMn2O5.5
Structures can be described as connections of polyhedra
that share:
Corners
Edges
Faces
The polyhedra are simplified for visual clarity.
Type of polyhedra:
Tetrahedra
Octahedra
Trigonal prismatic
…
Basically the same types of polyhedra as mention for sphere
packing
Limited units, Octahedra
Octahedra
Isolated octahedra
Dimer
Dimer
Dimer
Tetrahedra
MX6
M2X11
(Nb2F11-)
M2X10
(Nb2Cl10)
(U2Cl10)
M2X9
(Fe2(CO)9)
(I2O94-)
Connected by:
Corners
Edges
(Faces)
Connected by:
Corners
(Edges)
How these units connect will affect the chemical
composition, and vice versa.
Polymerization of MX6 octahedra
Corner sharing:
d(M-M) = 2*d(M-X)
Edge sharing:
d(M-M) = √2*d(M-X)
Face sharing:
d(M-M) = 1.16*d(M-X)
Infinite systems; octahedra by cornersharing
Number of corners shared in a given octahedra:
2, (3), 4, (5), 6
AX5 chains: (cis-, trans-)
cis-
VF5
trans-
BiF5
AX4 layers:
SnF4
K2NiF4
AX3 3D network:
ReO3
FeF3
ABX3
perovskite
Polymerization of MX4 tetrahedra
K2NiF4
ReO3
Corner sharing:
d(M-M) = 2*d(M-X)
2.0 only observed for SiO4
Si4+ - Si4+ repulsions
Edge sharing:
d(M-M) = 1.16*d(M-X)
Face sharing:
d(M-M) = 0.67*d(M-X)
Structures based on tetrahedas
Structures based on tetrahedas
No. Vertices only shared, Vertices common to two tetrahedra
shared Formula
Type of complex
Examples
vertices
No. Edges only shared, Edges common to two tetrahedra
shared Formula
Type of complex
edges
Examples
1 A2X7
Finite molecule or pyro ion
Cl2O7, S2O72-, etc.
1 A2X6
Finite dimer
Al2Cl6, Fe2Cl6
2 (AX3)n
Cyclic molecule,
or meta-ion
infinite chain
S3O9, Se4O12, (PNCl2)n
(P4O12)4-, (Si3O9)6-,
(SO3)n, (PO3)nn-
2 (AX2)n
Infinite chain
BeCl2, SiS2, Be(CH3)2
3 (A2X3)n
Infinite double chain
Cs(Cu2Cl3)
Finite polyhedral,
double chain,
layer or
3D structure
P4O10
Al[AlSiO5]
P2O5, Li2Si2O5
P2O5, La2[Be2O5]
4 (AX)
Infinite layer
LiOH, PbO
3D structures
Li2O, F2Ca
Layer,
double layer, or
3D structure
HgI2 (red)
CaSi2Al2O8 (hexag.)
SiO2 structures, GeS2
3 (A2X5)n
4 (AX2)n
Vertices common to three tetrahedra
3 (AX2)n
Infinite layer
AlOCl, GaOCl
n
6 (A2X)n
Vertices and edges shared
(AX)n
Double layer
La2O3, Ce2O2S, U2N2Sb
(AX)n
3D structure
β-BeO
Wurtsitt
ZnS4, SZn4
Sink blende
ZnS4, SZn4
CaF2
FCa4 - tetraheda
Na2O
NaO4 - tetraheda
Silicates:
SiO4 tetrahedas
Corner (vertice) sharing, never edge or face (too strong Si4+-Si4+ repulsions)
Only two SiO4 tetrahedra share a common corner
A4X11
Bridging oxygens count ½
Non-bridging count 1
A6X17
1:3.5
Rings
Double rings
Layer
Double layer
3D
1:4
AX3
A2X5
1:3
1:3
1:2.5
1:2.5
…
1:2