Transcript The tatle of lecture - Warsaw University of Technology

Ionic Conductors: Characterisation of Defect
Structure
Lectures 1-4
Introduction to Crystal Chemistry
Dr. I. Abrahams
Queen Mary University of London
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Crystal Chemistry
What is crystal chemistry?
The study of the structures of crystals including:
Description and classification of crystal structures
Factors that govern structure types adopted
Structure prediction
Structure-property relationships
What is a crystal?
A solid that shows a regularly repeating structure that can be
characterised by a basic repeating unit known as a unit cell.
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Unit Cells
Parallelepiped
The unit cell is generally chosen as the smallest repeating unit with the
highest symmetry.
The unit cell, when repeated in 3D, must cover all the space in the
crystal lattice.
Different crystal structures have different unit cells.
Unit cells are defined by six parameters in 3D.
a, b, c are the unit cell edges and ,  and  are the inter-axial angles.
(0 is the origin and its position is arbitrary).
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Crystal Systems
There are seven crystal systems. These can be distinguished by the
different unit cell shapes and their minimum intrinsic symmetry.
Crystal system
Triclinic
Unit cell shape
abc; 90
Minimum symmetry
None
Monoclinic
(standard setting)
abc; ==90, 90
1 two fold axis or
mirror plane
Orthorhombic
abc; ===90
Tetragonal
a=bc; ===90
Trigonal
(rhombohedral setting) a=b=c; ==90
(hexagonal setting)
a=bc; ==90=120
3 two fold axes or
mirror planes
1 four fold axis
1 three fold axis
Hexagonal
a=bc; ==90=120
1 six fold axis
Cubic
a=b=c; ===90
4 three fold axes
The symbol  used here refers to not necessarily equal to. In some cases
there is accidental equivalence, but the minimum symmetry is not present.
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Fractional Coordinates
The location of the origin is arbitrary, but is usually chosen to correspond to
a point of symmetry. It need not be an atom position.
Atoms positions can be defined with respect to the unit cell using fractional
coordinates x, y, z
x = X/a where X is the distance parallel to the a-axis
y = Y/b where Y is the distance parallel to the b-axis
z = Z/c where Z is the distance parallel to the c-axis
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Introduction to Crystal Chemistry
There are many crystalline solids, but only a few basic structures.
Many simple structures can be visualised in terms of close packing of
identical spheres, in some case with smaller spheres in the spaces
between the close packed spheres.
Atoms or ions can be regarded as “squashy” spheres. The squashy
character is a result of polarisation of the electronic cloud surrounding
these atoms or ions.
Different compounds with the same structure have the same geometry,
but different size, i.e. different ionic radii and bond lengths.
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e.g.
NaCl, MgO, LiI, TiC all exhibit the rocksalt structure
For compounds that adopt the rocksalt structure there is no direct
correlation between structure and bonding, i.e. the rocksalt structure
is adopted by ionic and covalent compounds.
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Close Packing (cp)
Identical spheres can pack in a number of ways. The closest way is
known as close packing. Consider some arrays of identical spheres.
1-D
cp, CN = 2
2-D
cp, CN = 6
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3-D
cp, CN = 12
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Hexagonal and Cubic Close Packing
There are two types of 3-D close packed arrays.
Hexagonal close packing
Cubic close packing
hcp
ccp
ABA…..
ABC…
A
A
B
B
A
C
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hcp and ccp Unit Cells
Like all crystalline solids hcp and ccp based solids can be described by
unit cells.
hcp
ccp
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Non-Close Packed Arrays
Compare two similar 2-D arrays.
2-D
cp, CN = 6
2-D
Non-cp, CN = 4
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3D Non-Close Packed Arrays
Body centred cubic (bcc) packing is a non-close packed array.
bcc CN = 8
Packing density
Even in close packed arrays there are spaces between the spheres. A
measure of how closely packed spheres are is the packing density
e.g in ccp
Volume of 4 spheres
Packing density 
Unit cell volume
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Packing Density
Look at a single unit cell face in ccp.
Diameter of sphere = 2r
 face diagonal = 4r
cell edge 
4r
2 2r
2
cell volume  16 2 r 3
4
volume of sphere   r 3
3
 volume of 4 spheres 
Packing
16 3
r
3
16  r 3

3
Density 

 0.7405 74%
3
16 2 r
3 2
Density
hcp
74%
ccp
74%
bcc
68%
Therefore the maximum packing density for identical spheres is 74% for a
cp array
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Metals
Metal atoms can be considered to be spherical and adopt structures that
exhibit high coordination numbers in order to achieve maximum overlap of
atomic orbitals.
Metallic elements
In metallic elements since all atoms are of the same type and size ccp,
hcp and bcc packing are typically adopted.
However, it should be noted that in some cases although a cp geometry is
adopted the packing density may be lower than 74% i.e. not truly close
packed.
eg ccp Ag, Au, Fe, Pb
hcp Be, Co,Mg
bcc Ba, Cr, K
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Alloys
Metallic compounds with more than one atom type . If the atom sizes are
similar then as with metallic elements ccp, hcp or bcc structures are
adopted.
e.g Cu:Au Alloy disordered ccp
Note at certain compositions Cu and Au can order over the lattice.
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Interstitial Sites
In order to describe inorganic compounds using close packing it is first
necessary to describe the interstitial sites present in a cp array. There are
two important types of interstitial site
1. Tetrahedral sites
Consider atoms from just two cp layers. Spheres in the top layer fit into
dips between 3 spheres in the bottom layer and vice versa. This gives a
tetrahedral interstitial site.
There are two types of
tetrahedral interstitial site
T+
A tetrahedron has 4 faces and
6 edges
T
(pointing up)
(pointing down)
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Interstitial Sites - Tetrahedra
Number of T+ sites = Number of T sites
The tetrahedral sites do not lie strictly between the cp layers.
T+ in layer below
T in layer above
The maximum radius rT of a sphere in a tetrahedral site is given by
rT = rcp  0.225
Where rcp is the radius of the close packed sphere.
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Interstitial Sites - Octahedra
2. Octahedral sites
Where dips in the top and bottom layers coincide we get an octahedral site.
An octahedron has 8 faces and 12 edges.
The maximum radius rO for an atom to fit into an octahedral site is
rO = rcp  0.414
i.e. much bigger than a tetrahedral site.
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Location of interstitial sites in cp
unit cells
1. hcp
Tetrahedral
Octahedral
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2. ccp
Tetrahedral
Octahedral
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Close packing described by polyhedra
One can view cp structures as built up from polyhedra (representing the
interstitial sites) that share faces, edges or corners.
Using this type of representation
(a)
The centre of the polyhedron represents the
interstitial site
(b)
The corner of the polyhedron represents the cp
atom
Polyhedral representations are very important as they emphasize the CN
of the interstitial ions, their relative positions and linkage.
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Interstitial sites in cp structures
1. hcp
Interstitial sites between cp
layers 1 and 2, and 2 and 3 are
identical and stacked one
above the other resulting in
mirror symmetry about B.
(a) Octahedra
Octahedra share faces perpendicular () to cp planes
Octahedra share edges parallel (  ) to cp planes
Results in columns of octahedra perpendicular to cp
planes.
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(b) Tetrahedra
Tetrahedra share faces and corners  to cp layers
Tetrahedra share edges  to cp layers
T+ shares a face with T in layer below
T shares a face with T+ in layer above
T+ shares edges with T within cp layer
T shares edges with T+ within cp layer
(c) Inter-polyhedral linkages
T+ and T sharing faces
gives a trigonal bipyramidal
site CN = 5
Unique to hcp.
Octahedra and tetrahedra share faces
within cp layer.
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2. ccp
Orientation of layers 1 and 2 and 2
and 3 now different.
Octahedra are not above octahedra
Tetrahedra are not above
tetrahedra.
Octahedra share only edges  to cp planes
Octahedra share only edges  to cp planes
Tetrahedra share only edges  to cp planes
Tetrahedra share only edges  to cp planes
T+ shares edges with T only
T shares edges with T+ only
Comparison of oct and & tet linkages in hcp and ccp
Oct shares face with oct in hcp only
Tet shares faces with tet in hcp only
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Interstitial Sites Summary
cp
cp atoms
per cell
Tet sites
per cell
Oct sites
per cell
Tet sites
per cp
atom
Oct sites
per cp
atom
hcp
2
4
3
2
1
ccp
4
8
4
2
1
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Important Inorganic Structures
Based on cp
Many inorganic structures are based on close packing of spheres and
can be described by close packing of one ion sublattice with counter
ions in all or part of the interstitial sites.
While these structures are not truly close packed (i.e. the ions do not
touch each other), their geometry can be described as close packed.
In the case of ionic conducting inorganic solids, many adopt ordered or
disordered forms of the classic inorganic structural types.
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ccp Based Structures
1. Cubic close packed structures
(a) Li3Bi
Li3Bi is an intermetallic compound and can be described as ccp Bi with
Li in all the octahedral and tetrahedral sites.
The Li3Bi structure therefore shows the complete filling of all interstitial
sites.
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(b) NaCl
ccp Cl with Na+ in all the octahedral sites.
ccp Cl at corners and face centres of unit cell
Na+ in oct sites at centre of cube and mid point of
each edge.
NB Tet sites empty
Unit cell
NaCl6 oct
Cl cp planes are  to body
diagonal.
NaCl6 oct share all 12 edges
with other NaCl6 oct.
Shared edge
NaCl6 oct share faces with
empty tet sites.
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rNa 
 0.71
Radius ratio 

rCl
Remember maximum ratio for octahedral coordination in cp system is
0.414
Therefore Cl ions in NaCl are not close packed, but do have cp
geometry with an fcc unit cell.
Note each Cl is surrounded by 6 Na+ ions (and each Na+ is surrounded
by 6 Cl ions).
Many binary compounds exhibit the rocksalt structure. All are
isostructural, but have different properties and bonding.
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Compound
a(Å)
Compound
a(Å)
MgO
4.213
LiF
4.0270
CaO
4.8105
NaF
4.64
SrO
5.160
NaCl
5.6402
BaO
5.539
AgF
4.92
NiO
4.1769
AgCl
5.549
TiO
4.177
AgBr
5.7745
MnO
4.445
MgS
5.200
FeO
4.307
CaS
5.6948
UC
4.955
LaN
5.30
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(c) Zinc blende or sphalerite (ZnS)
ccp S2 with Zn2+ in half the tet sites.
Tet sites all T+(or T) avoiding edge sharing.
Each S2 is surrounded by 4 Zn2+ and each Zn2+ surrounded by 4 S2.
ZnS
Many other structures
can be derived from zinc
blende
C (diamond)
GaAs
GaP
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Si
(d) Fluorite (CaF2)
ccp Ca2+ with F in all the tet sites. (Oct empty).
Both T+ and T occupied. Therefore tet share edges and corners, but
Ca2+ large and not cp.  tet centres are far apart.
Total 4 Ca2+ per cell and 8 F per cell  Ca:F = 1:2 i.e. CaF2
CaF8 = cubic coordination
Antifluorite
ccp anions with cations in all tet sites. e.g. Na2O
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hcp based structures
2. Structures based on hcp
(a) Nickel Arsenide (NiAs)
hcp As with Ni in all the oct sites. (tet empty).
Ni at 2/3, 1/3, 1/4 and 2/3, 1/3, 3/4
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NiAs6 octahedra
AsNi6 also 6 coordinate but not oct (trigonal prismatic)
Each NiAs6 oct shares 2 faces with other NiAs6 oct
resulting in columns of face sharing oct.
NiAs is the hcp analogue of NaCl(ccp), but with face
sharing.
NaCl: Na+  Na+ repulsions favour ccp.
NiAs: Ni2+ Ni2+ repulsion reduced due to
covalence and Ni-Ni bonding  hcp
favoured.
Structure adopted by FeS, NiS and CoS
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(b) Wurtzite (ZnS)
hcp S2 with Zn2+ in half the tet sites. (Oct empty).
Zn2+ on edges 0,0,5/8
Zn2+ in cell at 1/3, 2/3, 1/8
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Only T+ or T occupied.
Avoids tet sharing faces which is energetically unfavourable.
Also avoids tet sharing edges.
ZnS4 tet corner sharing only
SZn4 also tet
ZnS either wurtzite or zinc blende
Both tet ZnS4
Both corner share
Wurtzite more ionic
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Layered structures
(a) CdCl2 and CdI2
The structures of CdCl2 and CdI2 can be described as being based on
ccp and hcp halide lattices respectively with Cd2+ filling octahedral sites
in alternate layers.
This results in layered
compounds with
alternate layers held
together by van der
Waals forces.
In both structures CdX6
octahedra share edges
with other octahedra in
same layer
Cdl2
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CdCl2
(b) CrCl3 and BiI3
The structures of CrCl3 and BiI3 can be described as being based on
ccp and hcp halide lattices respectively. In both structures 1/3 of the
available oct sites are occupied. 2/3 of the oct sites in alternate layers
are filled by cations resulting in layered structures.
Each octahedron shares
edges with 3 other
octahedra within a layer.
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Other Important Structures
(a) Rutile (TiO2)
Essentially distorted hcp O2 with Ti4+ in half the oct
sites. Every alternate octahedron is filled resulting in
chains of edge sharing TiO6 octahedra.
Columns of octahedra with alternate columns empty. Columns corner
share with neighbouring columns. The columns run parallel to the cp
layers
OTi3 trigonal planar O2 coordination.
Other examples MnO2, SnO2, CrO2, MnF2.
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(b) Corundum (-Al2O3)
hcp O2 with Al3+ in 2/3 of the oct sites. Al3+ displaced resulting in distorted
tet coordination for O2. Corundum is noted for its hardness. Doping with Cr
or Ti results in the gemstones ruby and sapphire.
Other examples
Ti2O3
V2O3
Cr2O3
Ga2O3.
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(c) ReO3
ccp O2 with ¼ of the O2 ions missing. Re6+ locate in ¼ of the the
octahedral sites. The resulting structure is a 3-dimensional array of corner
sharing ReO6 octahedra. Each ReO6 octahedron shares all six corners
with other ReO6 octahedra and linear Re-O-Re linkages.
Other examples
ScF3
NbF3
TaF3
MoF3
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(d) Perovskite (CaTiO3)
Closely related to ReO3.. A ccp array of O2 with ¼ of the O2 ions
missing. Ti4+ located in ¼ of the the octahedral sites. Ca2+ is located in
the oxide ion vacancy. TiO6 octahedra share corners to give the 3-D
framework, with Ca2+ in essentially a 12 CN site. However distortion
lowers the coordination number to 8.
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(e) Spinel (MgAl2O4)
ccp array of O2 with Al3+ located in 1/2 of the the octahedral sites and Mg2+
in 1/8 of the tetrahedral sites. The structure consists of columns of edge
sharing octahedra which share edges with parallel columns. The tetrahedra
share corners with the octahedra.
Inverse spinel
Fe2MgO4 adopts an inverse spinel structure. With half the Fe ions (Fe3+) in
tetrahedral sites and the other half in octahedral sites with Mg2+.
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