ionic conductors: characterisation of defect structure lectures 1-4 introduction to crystal...

Ionic Conductors: Characterisation of Defect

Structure

Lectures 1-4

Introduction to Crystal Chemistry

Dr. I. AbrahamsQueen Mary University of London

Lectures co-financed by the European Union in scope of the European Social Fund


Crystal Chemistry

What is crystal chemistry?The study of the structures of crystals including:

Description and classification of crystal structuresFactors that govern structure types adoptedStructure predictionStructure-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.


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).

Unit Cells


Crystal Systems

There are seven crystal systems. These can be distinguished by the different unit cell shapes and their minimum intrinsic symmetry.

Crystal system Unit cell shape Minimum symmetryTriclinic abc; 90 None

Monoclinic 1 two fold axis or(standard setting) abc; ==90, 90 mirror plane

Orthorhombic abc; ===90 3 two fold axes or mirror planes

Tetragonal a=bc; ===90 1 four fold axis

Trigonal 1 three fold axis(rhombohedral setting) a=b=c; ==90(hexagonal setting) a=bc; ==90=120

Hexagonal a=bc; ==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.


Fractional CoordinatesThe 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-axisy = 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


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.


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.


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

2-D

cp, CN = 2

cp, CN = 6


3-D cp, CN = 12


Hexagonal and Cubic Close PackingThere are two types of 3-D close packed arrays.

Hexagonal close packing

hcp

ABA…..

Cubic close packing

ccp

ABC…

A

A

A

B B

C


hcp and ccp Unit Cells

Like all crystalline solids hcp and ccp based solids can be described by unit cells.

hcp

ccp


Non-Close Packed ArraysCompare two similar 2-D arrays.

2-D

2-D

cp, CN = 6

Non-cp, CN = 4


3D Non-Close Packed ArraysBody 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

volumecellUnit

spheres4ofVolumedensityPacking


Packing Density

Look at a single unit cell face in ccp.

Therefore the maximum packing density for identical spheres is 74% for a cp array

Diameter of sphere = 2r face diagonal = 4r

%747405.023216

316

3

163

4

216

222

4

3

3

3

3

3

r

r

r

r

r

rr

Density

spheres4ofvolume

sphereofvolume

volumecell

edgecell

Packing Density

hcp 74%

ccp 74%

bcc 68%


MetalsMetal 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 elementsIn 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


AlloysMetallic 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.


Interstitial SitesIn 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 sitesConsider 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.

A tetrahedron has 4 faces and 6 edges

T+

T

(pointing up)

(pointing down)

There are two types of tetrahedral interstitial site


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

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.

T in layer above


Interstitial Sites - Octahedra2. Octahedral sitesWhere 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.


Location of interstitial sites in cp unit cells

1. hcp

Tetrahedral Octahedral


2. ccp

Tetrahedral Octahedral


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.


Interstitial sites in cp structures1. hcp

(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.

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.


(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+ and T sharing faces gives a trigonal bipyramidal site CN = 5

Unique to hcp.

T+ shares edges with T within cp layer

T shares edges with T+ within cp layer

(c) Inter-polyhedral linkages

Octahedra and tetrahedra share faces within cp layer.


2. ccp

Octahedra share only edges to cp planesOctahedra share only edges to cp planes

Tetrahedra share only edges to cp planesTetrahedra share only edges to cp planesT+ shares edges with T onlyT shares edges with T+ only

Comparison of oct and & tet linkages in hcp and ccpOct shares face with oct in hcp onlyTet shares faces with tet in hcp only

Orientation of layers 1 and 2 and 2 and 3 now different.

Octahedra are not above octahedra

Tetrahedra are not above tetrahedra.


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


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.


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.

ccp Based Structures


(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

Cl cp planes are to body diagonal.

NaCl6 oct share all 12 edges with other NaCl6 oct.

NaCl6 oct share faces with empty tet sites.

Unit cellNaCl6 oct

Shared edge


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.

Radius ratio 71.0Cl

Na

r

r


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


(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.

Many other structures can be derived from zinc blende

ZnS C (diamond) Si

GaAs

GaP


(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


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


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


(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


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


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 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.


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.


(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.


(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


(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.


(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+.

ionic conductors: characterisation of defect structure lectures 1-4 introduction to crystal...

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