Material Science I Ceramic Materials Chapter 2: Crystal ... · packed spheres = highest atomic packing density for ionic bonding •Cationen take a maximum distance from each other,

1Ceramics: Crystal Chemistry, Chap 2

Material Science I

http://www.mineralienatlas.de/ http://webmineral.comhttp://www.uniterra.de

Ceramic Materials

F. Filser & L.J. Gauckler

ETH-Zürich, Departement Materials

[email protected]

HS 2007

Chapter 2: Crystal Chemistry

http://www.mineralienatlas.de/

http://webmineral.com/

http://www.uniterra.de/

mailto:[email protected]


Material Science I

Order of the atoms in a solid

• type, strength and direction of the bonds determines the atom‟s

spatial order in a solid.

• the strength of a bond is determined by the potential well.

• The order of the atoms in a solid determines its crystal structure.

• The crystal structure (spatial filling) is determined by

(a) the stoichiometry (chemical composition),

(b) the ratio of radii of the ions and

(c) the type of the bond (its tendency towards covalent bonding).


Material Science I

Bond types in Solids

• metallic

• ionic

• covalent

• attraction holds the solid together

• no prefered direction

• charge neutrality of the bond

• ionic bonds unfavorable

charge neutrality of the bonds

• direction of the bond is very important, and

• prevails the principle of achieving max. packing

density

• no preferred spatial direction of the bond

• delocalized electrons and conduction bands ->

charge neutrality of a bond is not required

• maximum coordination, densest packing

• metallic

• ionic

• covalent

Ceramic


Material Science I

Mechanism of Bond Formation: Ionic Bonding

electron transferordering of the ions in a

crystal structureRocksalt crystal

electron

reception

electron

donation


Material Science I

Equilibrium of Attraction and Repulsion:

Ionic Bonding

Sum of attracting and repelling potential

bringing together a cat-ion and an an-ion

Ion‟s Distance

Potential

Eattraction

E repulsion

r0 = equilibrium distance

attr

acti

ng

rep

elli

ng

Sum

- +r0


Material Science I

Equations for the Potentials: Ionic Bond

2

1 2

0

2

1 2

0

4

4

net att rep

att

rep n

net n

E E E

z z eE

r

BE

r

z z e BE

r r


Material Science I

Equilibrium Distance and Energy of a Bond:

Ionic Bond

0

2

1 2

2 1

0 0 0

2

1 2

0 0

04

11

4

net

n

r r

bond

dE z z e n B

dr r r

z z eE

r n

What can we do with the knowledge of Ebond ?


Material Science I

Lattice Energy: Ionic BondingExample: Structure type AB (NaCl)

NaCl lattice structure (Rocksalt)


Material Science I

Lattice Energy: Ionic BondingInteraction of charges within the lattice structure (NaCl)

(Equation is only valid for ions of equal charge)

2

1 2

0 0

2

1 2

0 0

2

1 2

0 0

11

4

11

4

11

4

6 12 8 6 24....

1 2 3 4 5sum

sum

Lattice Av

z z eE

r n

z z eE

r n

z z e

r nE N

2

1 2

0 0 04sum n

z z e BE

r r


Material Science I

Madelung Constant

Structure type Stoichiometry

Rocksalt NaCl AB 1.74756

Cesiumchloride CsCl AB 1.76267

Zinc blende ZnS AB 1.63806

Wurtzite ZnS AB 1.64132

Fluorite CaF2 AB2 5.0387

Rutile TiO2 AB2 4.816

Cadmiumiodide CdI2 AB2 4.383

Corundum Al2O3 A2B3 25.031

2.5190

2.4080

4.1719

2.4023

We find big, non-neglible differences in a simple

calculation of the Madelung constant like before

vs its precise calculation !!!


Material Science I

Literature on the Calculation of

the Madelung Constant

References for Madelung's Constant:

• M. L. Glasser and I. J. Zucker, Lattice sums, Theoretical Chemistry: Advances and Perspectives, v. 5, ed. D. Henderson, Academic Press, 1980.

• D. Borwein, J. M. Borwein and K. F. Taylor, Convergence of lattice sums and Madelung's constant, J. Math. Phys 26 (1985) 2999-3009; MR 86m:82047.

• J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Wiley, 1987; MR 99h:11147.

• I. J. Zucker and M. M. Robertson, Exact values for some two-dimensional lattice sums, J. Phys. A: Math. Gen. 8 (1975) 874-881; MR 54 #9515.

• K. F. Taylor, On Madelung's constant, J. Computat. Chem. 8 (1987) 291-295; MR 88h:82066.

• A. Hautot, A new method for the evaluation of slowly convergent series, J. Math. Phys 15 (1974) 1722-1727; MR 53 #9575.

• R. E. Crandall, New representations for the Madelung constant, Experim. Math. 8 (1999) 367-379.

http://www.ams.org/mathscinet-getitem?mr=86m:82047

http://www.ams.org/mathscinet-getitem?mr=99h:11147

http://www.ams.org/mathscinet-getitem?mr=54

http://www.ams.org/mathscinet-getitem?mr=88h:82066

http://www.ams.org/mathscinet-getitem?mr=53

http://www.expmath.org/expmath/volumes/8/8.html


Material Science I

Ionic Bonded Solids

• Ions are modeled as rigid and charged spheres.

• Ions possess an ionic radius which is a function of its atomic number and its

valency.

• Coulomb attraction is effective along the direct connection line of the spheres„

centerpoints.

• No ion overlap because of the high repulsion forces at short inter-ionic

distance.

• cations want to be surrounded by as much anions as possible closest

packed spheres = highest atomic packing density for ionic bonding

• Cationen take a maximum distance from each other, anions will do the same.

• Ionic bonds are isotrop, i.e. they are non-directional.

• The ionic ratio (cations to anions) determines the spatial structure of the ions.


Material Science I

Covalent Bonded Solids

• The direction of the bonds are the main factor.

• The atomic orbitals mainly determine the direction of

the bonds.

• Highest atomic packing density is sacrificed for the

direction of the bonds.

• A less dense packing for covalent bonded solids in

comparison to ionic bonded solids.


Material Science I

Metallic Bonded Solids

• Free electrons in metals, i.e. valency electrons can„t be allocated to

one atom and they move freely within the solid body. No

limitation because of charge neutrality.

• No limitations because of stoichiometry

• Ions are modeled as spheres, the bond is non-directional. All ions

and electrons possess the same attraction.

• In pure metals all ions are of the same size, therefore ions pack as

dense as possible, i.e. want to achieve highest packing density

• Simlarily for alloys and intermetallic phases. However, in some

cases the different radius of the atoms prevents it from achieving

the closest packing density.


Material Science I

Ionic Radius

• Each neutral atom possess a radius which is determined by its outer

electron orbit (Elektronenschale).

• Atomic radius decreases within a period of the periodic system

(horizontal from left to right).

• Ionisation of an atom (electron donation or reception) changes its radius.

• If valence electrons are donated (cation), then the remaining electrons will

be stronger attracted to the nucleus and the ionic radius decreases

Example: neutral charged Na atom: rNa = 1.86 Å, BUT rNa+ = 0.98 Å.

• If more than one electron is donated, then the ionic radius decreases.

• If valence electrons are received (anion), then the ionic radius increases

Example: neutral charged Cl atom, rCl = 1.07 Å, BUT rCl- = 1.81Å.

• If more than one electron is received then the ionic radius increases.


Material Science I

Packing Density in Ionic Bonded Solids

stable stable instable


Material Science I

rCation/RAnion Coordination

number CN

Name Geometry

0 - 0.155 2 linear

0.155 - 0.225 3 triangular

planar

0.225 - 0.414 4 tetrahedral

0.414 - 0.732 6 octahedral

0.732 - 1.0 8 cubic

1.0 12 12-

coordinated

Ionic Radius and Coordination Number

in Ionic Bonded Solids


Material Science I

Packing Density in Covalent Bonded Compounds

The coordination number is determined by:

• number of valence electrons in each atom

• number of valence electrons which participate in the bonding

• hybridisation of the orbitals (sp, sp2, sp3)

Atoms of Group IV A to VII A show number of bonding NB

assuming single bonds:

NB = 8 - NV

NB = number of bondings per atom

NV = number of valence electrons for that atom


Material Science I

Packing Density in Covalent Bonded Solids: Carbon

Diamond:

sp3 hybride: single bonds (s), hence the

coordination number is 4

Graphite:

sp2 hybride: single bonds (s), planar,

coordination number is 3

the free “unpaired” electron per C atom is

responsible for a weak bonding between the

platelike layers of the graphite.

more ceramic examples: SiC, Si3N4, AlN


Material Science I

Packing Density for Metals

pure metals: r/R=1

hence coordination number 12 => closest packed, highest packing density

fcc


Material Science I

fcc / cpp - metals

C

B

A

C

B

A

bcc - metals


Material Science I

Other Packing Density than the most Dense

Packing in Metalic Bonded Solids

Body-Centered Cubic (BCC)

Coordination number 8

Packing density 68 vol-%

Simple Cubic (SC) =

Coordination number 6

Packing density 52 vol-%


Material Science I

Why should metals have also

other packing densities than fcc?


Material Science I

Melting temperature: Melting temperature:

Gruppe IA (oC) Gruppe IIA (oC)

Li (181) Be (1290)

Na (98) Mg (650)

K (63) Ca (839)

Rb (39) Sr (769)

Cs (29) Ba (729)

Thermal Expansion Coefficient: Thermal Expansion Coefficient:

Gruppe IA (oC) Gruppe IIA (oC)

(x 10-6 cm/cm) (x 10-6 cm/cm)

Na (70) Be (12)

K (83) Mg (25)

E E

Why should metals have also



Material Science I

Group VB and VIB, transistion metal (V, Cr, Nb, Mo, Ta, W, Fe)

• These elements possess partial filled d orbitals in their base state.

• The d-electrons are split-up in either bonding or antibonding orbitals.

• This split favors the bcc structure over a closest packed structure (hcp,

fcc)

Why should metals also have



Material Science I

Atomic and Ionic Radii

From this section you should learn:

the concept of atomic radii

The concept of ionic radii and how they change with:

• the atomic number in the periodic system

• the coordination number

• the oxidation state / oxidation number

• for coordination numbers of CN 6 and 8, respectively


Material Science I

Atomic Radii

The periodic system of the elements: http://www.uniterra.de/

http://www.uniterra.de/


Material Science I

Ionic Radius = Bond Length

Ionic radius can„t be measured isolated, but it can be

derived from the bond length in elements and compounds. (see - Shannon, Acta Cryst. (1976) A32 751)

Oxygen ion is assumed to: r0 = 1.26 Å


Material Science I

Different Ionic Radii:

Ions can be approximated as rigid spheres.

Element or

Compound

Elements or

Compounds, („Alloys“)Pure Ionic bonding

Metals atomic radius = d/2 in the element (metalic radius)

covalent radius = d/2 in simple bonding (s)

Nonmmetals atomic radius = d/2 in the element

covalent radius = d/2 in simple bonding (s)


Material Science I

Radius of Anions & Cations in

Periods and Groups of the Periodic System

incre

asin

g ra

diu

s

dec inc

decreasing radius

incre

asin

g ra

diu

s

dec dec inc


Material Science I

Ionic Radius

The ionic radius (in pm) of iso-charged ions grew

with increasing nucleus charge (atomic number)


Material Science I

Determination of the Ionic Radius

electron density map of NaCl crystals

electron

density


Material Science I

Ionic Radius

References:

• Krug et al. Zeit. Phys Chem.

Frankfurt 4 36 (1955)

• Krebs, Fundamentals of Inorganic

Crystal Chemistry, (1968)


Material Science I

Coordination Number CN

Bonding CN Length (Å)

C-O 3 1.32

Si-O 4 1.66

Si-O 6 1.80

Ge-O 4 1.79

Ge-O 6 1.94

SnIV

-O 6 2.09

PbIV

-O 6 2.18

PbII-O 6 2.59

The ionic radius of an element increases with increasing CN.


Material Science I

Variation of the Ionic Radius with its CN


Material Science I

Valency


The ionic radius decreases with increasing valency.

Bonding CN Length (Å)

C-O 3 1.32

Si-O 4 1.66

Si-O 6 1.80

Ge-O 4 1.79

Ge-O 6 1.94

SnIV

-O 6 2.09

PbIV

-O 6 2.18

PbII-O 6 2.59


Material Science I

Main Group in the Periodic System


The ionic radius decreases with increasing valency.

The ionic radius increases within a main group of periodic system from

top to down (increasing atomic number)

Anions are often larger than cations.

Bonding LC-O 3 1.32Si-O 4 1.66Si-O 6 1.80Ge-O 4 1.79Ge-O 6 1.94Sn

IV-O 6 2.09

PbIV

-O 6 2.18Pb

II-O 6 2.59

CN Length (Å)C-O 3 1.32Si-O 4 1.66Si-O 6 1.80Ge-O 4 1.79Ge-O 6 1.94Sn

IV-O 6 2.09

PbIV

-O 6 2.18Pb

II-O 6 2.59


Material Science I

Interstices in Crystalline Solids


Material Science I

Tetrahedral Hole (Interstice)

Spatial space of the tetrahedral interstice.


Material Science I

Octahedral Hole (Interstice)

RX

RA

cross section of the octahedral interstice


Material Science I

Rules for the Ionic Radii Ratio(Coordination Number CN =6)

Calculation of the ionic radii ratio in case of

octahedral coordination (CN = 6)

R= radius of the large ions

r = radius of the small ions

2

145cos

rR

R

rRR 2

rR )12(

414.0R

r


Material Science I

For coordination number CN = 8 :

unit cell length a = 2R

ions are in touch along the room diagonal of the cell:

a3 = 2(R+r)

division: 3 = (R+r)/R

multiplication: 3R = R+r

then: R(3 -1) = r

r/R = 3 -1 = 0.732

Rules for the Ionic Radii Ratio(Coordination Number CN = 8)


Material Science I

Radius ratio: Limits of a Coordination Configuration

If r/R < 0.414, then the cation is too small and “wiggles” in the octahedral hole

If r/R > 0.414, then the anions will be moved away from each other

If r/R << or >> 0.414, then the coordination number changes

This simple rules works very often, however … not in all the cases!

Coordination smallest r/R ratio

linear, 2 -

triangular planar, 3 0.155

tetrahedral, 4 0.225

octahedral, 6 0.414

cubic, 8 0.732

closest packing, 12 1.000


Material Science I

Additional means for the classification of crystals

1) Maps for crystal structures (structure maps)

e.g. for AxByOz compounds:

draw a diagram showing radius of A ions vs radius of B ions and mark

the limits for the different structures (properties) in that map!

2) Mooser-Pearson Graphs

Focus on the amount of the convalent bond. Draw a graph showing the

difference of electronegativity of the elements versus the principal

quantum number and determine the limits for the different structures.

3) “Structure - Property - Maps”

e.g. for Perovskites ABO3

draw a graph using the polarisation of A ion vs the polarization of B

ion and mark the limits of a property of the compound.


Material Science I

A2BO4 Compounds


Material Science I

Mooser-Pearson Graph – AB Compounds


Material Science I

ABO3 “Structure – Property - Map”

LaMnO3

Perovskiktes: ABO3

LaCoO3

Kamata, K.,Nakamura, T., Sata, T.(1974):“ On the State of d-

electrons in perovskite-tape compounds ABO3“, Bulletin of Tokyo

Institute of Technology 120:73

Doshi, R., et al. (1999):“Development of Solid Oxide Fuel Cells

that operate at 500°C“ Journal of the Electrochemical Society:

146(4):1273

T

T

T


Material Science I

Ionic Radius: Summary

The ionic radius of an element increases with increasing

coordination number and decreasing valency.

The ionic radius increases with increasing atomic

number within the main groups in the periodic system

The ionic radius ratio can be calculated and in lots of

cases to predict the ionic coordination with its help.


Material Science I

Principles of coordination I

Coordination-scheme polyhedron examples

cubic close

packing ccp

(Cu, Ne, etc)

hexagonal

close packing

hcp

(Mg, He, etc)


Material Science I

Principles of coordination II


Material Science I

Propensity towards Tetrahedric Coordination

Many compounds show a tetrahedric coordination despite their ionic

radius ratio (rc/rA).

For example, many compounds with ionic radius ratio of rc/rA=0.414

crystalize in a tetrahedric coordination like zinc blende or wurtzite. This

is due if the covalent character of the bonds is pronounced, for example,

if :

1) cations with a high polarizing ability, i.e. Cu2+, Al3+, Zn2+, Hg2+ are

combined with anions which are readily to polarize, i.e. I-, S2-, Se2-.

and:

2) atoms are used which are likely to become sp3 -hybridized orbitals, i.e. Si, C

and Ge.


Material Science I

Ionic Bonded Solids

radius ratio rC/rX0.3 0.4 0.5 0.6 0.7 0.8 0.9

stoichiometry

AX ZnS NaCl CsCl

AX2SiO2 TiO2 CaF2

A2X3

Corundum

(-Al2O3)

ABX3CaCO3 and Perovskites

A3X4

AB2X4

Spinels


Material Science I

AX: Compilation of possible Structures

Compound ZnS NaCl CsCl

rCation/RAnion 0.40 0.53 0.92

Coordination Number 4 6 8

Zin

c b

lende

Wurt

zite

Rocksalt


Material Science I

AX: Zinc blende

Representative Ionic Radius Ratio ZnS

ZnS, -SiC, GaAs 0.40

Cation Zn2+

Coordination Number C.-Polyhedron Ionic Radius [Å]

4 Tetrahedron 0.74

Anion S2-

Coordination number C.-Polyhedron Ionic Radius [Å]

4 Tetrahedron 1.84


Material Science I

AX: Zinc blende (ZnS)

Zinc blende


Material Science I

AX: Wurtzite

Representative Ionic Radius Ratio ZnS

ZnS, AlN, BeO, ZnO 0.40

Cation Zn+


4 Tetrahedron 0.74

Anion S-


4 Tetraedron 1.84


Material Science I

AB: Wurtzite

Wurtzit

..\vrml\VMRL Animationen der Kristallstrukturen.ppt


Material Science I

AX: NaCl

Representative Ionic Radius Ratio NaCl

NaCl, CaO, MgO, FeO 0.54

Cation Na+


6 Octahedron 0.97

Anion Cl-

Koordinationszahl C.-Polyhedron Ionic Radius [Å]

6 Octahedron 1.81


Material Science I

AX: NaCl

Structure of Rocksalt

..\..\..\..\Vorlesungen\IGS\WS-03-04\DiamondFil\CaO-1.DSF

..\vrml\VMRL Animationen der Kristallstrukturen.ppt


Material Science I

AX2: Compilation of Structures

Compound SiO2 TiO2 CaF2

rCation/RAnion 0.32 0.52 0.74

Coordination Number 4 6 8

Quart

z T

ype

Rutile

Type

Flu

ori

te T

ype


Material Science I

AX2: Quartz

Representative Ionic Radius Ratio SiO2

SiO2 0.32

Cation Si4+


4 Tetrahedron 0.42

Anion O2-


2 linear Coord. 1.32


Material Science I

AX2: Quartz

high cristobalite


Material Science I

AX2: Rutile

Representative Ionic Radius Ratio TiO2

TiO2, PbO2, GeO2 0.52

Cation Ti4+


6 Octahedron 0.68

Anion O2-


3 planar 3-coord. 1.32


Material Science I

AX2: Rutile

Structure of Rutile


Material Science I

AX2: Fluorite

Representative Ionic Radius Ratio CaF2

CaF2, ZrO2, CeO2 0.74

Cation Ca2+


8 Cube 0.99

Anion F-


4 Tetrahedron 1.33


Material Science I

AX2: Fluorite

Structure of Fluorite


Material Science I

A2X3: Corundum

Representative Ionic Radius Ratio Al2O3

Al2O3, Fe2O3, Cr2O3, B2O3 0.39

Cation Al3+


6 Octahedron 0.51

Anion O2-


4 Octahedron with 2

empty vertices

1.32


Material Science I

A2X3: Corundum

Corundum

- the large circles represent the oxygen ions- and

- the small circles represent the alumium ions.

C

B

A

C

B

A


Material Science I

ABX3: Calcite

Representative Ionic Radius Ratio CaCO3

CaCO3 Ca2+: [CO3]2- = 0.36

Cation Ca2+


6 Octahedron 0.99

Anion [CO3]2-

Coordination number C.-Polyhedron Ionic Radius [Å]a

6 Octahedron 2.72

a The bond length C-O in CO3-complex is 1.36 Å.


Material Science I

ABX3: Calcite

Calcite

Ca


Material Science I

ABX3: Calcite

Calcite

Ca


Material Science I

ABX3: Calcite

Calcite

O

C

planar CO32-


Material Science I

ABX3: Calcite

Calcite

Ca

O

C


Material Science I

ABX3: Perovskite

Representative Ionic Radius Ratio CaTiO3

CaTiO3, BaTiO3 Ca2+:O2-=0.75; Ti4+:O2-=0.52

Cation Ca2+


12 cuboctahedron 0.99

Anion O2-


4 Planare 4-Coordination 1.32

Cation Ti4+


6 Octahedron 0.68


Material Science I

ABX3: Perovskite

Perovskite


Material Science I

ABX3: Perovskite

Perowskite

- TiO6 – octahedron

- CaO12 – cuboctahedron

(Ca2+ and O2- possesses a

cubic close packing)

mainly ferroelectrika, superconductors,

etc.


Material Science I

A2BX4: Spinel

Representative Ionic Radius Ratio MgFe2O4

Fe2MgO4, Mg2TiO4 Mg2+:O2-=0.50; Fe3+:O2-=0.48

Cation Mg2+


4 Tetrahedron 0.66

Anion O2-

Coordination Number K.-Polyhedron Ionic Radius [Å]

6 Octahedron 1.32

Cation Fe3+


6 Octahedron 0.64


Material Science I

A2BO4 Spinel


Material Science I

A2BO4 Spinel


Material Science I

A2BO4 Spinel


Material Science I

A2BO4 Spinel


Material Science I

A2BO4 Spinel


Material Science I

A2BO4 Spinel


Material Science I

A2BO4 Spinel


Material Science I

Spinel

B2AX4: Spinel

O2-

Mg2+

Fe3+

Example: MgO x Fe2O3 = MgFe2O4


Material Science I

Summary

Factors determining the crystal structure:

Three major factors influence the crystal structure in

solid compounds:

1.) the stoichiometry,

2.) the ratio of cation radius and anion radius

and

3.) the propensity for the convalent bond type (sp3-

hybridized bonding).


Material Science I

Additional Slides


Material Science I

What types of bonds do you know ?

Primary bonds:

metallic, ionic or covalent bonds.

Secondary bonds:

Van-der-Waals, hydrogen or coordinative

bonds.

Primary bonds are generally much stronger then

secondary bonds.

Documents

Material Science I Ceramic Materials Chapter 2: Crystal ... · packed spheres = highest atomic packing density for ionic bonding •Cationen take a maximum distance from each other,