Solid State Chemistry PDF

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Introduction

Introduction to

Solid State Chemistry

Anorganische Chemie III (Festkörperchemie)

International Graduate Studies

Chemistry of Materials

SS 2002

(Wednesday 08-09 and Thursday 08-10

Room SR 01.132)

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Introduction

Textbooks: Introduction to Solid State Chemistry (English) Textbooks: Introduction to Solid State Chemistry (English)

1. Doughlas, McDaniel, Alexander,

Concepts and Models ofConcepts and Models of Inorganic Chemistry

Inorganic Chemistry

, Wiley, 1992

2. Shriver, Atkins,

Inorganic Chemistry Inorganic Chemistry

(3rd ed, 1999)

W.H. Freeman and Company (Chs. 3, 18 ...)

3. L. Smart, E. Moore,

Solid State Chemistry Solid State Chemistry

, 2nd Ed.

Chapman & Hall, London, 1995

4. P.A. Cox,

The Electronic Structure and Chemistry ofThe Electronic Structure and Chemistry of Solids

Solids

, Oxford University Press, 1987

5. U. Müller,

Inorganic Structural Chemistry Inorganic Structural Chemistry

Wiley, Chichester, 1993

6. A.R. West,

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Introduction

Textbooks: Introduction to Solid State Chemistry (English) Textbooks: Introduction to Solid State Chemistry (English)

1. Doughlas, McDaniel, Alexander,

Concepts and Models ofConcepts and Models of Inorganic Chemistry

Inorganic Chemistry

, Wiley, 1992

2. Shriver, Atkins,

Inorganic Chemistry Inorganic Chemistry

(3rd ed, 1999)

W.H. Freeman and Company (Chs. 3, 18 ...)

3. L. Smart, E. Moore,

Solid State Chemistry Solid State Chemistry

, 2nd Ed.

Chapman & Hall, London, 1995

4. P.A. Cox,

The Electronic Structure and Chemistry ofThe Electronic Structure and Chemistry of Solids

Solids

, Oxford University Press, 1987

5. U. Müller,

Inorganic Structural Chemistry Inorganic Structural Chemistry

Wiley, Chichester, 1993

6. A.R. West,

Solid State Chemistry and its Solid State Chemistry and its ApplicationApplicationss

Wiley, New York, 1984

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Introduction

Textbooks II: Solid State Chemistry (German) Textbooks II: Solid State Chemistry (German)

1. E. Riedel:

Moderne Anorganische Chemie,Moderne Anorganische Chemie,

de Gruyter

1999,

1999, S. 32

S. 329 –

9 – 523 („Festk

523 („Festkörperchemie“)

örperchemie“)

2. L. Smart, E. Moore:

Einführung in die Einführung in die FestkörperchemieFestkörperchemie

,,

Vieweg 1997

3. A.R. West:

Grundlagen der FestkörperchemiGrundlagen der Festkörperchemiee

, VCH 1992

4. U. Müller:

Anorganische Strukturchemie Anorganische Strukturchemie

Teubner

Studienbücher 1991

5. E. Riedel:

Moderne Anorganische ChemieModerne Anorganische Chemie

de

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Fundamental Aspec

Fundamental Aspects of

ts of Solids & Sphere

Solids & Sphere Packing

Packing

1. Why Study Solids? 1. Why Study Solids?

2. Some crystallographic ideas 2. Some crystallographic ideas

•lattice (lattice types)

•motif (basis)

•crystal structure

•unit cell (counting atoms in

•unit cell (counting atoms in unit cells)

unit cells)

•fractional coordinates

•coordination number

3. Representation of structures 3. Representation of structures

•Perspective (Clinographic)

•Projection (Plan) diagrams

4. Close packing of spheres 4. Close packing of spheres

•hexagonal close packing (hcp)

•cubic close packing (ccp)

5. Structures of

5. Structures of metallic elementsmetallic elements

6. Interstitial sites in close-packed arrangements 6. Interstitial sites in close-packed arrangements

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Aims of the lecture

After studying this lecture you should be able to:

1. Define the Terms

• Lattice, motif, crystal structure, unit cell, coordination number

2. Interpret a Plan or Perspective structure diagram of a Unit Cell & determine

• Lattice type, number of atoms in unit cell, coordination number/geometry of atoms

3. For ccp, hcp and bcc structures

• Describe the Sphere Packing • Identify symmetry of unit cell

• Draw unit cell in plan or perspective

• State Coordination Number/Geometry of a sphere & the degree of space-filling

• Give some examples of metallic elements which adopt each structure • Draw on a plan or perspective view of the unit cell the positions of any

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What is special about the solid compared to molecules?

Covalent solids

Discrete molecules

C H H H H aromatic aliphatic conducting insulating, thermally conducting hard C₆₀ aromatic

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What is special about the solid compared to molecules?

Molecules such as HCl, C

₆

H

₆

or C

₆₀

are identical and

s

tochiometric

Solid state compounds may be

nonstochiometric

Example: „FeO“ is really Fe

_1-x

O

! Fe

_1-x

O

≠

FeO

_1+x

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What is special about the solid compared to molecules?

Solid solutions: c

omposition determines the properties

Examples:

Doped semiconductors n-Si/p-Si

LASERs Al

₂

O

₃

(Cr

3+

₎

Dielectrics Ba

_1-x

Ca

_x

TiO

₃

Pigments (TiO

_2-x

F

_x

, CdS/CdSe)

Steel (Fe/C)

„mailbox yellow“

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What is special about the solid compared to molecules?

Periodicity

leads to

Localization-delocalization

of charge carriers

(metallic conductivity, polarons ...)

Collective properties

such as magnetism (e.g. Fe,

Co, NiO, CoFe

₂

O

₄

)

Order-disorder effects (e.g. in intermetallic compounds)

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What is special about the solid compared to molecules?

Extended structures

⇒

no localized chemical reactivity

⇒

low mobility of constituents

large activation energies for reactions

⇒

high reaction temperatures/thermodynamic control

⇒

phase diagrams important for chemical reactions

⇒

experiments under extreme physical conditions:

very high pressures and temperatures

⇒

Description in terms of structure types

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What is special about the solid compared to molecules?

Frequently used classifications for solids

a) elements: new modifications (e.g. fullerenes), structure comparisons,

growth of high purity crystals ...

b) crystalline materials: synthesis and crystal growth, defects,

structure – chemical bonding – physical properties ...

c) noncrystalline/amorphous materials: special physical and chemical

properties ...

d) (quasi)molecular solids (e.g. phosphides): structural relations ... e) covalent solids (e.g. diamond, boron nitride):

extreme hardness, very high melting points ...

f) ionic solids (e.g. NaCl): structural chemistry, ionic conductivity ...

g) metals, semiconductors, isolators: close packings of spheres, alloys,

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Some Basic Definitions

LATTICE

An infinite array of points in space, in which each point has identical surroundings to all others.

CRYSTAL STRUCTURE

The periodic arrangement of atoms in the crystal.

It can be described by associating with each lattice point a group of

atoms called the MOTIF (BASIS)

UNIT CELL

The smallest component of the crystal, which when stacked together with pure translational repetition reproduces the whole crystal, Primitive (P)unit cells contain only a single lattice point

Don't mix up atoms with lattice points

Lattice points are infinitesimal points in space Atoms are physical objects

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Some Basic Definitions

The unit cell

“The smallest repeat unit of a crystal structure, in 3D,

which shows the full symmetry of the structure”

The unit cell is a box

with:

• 3 sides - a, b, c

• 3 angles -

α

,

β

,

γ

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Some Basic Definitions

Counting Lattice Points/Atoms in 2D Lattices

Unit cell is Primitive (1 lattice point) but contains TWO atoms in the motif Atoms at the corner of the 2D unit cell contribute only 1_/

4 to unit cell count

Atoms at the edge of the 2D unit cell contribute only 1_/

2 to unit cell count

Atoms within the 2D unit cell contribute 1 (i.e. uniquely) to that unit cell

Counting Atoms in 3D Cells

Atoms in different positions in a cell are shared by differing numbers of unit cells Vertex atom shared by 8 cells ⇒ 1_/

8 atom per cell Edge atom shared by 4 cells ⇒ 1_/

4 atom per cell Face atom shared by 2 cells ⇒ 1_/

2 atom per cell Body unique to 1 cell ⇒ 1 atom per cell

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Some Basic Definitions

1-dimensional lattice symmetries are found in many ornaments

2-dimensional lattice symmetries were famously exploited by the artist

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Close packing

1926 Goldschmidt proposed atoms could be considered as packing in solids as hard spheres

This reduces the problem of examining the packing of like atoms to that of examining the most efficient packing of any spherical object

- e.g. have you noticed how oranges are most effectively packed in displays at

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Principles of close packings of spheres

A single layer of spheres is closest-packed with

a HEXAGONAL coordination of each sphere

A second layer of spheres is placed in the indentations left by the first layer

•space is trapped between the layers that is not filled by the spheres

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Principles of close packings of spheres

When a third layer of spheres is placed in the

indentations of the second layer there are TWO choices 1. The third layer lies in indentations directly in line

(e c l i p s e d ) with the 1st layer

⇒ Layer ordering may be described as ABA

2. The third layer lies in the alternative indentations leaving it staggered with respect to both

previous layers

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Principles of close packings of spheres

(polytypism !!) A second layer of spheres is placed in the indentations left by the first layer

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Principles of close packings of spheres

hexagonal close packing (hcp)

cubic close packing (ccp or fcc)

Common unit cells for hcp and ccp

Be, Mg

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Principles of close packings of spheres

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Principles of close packings of spheres

Close-Packed Structures

The most efficient way to fill space with spheres

Is there another way of packing spheres that is more space-efficient? In 1611 Kepler asserted that there was no way of packing equivalent spheres at a greater density than that of a face-centred cubic

arrangement. This is now known as the Kepler Conjecture.

This assertion has long remained without rigorous proof. In 1998 Hales announced a computer-based solution. This proof is contained in over 250 manuscript pages and relies on over 3 gigabytes of computer files and so it will be some time before it has been checked rigorously by the scientific community to ensure that the Kepler Conjecture is indeed proven!

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Principles of close packings of spheres

Features of Close-Packing

1. Co ord in atio n Nu m ber = 12 2. 74% o f s p a c e is o c c u p i e d

Space filling R_fill = 4/3 π (Σ_iZ_ir _i3)/V_cell

Z_i = No. atoms/unit cell, V_cell = unit cell volume example:

4r = v2 aV_cell=a3=(4/v2 r)3=16 v2 r 3 a = 4/ v2 r V_atom = Z_i 4/3 pr 3

R_E=V_atom/V_cell=(16 pr 3)/ (3·16 v2 r 3)= p/3 v2 = 0.74

3. o c tah ed ral ( O ) ( r = 0.414 ) ~ 1 p e r s p h e r e tetrahedral ( T± ) (r = 0.225 ) ~ 2 p e r s p h e r e

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Principles of close packings of spheres

Density Calculations

In the fcc cell the atoms touch along the face diagonals, but not along the cell edge: length face diagonal = a(2)1/2 _{= 4r} _.

Al has a ccp arrangement of atoms. The radius of Al = 1.423Å. Calculate the lattice parameter of the unit cell and the density of solid Al.

Solution: fcc lattice ⇒ 4 atoms/cell [8 at corners (each 1/8), 6 in faces

(each 1/2)]

Lattice parameter: atoms in contact along face diagonal, therefore

4r _Al = a(2)1/2 _⇒ _{a = 4(1.432Å)/(2)}1/2 _{= 4.050Å.}

ρ_Al = M_cell/V_cell

M_cell/ 4 x m_Al = (26.98)(g/mol)(1mol/6.022x1023_{atoms)(4 atoms/unit cell) =}

1.792 x 10-22 _{g/unit cell}

V_cell = a3 _{= (4.05x10}-8_cm)3 _{= 66.43x10}-24 _cm3_{/unit cell}

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Principles of close packings of spheres

ABABAB.... repeat gives Hexagonal Close-Packing (HCP)

Unit cell showing the full symmetry of the arrangement is Hexagonal

2 atoms in the unit cell: (0, 0, 0) (2_/

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Principles of close packings of spheres

ABCABC.... repeat gives Cubic Close-Packing (CCP)

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Principles of close packings of spheres

68% of space is occupied C o o r d i n a t i o n N u m b e r ?

8 Nearest Neighbours at a =0.87a 6 Next-Nearest Neighbours at 1a

3 2 1

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Principles of close packings of spheres

3 2 1

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Principles of close packings of spheres

Polymorphism:

• Some metals exist in different structure types at ambient temperature & pressure

• Many metals adopt different structures at different temperature/pressure • Not all metals are close-packed

• Why different structures?

residual effects from some directional effects of atomic orbitals

• Difficult to predict structures

BCC clearly adopted for l o w n u m b e r of valence electrons

Best explanations are based on Band Theory of Metals (later)

In cases of polymorphism BCC is the structure adopted at h i g h e r temperatures

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Principles of close packings of spheres

More Complex close-packing sequences than simple HCP & CCP are

possible

• HCP & CCP are merely the simplest close-packed stacking sequences, others are possible!

All spheres in an HCP or CCP structure have identical environments

• Repeats of the formABCB.... are the next simplest

• There are two types of sphere environment

• surrounding layers are both of the same type

(i.e. anti-cuboctahedral coordination) like HCP, so labelled h • surrounding layers are different (i.e. cuboctahedral

coordination) like CCP, so labelled c

• Layer environment repeat is thus hchc...., so labelled hc

• U n i t c e l l is alternatively labelled 4 H

• has 4 layers in the c -direction

• hexagonal

•The h c (4 H) structure is adopted by early lanthanides

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Principles of close packings of spheres

• Non-Ideality of Structures

• Cobalt metal that has been cooled from T > 500°C has a close-packed

structure with a random stacking sequence

• "Normal" HCP cobalt is actually 90% AB... & 10% ABC...

- i.e. non-ideal HCP

•Many metals deviate from perfect HCP by „axial compression"

• e.g. for Beryllium (Be) c/_a = 1.57 (c.f. ideal c/_a = 1.63)

• Coordination is now [6 + 6] with slightly shorter distances to neighbours in adjacent layers

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Principles of close packings of spheres

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Principles of close packings of spheres

Location of interstitial holes in close-packed structures

The HOLES in close-packed arrangements may be filled with atoms of a

different sort.

It is therefore important to know:

• How holes are displaced in space relative to the positions of the spheres • How holes are displaced relative to each other

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