solid-state chemistry.pdf

Tremel: Solid State Chemistry

Introduction

Introduction toSolid 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)


Introduction

Textbooks: Introduction to Solid State Chemistry (English)

1. Doughlas, McDaniel, Alexander, Concepts and Models of Inorganic Chemistry, Wiley, 1992

2. Shriver, Atkins, Inorganic Chemistry (3rd ed, 1999)W.H. Freeman and Company (Chs. 3, 18 ...)

3. L. Smart, E. Moore, Solid State Chemistry, 2nd Ed.Chapman & Hall, London, 1995

4. P.A. Cox, The Electronic Structure and Chemistry of Solids, Oxford University Press, 1987

5. U. Müller, Inorganic Structural ChemistryWiley, Chichester, 1993

6. A.R. West, Solid State Chemistry and its ApplicationsWiley, New York, 1984


Introduction

Textbooks II: Solid State Chemistry (German)

1. E. Riedel: Moderne Anorganische Chemie, de Gruyter 1999, S. 329 – 523 („Festkörperchemie“)

2. L. Smart, E. Moore: Einführung in die Festkörperchemie, Vieweg 1997

3. A.R. West: Grundlagen der Festkörperchemie, VCH 1992

4. U. Müller: Anorganische Strukturchemie Teubner Studienbücher 1991

5. E. Riedel: Moderne Anorganische Chemiede Gruyter 1999, S. 329 – 523 („Festkörperchemie“)


Contents

Fundamental Aspects of Solids & Sphere Packing1. Why Study Solids?2. Some crystallographic ideas

•lattice (lattice types) •motif (basis) •crystal structure •unit cell (counting atoms in unit cells) •fractional coordinates •coordination number

3. Representation of structures•Perspective (Clinographic) •Projection (Plan) diagrams

4. Close packing of spheres•hexagonal close packing (hcp) •cubic close packing (ccp)

5. Structures of metallic elements6. Interstitial sites in close-packed arrangements


Aims of the lecture

After studying this lecture you should be able to:1. Define the Terms

• Lattice, motif, crystal structure, unit cell, coordination number2. 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

octahedral &/or tetrahedral interstitial holes


What is special about the solid compared to molecules?

Covalent solidsDiscrete molecules

C

H

H

H

H

aromatic aliphatic

conducting

insulating, thermally conductinghard

C60 aromatic



Molecules such as HCl, C6H6or C60 are identical and stochiometric

Solid state compounds may be nonstochiometric

Example: „FeO“ is really Fe1-xO

! Fe1-xO ≠ FeO1+x

Other examples: NaxWO3, WO3-x



Solid solutions: composition determines the properties

Examples:

Doped semiconductors n-Si/p-SiLASERs Al2O3(Cr3+)Dielectrics Ba1-xCaxTiO3Pigments (TiO2-xFx, CdS/CdSe)Steel (Fe/C)

„mailbox yellow“

„blue NaCl“



Periodicity leads to

Localization-delocalization of charge carriers (metallic conductivity, polarons ...)

Collective properties such as magnetism (e.g. Fe, Co, NiO, CoFe2O4)

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

FeAl or CuZn



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

Packing/coordination polyhedra vs. molecular units



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, mechanical and electrical properties, chemical bonding ...


Some Basic Definitions

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

CRYSTAL STRUCTUREThe 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 CELLThe 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 Lattice Points do not necessarily lie at the centre of atoms



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



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 CellsAtoms 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



1-dimensional lattice symmetries are found in many ornaments

2-dimensional lattice symmetries were famously exploited by the artist M.C. Escher in many patterns


Close packing

1926 Goldschmidt proposed atoms could be considered as packing in solids as hard spheresThis 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 your local shop?


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•TWO different types of HOLES (INTERSTITIAL sites) are left

•OCTAHEDRAL (O) holes with 6 nearest sphere neighbours •TETRAHEDRAL (T±) holes with 4 nearest sphere neighbours



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

(eclipsed) 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 ⇒ Layer ordering may be described as ABC



h exagonal close pack ing (h cp) and cubic close pack ing(ccp)

(polytypism !!)

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



hexagonal close packing(hcp)

cubic close packing (ccp or fcc)

Common unit cells for hcp and ccp

Be, Mg

Ca, Sr, Cu, Ag, Au



Holes in sphere packings

Calculation of the size of holes



Close-Packed StructuresThe most efficient way to fill space with spheresIs 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!(Simon Singh, Daily Telegraph, Aug. 13th 1998)



Features of Close-Packing1. Coordination Number = 122. 74% of space is occupied

Space filling Rfill = 4/3 π(ΣiZiri3)/Vcell

Zi = No. atoms/unit cell, Vcell = unit cell volume

example:

4r = v2 aVcell=a3=(4/v2 r)3=16 v2 r3

a = 4/ v2 r Vatom = Zi 4/3 pr3

RE=Vatom/Vcell=(16 pr3)/ (3·16 v2 r3)= p/3 v2 = 0.743. octahedral (O) ( r = 0.414) ~ 1 per sphere

tetrahedral (T±) (r = 0.225) ~ 2 per sphere



Density CalculationsIn the fcc cell the atoms touch along the face diagonals, but not alongthe 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 4rAl = a(2)1/2 ⇒ a = 4(1.432Å)/(2)1/2 = 4.050Å. ρAl = Mcell/Vcell

Mcell/ 4 x mAl = (26.98)(g/mol)(1mol/6.022x1023atoms)(4 atoms/unit cell) = 1.792 x 10-22 g/unit cell Vcell = a3 = (4.05x10-8cm)3 = 66.43x10-24 cm3/unit cell ρAl = {1.792x10-22g/unit cell}/{66.43x10-24 cm3/unit cell} = 2.698 g/cm3



ABABAB.... repeat gives Hexagonal Close-Packing (HCP) Unit cell showing the full symmetry of the arrangement is Hexagonal2 atoms in the unit cell: (0, 0, 0) (2/3, 1/3, 1/2)



ABCABC.... repeat gives Cubic Close-Packing (CCP) Unit cell showing the full symmetry of the arrangement is Face-Centred Cubic4 atoms in the unit cell: (0, 0, 0) (0, 1/2, 1/2) (1/2, 0, 1/2) (1/2, 1/2, 0)



68% of space is occupiedCoordination Number ?

8 Nearest Neighbours at a =0.87a6 Next-Nearest Neighbours at 1a

321



321



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 low number of valence electrons Best explanations are based on Band Theory of Metals (later)In cases of polymorphism BCC is the structure adopted at higher temperatures



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 form ABCB.... 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•Unit cell is alternatively labelled 4 H

•has 4 layers in the c-direction •hexagonal

•The hc (4 H) structure is adopted by early lanthanides•Samarium (Sm) has a 9-layer chh repeat sequence



•Non-Ideality of Structures •Cobalt metal that has been cooled from T > 500°C has a close-packedstructure 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



Other systems may be classified as having similar structures



Location of interstitial holes in close-packed structuresThe 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

solid-state chemistry.pdf

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