The Crystalline Solid State

Chapter 7

Crystalline Solid State

• Many more “molecules” in the solid state.– We will focus on crystalline solids composed of

atoms or ions.

• Unit cell – structural component that, when repeated in all directions, results in a macroscopic (observable) crystal.– 14 possible crystal structures (Bravais lattices)– Discuss positions of atoms in the unit cell.

The Cubic Unit Cell (or Primitive)

• 1 atom per unit cell (how?).

• What is the coordination number? Volume occupied?

• Let’s calculate the length of the edge. What size of sphere would fit into the hole?

The Body-Centered Cubic

• How many atoms per unit cell?

• What is the length of the edge? This is a more complicated systems than the simple cubic.

Close-Packed Structures

• How many atoms is each atom surrounded by in the same plane?

• What is the coordination number?• Hexagonal close packing (hcp) – discuss the third

layer (ABA).• Cubic close packing (ccp) or face-centered cubic

(fcc) – discuss the third layer (ABC).• Two tetrahedral holes and one octahedral hole per

atom. Can you see them?

Close-Packed Structures

• The hcp has hexagonal prisms sharing vertical faces (Figure).– How many atoms per unit cell in the hcp structure?– What is the length of the cell edge?

• The unit cell for the ccp or fcc is harder to see.– Need four close-packed layers to complete the cube.– What is the length of the cell edge?

• In both close-packed structures, 74.1% of the total volume is occupied.

Ionic Crystals

• The tetrahedral and octahedral holes can have varying occupancies.

• Holes are generally filled by smaller ions.– Tetrahedral holes– Octahedral holes

• NaCl structure

Metallic Crystals

• Most crystalize in bcc, ccp, and hcp structures.

• Hard sphere model does not work well.– Depends on electronic structure.

• Properties– Conductivity– Dislocations

Diamond

• Each carbon atom is bonded tetrahedrally to four nearest neighbors (Figure).– Essentially the same strength in all directions.

Structures of Binary Compounds

• Close-packed structures are generally defined by the larger ions (usually anions). The oppositely-charged ions occupy the holes.

• Two important factors in considering the structure– Radius ratio (r+/r-)– Relative number of combining cations and anions.

NaCl Crystal Structure

• Face-centered cubes of both ions offset by a half a unit cell in one direction.

• Many alkali metals have this same geometry.

• What is the coordination number (nearest neighbor)?

CsCl Crystal Structure

• Chloride ions form simple cubes with cesium ions in the center (Figure 7-7).

• The cesium ion is able to fit in to center hole. How?

• Other crystal structures.

TiO2 (the rutile structure)

• Distorted TiO6 octahedra.– Ti has a C.N. of 6,

octahedral coordination

– O has a C.N. of 3

Rationalization of Structure of Crystalline Solids

• Predicting coordination number from radius ratio (r+/r-).– A hard sphere treatment of the ions.– Treats bonding as purely ionic.– Simply, as as the M+ ratio increases, more

anions can pack around it.• Table 7-1.

Let’s look at a few (NaCl, CaF2, and CaCl2).

Thermodynamics of Ionic Crystal Formation

• A compound tends to adopt the crystal structure corresponding to lowest Gibbs energy.M+(g) + X-(g) MX(s)G = H - TS (standard state), 2nd term can be

ignored

• Lattice enthalpyMX(s) M+(g) + X-(g) HL (standard molar

enthalpy change)Currently, we are interested in lattice formation.

The Born-Haber Cycle

• A special thermodynamic cycle that includes lattice formation as one step.

• The cycle has to sum up to zero if written appropriately.

• Write down values for KCl.

The Born-Haber Cycle

• Calculate the lattice enthalpy for MgBr2.

• A discrepancy between this value and the real value may indicate the degree of covalent character.– We have assumed Coulombic interactions

between ions.

– The actual values for KCl and MgBr2 are 701 and 2406 kJ/mol (versus 720 and 2451).

Lattice Enthalpy Calculations

• Considering only Coulombic contributions– The electrostatic potential energy between each pair.

zA, zB = ionic charges in electron units

r0 = distance between ion centerse = electronic charge

4o = permittivity of a vacuum

e2/ 4o = 2.307 10-28 J m

Calculation would be performed on each cation/anion pair (nearest neighbor).

o

2

0

BA

4

e

r

zzU

Lattice Enthalpy Calculations

• A more accurate equation depicts the Coulombic interactions over the entire crystal.

NA = Avogadro’s constant

A = Madelung’s constant, value specific to a crystal type (in table). This is a sum of all the geometric factors carried out until the interaction become infinitesimal.

0

BA

o

2A

r

zz

4

eNU

Lattice Enthalpy Calculations

• Repulsions between ions in close proximity term.

C’ = constant (will cancel out when finding the minimum) = compressibility constant, ~ 30 pm

• Combining terms

0r

'A eCNU

0r

A0

BA

o

2A e'CNA

r

zz

4

eNU

Lattice Enthalpy Calculations

• Finding the minimum energy– dU/dr0 = O

• A negative of this value may be defined as the lattice enthalpy.

Ar

1r4

ezzNU

00o

2BAA

Lattice Enthalpy Calculations

• As the polarizability of the resultant ions increase the agreement with this ionic model worsens.– Polarizibility generally indicates more covalent

character.

Calculations

NaCl and CaBr2

Molecular Orbitals in Solids

• A very large number of atoms are used to generate molecular orbitals.– One-dimensional model.

– Creation of bands that are closely spaced.

– Factors affecting the width of the band.

This would be called an ‘s band’. A similar model can be constructed for the p-orbitals and d-orbitals.

The Bonding Picture in Solids

Molecular Orbitals in Solids

• Band gap – separation between bands in which no MOs exist (Figure 7-13).

• Valence band – highest energy band containing electrons.

• Conduction band – the band immediately above the valence band in energy.

Metals and Insulators

• Metals– Partially filled valence band (e.g. s band)

• Electrons move to slightly higher energy levels by applying a small voltage. Electrons and ‘holes’ are both free to move in the metal.

– Overlapping bands (e.g. s and p bands)• If the bands are close enough in energy (or

overlapping) an applied voltage can cause the electrons to jump into the next band (conduction band).

Density of States

• Concentration of energy levels within a band.

• Helps to describe bonding/reactivity in solids.

dE

))E(N(d

Conductivity of Solids Versus Temperature

• Metals – decrease with temperature.

• Semiconductors – increase with temperature.

• Insulators – increase with temperature (if measurable).

Semiconductor Types

• Intrinsic semiconductors – pure material having semiconductive properties.

• Doped semiconductors – semiconductors that are fabricated by adding a small amount of another element with energy levels close to the pure state material.– n-type semiconductors– p-type semiconductors (look at

figure)

Semiconductors

• Fermi-level (semiconductor) – the energy at which an electron is equally likely to be in each of two levels (Figure).

• Effects of dopants on the Fermi level.– n-type and p-type.

Diodes (creating p-n junctions)

• Migration of electrons from the n-type material to the p-type material.– Equilibrium is established due to charge transfer.

• Application of a negative potential to the n-type material and a positive potential to the p-type material.– Discuss (Figure 7-16).

Superconductivity

• No resistance to flow of electrons.– Currents started in a loop will continue to flow indefinitely.

• Type I superconductors – expel all magnetic fields below a critical temperature, Tc (Meisner effect).

• Type II superconductors – below a critical temperature exclude all magnetic fields completely. Between this temperature and a second critical temperature, they allow partial penetration by the magnetic field.– Levitation experiment works well.

Theory of Superconducting

• Cooper pair theory– Bardeen, Cooper, and

Schrieffer

– Electrons travel through the material in pairs.

– The formation and propagation of these pairs is assisted by small vibrations in the lattice.

• discuss

YBa2Cu3O7 High-Temperature Superconductors

• Discovered in 1987 and has a Tc of 93 K.– N2(l) can be used

• Type II superconductor.

• Difficult to work with.

• Possesses copper oxide planes and chains.

Bonding in Solid State Structures

• The hard-sphere model is too simplistic.– Deviations are observed in ion sizes.– Sharing of electrons (or transfer back to the

cation) can vary depending upon the polarizability.

• LiI versus NaCl (which structure would exhibit more covalent character?)

Bonding in TiO2

• The crystal has a rutile structure.– Each titanium has ___ nearest

neighbors and each oxygen atom has ___ nearest neighbors.

• There is no effective O···O or Ti···Ti interactions (only Ti···O interactions). Why?

• The structure consists of TiO6 fragments (discuss).

Bonding in TiO2

For a TiO6 monomer (no significant -bonding).

An approximation of the ‘bands in the solid structure.

Bonding in TiO2

• The calculated DOS curve in 3-d space is slightly more complicated.

• The O 2s, O2p, Ti t2g, and eg bands are well separate. The separation predicts that this material has ‘insulator-like’ properties.

Bonding in TiO

• Several of the 3d monoxides illustrate high conductivity that decreases with temperature.– TiO and VO (positioning in the

table).

• TiO adopts the rocksalt structure (NaCl).– Discuss geometry and

consequences on bonding.

Bonding in TiO

• The titanium atoms are close enough to form a ‘conduction’ band.– Overlap of t2g orbitals of

the metal ions in neighboring octahedral sites.

– Illustrated for dxy orbitals.

Bonding in TiO

• The calculated DOS curve for TiO reveals that the bonds aren’t well separated.– Diffuse bands indicate

more conductive behavior.

• Why is TiO2 different than TiO?

Bonding in TiO

• MnO, FeO, CoO, and NiO do not conduct, but they have the same basic structure. Why?

Imperfections in Solids

• All crystalline solids possess imperfections.– Crystal growth occurring at many sites causes

boundaries to form.– Vacancies and self-interstitials– Substitutions– Dislocations

Silicates• The earth’s crustal rocks (clays, soils, and sands)

are composed almost entirely (~95%) of silicate minerals and silica (O, Si, and Al).– There exist many structural types with widely

varying stoichiometries (replacement of Si by Al is common). Consequences?

• Common to all:– SiO4 tetrahedra units

• Si is coordinated tetrahedrally to 4 oxygens

http://www.soils.wisc.edu/virtual_museum/displays.html

http://mineral.galleries.com/minerals/silicate/class.htm

The Tetrahedral SiO4 Unit

Cheetham and Day

Structures with the SiO4 Unit

• Discrete structural units which commonly contain cations for charge balance.

• Corner sharing of O atoms into larger units.– O lattice is usually close-packed (near)– Charge balance is obtained by presence of

cations.

Individual units, chains, multiple chains (ribbons), rings, sheets and 3-d networks.

Structure Containing Discrete Units

• Nesosilicates – no O atoms are shared.– Contain individual SiO4

4- units.– ZrSiO4 (zircon) – illustrate with softwares

• Stoichiometry dictates 8-fold coordination of the cation.

– (Mg3 or Fe3)Al2Si3O12 (garnet) – illustrate with softwares

• 8-fold coordination for Mg or Fe and 6-fold coordination for the Al.

Structure Containing Discrete Units

• The sorosilicates (disilicates) – 1 O atom is shared.– Contain Si2O7

6- units

– Show Epidote (Ca2FeAl2(SiO4)(Si2O7)O(OH)) with softwares.

• Epidote contains SiO44- and Si2O7

6- units

– Near linear Si-O-Si bond angle between tetrahedra.

Cyclosilicates (discrete cyclic units)

• Each SiO4 units shares two O atoms with neighboring SiO4 tetrahedra.– Formula – SiO3

2- or [(SiO3)n]2n- (n=3-6 are the most common.

– Beryl – six-linked SiO4 tetrahedra (show with softwares). • Be3Al2(SiO3)6 – contains Si6O18

12- cyclic units• The impurities produce its colors.

– Wadeite – three-linked SiO4 tetrahedra (don’t have an actual picture)

• K2ZrSi3O9

Silicates with Chain or Ribbon Structures

• Corner sharing of SiO4 tetrahedra (SiO3

2-)

– Very common (usually to build up more complicated silicate structures).

• Differing conformations can be adopted by linked tetrahedra.– Changes the repeat distance.

– The 2T structure is the most common (long).

Silicates with Chain or Ribbon Structures

• The chains are usually packed parallel to provide sites of 6 and 8 coordination for the cations.– Jadeite [NaAlSi2O6]

• Illustrate the different repeat units.

• What is the repeat unit?

Silicate Chains Linking Together

• Can form double or triple chains/ribbons linked together (or more).

• Depends on the repeat unit in the chain.

• Tremolite [Ca2Mg5(Si4O11)2(OH)2

(illustrate with softwares)

Asbestos mineral (fibrous)

• Triple chain

Phyllosilicates (Silicates with Layer Structures)

• Clay minerals, micas, talc, soapstone.• Individual layers are formed by sharing 3 of

the 4 atoms of each tetrahedron.• Simplest structure is made up of a 2T

network of silicate chains to give a network composition of Si2O5

2-.– This is exhibited with kaolinite (illustrate the

silicon tetrahedral layer).

Creation of Layers in the Phyllosilicates

• Can be formed by sharing the fourth O atom between pairs of tetrahedra.– Produces an SiO2

stoichiometry (neutral)

– Replacing Si with Al• Al2Si2O8

2-; requires charge balance. The cations connect the double layers.

Creation of Layers in the Phyllosilicates

• Double layers can be produced by interleaving layers of the gibbsite Al(OH)3 or brucite Mg(OH)2 structure.– Incorporation of gibbsite produces kaolinite,

[Al2(OH)4Si2O5] (China clay); illustrate with software the different layers present.

– Placing a SiO layer on the other side of the AlO layer produces pyrophyllite, [Al2(OH)2Si4O10].

• Illustrate both with software.

More Layered Structures

• The Al can be replaced by Mg (2:3) ratio.– Kaolinite serpentine asbestos

– Pyrophyllite talc

• Charged layers can also result by replacing the framework Si with Al or other cations. For charge balance these layers can be interleaved with M(+1) or M(+2) to give micas (illite) or by layers of hydrated cations to give montmorillonite.– Illustrate both.

The Tectosilicates

• Each oxygen atom is shared by 2 tetrahedra (SiO2 formula).

• Silica (-quartz; one crystalline form)– Si-O-Si bond angles are ~144 degrees.

– Contains helical chains of SiO4.• Six combine to form hexagonal shape (illustrate).

The Tectosilicates (Zeolites - aluminosilicates)

• A large fraction of the Si atoms are replaced with Al (other metals can also be used). – Charge balance will be required (Si,Al)nO2n.

• Contain cavities that allow molecules to enter.– Able to tailor electronic and physical properties.

• Pore structure and cation exchange.

• Illustrate with software.