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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.

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crystalline solid state
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
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
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
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 structures6
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
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
Metallic Crystals
  • Most crystalize in bcc, ccp, and hcp structures.
  • Hard sphere model does not work well.
    • Depends on electronic structure.
  • Properties
    • Conductivity
    • Dislocations
  • Each carbon atom is bonded tetrahedrally to four nearest neighbors (Figure).
    • Essentially the same strength in all directions.
structures of binary compounds
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
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
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.
tio 2 the rutile structure
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
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
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 - TS (standard state), 2nd term can be ignored

  • Lattice enthalpy

MX(s)  M+(g) + X-(g) HL (standard molar enthalpy change)

Currently, we are interested in lattice formation.

the born haber cycle
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 cycle17
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
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 centers

e = electronic charge

4o = permittivity of a vacuum

e2/ 4o = 2.307  10-28 J m

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

lattice enthalpy calculations19
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.

lattice enthalpy calculations20
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
lattice enthalpy calculations21
Lattice Enthalpy Calculations
  • Finding the minimum energy
    • dU/dr0 = O
  • A negative of this value may be defined as the lattice enthalpy.
lattice enthalpy calculations22
Lattice Enthalpy Calculations
  • As the polarizability of the resultant ions increase the agreement with this ionic model worsens.
    • Polarizibility generally indicates more covalent character.


NaCl and CaBr2

molecular orbitals in solids
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.

molecular orbitals in solids25
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 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
Density of States
  • Concentration of energy levels within a band.
  • Helps to describe bonding/reactivity in solids.
conductivity of solids versus temperature
Conductivity of Solids Versus Temperature
  • Metals – decrease with temperature.
  • Semiconductors – increase with temperature.
  • Insulators – increase with temperature (if measurable).
semiconductor types
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)
  • 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
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).
  • 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
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
yba 2 cu 3 o 7 high temperature superconductors
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
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 tio 2
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 tio 237
Bonding in TiO2

For a TiO6 monomer (no significant -bonding).

An approximation of the ‘bands in the solid structure.

bonding in tio 238
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
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 tio40
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 tio41
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 tio42
Bonding in TiO
  • MnO, FeO, CoO, and NiO do not conduct, but they have the same basic structure. Why?
imperfections in solids
Imperfections in Solids
  • All crystalline solids possess imperfections.
    • Crystal growth occurring at many sites causes boundaries to form.
    • Vacancies and self-interstitials
    • Substitutions
    • Dislocations
  • 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

structures with the sio 4 unit
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
Structure Containing Discrete Units
  • Nesosilicates – no O atoms are shared.
    • Contain individual SiO44- 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 units48
Structure Containing Discrete Units
  • The sorosilicates (disilicates) – 1 O atom is shared.
    • Contain Si2O76- units
    • Show Epidote (Ca2FeAl2(SiO4)(Si2O7)O(OH)) with softwares.
      • Epidote contains SiO44- and Si2O76- units
    • Near linear Si-O-Si bond angle between tetrahedra.
cyclosilicates discrete cyclic units
Cyclosilicates (discrete cyclic units)
  • Each SiO4 units shares two O atoms with neighboring SiO4 tetrahedra.
    • Formula – SiO32- or [(SiO3)n]2n- (n=3-6 are the most common.
    • Beryl – six-linked SiO4 tetrahedra (show with softwares).
      • Be3Al2(SiO3)6 – contains Si6O1812- 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
Silicates with Chain or Ribbon Structures
  • Corner sharing of SiO4 tetrahedra (SiO32-)
    • 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 structures51
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
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
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 Si2O52-.
    • This is exhibited with kaolinite (illustrate the silicon tetrahedral layer).
creation of layers in the phyllosilicates
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
      • Al2Si2O82-; requires charge balance. The cations connect the double layers.
creation of layers in the phyllosilicates55
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
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
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
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.