Document 7249489

Download Report

Transcript Document 7249489

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
• 1 atom per unit cell (how?).
• What is the coordination number? Volume
• 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
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
• 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 oppositelycharged ions occupy the holes.
• Two important factors in considering the
– 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
• Many alkali metals have
this same geometry.
• What is the coordination
number (nearest
CsCl Crystal Structure
• Chloride ions form
simple cubes with cesium
ions in the center (Figure
• The cesium ion is able to
fit in to center hole.
• Other crystal structures.
TiO2 (the rutile structure)
• Distorted TiO6
– 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
• 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
• Lattice enthalpy
MX(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
• Write down values for
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.
z A z B  e2 
U 
r0  4o 
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 Calculations
• A more accurate equation depicts the Coulombic
interactions over the entire crystal.
N Ae2  z A z B 
U 
4o  r0 
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 Calculations
• Repulsions between ions in close proximity term.
 r0
U  N A C ' e
C’ = constant (will cancel out when finding the minimum)
 = compressibility constant, ~ 30 pm
• Combining terms
U 
 zAzB 
A  N A C' e
 r0 
 r0
Lattice Enthalpy Calculations
• Finding the minimum
– dU/dr0 = O
N A z A z Be 2   
1  A
U 
4o r0  r0 
• A negative of this value
may be defined as the
lattice enthalpy.
Lattice Enthalpy Calculations
• As the polarizability of the resultant ions
increase the agreement with this ionic
model worsens.
– Polarizibility generally indicates more covalent
NaCl and CaBr2
Molecular Orbitals in Solids
• A very large number of atoms
are used to generate molecular
– One-dimensional model.
– Creation of bands that are closely
– Factors affecting the width of the
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
• 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
Density of States
• Concentration of
energy levels within a
d ( N(E))
• Helps to describe
bonding/reactivity in
Conductivity of Solids Versus
• Metals – decrease with
• Semiconductors – increase
with temperature.
• Insulators – increase with
temperature (if
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
• 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 ntype material and a positive potential to the ptype material.
– Discuss (Figure 7-16).
• No resistance to flow of electrons.
– Currents started in a loop will continue to flow
• 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
– 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
• Discovered in 1987
and has a Tc of 93 K.
– N2(l) can be used
• Type II
• Difficult to work
• Possesses copper
oxide planes and
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
• LiI versus NaCl (which structure would exhibit
more covalent character?)
Bonding in TiO2
• The crystal has a rutile
– Each titanium has ___ nearest
neighbors and each oxygen
atom has ___ nearest
• 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
– TiO and VO (positioning in the
• 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
– 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
• Why is TiO2 different
than TiO?
Bonding in TiO
• MnO, FeO, CoO, and NiO do not conduct,
but they have the same basic structure.
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
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
Individual units, chains, multiple chains
(ribbons), rings, sheets and 3-d networks.
Structure Containing Discrete
• Nesosilicates – no O atoms are shared.
– Contain individual SiO44- units.
– ZrSiO4 (zircon) – illustrate with softwares
• Stoichiometry dictates 8-fold coordination of the
– (Mg3 or Fe3)Al2Si3O12 (garnet) – illustrate with
• 8-fold coordination for Mg or Fe and 6-fold
coordination for the Al.
Structure Containing Discrete
• The sorosilicates (disilicates) – 1 O atom is
– 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
Cyclosilicates (discrete cyclic
• Each SiO4 units shares two O atoms with
neighboring SiO4 tetrahedra.
– Formula – SiO32- or [(SiO3)n]2n- (n=3-6 are the most
– Beryl – six-linked SiO4 tetrahedra (show with
• 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
• Corner sharing of SiO4
tetrahedra (SiO32-)
– Very common (usually to
build up more complicated
silicate structures).
• Differing conformations can
be adopted by linked
– Changes the repeat distance.
– The 2T structure is the most
common (long).
Silicates with Chain or Ribbon
• The chains are usually packed parallel to
provide sites of 6 and 8 coordination for the
– 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
(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 Si2O52-.
– This is exhibited with kaolinite (illustrate the
silicon tetrahedral layer).
Creation of Layers in the
• Can be formed by
sharing the fourth O
atom between pairs of
– Produces an SiO2
stoichiometry (neutral)
– Replacing Si with Al
• Al2Si2O82-; requires charge
balance. The cations
connect the double layers.
Creation of Layers in the
• 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.