P1247676909HJwlk

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Stacked 2D hexagonal arrays
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Packing efficiency

• It can be easily shown that all close-packed
  arrays have a packing efficiency (Vocc/Vtot) of
  0.74
• This is the highest possible value for same-sized
  spheres, though this is hard to prove

And supposethat there were one form, which we
will call ice-nine - a crystal as hard as this
desk - with a melting point of, let us say,
one-hundred degrees Fahrenheit, or, better still,
one-hundred-and-thirty degrees. Kurt Vonnegut,
Jr. Cats Cradle
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Close-packing of polymer microspheres
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hcp vs ccp
Also close-packed (ABAC)n (ABCB)n Not close
packed (AAB)n (ABA)n Why not ? (ACB)n
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Unit cells for hcp and fcc
Unit cells, replicated and translated, will
generate the full lattice
Hexagonal cell hcp
Cubic cell ccp fcc
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Generating lattices
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Oh and Td sites in ccp
fcc lattice showing some Oh and Td sites
4 spheres / cell 4 Oh sites / cell 8 Td sites /
cell
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Ionic radii are related to coordination number
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Element Structures at STP
(ABCB)n
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Ti phase transitions

• RT ? 882C hcp
• 882 ? 1667 bcc
• 1667 ? 3285 liquid
• 3285 ? gas

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Classes of Alloys

• Substitutional
• Interstitial
• intermetallic

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Some alloys

• Alloy Composition
• Cu, Ni any
• Cu and Ni are ccp, r(Cu) 1.28, r(Ni) 1.25 Å
• Cast iron Fe, C (2 ), Mn, Si
• r(Fe) 1.26, r(C) 0.77
• Stainless Steels Fe, Cr, Ni, C
• Brass CuZn (b) bcc
• r(Zn) 1.37, hcp

substitutional
interstitial
intermetallic
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A few stainless steels
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Zintl phases
KGe
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NaCl (rocksalt)

• fcc anion array with all Oh sites filled by
  cations
• the stoichiometry is 11 (AB compound)
• CN 6,6
• Look down the body diagonal to see 2D hex arrays
  in the sequence (AcBaCb)n
• The sequence shows coordination, for example the
  c layer in AcB Oh coordination

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CaC2
Tetragonal distortion of rocksalt structure (a
b ? c) Complex anion also decreases (lowers)
symmetry
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Other fcc anion arrays
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Antifluorite / Fluorite

• Antifluorite is an fcc anion array with cations
  filling all Td sites
• 8 Td sites / unit cell and 4 spheres, so this
  must be an A2B-type salt.
• Stacking sequence is (AabBbcCca)n
• CN 4,8. Anion coordination is cubic.
• Fluorite structure reverses cation and anion
  positions. An example is the mineral fluorite CaF2

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Sphalerite (ZnS)

• fcc anion array with cations filling ½ Td sites
• Td sites are filled as shown
• Look down body diagonal of the cube to see the
  sequence (AaBbCc)n
• If all atoms were C, this is diamond structure.

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Sphalerite
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Semiconductor lattices based on diamond /
sphalerite

• Group 14 C, Si, Ge, a-Sn, SiC
• 3-5 structures cubic-BN, AlN, AlP, GaAs, InP,
  InAs, InSb, GaP,
• 2-6 structures BeS, ZnS, ZnSe, CdS, CdSe, HgS
• 1-7 structures CuCl, AgI

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Structure Maps
more ionic
incr. radius, polarizability
more covalent
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Lattices with hcp anion arrays
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NiAs

• hcp anion array with cations filling all Oh sites
• cation layers all eclipsing one another
• stacking sequence is (AcBc)n
• CN 6,6
• AcB and BcA gives Oh cation coordination, but cBc
  and cAc gives trigonal prismatic (D3h) anion
  coordination

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CdI2

• hcp anion array with cations filling ½ Oh sites
  in alternating layers
• Similar to NiAs, but leave out every other cation
  layer
• stacking sequence is (AcB)n
• CN (6, 3)
• anisotropic structure, strong bonding within AcB
  layers, weak bonding between layers
• the layers are made from edge-sharing CdI6
  octahedra

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LiTiS2
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LDH structures
Mg(OH)2 (brucite) MgxAl1-x(OH)2.An
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Rutile (TiO2)

• hcp anion array with cations filling ½ Oh sites
  in alternating rows
• the filled cation rows are staggered
• CN 6, 3
• the filled rows form chains of edge-sharing
  octahedra. These chains are not connected within
  one layer, but are connected by the row of
  octahedra in the layers above and below.
• Lattice symmetry is tetragonal due to the
  arrangement of cations.

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Rutile
TiO2-x and SiO2
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Wurtzite (ZnS)

• hcp anion array with cations filling ½ Td sites
• Stacking sequence (AaBb)n
• CN 4, 4
• wurtzite and sphalerite are closely related
  structures, except that the basic arrays are hcp
  and ccp, respectively.
• Many compounds can be formed in either structure
  type ZnS, has two common allotropes, sphalerite
  and wurtzite

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ReO3

• Re is Oh, each O is shared between 2 Re, so there
  are ½ 6 3 O per Re, overall stoichiometry is
  thus ReO3
• Neither ion forms a close-packed array. The
  oxygens fill 3/4 of the positions for fcc
  (compare with NaCl structure).
• The structure has ReO6 octahedra sharing all
  vertices.

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Perovskite (CaTiO3)

• Similar to ReO3, with a cation (CN 12) at the
  unit cell center.
• Simple perovskites have an ABX3 stoichiometry. A
  cations and X anions, combined, form a
  close-packed array, with B cations filling 1/4 of
  the Oh sites.

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Superconducting copper oxides

• Many superconducting copper oxides have
  structures based on the perovskite lattice. An
  example is
• YBa2Cu3O7. In this structure, the perovskite
  lattice has ordered layers of Y and Ba cations.
  The idealized stoichiometry has 9 oxygens, the
  anion vacancies are located mainly in the Y
  plane, leading to a tetragonal distortion and
  anisotropic (layered) character.

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Charged spheres
Assumes a uniform charge distribution
(unpolarizable ions). With softer ions, higher
order terms (d-2, d-3, ...) can be included.

• For 2 spherical ions in contact, the
  electrostatic interaction energy is
• Eel (e2 / 4 p e0) (ZA ZB / d)
• e e- charge 1.602 x 10-19 C
• e0 vac. permittivity 8.854 x 10-12 C2J-1m-1
• ZA charge on ion A
• ZB charge on ion B
• d separation of ion centers

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Infinite linear chains

• Consider an infinite linear chain of alternating
  cations and anions with charges e or e
• The electrostatic terms are
• Eel (e2/4pe0)(ZAZB/ d) 2(1) - 2(1/2) 2(1/3)
  - 2(1/4)
• (e2/4pe0)(ZAZB/d) (2 ln2)

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Madelung constants

• Generalizing the equation for 3D ionic solids,
  we have
• Eel (e2 / 4 p e0) (ZA ZB / d) A
• where A is called the Madelung constant and is
  determined by the lattice geometry

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Madelung constants
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Born-Meyer model

• Electrostatic forces are net attractive, so d ? 0
  (the lattice collapse to a point) without a
  repulsive term
• Add a pseudo hard-shell repulsion C e-d/d
• where C' and d are scaling factors (d has been
  empirically
• fit as 0.345 Å)
• Vrep mimics a step function for hard sphere
  compression (0 where d gt hard sphere radius,
  very large where d lt radius)

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Born-Meyer eqn

• The total interaction energy, E
• E Eel Erepulsive
• (e2 / 4pe0)(NAZAZB /d) NC'e-d/d
• Since E has a single minimum d, set dE/dd 0
  and solve for C

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Further refinements

• Eel include higher order terms
• Evdw NCr-6 instantaneous polarization
• EZPE Nhno lattice vibrations
• For NaCl
• Etotal Eel Erep Evdw EZPE
• -859 99 - 12 7 kJ/mol

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Kapustinskii approximation

• The ratio A/n is approximately constant, where n
  is the number of ions per formula unit (n is 2
  for an AB - type salt, 3 for an AB2 or A2B - type
  salt, ...)
• Substitute the average value into the B-M eqn,
  combine constants, to get the Kapustinskii
  equation
• DHL -1210 kJÅ/mol (nZAZB / d0) (1 - d/d0)
• with d0 in Å

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Kapustinskii eqn

• Using the average A / n value decreases the
  accuracy of calculated Es. Use only when lattice
  structure is unknown.
• DHL (ZA,ZB,n,d0). The first 3 of these parameters
  are given from in the formula unit, the only
  other required info is d0.
• d0 can be estimated for unknown structures by
  summing tabulated cation and anion radii. The
  ionic radii depend on both charge and CN.

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Example

• Use the Kapustiskii eqn to estimate DHL for MgCl2
• ZA 2, ZB -1, n 3
• r(Mg2) CN 8 1.03 Å
• r(Cl-) CN 6 1.67 Å
• d0 r r- 2.7 Å
• DHL(Kap calc) 2350 kJ/mol
• DHL(best calc) 2326
• DHL(B-H value) 2526

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Unit cell volume relation

• Note that d/d0 is a small term for most salts,
  so (1 - d/d0) 1,
• Then for a series of salts with the same ionic
  charges and formula units
• DHL 1 / d0
• For cubic structures
• DHL 1 / V1/3
• where V is the unit cell volume

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DHLvs V-1/3 for cubic lattices

• V1/3 is proportional to lattice E for cubic
  structures. V is easily obtained by powder
  diffraction.

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Born Haber cycle
½ D0
DHf KCl(s) DH K(s) ½Cl2(g) ? KCl(s)
Ea
DHf KCl(s) DHsub(K) I(K) ½
D0(Cl2) Ea(Cl) - DHL

I
-DHL
DHsub
All enthalpies are measurable except DHL Solve
to get DHL(B-H)
-DHf
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Is MgCl3 stable ?
DHf DHat,Mg 3/2 D0(Cl2) I(1)Mg I(2)Mg
I(3)Mg - 3 Ea(Cl) - DHL 151
3/2 (240) 737 1451 7733 - 3
(350) - 5200 4000 kJ/mol

• DHL is from the Kapustinskii eqn, using d0 from
  MgCl2
• The large positive DHf means it is not stable.
• I(3) is very large, there are no known stable
  compounds containing Mg3. Energies required to
  remove core electrons are not compensated by
  other energy terms.

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Entropic contributions

• DG DH - TDS
• Example Mg(s) Cl2(g) ? MgCl2(s)
• DS sign is usually obvious from phase changes. DS
  is negative (unfavorable) here due to conversion
  of gaseous reactant into solid product.
• Using tabulated values for molar entropies
• DS0rxn DS0(MgCl2(s)) - DS0(Mg(s)) - DS0(Cl2(g))
• 89.6 - 32.7
  - 223.0
• -166 J/Kmol
• -TDS at 300 K 50 at 600 K 100 kJ/mol
• Compare with DHf MgCl2(s) -640 kJ/mol
• DS term is usually a corrective term at moderate
  temperatures. At high T it can dominate.

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Thermochemical Radii

• What are the radii of polyatomic ions ?
• (Ex CO32-, SO42-, PF6-, B(C6H6)-, N(Et)4)
• If DHL is known from B-H cycle, use B-M or Kap
  eqn to determine d0.
• If one ion is not complex, the complex ion
  radius can be calculated from
• d0 rcation ranion
• Tabulated thermochemical radii are averages from
  several salts containing the complex ion.
• This method can be especially useful when for
  ions with unknown structure, or low symmetry.

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Thermochemical Radii

• Example
• DHL(BH) for Cs2SO4 is 1658 kJ/mol
• Use the Kap eqn
• DHL 1658 1210(6/d0)(1-0.345/d0)
• solve for d0 4.00 Å
• Look up r (Cs) 1.67 År (SO42-) 4.00 - 1.67
  2.33 Å
• The tabulated value is 2.30 Å (an
• avg for several salts)

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Predictive applications

• O2 (g) PtF6 (l) ? O2PtF6 (s)
• Neil Bartlett (1960) side-reaction in preparing
  PtF6
• Ea(PtF6) 787 kJ/mol. Compare Ea(F) 328
• I(Xe) I(O2), so XePtF6-(s) may be stable if
  DHL is similar. Bartlett reported the first noble
  gas compound in 1962.

O2(g) ? O2(g) e- 1164 kJ/mol e-
PtF6(g) ? PtF6-(g) - 787 O2(g)
PtF6-(g) ? O2PtF6(s) - 470 O2(g) PtF6(g) ?
O2PtF6(s) - 93
Estimated from the Kap eqn
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Some consequences of DHL

• Ion exchange / displacement
• Thermal / redox stabilities
• Solubilities

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Exchange / Displacement

• Large ion salt small ion salt is better than
  two salts with large and small ions combined.
• Example Salt DHL sum
• CsF 750
• NaI 705 1455 kJ/mol
• CsI 620
• NaF 926 1546
• This can help predict some reactions like
  displacements, ion exchange, thermal stability.

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Thermal stability of metal carbonates

• An important industrial reaction involves the
  thermolysis of metal carbonates to form metal
  oxides according to
• MCO3 (s) ? MO (s) CO2 (g)
• DG must be negative for the reaction to proceed.
  At the lowest reaction temp
• DG 0 and Tmin DH / DS
• DS is positive because gas is liberated. As T
  increases, DG becomes more negative (i.e. the
  reaction becomes more favorable). DS depends
  mainly on DS0CO2(g) and is almost independent
  of M.

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Thermal stability of metal carbonates

• MCO3 (s) ? MO (s) CO2 (g)
• Tmin almost directly proportional to DH.
• DHL favors formation of the oxide (smaller anion)
  for smaller cations.
• So Tmin for carbonates should increase with
  cation size.

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Solubility

• MX (s) --gt M (aq) X - (aq)
• DS is positive, so a negative DH is not always
  required for a spontaneous rxn. But DH is usually
  related to solubility.
• Use a B-H analysis to evaluate the energy terms
  that contribute to dissolution
• MX(s) ? M(g) X-(g) DHlat
• M(g) n L ? ML'n(aq) DHsolv, M
• X-(g) m L ? XL'm-(aq) DHsolv, X
  
• L'n L'n ? (n m) L DH L-L
• MX(s) ? M(aq) X-(aq) DHsolution, MX

Driving force for dissolution is ion solvation,
but this must compensate for the loss of lattice
enthalpy.
LiClO4 and LiSO3CF3 deliquesce (absorb water
from air and dissolve) due to dominance of DHsolv
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Solubility
The energy balance favors solvation for
large-small ion combinations, salts of ions with
similar sizes are often less soluble.
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Solubility

• Some aqueous solubilities at 25C
• DHsolution solubility
• salt (kJ/mol) (g /100 g H2O)
• LiF 5 0.3
• LiCl - 37 70
• LiI - 63 180
• MgF2 0.0076
• MgO 0.00062

• DHL terms dominate when ions have higher
  charges these salts are usually less soluble.

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Orbitals and Bands
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Band and DOS diagrams
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s vs T
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Intrinsic Semiconductors

• s n q m
• s conductivity
• n carrier density
• q carrier charge
• m carrier mobility
• P electron population
• e-(Eg)/2kT

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Bandgap vs Dc
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Arrhenius relation
Arrhenius relation s s0 e-Eg/2kT
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Extrinsic Semiconductors
n-type
p-type
n-type example P-doped Si
p-type example B-doped Si