1. 1. The Crystalline Solid State
Formulas and Structures
Simple Structures
Unit Cell = simplest repeating unit of the regular crystalline array
Bravais Lattices = 14 possible basic crystal structure unit cell types

2. Atoms on corners and edges are shared between unit cells
Rectangular corners shared by 8 unit cells (8 x 1/8 = 1 total atom in cell)
Other corners shared unequally, but still contribute 1 atom to unit cell
Edges shared by 4 cells, contribute (4 x ¼ = 1 atom to unit cell)
Faces shared by 2 cells, contribute ½ atom to each
Angles and Dimensions can vary: triclinic has all different lengths and angles
Lattice Points = positions of atoms needed to generate the whole crystal
Body Centered Cubic: (0,0,0) = origin and (½ , ½, ½) = center
All other atoms can be generated from these 2 by moving them exactly one cell length
Cubic Structures
Primitive Cubic is the simplest type
To fully describe: length of side, 90o angles, (0,0,0) lattice point
8 x 1/8 = 1 atom in the unit cell
each atom surrounded by 6 others (Coordination Number = CN = 6)
Not efficiently packed; only 52.4% of volume is occupied (74.1% max)
Vacant space at center with CN = r sphere would fit here

3. Body-Centered Cubic (bcc)
One more atom is added to the center of the cube
Size of unit cell must increase over simple cubic
Diagonal across center = 4r (r = radius of one atom)
Corner atoms not in contact with each other due to cell size expansion
Side = 2.31r (Calculate this in Ex. 7-1)
Unit cell contains 1(1) + 8(1/8) = 2 atoms
Lattice Points: (0,0,0) and ( ½ , ½ , ½ )
Close-Packed Structures
Spheres will arrange to take up the least space, not cubic or bcc
Two structures have almost identical packing efficiency (74.1%)
Hexagonal Close Packing = (hcp)
Cubic Close Packing = (ccp) = Face-Centered Cubic = (fcc)
Both have CN = 12 for each atom: 6 in its layer, 3 above, and 3 below
HCP has 3rd layer in identical position as first ABA; second layer above holes; CCP has 3rd layer displaced so above holes in first layer ABC
Both have 2 tetrahedral (Td) holes per atom (CN = 4) formed by 3 atoms in one layer and one atom above/below
Both have 1 octahedral (Oh) hole per atom (CN = 6) formed by 3 atoms in one layer and 3 atoms above/below 4r l l

5. HCP unit cell is smaller than the hexagonal prism
Take four touching atoms in 1st layer and extend lines up to 3rd layer
8(1/8) = 2 atoms in unit cell
Dimensions = 2r, 2r, and 2.83r
Angles = 120o, 90o, and 90o
Lattice Points = (0,0,0,) and ( 1/3, 2/3, 1/2)

6. CCP unit cell
One corner from layer 1, opposite corner from layer 4
6 atoms each from layers 2 and 3 in the unit cell
Unit cell is fcc
8(1/8) + 6(1/2) = 4 atoms in the unit cell
Lattice Points = (0,0,0), ( ½ , 0 , ½), ( ½ , ½, 0), (0 , ½ , ½)

7. Holes
2 Td and 1 Oh hole per atom in both (hcp) and (ccp)
If ionic compound, small cations can fill these holes
Td hole = 0.225r Oh hole = 0.414r
NaCl has Cl- in (ccp) with Na+ (ccp) but also in the Oh holes
Na+ = 0.695r, so it forces the Cl- to be farther apart, but still CN = 6
Metallic Crystals
Most metals crystallize in (bcc), (ccp), or (hcp) structures
Changing pressure or temperature can interchange these forms for a metal
Must consider bonding, not just geometric packing only
Soft and malleable metals usually have (ccp) structure (copper)
Harder and more brittle metals usually have the (hcp) structure (zinc)
Most metals can be bent due to non-directional bonding
Weak bonding to all neighbors, not strong bonding to any single one
Atoms can slide past each other and then realign into crystal form
Dislocations = imperfections in lattice; make it easier to bend
Impurities = other elements; allow slippage of layers
Work Hardening = hammer until impurities are together
Heat: can soften by dispersing impurities or harden if control cooling

8. Diamond
Each atom is tetrahedrally bonded to 4 other carbon atoms
Directional single bonds, unlike metals, cause hardness
Binary Compound Structures
Simplest structures just have 2nd element in holes of the first element’s lattice
Small cations can fit in Td / Oh holes of large anions
Large cations may only be able to fit in Oh holes
Even larger cations may cause changes in structure if they don’t fit in holes
Relative Number of Cations/Anions
M2X won’t allow close-packing of anion lattice with cations in Oh holes because there are more cations than holes
Alternatives: cations in Td holes, vacancies, not close-packed

9. Sodium Chloride Structure: NaCl
Na+ in (fcc) and Cl- in (fcc)
Offset by ½ unit cell length
Na+ are centered in edges of Cl- lattice (or vice versa)
Most alkali halides have this same structure
Large size difference of ions facilitate this structure
Each Cl- has CN = 6 Na+ Each Na+ has CN = 6 Cl-
Cesium Chloride Structure: CsCl
Cs in simple cubic structure with Cl- in center (or vice versa)
Cl- = 0.83Cs+ size (0.73r in center is ideal)
Rare structure, need big cation (Cs, Tl only cations known with this structure)

10. Zinc Blende Structure: ZnS
Same as diamond structure with alternating Zn and S atoms
Alternate: Zn and S each in (fcc) lattices combined so each ion is in a Td hole of the other lattice
Stoichiometry: only ½ of the Td holes are occupied and ½ are vacant
Wurtzite Structure: ZnS
Rarer than Zinc Blende structure for ZnS; formed at higher Temperatures
Zn and S each in (hcp) lattices combined so each ion is in a Td hole of the other lattice
Again, ½ of the Td holes are vacant

11. Fluorite Structure: CaF2
Ca2+ in (ccp) lattice with 8 F- surrounding each and occupying all Td holes
Alternate: F- in simple cubic lattice with Ca2+ in alternate body centers
Nearly perfect radius fits for this structure
Antifluorite Structure
Reverse stoichiometry compounds like Na2O
Every Td hole in the anion lattice is occupied by a cation

12. Nickel Arsenide Structure: NiAs
As atoms in close packed layers exactly above each other
Ni atoms in all the Oh holes
Both Ni and As have CN = 6
Alternate: Ni atoms occupy all Oh holes of (hcp) As lattice
Usual for MX compounds where X = Sn, As, Bi, S, Se, Te
Rutile Structure: TiO2
Distorted TiO6 octahedra forming columns by sharing edges
Ti CN = 6; O CN = 3
Adjacent columns connected by sharing corners of octahedra
Unit cell has Ti at corners and in the body center, 4 O in the faces, and 2 O in the plane of the body center Ti
MgF2, ZnF2 are other examples

13. The Radius Ratio
We can crudely predict CN by using the ratio r+/r-
This assumes atoms are just packing as hard spheres (not really all that occurs)
Examples:
NaCl r+/r- = 113/167 (CN = 4) = 0.667
r+/r- = 116/167 (CN = 6) = Fits best with CN = 6
ZnS r+/r- = 74/170 (CN = 4) = Fits best with CN = 4
r+/r- = 88/170 (CN = 6) = CN = 4 in actual structure
4) Exercise 7-2 CaF2 has fluoride ions in a simple cubic array and calcium ions in alternate body centers, with r+/r- = What are the coordination numbers of the two ions predicted by r+/r- ? What are the coordination numbers observed? Predict coordination numbers of Ca2+ in CaCl2 and CaBr2.

14. 2. Solid State Bonding and Applications
Thermodynamics of Ionic Crystal Formation
The Born-Haber Cycle = series of elementary steps leading to an overall reaction
Used to determine electron affinity when all other reactions experimentally known
Today, we can measure EA’s easily, so Cycle is used to find Lattice Enthalpies
Sample Cycle
Li(s) ----> Li(g) DHsub = 161 kJ/mol
½ F2(g) ----> F(g) DHdis = 79 kJ/mol
Li(g) ----> Li+(g) + e- DHion = 531 kJ/mol
F(g) + e > F-(g) DHioin = -328 kJ/mol
Li+(g) + F-(g) ----> LiF(s) DHxtal = kJ/mol
Li(s) + ½ F2(g) -- LiF(s) DHform = -769 kJ/mol
The Madelung Constant
Calculating Lattice Enthalpy appears straightforward: calculate attractions
e = x C
4peo = 1.11 x C2N-1m-2
[ ] = x J m

15. Problem: Long-range interactions change the Lattice Enthalpy
NaCl Na+ has 6 Cl- nearest neighbors at ro = ½ a (accounted for in equation)
Na+ also has 12 Na+ neighbors at r = a (not accounted for)
Na+ also has many other Na+ and Cl- neighbors at longer distances
M = Madelung Constant takes into account all attractions
Born-Mayer Equation incorporates Madelung Constant as well as accounting for repulsions (much more complicated than attractions)
r = constant = 30 pm works well
Increase in charge causes corresponding increase in Lattice Enthalpy
2+/2+ charges would give 4 x Lattice Enthalpy

16. Solubility, Size, and HSAB
We can use a Born-Haber Cycle to calculate dissolution energies
AgCl(s) ----> Ag+(g) + Cl-(g) DHLattEnth = 917 kJ/mol
Ag+(g) + H2O -- Ag+(aq) DHsolvation = -475 kJ/mol
Cl-(g) + H2O ----> Cl-(aq) DHsolvation = -369 kJ/mol
AgCl(s) + H2O ----> Ag+(aq) + Cl-(aq) DHdissolution = 73 kJ/mol
Factors effecting solubility
Size
Small ions have strong attractions; larger ions have small attractions
Large ions may have more water molecules surrounding them
Large/Large and Small/Small salts are less soluble than Large/Small
LiF = CsI < LiI = CsF Lattice Energy HSAB
Reaction: LiI(s) + CsF(s) ----> CsI(s) + LiF(s) DH = 917 kJ/mol
Cation
Hydr. E.
Anion
Hydr. E
Latt. Enth
Net Enth.
Li+
-519 F-
-506
-1025 I-
-293
-745
-67
Cs+
-276
-724
-58
-590
+21

17. Molecular Obitals and Band Structure
Band Formation
Overlap of 2 AO’s gives 2 MO’s
Overlap of n AO’s gives n MO’s
Solids have very large values of n, sometimes multiples of N
Band = many closely spaced MO’s of nearly continuous energy
Valence Band = highest occupied band (HOMO)
Conduction Band = next highest empty band (LUMO)
Insulator = large energy gap between Valence and Conduction bands
Electrons can’t move through material
Electron motion is what allows conduction of electricity and heat
Conductor = partly filled Valence and Conduction bands (Most Metals)
Little energy required for electron movement
Hole = electron vacancy that can also “move” in partially occupied bands
holes
Insulator Conductor w/ no Voltage Conductor with applied Voltage

18. Density of State = concentration of E levels in a Band = N(E)
Temperature Effect on Conductors (Metals)
Vibrations increase as temperature increases
Vibrations interfere with electron movement, slow conductance (increase resistance)
Semiconductors= full Valence Band, Empty Conductance Band, close together
Energy gap < 2 eV
Si, Ge are common pure substances that are semiconductors
At low temperature they are insulators (not as good as true insulators)
At higher temperature they are conducturs (not as good as true conductors)
Opposite temperature effects as metals (true conductors)

19. Doped Semiconductors
We can closely control on/off properties of semiconductors: this has led to the entire field of solid state electronics (computers)
Intrinsic Semiconductor = pure form is semiconductor (Si, Ge)
Doped Semiconductor = small amount of impurity effects semiconduction
n-type semiconductor = dopant has more e- than host (P in Si)
p-type semiconductor = dopant has less e- that host (Al in Si)
Careful doping results in tailored materials
Fermi Level = EF = Energy at which e- equally likely to be in either band
Intrinsic: EF about in middle of gap
n-type: EF raised above new band
p-type: EF lowered below new band
n-type semiconductor p-type semiconductor

20. Devices Using Semiconductors
p-n Junction
Diode = device where current flows only in one direction
At equilibrium a few e- have moved from n-type to p-type (n is +, p is -)
EF is at same level in n-type and p-type at equilibrium
Apply negative pot. to n-type and positive pot. to p-type
Forward Bias
Extra electrons allow e- in n-type to flow to p-type holes
Holes move to junction from p-type side
Current flows readily
Reverse Bias: holes and electrons move away from juction = no current flows

21. Photovoltaic Cells
A p-n Junction with the energy gap = hn of a light source
Light causes e- to jump to p-type even under reverse bias conditions
Light detector; Calculator battery
LED = Light Emitting Diode
A p-n Junction with forward bias
Electrons move from n to p and release energy in the form of light
Lower temp increases efficiency by decreasing vibrations
4) Laser = Light Amplification by Stimulated Emission of Radiation
LED with large band gap in p-type to prevent e- from escaping middle band
If Length = l(n/2), reflection by edges stimulate more in-phase photons

22. Superconductivity The Phenomenon
The conductivity of some metals changes abruptly
around 10 K (Critical Temperature = TC)
2) Superconductor = no resistance to e- flow
Kammerling and Onnes 1911 discover
for Hg cooled by liquid He
Major use today is superconducting magnets for NMR
Low Temperature Alloys
Type I Superconductors are often Nb-Ti alloys
Abrupt change between superconducting and normal conduction
Meissner Effect = no magnetic flux can enter superconductor below TC
“Floating Magnets” demonstration
Strong magnetic fields destroy superconduction above HC
Nb3Ge has highest TC = 23.3 K for this type of superconductor
Type II Superconductors work below TC1 and are complex between TC1 and TC2
Some magnetic flux can enter them in complex region (Floating Magnets)
Possible to use for Levitated Trains

23. Theory = Cooper Pairs
1950’s Bardeen, Cooper, and Schrieffer propose BCS Theory
Electrons travel through superconductors in pairs (Cooper Pairs)
Opposite spin electrons are slightly attracted at low temperatures
As one e- moves past +nucleus in metal, the next +nucleus attracts it
This increases the charge density, so the second e- of the pair is attracted to
Cooper pairs move through metal like a wave
Lattice helps push/pull e- through; no resistance because energetically favorable
When T > TC the thermal motion of the +nuclei disrupt the “wave”
High Temperature Superconductors
(La2-xSrX)CuO4 found to have TC = 30 K in 1986
YBa2Cu3O7 found to have TC = 92 K in 1987
Type II superconductors
Cool with N2(l) [cheap, bp = 77K] instead of He(l) [expensive, bp = 3K]
Ceramic = brittle; can’t easily make into wire, etc…
Structure: copper oxide planes and chains
Theory: BCS Theory applies, but not completely understood

24. Bonding in Ionic Crystals
Simplest Idea = hard spheres with only electrostatic interactions
Deviations form Simple Idea
Ionic size is difficult to measure
Pauling Li+ = 60 pm (from calculations)
Shannon Li+ = 90 pm (from crystal structures)
Sharing of electrons back from anion effects the size of the cation
Covalent Interactions very important as well: ZnS is strongly covalent
Complex theory involving MO’s = Crystal Field Theory (Chapter 10)
Imperfections in Solids
Crystal Growth
Slowly grown crystals are more perfect
Quickly grown crystals are usually made up of many small crystals which have run into each other = grain boundaries
Even “perfect crystals” have impurities and vacancies
Vacancies and Self-Interstitials
Vacancy = missing atom, ion, or molecule in the crystal = simplest imperfection
More formed at higher T, but only 1/10,000 even near the melting point
Effect is small; localized in one unit cell and/or layer of the crystal

25. Self-Interstitials = atoms/ions/molecules in the wrong place
Effect is felt for several layers of the crystal
Usually much rarer than vacancies
Substitutions = one element/ion in place of the expected element/ion
Common occurrence leading to a “Solid Solution”
Ni/Cu similar in size and electronegativity; both have (fcc) structure
Mixtures stable in any proportion = alloy
Random arrangement of atoms since they are so similar
Small atoms in holes between larger ones
Occupying a hole usually has small effect on the rest of the structure
May have large effect on properties of the material (C in Fe makes steel)
If impurity is large than hole: lattice strain or new solid phase
Dislocations
Atoms in one layer don’t match up with the next
Distances and angles effected for several layers in each direction
Screw Dislocation = one layer shifted a fraction of unit cell
Rapid growth location (more attraction into solution) = helical result
Mechanically weak; Electronically inefficient

26. The Silicates Abundance O, Si = 80% of Earth’s Crust
Many compounds and minerals formed, some industrially important
Silica = SiO2
Three crystalline forms: Quartz (low T form), Tidymite, and Cristobalite
Molten SiO2 often forms glasses instead of crystalline forms
Glass = disordered solid structure
Actually a solution that continues to “flow” but very slowly
SiO4 tetrahedra in all crystal forms with Si—O—Si angle = 143.6o
Quartz
Most common form
Helical chains of SiO4 tetrahedra make it chiral
Each full turn of the helix has 3 Si and 3 O atoms
Six helices combine to give a hexagonal structure

27. Other Silicates also have SiO4 units in chains, sheets, rings, arrays, etc…
Al3+ can substitute for Si4+ = Aluminosilicates
Other cations needed to balance charge
Al3+, Mg2+, Fe2+, Ti3+ common cations; occupy holes
Kaolinite = Al2(OH)2Si4O10
Talc = 3 Mg substitute for 2 Al Mg3 (OH)2Si4O10
OH- bridge between Al or Mg and Si

28. Mica
Layers of K+ ions between silicate and aluminate layers
Al in about 25% of the Si positions
Can be cleaved into incredibly smooth, flat sheets
Asbestos = fibrous mineral
Chrysotile = Mg3(OH)4Si2O5
Mg and Silicate layers differ in size leading to curling fibrous structure
Zeolites = mixed aluminosilicates with:
(Si,Al)nO2n frameworks and cations added to balance the charge
Used as water softeners before polymer resins developed (Cation exchange)
Cavities large enough for molecules to enter; may make stable complex
Water removal from organic solvent = Molecular Sieves
Cat litter and oil absorbant
Catalysts and catalyst support for petroleum cracking