CHEM 548

33 stops across the 7 instructional units of CHEM 548

CHEM 548 course roadmap

Work the units in teaching order: structural language and structure representation first, then crystal chemistry, defects and phases, bonding, bands, and transport.

Course skills

What you can do by the end

  1. Space-group notation. Unpack any space-group symbol: from P6₃/mmc alone, name the lattice type, every operation encoded, the crystal system, and the point group.
  2. The International Tables. Read a full International Tables page: both diagrams, the coordinate list, the Wyckoff table, and say what each panel claims about any structure in that group.
  3. Cells from asymmetric units. Expand a space group and its asymmetric unit into the full cell, by hand for a simple case and in VESTA generally, and collapse a full cell back to its asymmetric unit, counting Z along the way.
  4. VESTA. Drive VESTA on a real published structure: enter it from a paper's table, generate the symmetry copies, draw polyhedra, measure bonds and angles.
  5. Network accounting. Determine a structure's formula and connectivity from its coordination polyhedra and bond graph.
  6. Defect balance. Write and interpret Kröger–Vink symbols for vacancies, interstitials, and aliovalent substitutions.
  7. Spin-state prediction. Predict a transition-metal complex's high- or low-spin configuration from crystal-field splitting and pairing energy.
  8. Reciprocal-space reading. Construct a reciprocal lattice and interpret a band path through its special k-points.
  9. Semiconductor transport. Calculate a semiconductor's conductivity from electron and hole concentrations and mobilities.

Unit

Structural language

4 stops

Stop 1

Plane lattices and two-dimensional cells

Lecture 1 works in two dimensions, from the first drawing, a crystal as packed spheres, to a classification: four crystal systems told apart by their cell metrics and symmetry, five plane lattices, and a Lecture 2 practice set that derives both counts.

You should be able to

  • Name the four two-dimensional crystal systems and the cell conditions that define each.
  • Derive the four crystal systems and the five plane lattices, as the Lecture 2 practice questions do.
  • Describe the centered rectangular lattice both ways: one point set, a conventional centered cell and a primitive cell.

Read more

  • Crystalline materials repeat atomic arrangements periodically in three dimensions. Solid State Materials Chemistry, ch. 1 introduction.
  • The five plane lattices are oblique p, rectangular p, rectangular c, square p, and hexagonal p, with motif symmetry constraining the lattice. The Basics of Crystallography and Diffraction (Hammond), ch. 2, §2.4.
  • Choose a unit-cell origin for clarity, favouring atoms at corners or edge centres and an evident symmetry. Solid State Chemistry and its Applications (West), §1.1.

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Stop 2

Three dimensions: crystal systems and the 14 Bravais lattices

The classification repeats one dimension up, and the work is in the pruning: primitive, body-, base-, and face-centered cells are compared, then the redundant combinations are reduced by hand, monoclinic F to a smaller monoclinic C, base-centered cubic to a smaller primitive tetragonal cell.

You should be able to

  • Distinguish primitive, body-centered, base-centered, and face-centered cells.
  • Reduce a redundant centering to an equivalent smaller cell, as tetragonal C reduces to a smaller tetragonal P.
  • Show that a base-centered cubic cell describes a primitive tetragonal lattice.
  • Explain why the four-point FCC conventional cell is kept over its one-point primitive cell.

Read more

  • There are 14 Bravais lattices, the allowed kinds of translational symmetry in three dimensions. Solid State Materials Chemistry, §1.1.4.
  • A consistent right-handed crystallographic axial convention avoids ambiguity in coordinates and distinguishes handed coordinate systems. The Basics of Crystallography and Diffraction (Hammond), ch. 3, §3.2.
  • Define a Bravais lattice by point-group symmetry and centring mode, and locate it as a space group’s translational subgroup. International Tables for Crystallography, Volume A, §9.1.6, p. 744.

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Stop 3

Point symmetry and the crystallographic point groups

Lecture 3 opens by unpacking two symbols from the lattice table: in 4mm and 6mm, mm means mirror planes that contain the rotation axis, four in two perpendicular families or six in two alternating ones, and both groups are non-centrosymmetric and polar along the principal axis.

You should be able to

  • State the minimum and maximum point symmetry a lattice supports, as fourfold against 4mm for the square lattice.
  • Expand 4mm and 6mm into their rotation axes and vertical mirror-plane families.
  • Decide whether a point group is centrosymmetric and whether it is polar, as worked for 4mm and 6mm.

Read more

  • A point group is a set of operations on an isolated object satisfying closure, associativity, identity, and inverses. Solid State Materials Chemistry, §1.1.2.
  • Orthorhombic symbols refer to x, y, and z symmetry, monoclinic symbols to y, and tetragonal, hexagonal, and trigonal symbols begin with the unique z axis. The Basics of Crystallography and Diffraction (Hammond), ch. 4, §4.3.
  • Read a Hermann–Mauguin point-group symbol by its representative lattice-symmetry directions and rotation or improper-rotation symbols. International Tables for Crystallography, Volume A, §12.1.4, p. 818.

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Stop 4

Space groups: screw axes, glides, and the International Tables

Space groups combine the 14 Bravais lattices with the 32 crystallographic point groups, and the combining does real work: a twofold rotation plus C-centering already generates the half-cell shift of a 2₁ screw, so C2₁ earns no symbol of its own, and any group carrying a screw axis or glide plane is nonsymmorphic.

You should be able to

  • State what space-group symmetry combines: the 14 Bravais lattices with the 32 crystallographic point groups.
  • Show why C2₁ is not a distinct label: the C-centering translation already supplies the screw partner.
  • Classify a space group as nonsymmorphic from its screw axes and glide planes.

Read more

  • A space group combines Bravais-lattice translation with point-group rotation; there are 230 total. Solid State Materials Chemistry, §1.1.5.
  • External symmetry cannot distinguish a screw from its corresponding rotation, and optical activity can arise from an enantiomorphous screw arrangement or a chiral molecule. The Basics of Crystallography and Diffraction (Hammond), ch. 4, §4.5.
  • Distinguish a concrete space group from its space-group type and relate an observed structure to its International Tables entry. International Tables for Crystallography, Volume A, §8.2.2, p. 726.

In the lectures

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  • Stop 5 · Reading published structure dataA published structure starts from the space group and Wyckoff information that this stop teaches students to read.
  • Space-group notationThis stop explicitly makes P6₃/mmc a readable list of lattice, system, and symmetry operations.
  • The International TablesIts International Tables checkpoint requires locating a group and reading its general positions.

Unit

Structure representation

2 stops

Stop 5

Reading published structure data

A published structure is a short table, and the lecture spends all of it: fluorite arrives as Fm-3m, one lattice parameter, and two Wyckoff sites, multiplicity and the face centering generate the positions, coordinates like 1¼ fold back to ¼, and the finish is a calculated theoretical density for CaF₂.

You should be able to

  • Read a structure table: space group, lattice parameter, and Wyckoff sites.
  • Generate the fluorite positions from Wyckoff multiplicity and the face centering.
  • Reduce a generated fractional coordinate modulo one to bring it back into the cell.
  • Calculate a theoretical density from unit-cell data, as worked for CaF₂.

Read more

  • Fractional coordinates obey r = xa + yb + zc and wrap into the 0–1 interval by translation. Solid State Materials Chemistry, §1.1.1.
  • In an International Tables entry, Wyckoff letters label arbitrary position sets, and systematic absences appear as conditions on hkl reflection indices. The Basics of Crystallography and Diffraction (Hammond), ch. 4, §4.6.
  • Generate P1̅ equivalents by inversion, fold coordinates back into the cell, and read multiplicity, Wyckoff letters, and site symmetry. Solid State Chemistry and its Applications (West), §1.18.5.1.

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Stop 6

VESTA: driving real structures

The drawing tools arrive beside the structures they will draw: VESTA, CrystalMaker, and Diamond are introduced, fluorite is rendered as calcium and fluoride coordination polyhedra, hcp is developed as ABAB stacking with its screw-axis relation, and CsCl and Cu₃Au come out of bcc and fcc as ordered intermetallics; hands-on VESTA practice then starts at the ternary-oxides lecture.

You should be able to

  • Say what VESTA and its fellow structure-drawing programs are for.
  • Read a polyhedral drawing, as fluorite is drawn with calcium and fluoride polyhedra.
  • Describe the hcp cell through its ABAB stacking and screw-axis relation.
  • Derive the CsCl and Cu₃Au intermetallics from the bcc and fcc lattices.

Read more

  • Structures arise from efficient packing, directional-covalent networks, or local bonding that dictates coordination polyhedra. Solid State Materials Chemistry, §1.4.
  • Jagodzinski-Wyckoff notation labels local cubic or hexagonal layers c/h; a Ramsdell symbol gives layers per cell plus H/R/C, such as α-La = 4H. Solid State Materials Chemistry, §1.4.1.
  • Build cubic and hexagonal close-packed structures from A, B, and C layers and compare their stacking. Solid State Chemistry and its Applications (West), §1.10.

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Unit

Crystal chemistry and networks

5 stops

Stop 7

Close packing and the filling of holes (reading 1.4)

One description covers a large family of ionic structures: anions close-packed, cations in the holes between them; Lecture 4 introduces it, lays the structure types out as a eutactic grid, and lands on the semiconductor case, sphalerite from a ccp anion array with half its tetrahedral holes filled.

You should be able to

  • Describe an ionic structure type as a close packing of anions with cations in its holes.
  • Derive sphalerite (zinc blende) from a ccp anion array with half the tetrahedral holes filled.
  • Describe the reviewed structure types in eutactic terms.

Read more

  • Closest packing contains tetrahedral holes surrounded by four spheres and octahedral holes surrounded by six; each sphere has two tetrahedral and one octahedral hole. Solid State Materials Chemistry, §1.4.2.
  • All tetrahedral filling gives anti-fluorite Li₂O on ccp; half filling gives wurtzite and sphalerite ZnS; CaF₂ is fluorite. Solid State Materials Chemistry, §1.4.2.
  • Recognize rock salt as close-packed anions with all octahedral sites occupied by cations. Solid State Chemistry and its Applications (West), §1.17.1.1.

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Stop 8

Electron counting and Zintl phases (reading 1.3)

Electron bookkeeping starts at Lewis dot symbols and generalizes: the generalized octet rule is worked for SiO₂ and for NaTl, and NaTl returns with SrGa₂ to introduce the Zintl phases.

You should be able to

  • Apply the octet rule from Lewis dot symbols.
  • Work the generalized octet rule for SiO₂ and for NaTl.
  • Recognize NaTl and SrGa₂ as Zintl phases.

Read more

  • The 8−N rule says an electronegative sp atom forms enough bonds to complete eight valence electrons. Solid State Materials Chemistry, §1.3.2.
  • The generalized rule is VEC(A) = 8 + CC(m/n) − AA: VEC(A) above eight gives cation bonds or lone pairs, below eight gives anion–anion bonds. Solid State Materials Chemistry, §1.3.2.
  • CdSb has Sb₂⁴⁻ dumbbells at VEC(A) = 7; CaC₂ has triple-bonded C₂²⁻ at VEC(A) = 5. Solid State Materials Chemistry, §1.3.2.
  • NaTl and SrGa₂ have VEC(A) = 4 diamond-like and graphite-like polyanions; KGe has Ge₄⁴⁻ cages at VEC(A) = 5; CaSi has Si²⁻ chains at VEC(A) = 6. Solid State Materials Chemistry, §1.5.6.

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Stop 9

Crystal-chemical formulas and polyhedral connectivity

The formula notation carries the chemistry: four oxide structures pose how stoichiometry relates to structure, Pauling's parsimony and maximum-symmetry principles constrain the answers, and Lecture 6 delivers the general algorithm for writing a crystal-chemical formula, the notation behind the network-accounting course skill.

You should be able to

  • Write a crystal-chemical formula by the general algorithm.
  • Work a bond graph, as done for SiO₂, TiO₂, and CaF₂.
  • Apply parsimony and maximum symmetry when judging candidate structure descriptions.
  • Explain how an aluminosilicate framework joins Al and Si through a bent oxygen bridge near 145 degrees rather than a direct bond.

Read more

  • A crystal-chemical formula lists each crystallographically inequivalent atom and its coordination in superscript brackets. Solid State Materials Chemistry, §1.3.
  • Multi-site formulas can solve distinct coordination contributions, as for tetrahedral Mn²⁺ and octahedral Mn³⁺ in hausmannite Mn₃O₄. Solid State Materials Chemistry, §1.3.1.
  • A Niggli formula lists each polyhedral vertex by connectivity, so SiO₄/₂ means four two-connected oxygen vertices. Solid State Materials Chemistry, §1.4.4.
  • Edge and face sharing are more common for octahedra because they bring cations less close than they would in tetrahedra; tetrahedra do not face-share. Solid State Materials Chemistry, §1.4.4.

In the lectures

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Leads to

  • Stop 10 · Nets, frameworks, and the silicates (reading 1.4)The vertex and linker abstraction begins from the polyhedral connectivity that a Niggli formula records.
  • Network accountingCrystal-chemical and Niggli formulas turn coordination polyhedra and bond graphs directly into formula and connectivity accounting.

Stop 10

Nets, frameworks, and the silicates (reading 1.4)

Lecture 6 abstracts structures to nets: vertices, linkers, and the uninodal and binodal vocabulary are defined, corundum is examined through two representations, and the sodalite framework with its cage leads into the silicates and microporous solids.

You should be able to

  • Describe a structure in net vocabulary: vertices, linkers, uninodal or binodal.
  • Read corundum across its two structural representations.
  • Trace the sodalite framework and its cage.

Read more

  • A uninodal network has one vertex type and N-connected vertices; graphene is a 3-connected example. Solid State Materials Chemistry, §1.4.3.
  • Network expansion inserts a linker between vertices, such as oxygen between Si atoms in diamond-type silicon to form cristobalite. Solid State Materials Chemistry, §1.4.3.
  • Silicates are predominantly corner-sharing SiO₄ tetrahedra, often with Al substitution; oxygen is usually two-coordinate and silicates comprise about a third of inorganic structure types. Solid State Materials Chemistry, §1.5.4.
  • Classify silicate structures by tetrahedral linking into isolated, dimer, chain, sheet, and framework anions. Solid State Chemistry and its Applications (West), §1.17.16.

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Stop 11

Ternary oxides: spinel, perovskite, and garnet (reading 1.5)

Two cations enter one oxide: spinel arrives as the magnesium aluminate structure, perovskite coordination environments are worked, and SrTiO₃, CaTiO₃, and MgTiO₃ are compared as the tolerance factor falls below one; slide 14 is also where VESTA practice starts, regenerating fluorite from its space-group and Wyckoff-site data.

You should be able to

  • Describe the spinel structure of magnesium aluminate.
  • Work the coordination environments of a perovskite.
  • Compare SrTiO₃, CaTiO₃, and MgTiO₃ and tie their differences to tolerance factors below one.
  • Generate the fluorite positions from space-group and Wyckoff-site data in VESTA.

Read more

  • Spinel MgAl₂O₄ is [tet]₁[oct]₂O₄, filling one eighth of tetrahedral and one half of octahedral holes in ccp oxygen. Solid State Materials Chemistry, §1.5.1.
  • Glazer notation encodes octahedral rotations about three axes with letter magnitudes and +, −, or 0 phase relations; the simple combinations total 23. Solid State Materials Chemistry, §1.5.3.
  • Use the perovskite framework, site occupancies, and octahedral tilts as a structural family for real materials. Solid State Chemistry and its Applications (West), §1.17.7.
  • Use garnet site types and coordination environments to interpret a complex oxide structure. Solid State Chemistry and its Applications (West), §1.17.13.

In the lectures

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Leads to

  • Stop 27 · Band structures of transition-metal oxidesPerovskite structure and octahedral tilts supply the structural family whose band structure is interpreted here.
  • VESTARebuilding fluorite from its space group in VESTA is the stated course revisit that practices the VESTA tool skill.

Unit

Defects and phases

6 stops

Stop 12

Intrinsic point defects

Lecture 8 retires the perfect crystal: real ones always contain intrinsic defects, counted concretely as 17 places to put one vacancy among 16 atoms, and read for consequences, an electron trapped at an anion vacancy appearing as a color center, and defect concentrations that record the higher temperature at which diffusion last kept pace.

You should be able to

  • Explain why real crystals always contain intrinsic defects.
  • Count the positions available to one vacancy, as the 16-atom example counts 17.
  • Explain a color center as an electron trapped at an anion vacancy.
  • Explain why a measured defect concentration reflects a higher temperature than the measurement: diffusion slows as the crystal cools.

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Stop 13

Extrinsic defects and Kröger–Vink notation

Doping is deliberate defect chemistry: a substituent either matches the host ion's charge or does not, the aliovalent case is where the accounting lives, and Kröger-Vink notation is worked on lithium vacancies, oxygen interstitials, and strontium substitutions, which is the defect-balance course skill; silicon's donor and acceptor doping points ahead to the semiconductor lectures.

You should be able to

  • Distinguish isovalent from aliovalent substitution by whether the substituent's charge matches the host's.
  • Write and interpret Kröger-Vink symbols, as worked for lithium vacancies, oxygen interstitials, and strontium substitutions.
  • Connect donor and acceptor doping of silicon forward to the semiconductor lectures.

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Stop 14

Solid solutions and Vegard's law

Continuous substitution turns composition into a control knob: the unit-cell size varies linearly between the end members, the band gap follows its own weighted-sum line, and CdS-CdSe pigments show the knob at work, color shifting continuously with composition; the summary also collects the deviations from linearity.

You should be able to

  • Use Vegard's law to convert between lattice parameter and composition.
  • Predict a band gap, or the composition that yields one, from the weighted sum of the end members.
  • Connect a CdS-CdSe composition to its pigment color.
  • Recognize deviations from the linear laws.

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Stop 15

Binary and ternary phase diagrams

The map is read before it is complicated: lecture 9 opens the stop with the Gibbs phase rule, the liquid and two-phase fields of a binary system that forms no compounds, the eutectic point where three phases meet, the liquidus and the solidus that bound those fields, and the lever rule that turns a tie line into the fraction of each phase, on real pairs from diopside and anorthite to magnesium oxide and calcium oxide. Lecture 10 then reads the harder maps and rebuilds one: intermediate compounds complicate the binary diagrams, a ternary composition is read from a triangle whose flanking scales measure different components, the triangle rule turns segment lengths into phase fractions, and repeated common-tangent constructions generate a binodal diagram outright.

You should be able to

  • State the Gibbs phase rule and apply it at one atmosphere, where the degrees of freedom are the components plus one minus the phases.
  • Read a binary phase diagram that forms no compounds: tell the liquidus from the solidus, locate the eutectic point, and apply the lever rule to a tie line to get the fraction of each phase.
  • Say what intermediate-compound formation adds to a binary phase diagram.
  • Read a ternary composition against the correct flanking scale, where the left scale measures C rather than A.
  • Apply the triangle rule: each phase fraction is its opposite segment over the total line length.
  • Generate a binodal phase diagram from repeated common-tangent constructions.

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Stop 16

Free energy of mixing and the origin of the diagram

The diagram gets derived instead of read: entropy and enthalpy compete as temperature changes, the free-energy curves reshape through the consolute temperature, silicon-germanium works the ideal case, and at the opposite extreme a stoichiometrically sharp minimum pins a line compound.

You should be able to

  • Explain how entropy and enthalpy trade off as temperature changes.
  • Describe how the free-energy curves change near the consolute temperature.
  • Work the ideal silicon-germanium phase diagram.
  • Explain why a sharp stoichiometric minimum gives a line compound.

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Stop 17

Lattice energy, ionic radii, and radius ratios

Ionic bonding gets its number: Coulomb attraction and repulsion between point charges build up to the Born-Mayer lattice energy, worked end to end for KCl, and the radius-ratio rule turns radii into coordination cutoffs that the halide data then demote to guidelines, since ions are not hard spheres.

You should be able to

  • Distinguish attraction from repulsion in the Coulomb energy of point charges.
  • Evaluate the Born-Mayer equation, as worked for KCl.
  • Apply the radius-ratio rule to predict a coordination number cutoff.
  • Explain why the cutoffs are guidelines rather than rules: crystal radii move the crossover from halide to halide.

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Unit

Bonding

5 stops

Stop 18

Atomic orbitals, nodes, and size trends

The bonding unit restarts at the atom: quantum numbers and the filling rules, a 2p wavefunction changing sign across its nodal plane, a table of orbital sizes in angstroms, and a worked question tying the trends together, that second-period elements form strong π bonds because their 2s and 2p orbitals are so similar in size.

You should be able to

  • Assign quantum numbers and apply the Pauli and Hund filling rules.
  • Sketch a 2p wavefunction and mark the nodal plane where it changes sign.
  • Read the orbital-size table's trends, including the contracted second-period 2p and the transition-metal 3d.
  • Explain strong second-period π bonding from the similar sizes of 2s and 2p.

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Stop 19

Molecular orbitals and symmetry-adapted linear combinations

Two worked diagrams carry the machinery: BeH₂ is built from its ligand SALCs with the nonbonding level and the HOMO and LUMO labeled, O₂ adds σ-π splitting and s-p mixing, and a check-yourself question asks for the SALC construction rule itself.

You should be able to

  • Build the BeH₂ diagram from ligand SALCs and label its nonbonding level, HOMO, and LUMO.
  • Work the O₂ diagram, including σ-π splitting and s-p mixing.
  • State what a symmetry-adapted linear combination is and apply the construction rule.

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Stop 20

Molecular orbitals of octahedral and tetrahedral complexes

Lecture 13 assembles a real complex: CrCl₆³⁻ is set up orbital by orbital and completed into a full diagram with bonding, nonbonding, and antibonding sets, the methane diagram returns as the template for the metal's s and p interactions, and a closing question compares octahedral against tetrahedral field splitting.

You should be able to

  • Identify the metal and ligand orbitals that enter the CrCl₆³⁻ complex.
  • Build the complete octahedral diagram and label its bonding, nonbonding, and antibonding sets.
  • Reuse the methane MO diagram as the template for the metal s and p interactions.
  • Compare octahedral and tetrahedral field splitting.

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Stop 21

Spin states and Jahn-Teller distortions

Splitting against pairing energy decides high spin or low spin, and that decision is the spin-state-prediction course skill; the degenerate cases then reshape their surroundings, Cu²⁺ d9 octahedra elongating rather than compressing, d8 NiO and PtO dividing between octahedral and square-planar, and PbO sliding from CsCl to litharge as the second-order Jahn-Teller example.

You should be able to

  • Predict a high-spin or low-spin configuration from crystal-field splitting against pairing energy.
  • Explain why Cu²⁺ d9 octahedra elongate rather than compress.
  • Use the d8 splitting diagrams of NiO and PtO to explain octahedral against square-planar preference.
  • Recognize the PbO distortion from CsCl to litharge as second-order Jahn-Teller.

In the lectures

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  • Spin-state predictionThis stop explicitly predicts high or low spin from crystal-field splitting and pairing energy.

Stop 22

Bond valence: bonding as a structural check

Bond valence turns a measured structure into a check on itself: Pauling's electrostatic balances are worked on the Li₂O and SrTiO₃ bond graphs, measured SrTiO₃ distances become valences that sum site by site, and in diaspore the sums single out the oxygen that carries the hydrogen.

You should be able to

  • Apply Pauling's electrostatic valence balance to the Li₂O and SrTiO₃ bond graphs.
  • Compute bond valences from measured distances and sum them by site, as worked for SrTiO₃.
  • Use a valence sum to locate hydrogen, as the diaspore sums do.
  • State the method's rules, uses, and advantages.

In the lectures

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  • Network accountingTurning measured distances into valences and testing their sums is a structural check on coordination and bond-graph accounting.

Unit

Band theory

5 stops

Stop 23

From molecular orbitals to bands

An infinite crystal cannot be treated as a giant molecule, so the molecular-orbital diagram gives way to a band, built on the infinite one-dimensional hydrogen chain: the wavevector k names the phase pattern, at k = π/2a the crystal orbital repeats over four lattice spacings with nodes every two atoms, and bonding character and energy follow from k. Pairing those atoms into a chain of H₂ molecules doubles the unit cell, and because the number of orbitals in the cell is the number of bands, the one band becomes two: the filled band is stabilized and the empty one destabilized, so a gap opens at the zone edge and the dimerization pays for itself, which is the Peierls distortion. The gap is then read as a semiconductor reads it, valence band to conduction band, with the two extrema either at the same k or at different ones.

You should be able to

  • Explain why an infinite crystal needs a band description rather than a molecular-orbital diagram.
  • Write the crystal orbital of the one-dimensional hydrogen chain at a given k.
  • Work the k = π/2a case: a wavelength of four lattice spacings and nodes every two atoms.
  • Relate k to bonding character and orbital energy.
  • Count the bands of a chain from the orbitals in its unit cell, and argue the Peierls distortion the chain of H₂ molecules shows: pairing the atoms opens a gap at the zone edge and only the filled band gains.
  • Distinguish a direct from an indirect band gap by where the two band extrema sit in k.

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Stop 24

The reciprocal lattice and the Brillouin zone

Reciprocal space gets its vocabulary: the defining relation of the reciprocal-lattice vectors, lengths that scale inversely with their real-space partners, and the first Brillouin zone constructed in a two-dimensional reciprocal lattice, the frame in which the two-dimensional hydrogen sheet's Γ-X-M band path is read.

You should be able to

  • Construct reciprocal-lattice vectors and state their defining relation.
  • Construct the first Brillouin zone of a two-dimensional reciprocal lattice.
  • Predict how a reciprocal vector's length responds when its real-space vector shortens.
  • Follow a band path through labeled k-points, as the hydrogen sheet's Γ-X-M path is worked.

Read more

  • A Voronoi, Wigner–Seitz, or Dirichlet polyhedron defines the domain around a lattice point by planes equidistant from neighbours. The Basics of Crystallography and Diffraction (Hammond), ch. 3, §3.4.
  • The cubic P, I, and F Voronoi polyhedra are respectively a cube, truncated octahedron, and rhombic dodecahedron. The Basics of Crystallography and Diffraction (Hammond), ch. 3, §3.4.

In the lectures

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Stop 25

Several orbitals per cell: bands and the density of states

With several orbitals in the unit cell each band needs its own argument: the fluorine chain is the one-dimensional warm-up, four orbitals on one atom worked into four bands, the dxy basis is set up, the dx2-y2 band is worked at M into its most antibonding σ* crystal orbital, and the band that looked nonbonding turns out to lean on oxygen 2s mixing at Γ, with the density of states as the running summary.

You should be able to

  • Set up the basis for a d band in a multi-orbital unit cell.
  • Work the dx2-y2 band at M and identify the most antibonding σ* crystal orbital.
  • Test a nominally nonbonding band for weak mixing, as oxygen 2s mixes with Cu dx2-y2 at Γ.
  • Read a density of states alongside its band structure.

In the lectures

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Stop 26

Band structures of the elements

The cartoons give way to published band structures: reciprocal vectors go three-dimensional through the cross-product definition, aluminum's metallic behavior reads off its wide, partially filled bands, and in lead the antibonding s band drops below the bonding p states, destabilizing diamond-type tetrahedral coordination outright.

You should be able to

  • Construct three-dimensional reciprocal vectors from unit-cell cross products.
  • Relate hexagonal real and reciprocal vectors through perpendicularity: a* perpendicular to b.
  • Explain aluminum's metallic behavior from its wide bands crossing the Fermi level.
  • Explain why lead abandons the diamond structure: its antibonding s band falls below the bonding p states.

In the lectures

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Stop 27

Band structures of transition-metal oxides

The ligand-field diagram and the band structure meet in the transition-metal oxides: a π* band's overlap is worked from Γ to M, the t₂g-to-eg gap read at R is the octahedral ligand-field splitting in band form, and the three t₂g bands stay degenerate at Γ and R because every unit-cell translation shifts their phases equivalently.

You should be able to

  • Work a π* band's overlap pattern from Γ to M.
  • Read the t₂g-to-eg splitting at R as the octahedral ligand-field splitting energy.
  • Explain why the three t₂g bands remain degenerate at Γ and R.

In the lectures

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Unit

Transport and superconductivity

6 stops

Stop 28

The Drude model and mobility

A table of measured conductivities in siemens per meter opens the unit and asks what makes materials differ so widely; the classical answer runs through drift, the net motion a field imposes on carriers, with mobility defined as drift velocity over field, and the model's limitations flagged from the start.

You should be able to

  • Define drift velocity as the net carrier motion that carries the current.
  • Derive mobility as drift velocity over electric field.
  • Compare measured conductivities across materials and say what a transport model must explain.
  • State the Drude model's limitations.

In the lectures

Builds on

Leads to

Stop 29

The free-electron model and the Fermi surface

The quantum model replaces the classical gas: the purely parabolic free-electron band is compared against a tight-binding band to mark where the model holds, the Fermi velocity follows from the Fermi energy and effective mass, and silver is the worked case, filled 4d bands below the Fermi level and a wide 5s band carrying the high mobility.

You should be able to

  • Compare the free-electron parabola with a tight-binding band and mark where they agree.
  • Derive the Fermi velocity from the Fermi energy and the effective mass.
  • Explain silver's high mobility from its wide 5s band and its buried, filled 4d bands.

In the lectures

Builds on

Leads to

Stop 30

Semiconductors: doping, carriers, and devices

Semiconductor transport gets calculated: conductivity combines electron and hole concentrations with their mobilities, donor and acceptor levels sit in the gap at their ionization energies, a photovoltaic cell separates the photo-created pair with its built-in field, and the closing slide poses the NiO-TiO conductivity disparity for the transition-metal section that follows.

You should be able to

  • Calculate a semiconductor's conductivity from electron and hole concentrations and mobilities.
  • Place donor and acceptor levels in the band gap with their ionization energies.
  • Explain how a photovoltaic cell's built-in field separates the photo-created electron-hole pair.
  • State the NiO-TiO disparity that a simple band picture leaves unexplained.

In the lectures

Builds on

Leads to

  • Stop 31 · Correlated oxides: bandwidth against Hubbard UThe semiconductor stop explicitly flags NiO as a case where a simple band picture fails before Hubbard U explains why.
  • Stop 33 · SuperconductivityCarrier concentrations and mobility provide the ordinary transport baseline that zero resistance overturns.
  • Semiconductor transportIts first checkpoint calculates semiconductor conductivity from electron and hole concentrations and mobilities.

Stop 31

Correlated oxides: bandwidth against Hubbard U

The rock-salt series from TiO to NiO ties contracting d orbitals to narrowing bandwidth and rising on-site repulsion U, the Hubbard repulsion the lecture's objectives put first; bandwidth itself is read off a d band's dispersion from bonding to antibonding, as worked for the Ti dxy band of a square TiN plane.

You should be able to

  • State what the Hubbard repulsion U is.
  • Read a bandwidth from a d band's bonding-to-antibonding dispersion, as worked for the Ti dxy band of a square TiN plane.
  • Explain the TiO-to-NiO trend: contracting d orbitals, narrower bands, higher U.

In the lectures

Builds on

Leads to

Stop 32

Conducting polymers and carbon nanotubes

Teflon and doped polyacetylene bracket the conductivity comparison that opens the lecture; PEDOT:PSS supplies the working pair, positively charged thiophene rings on one polymer and a sulfonated polystyrene chain on the other, a chiral (10,5) nanotube is wrapped by hand, and the applications close with market estimates, nanotube composites alone at 1.2 to 1.5 billion dollars.

You should be able to

  • Place Teflon and doped polyacetylene at the two ends of the polymer conductivity comparison.
  • Distinguish PEDOT from PSS by structure: charged thiophene rings against a sulfonated polystyrene chain.
  • Construct a chiral nanotube by wrapping a strip, as worked for the (10,5) tube.
  • Name the nanotube application markets and their estimated sizes.

In the lectures

Builds on

Leads to

Stop 33

Superconductivity

The course ends on superconductivity whole: MRI, maglev, and tokamak magnets anchor the applications, the Meissner effect makes a superconductor a perfect diamagnet below its critical field, the BCS critical-temperature expression turns on Debye frequency and electron-phonon coupling, and the closing summary reports that simple electronic models now reproduce superconductivity.

You should be able to

  • Name the applications anchored by superconducting magnets: MRI, maglev, and tokamaks.
  • State the Meissner effect: perfect diamagnetism below a critical field.
  • Read the BCS critical-temperature expression and say what the Debye frequency and the electron-phonon coupling control.
  • State the closing conclusion: simple electronic models now reproduce superconductivity.

In the lectures

Builds on

Leads to

  • Semiconductor transportThe zero-resistance criterion supplies the limiting contrast that makes a semiconductor conductivity calculation physically legible.