Exam 2 Lecture Summary

Names (and terms) List

The Sommerfeld Model: An improvement on the Drude model, including the effect of the fact that electrons are fermions.

Equations and derivations:

• Relation between solutions to the time-dependent and time independent Schrödinger equations
• Plane-wave solutions to the Schrödinger equations, using Born-von Karman (periodic) boundary conditions
• Allowed states in reciprocal space, including their density
• Obtaining the Fermi energy from the electron density
• The Maxwell-Boltzman distribution (aka, the meaning of the Boltzmann factor)
• The Fermi-Dirac distribution (the Fermi function)
• Obtaining the density of states in the Sommerfeld model (any dimensionality)
• Electronic specific heat:
• The exact form, an integral that is too hard to evaluate
• The approximate argument, leading to the correct power of temperature

Concepts:

• Representation of allowed plane-wave states in reciprocal space (“k-space”).
• The FERMI SPHERE (very important!), and all its associated parts (F. surface, F. velocity, F. energy, …).
• Macrostates, microstates, momentum states (AKA k states), and one-electron states.
• Relationship between Fermi sphere energies, work function, and truly free electrons.
• Concepts from the derivation of the Fermi-Dirac distribution function n_F;i;N( ε_i) (given N electrons, the mean number of electrons found in one-electron state i which has energy ε_i).  I do not expect you to be able to reproduce the derivation.
• The general shape of the Fermi distribution function, how it depends on temperature T and chemical potential μ. What is the "purpose" of μ in the distribution?
• What does the density of states describe?
• DC current as described by Sommerfeld model in reciprocal space.

Types of Condensed Matter Cohesion

• How orbitals fill in elements, resulting in the shape of the periodic table
• Ionic bonding makes salts
  • Charge is transferred between different species, and then ions electrostatically attract.
    • Ionization potential and Electron affinity
  • Long range interaction requires consideration of more than nearest neighbors to get accurate binding energy
  • Non-directional interaction leads to moderately high densities, but not too high because can't have triangular coordination
  • Typical binding energies 2.5–5eV/ion → high melting temps
• Covalent bonding involves bonds, as in a molecule
  • Existence of bonds is a purely quantum mechanical phenomenon. Having multiple atom nuclei allows electrons the flexibility to find lower-energy wave functions
  • Bonds are directional, often leading to low densities/packing fractions
  • Typical energies 3–5eV/ion → high melting temps (e.g., ceramics), although many chemically react first
• Metallic bonding occurs when some electrons cease to be localized in atoms or bonds
  • Work function
  • Non-directional interaction leads to high densities
  • Typical binding energies 1–3eV/ion → moderately high melting temp (400–3500K)
• Hydrogen bonding: a sort of a permanent dipole-permanent dipole attraction
  • Only hydrogen creates dipoles this strong, when covalently bound to other atoms, due to having only a single electron
  • Typical binding energies 0.1eV/Hydrogen
• van der Waals force binding makes molecular crystals
  • fluctuating dipole-dipole attraction varies as 1/r⁶; often modeled with Lennard-Jones 6-12 potential
  • Typical binding energies 0.01eV/molecule → low melting temperatures, 25–250K

Vibrational modes of Solids

Equations and derivations:

• Force on an atom in a 1D chain of atoms & springs, and equations of motion from Newton's law
• Wave ansatz, exponential form
• Dispersion relation for 1D chain of atoms

Question types:

• Finding dispersion relations from equations of motion
• More finding densities of modes

Concepts:

• Born-Oppenheimer approximation: because electrons are lighter and faster, they rapidly equilibrate to whatever motion the ions make. Therefore, we can think of the electrons as providing an effective force between the ions (regardless of which category above). We do not have to consider the electrons as separate things; they just move with the ions.
• 1D chain of identical ions, assuming nearest neighbor forces
  • As long as ion displacements are not large, the forces between them will be approximately quadratic in position: Hooke’s Law!  based on Taylor expansion of the force
  • Plane waves satisfy these equations of motion as long as the appropriate dispersion relation is obeyed
• Due to aliasing, only k's in the 1^st Brillouin zone are physically useful. By counting degrees of freedom, we know that these plane wave solutions are a complete set.

Phonons

• Quantum Mechanics involves 2 quantizations: quantizing the available states (with different energies), and quantizing the number of things in those states (quanta).
• In this case, the energy in each vibrational mode is quantized in the usual harmonic oscillator way. The n_k is interpreted as the number of phonons in that mode with wave vector k.
• Since phonons are bosons with non-conserved number, their population depends on temperature according to the Bose distribution.
• Phonons with a specific wave vector k are "infinite" in size (that is they fill whole sample).  But we can make wave packets, which can travel through sample.
• difference between phase velocity and group velocity

Concepts to know:

• Interpretation of the Bose occupation factor as a particle number