在最绝望的等offer的时间最绝望的复习。

momentum p = h_bar * k

Drude model

DC conductivity:
conductivity σ and resistivity ρ defined by:

E = ρj, j = σE

j - Current density

The electric current per unit cross-sectional area

AC conductivity:

dp/dt = -p/τ  - eE_0 e^(-iωt)

σ = σ_0 / 1 - iωt

Screening:

∇^2(ϕ) = -4pi ρ(r), where ρ is the charge density
ϕ(r) = Q/r e ^(−r/λ_TF)
λ_TF ~ 1/ sqrt(D(E_F)), where D(E_F) is the density of states at the Fermi energy of a 3D electron gas with no ion

large n → smaller λTF → metal
small n → larger λTF → insulator

States

Let N(E) denotes the total number of states, then the density of states (per spin) is:

 - D(E) = dN / dE
 - N(E) = integral_dE (D(E))

Total number of states in k space:

 - N(k) = (pi * k_F^2) / (D(k))

Whereas when N denotes the total number of fermions (electrons, neutrons):

 - N = integral_dE (n_F * D(E))

where n_F is the Fermi–Dirac distribution: probability that a state of energy E is occupied by a fermion

When n denotes the density of electrons, and N denotes the total number of electrons:

D(k) (4/3 pi k_f^2) = N/2 
 - n = N / (L^d), d is the dimension

For X-ray: f_j (structure form) ~ atomic number Z_j

Equipartition theorem: U = 3Nk_bT

Energy representation to remember

 - E_electrons = h_bar^2 * k^2 / 2m

 - E_phonons = h_Barω(n_B + 1/2)

   - n_B is the Bose distribution
 - D(E) = L^d / (2pi)^d [1/ gradient(E(k))] integral_dS

Where integral_dS is the total length/ area / volume, corresponding to k, 2pik, 4pik^2

Debye model conclusion:

if phonon dispersion ~ k^m
d denotes the dimension

 - D(E) ~ k^(d-1) / k^(m-1)
 - Cv ~ T^(d/m)

Questions to review

Double the unit cell, what happens to the diffraction pattern?