Free Electron Model of Metals Flashcards
What is the equation for N(E) in the 3D box problem?
N(E) = Vk^2/3π^2 = V/3π^2 *(2mE/ћ^2)^3/2
What is the equation for the density of states n(E) in the 3D box problem?
n(E) = dN/dE = V/2π^2 *(2m/ћ^2)^3/2 *E^1/2
What do we get a picture of from n(E) and N(E) in the 3D problem?
A picture of a “Fermi Sphere” of radius k(F) filled up with gas of N electrons.
How can we use the density of states to calculate the total energy in the 3D problem?
Etot = integral from 0 to E(F) of E*n(E) dE = V/5π^2 *(2m/ћ^2)^3/2 *E(F)^5/2
How do we get the average energy per electron in the 3D problem?
Etot/N = 3/5 *E(F)
What is the equation for the Fermi velocity?
v(F) = ћ*k(F)/m
What is the equation for the fermi energy in terms of temperature?
E(F) = k(B)*T(F)
For a gas of free electrons at T=0K, what does the graph of f(E) against E look like?
Straight line up to E(F) where it cuts down to 0, creating a square.
What is the equation for the Fermi-Dirac function?
f(E, μ, T) = 1/(exp((E-μ)/k(B)T)+1)
What does the graph f the Fermi-Dirac function look like for T2? How does it differ for T1
Starts off flat then curves own and tends to zero. FOr T1
What is the equation for N including the fermi-dirac function?
N = integral from 0 to inf of f(E, μ, T)*n(E) dE
What is the equation for the total energy in terms of the Fermi-dirac function?
Etot = integral from 0 to inf of f(E, μ, T)En(E) dE
What is the total energy change proportional to?
Total energy change proportional to n(E(F))*k(B)T
How is the specific heat Cv related to the temperature and energy?
Cv = dEtot/dT which is proportional to T.
What is magnetic susceptibility?
If a material is placed in a magnetic field with spin up, it will be affected differently from those with spin down.
What is the equation for up spin states for energy including μ(B)?
E = ћ^2 k^2/2m + Begћ/4m = ћ^2 k^2/2m + μ(B)B
What is the equation for down spin states for energy including μ(B)?
E = ћ^2 k^2/2m - μ(B)B
What do we find from these energies for up and down spin states?
That the total number of up electron states occupied is not equal to the total number of down spin states.
What is the equation for the magnetisation M?
M = μ(B)(N(down)-N(up)) ~ μ(B)B*n(E(F))
What is the equation for the susceptibility χ?
χ = dM/dB = μ(B)*n(E(F))
For conduction, what analogy can we look at?
Conservation of mass flow in a fluid: flow through surface = rate of decrease of mass inside.
What is the equation for the conservation of mass flow in a fluid (integrals)?
Closed double integral over S of j.dS = - triple integral over V of dρ/dt dV, so ∇.j = -dρ/dt
Looking at the probability current in QM, what do we do to the Schrodinger equation and why?
We want the density ρ = Ψ* Ψ, so multiply the Schrodinger equation through by Ψ, and we look at the complex conjugate of the Schrodinger equation separately and multiply through by Ψ. Then subtract these 2 equations from eachother and rearrange for d(Ψ* *Ψ)/dt
What do we do with the equation for d(Ψ* *Ψ)/dt? What happens in 2D and 3D?
Use ρ = Ψ* *Ψ, so this equation equals -∇.j, so we can find j from this. In 2D its just d/dx in bracket, but in 3D can replace with ∇.
How can we check the equation for j?
Check for a plane wave description Ψ = exp(ik.r), and sub this into the equation for j.
Thinking of lattice vectors R(n1,n2,n3) = n1a+n2b+n3c, what does each unit cell go between?
-a2 <= x <= a/2
Thinking of the reciprocal lattice G(h,k,l) = hA+kB+lC, what is the spacing between each unit cell?
2π/a, so Brillouin zone is -π/a <= k <= π/a
