Free Electron Model of Metals

Question 1

What is the equation for N(E) in the 3D box problem?

Answer 1

N(E) = Vk^2/3π^2 = V/3π^2 *(2mE/ћ^2)^3/2

Question 2

What is the equation for the density of states n(E) in the 3D box problem?

Answer 2

n(E) = dN/dE = V/2π^2 *(2m/ћ^2)^3/2 *E^1/2

Question 3

What do we get a picture of from n(E) and N(E) in the 3D problem?

Answer 3

A picture of a “Fermi Sphere” of radius k(F) filled up with gas of N electrons.

Question 4

How can we use the density of states to calculate the total energy in the 3D problem?

Answer 4

Etot = integral from 0 to E(F) of E*n(E) dE = V/5π^2 *(2m/ћ^2)^3/2 *E(F)^5/2

Question 5

How do we get the average energy per electron in the 3D problem?

Answer 5

Etot/N = 3/5 *E(F)

Question 6

What is the equation for the Fermi velocity?

Answer 6

v(F) = ћ*k(F)/m

Question 7

What is the equation for the fermi energy in terms of temperature?

Answer 7

E(F) = k(B)*T(F)

Question 8

For a gas of free electrons at T=0K, what does the graph of f(E) against E look like?

Answer 8

Straight line up to E(F) where it cuts down to 0, creating a square.

Question 9

What is the equation for the Fermi-Dirac function?

Answer 9

f(E, μ, T) = 1/(exp((E-μ)/k(B)T)+1)

Question 10

What does the graph f the Fermi-Dirac function look like for T2? How does it differ for T1

Answer 10

Starts off flat then curves own and tends to zero. FOr T1

Question 11

What is the equation for N including the fermi-dirac function?

Answer 11

N = integral from 0 to inf of f(E, μ, T)*n(E) dE

Question 12

What is the equation for the total energy in terms of the Fermi-dirac function?

Answer 12

Etot = integral from 0 to inf of f(E, μ, T)En(E) dE

Question 13

What is the total energy change proportional to?

Answer 13

Total energy change proportional to n(E(F))*k(B)T

Question 14

How is the specific heat Cv related to the temperature and energy?

Answer 14

Cv = dEtot/dT which is proportional to T.

Question 15

What is magnetic susceptibility?

Answer 15

If a material is placed in a magnetic field with spin up, it will be affected differently from those with spin down.

Question 16

What is the equation for up spin states for energy including μ(B)?

Answer 16

E = ћ^2 k^2/2m + Begћ/4m = ћ^2 k^2/2m + μ(B)B

Question 17

What is the equation for down spin states for energy including μ(B)?

Answer 17

E = ћ^2 k^2/2m - μ(B)B

Question 18

What do we find from these energies for up and down spin states?

Answer 18

That the total number of up electron states occupied is not equal to the total number of down spin states.

Question 19

What is the equation for the magnetisation M?

Answer 19

M = μ(B)(N(down)-N(up)) ~ μ(B)B*n(E(F))

Question 20

What is the equation for the susceptibility χ?

Answer 20

χ = dM/dB = μ(B)*n(E(F))

Question 21

For conduction, what analogy can we look at?

Answer 21

Conservation of mass flow in a fluid: flow through surface = rate of decrease of mass inside.

Question 22

What is the equation for the conservation of mass flow in a fluid (integrals)?

Answer 22

Closed double integral over S of j.dS = - triple integral over V of dρ/dt dV, so ∇.j = -dρ/dt

Question 23

Looking at the probability current in QM, what do we do to the Schrodinger equation and why?

Answer 23

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

Question 24

What do we do with the equation for d(Ψ* *Ψ)/dt? What happens in 2D and 3D?

Answer 24

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 ∇.

Question 25

How can we check the equation for j?

Answer 25

Check for a plane wave description Ψ = exp(ik.r), and sub this into the equation for j.

Question 26

Thinking of lattice vectors R(n1,n2,n3) = n1a+n2b+n3c, what does each unit cell go between?

Answer 26

-a2 <= x <= a/2

Question 27

Thinking of the reciprocal lattice G(h,k,l) = hA+kB+lC, what is the spacing between each unit cell?

Answer 27

2π/a, so Brillouin zone is -π/a <= k <= π/a