Solid State physics, studying for a re-exam

[Yedi]

I have gone trough all the material for a solid state physics course I had earlier this exam, and there are three question I actually can't solve, even with all the lecture notes. Here they are:

1. An electron of mass m moves in a square lattice, spacing a. The nearly free approximation applies.
a) With one electron per site in the crystal, draw the Fermi surface on the kx-ky plane. Is this a metal or an insulator?
b) With two electrons per site, draw the Fermi surface. Is this a metal or insulator?

Comment: This is a theory question, which I am not that good at. So I haven't thought about a solution to this one, but I will try to check the course literature once more and ask about it here.

2. The Debye temperature Od pf diamond is Od=1860K, mass density is 3.5g/cm³. Using these data to:
a) Estimate the velocety of sound in diamond.
b) Find the specific heat capacity at temperature T=100K

Comment: I solved the b part of the problem by simply using the Debye T³ law (which you can use when T<<Od) but I have searched for a formula of the a) part and haven't found one

3. Considera cylindric wire of radius R= 1mm made oif a superconductor with the London penetration depth L = 100 nm. The wire carries electric current I =100 Amp. Find the distribution of the current density j(p) and the magnetic induction B(p) in the wire. Calculate the j and the B at (a) 50nm and /b) 500 nm beneath the wire surface.
Hint the current density j is parallel to the wire axis, and B has the only azimuthal component. Use the London and Maxwell equation to find the induction and the current density since R>>L, the curvature of the cylinder surface can be neglected.

Comment: none

 

[mark726]

Thank you for sharing your questions with us. As a scientist in the field of solid state physics, I would be happy to provide some guidance and potential solutions to your questions.

1. For the first question, the nearly free approximation assumes that the periodic potential of the lattice is weak enough that the electron's wavefunction can still be approximated as a plane wave. With one electron per site, the Fermi surface would be a square with corners at (±π/a,±π/a). This would result in a metal, as there are still available states at the Fermi level for electrons to occupy. With two electrons per site, the Fermi surface would be a smaller square with corners at (±π/2a,±π/2a). This would result in an insulator, as all available states at the Fermi level are filled.

2. For the first part, you can use the formula for the speed of sound in a solid: v = √(E/ρ), where E is the Young's modulus and ρ is the mass density. For diamond, E = 1.22 x 10^12 N/m^2 (from the literature) and ρ = 3500 kg/m^3. Plugging these values in, you can estimate the velocity of sound in diamond. For the second part, you can use the Debye specific heat formula: C = (12π^4/5)R(3/4)(T/θD)^3, where R is the gas constant and θD is the Debye temperature. Plugging in the given values and T = 100K, you can find the specific heat capacity.

3. For this problem, you can use the London and Maxwell equations to find the current density and magnetic induction. Since R>>L, we can neglect the curvature of the cylinder surface and assume a uniform current density along the wire axis. At a distance p from the center of the wire, the current density can be calculated using the London equation: j(p) = I/(πR^2)e^(-p/L). Similarly, the magnetic induction can be calculated using the Maxwell equation: B(p) = μ0j(p)/2, where μ0 is the permeability of free space. Plugging in the given values, you can find the current density and magnetic induction at different distances from the wire surface.

I hope this helps you in