Find the density of energy state and for what ab ratio of

Question

(1) Consider a 2D crystal of monovalent atoms with a simple rectangular lattice spacing in the x-direction and b = 1.25a in the y-direction.

The size of the crystal is L x L (in units of distance).

(a) Do the allowed values for k_x and k_y differ from a simple square crystal of the same size?
(b) Sketch the shape of the first Brillouin zone and indicate its dimensions.
(c) What is the size of the free electron Fermi "circle"? Show it on your sketch for (b).
(d) For what a/b ratio (of the rectmygulm lattice spacing) would the Fermi circle just touch the edge of the firer Brillouin zone?
(e) For the a/I ratio you found in (d), suppose that the atoms are now divalent (each contributes two electrons). Again, sketch the free electron Fermi circle.
(f) How would this free electron Fermi surface you drew M (d) be modified by the existence of periodic potential? Sketch two new Fermi contours, one corresponding to the presence of a weak potential, and the other corresponding to a strong potential.
(g) For the divalent case that you showed in (e), sketch both the free electron Fenui circle and the Fermi contour as modified by the existence of a weak periodic potential, and a stronger periodic

2. Liquid 'He can be considered as a gas of free Fermium with mass density p = 0.081 gm/cm³. Calculate the effective Fermi temperature, TF (in Kelvin) (MCI the free-Fermion linear coefficient of specific he. (7, in erg/K,cm³), where C_v = γT.

3) Find the density of energy state, with respect to 1, for an ideal ....interacting Fermi gas in one and two dimensions.

4. For a two-dimensional free electron gas,

(a) show that g(E) = m/Πh² , E > 0

(b) Use the result from (a) to show that

μ =  kBT ln[exp(Πnh²/mk_BT) - 1]