1. Calculate the energy at which the maximum density of occupied states occurs for free electrons in a metal with Fermi energy of 4 eV at 300K.

2. Suppose that the energy band for a particular state in a 2-dimensional square lattice with lattice parameter a could be written as :

E(k) = E0 – E’ cos kxa cos kya

(a) Sketch the dependence of E on k from k = 0 to the edge of the Brillouin zone in the (10) and (11) directions]
(b) If this E(k) corresponds to a conduction band, sketch the equal energy surfaces correspond to the first electrons to appear in the band minima of (a).
(c) If this E(k) corresponds to a valence band, sketch the equal energy surfaces corresponding to the first holes to appear in the band maxima of (a)
 

3. prob. 3.4 below

4  Consider a one-dimensional crystal with a total of six atoms, each with mass m, separated by a lattice constant of 0.5 nm, with the two end atoms fixed.
(a) How many allowed modes are there?
(b) What is the ratio of the maximum allowed wavelength to the minimum allowed wavelength?
(c) Calculate the relative displacements of each atom for a vibration with wavelength of 5/3 nm (i.e. 1.6666… nm).