Assignment #5

1. As an exercise to interpret phonon spectra, run the PHBAND program, discussed in Assignment #4, to answer the following questions about phonon spectrum of a face-centered cubic lattice. Assume that the conventional unit cell of the fcc lattice is aligned with the axes of the Cartesian coordinate system (check that this is indeed the case in the template file fcc.phin).

(a) (2 pt.) In which space direction is the sound velocity lowest?

(b) (2 pt.) As which k-point in the Brillouin zone can we find the highest phonon frequency?

(c) (2 pt.) By analyzing the eigenvectors of the dynamical matrix D_mn (k) (use option IEXIT=0 in the file fcc.phin), characterize this phonon mode using standard notation (acoustic or optical, longitudinal or transverse).

(d) (2 pt.) By doing an explicit calculation for k along the GX direction, show that the phonon spectrum/dispersion relation w(k) repeats itself beyond the boundary of the first Brillouin zone, i.e. w(k+K)=w(k).

2. Answer the following questions related to the Sommerfeld model of the electron gas.

(a) (4 pt.) Taking the temperature derivative of the total energy of the homogeneous electron gas in the Sommerfeld model

U= ó õ ¥ 0  dE E D(E) f(E,T) ,

derive the expression for the specific heat

cel= dU dT = ó õ ¥ 0  dE (E-EF)  df(E,T) dT  D(E) .

Here, D(E) is the electronic density of states, f(E,T) is the Fermi-Dirac function, and E_F is the Fermi level.

(b) (3 pt.) Show that c_el=gT as the temperature T approaches 0, and determine g.