Problem: Quasi-One-Dimensional Metal

In a quasi-one-dimensional metal, the overlap of the partially occupied electronic orbits is large along one direction, but much smaller in the other two directions. Assume a band that is described by

 

where , , and are the components of the electron momentum and . How will the constant energy cuts look on the plane? Show that for a nearly half-filled band the DOS (and therefore the thermodynamic properties like total energy, total number of electrons, and specific heat) of this system are similar to a truly one-dimensional electron gas! (Note: Materials like TaS and NbSe can be viewed this way.)