Fermi Vector in Ferromagnetic Material

[atomicpedals]

Homework Statement

Consider an electron gas of density $n_{0}$ in three dimensions that is completely ferromagnetic: all electron spins point in the same direction. Derive:
a) The Fermi wave vector in terms of $n_{0}$.
b) The parameter r as the radius in atomic unites that encloses one unit of charge.
c) The average kinetic energy per electron.

2. The attempt at a solution
a) My attempt is to simply solve as follows
$$n_{0}=\frac{k^{3}}{3\pi^{2}}$$
$$k=(3 \pi^{2} n_{0})^{1/3}$$
however I don't think this accounts for the fact that all spins point in the same direction. How do I account for this? Not being certain on this point of course follows through to parts b and c.

b) I'm tempted to simply solve for r
$$1 = \frac{4 \pi n_{0}}{3} (a_{0} r)^{3}$$
$$r = \frac{6^{1/3}}{2 a_{0} (n \pi)^{1/3}}$$

c) Same general tack but not sure how to account for the spins;
$$E = \int EdEN(E) = \frac{\mu}{5 \pi^{2}} (\frac{2 m \mu}{\hbar^{2}})^{3/2}= \frac{3}{5} n_{0} \mu c$$

 

[_coyote_]

^{2}where c is the speed of light. I think I'm missing something in parts b and c, but not sure what. Any advice would be much appreciated! Thanks in advance.