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## Homework Statement

Consider an electron gas of density [itex]n_{0}[/itex] in three dimensions that is completely ferromagnetic: all electron spins point in the same direction. Derive:

a) The Fermi wave vector in terms of [itex]n_{0}[/itex].

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

[tex]n_{0}=\frac{k^{3}}{3\pi^{2}}[/tex]

[tex]k=(3 \pi^{2} n_{0})^{1/3}[/tex]

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

[tex]1 = \frac{4 \pi n_{0}}{3} (a_{0} r)^{3}[/tex]

[tex]r = \frac{6^{1/3}}{2 a_{0} (n \pi)^{1/3}}[/tex]

c) Same general tack but not sure how to account for the spins;

[tex]E = \int EdEN(E) = \frac{\mu}{5 \pi^{2}} (\frac{2 m \mu}{\hbar^{2}})^{3/2}= \frac{3}{5} n_{0} \mu c[/tex]