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Homework Help: Energy per particle of polarized electron gas

  1. Sep 26, 2010 #1
    1. The problem statement, all variables and given/known data

    Calculate the total internal energy per electron at zero temperature of a free noninteracting gas of electrons of density [itex]n[/itex], in the following two cases.

    a) First assume that states with both spin directions are populated equally.

    b) Now assume that the gas is fully polarized: only states corresponding to one spin direction, say "up", are populated.

    3. The attempt at a solution

    Know internal energy per particle is given by,

    [tex]\frac{U}{N} = \frac{1}{N} \sum_{k\sigma} \epsilon_k n_F(\epsilon_k)[/tex]

    where [itex]\sigma[/itex] is summing spin states giving me a factor of one or two.

    Eventually I get to,

    [tex]\frac{U}{N} = \frac{1}{n} \int_{0}^{\inf} g(\epsilon})\epsilon n_F(\epsilon) d\epsilon[/tex]

    where [itex]\sigma[/itex] is wrapped into the density of states [itex]g(\epsilon)[/itex] and [itex]n_F(\epsilon)[/itex] is the Fermi-Dirac distribution.

    So is it correct that the last difference between (a) and (b) (that isn't carried by [itex]\sigma[/itex]) is the chemical potential [itex]\mu = \epsilon_F[/itex] in [itex]n_F(\epsilon)[/itex], where the polarized spins will have a Fermi radius twice as large as the non-polarized equally populated spins?

    Trying to understand setting up the calculation right now more than performing it...
    Last edited: Sep 26, 2010
  2. jcsd
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