1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted