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Math Help - Invert expression to find the Chemical potential of an Ideal Bose Gas

  1. #1
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    May 2008
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    Invert expression to find the Chemical potential of an Ideal Bose Gas

    Hi I'm having trouble with this step of the derivation of the chemical potential for an ideal fermi gas in the low temperature limit

    I've got as far as to find
    $<br />
E_F = \mu \left(1+\frac{\pi^2}{12} \left(\frac{kT}{\mu}\right)^2 + ...\right)<br />

    and need to invert this expression to get the chemical potential as a function of fermi energy

    I've tried writing this as (which is ok as this is the same as above to first order)
    $<br />
E_F = \frac{\mu}{1-\frac{\pi^2}{12} \left(\frac{kT}{\mu}\right)^2}<br />

    but I cant seem to get it into a form which is easily invertable any chance of a hand here please?

    NB the answer I'm trying to get to is
    $<br />
\mu = E_F \left(1-\frac{\pi^2}{12} \left(\frac{kT}{E_F}\right)^2 + ...\right)<br />
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  2. #2
    Senior Member
    Joined
    Mar 2010
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    280
    <br />
\displaystyle<br />
E=\mu(1+\frac{a}{\mu^2})<br />

    <br />
\displaystyle<br />
\mu^2-E\mu+a=0<br />

    <br />
\displaystyle<br />
\mu=\frac{E \pm \sqrt{E^2-4a}}{2}=\frac{E \pm E \sqrt{1-4a/E^2}}{2}=<br />

    <br />
\displaystyle<br />
=\frac{E \pm E (1-2a/E^2)}{2}=E(1-a/E^2)<br />
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  3. #3
    Member
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    May 2008
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    ok that makes me feel stupid for not spotting it

    Cheers!
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