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Fermi energy condensed matter exam problem

  1. Aug 1, 2014 #1

    SMC

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    1. The problem statement, all variables and given/known data

    Hello, I am preparing a condensed matter exam and I was wondering if I could get some help on the following question from a past exam paper:

    Show that for the free electron gas at zero temperature the Fermi energy is given by:

    [tex]ε_{F}=\frac{\hbar^{2}}{2m}(3π^{2}N)^{2/3}[/tex]

    and the fermi wave number by:

    [tex]k_{F}=(3π^{2}N)^{1/3}[/tex]

    where N is the volume density of electrons.

    2. Relevant equations

    the previous question was to derive the density of states:

    [tex]D(ε) = \frac{V}{2π^{2}}(\frac{2m}{\hbar^{2}})^{3/2} ε^{1/2}[/tex]

    but I don't know if I have to use that or not


    3. The attempt at a solution

    how I would have done it is derive [itex]k_{F}[/itex] from [tex]N=\frac{VK_{F}^{3}}{3π^{2}}[/tex] which is obtained by saying number of electrons N is equal to volume of fermi sphere over k-space volume element [itex](2π/L)^{3}[/itex].

    and then state that the dispersion for a free electron gas is
    [tex]ε(k) = \frac{\hbar^{2}k^{2}}{2m}[/tex] to get fermi energy.

    but this method gives a result with the V present in the equation while the equations in the question don't have the V for some reason. Also I derived wavenumber first then energy while the question asks to derive energy first then wavenumber.
     
    Last edited: Aug 1, 2014
  2. jcsd
  3. Aug 1, 2014 #2
    SMC - I think the [itex]N[/itex] in the question is the number of particles per unit volume. Otherwise their equations aren't dimensionally correct.
     
  4. Aug 1, 2014 #3

    SMC

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    yes ok that makes sense I should of noticed that, thank you Oxvillan. but I still don't understand why it asks me to derive fermi energy before wavenumber. is there a way of deriving fermi energy without deriving wavenumber first?

    or maybe the questions are just in the wrong order
     
  5. Aug 1, 2014 #4
    Actually I would have done the wavenumber first too :smile:

    But you can also integrate the density of states in energy space from zero to whatever the Fermi energy is and set that equal to [itex]N[/itex].
     
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