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Bose-Einstein Condensation & de Broglie wavelength

  1. Apr 11, 2012 #1
    1. The problem statement, all variables and given/known data
    Bose-Einstein condensation of a fluid occurs when the de Broglie wavelength of a "typical" particle becomes greater than the average nearest-neighbor distance. One can interpret the momentum in the de Broglie equation as
    [itex]p=\sqrt{<p^{2}>}[/itex]
    where [itex] <p^{2}>[/itex] means the thermal average for a single particle

    (a) Write down the de Broglie equation and use the above to obtain an approximate expression for the Bose-Einstein condensation temperature [itex]T_{B}[/itex] in terms of the particle density (N/V), the particle mass m, Boltzmann's constant k and Planck's constant h

    (b)For temperatures T less than the transition temperature [itex]T_{B}[/itex], the fraction, f, of particles in the ground state (f=N(0)/N) is a number between zero and one. Find the chemical potential, [itex]\mu[/itex] to order 1/N as a function of T when [itex] T <<T_{B} [/itex]. Assume the ground state energy is zero.

    2. Relevant equations
    de Broglie momentum: [itex] p= \frac{h}{\lambda} [/itex]
    Boltzmann distribution function: [itex] <n_{s}> = \frac{1}{e^{(\mu - \epsilon)\beta} - 1} [/itex]
    where [itex]\beta = \frac{1}{kT}[/itex]
    Energy: [itex] E = \frac{p^{2}}{2m} [/itex]
    Expectation value: [itex] <A> = \Sigma_{s} A <n_{s}> [/itex]


    3. The attempt at a solution

    I interpreted the average nearest-neighbor distance to be the ratio of the number of particles, N, and the volume, V, to the 1/3 power, [itex] \lambda > \frac{N}{V^{1/3}} [/itex].
    Put this into our equation for momentum:
    [itex] p < \frac{h V^{1/3}}{N} [/itex]
    But we have been told [itex]p=\sqrt{<p^{2}>}[/itex], thus [itex] <p^{2}> <(\frac{h V^{1/3}}{N})^{2} [/itex].
    Now we can use the energy-momentum relationship to find [itex] <E> < \frac{1}{2m}(\frac{h V^{1/3}}{N})^{2} [/itex].
    So the question comes down to how to approximate the expected energy for a single particle in state with possible states s. Normally, I would use the expected value function above and go through the approximation using density of states and find a function for the chemical potential and then finally solve for temperature. I just can't kick the feeling that there is an easier way. This was taken from a previous comprehensive exam, so I should be able to do it without notes in about an hour (or at least most of the problem). Is there a trick I'm missing?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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