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Electrons in empty space

  1. Mar 27, 2008 #1
    Hi,

    I understand that electrons in a solid (eg. metal), can be treated as Fermi gas obeying Fermi-Dirac statistics, which incorporates the exclusion principle. This differs from a normal gas because a regular (ideal) gas of atoms or molecules can have atoms occuppying exactly the same energy states, whereas with a Fermi gas this is not allowed.....

    OK, that's my basic understanding, now here's my question;

    If the electrons were not in a solid, e.g. in a vacuum (plasma etc.) would they obey Fermi Dirac stats? Or would a regular Maxwell-Bolztmann treatment suffice?

    In other words, when electrons are no longer associated in a common lattice, but in a free gas in vacuo, do they obey the exlusion principle, and hence have energy bands etc?

    Thanks guys

    Gareth
     
  2. jcsd
  3. Mar 27, 2008 #2

    Astronuc

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    In a plasma, the 'free' electron velocity distribution is generally described by a Maxwell-Boltzmann distribution. Of course, there may be recombination and ionization occuring in the plasma.
     
  4. Mar 27, 2008 #3
    Thanks Astronuc,

    So fundementally speaking, the exclusion principle applies for electrons bound by atoms or a solid (periodic potentials in the case of a solid) and hence you get spectral lines (atoms) or bands (solids).

    So, when these electrons are no longer asscociated with a potential and no longer have orbitals, but are associated via electron electron collisions i.e. in a free electron gas, they act as regular particles and two electrons can possess exactly the same quantum numbers.

    And hence can be treated with regular kinetic equations (Mawell-Boltzmann).

    Does this sound ok?
     
  5. Mar 27, 2008 #4

    ZapperZ

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    Actually, if you look at the Drude model, it DOES treats the electron gas in metals as almost a classical gas, without any FD statistics restriction.

    For free electrons in vacuum and in a Drude model, we treat them as if they are classical MB gas because there is no substantial overlap in their wavefunctions. That's why in accelerator physics, we model the charge particle dynamics via classical E&M. They are far enough away from each other that quantum effects are not perceivable.

    Zz.
     
    Last edited: Mar 27, 2008
  6. Mar 27, 2008 #5

    Astronuc

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    I think that's the key - and is certainly the case in a plasma/gas.
     
  7. Mar 27, 2008 #6
    Thanks for the replies,

    My understanding was that the Drude model was a relatively incomplete representation of conduction electrons in a metal, and that Fermi-Dirac statistics overcame certain shortcomings of the theory.

    But I'm still having a little trouble coming to terms with some aspects of this.

    Here are my thoughts;

    We have electrons which fill certain energy states when they are in orbit around an atom.

    These energy states are not a continuum but are quantised or discrete. Certain energy states can be occupied and certain ones can not. The reason some cannot be occuppied is due to the fixed number of states available in each orbital (why this is I still don't know, it's because of the exclusion principle, but I don't understand the reason).

    OK, so the electrons in orbit around an atom can't have the same wavefunctions, agreed.

    But is this because of their proximity to each other, or has it got to with their orbital/shell etc.

    So if we had a gas of electrons in very close proximity, (like in a metal), but they were a real gas and not part of any regular lattice (free from ionic cores) and not in the solid state, would quantum effects come into play because of their proximity?

    (sorry for the barrage of questions!)
     
  8. Mar 27, 2008 #7

    Astronuc

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    Not at all. Great questions! That's the way to learn.

    See these -
    http://www.pha.jhu.edu/~jeffwass/2ndYrSem/slide6.html
    http://www.pha.jhu.edu/~jeffwass/2ndYrSem/slide7.html

     
  9. Mar 27, 2008 #8

    ZapperZ

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    To suppliment what Astronuc has given in the link, you will note that Ohm's Law can be derived directly from the simple Drude model. So even with such "incomplete representation", it produced something that is widely used in electrical engineering.

    Zz.
     
  10. Mar 27, 2008 #9
    Thanks for the replies guys.

    Ok, so the Drude model is a pretty good approximation, especially for it's time.

    This brings me to my main challenge:

    Understanding under what circumstances the exclusion principle comes into play.

    If we have a gas of electrons (an actual gas, not a solid), equal in density to that of a Drude electron gas in a metal, will these particles be subject to Pauli's principle?

    So to get down to the most basic form of the question, whats determines the fact that no two electrons can have the same quantum numbers?

    Is it the fact that they are bound to an ion?

    Or is it the fact that they are in such close proximity?

    (obviously, if two electrons are bound by the same ion they will be in close proximity anyway, but try to give me some ideas here please)

    Thanks again!
     
  11. Mar 27, 2008 #10

    f95toli

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    The exclusion principle is ALWAYWS valid. In a gas it is usually unimportant because the probability of two electrons ending up in the same state is so small that the effects can be neglected.
    In e.g. an atom we can "label" the available states using simple quantum numbers and since there are relatively few states the exclusion principle is very important. But this is obviously not the case for a gas where two electrons with e.g. different momentum also are in different states.
    Note that the same is obviously true for electrons bound to atoms, there is nothing preventing two electrons that belong to two different atoms to be in states with the same atomic quantum numbers. The fact that the electrons are spatially separated also means that they are not in the same quantum state "relative" to the whole universe.
     
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