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Periodic Boundary Conditions proof

  1. May 28, 2015 #1
    Hi! When we model bloch-waves in a solid we assume that there exist some kind of periodic boundary conditions such that the wave function is periodic. In 1D, ##\psi(x)## repeats itself for every ##L##, ##\psi(x) = \psi(x+L)##, such as here: ?temp_hash=0dc795db7139072038dac60b394bf8b6.png

    OK, fine, we get pretty wave solutions if we assume the existence of the PBC. But what ##L##? As far as I know the only repeating unit in a crystal is the Wieger Seitz cell, which is sized on the atomic scale.

    Is ##L## just the wavelength?
     

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  2. jcsd
  3. May 28, 2015 #2

    DrDu

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    No, L is the size of your crystal, so e.g. 1cm. In the end, this size doesn't matter to much and you often take ##L \to \infty##.
     
  4. May 28, 2015 #3
    How can that be? if that is true, it means ##\psi## is a standing wave... But bloch waves are traveling!
     
  5. May 28, 2015 #4

    DrDu

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    That's why you use periodic boundary conditions, the waves then move on a circle and you can have left and right moving waves.
     
  6. May 28, 2015 #5
    this makes no sense physically. if you have a crystal cube of 1cm, you are assuming that its left side is connected to its right?
     
  7. May 28, 2015 #6

    DrDu

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    No, it doesn't, but there is a theorem by Wigner that the influence of the boundary conditions on the states vanishes like 1/N, where N is the number of particles (or elementary cells). So if you are interested in the bulk states only, you can pick the boundary conditions as seem convenient.
     
  8. May 28, 2015 #7
    Ah, OK, I see. So you can pick whatever PBC you like? What about if L = length of unit cell?
     
  9. May 28, 2015 #8

    DrDu

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    No, I said that the boundary conditions become unimportant when L is much larger than the elementary cell.
     
  10. May 28, 2015 #9
    Yes you did, my apologies. And thanks for the help!
     
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