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Umklapp processes

  1. Mar 27, 2008 #1
    1. The problem statement, all variables and given/known data
    Can someone please explain to me how umklapp processes K_1 +K_2 = K_3 +G where G is nonzero conserve momentum? I have read the explanation in Kittel over and over and I just don't understand. I also read https://www.physicsforums.com/showthread.php?t=165385 so don't just give me a link to that.


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 27, 2008 #2

    olgranpappy

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    Homework Helper

    your K's are not momentum--they are quasi-momentum. They enter into the theory via bloch's theorem which presumes the existance of an external periodic potential. because of this potential, eigenstates of the hamiltonian (labelled by their quasi-momentum) are not eigenstates of the momentum operator.

    Quasi-momentum is not conserved.

    Similarly, *in the presence of an external potential* true momentum is not conserved either.



     
  4. Mar 27, 2008 #3
    OK. So, is there a conservation of momentum equation associated with a given umklapp collision that we can write down or is that not part of the theory?
     
  5. Mar 27, 2008 #4

    olgranpappy

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    the relevant equation is the one you wrote down where quasi-momentum is not-conservered, but is "conserved modulo a reciproal lattice vector". So, for example, if I scatter a particle of energy E and (quasi) momentum [itex]\vec p[/itex] by absorbing a phonon of energy [itex]\omega[/itex] and wave-vector [itex]\vec q[/itex], then I have for conservation of energy and (non) conservation of quasi-momentum
    [tex]
    E_{\rm final}=E(p)+\omega(q)
    [/tex]
    and
    [tex]
    \vec p_{\rm final}=\vec p + \vec q + \vec Q
    [/tex]
    where Q is a vector of the reciprocal lattice.
     
  6. Mar 28, 2008 #5
    Perhaps another way to look at it is that conservation of momentum is a result of translation invariance -- application of Noether's theorem. So if I have a non-uniform potential through space I should not expect momentum to be conserved. Here, we have the slightly perculiar feature that spatial translation is invariant if you do it by a lattice vector. So we have a variable k which is "conserved up to a reciprocal lattice vector".
     
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