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Umklapp processes

  • Thread starter ehrenfest
  • Start date
1,996
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1. Homework Statement
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. Homework Equations



3. The Attempt at a Solution
 

Answers and Replies

olgranpappy
Homework Helper
1,271
3
1. Homework Statement
Can someone please explain to me how umklapp processes K_1 +K_2 = K_3 +G where G is nonzero conserve momentum?
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.



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. Homework Equations



3. The Attempt at a Solution
 
1,996
1
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.
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?
 
olgranpappy
Homework Helper
1,271
3
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.
 
979
1
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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