umklapp processes

[ehrenfest]

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 http://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

[olgranpappy]

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?

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 http://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

[ehrenfest]

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]

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 \vec p by absorbing a phonon of energy \omega and wave-vector \vec q, then I have for conservation of energy and (non) conservation of quasi-momentum

E_{\rm final}=E(p)+\omega(q)

and

\vec p_{\rm final}=\vec p + \vec q + \vec Q

where Q is a vector of the reciprocal lattice.

[genneth]

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".