Umklapp Process: K_1+K_2=K_3+G Explained

[ehrenfest]

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.

Homework Equations

The Attempt at a Solution

 

[olgranpappy]

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 existence 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.

Homework Equations

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 existence 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".