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With a plane wave, I always took the direction of the wavevector, $k$, as the direction of propogation (magnitude proportional to the inverse wavelength). Alternatively, it could represent the momentum (minus a factor $\hbar$) of a particle.

However inside a crystal, the electron wavevector and the electron velocity are not necessarily in the same direction. I'm thinking here of a 2D material with a cylindrical Fermi surface where the momentum may have a z component, but the Fermi velocity does not. In everyday cases you would expect momentum and velocity to be in the same direction, moreover I considered the propogation of the wave to be in the same direction as its particle analogue.

I realise that inside a crystal the electrons are no longer simple plane waves, but what then does the $k$ vector mean?

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Some more reading into Fermi-liquid theory tells me that the quasiparticles have their momentum enhanced, so it appears that maybe the Fermi velocity is the velocity of the electron, whereas the $k$ vector corresponds to the momentum of the quasiparticle. However if this is the case, it still leaves the question as to what the physical meaning of the quasiparticle $k$ values are and why we are so interested in them (instead of the electrons). –  Brendan Nov 30 '11 at 11:55
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3 Answers

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k points in the direction of phase velocity (i.e., normal to the surfaces of constant phase, also called wavefronts). The electron moves in the direction of group velocity (i.e., a localized electron would be made of a wavepacket, and the electron moves as the wavepacket center moves). (Formula for electron velocity is the same as the usual group velocity formula: $\nabla_k \omega(k) = \nabla_k E(k)/\hbar$.)

Your question can be rephrased: "Why is the phase velocity not parallel to the group velocity?" Or more generally: "Why is the phase velocity different from the group velocity?" The answer to the latter question (both mathematically and intuitively) can be found in pretty much any introductory physics book that defines and discusses the concept of group velocity.

In one dimension, of course the group velocity and phase velocity have to be parallel, but in 2D or 3D, whenever waves are propagating with a non-spherically-symmetric dispersion relation, the group velocity and phase velocity will usually point in different directions. So this also happens with light waves in crystals, with sound waves in crystals, with sound waves in sedimentary rock, etc.

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This makes a lot of sense, however it isn't necessarily a quasiparticle concept is it? I've written what I think it is at this address (basically Bloch wavefunctions cause the modulation) - would you say that is right? brendan.sdf-eu.org/phy/… –  Brendan Dec 11 '11 at 22:19
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Momentum conservation occurs because of spatial translation invariance. Similarly in quantum mechanics, we have quantum numbers that correspond to eigenvalues of operators that commute with the hamiltonian. For a crystal, our hamiltonian only has discrete translation invariance (everything looks the same only if you move the entire system by a Bravais lattice vector). This discrete translation operator is well defined, and its eigenvalues do not correspond to the elementary excitations of a Fermi liquid, but rather the basis of single-particle states in Fourier space.

(Which is not to say that the same physics doesn't exist when you go to a Fermi liquid, just that it's entirely beside the point. "Crystal momentum" has only to do with tracking the translation symmetry of the environment that a wave is trying to propagate in.)

As for what that means for electron motion, Steve B gives a good answer

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It seems that you already understand, that the wave vector of an "electron" quasiparticle in a crystal (or in any periodic potential) is not the same thing as the actual momentum of "the real" electron. So now you are kinda questioning the "meaning" of the concept of a quasiparticle itself. And I hope that you already know that "meanings" of many things in quantum mechanics are quite elusive.

Anyway, I'll try to say something that hopefully will make sense. First of all -- the wave vector $k$ is called "a crystal momentum" or "quasimomentum" of your quasiparticle. The basic meaning for that quantity is the "number" of energy level in the spectrum of elementary excitations of your system. You can think of it as a (continuous) index $k$ that numbers your energy level $E_k$. Actually you can look at "ordinary" momentum from the same point of view -- just an "index" denoting your state in continuous spectrum.

Second -- many problems in quantum mechanics are formulated in terms of scattering problems: you prepare an initial state and you have to calculate probability of some final state. So, we prepare bunch of quasiparticles in an initial state $i =(E_{k_1},E_{k_2},...,E_{k_n})$ and look at a transition to a final state with other set of quasiparticles $f =(E_{q_1},E_{q_2},...,E_{q_m})$. It turns out that in many cases it is a very good approximation, to state that transitions occur only when quasimomentum is conserved: $$P(i\to f) \sim \delta(k_1+k_2+...+k_n - q_1-q_2-...-q_m) $$ The statement is absolutely correct for "ordinary" particles -- it reflects the usual momentum conservation. And for the quasiparticles the statement is only an approximation. So we have (approximate) quasimomentum conservation.

Finally -- often the dependence of energy of your quasiparticle on the quasimomentum is quadratic or close to quadratic: $E_k \simeq Ak^2$. That allows you to rewrite it in a familiar form: $E_k \simeq \frac{\hbar^2k^2}{2m^*}$ -- so it looks like an energy of a "ordinary" free particle with a mass $m^*$. And that mass is called an "effective mass".

All this similarities allows us to talk about quasiparticles as if they are free particles with some masses, which freely fly around inside your solid body, sometimes colliding with each other. This picture turns out to be very useful.

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