So in our model that predicts the various Landau energy levels for a crystal in a uniform magnetic field, so far no consideration of electron spin has been introduced. As it turns out, there are two ways in which spin affects the system. According to the book, the major effect is that the Landau levels will be either raised or lowered a uniform amount, corresponding to
g*e*hbar*H/(4*m*c)
Where, Ashcroft and Mermin have once again reverted to using the H field rather than the B field. Whether the levels are raised or lowered depends on the orientation of the spin to field; raised for along the field, and lowered for opposite it.
The density of states is altered as a result, and if g0(energy) is the density of states when spin is ignored, the new expression is
g(energy) = 0.5*g0(energy + g*e*hbar*H/(4*m*c)) + 0.5*g0(energy - g*e*hbar*H/(4*m*c))
I know when I first read this in the book, I was a bit confused as it seemed that the two factors would cancel out. That was until I realised that they were functions of g0, not values inside a bracket multiplied by g0. Just found it a bit ambiguous at first when I read it.
The other consideration not taken care of in the model so far is that spin-orbit coupling hasn't been introduced. Whilst it maintains that any effect is small in lighter elements (it doesn't give a rigorous definition of either of those unfortunately) it has the effect of splitting further energy levels, reducing the degeneracy, and ultimately altering the density of states as before. From what I can gather, this can be ignored up until we deal with Diamagnetism and Paramagnetism, whereupon it is reintroduced, and added to the Hamiltonian in order to explain atomic susceptibilities.
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Thanks for the post!
ReplyDeleteLooking at your equation (p275 A&M), one would first suspect that there would be cancellation, but that would assume that the function g0 is linear.
By there not having any cancelling would mean that our density level function (g0) is not lineear, so we don't have evenly-spaced levels.
Something I find interesting is our consideration of angular momentum - we break it into spin and orbital angular momentum...it's just been popping up a lot this semester...
Sidenote: Perhaps a matter of convenience, but is there a convention for when to use the B or H field?
I suppose natural units ignore this problem
In terms of the B or H field, it just depends on whether magnetization is involved. Later on chapters seem to be about this, so I'm sure we'll have to be a little bit careful with them. But at this stage, I'd say the simple Bloch Model doesn't consider it, so it's just a proportionality thing and doesn't really matter. I know Ross usually uses B, so that's what I'm sticking with!
ReplyDeleteAnd trying to remember back to stat mech last year, but the density of states depended on what dimensions one was considering. So in 1D, it was proportional to sqrt(E), in 2D it was proportional to E, and in 3D it was proportional to E^3/2. So I have a feeling in 2D, the spins would just become degenerate and wouldn't affect it, but in 3D the linearity is lost and hence nondegeneracy.