Posted by Steve Simon on January 20, 2013, 9:26 am, in reply to "Pauli paramagnetism"
Hi Troubled,
Thanks for your question. I think I understand your question and there is indeed some subtlety in the energetics of magnetism -- and also I now see that the notes are misleading!
The key thing is to think carefully about thermodynamics (and indeed my book may need another footnote to explain this!). Recall that
dU = T dS + B dM
Lets assume we are at T=0, so that S=0 also. Now assume that there is a linear relation between M and B, given by
M = (chi/mu_0) B
So we have
dU = (chi/mu_0) B dB
If we integrate this we get an energy
U = (1/2) (chi/mu_0) B^2
note the 1/2 here. And we also have correctly that
M = dU/dB
However, note the total energy is of the form
U = (1/2) M B
Now, what was wrong in the notes: If we look at the "helmholtz" free energy (i.e., the energy where B is controled and M is the dependent variable)
dH = T dS - M dB
We can get between these free energies by the usual transform
H = U - B M
So that
H = -(1/2) M B = -(1/2) (chi/mu_0) B^2
And this is the correct type of (free) energy that is properly differentiated in Eq. 4.13 of the notes (so that we get the sign right). It is also the correct type of free energy differentiated in Eq. 19.8, and 19.14 later on (since B is held fixed). I am going to make a note of this on the known-error list!
Now, for a single up electron where B is the controlled, it is clear that
moment = -dH/dB = -mu_B
But for the entire system of many electrons, you might worry that you don't get
moment = -dH/dB
anymore since there could be a contribution from the kinetic energy. However, with a bit of thought (and looking at Fig 4.2) you can see that the total kinetic energy is unchanged with B, so it still holds that
moment = -mu_B (#up - #down)
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