Wednesday, January 28, 2015
Results\Progress from Class
In the above expression the first part of the sum is the energy assuming constant density of states and one energy band that starts from 0.
Considering the only first part we showed that differentiating with respect to \( N_+ \) we get \( N_+=N/2 \) for minimum energy. Therefore the equilibrium dictated equal distribution of spin up and spin down.
The next two terms in the sum above where derived by counting. The first term is for the up up or down down spin and the second for the up down spins. As you can see we followed Zack's instructions for the energy contribution of each configuration.
We did not have time to graph or to take the derivative of the expression to find the new equilibrium as the counting took some time but I think everyone is on the same page!
Rationalization of the expression above:
\(N_{\pm} \) C 2 yields the number of possible up-up or down-down pairs
\( N_+*(N-N_+) \) yields the number of up down spins.
A sanity check could be seeing whether the total pairs N C 2
N C 2= \(N_+ \) C 2 + \(N_- \) C 2 + \( N_+*(N-N_+) \)
*C means choose
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What is this the energy of? I suggested it was the energy of the crystal, but that's admittedly vague and nobody else seemed to like it either.
ReplyDeleteWell it is the total energy of the electrons in the highest (partially) filled band, which is sort of an important energy. The interesting thing is what it tells us about the state of the system.
DeleteMaybe you can do a change of variables using Nup-Ndown as one and Nup+Ndown as the other. (Instead of using Nup and Ndown).
ReplyDelete