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Question: conduction electron ferromagnetism we approximate the effect of exchange interactions...

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Conduction electron ferromagnetism

We approximate the effect of exchange interactions among the conduction electrons if we assume that electrons with parallel spins interact with one another with energy -U, and U is positive, while electrons with antiparallel spins do not interact with each other.

1) Show that the total energy of the spin-up band is given by E^+ = E_0(1+\zeta)^{5/3}-\frac{1}{8}UN^2(1+\zeta)^2-\frac{1}{2}N\mu _B B_0(1+\zeta) , where N is the total number of electrons. Find a similar expresion for E^-. Hint: consider that the number of pairs of electrons with parallel spin-up and spin-downs is \frac{1}{2}(N^+)^2 and \frac{1}{2}(N^-)^2 respectively.

2) Minimize the total energy density E_T = E^+ + E^- , then solve for \zeta \ll 1. Show the magnetization is M = \frac{3n\mu_B^2}{2E_F-\frac{3}{2}UN}B_0 , where n is the total electron density, so that the exchange interaction enhances the susceptibility.

3) Finally, show that with B_0 = 0, the total energy is unstable at \zeta = 0 when U>4E_F/(3N).

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