Question: conduction electron ferromagnetism we approximate the effect of exchange interactions...
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 , and 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 , where is the total number of electrons. Find a similar expresion for . Hint: consider that the number of pairs of electrons with parallel spin-up and spin-downs is and respectively.
2) Minimize the total energy density , then solve for . Show the magnetization is , where is the total electron density, so that the exchange interaction enhances the susceptibility.
3) Finally, show that with , the total energy is unstable at when .