Physics of Magnetism

64 CHAPTER 7. ITINERANT-ELECTRON MAGNETISM

level is high. After the transfer, there will be more spin-up electrons than spin-down

electrons, and the magnetic moment, which has arisen, will be equal to

We will first derive a simple band model, which accounts for the existence of ferro­

magnetism. The interaction Hamiltonian, following the above definition of can be

written as

where and represent the number of electrons per atom for each spin state, and where
the total number of 3d electrons per atom equals Because is a positive
quantity, Eq. (7.1.2) will lead to the lowest energy if the product is as small as possible.
For equally populated subbands, this product has its maximum value and hence the highest
energy. Consequently, electron transfer is always favorable for the lowering of the exchange
energy, and this electron transfer will come to an end only if one of the two spin subbands
is empty or has become completely filled up.
We define N(E) as the density of states per spin subband, and p as the fraction of
electrons that has moved from the spin-down band to the spin-up band. This means

Let us assume that the interaction Hamiltonian (Eq. 7.1.2) leads to an increase in the number

of spin-up electrons at the cost of the number of spin-down electrons. The corresponding

gain in magnetic energy is then

This energy gain is accompanied by an energy loss in the form of the amount of energy

needed to fill the states of higher kinetic energy in the band. For a small displacement

(see Fig. 7.1.1b), this kinetic-energy loss can be written as

The total energy variation is then