Stoner criterion

The Stoner criterion is a condition to be fulfilled for the ferromagnetic order to arise in a simplified model of a solid.

It is named after Edmund Clifton Stoner.

Ferromagnetism ultimately stems from Pauli exclusion.

The simplified model of a solid which is nowadays usually called the Stoner model, can be formulated in terms of dispersion relations for spin up and spin down electrons, where the second term accounts for the exchange energy,

) is the dimensionless density[note 1] of spin up (down) electrons and

is the dispersion relation of spinless electrons where the electron-electron interaction is disregarded.

can be used to calculate the total energy of the system as a function of its polarization

If the lowest total energy is found for

, the system prefers to remain paramagnetic but for larger values of

state will spontaneously pass into a polarized one.

This is the Stoner criterion, expressed in terms of the

density of states[note 1] at the Fermi energy

The particle density operators are written as their mean value

and the product of spin-up and spin-down fluctuations is neglected.

We obtain[note 1] With the third term included, which was omitted in the definition above, we arrive at the better-known form of the Stoner criterion