User:John R. Brews/Sandbox

Interacting atoms

The outline of paramagnetism above ignores all interactions between atoms, and makes them all act individually. A natural question is: if the torque aligning atoms is due to the magnetic field in the atom's vicinity, shouldn't the field include the effect of the neighboring atoms upon the field?

Such a modified theory was proposed by Pierre-Ernest Weiss by introducing the notion of a molecular field, a magnetic field contribution that was proportional to the magnetization in the vicinity of an atom:

$\mathbf {H_{W}} =\gamma \mathbf {M} \,$

where H_W is the "Weiss field" and γ is the "molecular field constant". This contribution is added to the applied magnetic field H_A to obtain the total field H:

$\mathbf {H} =\mathbf {H} _{A}+\mathbf {H} _{W}\ .$

Given a method to determine M from H, we then find:

$\mathbf {M} =\mathbf {M} (\mathbf {H} _{A}+\gamma \mathbf {M} )\ ,$

an implicit determination of M for any given H_A. In particular, if we adopt the Brillouin approach based upon B_J as a function of

$x={\frac {gm_{B}J\mu _{0}H}{k_{B}T}}\ ,$

all that is needed is to replace H with the modified H above that includes the Weiss field.

Although this approach has some application to those paramagnetic and ferrimagnetic materials that are ionic solids with localized moments, it doesn't work for ferromagnetic materials because most are metals with itinerant electrons.