Set of supplementary notes.

9.2. MAGNETISM 143
Figure 9.11: . In metals, a local moment will polarise the conduction electron spins, producing
a spin density that decays away and oscillates in sign with period 1/2k F . The interaction of
the induced spin density with a neighbouring local moment produces the RKKY interaction
where, to keep things simple, we neglect the vector nature of H and M. In this equation
of state, a takes the role of the inverse susceptibility χ −1 = dH/dM and b ensures that the
magnetisation bends over towards saturation for high fields.
For a material to be ferromagnetic, we require a finite M, a remanent magnetisation, even
for zero H. This would appear to be possible only, if the parameter a in the equation of
state is negative. Systems of non-interacting electrons, however, do not exhibit a negative
susceptibility: the Curie law for isolated local moments gives a ∝ T , whereas in metals, the
Pauli susceptibility is positive and only very weakly temperature dependent. We therefore need
to introduce a further term which captures the effect of interactions between the electrons.
The simplest way to incorporate these interactions is to introduce an exchange molecular
field, h, into the equation of state: H + h = aM + bM 3 . The exchange molecular field is not a
real magnetic field, which could deflect a compass needle or induce voltages in a pick-up coil. It
is a way to represent, in a mean field sense, the effect of the exchange interaction produced by
other electrons on a test electron. If we assume that the exchange field is simply proportional
to the overall magnetisation (with constant of proportionality λ, this is the Weiss molecular
field concept), then we arrive at a feed-back equation:
H + λM = aM + bM 3 ,
which can be recast in the form of the original equation of state, with a modified linear coefficient
a ∗ = a − λ:
H = (a − λ)M + bM 3 = a ∗ M + bM 3
We see, then, that although the noninteracting susceptibility is finite, interactions between
the electrons give rise to a feed-back effect, which boosts the magnetic susceptibility χ = 1/a ∗ =
χ 0 /(1 − λ)χ 0 , where χ 0 = 1/a is the noninteracting susceptibility. This leads to a magnetic
instability, if
λχ 0 > 1 (9.37)
The above equation represents a more general form of the Stoner criterion (Eq. 9.31).