There are various ways to describe ferrimagnets, the simplest of which is with
mean-field theory. In mean-field theory the field acting on the atoms can be written as : \vec{H}=\vec{H}_0+\vec{H}_m, where \vec{H}_0 is the
applied magnetic field, and \vec{H}_m is field caused by the interactions between the atoms. The following assumption then is \vec{H}_m=\gamma\vec{M}. Here, \vec{M} is the average magnetization of the lattice, and \gamma is the molecular field coefficient. When \vec{M} and \gamma is allowed to be position- and orientation-dependent, it can then be written in the form : \vec{H}_i=\vec{H}_0+\sum_{k=1}^n\gamma_{ik}\vec{M}_k, where \vec{H}_i is the field acting on the
i-th substructure, and \gamma_{ik} is the molecular field coefficient between the
i-th and
k-th substructures. For a diatomic lattice, two types of sites can be designated,
a and
b. N can be designated the number of magnetic ions per unit volume, \lambda the fraction of the magnetic ions on the
a sites, and \mu=1-\lambda the fraction on the
b sites. This then gives : \vec{H}_{aa} = \gamma_{aa}\vec{M}_a, \quad \vec{H}_{ab} = \gamma_{ab}\vec{M}_b, \quad \vec{H}_{ba} = \gamma_{ba}\vec{M}_a, \quad \vec{H}_{bb} = \gamma_{bb}\vec{M}_b. It can be shown that \gamma_{ab}=\gamma_{ba} and that \gamma_{aa}\neq\gamma_{bb}, unless the structures are identical. \gamma_{ab}>0 favors a parallel alignment of \vec{M}_a and \vec{M}_b, while \gamma_{ab} favors an anti-parallel alignment. For ferrimagnets, \gamma_{ab}, so it will be convenient to take \gamma_{ab} as a positive quantity and write the minus sign explicitly in front of it. For the total fields on
a and
b this then gives : \vec{H}_a=\vec{H}_0+\gamma_{aa}\vec{M}_a-\gamma_{ab}\vec{M}_b, : \vec{H}_b=\vec{H}_0+\gamma_{bb}\vec{M}_b-\gamma_{ab}\vec{M}_a. Furthermore, the parameters \alpha=\gamma_{aa}/\gamma_{ab} and \beta=\gamma_{bb}/\gamma_{ab} will be introduced, which give the ratio between the strengths of the interactions. At last, the reduced magnetizations will be introduced : \vec{\sigma}_a=\vec{M}_a/\lambda Ng\mu_B S_a, : \vec{\sigma}_b=\vec{M}_b/\mu Ng\mu_BS_b with S_i the spin of the
i-th element. This then gives for the fields: : \vec{H}_a=\vec{H}_0+Ng\mu_B S_a\gamma_{ab}(\lambda\alpha\vec{\sigma}_a-\mu\vec{\sigma}_b), : \vec{H}_b=\vec{H}_0+Ng\mu_BS_b\gamma_{ab}(-\lambda\vec{\sigma}_a+\mu\beta\vec{\sigma}_b) The solutions to these equations (omitted here) are then given by : \sigma_a=B_{S_a}(g\mu_bS_aH_a/k_\text{B}T), : \sigma_b=B_{S_b}(g\mu_bS_bH_b/k_\text{B}T). where B_J(x) is the
Brillouin function. The simplest case to solve now is S_a=S_b=1/2. Since B_{1/2}(x)=\tanh(x), this then gives the following pair of equations: : \lambda\sigma_a=\frac{\tau F(\lambda,\alpha,\beta)}{\alpha\beta-1}(\beta\tanh^{-1}\sigma_a+\tanh^{-1}\sigma_b), : \mu\sigma_b=\frac{\tau F(\lambda,\alpha,\beta)}{\alpha\beta-1}(\tanh^{-1}\sigma_a+\alpha\tanh^{-1}\sigma_b) with \tau=T/T_\text{c} and F(\lambda,\alpha,\beta)=\frac{1}{2}\left(\lambda\alpha+\mu\beta+\sqrt{(\lambda\alpha-\mu\beta)^2+4\lambda\mu}\right). These equations do not have a known analytical solution, so they must be solved numerically to find the temperature dependence of \mu. ==Effects of temperature==