Analytical formula and numerical calculation of the second harmonic of dynamic susceptibility in concentrated ferrofluids

In this work, the second component of dynamic susceptibility of an ensemble of interacting magnetic particles is studied by using analytical and numerical methods. The configuration of superimposed magnetic fields is considered: alternating and parallel constant fields. Dipole-dipole interactions are taken into account within two-particle correlations using a modified first-order mean-field theory approach.
From the analytical solution of the Fokker–Planck equation, an expression for the second harmonic is obtained as a function of two parameters: the Langevin susceptibility $\chi_L$, which characterizes dipole-dipole interactions, and the Langevin parameter $\xi_0$, representing the ratio of magnetic energy to thermal energy.
The obtained expression for the second harmonic agrees with previously known results where interparticle interactions were neglected. This research has significant theoretical interest and can be used for more precise characterization of magnetic particle properties.