Quantum Statistical Derivation of Ferromagnetic-Resonance Equations

Characterization of Magnetic Phases in Nanostructured Ferrites by Electron Spin Resonance

“…This means that the equilibrium in the spin system is established by the strong exchange interaction, while the relaxation is conditioned by the weak dipolar interaction, so the equilibrium of the system is established by the strong exchange interaction much more quickly than the relaxation process. This assumption is confirmed by experimental data on the inelastic magnetic scattering of neutrons [9].…”

“…At room temperature, the EPR of γ-Fe 2 O 3 and BaFe 2 O 4 and nanoparticles show a tow line pattern as shown in Figure 6 [9], which is typical of superparamagnetic resonance in contrast with the BaFe 12 O 19 where the narrow line is more pronounced (as shown in Figure 6c). In these sample, the line width ΔH ∼120 Oe and g = 2.00; this value of the g factor is typical paramagnetic.…”

“…For temperatures above the Curie temperature, the magnetic moments in an external field obey the Bloch equations. These equations follow the Larmor theorem where the motion of the magnetic moments in a magnetic field originates a torque [6][7][8][9]:…”

“…From the theoretical point of view, the ferromagnetic resonance is described by the FMR equations. These are differential equations that connect time derivatives of the magnetization components with the components of the magnetization, the external field, the static magnetic susceptibility, and the time relaxation [9,27]. The FMR equations are different depending on the temperature regime which one is working in.…”

“…Below the Curie temperature, the FMR equations are called the Landau-Lifshitz equations. Over the Curie temperature, the FMR equations lead to the Bloch-type ferromagnetic-resonance equations [9]. These equations were obtained empirically; however, for a better understanding of the microscopic phenomena, a quantum statistical derivation was developed by O.…”