The equation of motion of a magnetic moment under an applied field is
Eq. (2.39) does not explain the absorption line in resonance experiments and the fact that the magnetic moment eventually aligns with field direction. The motion is modified by the interaction with crystal lattice vibrations, conduction electrons and other external sources. L.D Landau and E.M. Lifshitz [Landau 35] proposed to add a term proportional to , which conserves the magnitude, in order to obtain a phenomenological equation of the spin dynamics. Posteriorly, T. Gilbert suggested to add a viscous force to the equation of motion (2.39) [Gilbert 55]. The Gilbert and the Landau-Lifshitz equations are equivalent with the renormalization of the precession and dissipation terms. The Gilbert version is preferred because it predicts slower motion with increasing damping. The Gilbert equation, converted to Landau-Lifshitz form, is known as the Landau-Lifshitz-Gilbert(LLG) equation and has the expression:
The second term of right side of Eq. (2.41), the damping term, makes the magnetization rotate towards the direction of the effective field and eventually to be parallel to its direction, reaching the equilibrium. That represents a minimum of the energy. Accordingly, we can use the integration of LLG equation to minimize the energy [Berkov 93] and this is the method we mostly use in this thesis. For this purpose, it is better to use a large value of the damping constant and to remove the precession term.