From classical mechanics, torque is defined as the time rate of change of angular momentum
or, stated mathematically, Absent any other effects, this change in angular momentum would be realized through the dipole moment coming into rotation to align with the field.
However, the effect of a torque applied to an electron's magnetic moment must be considered in light of spin-orbit interaction.
Because the magnetic moment of an electron is a consequence of its spin and orbit and the associated angular momenta, the magnetic moment of an electron is directly proportional to its angular momentum through the gyromagnetic ratio
, such that The gyromagnetic ratio for a free electron has been experimentally determined as γe = 1.760859644(11)×1011 s−1⋅T−1.
Taking the derivative of the gyromagnetic ratio with respect to time yields the relationship, Thus, due to the relationship between an electron's magnetic moment and its angular momentum, any torque applied to the magnetic moment will give rise to a change in magnetic moment parallel to the torque.
Atomic-level dynamics involves interactions between magnetization, electrons, and phonons.
In a general sense, the differential equation governing precession can be rewritten to include this damping effect, such that,[4] Taking the Taylor series expansion about t, while noting that
, provides a linear approximation for the time delayed magnetic field, when neglecting higher order terms.
This approximation can then be substituted back into the differential equation to obtain where is called the dimensionless damping tensor.
The damping tensor is often considered a phenomenological constant resulting from interactions that have not yet been fully characterized for general systems.
The Landau-Lifshitz-Gilbert equation can also be written in terms of torques, where the damping torque is given by By way of the micromagnetic theory,[5] the Landau-Lifshitz-Gilbert equation also applies to the mesoscopic- and macroscopic-scale magnetization