Rotation Physics

A magnetic moment

Magnetic moment

The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...

  in the presence of a magnetic field

Magnetic field

A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

  experiences a torque

Torque

Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

  that attempts to bring the moment and field vectors into alignment. The classical expression for this alignment torque is given by,
and shows that the torque is proportional to the strengths of the moment and field and to the angle of misalignment between them.

From classical mechanics

Classical mechanics

In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

, torque is defined as the time rate of change of angular momentum

Angular momentum

In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

  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.

Precession

However, the effect of a torque applied to an electron's magnetic moment must be considered in light of spin-orbit interaction

Spin-orbit interaction

In quantum physics, the spin-orbit interaction is any interaction of a particle's spin with its motion. The first and best known example of this is that spin-orbit interaction causes shifts in an electron's atomic energy levels due to electromagnetic interaction between the electron's spin and...

. 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 . This value is very close to that used for Fe-based magnetic materials.

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.

Substituting the classical expression for torque on a magnetic dipole moment yields the differential equation,.

Specifying that the applied magnetic field is in the direction and separating the differential equation into its Cartesian components,,
it can be explicitly seen that the instantaneous change in magnetic moment occurs perpendicular to both the applied field and the direction of the moment, with no change in moment in the direction of the field .

Damping

While the transfer of angular momentum on a magnetic moment from an applied magnetic field is shown to cause precession of the moment about the field axis, the rotation of the moment into alignment with the field occurs through damping processes.

Atomic-level dynamics involves interactions between magnetization, electrons, and phonons. These interactions are transfers of energy generally termed relaxation. Magnetization damping can occur through energy transfer (relaxation) from an electron's spin to:

• Itinerant electrons (electron-spin relaxation)
• Lattice vibrations (spin-phonon relaxation)
• Spin waves, magnons (spin-spin relaxation)
• Impurities (spin-electron, spin-phonon, or spin-spin)

Damping results in a sort of magnetic field "viscosity," whereby the magnetic field under consideration is delayed by a finite time period . In a general sense, the differential equation governing precession can be rewritten to include this damping effect, such that,.
Taking the Taylor series

Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

 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. For most applications, damping can be considered isotropic, meaning that the damping tensor is diagonal,,
and can be written as a scalar, dimensionless damping constant,.

Landau-Lifshitz-Gilbert Equation

With these considerations, the differential equation governing the behavior of a magnetic moment in the presence of an applied magnetic field with damping can be written in the most familiar form of the Landau-Lifshitz-Gilbert equation,.
Since without damping is directed perpendicular to both the moment and the field, the damping term of the Landau-Lifshitz-Gilbert equation provides for a change in the moment towards the applied field. 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

Micromagnetism

Micromagnetics deals with the interactions between magnetic moments on sub-micrometre length scales. These are governed by several competing energy terms. Dipolar energy is the energy which causes magnets to align north to south pole...

, the Landau-Lifshitz-Gilbert equation also applies to the mesoscopic- and macroscopic-scale magnetization

Magnetization

In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material...

of a sample by simple substitution,.