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== Precession ==
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]] <math>\gamma</math>, such that
:<math>\mathbf{m}=-\gamma \mathbf{L}</math>.
The gyromagnetic ratio for a free electron has been experimentally determined as γ<sub>e</sub> = {{val|1.760859644e11|(11)|u=s<sup>−1</sup>⋅T<sup>−1</sup>}}.<ref>[http://physics.nist.gov/cgi-bin/cuu/Value?gammae CODATA Value: electron gyromagnetic ratio], ''The NIST Reference on Constants, Units, and Uncertainty''</ref> This value is very close to that used for Fe-based magnetic materials.
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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]],
:<math>\frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}=-\gamma\mu_0 \left(\mathbf{m} \times \mathbf{H}\right)</math>.
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Damping results in a sort of magnetic field "viscosity," whereby the magnetic field <math>H_{eff}</math> under consideration is delayed by a finite time period <math>\delta{t}</math>. In a general sense, the differential equation governing precession can be rewritten to include this damping effect, such that,<ref>M. L. Plumer, J. van Ek, and D. Weller (Eds.), ''The Physics of Ultra-High-Density Magnetic Recording,'' Berlin: Springer-Verlag, 2001.</ref>
:<math>\frac{\mathrm{d}\mathbf{m}\left(t\right)}{\mathrm{d}t}=-\gamma\mu_0 \mathbf{m}\left(t\right) \times \mathbf{H_{eff}}\left(t-\delta t\right)</math>.
Taking the [[Taylor series]] expansion about ''t'', while noting that <math>\tfrac{\mathrm{d}\mathbf{H_{eff}}}{\mathrm{d}t}=\tfrac{\mathrm{d}\mathbf{H_{eff}}}{\mathrm{d}\mathbf{m}}\tfrac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}</math>, provides a [[linear approximation]] for the time delayed magnetic field,
:<math>\mathbf{H_{eff}}\left(t-\delta t\right)=\mathbf{H_{eff}}\left(t\right)-\delta t \frac{\mathrm{d}\mathbf{H_{eff}}}{\mathrm{d}\mathbf{m}}\frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}+\dots</math>,
when neglecting higher order terms. This approximation can then be substituted back into the differential equation to obtain
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