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T2* relaxation

In practice however, there is an additional dephasing of moments caused by the inhomogeneities of the external field itself that also reduces net magnetization. Therefore the total reduction of the transverse component of magnetization over time is the resultant of the above described $T_2$ relaxation and this new process caused by the external field inhomogeneities. The latter is called $T'_2$ relaxation and usually also acts as an exponential reduction with a time constant $T'_2$. The sum of these two effects results in another exponential decay with characteristic time $T^*_2$ given by the following:

\frac{1}{T^*_2} = \frac{1}{T_2} + \frac{1}{T'_2}(1)

In the formula above the reciprocals of characteristic times are usually referred to as relaxation rates, so (1) says that the relaxation rates simply add up.

Now that we introduced the relaxation effects, we can construct the total system of equations of motion for a static field. Assume we have a static and perfectly homogeneous magnetic field (so there is no $T'_2$ dephasing) points to the $z$ direction. Then the equations in the laboratory frame:

\frac{ \mathrm{d} M_x } { \mathrm{d} t } = \omega_0 M_y - \frac{M_x}{T_2}(2)
\frac{ \mathrm{d} M_y } { \mathrm{d} t } = - \omega_0 M_x - \frac{M_y}{T_2}(3)
\frac{ \mathrm{d} M_z } { \mathrm{d} t } = \frac{M_0 - M_z}{T_1}(4)

By using the complex formalism introduced at the classical description of precession, the first two coupled equation will become one complex equation:

M_+ = M_x + \mathrm{i} M_y(5)
\frac{ \mathrm{d} M_+ } { \mathrm{d} t } = - \mathrm{i} \omega_0 M_+ - \frac{M_+}{T_2}(6)

And the solution with this complex description will be

M_+ (t) = M_+ (0) \mathrm{e}^{ - \mathrm{i} \omega_0 t - \tfrac {t}{T_2}}(7)
M_z (t) = M_z (0) \mathrm{e}^{ - \tfrac{t}{T_1}} + M_0 \left ( 1 - \mathrm{e}^{ - \tfrac{t}{T_1}} \right )(8)

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