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T2 relaxation

The other relaxation process called transverse or $T_2$ relaxation will be presented here intuitively in a classical model. Assume we have a bunch of magnetic moments originally point to the direction of the external field, then we rotate them to the transverse plane with an RF pulse. The local magnetic field a specific spin experiences is the sum of the external field and the small fields of the neighbouring spins. The latter small fields vary in space that leads to different local precession frequencies. Therefore the spins tend to fan out in time as shown in Figure 1, this is usually called dephasing. Since the measureable magnetization is the vector (or complex) sum of the individual magnetic moments, the net magnetization decreases over time. This reduction brings forth another exponential decay with characteristic time $T_2$ that adds to the time derivative of the transverse component of the magnetization. We can express it by adding an exponential term to the transverse component of the magnetization. In the laboraroty frame:

\frac{ \mathrm{d} \mathbf{M_{\bot}} } { \mathrm{d} t } \Bigg |_{\mathrm{lab}} = \gamma \mathbf{M_{\bot}} \times \mathbf{B} - \frac{\mathbf{M_{\bot}}}{T_2}(1)

And in the rotating system:

\frac{ \mathrm{d} \mathbf{M_{\bot}} } { \mathrm{d} t } \Bigg |_{\mathrm{rot}} = \gamma \mathbf{M_{\bot}} \times \mathbf{B_{eff}} - \frac{\mathbf{M_{\bot}}}{T_2}(2)

If the frequency of the rotating frame equals to the Larmor frequency and there are no other magnetic fields like RF excitation then the effective field becomes zero and the solution of (2) become a simple exponential decay:

M_{\bot} (t) = M_{\bot} (0) \mathrm{e}^{- \tfrac{t}{T_2}}(3)


Figure 1. Dephasing of the individual moments in the rotating frame and the decreasing net magnetization as a result.

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