Inversion Recovery

While $T_2$ relaxation can be measured with a spin echo experiment, logitudinal relaxation is observed with a $T_1$ -dependent pulse sequence called inversion recovery. The sequence diagram is shown in Figure 1. The idea is the following. First we rotate the magnetization from its equilibrium state with 180°. After that the magnetization vector will only have longitudinal component equal to the equilibrium value pointing to the other direction, and will start to relax back with the longitudinal relaxation $T_1$ . We give it some time to relax, then apply a 90° pulse to rotate the remaining longitudinal component to the transverse field and start the acquisition. In this way we will get an FID signal which is proportional to the longitudinal component of the magnetization just prior the 90° RF pulse. Performing the experiment with different waiting time between the two pulses, that is, with different $T_I$ inversion times, one can measure the $T_1$ relaxation by fitting a suitable exponential curve to the FID signal amplitudes.

Figure 1. Sequence diagram of the inversion recovery experiment.

If we start the measurement of time at the 180° RF pulse assumed to be instantaneous then the dynamics of magnetization:

$\label{inversion_recovery1} M_z (0^+) = -M_0$ (1)

$\label{inversion_recovery2} M_z (t<T_I) = -M_0 \mathrm{e}^{-\tfrac{t}{T_1}} + M_0 \left ( 1 - \mathrm{e}^{-tfrac{t}{T_1}} \right) = M_0 \left ( 1 - 2 \mathrm{e}^{- \tfrac{t}{T_1}} \right)$ (2)

$\label{inversion_recovery3} M_{\bot} (t>T_I) = \left | M_0 \left ( 1 - 2 \mathrm{e}^{- \tfrac{t}{T_1}} \right) \right | \mathrm{e}^{-\tfrac{t-T_I}{T^*_2}}$ (3)

From the above equations one can calculate that the FID sign vanishes when $T_I = T_1 \mathrm{ln}(2)$ .

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