In MRI, I was thinking that the decrease in transverse magnetization results in the recovery of longitudinal magnetization.
However, recently, I found it was wrong. I saw a diagram
(McRobbie 2014. Fundamentals of MR Imaging)
and also saw many writings saying "T1 and T2 relaxation are independent." How is it possible? Where does the power emitted(?) from T2 decay go to?
(and also, all the animations I saw expressed them as dependent on each other.)
They are indeed independent. I don't know what animations you're referring to, but this independence is clear from their defining equations, which do not depend on each other:
$M_z(t)=M_z(0)(1-e^{-t/T_1})$
$M_{xy}(t)=M_{xy}(0)e^{-t/T_2}$
T1 is the longitudinal relaxation time and involves interaction of spins with the lattice (read: surroundings). T2 is the transverse relaxation time and involves interactions among spins with no energy exchanged with the lattice.
The energy associated with T2 decay does not go into the lattice but results in a loss of coherence among the spins. This is called dephasing and is the reason why the transverse magnetization is observed to decrease.
The first thing you have to note is that: no, the transverse $M_{xy}$ component does not exchange energy with the longitudinal $M_z$ component for the magnetization $\vec{M}$ to return to equilibrium. Remember that the energy of a magnetic dipole in a static field $\vec{B}=B_z\hat{z}$ is given by $$ E=-\vec{\mu}\cdot \vec{B}=-\mu_zB_z=-{M_z}B_zV $$ where $V$ is the volume of the sample (assuming the dipole distribution is uniform over the sample, for simplicity).
This should tell you that the energy of the sample only depends on $M_z$. The laws of physics do not require that the sample "regains" energy it "converted" to $M_{xy}$ as it relaxes, as you initially assumed. To address your question, the $T_2$-decay does NOT involve any energy transfer as the longitudinal alignment with respect to the static field does not change.
Indeed, the longitudinal $T_1$ and transversal $T_2$ relaxation occur independently through two different mechanisms. These mechanisms can even happen in totally different time scales. For example, the $T_2$ relaxation of a nitrogen-vacancy center happens within microseconds, while the $T_1$ relaxation takes milliseconds to finish. They are NOT dependent on each other, but they happen simultaneously. Once you tip $\vec{M}$ from its initial equilibrium (i.e. into the $xy$-plane), the following processes work to return $\vec{M}$ to the equilibrium.
The sample never loses any energy when you tip it toward the $xy$-plane. Instead, it gains energy from the control pulse you use on it. As the sample is no longer in equilibrium with its surroundings, it can "share" its extra energy (which it gains from the control pulse) with its surroundings via the $T_1$ relaxation. The "surroundings" here are usually the surrounding lattice, hence the name "spin-lattice relaxation". This is where the energy transfer occurs.
Meanwhile, the $T_2$ relaxation does not involve energy transfers. Due to inhomogeneity in magnetic fields, different parts of your sample may experience slightly different static field strengths. Now, you can break $\vec{M}$ down into the "constituting" magnetization of each part of the sample. By "part", I mean a chunk of your sample that experiences the same magnetic field strength but is different from the other parts of the sample. Each constituting magnetization undergoes Larmor precession at slightly different rates. Give it long enough, and they are dephased far enough from each other that they cancel out each other. This is what goes on during a $T_2$-relaxation---call it a "transversal component self-annihilation", if you will.
In conclusion, the decay of $M_{xy}$ does NOT result in the recovery of $M_z$. $M_{xy}$ decays because its constituting components are dephased from each other and cancel each other out, while $M_z$ recovers by giving its energy to its surroundings. The two processes happen simultaneously, independent of each other, and one process can finish faster than the others (as shown in your figure). The $T_2$ relaxation never involves energy emissions.
When you say the videos "show them dependant on each other", perhaps your confusion arises because they both evolve with time.
I think a simple classical explanation is best, so I won't go quantum.
You can imagine your substance as broken down into fractional volumes, each with its own characteristic magnetic vector. Summed together, they represent the total vector.
T1 relaxation is due to the torque from the large external field on these characteristic vectors. They are going to get torqued no matter the phase coherence between them.
T2 relaxation is due to different precession rates among the numerous characteristic environments. Each volume experiences a slightly different magnetic environment because the external magnetic is not perfectly homogenous and because the matter around any particular dipole can contribute small deviations to the external field. Dipoles with slightly different magnetic environments will precess at slightly different rates. This is where the phase decoherence comes from. The videos should actually show the total vector getting shorter as the contributing vectors spread out.
This shows that there are two different mechanisms causing relaxation. But details matter. If you want to consider different substances, then you can see a correlation between T1 and T2. The correlation can be seen here.
