Chemical Sciences: A Manual for CSIR-UGC National Eligibility Test for Lectureship and JRF/Relaxation (NMR)

Relaxation in the topic of Nuclear magnetic resonance phenomenology, describes the evolution of magnetizations separately in two directions:
 * longitudinal relaxation: The part of the magnetization vector M that is parallel to the main magnetic field B0 is usually called longitudinal magnetization, designated as Mz. The process for it to recover to the thermal equilibrium magnetization M0 is called longitudinal relaxation, which involves a time constant T1.
 * $$M_z(t) = M_0 - e^{-t/T_1}\cdot [M_0-M_z(t=0)] \,$$


 * transverse relaxation: The part of the magnetization vector M that is perpendicular to the main magnetic field B0 is usually called transverse magnetization, which can be written as Mxy, MT, or $$M_{\perp}$$. The process for it to decay to nearly zero is called transverse relaxation, which involves a time constant T2.
 * $$M_{xy}(t) = e^{-t/T_2}\cdot M_{xy}(t=0) \,$$

T1
In an ideal environment where strict conservation of angular momentum is true for the nuclei being observed, T1 would not exist. When you alter the magnetization of a nucleus, in the experimental pulse, it should maintain its precession. So the bulk magnetization which is set into a disequilibrium cannot equilibrate. However, in a real system, there is spin transfer between the observed nuclei and the environment. This allows for conservation-"forbidden" transitions to occur, and "relaxation" from "excited" state back to equilibrium.

T1 is by definition, the component of relaxation which occurs in the direction of the ambient magnetic field. This generally comes about by interactions between the nucleus of interest and unexcited nuclei in the environment, as well as electric fields in the environment (collectively known as the 'lattice'). Therefore T1 is known as "spin-lattice" relaxation.

T1 is measured as the time required for the magnetization vector M to be restored to 63% of its original magnitude. It varies with the magnetic field density B.

T2
In an idealized system, T2 would also not exist. However, in real systems, there is spin transfer amongst excited nuclei which disperses magnetization that is out of equilibrium.

T2 by definition is the component of 'true' relaxation (see T2*) to equilibrium that occurs perpendicular to the ambient magnetic field. Because of this, the relaxation is dominated by interactions between spinning nuclei which are already excited. For this reason, T2 relaxation is called "transverse" or "spin-spin" relaxation.

Since T2 processes follow an exponential decay, the quantity T2 is defined as the time required for the transverse Magnetization vector to drop to 37% of its original magnitude after its initial excitation.

Unlike T1, T2 does not vary with the field strength B. It is strictly a property of the tissue. Unlike T2*, it is not susceptible to gradient fluctuations.

T2*
In an idealized system, all nuclei in a given chemical environment in a magnetic field spin with the same frequency. However, in real systems, there are minor differences in chemical environment which can lead to a distribution of resonance frequencies around the ideal. Over time, this distribution can lead to a dispersion of the tight distribution of magnetic spin vectors, and loss of signal (Free Induction Decay). In fact, for most magnetic resonance experiments, this "relaxation" dominates.

However, this is not a true "relaxation" process. For most molecules, the deviation from ideal relaxation is consistent over time, and the signal can be recovered by performing a spin-echo experiment.

Unlike T2, T2* is influenced by magnetic field gradient irregularities. The T2* relaxation time is always less than the T2 relaxation time.

Why T1 is slower than T2
As a general rule, the following always holds true: T1 > T2 > T2*.

In order to get magnetization transfer, the energies and orientations of spins with magnetic entities in the lattice must be matched. In most setups, this is a relatively rare condition, compared to spin-spin interactions, which a priori are aligned with each other.

Local magnetic field inhomogeneity
Besides, due to the inhomogeneity in main field, there occurs intra-voxel dephasing resulting in a far faster decay of the signal from transverse magnetization than that predicted by T2 decay, which can be written as:
 * $$M_{xy}(t) = e^{-t/T_2^*}\cdot M_{xy}(t=0) ; \,$$
 * $$ T_2^*<<T_2 \,$$

The corresponding transverse relaxation time constant is thus T2*, which is much smaller than T2. The relation between them is:
 * $$\frac{1}{T_2^*}=\frac{1}{T_2}+\gamma \Delta B_0$$

where γ represents magnetogyric ratio, and ΔB0 the difference in strength of the locally varying field.

Common relaxation time constants in human tissues
Following is a table of the approximate values of the two relaxation time constants for nonpathological human tissues, just for simple reference.



Following is a table of the approximate values of the two relaxation time constants for chemicals that commonly show up in human brain, physiologically or pathologically.



Microscopic mechanism
In 1948, Nicolaas Bloembergen, Edward Mills Purcell, and R.V. Pound proposed the so-called Bloembergen-Purcell-Pound theory (BPP theory) to explain the relaxation constant of a pure substance in correspondence with its state, taking into account the effect of tumbling motion of molecules on the local magnetic field disturbance. The theory was in good agreement with the experiments for pure substance, but not for complicated environment such as human body.

From this theory, one can get T1、T2:
 * $$\frac{1}{T_1}=K[\frac{\tau_c}{1+\omega_0^2\tau_c^2}+\frac{4\tau_c}{1+4\omega_0^2\tau_c^2}]$$
 * $$\frac{1}{T_2}=\frac{K}{2}[3\tau_c+\frac{5\tau_c}{1+\omega_0^2\tau_c^2}+\frac{2\tau_c}{1+4\omega_0^2\tau_c^2}]$$,

where $$\omega_0$$ is the Larmor frequency in correspondence with the strength of the main magnetic field $$B_0$$. $$\tau_c$$ is the correlation time of the molecular tumbling motion. $$K=\frac{3\mu^2}{160\pi^2}\frac{\hbar^2\gamma^4}{r^6}$$ is a constant with μ being the magnetic dipole moment of the spin-1/2 nuclei, $$\hbar=\frac{h}{2\pi}$$ the reduced Planck constant, γ the gyromagnetic ratio of such species of nuclei, and r the distance between the two nuclei carrying magnetic dipole moment.

Taking for example the H2O molecules in liquid phase without the contamination of oxygen 17, the value of K is 1.02×1010 sec-2 and the correlation time $$\tau_c$$ is on the order of ps = $$10^{-12}$$ sec, while hydrogen nuclei 1H (protons) at 1.5 tesla carry an Larmor frequency of approximately 64 MHz. We can then estimate using τc = 5×10-12 sec:


 * $$\omega_0\tau_c = 3.2\times 10^{-5} $$(dimensionless)
 * $$T_1=(1.02\times 10^{10}[\frac{ 5\times 10^{-12} }{1 + (3.2\times 10^{-5} )^2} + \frac{ 4\cdot 5\times 10^{-12} }{1 + 4\cdot (3.2\times 10^{-5} )^2}])^{-1} $$= 3.92 sec
 * $$T_2=(\frac{1.02\times 10^{10}}{2}[3\cdot 5\times 10^{-12} + \frac{5\cdot 5\times 10^{-12} }{1 + (3.2\times 10^{-5} )^2} + \frac{ 2\cdot 5\times 10^{-12} }{1 + 4\cdot (3.2\times 10^{-5} )^2}])^{-1} $$= 3.92 sec,

which is close to the experimental value, 3.6 sec. Meanwhile, we can see that at this extreme case, T1 equals T2.