In MRI and NMR spectroscopy, an observable nuclear spin polarization (magnetization) is created by a homogeneous magnetic field. This field makes the magnetic dipole moments of the sample precess at the resonance (Larmor) frequency of the nuclei. At thermal equilibrium, nuclear spins precess randomly about the direction of the applied field. They become abruptly phase coherent when they are hit by radiofrequency (RF) pulses at the resonant frequency, created orthogonal to the field. The RF pulses cause the population of spin-states to be perturbed from their thermal equilibrium value. The generated transverse magnetization can then induce a signal in an RF coil that can be detected and amplified by an RF receiver. The return of the longitudinal component of the magnetization to its equilibrium value is termed spin-lattice relaxation while the loss of phase-coherence of the spins is termed spin-spin relaxation, which is manifest as an observed free induction decay (FID).^[1]

For spin=½ nuclei (such as ¹H), the polarization due to spins oriented with the field N₋ relative to the spins oriented against the field N₊ is given by the Boltzmann distribution:

${\displaystyle {\frac {N_{+}}{N_{-}}}=e^{-{\frac {\Delta E}{kT}}}}$

where ΔE is the energy level difference between the two populations of spins, k is the Boltzmann constant, and T is the sample temperature. At room temperature, the number of spins in the lower energy level, N−, slightly outnumbers the number in the upper level, N+. The energy gap between the spin-up and spin-down states in NMR is minute by atomic emission standards at magnetic fields conventionally used in MRI and NMR spectroscopy. Energy emission in NMR must be induced through a direct interaction of a nucleus with its external environment rather than by spontaneous emission. This interaction may be through the electrical or magnetic fields generated by other nuclei, electrons, or molecules. Spontaneous emission of energy is a radiative process involving the release of a photon and typified by phenomena such as fluorescence and phosphorescence. As stated by Abragam, the probability per unit time of the nuclear spin-1/2 transition from the + into the - state through spontaneous emission of a photon is a negligible phenomenon.^[2][3] Rather, the return to equilibrium is a much slower thermal process induced by the fluctuating local magnetic fields due to molecular or electron (free radical) rotational motions that return the excess energy in the form of heat to the surroundings.

T₁ and T₂

The decay of RF-induced NMR spin polarization is characterized in terms of two separate processes, each with their own time constants. One process, called T₁, is responsible for the loss of resonance intensity following pulse excitation. The other process, called T₂, characterizes the width or broadness of resonances. Stated more formally, T₁ is the time constant for the physical processes responsible for the relaxation of the components of the nuclear spin magnetization vector M parallel to the external magnetic field, B₀ (which is conventionally designated as the z-axis). T₂ relaxation affects the coherent components of M perpendicular to B₀. In conventional NMR spectroscopy, T₁ limits the pulse repetition rate and affects the overall time an NMR spectrum can be acquired. Values of T₁ range from milliseconds to several seconds, depending on the size of the molecule, the viscosity of the solution, the temperature of the sample, and the possible presence of paramagnetic species (e.g., O₂ or metal ions).

T₁

The longitudinal (or spin-lattice) relaxation time T₁ is the decay constant for the recovery of the z component of the nuclear spin magnetization, M_z, towards its thermal equilibrium value, ${\displaystyle M_{z,\mathrm {eq} }}$. In general,

${\displaystyle M_{z}(t)=M_{z,\mathrm {eq} }-e^{-t/T_{1}}}$

In specific cases:

• If M has been tilted into the xy plane, then ${\displaystyle M_{z}(0)=0}$ and the recovery is simply

${\displaystyle M_{z}(t)=M_{z,\mathrm {eq} }\left(1-e^{-t/T_{1}}\right)}$

i.e. the magnetization recovers to 63% of its equilibrium value after one time constant T₁.

• In the inversion recovery experiment, commonly used to measure T₁ values, the initial magnetization is inverted, ${\displaystyle M_{z}(0)=-M_{z,\mathrm {eq} }}$, and so the recovery follows

${\displaystyle M_{z}(t)=M_{z,\mathrm {eq} }\left(1-2e^{-t/T_{1}}\right)}$

T₁ relaxation involves redistributing the populations of the nuclear spin states in order to reach the thermal equilibrium distribution. By definition, this is not energy conserving. Moreover, spontaneous emission is negligibly slow at NMR frequencies. Hence truly isolated nuclear spins would show negligible rates of T₁ relaxation. However, a variety of relaxation mechanisms allow nuclear spins to exchange energy with their surroundings, the lattice, allowing the spin populations to equilibrate. The fact that T₁ relaxation involves an interaction with the surroundings is the origin of the alternative description, spin-lattice relaxation.

Note that the rates of T₁ relaxation (i.e., 1/T₁) are generally strongly dependent on the NMR frequency and so vary considerably with magnetic field strength B. Small amounts of paramagnetic substances in a sample speed up relaxation very much. By degassing, and thereby removing dissolved oxygen, the T₁/T₂ of liquid samples easily go up to an order of ten seconds.

Spin saturation transfer

Especially for molecules exhibiting slowly relaxing (T₁) signals, the technique spin saturation transfer (SST) provides information on chemical exchange reactions. The method is widely applicable to fluxional molecules. This magnetization transfer technique provides rates, provided that they exceed 1/T₁.^[4]

T₂

The transverse (or spin-spin) relaxation time T₂ is the decay constant for the component of M perpendicular to B₀, designated M_xy, M_T, or ${\displaystyle M_{\perp }}$. For instance, initial xy magnetization at time zero will decay to zero (i.e. equilibrium) as follows:

${\displaystyle M_{xy}(t)=M_{xy}(0)e^{-t/T_{2}}\,}$

i.e. the transverse magnetization vector drops to 37% of its original magnitude after one time constant T₂.

T₂ relaxation is a complex phenomenon, but at its most fundamental level, it corresponds to a decoherence of the transverse nuclear spin magnetization. Random fluctuations of the local magnetic field lead to random variations in the instantaneous NMR precession frequency of different spins. As a result, the initial phase coherence of the nuclear spins is lost, until eventually the phases are disordered and there is no net xy magnetization. Because T₂ relaxation involves only the phases of other nuclear spins it is often called "spin-spin" relaxation.

T₂ values are generally much less dependent on field strength, B, than T₁ values.

Hahn echo decay experiment can be used to measure the T₂ time, as shown in the animation below. The size of the echo is recorded for different spacings of the two applied pulses. This reveals the decoherence which is not refocused by the 180° pulse. In simple cases, an exponential decay is measured which is described by the ${\displaystyle T_{2}}$ time.

T₂* and magnetic field inhomogeneity

In an idealized system, all nuclei in a given chemical environment, in a magnetic field, precess 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. This results in dephasing.

However, decoherence because of magnetic field inhomogeneity is not a true "relaxation" process; it is not random, but dependent on the location of the molecule in the magnet. For molecules that aren't moving, the deviation from ideal relaxation is consistent over time, and the signal can be recovered by performing a spin echo experiment.

The corresponding transverse relaxation time constant is thus T₂^*, which is usually much smaller than T₂. The relation between them is:

${\displaystyle {\frac {1}{T_{2}^{*}}}={\frac {1}{T_{2}}}+{\frac {1}{T_{inhom}}}={\frac {1}{T_{2}}}+\gamma \Delta B_{0}}$

where γ represents gyromagnetic ratio, and ΔB₀ the difference in strength of the locally varying field.^[5][6]

Unlike T₂, T₂* is influenced by magnetic field gradient irregularities. The T₂* relaxation time is always shorter than the T₂ relaxation time and is typically milliseconds for water samples in imaging magnets.

Is T₁ always longer than T₂?

In NMR systems, the following relation holds absolute true^[7] ${\displaystyle T_{2}\leq 2T_{1}}$. In most situations (but not in principle) ${\displaystyle T_{1}}$ is greater than ${\displaystyle T_{2}}$. The cases in which ${\displaystyle 2T_{1}>T_{2}>T_{1}}$ are rare, but not impossible.^[8]

Bloch equations

Bloch equations are used to calculate the nuclear magnetization M = (M_x, M_y, M_z) as a function of time when relaxation times T₁ and T₂ are present. Bloch equations are phenomenological equations that were introduced by Felix Bloch in 1946.^[9]

${\displaystyle {\frac {\partial M_{x}(t)}{\partial t}}=\gamma (\mathbf {M} (t)\times \mathbf {B} (t))_{x}-{\frac {M_{x}(t)}{T_{2}}}}$

${\displaystyle \frac{\mathrm{\partial <!--\; \partial \; -->}{M}_y(t)}{\mathrm{\partial <!--\; \partial \; -->}t}=\mathrm{\gamma <!--\; \gamma \; -->}(\mathbf{M}(t)\times <!--\; \times \; -->\mathbf{B}}$Zdroj:https://en.wikipedia.org?pojem=Relaxation_(NMR)
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