In magnetic resonance imaging (MRI) and nuclear magnetic resonance spectroscopy (NMR), 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). For spin-1/2 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:
N + N − = e − Δ E k T {\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. 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, Mz, towards its thermal equilibrium value, M z , e q {\displaystyle M_{z,\mathrm {eq} }} . In general,
M z ( t ) = M z , e q − [ M z , e q − M z ( 0 ) ] e − t / T 1 {\displaystyle M_{z}(t)=M_{z,\mathrm {eq} }-[M_{z,\mathrm {eq} }-M_{z}(0)]e^{-t/T_{1}}}
In specific cases:
If M has been tilted into the xy plane, then M z ( 0 ) = 0 {\displaystyle M_{z}(0)=0} and the recovery is simply
M z ( t ) = M z , e q ( 1 − e − t / T 1 ) {\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.
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