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- 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:
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, . In general,
In specific cases: If M has been tilted into the xy plane, then and the recovery is simply
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, , and so the recovery follows
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.
T2
The transverse (or spin-spin) relaxation time T2 is the decay constant for the component of M perpendicular to B0, designated Mxy, MT, or . For instance, initial xy magnetization at time zero will decay to zero (i.e. equilibrium) as follows:
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 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:
where γ represents gyromagnetic ratio, and ΔB0 the difference in strength of the locally varying field.
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 trueMalcolm H. Levitt: Spin Dynamics: Basics of Nuclear Magnetic Resonance, 2nd edition, John Wiley & Sons, New York 2008, , Section 11.9.2 . In most situations (but not in principle) is greater than . The cases in which are rare, but not impossible.
Bloch equations
Bloch equations are used to calculate the nuclear magnetization M = (Mx, My, Mz) 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.F Bloch, Nuclear Induction, Physical Review 70, 460–473 (1946)
Where is the cross-product, γ is the gyromagnetic ratio and B(t) = (Bx(t), By(t), B0 + Bz(t)) is the magnetic flux density experienced by the nuclei. The z component of the magnetic flux density B is typically composed of two terms: one, B0, is constant in time, the other one, Bz(t), is time dependent. It is present in magnetic resonance imaging and helps with the spatial decoding of the NMR signal.
The equation listed above in the section on T1 and T2 relaxation are those in the Bloch equations.
Solomon equations
Solomon equations are used to calculate the transfer of magnetization as a result of relaxation in a dipolar system. They can be employed to explain the nuclear Overhauser effect, which is an important tool in determining molecular structure.
Common relaxation time constants in human tissues Following is a table of the approximate values of the two relaxation time constants for hydrogen nuclear spins in nonpathological human tissues.
+ At a main field of 1.5 T Tissue type Approximate T1 value in ms Approximate T2 value in ms Adipose tissues 240-250 60-80Whole blood (deoxygenated) 1350 50Whole blood (oxygenated) 1350 200Cerebrospinal fluid (similar to pure water) 4200 - 4500 2100-2300Gray matter of cerebrum 920 100White matter of cerebrum 780 90Liver 490 40Kidneys 650 60-75Muscles 860-900 50
Following is a table of the approximate values of the two relaxation time constants for chemicals that commonly show up in human brain magnetic resonance spectroscopy (MRS) studies, physiologically or pathologically.
+ At a main field of 1.5 T Signals of chemical groups Relative resonance frequency Approximate T1 value (ms) Approximate T2 value (ms) Creatine (Cr) and Phosphocreatine (PCr)Chemicals of brain relaxation time at 1.5T. Kreis R, Ernst T, and Ross BD "Absolute Quantification of Water and Metabolites in the Human Brain. II. Metabolite Concentrations" Journal of Magnetic Resonance, Series B 102 (1993): 9-19 3.0 ppm gray matter: 1150–1340,white matter: 1050–1360 gray matter: 198–207,white matter: 194-218N-Acetyl group (NA),mainly from N-acetylaspartate (NAA) 2.0 ppm gray matter: 1170–1370,white matter: 1220-1410 gray matter: 388–426,white matter: 436-519 —CH3 group ofLactateLactate relaxation time at 1.5 T. Isobe T, Matsumura A, Anno I, Kawamura H, Muraishi H, Umeda T, Nose T. "Effect of J coupling and T2 Relaxation in Assessing of Methyl Lactate Signal using PRESS Sequence MR Spectroscopy." Igaku Butsuri (2005) v25. 2:68-74. 1.33 ppm(doublet: 1.27 & 1.39 ppm) (To be listed) 1040
Relaxation in the rotating frame, T1ρ The discussion above describes relaxation of nuclear magnetization in the presence of a constant magnetic field B0. This is called relaxation in the laboratory frame.