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In MRI and
NMR spectroscopy Nuclear magnetic resonance spectroscopy, most commonly known as NMR spectroscopy or magnetic resonance spectroscopy (MRS), is a spectroscopic technique to observe local magnetic fields around atomic nuclei. The sample is placed in a magnetic fi ...
, an observable nuclear spin polarization (
magnetization In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or D ...
) is created by a homogeneous magnetic field. This field makes the magnetic dipole moments of the sample precess at the resonance (
Larmor Sir Joseph Larmor (11 July 1857 – 19 May 1942) was an Irish and British physicist and mathematician who made breakthroughs in the understanding of electricity, dynamics, thermodynamics, and the electron theory of matter. His most influen ...
) 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 radiofrequent (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=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: ::: \frac = e^ 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 Spontaneous emission is the process in which a quantum mechanical system (such as a molecule, an atom or a subatomic particle) transits from an excited energy state to a lower energy state (e.g., its ground state) and emits a quantized amount ...
. 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.


''T''1 and ''T''2

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 ''T''1, is responsible for the loss of resonance intensity following pulse excitation. The other process, called ''T''2, characterizes the width or broadness of resonances. Stated more formally, ''T''1 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). ''T''2 relaxation affects the coherent components of M perpendicular to B0. In conventional NMR spectroscopy, ''T''1 limits the pulse repetition rate and affects the overall time an NMR spectrum can be acquired. Values of ''T''1 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).


''T''1

The longitudinal (or spin-lattice) relaxation time ''T''1 is the
decay constant A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
for the recovery of the ''z'' component of the nuclear spin magnetization, ''Mz'', towards its thermal equilibrium value, M_. In general, :M_z(t) = M_ - _ - M_z(0)^ In specific cases: *If M has been tilted into the ''xy'' plane, then M_z(0)=0 and the recovery is simply :M_z(t) = M_\left( 1 - e^ \right) i.e. the magnetization recovers to 63% of its equilibrium value after one time constant ''T''1. *In the
inversion recovery Inversion recovery is an MRI sequence that provides high contrast between tissue and lesion. It can be used to provide high T1 weighted image, high T2 weighted image, and to suppress the signals from fat, blood, or cerebrospinal fluid (CSF). Fluid- ...
experiment, commonly used to measure ''T''1 values, the initial magnetization is inverted, M_z(0)=-M_, and so the recovery follows :M_z(t) = M_\left( 1 - 2e^ \right) ''T''1 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 Spontaneous emission is the process in which a quantum mechanical system (such as a molecule, an atom or a subatomic particle) transits from an excited energy state to a lower energy state (e.g., its ground state) and emits a quantized amount ...
is negligibly slow at NMR frequencies. Hence truly isolated nuclear spins would show negligible rates of ''T''1 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 ''T''1 relaxation involves an interaction with the surroundings is the origin of the alternative description, ''spin-lattice relaxation''. Note that the rates of ''T''1 relaxation (i.e., 1/''T''1) 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 Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements ...
, the ''T''1/''T''2 of liquid samples easily go up to an order of ten seconds.


Spin saturation transfer

Especially for molecules exhibiting slowly relaxing (''T''1) signals, the technique spin saturation transfer (SST) provides information on chemical exchange reactions. The method is widely applicable to
fluxional molecule In chemistry and molecular physics, fluxional (or non-rigid) molecules are molecules that undergo dynamics such that some or all of their atoms interchange between symmetry-equivalent positions. Because virtually all molecules are fluxional in ...
s. This magnetization transfer technique provides rates, provided that they exceed 1/''T''1.


''T''2

The transverse (or spin-spin) relaxation time ''T''2 is the decay constant for the component of M perpendicular to B0, designated Mxy, MT, or M_. For instance, initial ''xy'' magnetization at time zero will decay to zero (i.e. equilibrium) as follows: :M_(t) = M_(0) e^ \, i.e. the transverse magnetization vector drops to 37% of its original magnitude after one time constant ''T''2. ''T''2 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 Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In oth ...
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 ''T''2 relaxation involves only the phases of other nuclear spins it is often called "spin-spin" relaxation. ''T''2 values are generally much less dependent on field strength, B, than ''T''1 values. Hahn echo decay experiment can be used to measure the ''T''2 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 A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
is measured which is described by the T_2 time.


''T''2* 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: : \frac=\frac+\frac = \frac+\gamma \Delta B_0 where γ represents
gyromagnetic ratio In physics, the gyromagnetic ratio (also sometimes known as the magnetogyric ratio in other disciplines) of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol , gamma. Its SI u ...
, 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 ''T''1 always longer than ''T''2?

In NMR systems, the following relation holds absolute true T_2 \le 2 T_1 . In most situations (but not in principle) T_1 is greater than T_2. The cases in which 2 T_1 > T_2 > T_1 are rare, but not impossible.


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 ''T''1 and ''T''2 are present. Bloch equations are phenomenological equations that were introduced by Felix Bloch in 1946. :\frac = \gamma ( \mathbf (t) \times \mathbf (t) ) _x - \frac :\frac = \gamma ( \mathbf (t) \times \mathbf (t) ) _y - \frac :\frac = \gamma ( \mathbf (t) \times \mathbf (t) ) _z - \frac Where \times is the cross-product, γ is the gyromagnetic ratio and B(''t'') = (''B''''x''(''t''), ''B''''y''(''t''), ''B''0 + ''B''''z''(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, ''B''0, is constant in time, the other one, ''B''''z''(t), is time dependent. It is present in
magnetic resonance imaging Magnetic resonance imaging (MRI) is a medical imaging technique used in radiology to form pictures of the anatomy and the physiological processes of the body. MRI scanners use strong magnetic fields, magnetic field gradients, and radio wave ...
and helps with the spatial decoding of the NMR signal. The equation listed above in the section on ''T''1 and ''T''2 relaxation are those in the Bloch equations.


Solomon equations

Solomon equations are used to calculate the transfer of
magnetization In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or D ...
as a result of relaxation in a
dipolar In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways: *An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this syste ...
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. Following is a table of the approximate values of the two relaxation time constants for chemicals that commonly show up in human
brain A brain is an organ (biology), organ that serves as the center of the nervous system in all vertebrate and most invertebrate animals. It is located in the head, usually close to the sensory organs for senses such as Visual perception, vision. I ...
magnetic resonance spectroscopy (MRS) studies,
physiologically Physiology (; ) is the scientific study of functions and mechanisms in a living system. As a sub-discipline of biology, physiology focuses on how organisms, organ systems, individual organs, cells, and biomolecules carry out the chemical a ...
or
pathologically Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in ...
.


Relaxation in the rotating frame, ''T''

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. Another technique, called relaxation in the rotating frame, is the relaxation of nuclear magnetization in the presence of the field B0 together with a time-dependent magnetic field B1. The field B1 rotates in the plane perpendicular to B0 at the Larmor frequency of the nuclei in the B0. The magnitude of B1 is typically much smaller than the magnitude of B0. Under these circumstances the relaxation of the magnetization is similar to laboratory frame relaxation in a field B1. The decay constant for the recovery of the magnetization component along B1 is called the spin-lattice relaxation time in the rotating frame and is denoted ''T''. Relaxation in the rotating frame is useful because it provides information on slow motions of nuclei.


Microscopic mechanisms

Relaxation of nuclear spins requires a microscopic mechanism for a nucleus to change orientation with respect to the applied magnetic field and/or interchange energy with the surroundings (called the lattice). The most common mechanism is the
magnetic dipole-dipole interaction Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
between the magnetic moment of a nucleus and the magnetic moment of another nucleus or other entity (electron, atom, ion, molecule). This interaction depends on the distance between the pair of dipoles (spins) but also on their orientation relative to the external magnetic field. Several other relaxation mechanisms also exist. The chemical shift anisotropy (CSA) relaxation mechanism arises whenever the electronic environment around the nucleus is non spherical, the magnitude of the electronic shielding of the nucleus will then be dependent on the molecular orientation relative to the (fixed) external magnetic field. The spin rotation (SR) relaxation mechanism arises from an interaction between the nuclear spin and a coupling to the overall molecular rotational angular momentum. Nuclei with spin I ≥ 1 will have not only a nuclear dipole but a quadrupole. The nuclear quadrupole has an interaction with the electric field gradient at the nucleus which is again orientation dependent as with the other mechanisms described above, leading to the so-called quadrupolar relaxation mechanism. Molecular reorientation or tumbling can then modulate these orientation-dependent spin interaction energies. According to
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, time-dependent interaction energies cause transitions between the nuclear spin states which result in nuclear spin relaxation. The application of time-dependent
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
in quantum mechanics shows that the relaxation rates (and times) depend on
spectral density The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies ...
functions that are the Fourier transforms of the autocorrelation function of the fluctuating magnetic dipole interactions. The form of the spectral density functions depend on the physical system, but a simple approximation called the BPP theory is widely used. Another relaxation mechanism is the electrostatic interaction between a nucleus with an electric quadrupole moment and the electric field gradient that exists at the nuclear site due to surrounding charges. Thermal motion of a nucleus can result in fluctuating electrostatic interaction energies. These fluctuations produce transitions between the nuclear spin states in a similar manner to the magnetic dipole-dipole interaction.


BPP theory

In 1948,
Nicolaas Bloembergen Nicolaas Bloembergen (March 11, 1920 – September 5, 2017) was a Dutch- American physicist and Nobel laureate, recognized for his work in developing driving principles behind nonlinear optics for laser spectroscopy. During his career, he was a p ...
,
Edward Mills Purcell Edward Mills Purcell (August 30, 1912 – March 7, 1997) was an American physicist who shared the 1952 Nobel Prize for Physics for his independent discovery (published 1946) of nuclear magnetic resonance in liquids and in solids. Nuclear mag ...
, and Robert 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
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and b ...
s on the local magnetic field disturbance. The theory agrees well with experiments on pure substances, but not for complicated environments such as the human body. This theory makes the assumption that the autocorrelation function of the microscopic fluctuations causing the relaxation is proportional to e^, where \tau_c is called the correlation time. From this theory, one can get T1 > T2 for magnetic dipolar relaxation: :\frac=K\left frac+\frac\right/math> :\frac=\frac\left \tau_c+\frac+\frac\right/math>, 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\frac is defined for spin-1/2 nuclei and is a constant with \mu_0 being the magnetic permeability of free space of the \hbar=\frac the
reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
, γ the
gyromagnetic ratio In physics, the gyromagnetic ratio (also sometimes known as the magnetogyric ratio in other disciplines) of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol , gamma. Its SI u ...
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 A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, ...
without the contamination of oxygen-17, the value of ''K'' is 1.02×1010 s−2 and the correlation time \tau_c is on the order of
picosecond A picosecond (abbreviated as ps) is a unit of time in the International System of Units (SI) equal to 10−12 or (one trillionth) of a second. That is one trillionth, or one millionth of one millionth of a second, or 0.000 000 000&nbs ...
s = 10^ s, while hydrogen nuclei 1H (
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
s) at 1.5 tesla precess at a Larmor frequency of approximately 64 MHz (Simplified. BPP theory uses angular frequency indeed). We can then estimate using ''τ''c = 5×10−12 s: :\omega_0\tau_c = 3.2\times 10^ (dimensionless) :T_1=\left(1.02\times 10^\left frac + \frac\rightright)^ = 3.92 s :T_2=\left(\frac\left \cdot 5\times 10^ + \frac + \frac\rightright)^ = 3.92 s, which is close to the experimental value, 3.6 s. Meanwhile, we can see that at this extreme case, T1 equals T2. As follows from the BPP theory, measuring the T1 times leads to internuclear distances r. One of the examples is accurate determinations of the metal – hydride (M-H) bond lengths in solutions by measurements of 1H selective and non-selective T1 times in variable-temperature relaxation experiments via the equation:D. G. Gusev, A. B. Vymenits, V. I. Bakhmutov Short spin-lattice relaxation times of hydride ligands. Proton-metal dipole-dipole interactions Inorg. Chem., 1991, 30 (16), p. 3116. DOI: 10.1021/ic00016a003Inorg. Chem., 1993, 32 (15), p. 3270. :r(M-H) = C \left(\frac\right)^ :k = (f-1)/(0.5-f/3), with f = T_ / T_1 :C = 10^7 \left(\frac\right)^ where r, frequency and T1 are measured in Å, MHz and s, respectively, and ''IM'' is the spin of M.


See also

*
Nuclear magnetic resonance Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are perturbed by a weak oscillating magnetic field (in the near field) and respond by producing an electromagnetic signal with a ...
* Nuclear magnetic resonance spectroscopy * Nuclear magnetic resonance spectroscopy of carbohydrates *
Nuclear magnetic resonance spectroscopy of nucleic acids Nucleic acid NMR is the use of nuclear magnetic resonance spectroscopy to obtain information about the structure and dynamics of nucleic acid molecules, such as DNA or RNA. It is useful for molecules of up to 100 nucleotides, and as of 2003, nearl ...
* Nuclear magnetic resonance spectroscopy of proteins *
Protein dynamics Proteins are generally thought to adopt unique structures determined by their amino acid sequences. However, proteins are not strictly static objects, but rather populate ensembles of (sometimes similar) conformations. Transitions between these stat ...
*
Relaxation (physics) In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium. Each relaxation process can be categorized by a relaxation time τ. The simplest theoretical description of relaxation as function of time ' ...
* Relaxometry *
Spin–lattice relaxation During nuclear magnetic resonance observations, spin–lattice relaxation is the mechanism by which the longitudinal component of the total nuclear magnetic moment vector (parallel to the constant magnetic field) exponentially relaxes from a higher ...
* Spin–spin relaxation


References


External links


The Basics of NMR
RIT
Field-cycling NMR relaxometryrelax
Software for the analysis of NMR dynamics
Estimation of T1 and T2 relaxation parameters in MRI
{{quantum information Nuclear magnetic resonance Articles containing video clips Magnetic resonance imaging