Relaxation (NMR)
Encyclopedia
In nuclear magnetic resonance (NMR)
spectroscopy and magnetic resonance imaging (MRI)
the term relaxation describes several processes by which nuclear magnetization
prepared in a non-equilibrium state return to the equilibrium distribution. In other words, relaxation describes how fast spins
"forget" the direction in which they are oriented. The rates of this spin relaxation can be measured in both spectroscopy and imaging applications.
to the
external magnetic field, B0 (which is conventionally oriented along the z axis). These two principal relaxation processes are termed T1 and T2 relaxation respectively.
. In general,
In specific cases:
i.e. the magnetization recovers to 63% of its equilibrium value after one time constant T1.
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 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.
. For instance, initial xy magnetisation 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.
A 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.
). 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.
.
In most situations (but not in principle)
is greater than 
.
equations that were introduced by Felix Bloch
in 1946.
Where 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. M(t) × B(t) is the cross product
of these two vectors.
The equation listed above in the section on T1 and T2 relaxation can be derived from Bloch equations.
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
.
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.
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
, 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 quantum mechanics shows that the relaxation rates (and times) depend on spectral density
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.
, Edward Mills Purcell
, 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
s on the local magnetic field disturbance. The theory was in good agreement 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
, where 
is called the correlation time. From this theory, one can get T1、T2 for magnetic dipolar relaxation:
,
where
is the Larmor frequency in correspondence with the strength of the main magnetic field 
. 
is the correlation time of the molecular tumbling motion. 
is a constant with μ being the magnetic dipole moment of the spin-1/2 nuclei, 
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 s-2 and the correlation time
is on the order of picosecond
s =
s
, while hydrogen nuclei
1H (proton
s) at 1.5 teslas carry an Larmor frequency of approximately 64 MHz. We can then estimate using τc = 5×10-12 s:
(dimensionless)
= 3.92 s
= 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.
Nuclear magnetic resonance
Nuclear magnetic resonance is a physical phenomenon in which magnetic nuclei in a magnetic field absorb and re-emit electromagnetic radiation...
spectroscopy and magnetic resonance imaging (MRI)
Magnetic resonance imaging
Magnetic resonance imaging , nuclear magnetic resonance imaging , or magnetic resonance tomography is a medical imaging technique used in radiology to visualize detailed internal structures...
the term relaxation describes several processes by which nuclear magnetization
Magnetization
In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material...
prepared in a non-equilibrium state return to the equilibrium distribution. In other words, relaxation describes how fast spins
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
"forget" the direction in which they are oriented. The rates of this spin relaxation can be measured in both spectroscopy and imaging applications.
T1 and T2
Different physical processes are responsible for the relaxation of the components of the nuclear spin magnetization vector M parallel and perpendicularPerpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...
to the
external magnetic field, B0 (which is conventionally oriented along the z axis). These two principal relaxation processes are termed T1 and T2 relaxation respectively.
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
Boltzmann distribution
In chemistry, physics, and mathematics, the Boltzmann distribution is a certain distribution function or probability measure for the distribution of the states of a system. It underpins the concept of the canonical ensemble, providing its underlying distribution...
. By definition this is not energy conserving. Moreover, spontaneous emission
Spontaneous emission
Spontaneous emission is the process by which a light source such as an atom, molecule, nanocrystal or nucleus in an excited state undergoes a transition to a state with a lower energy, e.g., the ground state and emits a photon...
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 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
Oxygen is the element with atomic number 8 and represented by the symbol O. Its name derives from the Greek roots ὀξύς and -γενής , because at the time of naming, it was mistakenly thought that all acids required oxygen in their composition...
, the T1/T2 of liquid samples easily go up to an order of ten seconds.
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 magnetisation 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
Larmor precession
In physics, Larmor precession is the precession of the magnetic moments of electrons, atomic nuclei, and atoms about an external magnetic field...
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.
A Hahn echo
Spin echo
In magnetic resonance, a spin echo is the refocusing of precessing spin magnetisation by a pulse of resonant radiation. Modern nuclear magnetic resonance and magnetic resonance imaging rely heavily on this effect....
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 DecayFree induction decay
In Fourier Transform NMR, free induction decay is the observable NMR signal generated by non-equilibrium nuclear spin magnetisation precessing about the magnetic field ....
). In fact, for most magnetic resonance experiments, this "relaxation" dominates. This results in dephasing
Dephasing
Dephasing is a name for the mechanism that recovers classical behavior from a quantum system. It is an important effect in condensed matter physics, particularly in the study of mesoscopic devices...
.
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
Spin echo
In magnetic resonance, a spin echo is the refocusing of precessing spin magnetisation by a pulse of resonant radiation. Modern nuclear magnetic resonance and magnetic resonance imaging rely heavily on this effect....
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?
The following always holds true:.In most situations (but not in principle)
is greater than .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 phenomenologicalPhenomenology (science)
The term phenomenology in science is used to describe a body of knowledge that relates empirical observations of phenomena to each other, in a way that is consistent with fundamental theory, but is not directly derived from theory. For example, we find the following definition in the Concise...
equations that were introduced by Felix Bloch
Felix Bloch
Felix Bloch was a Swiss physicist, working mainly in the U.S.-Life and work:Bloch was born in Zürich, Switzerland to Jewish parents Gustav and Agnes Bloch. He was educated there and at the Eidgenössische Technische Hochschule, also in Zürich. Initially studying engineering he soon changed to physics...
in 1946.
Where 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. M(t) × B(t) is the cross product
Cross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
of these two vectors.
The equation listed above in the section on T1 and T2 relaxation can be derived from Bloch equations.
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.| Tissue Type | Approximate T1 value in ms Millisecond A millisecond is a thousandth of a second.10 milliseconds are called a centisecond.... |
Approximate T2 value in ms |
|---|---|---|
| Adipose tissue Adipose tissue In histology, adipose tissue or body fat or fat depot or just fat is loose connective tissue composed of adipocytes. It is technically composed of roughly only 80% fat; fat in its solitary state exists in the liver and muscles. Adipose tissue is derived from lipoblasts... s |
240-250 | 60-80 |
| Whole blood Whole blood Whole blood is a term used in transfusion medicine for human blood from a standard blood donation. The blood is typically combined with an anticoagulant during the collection process, but is generally otherwise unprocessed... (deoxygenated) |
1350 | 50 |
| Whole blood Whole blood Whole blood is a term used in transfusion medicine for human blood from a standard blood donation. The blood is typically combined with an anticoagulant during the collection process, but is generally otherwise unprocessed... (oxygenated) |
1350 | 200 |
| Cerebrospinal fluid Cerebrospinal fluid Cerebrospinal fluid , Liquor cerebrospinalis, is a clear, colorless, bodily fluid, that occupies the subarachnoid space and the ventricular system around and inside the brain and spinal cord... (similar to pure water Water Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a... ) |
4200 - 4500 | 2100-2300 |
| Gray matter Gray Matter "Gray Matter" is a short story by Stephen King, first published in the October 1973 issue of Cavalier magazine, and later collected in King's 1978 collection Night Shift. It is set in the same area as King's novel Dreamcatcher.-Setting:... of cerebrum |
920 | 100 |
| White matter White matter White matter is one of the two components of the central nervous system and consists mostly of myelinated axons. White matter tissue of the freshly cut brain appears pinkish white to the naked eye because myelin is composed largely of lipid tissue veined with capillaries. Its white color is due to... of cerebrum |
780 | 90 |
| Liver Liver The liver is a vital organ present in vertebrates and some other animals. It has a wide range of functions, including detoxification, protein synthesis, and production of biochemicals necessary for digestion... |
490 | 40 |
| Kidneys | 650 | 60-75 |
| Muscle Muscle Muscle is a contractile tissue of animals and is derived from the mesodermal layer of embryonic germ cells. Muscle cells contain contractile filaments that move past each other and change the size of the cell. They are classified as skeletal, cardiac, or smooth muscles. Their function is to... s |
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
Brain
The brain is the center of the nervous system in all vertebrate and most invertebrate animals—only a few primitive invertebrates such as sponges, jellyfish, sea squirts and starfishes do not have one. It is located in the head, usually close to primary sensory apparatus such as vision, hearing,...
magnetic resonance spectroscopy (MRS) studies, physiologically
Physiology
Physiology is the science of the function of living systems. This includes how organisms, organ systems, organs, cells, and bio-molecules carry out the chemical or physical functions that exist in a living system. The highest honor awarded in physiology is the Nobel Prize in Physiology or...
or pathologically
Pathology
Pathology is the precise study and diagnosis of disease. The word pathology is from Ancient Greek , pathos, "feeling, suffering"; and , -logia, "the study of". Pathologization, to pathologize, refers to the process of defining a condition or behavior as pathological, e.g. pathological gambling....
.
| Signals of Chemical Groups | Relative resonance frequency | Approximate T1 value (ms) | Approximate T2 value (ms) |
|---|---|---|---|
| Creatine (Cr) Creatine Creatine is a nitrogenous organic acid that occurs naturally in vertebrates and helps to supply energy to all cells in the body, primarily muscle. This is achieved by increasing the formation of Adenosine triphosphate... and Phosphocreatine (PCr) Phosphocreatine Phosphocreatine, also known as creatine phosphate or PCr , is a phosphorylated creatine molecule that serves as a rapidly mobilizable reserve of high-energy phosphates in skeletal muscle and brain.-Chemistry:... |
3.0 ppm | gray matter: 1150-1340, white matter: 1050-1360 |
gray matter: 198-207, white matter: 194-218 |
| N-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 of Lactate Lactic acid Lactic acid, also known as milk acid, is a chemical compound that plays a role in various biochemical processes and was first isolated in 1780 by the Swedish chemist Carl Wilhelm Scheele. Lactic acid is a carboxylic acid with the chemical formula C3H6O3... |
1.33 ppm (doublet: 1.27 & 1.39 ppm) |
(To be listed) | 1040 |
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 interactionMagnetic dipole-dipole interaction
Magnetic dipole–dipole interaction, also called dipolar coupling, refers to the direct interaction between two magnetic dipoles. The potential energy of the interaction is as follows:...
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
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, time-dependent interaction energies cause transitions between the nuclear spin states which result in nuclear spin relaxation. The application of time-dependent perturbation theory
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
in quantum mechanics shows that the relaxation rates (and times) depend on spectral density
Spectral density
In statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...
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
Quadrupole
A quadrupole or quadrapole is one of a sequence of configurations of—for example—electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity.-Mathematical...
moment and the electric field gradient
Electric field gradient
In atomic, molecular, and solid-state physics, the electric field gradient measures the rate of change of the electric field at an atomic nucleus generated by the electronic charge distribution and the other nuclei...
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 BloembergenNicolaas Bloembergen
Nicolaas Bloembergen is a Dutch-American physicist and Nobel laureate.He received his Ph.D. degree from University of Leiden in 1948; while pursuing his PhD at Harvard, Bloembergen also worked part-time as a graduate research assistant for Edward Mills Purcell at the MIT Radiation Laboratory...
, Edward Mills Purcell
Edward Mills Purcell
Edward Mills Purcell was an American physicist who shared the 1952 Nobel Prize for Physics for his independent discovery of nuclear magnetic resonance in liquids and in solids. Nuclear magnetic resonance has become widely used to study the molecular structure of pure materials and the...
, and Robert Pound
Robert Pound
Robert Vivian Pound was an American physicist who helped discover nuclear magnetic resonance and who devised the famous Pound-Rebka experiment supporting general relativity .Pound was born in Ridgeway, Ontario....
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
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
s on the local magnetic field disturbance. The theory was in good agreement 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
, where is called the correlation time. From this theory, one can get T1、T2 for magnetic dipolar relaxation:,where
is the Larmor frequency in correspondence with the strength of the main magnetic field . is the correlation time of the molecular tumbling motion. is a constant with μ being the magnetic dipole moment of the spin-1/2 nuclei, 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
Oxygen-17
Oxygen-17 is a low abundant isotope of oxygen . Being the only stable isotope of oxygen possessing a nuclear spin and the unique characteristic of field-independent relaxation it enables NMR studies of metabolic pathways of compounds incorporating oxygen at high magnetic fields Oxygen-17 is a low...
, the value of K is 1.02×1010 s-2 and the correlation time
is on the order of picosecondPicosecond
A picosecond is 10−12 of a second. That is one trillionth, or one millionth of one millionth of a second, or 0.000 000 000 001 seconds. A picosecond is to one second as one second is to 31,700 years....
s =
sSecond
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
, while hydrogen nuclei
Hydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...
1H (proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
s) at 1.5 teslas carry an Larmor frequency of approximately 64 MHz. We can then estimate using τc = 5×10-12 s:
(dimensionless)= 3.92 s= 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.
See also
- basics of NMR
- Relaxation in high-resolution NMR spectroscopy
- Field-cycling NMR relaxometry
- RelaxometryRelaxometryRelaxometry refers to the study and/or measurement of relaxation variables in Nuclear Magnetic Resonance and Magnetic Resonance Imaging.-Links:*...