Relaxation (NMR) - Citizendia

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. Magnetization is defined as the quantity of Magnetic moment per unit volume In other words, relaxation describes how fast spins "forget" the direction in which they are oriented. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin The rates of this spin relaxation can be measured in both spectroscopy and imaging applications.

1 T₁ and T₂
2 Common relaxation time constants in human tissues
3 Microscopic mechanism
4 References
5 See also

T₁ and T₂

Different physical processes are responsible for the relaxation of the components of the nuclear spin magnetization vector M parallel and perpendicular to the external magnetic field, B₀ (which is conventionally oriented along the z axis). In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent These two principal relaxation processes are termed T₁ and T₂ relaxation respectively.

T₁

Main article: Spin-lattice relaxation time

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, $M z,eq$ . Spin-lattice relaxation time, known as T1, is a time constant in Nuclear Magnetic Resonance and Magnetic Resonance Imaging. A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value In general,

$M_z(t) = M_{z,\mathrm{eq}} - [M_{z,\mathrm{eq}} - M_z(0)]e^{-t/T_1}$

In specific cases:

If M has been tilted into the xy plane, then $M z (0) = 0$ and the recovery is simply

$M_z(t) = M_{z,\mathrm{eq}}\left( 1 - e^{-t/T_1} \right)$

i. e. the magnetisation 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 magnetisation is inverted, $M z (0) = - M z,eq$ , and so the recovery follows

$M_z(t) = M_{z,\mathrm{eq}}\left( 1 - 2e^{-t/T_1} \right)$

T₁ relaxation involves redistributing the populations of the nuclear spin states in order to reach the thermal equilibrium distribution. WikipediaWikiProject Probability#Standards for a discussion of standards used for probability distribution articles such as this one By definition this is not energy conserving. Moreover, spontaneous emission is negligibly slow at NMR frequencies. Spontaneous emission is the process by which a light source such as an Atom, Molecule, Nanocrystal or nucleus in an Excited state 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 may vary considerably with magnetic field strength, B.

T₂

Main article: Spin-spin relaxation time

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 $M_{\perp}$ . Spin-spin relaxation time, known as T2, is a time constant in Nuclear Magnetic Resonance and Magnetic Resonance Imaging. For instance, initial xy magnetisation at time zero will decay to zero (i. e. equilibrium) as follows:

$M_{xy}(t) = M_{xy}(0) e^{-t/T_2} \,$

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. In Quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior Random fluctuations of the local magnetic field lead to random variations in the instantaneous NMR precession frequency of different spins. In Physics, Larmor precession (named after Joseph Larmor) refers to the Precession of the Magnetic moments of Electrons atomic 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.

T₂* and magnetic field inhomogeneity

In an idealized system, all nuclei in a given chemical environment in a magnetic field spin 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 Fourier Transform NMR, a free induction decay (FID is the observable NMR signal generated by non-equilibrium nuclear spin magnetisation precessing about the In fact, for most magnetic resonance experiments, this "relaxation" dominates. This results in intra-voxel dephasing. A voxel (a Portmanteau of the words Volumetric and Pixel) is a volume element representing a value on a Regular grid in Dephasing is a process in which coherence in a substance caused by perturbation decays over time and the system returns to the state before perturbation

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. In Nuclear magnetic resonance, spin echo refers to the refocusing of precessing nuclear spin magnetisation by a 180° pulse of resonant radiofrequency

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

$\frac{1}{T_2^*}=\frac{1}{T_2}+\frac{1}{T_{inhom}} = \frac{1}{T_2}+\gamma \Delta B_0$

where γ represents gyromagnetic ratio, and ΔB₀ the difference in strength of the locally varying field. 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

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.

The reason that T₁ is slower than T₂

As a general rule, the following always holds true: T₁ > T₂ > T₂*.

If T₂ were to be slower than T₁, then the magnetizations perpendicular to the initial direction would have not dephased by the time the sample had returned to equilibrium. This is physically impossible, as once the sample has returned to equilibrium, there is no magnetization perpendicular to the original direction. Hence, T₁ must be greater than or equal to T₂.

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.

**At a main field of 1. 5 T**
Tissue Type	Approximate T₁ value in ms	Approximate T₂ value in ms
Adipose tissues	240-250	60-80
Whole blood (deoxygenated)	1350	50
Whole blood (oxygenated)	1350	200
Cerebrospinal fluid (similar to pure water)	2200-2400	500-1400
Gray matter of cerebrum	920	100
White matter of cerebrum	780	90
Liver	490	40
Kidneys	650	60-75
Muscles	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. The tesla (symbol T) is the SI derived unit of Magnetic field B (which is also known as "magnetic flux density" and "magnetic A millisecond (from Milli- and Second; abbreviation ms is one thousandth of a Second. "Adipose" redirects here For the Doctor Who monster see " Partners in Crime " Whole Blood is the term used in Transfusion medicine for human Blood from a standard Blood donation. Whole Blood is the term used in Transfusion medicine for human Blood from a standard Blood donation. Blood is a specialized Bodily fluid that delivers necessary substances to the body's cells ��such as nutrients and oxygen—and transports Waste products Cerebrospinal fluid ( CSF) Liquor cerebrospinalis, is a clear Bodily fluid that occupies the Subarachnoid space and the Ventricular system Water is a common Chemical substance that is essential for the survival of all known forms of Life. The telencephalon (tɛlɛnˈsɛfəlɒn cerebrum, or forebrain is the most Anterior or especially in humans most Dorsal region of the White matter is one of the three main solid components of the Central nervous system. The telencephalon (tɛlɛnˈsɛfəlɒn cerebrum, or forebrain is the most Anterior or especially in humans most Dorsal region of the The liver is a vital organ in the human body and is present in Vertebrates and some other animals The kidneys are complicated organs that have numerous biological roles Muscle (from Latin musculus, diminutive of mus "mouse" is contractile tissue of the body and is derived from the The brain is the center of the Nervous system in animals All Vertebrates and the majority of Invertebrates have a brain In vivo (that is 'in the living organism' magnetic resonance spectroscopy (MRS is a specialised technique associated with magnetic resonance imaging (MRI Physiology (from Greek grc φύσις physis, "nature origin" and grc -λογία -logia) is the study of the mechanical physical Pathology (from Greek grc πάθος pathos, "fate harm" and grc -λογία -logia) is the study and

**At a main field of 1. 5 T**
Creatine (Cr) and Phosphocreatine (PCr) ^[1]	3. Creatine is Nitrogenous Organic acid that occurs naturally in Vertebrates and helps to supply energy to Muscle and nerve cells Phosphocreatine, also known as creatine phosphate or Pcr, is a phosphorylated Creatine molecule that is an important energy store in skeletal 0 ppm	gray matter: 1150-1340, white matter: 1050-1360	gray matter: 198-207, white matter: 194-218
Signals of Chemical Groups	Relative resonance frequency	Approximate T₁ value (ms)	Approximate T₂ value (ms)
N-Acetyl group (NA), mainly from N-Acetylaspartate (NAA) ^[2]	2. 0 ppm	gray matter: 1170-1370, white matter: 1220-1410	gray matter: 388-426, white matter: 436-519
—CH₃ group of Lactate ^[3]	1. Lactic acid ( IUPAC Systematic name: 2-hydroxypropanoic acid) also known as milk acid, is a Chemical compound that plays a role 33 ppm (doublet: 1. 27 & 1. 39 ppm)	(To be listed)	1040

Microscopic mechanism

In 1948, Nicolaas Bloembergen, Edward Mills Purcell, and R. Year 1948 ( MCMXLVIII) was a Leap year starting on Thursday (link will display the 1948 calendar of the Gregorian calendar. Nicolaas Bloembergen (born Dordrecht, March 11, 1920) is a Dutch -born American physicist and Nobel laureate Edward Mills Purcell ( August 30, 1912 &ndash March 7, 1997) was an American physicist who shared the 1952 Nobel Prize for Physics V. 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 molecules on the local magnetic field disturbance ^[4]. In Chemistry, a molecule is defined as a sufficiently stable electrically neutral group of at least two Atoms in a definite arrangement held together by The theory was in good agreement with the experiments for pure substance, but not for complicated environment such as human body.

From this theory, one can get T₁、T₂:

$\frac{1}{T_1}=K[\frac{\tau_c}{1+\omega_0^2\tau_c^2}+\frac{4\tau_c}{1+4\omega_0^2\tau_c^2}]$

$\frac{1}{T_2}=\frac{K}{2}[3\tau_c+\frac{5\tau_c}{1+\omega_0^2\tau_c^2}+\frac{2\tau_c}{1+4\omega_0^2\tau_c^2}]$ ,

where $ω 0$ is the Larmor frequency in correspondence with the strength of the main magnetic field $B 0$ . In Physics, Larmor precession (named after Joseph Larmor) refers to the Precession of the Magnetic moments of Electrons atomic $τ c$ is the correlation time of the molecular tumbling motion. $K=\frac{3\mu^2}{160\pi^2}\frac{\hbar^2\gamma^4}{r^6}$ is a constant with μ being the magnetic dipole moment of the spin-1/2 nuclei, $\hbar=\frac{h}{2\pi}$ 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. In Physics, Astronomy, Chemistry, and Electrical engineering, the term magnetic moment of a system (such as a loop of Electric current In Quantum mechanics, spin is an intrinsic property of all elementary particles. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. 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

Taking for example the H₂O molecules in liquid phase without the contamination of oxygen 17, the value of K is 1. Liquid is one of the principal States of matter. A liquid is a Fluid that has the particles loose and can freely form a distinct surface at the boundaries of There are three stable isotopes of oxygen that lead to Oxygen ( O) having a standard atomic mass of 15 02×10¹⁰ s^-2 and the correlation time $τ c$ is on the order of picoseconds = $10 - 12$ s, while hydrogen nuclei ¹H (protons) at 1. To help compare Orders of magnitude of different Times this page lists times between 10&minus12 seconds and 10&minus11 seconds (1 Pico The second ( SI symbol s) sometimes abbreviated sec, is the name of a unit of Time, and is the International System of Units Hydrogen (ˈhaɪdrədʒən is the Chemical element with Atomic number 1 The proton ( Greek πρῶτον / proton "first" is a Subatomic particle with an Electric charge of one positive 5 teslas carry an Larmor frequency of approximately 64 MHz. The hertz (symbol Hz) is a measure of Frequency, informally defined as the number of events occurring per Second. We can then estimate using τ_c = 5×10^-12 s:

$\omega_0\tau_c = 3.2\times 10^{-5}$ (dimensionless)

$T_1=(1.02\times 10^{10}[\frac{ 5\times 10^{-12} }{1 + (3.2\times 10^{-5} )^2} + \frac{ 4\cdot 5\times 10^{-12} }{1 + 4\cdot (3.2\times 10^{-5} )^2}])^{-1}$ = 3. 92 s

$T_2=(\frac{1.02\times 10^{10}}{2}[3\cdot 5\times 10^{-12} + \frac{5\cdot 5\times 10^{-12} }{1 + (3.2\times 10^{-5} )^2} + \frac{ 2\cdot 5\times 10^{-12} }{1 + 4\cdot (3.2\times 10^{-5} )^2}])^{-1}$ = 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.

References

^ 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
^ Lactate 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.
^ BPP theory. Bloembergen, E. M. Purcell, R. V. Pound "Relaxation Effects in Nuclear Magnetic Resonance Absorption" Physical Review (1948) v73. 7:679-746

Contents

T1 and T2

T1

T2

T2* and magnetic field inhomogeneity

The reason that T1 is slower than T2