|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Expert-subject|Spectroscopy|date=November 2008}}
| | [http://home-Hiv-tests.com/home-test-kits/std-tests Marvella] is what you at home std test can contact her but it's not the most female title out there. Hiring is her working day occupation now but [http://test.gydch.com/praise/919726 std home test] she's always needed at home std test her personal business. South Dakota is where me and my husband live and my family members loves it. One of the extremely [http://www.pornextras.info/blog/187567 www.pornextras.info] very best issues in the world for him is to collect badges but he is struggling to [http://Articles.Baltimoresun.com/2011-02-21/specialsection/bs-hs-std-testing-20110221_1_std-testing-gonorrhea-and-chlamydia-chlamydia-cases discover time] for it.<br><br>Feel free to visit my weblog :: home std test ([http://vip.akwacity.com/oxwall/blogs/post/10278 go to this site]) |
| | |
| In [[Nuclear magnetic resonance|nuclear magnetic resonance (NMR)]] spectroscopy and [[Magnetic resonance imaging|magnetic resonance imaging (MRI)]] the term '''relaxation''' describes several processes by which nuclear [[magnetization]] prepared in a non-equilibrium state returns to the equilibrium distribution. In other words, relaxation describes how fast [[spin (physics)|spins]] "forget" the direction in which they are oriented. The rates of this spin relaxation can be measured in both spectroscopy and imaging applications.
| |
| | |
| ==''T''<sub>1</sub> and ''T''<sub>2</sub>==
| |
| 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, '''B'''<sub>0</sub> (which is conventionally oriented along the ''z'' axis). These two principal relaxation processes are termed T<sub>1</sub> and T<sub>2</sub> relaxation respectively.
| |
| | |
| ===''T''<sub>1</sub>===
| |
| {{main | Spin-lattice relaxation time}}
| |
| | |
| The longitudinal (or spin-lattice) relaxation time ''T''<sub>1</sub> is the [[decay constant]] for the recovery of the ''z'' component of the nuclear spin magnetization, ''M<sub>z</sub>'', towards its thermal equilibrium value, <math>M_{z,\mathrm{eq}}</math>. In general,
| |
| :<math>M_z(t) = M_{z,\mathrm{eq}} - [M_{z,\mathrm{eq}} - M_z(0)]e^{-t/T_1}</math>
| |
| | |
| In specific cases:
| |
| * If '''M''' has been tilted into the ''xy'' plane, then <math>M_z(0)=0</math> and the recovery is simply
| |
| :<math>M_z(t) = M_{z,\mathrm{eq}}\left( 1 - e^{-t/T_1} \right)</math>
| |
| i.e. the magnetization recovers to 63% of its equilibrium value after one time constant ''T''<sub>1</sub>.
| |
| | |
| * In the [[inversion recovery]] experiment, commonly used to measure ''T''<sub>1</sub> values, the initial magnetization is inverted, <math>M_z(0)=-M_{z,\mathrm{eq}}</math>, and so the recovery follows
| |
| :<math>M_z(t) = M_{z,\mathrm{eq}}\left( 1 - 2e^{-t/T_1} \right)</math>
| |
| | |
| ''T''<sub>1</sub> relaxation involves redistributing the populations of the nuclear spin states in order to reach the [[Boltzmann distribution|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 ''T''<sub>1</sub> 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''<sub>1</sub> relaxation involves an interaction with the surroundings is the origin of the alternative description, ''spin-lattice relaxation''.
| |
| | |
| Note that the rates of ''T''<sub>1</sub> 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 ''T''<sub>1</sub>/''T''<sub>2</sub> of liquid samples easily go up to an order of ten seconds.
| |
| | |
| ===''T''<sub>2</sub>===
| |
| {{main|Spin-spin relaxation time}}
| |
| | |
| The transverse (or spin-spin) relaxation time ''T''<sub>2</sub> is the decay constant for the component of '''M''' perpendicular to '''B'''<sub>0</sub>, designated '''M'''<sub>xy</sub>, '''M'''<sub>T</sub>, or <math>M_{\perp}</math>. For instance, initial ''xy'' magnetization at time zero will decay to zero (i.e. equilibrium) as follows:
| |
| :<math>M_{xy}(t) = M_{xy}(0) e^{-t/T_2} \,</math>
| |
| i.e. the transverse magnetization vector drops to 37% of its original magnitude after one time constant ''T''<sub>2</sub>.
| |
| | |
| ''T''<sub>2</sub> 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 [[Larmor precession|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 ''T''<sub>2</sub> relaxation involves only the phases of other nuclear spins it is often called "spin-spin" relaxation.
| |
| | |
| ''T''<sub>2</sub> values are generally much less dependent on field strength, B, than ''T''<sub>1</sub> values.
| |
| | |
| A [[Spin echo|Hahn echo]] decay experiment can be used to measure the ''T''<sub>2</sub> 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 <math>T_2</math> time.
| |
| | |
| [[File:GWM HahnEchoDecay.gif]]
| |
| | |
| ===''T''<sub>2</sub>* 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 T<sub>2</sub><sup>*</sup>, which is usually much smaller than T<sub>2</sub>. The relation between them is:
| |
| : <math>\frac{1}{T_2^*}=\frac{1}{T_2}+\frac{1}{T_{inhom}} = \frac{1}{T_2}+\gamma \Delta B_0 </math> | |
| where γ represents [[gyromagnetic ratio]], and ΔB<sub>0</sub> the difference in strength of the locally varying field.
| |
| | |
| Unlike T<sub>2</sub>, T<sub>2</sub>* is influenced by magnetic field gradient irregularities. The T<sub>2</sub>* relaxation time is always shorter than the T<sub>2</sub> relaxation time and is typically milliseconds for water samples in imaging magnets.
| |
| | |
| ===Is ''T''<sub>1</sub> always longer than ''T''<sub>2</sub>?===
| |
| The following always holds true:<ref>Malcolm H. Levitt: ''Spin Dynamics: Basics of Nuclear Magnetic Resonance'', 2nd edition, John Wiley & Sons, New York 2008, ISBN 0-470-51117-6, Section 11.9.2</ref> <math>2 T_1 \ge T_2 \ge T_2^*</math>.
| |
| In most situations (but not in principle) <math>T_1</math> is greater than <math>T_2</math>.
| |
| | |
| ==Bloch equations==
| |
| {{Main|Bloch equations}}
| |
| | |
| Bloch equations are used to calculate the nuclear magnetization '''M''' = (''M''<sub>''x''</sub>, ''M''<sub>''y''</sub>, ''M''<sub>''z''</sub>) as a function of time when relaxation times ''T''<sub>1</sub> and ''T''<sub>2</sub> are present. Bloch equations are [[Phenomenology (science)|phenomenological]] equations that were introduced by [[Felix Bloch]] in 1946.<ref>F Bloch, ''Nuclear Induction'', Physical Review '''70''', 460-473 (1946)</ref>
| |
| | |
| :<math>\frac {\partial M_x(t)} {\partial t} = \gamma ( \bold {M} (t) \times \bold {B} (t) ) _x - \frac {M_x(t)} {T_2}</math>
| |
| :<math>\frac {\partial M_y(t)} {\partial t} = \gamma ( \bold {M} (t) \times \bold {B} (t) ) _y - \frac {M_y(t)} {T_2}</math>
| |
| :<math>\frac {\partial M_z(t)} {\partial t} = \gamma ( \bold {M} (t) \times \bold {B} (t) ) _z - \frac {M_z(t) - M_0} {T_1}</math>
| |
| | |
| Where γ is the gyromagnetic ratio and '''B'''(''t'') = (''B''<sub>''x''</sub>(''t''), ''B''<sub>''y''</sub>(''t''), ''B''<sub>0</sub> + ''B''<sub>''z''</sub>(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''<sub>0</sub>, is constant in time, the other one, ''B''<sub>''z''</sub>(t), is time dependent. It is present in [[MRI|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 ''T''<sub>1</sub> and ''T''<sub>2</sub> relaxation can be derived from Bloch equations.
| |
| | |
| ==Solomon equations==
| |
| {{main|Solomon equations}}
| |
| | |
| Solomon equations are used to calculate the transfer of [[magnetization]] as a result of relaxation in a [[dipole|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 nonpathological human tissues, just for simple reference.
| |
| | |
| <br style="clear:both;">
| |
| {| class="wikitable"
| |
| |- align=center border="1" cellpadding="3" cellspacing="0" bgcolor="#DDDDDD"
| |
| <caption><big>'''At a main field of 1.5 [[tesla (unit)|T]]'''</big></caption>
| |
| |Tissue Type
| |
| ! Approximate T<sub>1</sub> value in [[millisecond|ms]]
| |
| ! Approximate T<sub>2</sub> value in ms
| |
| |-
| |
| |[[Adipose tissue]]s
| |
| ! 240-250
| |
| ! 60-80
| |
| |-
| |
| |[[Whole blood]] ([[deoxygenated blood|deoxygenated]])
| |
| ! 1350
| |
| ! 50
| |
| |-
| |
| |[[Whole blood]] ([[oxygenated blood|oxygenated]])
| |
| ! 1350
| |
| ! 200
| |
| |-
| |
| |[[Cerebrospinal fluid]] (similar to pure [[water]])
| |
| ! 4200 - 4500
| |
| ! 2100-2300
| |
| |-
| |
| |[[Gray matter]] of [[cerebrum]]
| |
| ! 920
| |
| ! 100
| |
| |-
| |
| |[[White matter]] of [[cerebrum]]
| |
| ! 780
| |
| ! 90
| |
| |-
| |
| |[[Liver]]
| |
| ! 490
| |
| ! 40
| |
| |-
| |
| |[[Kidneys]]
| |
| ! 650
| |
| ! 60-75
| |
| |-
| |
| |[[Muscle]]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]] [[Magnetic resonance spectroscopy|magnetic resonance spectroscopy (MRS)]] studies, [[physiology|physiologically]] or [[pathology|pathologically]].
| |
| | |
| <br style="clear:both;">
| |
| {| class="wikitable"
| |
| |- align=center border="1" cellpadding="3" cellspacing="0" bgcolor="#DDDDDD"
| |
| <caption><big>'''At a main field of 1.5 T'''</big></caption>
| |
| | Signals of Chemical Groups
| |
| ! Relative resonance frequency
| |
| ! Approximate T<sub>1</sub> value (ms)
| |
| ! Approximate T<sub>2</sub> value (ms)
| |
| |-
| |
| ! [[Creatine|Creatine (Cr)]] and [[Phosphocreatine|Phosphocreatine (PCr)]]<ref name="brain">'''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</ref>
| |
| ! 3.0 ppm
| |
| ! gray matter: 1150-1340, <br>white matter: 1050-1360
| |
| ! gray matter: 198-207, <br>white matter: 194-218
| |
| |-
| |
| !N-Acetyl group (NA), <br />mainly from [[N-Acetylaspartate|N-Acetylaspartate (NAA)]]<ref name="brain"/>
| |
| ! 2.0 ppm
| |
| ! gray matter: 1170-1370, <br>white matter: 1220-1410
| |
| ! gray matter: 388-426, <br>white matter: 436-519
| |
| |-
| |
| ! —CH<sub>3</sub> group of <br />[[lactic acid|Lactate]]<ref>'''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.</ref>
| |
| ! 1.33 ppm <br />(doublet: 1.27 & 1.39 ppm)
| |
| ! (To be listed)
| |
| ! 1040
| |
| |}
| |
| | |
| ==Relaxation in the rotating frame, ''T''<sub>1ρ</sub>==
| |
| The discussion above describes relaxation of nuclear magnetization in the presence of a constant magnetic field '''B'''<sub>0</sub>. 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 '''B'''<sub>0</sub> together with a time-dependent magnetic field '''B'''<sub>1</sub>. The field '''B'''<sub>1</sub> rotates in the plane perpendicular to '''B'''<sub>0</sub> at the [[Larmor frequency]] of the nuclei in the '''B'''<sub>0</sub>. The magnitude of '''B'''<sub>1</sub> is typically much smaller than the magnitude of '''B'''<sub>0</sub>. Under these circumstances the relaxation of the magnetization is similar to laboratory frame relaxation in a field '''B'''<sub>1</sub>. The decay constant for the recovery of the magnetization component along '''B'''<sub>1</sub> is called the spin-lattice relaxation time in the rotating frame and is denoted ''T''<sub>1ρ</sub>.
| |
| 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]] 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.<ref>A. Abragam "Principles of Nuclear Magnetism" (Oxford University Press, 1961)</ref> The form of the spectral density functions depend on the physical system, but a simple approximation called the [[Relaxation (NMR)#BPP theory|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]], [[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.<ref>Bloembergen, E.M. Purcell, R.V. Pound "Relaxation Effects in Nuclear Magnetic Resonance Absorption" ''Physical Review'' (1948) v73. 7:679-746</ref> 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 <math> e^{-t/\tau_c}</math>, where <math> \tau_c </math> is called the correlation time. From this theory, one can get T<sub>1</sub>、T<sub>2</sub> for magnetic dipolar relaxation:
| |
| :<math>\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}]</math>
| |
| :<math>\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}]</math>,
| |
| where <math>\omega_0</math> is the [[Larmor frequency]] in correspondence with the strength of the main magnetic field <math>B_0</math>. <math>\tau_c</math> is the correlation time of the molecular [[Brownian motion|tumbling]] motion. <math>K=\frac{3\mu_0^2}{160\pi^2}\frac{\hbar^2\gamma^4}{r^6}</math> is defined for spin-1/2 nuclei and is a constant with <math>\mu_0</math> being the [[magnetic permeability of free space]] of the <math>\hbar=\frac{h}{2\pi}</math> 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 H<sub>2</sub>O molecules in [[liquid phase]] without the contamination of [[oxygen-17]], the value of ''K'' is 1.02×10<sup>10</sup> s<sup>−2</sup> and the correlation time <math>\tau_c</math> is on the order of [[picosecond]]s = <math>10^{-12}</math> [[second|s]], while [[hydrogen|hydrogen nuclei]] <sup>1</sup>H ([[proton]]s) at 1.5 teslas carry an Larmor frequency of approximately 64 [[MHz]]. We can then estimate using ''τ''<sub>c</sub> = 5×10<sup>−12</sup> s:
| |
| | |
| :<math>\omega_0\tau_c = 3.2\times 10^{-5} </math>(dimensionless)
| |
| :<math>T_1=\left(1.02\times 10^{10}\left[\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}\right]\right)^{-1} </math>= 3.92 s
| |
| :<math>T_2=\left(\frac{1.02\times 10^{10}}{2}\left[3\cdot 5\times 10^{-12} + \frac{5\cdot 5\times 10^{-12} }{1 + \left(3.2\times 10^{-5} \right)^2} + \frac{ 2\cdot 5\times 10^{-12} }{1 + 4\cdot (3.2\times 10^{-5} )^2}\right]\right)^{-1} </math>= 3.92 s,
| |
| | |
| which is close to the experimental value, 3.6 s. Meanwhile, we can see that at this extreme case, T<sub>1</sub> equals T<sub>2</sub>.
| |
| As follows from the BBP theory, measuring the T<sub>1</sub> 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 <sup>1</sup>H selective and non-selective T<sub>1</sub> times in variable-temperature relaxation experiments via equation. <sup>[7]</sup>
| |
| | |
| '''r(M-H) (Å ) = C((1.4k + 4.47) T<sub>1min</sub> / ν)<sup>1/6</sup>
| |
| | |
| k = (f-1)/(0.5-f/3) with f = T<sub>1s</sub>/T<sub>1</sub>
| |
| | |
| C = 10<sup>7</sup>(γH<sup>2</sup> γM<sup>2</sup> ħ<sup>2</sup> I(I +1) / 15 ) <sup>1/6</sup>
| |
| ''I'' is spin of M'''
| |
| | |
| ==See also==
| |
| * [[Nuclear magnetic resonance]]
| |
| * [[Nuclear magnetic resonance spectroscopy of carbohydrates]]
| |
| * [[Nuclear magnetic resonance spectroscopy of nucleic acids]]
| |
| * [[Nuclear magnetic resonance spectroscopy of proteins]]
| |
| * [[NMR spectroscopy]]
| |
| * [[Protein dynamics]]
| |
| * [[Relaxometry]]
| |
| | |
| ==References==
| |
| <references/>
| |
| 7. Gusev DG, Nietlishpach D, Vimenits AB, Bakhmutov VI, Berke H. “Synthesis and NMR T1 Realaxation Study of Rhenium and Manganese Hydride Complexes”. Inorganic Chemistry 1993 v. 32 3270-3276.
| |
| | |
| ==External links==
| |
| * [http://www.cis.rit.edu/htbooks/nmr/ basics of NMR]
| |
| * [http://chem.ch.huji.ac.il/nmr/techniques/other/t1t2/t1t2.html Relaxation in high-resolution NMR spectroscopy]
| |
| * [http://www.stelar.it/Papers/2004_Kimmich_Nmrd.pdf Field-cycling NMR relaxometry]
| |
| * [http://nmr-relax.com relax] Software for the analysis of NMR dynamics
| |
| * [http://www.visielab.ua.ac.be/sites/default/files/ijist99.pdf Estimation of T1 and T2 relaxation parameters in MRI]
| |
| | |
| [[Category:Nuclear magnetic resonance]]
| |