Relaxation in NMR Spectroscopy
Matthias Ernst
Physical Chemistry, ETH Zürich, Vladimir-Prelog-Weg 2, Zürich 8093 Switzerland
Search for more papers by this authorMatthias Ernst
Physical Chemistry, ETH Zürich, Vladimir-Prelog-Weg 2, Zürich 8093 Switzerland
Search for more papers by this authorP.K. Madhu
Search for more papers by this authorG. Rajalakshmi
Search for more papers by this authorAbstract
This chapter discusses the foundations of relaxation theory starting from a phenomenological description going to a semi-classical and finally to a quantum-mechanical description. Since the majority of relaxation work in NMR is based on semi-classical relaxation theory (mostly Redfield theory), the emphasis is put on this type of description. Besides the derivation, relaxation pathways in the simple example of a heteronuclear and a homonuclear two-spin system are discussed for dipolar-coupling and chemical-shielding anisotropy-based relaxation including cross-correlated relaxation pathways. Finally, the basics of relaxation in quadrupolar spin systems is briefly discussed.
References
- Abragam , A. The Principles of Nuclear Magnetism . ( 1961 ). Oxford University Press .
-
Ernst , R.R.
,
Bodenhausen , G.
,
Wokaun , A.
(
1990
).
Principles of Nuclear Magnetic Resonance in One and Two Dimensions
.
Oxford
:
Oxford University Press
.
10.1093/oso/9780198556473.001.0001 Google Scholar
- Levitt , M.H. ( 2013 ). Spin Dynamics . Basics of Nuclear Magnetic Resonance. John Wiley & Sons .
- Bloch , F. ( 1946 ). Nuclear induction . Phys Rev. 70 ( 7-8 ): 460 – 474 .
- Redfield , A.G. ( 1966 ). Intramolecular dipolar relaxation in multi-spin systems . Adv. Magn. Reson. 1 : 1 – 32 .
- Kubo , R. ( 1977 ). A stochastic theory of line shape . Adv. Chem. Phys. 15 : 101 – 127 .
- Werbelow , L.G. and Grant , D.M. ( 1977 ). Intramolecular dipolar relaxation in multi-spin systems . Adv. Magn. Reson. 9 : 189 – 275 .
- Vold , R.L. and Vold , R.R. ( 1978 ). Nuclear magnetic relaxation in coupled spin systems . Prog. NMR Spectr. 12 ( 2 ): 79 – 133 .
- Jeener , J. ( 1985 ). Superoperators in magnetic-resonance . Adv. Magn. Reson. 10 : 1 – 51 .
- Kowalewski , J. , Nordenskiold , L. , Benetis , N. and Westlund , P.O. ( 1985 ). Theory of nuclear-spin relaxation in paramagnetic systems in solution . Prog. NMR Spectr. 17 : 141 – 185 .
- Bull , T.E. ( 1992 ). Relaxation in the rotating frame in liquids . Prog. NMR Spectr. 24 : 377 – 410 .
- Fischer , M.W.F. , Majumdar , A. and Zuiderweg , E.R.P. ( 1998 ). Protein NMR relaxation: theory, applications and outlook . Prog. NMR Spectr. 33 ( 4 ): 207 – 272 .
- Sharp , R. , Lohr , L. and Miller , J. ( 2001 ). Paramagnetic NMR relaxation enhancement: recent advances in theory . Prog. NMR Spectr. 38 ( 2 ): 115 – 158 .
-
Luginbühl , P.
and
Wüthrich , K.
(
2002
).
Semi-classical nuclear spin relaxation theory revisited for use with biological macromolecules
.
Prog. NMR Spectr.
40
(
3
):
199
–
247
.
10.1016/S0079-6565(01)00043-7 Google Scholar
- Schneider , D.J. and Freed , J.H. ( 2009 ). Spin relaxation and motional dynamics . Adv. Chem Phys. 73: 387.
- Nicholas , M.P. , Eryilmaz , E. , Ferrage , F. , Cowburn , D. and Ghose , R. ( 2010 ). Nuclear spin relaxation in isotropic and anisotropic media . Prog. NMR Spectr. 57 ( 2 ): 111 – 158 .
- Schanda , P. and Ernst , M. ( 2016 ). Studying dynamics by magic-angle spinning solid-state NMR spectroscopy: principles and applications to biomolecules . Prog. NMR Spectr . 96 ( C ): 1 – 46 .
-
Caspers , W.J.
(
1964
).
Theory of Spin Relaxation
.
Interscience Publishers
.
10.1063/1.3051869 Google Scholar
- McConnell , J. ( 2009 ). The Theory of Nuclear Magnetic Relaxation in Liquids . Cambridge University Press .
-
Gamliel , D.
and
Levanon , H.
(
1995
).
Stochastic Processes in Magnetic Resonance
.
World Scientific
.
10.1142/9789812831040 Google Scholar
- Cowan , B. ( 2005 ). Nuclear Magnetic Resonance and Relaxation . Cambridge University Press .
- Bakhmutov , V.I. ( 2005 ). Practical Nuclear Magnetic Resonance Relaxation for Chemists . John Wiley & Sons .
-
Kowalewski , J.
and
Mäler , L.
(
2006
).
Nuclear Spin Relaxation in Liquids
. Theory, experiments, and applications.
CRC Press
.
10.1201/9781420012194 Google Scholar
- Kruk , D. ( 2007 ). Theory of Evolution and Relaxation of Multi-Spin Systems . Application to nuclear magnetic resonance and electron spin resonance . Suffolk : Arima Publishing .
- Lindblad , G. ( 1976 ). On the generators of quantum dynamical semigroups . Commun. Math. Phys. 48 ( 2 ): 119 – 130 .
-
Bengs , C.
and
Levitt , M.H.
(
2020
).
A master equation for spin systems far from equilibrium
.
J. Magn. Reson.
310: 106645.
10.1016/j.jmr.2019.106645 Google Scholar
- Bengs , C. ( 2021 ). Markovian exchange phenomena in magnetic resonance and the Lindblad equation . J. Magn. Reson. 322: 106868.
-
Pell , A.J.
(
2021
).
A method to calculate the NMR spectra of paramagnetic species using thermalized electronic relaxation
.
J. Magn. Reson.
326: 106939.
10.1016/j.jmr.2021.106939 Google Scholar
- Kurzbach , D. and Jannin , S. ( 2019 ). Dissolution dynamic nuclear polarization methodology and instrumentation . eMagRes. 7 .
-
Barbara , T.M.
(
2021
).
The Lindbladian form and the reincarnation of Felix Bloch's generalized theory of relaxation
.
Magn. Reson.
2
(
2
):
689
–
698
.
10.5194/mr-2-689-2021 Google Scholar
- Rodin , B.A. and Abergel , D. ( 2021 ). Spin relaxation: under the sun, anything new? Magn. Reson. Discus. 3 , 1 – 22 .
- Kubo , R. ( 1963 ). Stochastic liouville equations . J. Math. Phys. 4 ( 2 ): 174 – 183 .
- Freed , J.H. and Fraenkel , G.K. ( 1963 ). Theory of linewidths in electron spin resonance spectra . J. Chem. Phys. 39 ( 2 ): 326 – 348 .
- Vega , A.J. and Fiat , D. ( 1974 ). Stochastic liouville equation and approach to thermal equilibrium . Pure Appl. Chem. 40 ( 1-2 ): 181 – 192 .
- Moro , G.J. and Freed , J.H. ( 1980 ). Efficient computation of magnetic resonance spectra and related correlation functions from stochastic Liouville equations . J. Phys. Chem. 84 ( 22 ): 2837 – 2840 .
- Meirovitch , E. , Shapiro , Y.E. , Pohmeno , A. and Freed , J.H. ( 2010 ). Structural dynamics of bio-macromolecules by NMR: the slowly relaxing local structure approach . Prog. NMR Spectr. 56 ( 4 ): 360 – 405 .
- Bain , A.D. ( 2003 ). Chemical exchange in NMR . Prog. NMR Spectr. 43 ( 3-4 ): 63 – 103 .
- Bloembergen , N. , Purcell , E.M. and Pound , R.V. ( 1948 ). Relaxation effects in nuclear magnetic resonance absorption . Phys. Rev. 73 ( 7 ): 679 – 712 .
- Solomon , I. ( 1955 ). Relaxation processes in a system of 2 spins . Phys. Rev. 99 ( 2 ): 559 – 565 .
- Wangsness , R.K. and Bloch , F. ( 1953 ). The dynamical theory of nuclear induction . Phys. Rev. 89 ( 4 ): 728 – 739 .
- Redfield , A.G. ( 1955 ). Nuclear magnetic resonance saturation and rotary saturation in solids . Phys. Rev. 98 ( 6 ): 1787 – 1809 .
- Redfield , A.G. ( 1957 ). On the theory of relaxation processes . IBM J. Res. Dev. 1 : 19 – 31 .
- Levitt , M.H. and Di Bari , L. ( 1992 ). Steady-state in magnetic-resonance pulse experiments . Phys. Rev. Lett. 69 ( 21 ): 3124 – 3127 .
- Levitt , M.H. and Di Bari , L. ( 1994 ). The homogeneous master equation and the manipulation of relaxation networks . Bull. Magn. Reson. 16 : 94 – 114 .
- Levante , T.O. and Ernst , R.R. ( 1995 ). Homogeneous versus inhomogeneous quantum-mechanical master-equations . Chem. Phys. Lett. 241 : 73 – 78 .
- Bloch , F. ( 1957 ). Generalized theory of relaxation . Phys. Rev. 105 ( 4 ): 1206 .
- Hubbard , P.S. ( 1961 ). Quantum-mechanical and semiclassical forms of the density operator theory of relaxation . Rev. Mod. Phys. 33 ( 2 ): 249 – 264 .
- Breuer , H.P. and Petruccione , F. ( 2002 ). The Theory of Open Quantum Systems . Oxford University Press on Demand .
- Gupta , A. , Strait-Gardner , T. and Price , W.S. ( 2021 ). Is it time to forgo the use of the terms “Spin-Lattice” and “Spin-Spin” relaxation in NMR and MRI? . J. Phys. Chem. Lett. 12 : 6305 – 6312 .
-
Traficante , D.D.
(
1991
).
Relaxation. Can T
2
, be longer than T
1
?
Concepts. Magn. Reson.
3
:
171
–
177
.
10.1002/cmr.1820030305 Google Scholar
- Keeler , J. ( 2011 ). Understanding NMR Spectroscopy . John Wiley & Sons .
- Jeener , J. , Meier , B.H. , Bachmann , P. and Ernst , R.R. ( 1979 ). Investigation of exchange processes by two-dimensional NMR spectroscopy . J. Chem. Phys. 71 ( 11 ): 4546 – 4553 .
- Kumar , A. , Ernst , R.R. and Wüthrich , K. ( 1980 ). A two-dimensional nuclear Overhauser enhancement (2D NOE) experiment for the elucidation of complete proton-proton cross-relaxation networks in biological macromolecules . Biochem. Bioph. Res. Comm. 95 ( 1 ): 1 – 6 .
- Neuhaus , D. and Williamson , M.P. ( 2000 ). The nuclear Overhauser effect in structural and conformational analysis . VCH Verlagsgesellschaft Mbh .
- Goldman , M. ( 2001 ). Formal theory of spin-lattice relaxation . J. Magn. Reson. 149 ( 2 ): 160 – 187 .
- Andrew , E.R. , Bradbury , A. and Eades , R.G. ( 1958 ). Nuclear magnetic resonance spectra from a crystal rotated at high speed . Nature. 182 ( 4650 ): 1659 – 1659 .
- Andrew , E.R. , Bradbury , A. and Eades , R.G. ( 1959 ). Removal of dipolar broadening of nuclear magnetic resonance spectra of solids by specimen rotation . Nature. 183 ( 4678 ): 1802 – 1803 .
- Lowe , I.J. ( 1959 ). Free induction decays of rotating solids . Phys. Rev. Lett. 2 ( 7 ): 285 – 287 .
- Polimeno , A. and Freed , J.H. ( 1995 ). Slow motional ESR in complex fluids: the slowly relaxing local structure model of solvent cage effects . J. Phys. Chem. 99 ( 27 ): 10995 – 11006 .
-
Polimeno , A.
,
Barone , V.
and
Freed , J.H.
(
2011
).
Stochastic methods for magnetic resonance spectroscopies
. In:
Computational Strategies for Spectroscopy: From Small Molecules to Nano Systems
. (ed.
V. Barone
),
549-582
.
John Wiley & Sons, Ltd.
10.1002/9781118008720.ch12 Google Scholar
- Mehring , M. ( 1983 ). Principles of High Resolution NMR in Solids . Springer-Verlag .
-
Werbelow , L.G.
(
2007
).
Dynamic Frequency Shift
.
Chichester, UK
:
John Wiley & Sons, Ltd.
10.1002/9780470034590.emrstm0138 Google Scholar
- Smith , S.A. , Palke , W.E. and Gerig , J.T. ( 1994 ). Superoperator propagators in simulations of NMR spectra . J. Magn. Reson. Ser. A. 106 ( 1 ): 57 – 64 .
- Allard , P. , Helgstrand , M. and Hard , T. ( 1997 ). A method for simulation of NOESY, ROESY, and off-resonance ROESY spectra . J. Magn. Reson. 129 ( 1 ): 19 – 29 .
- Allard , P. , Helgstrand , M. and Hard , T. ( 1998 ). The complete homogeneous master equation for a heteronuclear two-spin system in the basis of Cartesian product operators . J. Magn. Reson. 134 ( 1 ): 7 – 16 .
- Ghose , R. ( 2000 ). Average Liouvillian theory in nuclear magnetic resonance—principles, properties, and applications . Concepts Magn. Reson. 12 : 152 – 172 .
- Tropp , J.A. ( 1993 ). Dipolar cross correlation and the nuclear Overhauser effect in systems with strong scalar coupling . J. Magn. Reson. Ser. A. 103 ( 1 ): 90 – 91 .
- Dorai , K. and Kumar , A. ( 2000 ). Cross correlations in the longitudinal relaxation of strongly coupled spins . J. Magn. Reson. 145 ( 1 ): 8 – 17 .
- D'Silva , L. , Pola , A. , Dutta , P. , Martinez , G.V. , Sprenger , P. , Gillies , R.J. et al. ( 2012 ). Slow relaxation of longitudinal multispin orders in weakly and strongly coupled two-spin systems . Magn. Reson. Chem. 50 ( 6 ): 443 – 448 .
- Morris , G.A. and Freeman , R. ( 1979 ). Enhancement of nuclear magnetic-resonance signals by polarization transfer . J. Am. Chem. Soc. 101 ( 3 ): 760 – 762 .
- Bothner-By , A.A. , Stephens , R.L. , Lee , J.M , Warren , C.D. and Jeanloz , R.W. ( 1984 ). Structure determination of a tetrasaccharide: Transient nuclear overhauser effects in the rotating frame . J. Am. Chem. Soc. 106 ( 3 ): 811 – 813 .
- Bax , A. and Davis , D.G. ( 1985 ). Practical aspects of two-dimensional transverse NOE spectroscopy . J. Magn. Reson. 63 ( 1 ): 207 – 213 .
- Carravetta , M. , Johannessen , O.G. and Levitt , M.H. ( 2004 ). Beyond the T 1 limit: singlet nuclear spin states in low magnetic fields . Phys. Rev. Lett. 92 ( 15 ): 153003 .
-
Levitt , M.H.
(
2010
).
Singlet and other states with extended lifetimes
.
eMagRes.
10.1002/9780470034590.emrstm1036 Google Scholar
- Pileio , G. ( 2017 ). Singlet NMR methodology in two-spin-1/2 systems . Prog. NMR Spectr. 98-99 : 1 – 19 .
- Teleanu , F. , Sadet , A. and Vasos , P.R. ( 2021 ). Symmetry versus entropy: long-lived states and coherences . Prog. NMR Spectr. 122 : 63 – 75 .
- Pervushin , K. , Riek , R. , Wider , G. and Wüthrich , K. ( 1997 ). Attenuated T 2 relaxation by mutual cancellation of dipole–dipole coupling and chemical shift anisotropy indicates an avenue to NMR structures of very large biological macromolecules in solution . Proc. Natl. Acad. Sci. USA. 94 ( 23 ): 12366 – 12371 .
- Fernández , C. and Wider , G. ( 2003 ). TROSY in NMR studies of the structure and function of large biological macromolecules . Curr. Opin. Struct. Biol. 13 ( 5 ): 570 – 580 .
- Xu , Y. and Matthews , S. ( 2013 ). TROSY NMR spectroscopy of large soluble proteins . In: Topics in Current Chemistry 97 – 119 .
- Schütz , S. and Sprangers , R. ( 2020 ). Methyl TROSY spectroscopy: a versatile NMR approach to study challenging biological systems . Prog, NMR Spectr. 116 : 56 – 84 .
-
Werbelow , L.G.
(
2007
).
Relaxation Theory for Quadrupolar Nuclei
.
Chichester, UK
:
John Wiley & Sons, Ltd.
10.1002/9780470034590.emrstm0464 Google Scholar
- Werbelow , L.G. and Kowalewski , J. ( 1997 ). Nuclear spin relaxation of spin one-half nuclei in the presence of neighboring higher-spin nuclei . J. Chem. Phys. 107 ( 8 ): 2775 – 2781 .
- Werbelow , L.G. ( 2006 ). Effect of rapidly relaxed electron spin anisotropically coupled to nearby nuclear partners . Magn. Reson. Chem. 44 ( 3 ): 249 – 254 .
- Hubbard , P.S. ( 1963 ). Theory of nuclear magnetic relaxation by spin-rotational interactions in liquids . Phys. Rev. 131 ( 3 ): 1155 – 1165 .
- Matson , G.B. ( 1977 ). Methyl NMR relaxation: the effects of spin rotation and chemical shift anisotropy mechanisms . J. Chem. Phys. 67 ( 11 ): 5152 – 5161 .