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<keywords content="diffusion rates{{Caution|If hard core NMR or physics theory is not to your taste, dimensionless units, hidden units, radian, relaxation rates, rotational correlation times, SI supplementary units, spherical harmonics" />please do not read any further!}}

'''Warning:''' If hard core The concept of the hidden radian unit is quite important for understanding relaxation in NMR or , in fact, any rotational process in physics theory . For example it is important for understanding the model-free equations, for reduced spectral density mapping, for SRLS, and for relaxation dispersion. In these cases, the reason is because relaxation rates are measured in rad/s. As described in [[#SI supplementary units]], the radian unit can be dropped because it is plainly obvious that NMR and relaxation is not an angular process and hence radian units are implied (if you didn't get that, that was sarcasm). Hence R2 can be said to your tastebe in units of 1/s, but never, ever Hz. Also note that because of the SI conventions described below, please do describing the correlation time in s units does not read any further!imply that there are no radian units. But reporting rates as Hz implies no radian units whereas reporting as 1/s instead indicates that radian units are present.

This concept is quite important for understanding relaxation in NMR or, in fact, any rotational process in physics. It is important for understanding also a follow on from the model-free equations, for reduced spectral density mapping, for SRLS, and for relaxation dispersion. The reason is because relaxation rates are measured in rad[https://smail. As described in [[#SI supplementary units]], the radian unit can be dropped because it is plainly obvious that NMR and relaxation is an angular process and hence radian units are implied (that was sarcastic)gna. Hence R2 can be said to be in units of 1org/public/relax-devel/2007-06/s, but never, ever Hzmsg00012. Also note that because html comprehensive unit analysis of the SI conventions described below, describing the correlation time in Abragam's units does not imply that there are no radian units. But reporting rates as Hz implies no radian units whereas reporting as 1/s instead indicates that radian units are presentrelaxation equations].

This is also a follow on The text below uses quotes from many different sources to demonstrate that the [https://mail.gna.org/public/relax-devel/2007-06/msg00012.html comprehensive hidden radian unit analysis of Abragam's relaxation equations]is actually very much present in everything we do in NMR.

"<blockquote>(b) The radian and steradian are special names for the number one that may be used to convey information about the quantity concerned. In practice the symbols rad and sr are used where appropriate, but the symbol for the derived unit one is generally omitted in specifying the values of dimensionless quantities."</blockquote> 

"<blockquote>In a few cases, however, a special name is given to the unit one, in order to facilitate the identification of the quantity involved. This is the case for the radian and the steradian. The radian and steradian have been identified by the CGPM as special names for the coherent derived unit one, to be used to express values of plane angle and solid angle, respectively, and are therefore included in Table 3."</blockquote> 

"<blockquote>As discussed in Section 2.2.3, the coherent SI unit for dimensionless quantities, also termed quantities of dimension one, is the number one, symbol 1. Values of such quantities are expressed simply as numbers. The unit symbol 1 or unit name "one" are not explicitly shown, nor are special symbols or names given to the unit one, apart from a few exceptions as follows. For the quantity plane angle, the unit one is given the special name radian, symbol rad, and for the quantity solid angle, the unit one is given the special name steradian, symbol sr. For the logarithmic ratio quantities, the special names neper, symbol Np, bel, symbol B, and decibel, symbol dB, are used (see 4.1 and Table 8, p. 127)."</blockquote>

Quotes from the PDF linked below (page 67) from the appendix section titled "SI supplementary units (radian and steradian)": <blockquote>...the units radian and steradian are usually introduced into expressions for units when there is need for clarification...</blockquote>  Quote from the PDF linked below (page 67) from the appendix sectiontitled "Elimination of the class of supplementary units in the SI" forresolution 8 of the CGPM conference:"<blockquote>decides..."</blockquote> "<blockquote>to interpret the supplementary units in the SI, namely the radian andthe steradian, as dimensionless derived units, the names and symbolsof which may, but need not, be used in expressions for other SIderived units, as is convenient,"</blockquote> "<blockquote>and, consequently, to eliminate the class of supplementary units as aseparate class in the SI."</blockquote>

This reference explains a bit more clearly why the radian unit isinvisible in most situations.

where <math alt="theta''θ(t)">\theta(t)</math> '' and <math alt="phi''ϕ(t)">\phi(t)</math> '' are the time dependent spherical angles in the dimensionless radian units. The time ''t '' is normal time and hence has no hidden radian units. Spherical harmonics are the angular portion of the solution to Laplace's equation and, because it is angular, it is using the radian angular SI unit.  

;Title: Physical Properties of Lipids;Authors: Alejandro G. Marangoni, Suresh Narine;Subject: Fluorescence;Year: 2002;Link: http://books.google.com/books?id=OCBav13l_MsC&pg=PA166&dq=rotational+correlation+time+radian&lr=;Quote (page 166): "<blockquote>The rotational correlation time [phi] is the timerequired by the fluorophore to rotate through an arc of 1 radian (phi= 1/(2.pi.nu))." Title: BiophysicsAuthors: Gerald Ehrenstein, Harold LecarSubject: NMR spin relaxationYear: 1982Link: http:<//books.google.com/books?id=rThFVFmAdDAC&pg=PA14&dq=rotational+correlation+time+radianQuote (page 14): "The value of tau_c can be approximated as the timerequired for the molecule containing the resonant nucleus to eitherrotate 1 radian (rotational correlation time) or diffuse a distanceequivalent to its own dimensions (translational correlation time)."blockquote>

;Title: Protein NMR Spectroscopy (second edition)Biophysics;Authors: John Cavanagh, Wayne J. Fairbrother, Arthur G. Palmer, IIIGerald Ehrenstein,Harold LecarNicholas J. Skelton, Mark Rance;SubjectSubject: NMR spin relaxation;Year: 20071982;Link: http://books.google.com/books?id=2-LqLHOLHZwCrThFVFmAdDAC&pg=PA366PA14&dq=rotational+correlation+time+radian;Quote (page 36614): "...in which the correlation time, <blockquote>The value of tau_c, isapproximately can be approximated as the average time required for the molecule containing the resonant nucleus to either rotate by 1radian(rotational correlation time) or diffuse a distance equivalent to its own dimensions (translational correlation time)."</blockquote>

;Title: Hydration Processes in Biology: Theoretical and Experimental Protein NMR Spectroscopy (second edition)Approaches;AuthorsAuthor: Marie-Claire Bellissent-FunelJohn Cavanagh, Wayne J. Fairbrother, Arthur G. Palmer, III, Nicholas J. Skelton, Mark Rance;Subject: Water motionNMR relaxation;Year: 19992007;Link: http://books.google.com/books?id=9tJaB00wXhgC2-LqLHOLHZwC&pg=PA243PA366&dq=rotational+correlation+time+radian&lr=;Quote (page 243366): "For such sites, the rotational and translationaldiffusion of water should both be rate-limited by H-bondrearrangements and it can therefore be argued that the residence time(the time taken to diffusion ca<blockquote>... 3 Angstrom) should be close to in which thefirst-rank rotational correlation time (, tau_c, is approximately the average time taken for the molecule to rotatethrough one by 1 radian), i.e., tau_W ~= 3 tau_s (where tau_s is thesecond-rank rotational correlation time)."(Interesting that the factor of 3 is only approximate here!!! Nils,do you have a citation where the equation is not appriximate?)</blockquote>

;Title:Hydration Processes in Biology: Theoretical and Experimental NMR of Macromolecules: A Practical ApproachApproaches;Author: Gordon Carl Kenmure RobertsMarie-Claire Bellissent-Funel;Subject: NMR relaxationWater motion;Year: 19931999;Link: http://books.google.com/books?id=K7n7SnmDbSAC9tJaB00wXhgC&pg=PA9PA243&dq=rotational+correlation+time+radian&lr=;Quote (page 9243): "The <blockquote>For such sites, the rotational and translational diffusion of water should both be rate-limited by H-bond rearrangements and it can therefore be argued that the residence time (the time taken to diffusion ca. 3 Angstrom) should be close to the first-rank rotational correlation time, tau_c, is (the timetaken for the particle to rotate through an angle of one radian), i.e., tau_W ~= 3 tau_s (57°where tau_s is the second-rank rotational correlation time)."</blockquote>

;Title: Fundamentals NMR of Protein NMR SpectrosopyMacromolecules: A Practical Approach;AuthorAuthors: Gordon S. Rule, T. Kevin HitchensCarl Kenmure Roberts;Subject: NMR relaxation;Year: 20061993;Link: http://books.google.com/books?id=8vmf5y6Jf84CK7n7SnmDbSAC&pg=PA441PA9&dq=rotational+correlation+time+radian&lr=;Quote (page 4419): "[<blockquote>The rotational correlation time, tau_c] , is the time required taken for a molecule the particle torotate, on average, 1 through an angle of one radian(57°)."</blockquote>

;Title: Nuclear Magnetic Resonance in Biochemistry: Principles and Fundamentals of Protein NMR SpectrosopyApplications;AuthorsAuthor: Thomas LGordon S. Rule, T. JamesKevin Hitchens;Subject: NMRrelaxation;Year: 19752006;Link: http://books.google.com/books?id=iItqAAAAMAAJ8vmf5y6Jf84C&qpg=rotational+correlation+time+radianPA441&dq=rotational+correlation+time+radian&pgis=1;Quote (page ?441): "The rotational correlation <blockquote>[tau_c] is the time (rc or rr) providesrequired for a ... having many molecular collisions before it turns molecule to rotate, on average, 1 radian."</blockquote>

;Title: Biophysical ChemistryNuclear Magnetic Resonance in Biochemistry: Principles, Techniques, and Applications;Author: Alan GThomas L. MarshallJames;Subject: Rotational diffusion (for fluorescence)NMR;Year: 19781975;Link: http://books.google.com/books?id=PJhqAAAAMAAJiItqAAAAMAAJ&q=rotational+correlation+time+radian&dq=rotational+correlation+time+radian&pgis=1;Quote (page 720?): "...decay of the correlation function for<blockquote>The rotational diffusion, tau_rot may also be thought of as thecharacteristic correlation time it takes for a typical macromolecule to rotate(diffusionallyrc or rr) through an angle of the order of provides a radian..."having many molecular collisions before it turns 1 radian.</blockquote>

;Title:Biophysical Chemistry: Practical NMR Relaxation for ChemistsPrinciples, Techniques, and Applications;Author: Vladimir IAlan G. BakhmutovMarshall;Subject: NMR relaxationRotational diffusion (for fluorescence);Year: 20041978;Link: http://books.google.com/books?id=_gIh9KHIOx4CPJhqAAAAMAAJ&pgq=PA13rotational+correlation+time+radian&dq=rotational+correlation+time+radian&lrpgis=1;Quote (page 13720): "However, a more correct definition <blockquote>...decay of the tau_c isconnected with thecorrelation function for rotational diffusion, so-called autocorrelation function in the theorytau_rot may also be thought of nuclear relaxation where as the tau_c is an average characteristic time it takes for themolecule a typical macromolecule to progresses rotate (sic.diffusionally) through one an angle of the order of a radian."..</blockquote>

;Title: Modern Protein Chemistry: Practical AspectsNMR Relaxation for Chemists;AuthorAuthors: Gary C. Howard, William EVladimir I. BrownBakhmutov;Subject: NMR relaxation;Year: 20012004;Link: http://books.google.com/books?id=MIxdC7GPz0sC_gIh9KHIOx4C&pg=PA45PA13&dq=rotational+correlation+time+radian&lr=;Quote (page 4513): "[<blockquote>However, a more correct definition of the tau_c] is roughly equal to connected with the, so-called autocorrelation function in the theory of nuclear relaxation where the tau_c is an average time it takes afor the molecule to rotate 1 progresses (sic.) through one radian while undergoing random rotationalmotion."</blockquote>

;Title: MRS of the Brain and Neurological DisordersModern Protein Chemistry: Practical Aspects;Authors: Koji TeradaGary C. Howard, Akihiro Igata, Toshiro Fujimoto, TetsuhikoWilliam E. BrownAsakura, Institute of Advanced Medical Technology;SubjectSubject: ImagingNMR relaxation;Year: 20002001;Link: http://books.google.com/books?id=kF2dw7c33cACMIxdC7GPz0sC&pg=PA41PA45&dq=rotational+correlation+time+radian&lr=#PPA43,M1;Quote (page 4145): "...Brownian motion. This has a time scale, therotational correlation time (<blockquote>[tau_c) defined as ] is roughly equal to the time taken onaverage for it takes a solute molecule to rotate by one 1 radian or roughly thereciprocal of the rate of tumbling in solution of the relevant pieceof the moleculewhile undergoing random rotational motion."</blockquote>

;Title: Structural Biology: Practical NMR ApplicationsMRS of the Brain and Neurological Disorders;AuthorsAuthor: Quincy TengKoji Terada, Akihiro Igata, Toshiro Fujimoto, Tetsuhiko Asakura, Institute of Advanced Medical Technology;Subject: NMR relaxationImaging;Year: 20052000;Link: http://books.google.com/books?id=dRmmGFkummICkF2dw7c33cAC&pg=PA36PA41&dq=rotational+correlation+time+radian&lr=#PPA36PPA43,M1;Quote (page 3641): <blockquote>...Brownian motion. "The This has a time scale, the rotational correlation time is used to describe the rateof random motions and is expressed (tau_c) defined as the time taken on average time betweencollisions for translational motions or the time for a solute molecule torotate by one radian or roughly the reciprocal of the rate of tumbling in rotational motionsolution of the relevant piece of the molecule."</blockquote>

;Title:Structural Biology: High-resolution Practical NMR Techniques in Organic ChemistryApplications;Author: Timothy D. W. ClaridgeQuincy Teng;Subject: NMR relaxation;Year: 19992005;Link: http://books.google.com/books?id=9srIkkL-YMkCdRmmGFkummIC&pg=PA283PA36&dq=rotational+correlation+time+radian&lr=#PPA284PPA36,M1;Quote (page 28336): "... its rotational <blockquote>The correlation time, tau_c. Thisis usually taken used to define describe the rate of random motions and is expressed as the average time required between collisions for translational motions or the time for a moleculeto rotate through an angle of 1 one radian about any axis, in rotational motion..."</blockquote>

;Title: A Dictionary of Concepts High-resolution NMR Techniques in NMROrganic Chemistry;Author: STimothy D. W. HomansClaridge;Subject: NMR relaxation;Year: 19891999;Link: http://books.google.com/books?id=wpggNxUrzSMC9srIkkL-YMkC&pg=PA72PA283&dq=rotational+correlation+time+radian&lr=#PPA284,M1;Quote (page 72283): "For example, in the case of random translationalmotions<blockquote>... its rotational correlation time, tau_c . This is defined as the mean time between collisions, whereasin the case of reorientational (rotational) motion, it is defined asusually taken to define the average time required for the molecule to rotate by one through an angle of 1 radianabout any axis, ..."Note this book later on page 72 makes the mistake (according to me) ofsaying that 1</tau_c is in Hertz.blockquote>

;Title: Molecular Crystals and Liquid CrystalsA Dictionary of Concepts in NMR;Author: Gordon and Breach Science PublishersS. W. Homans;Subject: CrystalsNMR relaxation;Year: 19741989;Link: http://books.google.com/books?id=bTW3AAAAIAAJwpggNxUrzSMC&qpg=rotational+correlation+time+radianPA72&dq=rotational+correlation+time+radian&lr=&pgis=1;Quote (page ?72): "The rotational correlation time may be computed from<blockquote>For example, in the linewidths case of random translational motions, tau_c is defined as the mean time between collisions, whereas in the ... roughly case of reorientational (rotational) motion, it is defined as the average time required for the radical molecule toreorient rotate by 1 one radian is given by ..."</blockquote>

Title: Industrial Research/developmentAuthor: Technical Pub. Co.Subject: NMR relaxationYear: 1978Link: http://books.google.com/books?id=EstVAAAAMAAJ&q=rotational+correlation+time+radian&dq=rotational+correlation+time+radian&lr=&pgis=1Quote Note this book later on page 72 makes the mistake (page ?according to me): " is the Larmor angular frequency in radiansof saying that 1/sec andtau_c is the rotational correlation time of the nuclei in sec/radianHertz."

;Title:Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications;Authors:Józef Kowalewski, Lena Mäler;Subject: NMR relaxation;Year:2006;Link: http://books.google.com/books?id=MiUfcE1C9CQC&pg=PA14&dq=relaxation+rate+radian&lr=#PPA19,M1;Quote (page 15)<blockquote>Because the natural unit for the angular frequency is radians per second, the relaxation rate, or the inverse of relaxation time, R2 = 1/T2, should indeed also be expressed in these units. Usually, relaxation times are given in seconds (the rates are given in 1/s), which tacitly implies that the radians can be omitted; we note in parenthesis that the radian is considered a dimensionless unit in physics.</blockquote>;Quote 2 (page 15)<blockquote>The Fourier transform of an exponential decay is Lorentzian centered at zero frequency, with the full width at half-height (in Hertz) equal to Delta_nu = 1/(pi.T2)...</blockquote> ;Title:Practical NMR Relaxation for Chemists;Author: Vladimir I. Bakhmutov;Subject: NMR relaxation;Year: 2004;Link: http://books.google.com/books?id=_gIh9KHIOx4C&pg=PA13&dq=rotational+correlation+time+radian&lr=;Quote (page 9): "<blockquote>...linewidths, Delta_nu, measured in Hz, aredirectly controlled by T1 and T2 relaxation times according to: Delta_nu = 1/(pi T1,2)</blockquote> ;Title:Modern Protein Chemistry: Practical Aspects;Authors:Gary C. Howard, William E. Brown;Subject:NMR relaxation;Year:2001;Link:http://books.google.com/books?id=MIxdC7GPz0sC&pg=PA45&dq=rotational+correlation+time+radian&lr=;Quote (page 45)<blockquote>The actual relationship between the spin-spin relation rate and the lines width (Delta_nu) is given by R2, the rate of spin-spin relaxation; T2 is the time constant for spin-spin relaxation, Delta_nu = 1/pi . R2 = 1/(pi T1.T2).</blockquote> ;Title:Structural Biology: Practical NMR Applications;Author:Quincy Teng;Subject:NMR relaxation;Year:2005;Link:http://books.google.com/books?id=dRmmGFkummIC&pg=PA36&dq=rotational+correlation+time+radian&lr=#PPA36,2M1;Quote (page 37)"<blockquote>...T1 relaxation is inversely proportional to correlation time tau_c...</blockquote> 

Title: Modern Protein Chemistry: Practical AspectsAuthors: Gary C. Howard, William E. BrownSubject: NMR relaxationYear: 2001Link: http://books.google.com/books?id=MIxdC7GPz0sC&pgSee also =PA45&dq=rotational+correlation+time+radian&lr=Quote (page 45): "The actual relationship between the spin-spinrelation rate and the lines width (Delta_nu) is given by R2, the rateof spin-spin relaxation; T2 is the time constant for spin-spinrelaxation, Delta_nu = 1/pi . R2 = 1/(pi.T2)."

Title[[Category: Structural Biology: Practical NMR ApplicationsAuthor: Quincy TengSubject: NMR relaxationYear: 2005Link: http://books.google.com/books?id=dRmmGFkummIC&pg=PA36&dq=rotational+correlation+time+radian&lr=#PPA36,M1Quote (page 37): "...T1 relaxation is inversely proportional tocorrelation time tau_c..."Theory]]