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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 equationand, and I would assume that because it is angular, it is using the radian angular SI unit.

My opinion here is that the The rotational correlation time is adescriptor of the change of angles - and these angles are in thehiddenidden, dimensionless radian units. Hence the correlation time ismeasured in s/rad or in the hidden supplementary unit notation simplys. But I prefer it is better to think of the concept as the diffusion rate, ameasure of the rate of rotational Brownian diffusion.

;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))."</blockquote>

;Title: Biophysics;Authors: Gerald Ehrenstein, Harold Lecar;Subject: NMR spin relaxation;Year: 1982;Link: http://books.google.com/books?id=rThFVFmAdDAC&pg=PA14&dq=rotational+correlation+time+radian;Quote (page 14): "<blockquote>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);Authors: John Cavanagh, Wayne J. Fairbrother, Arthur G. Palmer, III,Nicholas J. Skelton, Mark Rance;Subject: NMR relaxation;Year: 2007;Link: http://books.google.com/books?id=2-LqLHOLHZwC&pg=PA366&dq=rotational+correlation+time+radian;Quote (page 366): "<blockquote>...in which the correlation time, tau_c, isapproximately the average time for the molecule to rotate by 1radian."</blockquote>

;Title: Hydration Processes in Biology: Theoretical and Experimental Approaches;Author: Marie-Claire Bellissent-Funel;Subject: Water motion;Year: 1999;Link: http://books.google.com/books?id=9tJaB00wXhgC&pg=PA243&dq=rotational+correlation+time+radian&lr=;Quote (page 243): "<blockquote>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. 3 Angstrom) should be close to thefirst-rank rotational correlation time (the time taken to rotatethrough one 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: NMR of Macromolecules: A Practical Approach;Author: Gordon Carl Kenmure Roberts;Subject: NMR relaxation;Year: 1993;Link: http://books.google.com/books?id=K7n7SnmDbSAC&pg=PA9&dq=rotational+correlation+time+radian&lr=;Quote (page 9): "<blockquote>The rotational correlation time, tau_c, is the timetaken for the particle to rotate through an angle of one radian(57°)."</blockquote>

;Title: Fundamentals of Protein NMR Spectrosopy;Authors: Gordon S. Rule, T. Kevin Hitchens;Subject: NMR relaxation;Year: 2006;Link: http://books.google.com/books?id=8vmf5y6Jf84C&pg=PA441&dq=rotational+correlation+time+radian;Quote (page 441): "<blockquote>[tau_c] is the time required for a molecule torotate, on average, 1 radian."</blockquote>

;Title: Nuclear Magnetic Resonance in Biochemistry: Principles and Applications;Author: Thomas L. James;Subject: NMR;Year: 1975;Link: http://books.google.com/books?id=iItqAAAAMAAJ&q=rotational+correlation+time+radian&dq=rotational+correlation+time+radian&pgis=1;Quote (page ?): "<blockquote>The rotational correlation time (rc or rr) providesa ... having many molecular collisions before it turns 1 radian."</blockquote>

;Title: Biophysical Chemistry: Principles, Techniques, and Applications;Author: Alan G. Marshall;Subject: Rotational diffusion (for fluorescence);Year: 1978;Link: http://books.google.com/books?id=PJhqAAAAMAAJ&q=rotational+correlation+time+radian&dq=rotational+correlation+time+radian&pgis=1;Quote (page 720): "<blockquote>...decay of the correlation function forrotational diffusion, tau_rot may also be thought of as thecharacteristic time it takes for a typical macromolecule to rotate(diffusionally) through an angle of the order of a radian..."</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 13): "<blockquote>However, a more correct definition of the tau_c isconnected with the, so-called autocorrelation function in the theoryof nuclear relaxation where the tau_c is an average time for themolecule to progresses (sic.) through one radian."</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>[tau_c] is roughly equal to the time it takes amolecule to rotate 1 radian while undergoing random rotationalmotion."</blockquote>

;Title: MRS of the Brain and Neurological Disorders;Authors: Koji Terada, Akihiro Igata, Toshiro Fujimoto, TetsuhikoAsakura, Institute of Advanced Medical Technology;Subject: Imaging;Year: 2000;Link: http://books.google.com/books?id=kF2dw7c33cAC&pg=PA41&dq=rotational+correlation+time+radian&lr=#PPA43,M1;Quote (page 41): "<blockquote>...Brownian motion. This has a time scale, therotational correlation time (tau_c) defined as the time taken onaverage for a solute molecule to rotate by one radian or roughly thereciprocal of the rate of tumbling in solution of the relevant pieceof the molecule."</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,M1;Quote (page 36): "<blockquote>The correlation time is used to describe the rateof random motions and is expressed as the average time betweencollisions for translational motions or the time for a molecule torotate one radian in rotational motion."</blockquote>

;Title: High-resolution NMR Techniques in Organic Chemistry;Author: Timothy D. W. Claridge;Subject: NMR relaxation;Year: 1999;Link: http://books.google.com/books?id=9srIkkL-YMkC&pg=PA283&dq=rotational+correlation+time+radian&lr=#PPA284,M1;Quote (page 283): "<blockquote>... its rotational correlation time, tau_c. Thisis usually taken to define the average time required for the moleculeto rotate through an angle of 1 radian about any axis, ..."</blockquote>

;Title: A Dictionary of Concepts in NMR;Author: S. W. Homans;Subject: NMR relaxation;Year: 1989;Link: http://books.google.com/books?id=wpggNxUrzSMC&pg=PA72&dq=rotational+correlation+time+radian&lr=;Quote (page 72): "<blockquote>For example, in the case of random translationalmotions, tau_c is defined as the mean time between collisions, whereasin the case of reorientational (rotational) motion, it is defined asthe average time for the molecule to rotate by one radian."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 CrystalsAuthor: Gordon and Breach Science PublishersSubject: CrystalsYear: 1974Link: http://books.google.com/books?id=bTW3AAAAIAAJ&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): "The rotational correlation time may be computed fromthe linewidths of the ... roughly the time required for the radical toreorient by saying that 1 radian is given by ..." 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 (page ?): " is the Larmor angular frequency in radians/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]]