Changes

WarningA major point of confusion in the NMR field is that of the dimensionless and hidden radian unit. This often results in people mistakenly writing Hertz units when inverse seconds (or rad/s) should have been used. This issue can be traced back to the [http: //en.wikipedia.org/wiki/International_System_of_Units SI] organisation itself. If hard core NMR or physics theory Specifically the SI supplementary units definitions whereby the assumption is made that if a process is not to your tasterotational, please do not read any further!a physicist should know that radian units are implicit.

This post is mainly for later reference, but == Introduction == The concept of the hidden radian unit is quite important for understanding the relaxation equations in NMRor, in fact, and actually any rotational physical processin physics. It For example it is important for understanding the model-free equations, for reduced spectral density mapping, for SRLS, and for relaxation dispersion. The In these cases, the reason is because R1 and R2 relaxation rates are measured in rad/s. As I describe described in section 1[[#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 (if you didn't get that, that was sarcasticsarcasm). Hence R2 can be said to be in units of 1/s, but never, ever Hz. Also note that because of the SI conventions described below, describing the correlation time in s units does not prove imply that there are no radian units. But reporting rates as Hz implies no radian units whereas reporting as 1/s instead often means indicates that radian units are present.

Keywords (for finding this post at a later date): diffusion rates,dimensionless units, The text below uses quotes from many different sources to demonstrate that the hidden units, radian, relaxation rates,rotational correlation times, SI supplementary units, sphericalharmonicsunit is actually very much present in everything we do in NMR.

 == Source ==

== SI supplementary units ==

=== SI supplementary units (radian and steradian) ===

"<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>

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

This is the part meaning that radians <blockquote>The units radian (rad) and steradian (sr), for plane angle and solid angle respectively, are implied if you are doinganything angulardescribed as 'SI supplementary units' [3]. I don't know what Since they mean by clarity because byomitting them it complicates thingsare of dimension 1 (i.e. Maybe you have to dimensionless), they may be included if appropriate, or they may be a physicistbefore you can see this omitted if clarityis not lost thereby, in expressions for derived SI units.</blockquote>

Y_ml(thetawhere ''θ(t), phi'' and ''ϕ(t))'' 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.

= Rotational correlation times == Book quotations ===

My opinion here is that the rotational correlation time is a;Titledescriptor :Physical Properties of the change of angles - and these angles are in theLipids;Authorshidden:Alejandro G. Marangoni, dimensionless radian unitsSuresh Narine;Subject:Fluorescence;Year:2002;Link:http://books.google. Hence the com/books?id=OCBav13l_MsC&pg=PA166&dq=rotational+correlation +time is+radian&lr=measured in s/rad or in the hidden supplementary unit notation simply;Quote (page 166)s. But I prefer to think of <blockquote>The rotational correlation time [phi] is the concept as time required by the diffusion rate, ameasure fluorophore to rotate through an arc of the rate of rotational Brownian diffusion1 radian (phi = 1/(2.pi.nu)).</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= Book quotations =PA366&dq=rotational+correlation+time+radian;Quote (page 366)<blockquote>...in which the correlation time, tau_c, is approximately the average time for the molecule to rotate by 1 radian.</blockquote>

;Title:Hydration Processes in Biology: Theoretical and Experimental Physical Properties of LipidsApproaches;AuthorAuthors: Alejandro G. Marangoni, Suresh NarineMarie-Claire Bellissent-Funel;Subject: FluorescenceWater motion;Year: 20021999;Link: http://books.google.com/books?id=OCBav13l_MsC9tJaB00wXhgC&pg=PA166PA243&dq=rotational+correlation+time+radian&lr=;Quote (page 166243)<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): "The should be close to the first-rank rotational correlation time [phi] is (the timerequired by the fluorophore taken to rotate through an arc of 1 one radian (phi), i.e., tau_W ~= 1/3 tau_s (2.pi.nu)where tau_s is the second-rank rotational correlation time)."</blockquote>

;Title: BiophysicsNMR of Macromolecules: A Practical Approach;AuthorAuthors: Gerald Ehrenstein, Harold LecarGordon Carl Kenmure Roberts;Subject: NMR spin relaxation;Year: 19821993;Link: http://books.google.com/books?id=rThFVFmAdDACK7n7SnmDbSAC&pg=PA14PA9&dq=rotational+correlation+time+radian&lr=;Quote (page 149): "<blockquote>The value of rotational correlation time, tau_c can be approximated as , is the timerequired taken for the molecule containing the resonant nucleus particle to eitherrotate 1 through an angle of one radian (rotational correlation time) or diffuse a distanceequivalent to its own dimensions (translational correlation time57°)."</blockquote>

;Title: Fundamentals of Protein NMR Spectroscopy (second edition)Spectrosopy;Authors: John Cavanagh, Wayne JGordon S. FairbrotherRule, Arthur GT. Palmer, III,Kevin HitchensNicholas J. Skelton, Mark Rance;SubjectSubject: NMR relaxation;Year: 20072006;Link: http://books.google.com/books?id=2-LqLHOLHZwC8vmf5y6Jf84C&pg=PA366PA441&dq=rotational+correlation+time+radian;Quote (page 366441): "...in which the correlation time, <blockquote>[tau_c, ] isapproximately the average time required for the a molecule to rotate by , on average, 1radian."</blockquote>

;Title: Hydration Processes Nuclear Magnetic Resonance in BiologyBiochemistry: Theoretical Principles and Experimental ApplicationsApproaches;AuthorAuthor: Marie-Claire Bellissent-FunelThomas L. James;Subject: Water motionNMR;Year: 19991975;Link: http://books.google.com/books?id=9tJaB00wXhgCiItqAAAAMAAJ&pgq=PA243rotational+correlation+time+radian&dq=rotational+correlation+time+radian&lrpgis=1;Quote (page 243): "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 <blockquote>The rotational correlation time (the time taken to rotatethrough one radianrc or rr), iprovides a .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?)having many molecular collisions before it turns 1 radian.</blockquote>

;Title: NMR of MacromoleculesBiophysical Chemistry: A Practical ApproachPrinciples, Techniques, and Applications;Author: Gordon Carl Kenmure RobertsAlan G. Marshall;Subject: NMR relaxationRotational diffusion (for fluorescence);Year: 19931978;Link: http://books.google.com/books?id=K7n7SnmDbSACPJhqAAAAMAAJ&pgq=PA9rotational+correlation+time+radian&dq=rotational+correlation+time+radian&lrpgis=1;Quote (page 9720): "The <blockquote>...decay of the correlation function for rotational correlation timediffusion, tau_c, is tau_rot may also be thought of as the characteristic timetaken it takes for the particle a typical macromolecule to rotate (diffusionally) through an angle of one the order of a radian(57°)."..</blockquote>

;Title: Fundamentals of Protein Practical NMR SpectrosopyRelaxation for Chemists;AuthorAuthors: Gordon SVladimir I. Rule, T. Kevin HitchensBakhmutov;Subject: NMR relaxation;Year: 20062004;Link: http://books.google.com/books?id=8vmf5y6Jf84C_gIh9KHIOx4C&pg=PA441PA13&dq=rotational+correlation+time+radian&lr=;Quote (page 44113): "[<blockquote>However, a more correct definition of the tau_c] is connected with the, so-called autocorrelation function in the theory of nuclear relaxation where the tau_c is an average time required for a the molecule torotate, on average, 1 progresses (sic.) through one radian."</blockquote>

;Title: Nuclear Magnetic Resonance in BiochemistryModern Protein Chemistry: Principles and Practical AspectsApplications;AuthorsAuthor: Thomas LGary C. Howard, William E. JamesBrown;Subject: NMRrelaxation;Year: 19752001;Link: http://books.google.com/books?id=iItqAAAAMAAJMIxdC7GPz0sC&qpg=rotational+correlation+time+radianPA45&dq=rotational+correlation+time+radian&pgislr=1;Quote (page ?45): "The rotational correlation <blockquote>[tau_c] is roughly equal to the time (rc or rr) providesit takes a ... having many molecular collisions before it turns molecule to rotate 1 radianwhile undergoing random rotational motion."</blockquote>

;Title: Biophysical ChemistryMRS of the Brain and Neurological Disorders;Authors: PrinciplesKoji Terada, Akihiro Igata, TechniquesToshiro Fujimoto, and ApplicationsTetsuhiko Asakura, Institute of Advanced Medical TechnologyAuthor: Alan G. Marshall;SubjectSubject: Rotational diffusion (for fluorescence)Imaging;Year: 19782000;Link: http://books.google.com/books?id=PJhqAAAAMAAJkF2dw7c33cAC&qpg=rotational+correlation+time+radianPA41&dq=rotational+correlation+time+radian&pgislr=1#PPA43,M1;Quote (page 72041): "<blockquote>...Brownian motion.decay of This has a time scale, the rotational correlation function forrotational diffusion, tau_rot may also be thought of time (tau_c) defined as thecharacteristic time it takes taken on average for a typical macromolecule solute molecule to rotate(diffusionally) through an angle by one radian or roughly the reciprocal of the rate of tumbling in solution of the order relevant piece of a radianthe molecule..."</blockquote>

;Title:Structural Biology: Practical NMR Relaxation for ChemistsApplications;Author: Vladimir I. BakhmutovQuincy Teng;Subject: NMR relaxation;Year: 20042005;Link: http://books.google.com/books?id=_gIh9KHIOx4CdRmmGFkummIC&pg=PA13PA36&dq=rotational+correlation+time+radian&lr=#PPA36,M1;Quote (page 1336): "However, a more correct definition of the tau_c <blockquote>The correlation time isconnected with used to describe the, so-called autocorrelation function in the theoryrate of nuclear relaxation where random motions and is expressed as the tau_c is an average time between collisions for translational motions or thetime for a molecule to progresses (sic.) through rotate one radianin rotational motion."</blockquote>

;Title: Modern Protein High-resolution NMR Techniques in Organic Chemistry: Practical AspectsAuthors;Author: Gary CTimothy D. Howard, William EW. BrownClaridge;Subject: NMR relaxation;Year: 20011999;Link: http://books.google.com/books?id=MIxdC7GPz0sC9srIkkL-YMkC&pg=PA45PA283&dq=rotational+correlation+time+radian&lr=#PPA284,M1;Quote (page 45283): <blockquote>... its rotational correlation time, tau_c. "[tau_c] This is roughly equal usually taken to define the average time it takes arequired for the molecule to rotate through an angle of 1 radian while undergoing random rotationalmotionabout any axis, ..."</blockquote>

;Title: MRS A Dictionary of the Brain and Neurological DisordersConcepts in NMR;AuthorAuthors: Koji Terada, Akihiro Igata, Toshiro Fujimoto, TetsuhikoS. W. HomansAsakura, Institute of Advanced Medical Technology;SubjectSubject: ImagingNMR relaxation;Year: 20001989;Link: http://books.google.com/books?id=kF2dw7c33cACwpggNxUrzSMC&pg=PA41PA72&dq=rotational+correlation+time+radian&lr=#PPA43,M1;Quote (page 4172): "...Brownian motion. This has a <blockquote>For example, in the case of random translational motions, tau_c is defined as the mean time scalebetween collisions, whereas in thecase of reorientational (rotational correlation time (tau_c) motion, it is defined as the average time taken onaverage for a solute the 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 ApplicationsAuthor: Quincy TengSubject: NMR relaxationYear: 2005Link: http://books.google.com/books?id=dRmmGFkummIC&pg=PA36&dq=rotational+correlation+time+radian&lr=#PPA36,M1Quote Note this book later on page 72 makes the mistake (page 36according to me): "The correlation time is used to describe the rateof random motions and saying that 1/tau_c is expressed as the average time betweencollisions for translational motions or the time for a molecule torotate one radian in rotational motionHertz."

;Title: High-resolution NMR Techniques in Organic ChemistryMolecular Crystals and Liquid Crystals;Author: Timothy D. W. ClaridgeGordon and Breach Science Publishers;Subject: NMR relaxationCrystals;Year: 19991974;Link: http://books.google.com/books?id=9srIkkL-YMkCbTW3AAAAIAAJ&pgq=PA283rotational+correlation+time+radian&dq=rotational+correlation+time+radian&lr=#PPA284,M1&pgis=1;Quote (page 283?): "... its <blockquote>The rotational correlation time, tau_cmay be computed from the linewidths of the .. Thisis usually taken to define . roughly the average time required for the moleculeradical to rotate through an angle of reorient by 1 radian about any axis, is given by ..."</blockquote>

;Title: A Dictionary of Concepts in NMRIndustrial Research/development;Author: STechnical Pub. WCo. Homans;Subject: NMR relaxation;Year: 19891978;Link: http://books.google.com/books?id=wpggNxUrzSMCEstVAAAAMAAJ&pgq=PA72rotational+correlation+time+radian&dq=rotational+correlation+time+radian&lr=&pgis=1;Quote (page 72?): "For example, in the case of random translationalmotions, tau_c <blockquote> is defined as the mean time between collisions, whereasLarmor angular frequency in radians/sec and tau_c is the case of reorientational (rotational) motion, it is defined asthe average correlation time for of the molecule to rotate by one nuclei in sec/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&dqRelaxation rates =rotational+correlation+time+radian&lr=&pgis=1Quote (page ?): "The rotational correlation time may be computed fromthe linewidths of the ... roughly the time required for the radical toreorient by 1 radian is given by ..."

Title: Industrial Research/developmentAuthor: Technical Pub. Co.Subject: NMR The units for relaxationYear: 1978Link: http:rates are in rad//books.googles.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 equations from the nuclei in secbook quotations hopefully show this conversion from Hz to rad/radians."

Here;Title:Nuclear Spin Relaxation in Liquids: Theory, Experiments, my opinion 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 that radians per second, the relaxation rate, or the R1 and 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 radgiven 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 equationsfrom Fourier transform of an exponential decay is Lorentzian centered at zero frequency, with the book quotations hopefully show this conversion from Hz full width at half-height (in Hertz) equal toradDelta_nu = 1/s(pi.T2)...</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= Book quotations 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.T2).</blockquote>

;Title: Nuclear Spin Relaxation in LiquidsStructural Biology: Theory, Experiments, andPractical NMR ApplicationsApplications;AuthorAuthors: Józef Kowalewski, Lena MälerQuincy Teng;Subject: NMR relaxation;Year: 20062005;Link: http://books.google.com/books?id=MiUfcE1C9CQCdRmmGFkummIC&pg=PA14PA36&dq=relaxationrotational+correlation+ratetime+radian&lr=#PPA19PPA36,M1;Quote (page 1537): "Because the natural unit for the angular frequencyis radians per second, the relaxation rate, or the inverse ofrelaxation time, R2 = 1/T2, should indeed also be expressed in theseunits<blockquote>... Usually, T1 relaxation times are given in seconds (the rates aregiven in 1/s), which tacitly implies that the radians can be omitted;we note in parenthesis that the radian is considered a dimensionlessunit in physics."Quote 2 (page 15): "The Fourier transform of an exponential decay isLorentzian centered at zero frequency, with the full width athalf-height (in Hertz) equal inversely proportional to Delta_nu = 1/(pi.T2)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]]