Hi Ed, After months of leaving this message in my Inbox for a further in-depth read, I finally got time and motivation to go through this discussion on "hard core NMR" and "physics theory", as you described it yourself...
This text is very interesting and instructive ! Thank you ! Séb :) Edward d'Auvergne wrote: > Hi, > > Warning: If hard core NMR or physics theory is not to your taste, > please do not read any further! > > This post is mainly for later reference, but is quite important for > understanding the relaxation equations in NMR, and actually any > rotational physical process. It is important for the model-free > equations, for reduced spectral density mapping, for SRLS, and for > relaxation dispersion. The reason is because R1 and R2 are measured > in rad/s. As I describe in section 1, 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). > 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 that there are no > radian units. But reporting rates as Hz implies no radian units > whereas reporting as 1/s instead often means radian units are present. > > This is also a follow on from the comprehensive unit analysis of > Abragam's relaxation equations at > https://mail.gna.org/public/relax-devel/2007-06/msg00012.html. > > Keywords (for finding this post at a later date): diffusion rates, > dimensionless units, hidden units, radian, relaxation rates, > rotational correlation times, SI supplementary units, spherical > harmonics. > > > > 1 SI supplementary units. > > 1.1 SI supplementary units (radian and steradian). > > Quote from the PDF linked below (page 26) in the table titled "Table > 3. Coherent derived units in the SI with special names and symbols" > about the 'SI coherent derived unit' for the 'plane angle' unit of > radian: > "(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." > > Quote from the PDF linked below (page 28) in the section titled "2.2.3 > Units for dimensionless quantities, also called quantities of > dimension one": > "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." > > Quote from the PDF linked below (page 42) in the section titled "5.3.7 > Stating values of dimensionless quantities, or quantities of dimension > one". This is not very clear but explains why the rad unit is many > times hidden, and why the other dimensionless units such as % and ppm > must be stated (need to read the whole section for that): > "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)." > > Quotes from the PDF linked below (page 67) from the appendix section > titled "SI supplementary units (radian and steradian)": > "...the units radian and steradian are usually introduced into > expressions for units when there is need for clarification..." > > Quote from the PDF linked below (page 67) from the appendix section > titled "Elimination of the class of supplementary units in the SI" for > resolution 8 of the CGPM conference: > "decides..." > "to interpret the supplementary units in the SI, namely the radian and > the steradian, as dimensionless derived units, the names and symbols > of which may, but need not, be used in expressions for other SI > derived units, as is convenient," > "and, consequently, to eliminate the class of supplementary units as a > separate class in the SI." > > Links: > http://www.bipm.org/en/si/si_brochure/ > http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf > > > 1.2 IUPAC report. > > This reference explains a bit more clearly why the radian unit is > invisible in most situations. > > Title: Quantities, units, and symbols in physical chemistry (second edition). > > Quote from page 11: > "The units radian (rad) and steradian (sr), for plane angle and solid > angle respectively, are described as 'SI supplementary units' [3]. > Since they are of dimension 1 (i.e. dimensionless), they may be > included if appropriate, or they may be omitted if clarity is not lost > thereby, in expressions for derived SI units." > > This is the part meaning that radians are implied if you are doing > anything angular. I don't know what they mean by clarity because by > omitting them it complicates things. Maybe you have to be a physicist > before you can see this clarity. > > > > 2 Spherical harmonics. > > The time dependent spherical harmonic can be written as > > Y_ml(theta(t), phi(t)), > > where theta(t) and phi(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 I would assume that > because it is angular, it is using the radian angular SI unit. > > > > 3 Rotational correlation times. > > My opinion here is that the rotational correlation time is a > descriptor of the change of angles - and these angles are in the > hidden, dimensionless radian units. Hence the correlation time is > measured in s/rad or in the hidden supplementary unit notation simply > s. But I prefer to think of the concept as the diffusion rate, a > measure of the rate of rotational Brownian diffusion. > > > 3.1 Book quotations. > > 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): "The rotational correlation time [phi] is the time > required by the fluorophore to rotate through an arc of 1 radian (phi > = 1/(2.pi.nu))." > > 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): "The value of tau_c can be approximated as the time > required for the molecule containing the resonant nucleus to either > rotate 1 radian (rotational correlation time) or diffuse a distance > equivalent to its own dimensions (translational correlation time)." > > 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): "...in which the correlation time, tau_c, is > approximately the average time for the molecule to rotate by 1 > radian." > > 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): "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 (the time taken to rotate > through one radian), i.e., tau_W ~= 3 tau_s (where tau_s is the > second-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?) > > 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): "The rotational correlation time, tau_c, is the time > taken for the particle to rotate through an angle of one radian > (57°)." > > 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): "[tau_c] is the time required for a molecule to > rotate, on average, 1 radian." > > 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 ?): "The rotational correlation time (rc or rr) provides > a ... having many molecular collisions before it turns 1 radian." > > 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): "...decay of the correlation function for > rotational diffusion, tau_rot may also be thought of as the > characteristic time it takes for a typical macromolecule to rotate > (diffusionally) through an angle of the order of a radian..." > > 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): "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 for the > molecule to progresses (sic.) through one radian." > > 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): "[tau_c] is roughly equal to the time it takes a > molecule to rotate 1 radian while undergoing random rotational > motion." > > Title: MRS of the Brain and Neurological Disorders > Authors: Koji Terada, Akihiro Igata, Toshiro Fujimoto, Tetsuhiko > Asakura, 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): "...Brownian motion. This has a time scale, the > rotational correlation time (tau_c) defined as the time taken on > average for a solute molecule to rotate by one radian or roughly the > reciprocal of the rate of tumbling in solution of the relevant piece > of the molecule." > > 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): "The correlation time is used to describe the rate > of random motions and is expressed as the average time between > collisions for translational motions or the time for a molecule to > rotate one radian in rotational motion." > > 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): "... its rotational correlation time, tau_c. This > is usually taken to define the average time required for the molecule > to rotate through an angle of 1 radian about any axis, ..." > > 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): "For example, in the case of random translational > motions, tau_c is defined as the mean time between collisions, whereas > in the case of reorientational (rotational) motion, it is defined as > the average time for the molecule to rotate by one radian." > Note this book later on page 72 makes the mistake (according to me) of > saying that 1/tau_c is in Hertz. > > Title: Molecular Crystals and Liquid Crystals > Author: Gordon and Breach Science Publishers > Subject: Crystals > Year: 1974 > Link: > http://books.google.com/books?id=bTW3AAAAIAAJ&q=rotational+correlation+time+radian&dq=rotational+correlation+time+radian&lr=&pgis=1 > Quote (page ?): "The rotational correlation time may be computed from > the linewidths of the ... roughly the time required for the radical to > reorient by 1 radian is given by ..." > > Title: Industrial Research/development > Author: Technical Pub. Co. > Subject: NMR relaxation > Year: 1978 > Link: > http://books.google.com/books?id=EstVAAAAMAAJ&q=rotational+correlation+time+radian&dq=rotational+correlation+time+radian&lr=&pgis=1 > Quote (page ?): " is the Larmor angular frequency in radians/sec and > tau_c is the rotational correlation time of the nuclei in sec/radian." > > > > 4 Relaxation rates > > Here, my opinion is that the R1 and R2 units are rad/s. The equations > from the book quotations hopefully show this conversion from Hz to > rad/s. > > > 4.1 Book quotations. > > 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): "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." > Quote 2 (page 15): "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)..." > > 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): "...linewidths, Delta_nu, measured in Hz, are > directly controlled by T1 and T2 relaxation times according to: > Delta_nu = 1/(pi T1,2)" > > 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): "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)." > > 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 37): "...T1 relaxation is inversely proportional to > correlation time tau_c..." > > > Apologies for the length, but this is designed to be a comprehensive > reference for future use. > > Regards, > > Edward > > _______________________________________________ > relax (http://nmr-relax.com) > > This is the relax-users mailing list > relax-users@gna.org > > To unsubscribe from this list, get a password > reminder, or change your subscription options, > visit the list information page at > https://mail.gna.org/listinfo/relax-users > > -- Sébastien Morin PhD Student S. Gagné NMR Laboratory Université Laval & PROTEO Québec, Canada _______________________________________________ relax (http://nmr-relax.com) This is the relax-users mailing list relax-users@gna.org To unsubscribe from this list, get a password reminder, or change your subscription options, visit the list information page at https://mail.gna.org/listinfo/relax-users
