Latest revision as of 14:31, 19 November 2015

Caution If hard core NMR or physics theory is not to your taste, please do not read any further!

A 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 SI organisation itself. Specifically the SI supplementary units definitions whereby the assumption is made that if a process is rotational, a physicist should know that radian units are implicit.

Introduction

The concept of the hidden radian unit is quite important for understanding relaxation in NMR or, in fact, any rotational process in physics. 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 an angular process and hence radian units are implied (if you didn't get that, that was sarcasm). 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 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 is also a follow on from the comprehensive unit analysis of Abragam's relaxation equations.

The text below uses quotes from many different sources to demonstrate that the hidden radian unit is actually very much present in everything we do in NMR.

Source

This was originally published on the relax users mailing list.

SI supplementary units

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: [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.

Spherical harmonics

The time dependent spherical harmonic can be written as:

[math] \Psi_{ml}(\theta(t), \phi(t)), [/math]

where θ(t) 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

The rotational correlation time is a descriptor of the change of angles - and these angles are in the idden, dimensionless radian units. Hence the correlation time is measured in s/rad or in the hidden supplementary unit notation simply s. But it is better to think of the concept as the diffusion rate, a measure of the rate of rotational Brownian diffusion.

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).

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.

Relaxation rates

The units for relaxation rates are in rad/s. The equations from the book quotations hopefully show this conversion from Hz to rad/s.

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...

Difference between revisions of "Hidden radian units"

Latest revision as of 14:31, 19 November 2015

Contents

Introduction

Source

SI supplementary units

SI supplementary units (radian and steradian)

IUPAC report

Spherical harmonics

Rotational correlation times

Book quotations

Relaxation rates

Book quotations

See also

Navigation menu

Personal tools

Namespaces

Variants

Views

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@@ Line 1: / Line 1: @@
-<keywords content="diffusion rates, dimensionless units, hidden units, radian, relaxation rates, rotational correlation times, SI supplementary units, spherical harmonics" />
+{{Caution|If hard core NMR or physics theory is not to your taste, please do not read any further!}}
 A 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.  Specifically the SI supplementary units definitions whereby the assumption is made that if a process is rotational, a physicist should know that radian units are implicit.
 == Introduction ==
-'''Warning:'''  If hard core NMR or physics theory is not to your taste, please do not read any further!
+The concept of the hidden radian unit is quite important for understanding relaxation in NMR or, in fact, any rotational process in physics.  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 an angular process and hence radian units are implied (if you didn't get that, that was sarcasm).  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 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 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/s.  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).  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 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 is also a follow on from the [https://mail.gna.org/public/relax-devel/2007-06/msg00012.html comprehensive unit analysis of Abragam's relaxation equations].
-This is also a follow on from the [https://mail.gna.org/public/relax-devel/2007-06/msg00012.html comprehensive unit analysis of Abragam's relaxation equations].
+The text below uses quotes from many different sources to demonstrate that the hidden radian unit is actually very much present in everything we do in NMR.
 == Source ==
@@ Line 57: / Line 55: @@
 === IUPAC report ===
-This reference explains a bit more clearly why the radian unit is
+This reference explains a bit more clearly why the radian unit is invisible in most situations.
-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.
+<blockquote>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.</blockquote>
+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.
 == Spherical harmonics ==
@@ Line 84: / Line 73: @@
 </math>
-where <math alt="theta(t)">\theta(t)</math> and <math alt="phi(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.
+where ''θ(t)'' 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 ==
@@ Line 95: / Line 82: @@
 === Book quotations ===
-Title:  Physical Properties of Lipids
+;Title
-Authors:  Alejandro G. Marangoni, Suresh Narine
+:Physical Properties of Lipids
-Subject:  Fluorescence
+;Authors
-Year:  2002
+:Alejandro G. Marangoni, Suresh Narine
-Link:
+;Subject
-http://books.google.com/books?id=OCBav13l_MsC&pg=PA166&dq=rotational+correlation+time+radian&lr=
+:Fluorescence
-Quote (page 166):  "The rotational correlation time [phi] is the time
+;Year
-required by the fluorophore to rotate through an arc of 1 radian (phi
+:2002
-= 1/(2.pi.nu))."
+;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 time required by the fluorophore to rotate through an arc of 1 radian (phi = 1/(2.pi.nu)).</blockquote>
-Title:  Biophysics
+;Title
-Authors:  Gerald Ehrenstein, Harold Lecar
+:Biophysics
-Subject:  NMR spin relaxation
+;Authors
-Year:  1982
+:Gerald Ehrenstein, Harold Lecar
-Link:
+;Subject
-http://books.google.com/books?id=rThFVFmAdDAC&pg=PA14&dq=rotational+correlation+time+radian
+:NMR spin relaxation
-Quote (page 14):  "The value of tau_c can be approximated as the time
+;Year
-required for the molecule containing the resonant nucleus to either
+:1982
-rotate 1 radian (rotational correlation time) or diffuse a distance
+;Link
-equivalent to its own dimensions (translational correlation time)."
+: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 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).</blockquote>
-Title:  Protein NMR Spectroscopy (second edition)
+;Title
-Authors:  John Cavanagh, Wayne J. Fairbrother, Arthur G. Palmer, III,
+:Protein NMR Spectroscopy (second edition)
-Nicholas J. Skelton, Mark Rance
+;Authors
-Subject:  NMR relaxation
+:John Cavanagh, Wayne J. Fairbrother, Arthur G. Palmer, III, Nicholas J. Skelton, Mark Rance
-Year:  2007
+;Subject
-Link:
+:NMR relaxation
-http://books.google.com/books?id=2-LqLHOLHZwC&pg=PA366&dq=rotational+correlation+time+radian
+;Year
-Quote (page 366):  "...in which the correlation time, tau_c, is
+:2007
-approximately the average time for the molecule to rotate by 1
+;Link
-radian."
+: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, is approximately the average time for the molecule to rotate by 1 radian.</blockquote>
-Title:  Hydration Processes in Biology: Theoretical and Experimental
+;Title
-Approaches
+:Hydration Processes in Biology: Theoretical and Experimental  Approaches
-Author:  Marie-Claire Bellissent-Funel
+;Author
-Subject:  Water motion
+:Marie-Claire Bellissent-Funel
-Year:  1999
+;Subject
-Link:
+:Water motion
-http://books.google.com/books?id=9tJaB00wXhgC&pg=PA243&dq=rotational+correlation+time+radian&lr=
+;Year
-Quote (page 243):  "For such sites, the rotational and translational
+:1999
-diffusion of water should both be rate-limited by H-bond
+;Link
-rearrangements and it can therefore be argued that the residence time
+:http://books.google.com/books?id=9tJaB00wXhgC&pg=PA243&dq=rotational+correlation+time+radian&lr=
-(the time taken to diffusion ca. 3 Angstrom) should be close to the
+;Quote (page 243)
-first-rank rotational correlation time (the time taken to rotate
+<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 (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).</blockquote>
-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
+;Title
-Author:  Gordon Carl Kenmure Roberts
+:NMR of Macromolecules: A Practical Approach
-Subject:  NMR relaxation
+;Author
-Year:  1993
+:Gordon Carl Kenmure Roberts
-Link:
+;Subject
-http://books.google.com/books?id=K7n7SnmDbSAC&pg=PA9&dq=rotational+correlation+time+radian&lr=
+:NMR relaxation
-Quote (page 9):  "The rotational correlation time, tau_c, is the time
+;Year
-taken for the particle to rotate through an angle of one radian
+:1993
-(57°)."
+;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 time taken for the particle to rotate through an angle of one radian (57°).</blockquote>
-Title:  Fundamentals of Protein NMR Spectrosopy
+;Title
-Authors:  Gordon S. Rule, T. Kevin Hitchens
+:Fundamentals of Protein NMR Spectrosopy
-Subject:  NMR relaxation
+;Authors
-Year:  2006
+:Gordon S. Rule, T. Kevin Hitchens
-Link:
+;Subject
-http://books.google.com/books?id=8vmf5y6Jf84C&pg=PA441&dq=rotational+correlation+time+radian
+:NMR relaxation
-Quote (page 441):  "[tau_c] is the time required for a molecule to
+;Year
-rotate, on average, 1 radian."
+: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 to rotate, on average, 1 radian.</blockquote>
-Title:  Nuclear Magnetic Resonance in Biochemistry: Principles and
+;Title
-Applications
+:Nuclear Magnetic Resonance in Biochemistry: Principles and  Applications
-Author:  Thomas L. James
+;Author
-Subject:  NMR
+:Thomas L. James
-Year:  1975
+;Subject
-Link:
+:NMR
-http://books.google.com/books?id=iItqAAAAMAAJ&q=rotational+correlation+time+radian&dq=rotational+correlation+time+radian&pgis=1
+;Year
-Quote (page ?):  "The rotational correlation time (rc or rr) provides
+:1975
-a ... having many molecular collisions before it turns 1 radian."
+;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) provides a ... having many molecular collisions before it turns 1 radian.</blockquote>
-Title:  Biophysical Chemistry: Principles, Techniques, and Applications
+;Title
-Author:  Alan G. Marshall
+:Biophysical Chemistry: Principles, Techniques, and Applications
-Subject:  Rotational diffusion (for fluorescence)
+;Author
-Year:  1978
+:Alan G. Marshall
-Link:
+;Subject
-http://books.google.com/books?id=PJhqAAAAMAAJ&q=rotational+correlation+time+radian&dq=rotational+correlation+time+radian&pgis=1
+:Rotational diffusion (for fluorescence)
-Quote (page 720):  "...decay of the correlation function for
+;Year
-rotational diffusion, tau_rot may also be thought of as the
+:1978
-characteristic time it takes for a typical macromolecule to rotate
+;Link
-(diffusionally) through an angle of the order of a radian..."
+: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 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...</blockquote>
-Title:  Practical NMR Relaxation for Chemists
+;Title
-Author:  Vladimir I. Bakhmutov
+:Practical NMR Relaxation for Chemists
-Subject:  NMR relaxation
+;Author
-Year:  2004
+:Vladimir I. Bakhmutov
-Link:
+;Subject
-http://books.google.com/books?id=_gIh9KHIOx4C&pg=PA13&dq=rotational+correlation+time+radian&lr=
+:NMR relaxation
-Quote (page 13):  "However, a more correct definition of the tau_c is
+;Year
-connected with the, so-called autocorrelation function in the theory
+:2004
-of nuclear relaxation where the tau_c is an average time for the
+;Link
-molecule to progresses (sic.) through one radian."
+: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 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.</blockquote>
-Title:  Modern Protein Chemistry: Practical Aspects
+;Title
-Authors:  Gary C. Howard, William E. Brown
+:Modern Protein Chemistry: Practical Aspects
-Subject:  NMR relaxation
+;Authors
-Year:  2001
+:Gary C. Howard, William E. Brown
-Link:
+;Subject
-http://books.google.com/books?id=MIxdC7GPz0sC&pg=PA45&dq=rotational+correlation+time+radian&lr=
+:NMR relaxation
-Quote (page 45):  "[tau_c] is roughly equal to the time it takes a
+;Year
-molecule to rotate 1 radian while undergoing random rotational
+:2001
-motion."
+;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 a molecule to rotate 1 radian while undergoing random rotational motion.</blockquote>
-Title:  MRS of the Brain and Neurological Disorders
+;Title
-Authors:  Koji Terada, Akihiro Igata, Toshiro Fujimoto, Tetsuhiko
+:MRS of the Brain and Neurological Disorders
-Asakura, Institute of Advanced Medical Technology
+;Authors
-Subject:  Imaging
+:Koji Terada, Akihiro Igata, Toshiro Fujimoto, Tetsuhiko Asakura, Institute of Advanced Medical Technology
-Year:  2000
+;Subject
-Link:
+:Imaging
-http://books.google.com/books?id=kF2dw7c33cAC&pg=PA41&dq=rotational+correlation+time+radian&lr=#PPA43,M1
+;Year
-Quote (page 41):  "...Brownian motion.  This has a time scale, the
+:2000
-rotational correlation time (tau_c) defined as the time taken on
+;Link
-average for a solute molecule to rotate by one radian or roughly the
+:http://books.google.com/books?id=kF2dw7c33cAC&pg=PA41&dq=rotational+correlation+time+radian&lr=#PPA43,M1
-reciprocal of the rate of tumbling in solution of the relevant piece
+;Quote (page 41)
-of the molecule."
+<blockquote>...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.</blockquote>
-Title:  Structural Biology: Practical NMR Applications
+;Title
-Author:  Quincy Teng
+:Structural Biology: Practical NMR Applications
-Subject:  NMR relaxation
+;Author
-Year:  2005
+:Quincy Teng
-Link:
+;Subject
-http://books.google.com/books?id=dRmmGFkummIC&pg=PA36&dq=rotational+correlation+time+radian&lr=#PPA36,M1
+:NMR relaxation
-Quote (page 36):  "The correlation time is used to describe the rate
+;Year
-of random motions and is expressed as the average time between
+:2005
-collisions for translational motions or the time for a molecule to
+;Link
-rotate one radian in rotational motion."
+: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 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.</blockquote>
-Title:  High-resolution NMR Techniques in Organic Chemistry
+;Title
-Author:  Timothy D. W. Claridge
+:High-resolution NMR Techniques in Organic Chemistry
-Subject:  NMR relaxation
+;Author
-Year:  1999
+:Timothy D. W. Claridge
-Link:
+;Subject
-http://books.google.com/books?id=9srIkkL-YMkC&pg=PA283&dq=rotational+correlation+time+radian&lr=#PPA284,M1
+:NMR relaxation
-Quote (page 283):  "... its rotational correlation time, tau_c.  This
+;Year
-is usually taken to define the average time required for the molecule
+:1999
-to rotate through an angle of 1 radian about any axis, ..."
+;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.  This is usually taken to define the average time required for the molecule to rotate through an angle of 1 radian about any axis, ...</blockquote>
-Title:  A Dictionary of Concepts in NMR
+;Title
-Author:  S. W. Homans
+:A Dictionary of Concepts in NMR
-Subject:  NMR relaxation
+;Author
-Year:  1989
+:S. W. Homans
-Link:
+;Subject
-http://books.google.com/books?id=wpggNxUrzSMC&pg=PA72&dq=rotational+correlation+time+radian&lr=
+:NMR relaxation
-Quote (page 72):  "For example, in the case of random translational
+;Year
-motions, tau_c is defined as the mean time between collisions, whereas
+:1989
-in the case of reorientational (rotational) motion, it is defined as
+;Link
-the average time for the molecule to rotate by one radian."
+:http://books.google.com/books?id=wpggNxUrzSMC&pg=PA72&dq=rotational+correlation+time+radian&lr=
-Note this book later on page 72 makes the mistake (according to me) of
+;Quote (page 72)
-saying that 1/tau_c is in Hertz.
+<blockquote>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.</blockquote>
-Title:  Molecular Crystals and Liquid Crystals
+Note this book later on page 72 makes the mistake (according to me) of saying that 1/tau_c is in Hertz.
-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."
+;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 ?)
+<blockquote>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 ...</blockquote>
+;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 ?)
+<blockquote> is the Larmor angular frequency in radians/sec and tau_c is the rotational correlation time of the nuclei in sec/radian.</blockquote>
 == Relaxation rates ==
@@ Line 280: / Line 299: @@
 === 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
+;Title
-Author:  Vladimir I. Bakhmutov
+:Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications
-Subject:  NMR relaxation
+;Authors
-Year:  2004
+:Józef Kowalewski, Lena Mäler
-Link:
+;Subject
-http://books.google.com/books?id=_gIh9KHIOx4C&pg=PA13&dq=rotational+correlation+time+radian&lr=
+:NMR relaxation
-Quote (page 9):  "...linewidths, Delta_nu, measured in Hz, are
+;Year
-directly controlled by T1 and T2 relaxation times according to:
+:2006
-    Delta_nu = 1/(pi T1,2)"
+;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, are directly 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.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,M1
+;Quote (page 37)
+<blockquote>...T1 relaxation is inversely proportional to correlation time tau_c...</blockquote>
-Title:  Modern Protein Chemistry: Practical Aspects
+= See also =
-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
+[[Category:Theory]]
-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..."