mailSI supplementary units - the dimensionless and hidden radian unit.

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Posted by Edward d'Auvergne on January 08, 2009 - 11:19:

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

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

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
"(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:
"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."


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
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/("

Title:  Biophysics
Authors:  Gerald Ehrenstein, Harold Lecar
Subject:  NMR spin relaxation
Year:  1982
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
Quote (page 366):  " which the correlation time, tau_c, is
approximately the average time for the molecule to rotate by 1

Title:  Hydration Processes in Biology: Theoretical and Experimental 
Author:  Marie-Claire Bellissent-Funel
Subject:  Water motion
Year:  1999
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
Quote (page 9):  "The rotational correlation time, tau_c, is the time
taken for the particle to rotate through an angle of one radian

Title:  Fundamentals of Protein NMR Spectrosopy
Authors:  Gordon S. Rule, T. Kevin Hitchens
Subject:  NMR relaxation
Year:  2006
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 
Author:  Thomas L. James
Subject:  NMR
Year:  1975
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
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
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
Quote (page 45):  "[tau_c] is roughly equal to the time it takes a
molecule to rotate 1 radian while undergoing random rotational

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
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
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
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
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
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
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

4.1  Book quotations.

Title:  Nuclear Spin Relaxation in Liquids: Theory, Experiments, and
Authors:  Józef Kowalewski, Lena Mäler
Subject:  NMR relaxation
Year:  2006
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
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
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
   Delta_nu = 1/pi . R2 = 1/(pi.T2)."

Title:  Structural Biology: Practical NMR Applications
Author:  Quincy Teng
Subject:  NMR relaxation
Year:  2005
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.



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