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

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.

### 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:

$\Psi_{ml}(\theta(t), \phi(t)),$

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
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
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)
...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
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
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
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
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
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 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
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.

## 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
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
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 relaxation, 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...