Posted by **Edward d'Auvergne** on June 19, 2007 - 12:03:

Hi, I have thoroughly checked all the units of the physical constants, spectral densities, rotational correlation times, and relaxation rates and have a few important corrections about the units which are used in relax and elsewhere. Please read below for the details and a long story about SI vs. CGS units, frequency vs. angular frequency, and how it all relates to angular momentum. On 6/15/07, Sebastien Morin <sebastien.morin.1@xxxxxxxxx> wrote:

Hi, Here are the different values I obtain for a residue with R1 = 1.1336 +- 0.0851 ; R2 = 12.9336 +- 0.9649 ; and NOE = 0.463921 +- 0.045 J(0) J(wN) J(wH) ===================== ===================== ===================== Here are the results with Leo Spyracopoulos's Mathematica notebook. 4.704231413115747e-9 2.664085520910741e-10 9.485555428699657e-12 Here are the results with relax without multiplying frequencies in Hz by 2 pi. 4.6978912534878238e-09 2.6603551824374712e-10 9.478993207668287e-12 ratio 0.9986522432526923179 0.9985997677461966745 0.9993081880043085706 Here are the results with relax when multiplying frequencies in Hz by 2 pi. 3.4634030539343071e-09 1.9612804482358541e-10 9.478993207668287e-12 ratio 0.7362314371436068543 0.7361927508863804185 0.9993081880043085706 The very small discrepancy we get (ratio 0.999...) is due to several small differences in the definitions of constants (e.g. gn = -2.7126 in relax and -2.7108 in Leo's notebook). This is not important. However, there is a non negligible difference with the use of either frequencies in Hz or frequencies transformed to rad/s. Now, what do we do with that ? The spectral densities are in units of rad / s = rad s^-1. (these are not SI units, however)

The units of radians per second, or radian Hertz, is the angular frequency (omega) rather than the frequency (nu), yet both are nevertheless SI units. For example see http://en.wikipedia.org/wiki/Angular_frequency or related sites. These SI units are the same in the CGS unit system. The reason that we use the angular frequency rather than frequency is because we are talking about angular momentum.

The rates (R1, R2 and sigmaNOE) are in units of s^-1. The constant 'd' is in units of s^-2.

This is not correct. The correct units are rad^2.s^-2. This can be found by doing a unit analysis on the SI dipolar constant with the mu0/4pi component (or alternatively doing the analysis in the CGS system). The reason for the radians being part of the equation is because we are using Dirac's constant (h_bar) rather than Planck's constant (h). While Planck's constant has the units of Joules per Hertz (or Joule seconds) because Dirac's constant is divided by 2pi its units are Joules per Hertz per radian. However in most cases the radian units of Dirac's constant are not stated as we are usually talking about angular momentum (the angular part means we use radians hence it is indirectly implied). The other reason is because the gyromagnetic ratio also has units of radian, and again this is usually not reported (for example see Table 1.1 of Cavanagh in which the gyromagnetic ratios are in fact in rad.s^-1.T^-1, although they are not reported as such). All of this is still in SI units - the radian components are independent of the SI or CGS systems. The relevant units in the SI system for the dipolar constant, defined as d = (mu0/4pi)^2 . (gH.gX.h_bar/<r>^3)^2, are mu0 -> kg.m.s^-2.A^-2, h_bar -> J.s.rad^-1, gx -> rad.s^-1.T^-1, r -> m, where tesla (T) is equal to the units kg.A^-1.s^-2. The SI units for the CSA constant, defined as c = (omegaX . csa / 3)^2, are omegaX -> rad.s^-1, csa -> unitless. Hence both constants have the units of rad^2.s^-2.

The constant 'c' is in units of s^-2 also (or rad^2 s^-2 if we use frequencies in rad s^-1, which are not SI units).

The constant c is defined by the angular frequency (omega) rather than the frequency (nu). Hence the units of this constant in the relaxation equations, in both SI and CGS units, is always rad^2.s^-2 as well.

I now hesitate between 3 views. 1. Should the spectral densities be in SI units (i.e. in s, as the frequencies are in s^-1 and the rates in s^-1) ?

relax currently reports the spectral densities in the SI units for angular frequency of radian Hertz. Although often reported as Hertz, the correct unit is radian Hertz. This is again because radians are implied, but this time because we are talking about rotations. The reason is as follows: The spectral density function for isotropic diffusion is J(w) = tm / (1 + (w.tm)^2). The units for w (or omega) is rad.s^-1. Because the product w.tm is unitless the units for tm are in reality s.rad^-1. Hence the units of the spectral density function J(w) is also s.rad^-1. Although not reported in text books such as Cavanagh, you can still see remnants of the radians. For example in that book tc (or tm) is described as being "approximately the average time for the molecule to rotate by one radian". All the external and internal correlation times are in units of s.rad^-1 as they are all rotational correlation times! But because the are rotational times, the radians are implied and do not need to be reported.

2. If the spectral densities are in s rad^-1, the frequencies used to calculate them should also be in rad s^-1, thus the constant 'c' should be in rad^2 s^-2 and the constant 'd' also (so we should calculate it without multiplying by the factor of (mu/4pi)^2). This would be about using old units instead of the SI units, but then the spectral densities would be in s rad^-1.

In SI units the spectral density function is in units of s.rad^-1 whereas the physical constants (dipolar, csa, etc.) are in rad^2.s^-2. Hence the units for the relaxation rates - hold on to your seat - is in radian Hertz! The R1 relaxation rate is in rad.s^-1. Or the T1 relaxation time is in s.rad^-1. The reason that all of this is hardly ever discussed is because the units of radians is implied by the fact that this all relates to angular momentum. In the rotational world, radians are ubiquitous. Yet they are silent because they are implied. Unfortunately this 'hiding' of radians, combined with the CGS vs. SI unit systems, creates large amounts of confusion.

3. We could calculate everything in SI units (as we do right now) and normalize to rs ad^-1 in the end. Either way, I think that Leo's notebook yields spectral densities s and not the usual s rad^-1 (as in the 2006 paper : JBNMR,36:215-224, for which I calculated back spectral density values and yielded the same as published where they say it's in s rad^-1, but the units deriving says it's s).

relax reports spectral densities in the SI (and CGS) units of s.rad^-1 as dictated by angular momentum. Hence the input frequency of Hz must be multiplied by 2pi. Otherwise the product w.tc is not unitless and hence its square cannot be added to 1 as J(w) = tm/(1+(w.tm)^2).

Can you please tell me if I'm right with those ideas before I write to Leo to report this apparent bug or typo or whatever...

I hope that what I've written clarifies a few of the problems. Sincerely, Edward

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