Re: SI supplementary units - the dimensionless and hidden radian unit.
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 bi
SI supplementary units - the dimensionless and hidden radian unit.
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 t
