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Non-SI units accepted for use with the SI, and units based on fundamental constants
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Summary
Non-SI units accepted for use with the SI, and units based on fundamental constants
Other non-SI units not recommended for use

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Related articles
Table 6: Non-SI units accepted for use with the SI
Table 7: Non-SI units whose values in SI units must be obtained experimentally
Table 8: Other non-SI units
Table 9: Non-SI units associated with the CGS and CGS-Gaussian system of units
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SI brochure, Section 4.1


The CIPM (2004) has revised the classification of non-SI units from that in the previous (7th) edition of this Brochure. Table 6 gives non-SI units that are accepted for use with the International System by the CIPM, because they are widely used with the SI in matters of everyday life. Their use is expected to continue indefinitely, and each has an exact definition in terms of an SI unit. Tables 7, 8 and 9 contain units that are used only in special circumstances. The units in Table 7 are related to fundamental constants, and their values have to be determined experimentally. Tables 8 and 9 contain units that have exactly defined values in terms of SI units, and are used in particular circumstances to satisfy the needs of commercial, legal, or specialized scientific interests. It is likely that these units will continue to be used for many years. Many of these units are also important for the interpretation of older scientific texts. Each of the Tables 6, 7, 8 and 9 is discussed in turn below.

Table 6 includes the traditional units of time and angle. It also contains the hectare, the litre, and the tonne, which are all in common everyday use throughout the world, and which differ from the corresponding coherent SI unit by an integer power of ten. The SI prefixes are used with several of these units, but not with the units of time.

Table 6. Non-SI units accepted for use with the International System of Units

Table 7 contains units whose values in SI units have to be determined experimentally, and thus have an associated uncertainty. Except for the astronomical unit, all other units in Table 7 are related to fundamental physical constants. The first three units, the non-SI units electronvolt, symbol eV, dalton or unified atomic mass unit, symbol Da or u, respectively, and the astronomical unit, symbol ua, have been accepted for use with the SI by the CIPM. The units in Table 7 play important roles in a number of specialized fields in which the results of measurements or calculations are most conveniently and usefully expressed in these units. For the electronvolt and the dalton the values depend on the elementary charge e and the Avogadro constant NA, respectively.

There are many other units of this kind, because there are many fields in which it is most convenient to express the results of experimental observations or of theoretical calculations in terms of fundamental constants of nature. The two most important of such unit systems based on fundamental constants are the natural unit (n.u.) system used in high energy or particle physics, and the atomic unit (a.u.) system used in atomic physics and quantum chemistry. In the n.u. system, the base quantities for mechanics are speed, action, and mass, for which the base units are the speed of light in vacuum c0, the Planck constant h divided by 2pi, called the reduced Planck constant with symbol , and the mass of the electron me, respectively. In general these units are not given any special names or symbols but are simply called the n.u. of speed, symbol c0, the n.u. of action, symbol , and the n.u. of mass, symbol me. In this system, time is a derived quantity and the n.u. of time is a derived unit equal to the combination of base units /mec02. Similarly, in the a.u. system, any four of the five quantities charge, mass, action, length, and energy are taken as base quantities. The corresponding base units are the elementary charge e, electron mass me, action , Bohr radius (or bohr) a0, and Hartree energy (or hartree) Eh, respectively. In this system, time is again a derived quantity and the a.u. of time a derived unit, equal to the combination of units /Eh. Note that a0 = alpha/(4piRinfinity), where alpha is the fine-structure constant and Rinfinity is the Rydberg constant; and Eh = e2/(4piepsilon0a0) = 2Rhc0 = alpha2mec02, where epsilon0 is the electric constant and has an exact value in the SI.

For information, these ten natural and atomic units and their values in SI units are also listed in Table 7. Because the quantity systems on which these units are based differ so fundamentally from that on which the SI is based, they are not generally used with the SI, and the CIPM has not formally accepted them for use with the International System. To ensure understanding, the final result of a measurement or calculation expressed in natural or atomic units should also always be expressed in the corresponding SI unit. Natural units (n.u.) and atomic units (a.u.) are used only in their own special fields of particle and atomic physics, and quantum chemistry, respectively. Standard uncertainties in the least significant digits are shown in parenthesis after each numerical value.

Table 7. Non-SI units whose values in SI units (the natural unit of speed excepted) must be obtained experimentally

Tables 8 and 9 contain non-SI units that are used by special interest groups for a variety of different reasons. Although the use of SI units is to be preferred for reasons already emphasized, authors who see a particular advantage in using these non-SI units should have the freedom to use the units that they consider to be best suited to their purpose. Since, however, SI units are the international meeting ground in terms of which all other units are defined, those who use units from Tables 8 and 9 should always give the definition of the units they use in terms of SI units.

Table 8 also gives the units of logarithmic ratio quantities, the neper, bel, and decibel. These are dimensionless units that are somewhat different in their nature from other dimensionless units, and some scientists consider that they should not even be called units. They are used to convey information on the nature of the logarithmic ratio quantity concerned. The neper, Np, is used to express the values of quantities whose numerical values are based on the use of the neperian (or natural) logarithm, ln = loge. The bel and the decibel, B and dB, where 1 dB = (1/10) B, are used to express the values of logarithmic ratio quantities whose numerical values are based on the decadic logarithm, lg = log10. The way in which these units are interpreted is described in footnotes (g) and (h) of Table 8. The numerical values of these units are rarely required. The units neper, bel, and decibel have been accepted by the CIPM for use with the International System, but are not considered as SI units.

The SI prefixes are used with two of the units in Table 8, namely, with the bar (e.g. millibar, mbar), and with the bel, specifically for the decibel, dB. The decibel is listed explicitly in the table because the bel is rarely used without the prefix.

Table 8. Other non-SI units

Table 9 differs from Table 8 only in that the units in Table 9 are related to the older CGS (centimetre-gram-second) system of units, including the CGS electrical units. In the field of mechanics, the CGS system of units was built upon three quantities and their corresponding base units: the centimetre, the gram, and the second. The CGS electrical units were still derived from only these same three base units, using defining equations different from those used for the SI. Because this can be done in different ways, it led to the establishment of several different systems, namely the CGS-ESU (electrostatic), the CGS-EMU (electromagnetic), and the CGS-Gaussian unit systems. It has always been recognized that the CGS-Gaussian system, in particular, has advantages in certain areas of physics, particularly in classical and relativistic electrodynamics (9th CGPM, 1948, Resolution 6). Table 9 gives the relations between these CGS units and the SI, and lists those CGS units that were assigned special names. As for the units in Table 8, the SI prefixes are used with several of these units (e.g. millidyne, mdyn; milligauss, mG, etc.).

Table 9. Non-SI units associated with the CGS and the CGS-Gaussian system of units


Related articles

Table 6: Non-SI units accepted for use with the SI
Table 7: Non-SI units whose values in SI units must be obtained experimentally
Table 8: Other non-SI units
Table 9: Non-SI units associated with the CGS and CGS-Gaussian system of units
     

Note: For the official text, please refer to the PDF files available at:
  • http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf (in English) and
  • http://www.bipm.org/utils/common/pdf/si_brochure_8_fr.pdf (in French).
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