# fundamental constants

## Fundamental constants

That group of physical constants which play a fundamental role in the basic theories of physics. These constants include the speed of light in vacuum, *c*; the magnitude of the charge on the electron, *e*, which is the fundamental unit of electric charge; the mass of the electron, *m _{e}*; Planck's constant, ℏ; and the fine-structure constant, α.

Relative | ||||
---|---|---|---|---|

Numerical | uncertainty, | |||

Quality | Symbol | value* | Units^{†} | ppm |

Speed of light in vacuum | c | 299792458 | m/s | (defined) |

Constant of gravitation | G | 6.67259(85) | 10^{-11} m^{3}/(kg · s^{2}) | 128 |

Planck constant | h | 6.6260755(40) | 10^{-34} J · s | 0.60 |

Elementary charge | e | 1.60217733(49) | 10^{-19}C | 0.30 |

Magnetic flux quantum, h/(2e) | Φ_{0} | 2.06783461(61) | 10^{-15} Wb | 0.30 |

Fine-structure constant, | α | 7.29735308(33) | 10^{-3} | 0.045 |

μ_{0}ce^{2}/(2h) | α^{-1} | 137.0359895(61) | 0.045 | |

Electron mass | m_{e} | 9.1093897(54) | 10^{-31} kg | 0.59 |

Proton mass | m_{p} | 1.6726231(10) | 10^{-27} kg | 0.59 |

Neutron mass | m_{n} | 1.6749286(10) | 10^{-27} kg | 0.59 |

Proton-electron mass ratio | m/_{p}m_{e} | 1836.152701(37) | 0.020 | |

Rydberg constant, mα_{e}c^{2}/(2h) | R_{∞} | 1.0973731534(13) | m^{-1} | 0.0012 |

Bohr radius, α/(4&pgr;R_{∞}) | a_{0} | 5.29177249(24) | 10^{-11} m | 0.045 |

Compton wavelength of | ||||

the electron, h/(m_{e}c) = α^{2}/(2R_{∞}) | λ_{c} | 2.42631058(22) | 10^{-12} m | 0.089 |

Classical electron radius, | ||||

μ_{0}e^{2}/(4&pgr;m) = α_{e}^{3}/(4&pgr;R_{∞}) | r_{e} | 2.81794092(38) | 10^{-15} m | 0.13 |

Bohr magneton, eh/(4&pgr;m)_{e} | μ_{B} | 9.2740154(31) | 10^{-24} J/T | 0.34 |

Electron magnetic moment | ||||

in Bohr magnetons | μ_{e}/μ_{B} | 1.001159652193(10) | 10^{-5} | |

Nuclear magneton, eh/(4&pgr;m)_{p} | μ_{N} | 5.0507866(17) | 10^{-27} J/T | 0.34 |

Proton magnetic moment | ||||

in nuclear magnetons | μ_{p}/μ_{N} | 2.792847386(63) | 0.023 | |

Boltzmann constant | k | 1.380658(12) | 10^{-23} J/K | 8.5 |

Avogadro constant | N_{A} | 6.0221367(36) | 10^{-23}/mol | 0.59 |

Fataday constant, N_{A}e | F | 96485.309(29) | C/mol | 0.30 |

Molar gas constant, N_{A}k | R | 8.314510(70) | J/(mol-K) | 8.4 |

*The digits in parentheses represent the one-standard-deviation uncertainties in the last digits of the quoted value. ^{†}C = coulomb, J = joule, kg = kilogram, m = meter, mol = mole, s = second, T = tesla, Wb = weber. |

These five quantities typify the different origins of the fundamental constants: *c* and ℏ are examples of quantities which appear naturally in the mathematical formulation of certain physical theories---Einstein's theories of relativity, and quantum theory, respectively; *e* and *m _{e}* are examples of quantities which characterize the elementary particles of which all matter is constituted; and α, the fundamental constant of quantum electrodynamics (QED), is an example of quantities which are combinations of other fundamental constants, but are actually constants in their own right since the same combination always appears together in the basic equations of physics.

Reliable numerical values for the fundamental physical constants are required for two main reasons. First, they are necessary if quantitative predictions from physical theory are to be obtained. Second, and even more important, the self-consistency of the basic theories of physics can be critically tested by a careful intercomparison of the numerical values of fundamental constants obtained from experiments in the different fields of physics. In general, the accuracy of fundamental constants determinations has continually improved over the years. Whereas in the past, 100 ppm (0.01%) and even 1000 ppm (0.1%) measurements were commonplace, today 0.01 ppm and better determinations are not unusual (ppm = parts per million).

Complex relationships can exist among groups of constants and conversion factors, and a particular constant may be determined either directly by measurement or indirectly by an appropriate combination of other directly measured constants. If the direct and indirect values have comparable accuracy, then both must be taken into account in order to arrive at a best value for that quantity. (By best value is meant that value believed to be closest to the true but unknown value.) Generally, each of the several routes which can be followed to a particular constant, both direct and indirect, will give a slightly different numerical value. Such a situation may be satisfactorily handled by the mathematical method known as least-squares. This technique provides a self-consistent procedure for calculating best “compromise” values of the constants from all of the available data. It automatically takes into account all possible routes and determines a single final value for each constant being calculated. It does this by weighting the different routes according to their relative uncertainties. The appropriate weights follow from the uncertainties assigned the individual measurements constituting the original set of data.

The 1986 least-squares adjustment, carried out under the auspices of the CODATA Task Group on Fundamental Constants, succeeded a CODATA adjustment in 1973 by E. R. Cohen and B. N. Taylor; CODATA, the Committee on Data for Science and Technology, is an interdisciplinary committee of the International Council of Scientific Unions. Recommended values are shown in the table.