Physical constant

A physical constant or natural constant (occasionally also elementary constant) is a physical quantity whose value cannot be influenced and does not change in space or time.

Fundamental constants of nature are those constants that relate to general properties of space, time, and physical processes that apply equally to every kind of particle and interaction. These are the speed of light, Planck's quantum of action, and the gravitational constant (see also Natural Units).

Other elementary (or fundamental) constants of nature relate to the individual types of particles and interactions, e.g. their masses and charges. Derived natural constants can be calculated from the fundamental and elementary constants. For example, Bohr's radius, a constant relevant to atomic physics, can be calculated from Planck's quantum of action, the speed of light, the elementary charge and the mass of the electron.

Sometimes parameters or coefficients which are constant only in a certain arrangement or constellation are called constants, e.g. Kepler's constant, the decay constant or the spring constant. Strictly speaking, however, they are not constants, but parameters of the arrangement under investigation.

Some natural sciences combine important constants into groups of fundamental constants, e.g. in astronomy and geodesy these are the exact reference values of earth and sun mass, the earth radius, the astronomical unit or the gravitational constant.

Reference values commonly used in practice, such as the duration of a year, the pressure of the standard atmosphere or the acceleration due to gravity, are not natural constants. They are useful to man in his earthly environment, but as a rule they have no significance of a fundamental nature beyond that, nor do they prove to be truly constant with increasing measurement accuracy. However, they served for the first definition of units of measurement (also e.g. for second, meter, kilogram). Modern efforts have been made to define the units of measurement by direct reference to (fundamental or elementary) natural constants. The natural constants selected for this purpose are thus given a fixed, unchanging numerical value. By the 26th General Conference on Weights and Measures, all units of the International System of Units were defined with effect from 20 May 2019 by three fundamental natural constants (c, h, e), one special atomic transition (νCs) and three arbitrarily determined constants (kB, NA, Kcd).

Table of some constants

The digits in parentheses after a numerical value denote the uncertainty in the last digits of the value. (Example: The so-called shorthand notation 6.674 30(15) is equivalent to 6.674 30 ±0.000 15). The uncertainty is given as the estimated standard deviation of the given numerical value from the actual value. The numerical values are based on CODATA 2018.

Designation of the constant

Symbol(s)

Value (SI)

Fundamental

Note

Electromagnetism

Speed of light in vacuum

c

299792458 {\displaystyle \textstyle \mathrm {\frac {m}{s}} }

Yes

Nk. F.

Elementary charge

e

1.602176634e-19 {\displaystyle \textstyle \mathrm {C} }

Yes

Nk. F.

Magnetic field constant

\mu _{0}

1.25663706212(19)e-6 {\displaystyle \textstyle \mathrm {\frac {H}{m}} }

Yes

fK. a. M.

Electric field constant

{\displaystyle \varepsilon _{0}={\frac {1}{\mu _{0}\,c^{2}}}}

8.8541878128(13)e-12 {\displaystyle \textstyle \mathrm {\frac {A\,s}{V\,m}} }

Yes

fK. a. M.

Coulomb constant

k_{C}={\frac {1}{4\pi \varepsilon _{0}}}

8.9875517922(14)e9 {\displaystyle \textstyle \mathrm {\frac {m}{F}} }

Yes

fK. a. M.

Wave impedance of the vacuum

{\displaystyle Z_{w0}=\mu _{0}\,c}

3.76730313667(57)e2 Ω {\displaystyle \textstyle \mathrm {\Omega } }

Yes

fK. a. M.

Gravity and cosmology

Gravitational constant

G

6.67430(15)e-11 {\displaystyle \textstyle \mathrm {\frac {m^{3}}{kg\,s^{2}}} }

Yes

Nk. M.

Planck mass

{\displaystyle m_{\text{Planck}}={\sqrt {\frac {\hbar \,c}{G}}}}

2.176434(24)e-8 {\displaystyle \textstyle \mathrm {kg} }

Yes

Planck length

{\displaystyle l_{\text{Planck}}={\frac {\hbar }{m_{\text{Planck}}\,c}}}

1.616255(18)e-35 {\displaystyle \textstyle \mathrm {m} }

Yes

Planck time

{\displaystyle t_{\text{Planck}}={\frac {l_{\text{Planck}}}{c}}}

5.391247(60)e-44 {\displaystyle \textstyle \mathrm {s} }

Yes

Gravitational coupling constant

{\displaystyle \alpha _{G}={\frac {G\,m_{\text{e}}^{2}}{\hbar \,c}}={\frac {m_{e}^{2}}{m_{\text{Planck}}^{2}}}}

1.751810(39)e-45

No

Nk. a. M.

Thermodynamics

Boltzmann constant

k_{B}

1.380649e-23 {\displaystyle \textstyle {\frac {\mathrm {J} }{\mathrm {K} }}}= 8.617333262...e-5 {\displaystyle \textstyle \mathrm {\frac {eV}{K}} }

Yes

fK. F.

Stefan-Boltzmann constant

{\displaystyle \sigma ={\frac {2\pi ^{5}k_{B}^{4}}{15\,h^{3}c^{2}}}}

5.670374419...e-8 {\displaystyle \textstyle \mathrm {\frac {W}{m^{2}\,K^{4}}} }

Yes

fK. a. F.

Vienna Constant

{\displaystyle b={\frac {hc}{4{,}965\,114\cdot k_{B}}}}

2.897771955...e-3 {\displaystyle \textstyle \mathrm {m\cdot K} }

Yes

fK. a. F.

Avogadro constant

N_A

6.02214076e23 {\displaystyle \textstyle {\frac {1}{\mathrm {mol} }}}

No

fK. F.

Faraday constant

{\displaystyle F=e\,N_{A}}

96485.3321233100184... {\displaystyle \textstyle \mathrm {\frac {C}{mol}} }

No

fK. a. F.

Gas constant

{\displaystyle R=N_{A}\,k_{B}}

8.31446261815324 {\displaystyle \textstyle \mathrm {\frac {J}{K\cdot mol}} }

No

fK. a. F.

Loschmidt constant at T0=273.15 K and p0=101.325 kPa

N_{L}or {\displaystyle n_{0}=N_{A}\cdot {\frac {p_{0}}{RT_{0}}}}

2.686780111...e25 {\displaystyle \textstyle \mathrm {\frac {1}{m^{3}}} }

No

fK. a. F.

Molar volume of an ideal gas

{\displaystyle V_{m_{0}}={\frac {R\,T_{0}}{p_{0}}}={\frac {N_{A}}{n_{0}}}}

0.02241396954... {\displaystyle \textstyle \mathrm {\frac {m^{3}}{mol}} }

No

fK. a. F.

Atomic Physics

Rydberg constant

{\displaystyle R_{\infty }={\frac {e^{4}\,m_{e}}{8\varepsilon _{0}^{2}h^{3}c}}={\frac {\alpha ^{2}}{2}}\,{\frac {m_{\mathrm {e} }c}{h}}}

1.0973731568160(21)e7 {\displaystyle \textstyle {\frac {1}{\mathrm {m} }}}

No

F. a. M.

Rydberg Energy

{\displaystyle R_{\infty }hc={\frac {E_{h}}{2}}={\frac {\alpha ^{2}}{2}}m_{\mathrm {e} }c^{2}}

13.605693122994(26) {\displaystyle \textstyle \mathrm {eV} }= 2.1798723611035(42)e-18 {\displaystyle \textstyle \mathrm {J} }

No

Rydberg frequency

R_{\infty }\,c

3.2898419602508(64)e15 {\displaystyle \textstyle \mathrm {Hz} }

No

Hartree Energy

{\displaystyle E_{h}={\frac {e^{4}\,m_{\mathrm {e} }}{4\,\varepsilon _{0}^{2}\,h^{2}}}=m_{\mathrm {e} }c^{2}\alpha ^{2}}

4.3597447222071(85)e-18 {\displaystyle \textstyle \mathrm {J} }

No

Quantum and particle physics

Planck's constant

h

6.62607015e-34 {\displaystyle \textstyle \mathrm {J\,s} }= 4.135667696...e-15 {\displaystyle \textstyle \mathrm {eV\,s} }

Yes

Nk. F.

Planck's reduced quantum of action

\hbar ={\frac {h}{2\pi }}

1.054571817...e-34 {\displaystyle \textstyle \mathrm {J\,s} }

Yes

Nk. a. F.

Spectral radiation constant

{\displaystyle c_{1L}={\frac {2hc^{2}}{sr}}}

1.191042972...e-16 {\displaystyle \textstyle \mathrm {\frac {W\,m^{2}}{sr}} }

Nk. a. F.

First radiation constant

{\displaystyle c_{1}=2\pi \,hc^{2}}

3.741771852...e-16 {\displaystyle \textstyle \mathrm {W\,m^{2}} }

Yes

Nk. a. F.

Second radiation constant

{\displaystyle c_{2}={\frac {hc}{k_{B}}}}

1.438776877...e-2 {\displaystyle \textstyle \mathrm {m\cdot K} }

Yes

fK. a. F.

Fine structure constant

{\displaystyle \alpha ={\frac {\mu _{0}\,e^{2}c}{2h}}}

7.2973525693(11)e-3 = (137.035999084(21))−1

No

F. a. M.

Nuclear Magneton

{\displaystyle \mu _{N}={\frac {e\,\hbar }{2\,m_{p}}}}

5.0507837461(15)e-27 {\displaystyle \textstyle \mathrm {\frac {J}{T}} }

No

Nk. a. M.

magnetic flux quantum

{\displaystyle \Phi _{0}={\frac {h}{2e}}}

2.067833848...e-15 {\displaystyle \textstyle \mathrm {Wb} }

No

Nk. a. F.

Josephson constant

{\displaystyle K_{J}={\frac {1}{\Phi _{0}}}={\frac {2e}{h}}}

4.835978484...e14 {\displaystyle \textstyle \mathrm {\frac {Hz}{V}} }

No

Nk. a. F.

Von Klitzing constant

{\displaystyle R_{K}={\frac {h}{e^{2}}}}

25812.80745... Ω {\displaystyle \textstyle \mathrm {\Omega } }

No

Nk. a. F.

Conductance quantum

{\displaystyle G_{0}={\frac {2e^{2}}{h}}}

7.748091729...e-5 {\displaystyle \textstyle \mathrm {\frac {s\,C^{2}}{m^{2}\,kg}} }

No

Nk. a. F.

Fermi constant

{\displaystyle G_{\rm {F}}^{0}={\frac {G_{\rm {F}}}{(\hbar c)^{3}}}={\frac {\sqrt {2}}{8}}{\frac {g^{2}}{m_{\text{W}}^{2}}}}

4.5437957(23)e14 {\displaystyle \textstyle \mathrm {\frac {1}{J^{2}}} }= 1.1663787(6)e-5 {\displaystyle \textstyle \mathrm {\frac {1}{GeV^{2}}} }

No

???

Electron

Electron mass

m_{{\mathrm e}}

9.1093837015(28)e-31 {\displaystyle \textstyle \mathrm {kg} }= 5.48579909065(16)e-4 {\displaystyle \textstyle \mathrm {u} }

No

Nk. M.

Compton wavelength of the electron

{\displaystyle \lambda _{C}={\frac {h}{m_{\mathrm {e} }c}}}

2.42631023867(73)e-12 {\displaystyle \textstyle \mathrm {m} }

No

Nk. a. M.

Drilling radius

{\displaystyle a_{0}={\frac {4\pi \varepsilon _{0}\hbar ^{2}}{e^{2}\,m_{\mathrm {e} }}}={\frac {1}{\alpha }}{\frac {\lambda _{C}}{2\pi }}={\frac {\hbar }{\alpha m_{\mathrm {e} }c}}}

5.29177210903(80)e-11 {\displaystyle \textstyle \mathrm {m} }

No

fK. a. M.

Classical electron radius

{\displaystyle r_{\mathrm {e} }={\frac {1}{4\pi \varepsilon _{0}}}\,{\frac {e^{2}}{m_{\mathrm {e} }c^{2}}}=\alpha ^{2}\,a_{0}}

2.8179403262(13)e-15 {\displaystyle \textstyle \mathrm {m} }

No

Nk. a. M.

Bohr's magneton

{\displaystyle \mu _{B}={\frac {e\,\hbar }{2\,m_{\mathrm {e} }}}}

9.2740100783(28)e-24 {\displaystyle \textstyle \mathrm {\frac {J}{T}} }

No

fK. a. M.

Magnetic moment of the electron

{\displaystyle \mu _{\mathrm {e} }}

-9.2847647043(28)e-24 {\displaystyle \textstyle \mathrm {\frac {J}{T}} }

No

???

Landé factor of the electron

{\displaystyle g_{\mathrm {e} }=-2{\frac {\mu _{\mathrm {e} }}{\mu _{B}}}}

-2.00231930436256(35)

No

???

Gyromagnetic ratio of the electron

{\displaystyle \gamma _{\mathrm {e} }=-2{\frac {\mu _{\mathrm {e} }}{\hbar }}={\frac {g_{\mathrm {e} }\mu _{B}}{\hbar }}}

1.76085963023(53)e11 {\displaystyle \textstyle {\frac {1}{\mathrm {s\,T} }}}

No

???

Specific charge of the electron

{\displaystyle {\frac {e}{m_{\mathrm {e} }}}}

-1.75882001076(53)e11 {\displaystyle \textstyle \mathrm {\frac {C}{kg}} }

No

Nk. a. M.

Neutron

Neutron mass

{\displaystyle m_{\mathrm {n} }}

1.67492749804(95)e-27 {\displaystyle \textstyle \mathrm {kg} }= 1.00866491595(49) {\displaystyle \textstyle \mathrm {u} }

No

Nk. M.

Gyromagnetic ratio of the neutron

{\displaystyle \gamma _{\mathrm {n} }}

1.83247171(43)e8 {\displaystyle \textstyle {\frac {1}{\mathrm {s\,T} }}}

No

???

Magnetic moment of the neutron

{\displaystyle \mu _{\mathrm {n} }}

-9.6623651(23)e-27 {\displaystyle \textstyle \mathrm {\frac {J}{T}} }

No

???

Proton

Proton mass

{\displaystyle m_{\mathrm {p} }}

1.67262192369(51)e-27 {\displaystyle \textstyle \mathrm {kg} }= 1.007276466621(53) {\displaystyle \textstyle \mathrm {u} }

No

Nk. M.

Gyromagnetic ratio of the proton

{\displaystyle \gamma _{\mathrm {p} }}

2.6752218744(11)e8 {\displaystyle \textstyle {\frac {1}{\mathrm {s\,T} }}}

No

???

Magnetic moment of the proton

{\displaystyle \mu _{\mathrm {p} }}

1.41060679736(60)e-26 {\displaystyle \textstyle \mathrm {\frac {J}{T}} }

No

???

Ratio of proton mass to electron mass

{\displaystyle {\frac {m_{\mathrm {p} }}{m_{\mathrm {e} }}}}

1836.15267343(11)

No

???

Abbreviation

Meaning

Nk. F.

Natural constant, determination of the measure

Nk. a. F.

Derived from natural constants only, determination of the measured value

fK. F.

freely defined constant with definition of the dimension number

fK. a. F.

user-defined derived constant with definition of the measured value

fK. a. M.

freely defined, derived constant, measured value

Nk. M.

Natural constant, measured value

Nk. a. M.

Derived from natural constants only, measured value
  1. ↑ a b c d e value is used to define SI units.
  2. ↑ a b c d Until the revision of the SI units in 2019, μ0 had the exact value 4π-10-7 H/m. Thus, ε0, kC, and Zw0 were also exactly fixed.
  3. ↑ from me and mPlanck
  4. ↑ Since 2019, the unit "kelvin" is no longer independent, but defined by the thermodynamic energy; the Boltzmann constant has since become an arbitrary conversion factor to the unit "joule".
  5. ↑ a b c d e f g h i j k l m n Derived value
  6. ↑ Avogadro's constant as well as pressure and temperature at standard conditions are not natural constants but arbitrarily determined.
  7. ↑ a b At standard conditions

Constancy of the constants of nature

Whether the constants of nature are also really constant over astronomical time periods is the subject of current research. For example, measurements of the spectral lines of quasars with the Keck telescope in Hawaii seemed to indicate a slight decrease in the fine structure constant by about one hundredth of a part per thousand over the course of ten billion years. This result was controversial from the beginning; on the one hand, researchers pointed out the uncertain error estimate of the data analysis, and on the other hand, there are data from the Oklo mine in West Africa, where about 2 billion years ago uranium had accumulated to such an extent and had such a high content of the isotope U-235 that a nuclear fission chain reaction took place. According to these data, the fine structure constant had the same numerical value then as it does today. Recent measurements of the spectral lines of quasars with the Very Large Telescope of the European Southern Observatory in Chile contradict the earlier results at the Keck telescope and point to the constancy of the fine structure constant.

In the meantime, precision measurements are possible that can verify any steady fluctuations of the order of magnitude suggested by the observations with the Keck telescope, even in the laboratory over short periods of time. Investigations by Theodor Hänsch and his research group at the Max Planck Institute of Quantum Optics prove the constancy of the fine structure constant with an accuracy of 15 decimal places over a period of four years.


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