Physical constant

Author: Leandro Alegsa Creation date: November 13, 2022

In science, a physical constant is a number (physical quantity) that does not change. This is different from a mathematical constant, which is not based on a real measurement.

There are many physical constants in science. Some famous ones are the speed of light, Planck's constant, the gravitational constant, and Avogadro's number.

Some scientists think that physical constants may decrease as the universe gets older. There has been no evidence to support this yet.

Constants that do not have units (like length or speed) are called fundamental physical constants.

Some believe that if the physical constants were different, there would be no life in the universe. They say that the universe seems to be fine-tuned for intelligent life.

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

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

Yes

Nk. F.

Elementary charge

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

Yes

Nk. F.

Magnetic field constant

$\mu _{0}$

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

Yes

fK. a. M.

Electric field constant

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

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

Yes

fK. a. M.

Coulomb constant

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

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

Yes

fK. a. M.

Wave impedance of the vacuum

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

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

Yes

fK. a. M.

Gravity and cosmology

Gravitational constant

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

Yes

Nk. M.

Planck mass

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

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

Yes

Planck length

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

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

Yes

Planck time

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

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

Yes

Gravitational coupling constant

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

1.751810(39)e-45

Nk. a. M.

Thermodynamics

Boltzmann constant

$k_{B}$

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

Yes

fK. F.

Stefan-Boltzmann constant

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

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

Yes

fK. a. F.

Vienna Constant

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

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

Yes

fK. a. F.

Avogadro constant

N_A

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

fK. F.

Faraday constant

$F=e\,N_{A}$

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

fK. a. F.

Gas constant

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

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

fK. a. F.

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

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

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

fK. a. F.

Molar volume of an ideal gas

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

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

fK. a. F.

Atomic Physics

Rydberg constant

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

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

F. a. M.

Rydberg Energy

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

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

Rydberg frequency

$R_{\infty }\,c$

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

Hartree Energy

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

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

Quantum and particle physics

Planck's constant

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

Yes

Nk. F.

Planck's reduced quantum of action

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

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

Yes

Nk. a. F.

Spectral radiation constant

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

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

Nk. a. F.

First radiation constant

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

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

Yes

Nk. a. F.

Second radiation constant

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

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

Yes

fK. a. F.

Fine structure constant

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

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

F. a. M.

Nuclear Magneton

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

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

Nk. a. M.

magnetic flux quantum

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

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

Nk. a. F.

Josephson constant

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

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

Nk. a. F.

Von Klitzing constant

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

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

Nk. a. F.

Conductance quantum

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

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

Nk. a. F.

Fermi constant

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

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

???

Electron

Electron mass

$m_{{\mathrm e}}$

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

Nk. M.

Compton wavelength of the electron

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

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

Nk. a. M.

Drilling radius

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

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

fK. a. M.

Classical electron radius

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

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

Nk. a. M.

Bohr's magneton

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

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

fK. a. M.

Magnetic moment of the electron

$\mu _{\mathrm {e} }$

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

???

Landé factor of the electron

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

-2.00231930436256(35)

???

Gyromagnetic ratio of the electron

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

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

???

Specific charge of the electron

${\frac {e}{m_{\mathrm {e} }}}$

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

Nk. a. M.

Neutron

Neutron mass

$m_{\mathrm {n} }$

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

Nk. M.

Gyromagnetic ratio of the neutron

$\gamma _{\mathrm {n} }$

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

???

Magnetic moment of the neutron

$\mu _{\mathrm {n} }$

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

???

Proton

Proton mass

$m_{\mathrm {p} }$

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

Nk. M.

Gyromagnetic ratio of the proton

$\gamma _{\mathrm {p} }$

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

???

Magnetic moment of the proton

$\mu _{\mathrm {p} }$

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

???

Ratio of proton mass to electron mass

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

1836.15267343(11)

???

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

↑ ^{a b c d e} value is used to define SI units.
↑ ^{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.
↑ from _me and mPlanck
↑ 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".
↑ ^{a b c d e f g h i j k l m n} Derived value
↑ Avogadro's constant as well as pressure and temperature at standard conditions are not natural constants but arbitrarily determined.
↑ ^{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.