List of physical constants

The constants listed here are known values of physical constants expressed in SI units; that is, physical quantities that are generally believed to be universal in nature and thus are independent of the unit system in which they are measured. Many of these are redundant, in the sense that they obey a known relationship with other physical constants and can be determined from them.

Table of physical constants

Symbol	Quantity	Value [lower-alpha 1] [lower-alpha 2]	Relative standard uncertainty	Ref [1]
${\displaystyle c}$	speed of light in vacuum	299792458 m⋅s−1	0	[2]
${\displaystyle h}$	Planck constant	6.62607015×10−34 J⋅Hz−1	0	[3]
${\displaystyle \hbar =h/2\pi }$	reduced Planck constant	1.054571817...×10−34 J⋅s	0	[4]
${\displaystyle \mu _{0}=4\pi \alpha \hbar /e^{2}c}$	vacuum magnetic permeability	1.25663706212(19)×10−6 N⋅A−2	1.5×10−10	[5]
${\displaystyle Z_{0}=\mu _{0}c}$	characteristic impedance of vacuum	376.730313668(57) Ω	1.5×10−10	[6]
${\displaystyle \varepsilon _{0}=1/\mu _{0}c^{2}}$	vacuum electric permittivity	8.8541878128(13)×10−12 F⋅m−1	1.5×10−10	[7]
${\displaystyle k,k_{\text{B}}}$	Boltzmann constant	1.380649×10−23 J⋅K−1	0	[8]
${\displaystyle G}$	Newtonian constant of gravitation	6.67430(15)×10−11 m3⋅kg−1⋅s−2	2.2×10−5	[9]
${\displaystyle k_{\text{e}}=1/4\pi \varepsilon _{0}}$	Coulomb constant	8.9875517923(14)×109 N⋅m2⋅C−2	1.5×10−10	[10]
${\displaystyle \Lambda }$	cosmological constant	1.089(29)×10−52 m−2	0.027	[11]
${\displaystyle \sigma =\pi ^{2}k_{\text{B}}^{4}/60\hbar ^{3}c^{2}}$	Stefan–Boltzmann constant	5.670374419...×10−8 W⋅m−2⋅K−4	0	[12]
${\displaystyle c_{1}=2\pi hc^{2}}$	first radiation constant	3.741771852...×10−16 W⋅m2	0	[13]
${\displaystyle c_{\text{1L}}=2hc^{2}/\mathrm {sr} }$	first radiation constant for spectral radiance	1.191042972...×10−16 W⋅m2⋅sr−1	0	[14]
${\displaystyle c_{2}=hc/k_{\text{B}}}$	second radiation constant	1.438776877...×10−2 m⋅K	0	[15]
${\displaystyle b}$ [lower-alpha 3]	Wien wavelength displacement law constant	2.897771955...×10−3 m⋅K	0	[16]
${\displaystyle b'}$ [lower-alpha 4]	Wien frequency displacement law constant	5.878925757...×1010 Hz⋅K−1	0	[17]
${\displaystyle b_{\text{entropy}}}$	Wien entropy displacement law constant	3.002916077...×10−3 m⋅K	0	[18]
${\displaystyle e}$	elementary charge	1.602176634×10−19 C	0	[19]
${\displaystyle G_{0}=2e^{2}/h}$	conductance quantum	7.748091729...×10−5 S	0	[20]
${\displaystyle G_{0}^{-1}=h/2e^{2}}$	inverse conductance quantum	12906.40372... Ω	0	[21]
${\displaystyle R_{\text{K}}=h/e^{2}}$	von Klitzing constant	25812.80745... Ω	0	[22]
${\displaystyle K_{\text{J}}=2e/h}$	Josephson constant	483597.8484...×109 Hz⋅V−1	0	[23]
${\displaystyle \Phi _{0}=h/2e}$	magnetic flux quantum	2.067833848...×10−15 Wb	0	[24]
${\displaystyle \alpha =e^{2}/4\pi \varepsilon _{0}\hbar c}$	fine-structure constant	7.2973525693(11)×10−3	1.5×10−10	[25]
${\displaystyle \alpha ^{-1}}$	inverse fine-structure constant	137.035999084(21)	1.5×10−10	[26]
${\displaystyle m_{\text{e}}}$	electron mass	9.1093837015(28)×10−31 kg	3.0×10−10	[27]
${\displaystyle m_{\mu }}$	muon mass	1.883531627(42)×10−28 kg	2.2×10−8	[28]
${\displaystyle m_{\tau }}$	tau mass	3.16754(21)×10−27 kg	6.8×10−5	[29]
${\displaystyle m_{\text{p}}}$	proton mass	1.67262192369(51)×10−27 kg	3.1×10−10	[30]
${\displaystyle m_{\text{n}}}$	neutron mass	1.67492749804(95)×10−27 kg	5.7×10−10	[31]
${\displaystyle m_{\text{t}}}$	top quark mass	3.0784(53)×10−25 kg	1.7×10−3	[32]
${\displaystyle m_{\text{p}}/m_{\text{e}}}$	proton-to-electron mass ratio	1836.15267343(11)	6.0×10−11	[33]
${\displaystyle m_{\text{W}}/m_{\text{Z}}}$	W-to-Z mass ratio	0.88153(17)	1.9×10−4	[34]
${\displaystyle \sin ^{2}\theta _{\text{W}}=1-(m_{\text{W}}/m_{\text{Z}})^{2}}$	weak mixing angle	0.22290(30)	1.3×10−3	[35]
${\displaystyle g_{\text{e}}}$	electron g-factor	−2.00231930436256(35)	1.7×10−13	[36]
${\displaystyle g_{\mu }}$	muon g-factor	−2.0023318418(13)	6.3×10−10	[37]
${\displaystyle g_{\text{p}}}$	proton g-factor	5.5856946893(16)	2.9×10−10	[38]
${\displaystyle h/2m_{\text{e}}}$	quantum of circulation	3.6369475516(11)×10−4 m2⋅s−1	3.0×10−10	[39]
${\displaystyle \mu _{\text{B}}=e\hbar /2m_{\text{e}}}$	Bohr magneton	9.2740100783(28)×10−24 J⋅T−1	3.0×10−10	[40]
${\displaystyle \mu _{\text{N}}=e\hbar /2m_{\text{p}}}$	nuclear magneton	5.0507837461(15)×10−27 J⋅T−1	3.1×10−10	[41]
${\displaystyle r_{\text{e}}=e^{2}k_{\text{e}}/m_{\text{e}}c^{2}}$	classical electron radius	2.8179403262(13)×10−15 m	4.5×10−10	[42]
${\displaystyle \sigma _{\text{e}}=(8\pi /3)r_{\text{e}}^{2}}$	Thomson cross section	6.6524587321(60)×10−29 m2	9.1×10−10	[43]
${\displaystyle a_{0}=\hbar ^{2}/k_{\text{e}}m_{\text{e}}e^{2}=r_{\text{e}}/\alpha ^{2}}$	Bohr radius	5.29177210903(80)×10−11 m	1.5×10−10	[44]
${\displaystyle E_{\text{h}}=\alpha ^{2}c^{2}m_{\text{e}}}$	Hartree energy	4.3597447222071(85)×10−18 J	1.9×10−12	[45]
${\displaystyle \mathrm {Ry} =hcR_{\infty }=E_{\text{h}}/2}$	Rydberg unit of energy	2.1798723611035(42)×10−18 J	1.9×10−12	[46]
${\displaystyle R_{\infty }=\alpha ^{2}m_{\text{e}}c/2h}$	Rydberg constant	10973731.568160(21) m−1	1.9×10−12	[47]
${\displaystyle G_{\text{F}}/(\hbar c)^{3}}$	Fermi coupling constant	1.1663787(6)×10−5 GeV−2	5.1×10−7	[48]
${\displaystyle N_{\text{A}},L}$	Avogadro constant	6.02214076×1023 mol−1	0	[49]
${\displaystyle R=N_{\text{A}}k_{\text{B}}}$	molar gas constant	8.31446261815324 J⋅mol−1⋅K−1	0	[50]
${\displaystyle F=N_{\text{A}}e}$	Faraday constant	96485.3321233100184 C⋅mol−1	0	[51]
${\displaystyle N_{\text{A}}h}$	molar Planck constant	3.9903127128934314×10−10 J⋅s⋅mol−1	0	[52]
${\displaystyle m({}^{12}{\text{C}})}$	atomic mass of carbon-12	1.99264687992(60)×10−26 kg	3.0×10−10	[53]
${\displaystyle M({}^{12}{\text{C}})=N_{\text{A}}m({}^{12}{\text{C}})}$	molar mass of carbon-12	11.9999999958(36)×10−3 kg⋅mol−1	3.0×10−10	[54]
${\displaystyle m_{\text{u}}=m({}^{12}{\text{C}})/12}$	atomic mass constant	1.66053906660(50)×10−27 kg	3.0×10−10	[53]
${\displaystyle M_{\text{u}}=M({}^{12}{\text{C}})/12}$	molar mass constant	0.99999999965(30)×10−3 kg⋅mol−1	3.0×10−10	[55]
${\displaystyle V_{\text{m}}({\text{Si}})}$	molar volume of silicon	1.205883199(60)×10−5 m3⋅mol−1	4.9×10−8	[56]
${\displaystyle \Delta \nu _{\text{Cs}}}$	hyperfine transition frequency of 133Cs	9192631770 Hz	0	[57]

Uncertainties

While the values of the physical constants are independent of the system of units in use, each uncertainty as stated reflects our lack of knowledge of the corresponding value as expressed in SI units, and is strongly dependent on how those units are defined. For example, the atomic mass constant ${\displaystyle m_{\text{u}}}$ is exactly known when expressed using the dalton (its value is exactly 1 Da), but the kilogram is not exactly known when using these units, the opposite of when expressing the same quantities using the kilogram.

Technical constants

Some of these constants are of a technical nature and do not give any true physical property, but they are included for convenience. Such a constant gives the correspondence ratio of a technical dimension with its corresponding underlying physical dimension. These include the Boltzmann constant ${\displaystyle k_{\text{B}}}$, which gives the correspondence of the dimension temperature to the dimension of energy per degree of freedom, and the Avogadro constant ${\displaystyle N_{\text{A}}}$, which gives the correspondence of the dimension of amount of substance with the dimension of count of entities (the latter formally regarded in the SI as being dimensionless). By implication, any product of powers of such constants is also such a constant, such as the molar gas constant ${\displaystyle R}$.

See also

• List of mathematical constants
• Physical constant
• List of particles

Notes

1. The values are given in the so-called concise form; the number in parentheses is the standard uncertainty and indicates the amount by which the least significant digits of the value are uncertain.
2. In some instances an exact value has been displayed, calculated from the defining expression, rather than the incomplete decimal expansion as given by the source.
3. ${\displaystyle b={\frac {hc}{k\left(5+W_{0}\left(-5e^{-5}\right)\right)}}}$, where ${\displaystyle W_{0}}$ is the principal branch of the Lambert W function.
4. ${\displaystyle b'=\left(3+W_{0}\left(-3e^{-3}\right)\right){\frac {k}{h}}}$, where ${\displaystyle W_{0}}$ is the principal branch of the Lambert W function.

References

1. Eite Tiesinga; Peter J. Mohr; David B. Newell; Barry N. Taylor (September 2021), "CODATA Recommended Values of the Fundamental Physical Constants: 2018" (PDF), J. Phys. Chem. Ref. Data, 50 (3), doi:10.1063/5.0064853
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5. "2018 CODATA Value: vacuum magnetic permeability". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
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15. "2018 CODATA Value: second radiation constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
16. "2018 CODATA Value: Wien wavelength displacement law constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
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18. Delgado-Bonal A (May 2017). "Entropy of radiation: the unseen side of light". Scientific Reports. 7 (1): 1642. Bibcode:2017NatSR...7.1642D. doi:10.1038/s41598-017-01622-6. PMC   5432030 . PMID   28490790.
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20. "2018 CODATA Value: conductance quantum". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
21. "2018 CODATA Value: inverse of conductance quantum". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
22. "2018 CODATA Value: von Klitzing constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
23. "2018 CODATA Value: Josephson constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
24. "2018 CODATA Value: magnetic flux quantum". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
25. "2018 CODATA Value: fine-structure constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
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27. "2018 CODATA Value: electron mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
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29. "2018 CODATA Value: tau mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
30. "2018 CODATA Value: proton mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
31. "2018 CODATA Value: neutron mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-23.
32. "t-quark mass". pdglive.lbl.gov. Particle Data Group. Retrieved 9 July 2022. t-quark mass (direct measurements): 172.69(30) GeV/c²
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34. "2018 CODATA Value: W to Z mass ratio". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-12-21.
35. "2018 CODATA Value: weak mixing angle". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
36. "2018 CODATA Value: electron g factor". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2020-03-13.
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38. "2018 CODATA Value: proton g factor". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-03-08.
39. "2018 CODATA Value: quantum of circulation". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
40. "2018 CODATA Value: Bohr magneton". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
41. "2018 CODATA Value: nuclear magneton". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
42. "2018 CODATA Value: classical electron radius". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
43. "2018 CODATA Value: Thomson cross section". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
44. "2018 CODATA Value: Bohr radius". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
45. "2018 CODATA Value: Hartree energy". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
46. "2018 CODATA Value: Rydberg constant times hc in J". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2023-12-31.
47. "2018 CODATA Value: Rydberg constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
48. "2018 CODATA Value: Fermi coupling constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
49. "2018 CODATA Value: Avogadro constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
50. "2018 CODATA Value: molar gas constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
51. "2018 CODATA Value: Faraday constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
52. "2018 CODATA Value: molar Planck constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
53. "2018 CODATA Value: atomic mass constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
54. "2018 CODATA Value: molar mass of carbon-12". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
55. "2018 CODATA Value: molar mass constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
56. "2018 CODATA Value: molar volume of silicon". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-06-23.
57. "2018 CODATA Value: hyperfine transition frequency of Cs-133". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-18.