Physical constant

A physical constant or natural constant (sometimes also an elementary constant ) is a physical quantity whose value cannot be influenced and which does not change spatially or temporally.

The constants that refer to general properties of space, time and physical processes that apply equally to every type of particle and interaction are called fundamental constants of nature . These are the speed of light , Planck's quantum of action and the gravitational constant (see also natural units ).

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

In some cases, parameters or coefficients that are only constant in a certain arrangement or constellation are referred to as "constants", such as the Kepler constant , the decay constant or the spring constant, etc. Strictly speaking, however, they are not constants, but parameters of the investigated Arrangement.

Some natural sciences combine important constants into groups of fundamental constants , e.g. B. in astronomy and geodesy these are the exact reference values ​​of the earth and solar mass , the earth's 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 humans in their earthly environment, but as a rule they do not have a fundamental meaning beyond that and do not prove to be really constant with increasing measurement accuracy. However, they were used for the initial definition of units of measurement (also e.g. for seconds , meters , kilograms ). Modern efforts have been aimed at defining the units of measurement as far as possible through direct reference to (fundamental or elementary) natural constants. The natural constants selected for this receive a firmly defined, unchangeable numerical value. From the 26th General Conference on Weights and Measures , all units of the International System of Units with effect from May 20, 2019 were replaced by four fundamental natural constants ( c , h , e , k _B ), another natural constant (ν _Cs ) and two arbitrarily determined constants ( N _A , K _cd ).

Table of some constants

The numbers in brackets after a numerical value indicate the uncertainty in the last digits of the value. (Example: The so-called abbreviated form 6.674 30 (15) is synonymous with 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 .

Name of the constant	Symbol (s)	Value ( SI )	annotation
Electromagnetism
Speed ​​of light in a vacuum	${\displaystyle c}$	299 792 458 ${\displaystyle \textstyle \mathrm {\frac {m}{s}} }$	Nk. F.
Elemental charge	${\displaystyle e}$	1.602 176 634e-19th ${\displaystyle \textstyle \mathrm {C} }$	Nk. F.
Magnetic field constant	${\displaystyle \mu _{0}}$	1.256 637 062 12 (19)e-6th ${\displaystyle \textstyle \mathrm {\frac {H}{m}} }$	fK. a. M.
Electric field constant	${\displaystyle \varepsilon _{0}={\frac {1}{\mu _{0}\,c^{2}}}}$	8th.854 187 812 8 (13)e-12 ${\displaystyle \textstyle \mathrm {\frac {A\,s}{V\,m}} }$	fK. a. M.
Coulomb's constant	${\displaystyle k_{C}={\frac {1}{4\pi \varepsilon _{0}}}}$	8th.987 551 792 2 (14)e9 ${\displaystyle \textstyle \mathrm {\frac {m}{F}} }$	fK. a. M.
Characteristic impedance of the vacuum	${\displaystyle Z_{w0}=\mu _{0}\,c}$	3.767 303 136 67 (57)e2 ${\displaystyle \textstyle \mathrm {\Omega } }$	fK. a. M.
Gravitation and cosmology
Gravitational constant	${\displaystyle G}$	6th.67430 (15)e-11 ${\displaystyle \textstyle \mathrm {\frac {m^{3}}{kg\,s^{2}}} }$	Nk. M.
Planck mass	${\displaystyle m_{\text{Planck}}={\sqrt {\frac {\hbar \,c}{G}}}}$	2.176 434 (24)e-8th ${\displaystyle \textstyle \mathrm {kg} }$
Planck length	${\displaystyle l_{\text{Planck}}={\frac {\hbar }{m_{\text{Planck}}\,c}}}$	1.616 255 (18)e-35 ${\displaystyle \textstyle \mathrm {m} }$
Planck time	${\displaystyle t_{\text{Planck}}={\frac {l_{\text{Planck}}}{c}}}$	5.391 247 (60)e-44 ${\displaystyle \textstyle \mathrm {s} }$
Gravitational coupling constant	${\displaystyle \alpha _{G}={\frac {G\,m_{\text{e}}^{2}}{\hbar \,c}}={\frac {m_{e}^{2}}{m_{\text{Planck}}^{2}}}}$	1.751 810 (39)e-45	Nk. a. M.
thermodynamics
Boltzmann's constant	${\displaystyle k_{B}}$	1.380 649e-23 ${\displaystyle \textstyle {\frac {\mathrm {J} }{\mathrm {K} }}}$ = 8.617 333 262 ...e-5 ${\displaystyle \textstyle \mathrm {\frac {eV}{K}} }$	fK. F.
Stefan-Boltzmann constant	${\displaystyle \sigma ={\frac {2\pi ^{5}k_{B}^{4}}{15\,h^{3}c^{2}}}}$	5.670 374 419 ...e-8th ${\displaystyle \textstyle \mathrm {\frac {W}{m^{2}\,K^{4}}} }$	fK. a. F.
Vienna constant	${\displaystyle b={\frac {hc}{4{,}965\,114\cdot k_{B}}}}$	2.897 771 955…e-3 ${\displaystyle \textstyle \mathrm {m\cdot K} }$	fK. a. F.
Avogadro's constant	${\displaystyle N_{A}}$	6th.022 140 76e23 ${\displaystyle \textstyle {\frac {1}{\mathrm {mol} }}}$	fK. F.
Faraday constant	${\displaystyle F=e\,N_{A}}$	96 485.332 123 310 018 4 ... ${\displaystyle \textstyle \mathrm {\frac {C}{mol}} }$	fK. a. F.
Gas constant	${\displaystyle R=N_{A}\,k_{B}}$	8th.314 462 618 153 24 ${\displaystyle \textstyle \mathrm {\frac {J}{K\cdot mol}} }$	fK. a. F.
Loschmidt constant at T 0 = 273.15 K and p 0 = 101.325 kPa	${\displaystyle N_{L}}$ or ${\displaystyle n_{0}=N_{A}\cdot {\frac {p_{0}}{RT_{0}}}}$	2.686 780 111 ...e25th ${\displaystyle \textstyle \mathrm {\frac {1}{m^{3}}} }$	fK. a. F.
Molar volume of an ideal gas	${\displaystyle V_{m_{0}}={\frac {R\,T_{0}}{p_{0}}}={\frac {N_{A}}{n_{0}}}}$	0.022 413 969 54… ${\displaystyle \textstyle \mathrm {\frac {m^{3}}{mol}} }$	fK. a. F.
Atomic physics
Rydberg's constant	${\displaystyle R_{\infty }={\frac {e^{4}\,m_{e}}{8\varepsilon _{0}^{2}h^{3}c}}={\frac {\alpha ^{2}}{2}}\,{\frac {m_{\mathrm {e} }c}{h}}}$	1.097 373 156 816 0 (21)e7th ${\displaystyle \textstyle {\frac {1}{\mathrm {m} }}}$	F. a. M.
Rydberg energy	${\displaystyle R_{\infty }hc={\frac {E_{h}}{2}}}$	13.605 693 122 994 (26) ${\displaystyle \textstyle \mathrm {eV} }$ =  2.179 872 361 103 5 (42)e-18th ${\displaystyle \textstyle \mathrm {J} }$
Rydberg frequency	${\displaystyle R_{\infty }\,c}$	3.289 841 960 250 8 (64)e15th ${\displaystyle \textstyle \mathrm {Hz} }$
Hartree energy	${\displaystyle E_{h}={\frac {e^{4}\,m_{e}}{4\,\varepsilon _{0}^{2}\,h^{2}}}}$	4th.359 744 722 207 1 (85)e-18th ${\displaystyle \textstyle \mathrm {J} }$
Quantum and Particle Physics
Planck's quantum of action	${\displaystyle h}$	6th.626 070 15e-34 ${\displaystyle \textstyle \mathrm {J\,s} }$ = 4.135 667 696 ...e-15th ${\displaystyle \textstyle \mathrm {eV\,s} }$	Nk. F.
Reduced Planck's quantum of action	${\displaystyle \hbar ={\frac {h}{2\pi }}}$	1.054 571 817 ...e-34 ${\displaystyle \textstyle \mathrm {J\,s} }$	Nk. a. F.
Spectral radiation constant	${\displaystyle c_{1L}={\frac {2hc^{2}}{sr}}}$	1.191 042 972…e-16 ${\displaystyle \textstyle \mathrm {\frac {W\,m^{2}}{sr}} }$	Nk. a. F.
First radiation constant	${\displaystyle c_{1}=2\pi \,hc^{2}}$	3.741 771 852…e-16 ${\displaystyle \textstyle \mathrm {W\,m^{2}} }$	Nk. a. F.
Second radiation constant	${\displaystyle c_{2}={\frac {hc}{k_{B}}}}$	1.438 776 877 ...e-2 ${\displaystyle \textstyle \mathrm {m\cdot K} }$	fK. a. F.
Fine structure constant	${\displaystyle \alpha ={\frac {\mu _{0}\,e^{2}c}{2h}}}$	7th.297 352 569 3 (11)e-3 = (137.035 999 084 (21) ) −1	F. a. M.
Nuclear magneton	${\displaystyle \mu _{N}={\frac {e\,\hbar }{2\,m_{p}}}}$	5.050 783 746 1 (15)e-27 ${\displaystyle \textstyle \mathrm {\frac {J}{T}} }$	Nk. a. M.
Magnetic flux quantum	${\displaystyle \Phi _{0}={\frac {h}{2e}}}$	2.067 833 848…e-15th ${\displaystyle \textstyle \mathrm {Wb} }$	Nk. a. F.
Josephson's constant	${\displaystyle K_{J}={\frac {1}{\Phi _{0}}}={\frac {2e}{h}}}$	4th.835 978 484 ...e14th ${\displaystyle \textstyle \mathrm {\frac {Hz}{V}} }$	Nk. a. F.
Von Klitzing's constant	${\displaystyle R_{K}={\frac {h}{e^{2}}}}$	25 812.80745 ... ${\displaystyle \textstyle \mathrm {\Omega } }$	Nk. a. F.
Conductance quantum	${\displaystyle G_{0}={\frac {2e^{2}}{h}}}$	7th.748 091 729…e-5 ${\displaystyle \textstyle \mathrm {\frac {s\,C^{2}}{m^{2}\,kg}} }$	Nk. a. F.
Fermi constant	${\displaystyle G_{\rm {F}}^{0}={\frac {G_{\rm {F}}}{(\hbar c)^{3}}}={\frac {\sqrt {2}}{8}}{\frac {g^{2}}{m_{\text{W}}^{2}}}}$	4th.543 795 7 (23)e14 ${\displaystyle \textstyle \mathrm {\frac {1}{J^{2}}} }$ = 1.166 378 7 (6)e-5 ${\displaystyle \textstyle \mathrm {\frac {1}{GeV^{2}}} }$	???
electron
Electron mass	${\displaystyle m_{e}}$	9.109 383 701 5 (28)e-31 ${\displaystyle \textstyle \mathrm {kg} }$ = 5.485 799 090 65 (16)e-4th ${\displaystyle \textstyle \mathrm {u} }$	Nk. M.
Compton wavelength of the electron	${\displaystyle \lambda _{C}={\frac {h}{m_{e}c}}}$	2.426 310 238 67 (73)e-12 ${\displaystyle \textstyle \mathrm {m} }$	Nk. a. M.
Bohr radius	${\displaystyle a_{0}={\frac {4\pi \varepsilon _{0}\hbar ^{2}}{e^{2}\,m_{e}}}={\frac {1}{\alpha }}{\frac {\lambda _{C}}{2\pi }}}$	5.291 772 109 03 (80)e-11 ${\displaystyle \textstyle \mathrm {m} }$	fK. a. M.
Classic electron radius	${\displaystyle r_{e}={\frac {1}{4\pi \varepsilon _{0}}}\,{\frac {e^{2}}{m_{e}c^{2}}}=\alpha ^{2}\,a_{0}}$	2.817 940 326 2 (13)e-15th ${\displaystyle \textstyle \mathrm {m} }$	Nk. a. M.
Bohr's magneton	${\displaystyle \mu _{B}={\frac {e\,\hbar }{2\,m_{e}}}}$	9.274 010 078 3 (28)e-24 ${\displaystyle \textstyle \mathrm {\frac {J}{T}} }$	fK. a. M.
Magnetic moment of the electron	${\displaystyle \mu _{e}}$	-9.284 764 704 3 (28)e-24 ${\displaystyle \textstyle \mathrm {\frac {J}{T}} }$	???
Landé factor of the electron	${\displaystyle g_{e}=-2{\frac {\mu _{e}}{\mu _{B}}}}$	-2.002 319 304 362 56 (35)	???
Gyromagnetic ratio of the electron	${\displaystyle \gamma _{e}=-2{\frac {\mu _{e}}{\hbar }}={\frac {g_{e}\mu _{B}}{\hbar }}}$	1.760 859 630 23 (53)e11 ${\displaystyle \textstyle {\frac {1}{\mathrm {s\,T} }}}$	???
Specific charge of the electron	${\displaystyle {\frac {e}{m_{e}}}}$	-1.758 820 010 76 (53)e11 ${\displaystyle \textstyle \mathrm {\frac {C}{kg}} }$	Nk. a. M.
neutron
Neutron mass	${\displaystyle m_{n}}$	1.674 927 498 04 (95)e-27 ${\displaystyle \textstyle \mathrm {kg} }$ = 1.008 664 915 95 (49) ${\displaystyle \textstyle \mathrm {u} }$	Nk. M.
Gyromagnetic ratio of the neutron	${\displaystyle \gamma _{n}}$	1.832 471 71 (43)e8th ${\displaystyle \textstyle {\frac {1}{\mathrm {s\,T} }}}$	???
Magnetic moment of the neutron	${\displaystyle \mu _{n}}$	-9.662 365 1 (23)e-27 ${\displaystyle \textstyle \mathrm {\frac {J}{T}} }$	???
proton
Proton mass	${\displaystyle m_{p}}$	1.672 621 923 69 (51)e-27 ${\displaystyle \textstyle \mathrm {kg} }$ = 1.007 276 466 621 (53) ${\displaystyle \textstyle \mathrm {u} }$	Nk. M.
Gyromagnetic ratio of the proton	${\displaystyle \gamma _{p}}$	2.675 221 874 4 (11)e8th ${\displaystyle \textstyle {\frac {1}{\mathrm {s\,T} }}}$	???
Magnetic moment of the proton	${\displaystyle \mu _{p}}$	1.410 606 797 36 (60)e-26th ${\displaystyle \textstyle \mathrm {\frac {J}{T}} }$	???
Ratio of proton to electron mass	${\displaystyle {\frac {m_{p}}{m_{e}}}}$	1 836.152 673 43 (11)	???
abbreviation meaning Nk. F. Natural constant, definition of the measure Nk. a. F. Derived only from natural constants, definition of the measure fK. F. Freely defined constant with definition of the dimension fK. a. F. Freely defined, derived constant with definition of the dimension fK. a. M. freely defined, derived constant, measured value Nk. M. Natural constant, measured value Nk. a. M. Derived only from natural constants, measured value
↑ a b c d e Value is used to define SI units. ↑ a b c d Until the new definition of the SI units in 2019, μ 0 had the exact value 4π · 10 −7 H / m. As a result, ε 0 , k C and Z w0 were also precisely defined. ↑ off and ${\displaystyle m_{e}}$ ${\displaystyle m_{\text{Planck}}}$ ↑ a b c d e f g h i j k l m n Derived value ↑ The Avogadro constant as well as pressure and temperature under normal conditions are not natural constants, but are determined arbitrarily. ↑ a b Under standard conditions

Constancy of the natural constants

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

Precision measurements are now possible, which can check any constant fluctuations in the order of magnitude suggested by observations with the Keck telescope, even in the laboratory in short periods of time. Investigations by Theodor Hänsch and his working group at the Max Planck Institute for Quantum Optics prove the constancy of the fine structure constant with an accuracy of 15 decimal places over a period of four years.

Changes to the information due to new measurements

The Committee on Data for Science and Technology , CODATA for short , records in documents how the information on the natural constants changes due to increasingly precise measurements . The National Institute of Standards and Technology ( NIST ) in the USA, which works closely with CODATA, has been publishing PDF documents online for some time with current estimates of the values ​​of the physical constants, including older documents that relate to e.g. B. record all changes between 1986 and 2014.

Fine tuning of the natural constants

→ Main article : Fine tuning of the natural constants

In order to explain the physical state of the observable universe, some authors postulate a fine-tuning of the natural constants . However, it is debatable whether this fine-tuning actually exists or whether it is only a result of insufficient understanding.

literature

• Harald Fritzsch : The absolutely unchangeable: the final riddle of physics . Piper, Munich / Zurich 2005, ISBN 978-3-492-04684-8
• John D. Barrow : The 1 × 1 of the universe: New knowledge about the natural constants . Rowohlt Taschenbuch Verlag, Reinbek bei Hamburg 2006, ISBN 978-3-499-62060-7
• PJ Mohr, BN Taylor: CODATA recommended values ​​of the fundamental physical constants: 1998 . In: Rev. Mod. Phys. , vol. 72 (2000), pp. 351–495 online (PDF; 1.1 MB)
• PJ Mohr, BN Taylor: CODATA recommended values ​​of the fundamental physical constants: 2002 . In: Rev. Mod. Phys. , vol. 77 (2005), pp. 1–107, doi: 10.1103 / RevModPhys. 77.1
• PJ Mohr, BN Taylor, DB Newell: CODATA recommended values ​​of the fundamental physical constants: 2006 . In: Rev. Mod. Phys. , vol. 80 (2008), 633-730, doi: 10.1103 / RevModPhys.80.633
• Brief Overview of the CODATA 2010 Adjustment of the Values ​​of the Constants . physics.nist.gov (PDF; 313 kB)
• PJ Mohr, BN Taylor, DB Newell: CODATA Recommended Values ​​of the Fundamental Physical Constants: 2010 . Preprint physics.nist.gov (PDF 1.1 MB)
• Scale 7 - The Immutable . (PDF; 3.7 MB) In: PTB magazine , September 2006 issue

Web links

Wiktionary: Natural constant  - explanations of meanings, word origins, synonyms, translations

• Leaflet: The legal units in Germany. (PDF; 1.6 MB) Physikalisch-Technische Bundesanstalt , June 2015, accessed on November 22, 2015 . 
• Compilation of many physical constants ( Memento from June 14, 2006 in the Internet Archive ) (PDF file; 100 kB)
• NIST database for physical constants (English)
• The forever unchangeable FAZ article on new research results on the constancy of natural constants
• Do natural constants vary? from the alpha-Centauri television series(approx. 15 minutes). First broadcast on Nov 8, 2006.
• Collection of physical constants of the Particle Data Group, PDG (English)

Individual evidence

1. ↑ Robert Rompe, Hans-Jürgen Treder: What are and what do the elementary constants 7 . In: Annals of Physics. 7th episode . tape 42 , issue 4-6. JA Barth, Leipzig 1985, p. 559-576 ( zs.thulb.uni-jena.de [PDF; 1.3 MB ; accessed on February 10, 2016]).
2. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for the speed of light).  
3. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (value for the elementary charge).  
4. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for the magnetic field constant).  
5. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for the electric field constant).  
6. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for the gravitational constant).  
7. ↑ CODATA Recommended Values. NIST , accessed April 20, 2020 (value for the Planck mass).  
8. ↑ CODATA Recommended Values. NIST , accessed April 20, 2020 (English, value for Planck length).  
9. ↑ CODATA Recommended Values. NIST , accessed on April 20, 2020 (value for Planck time).  
10. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (value for the Boltzmann constant in joules per Kelvin ).  
11. ↑ CODATA Recommended Values. NIST , accessed on April 20, 2020 (English, value for the Boltzmann constant in electron volts per Kelvin ).  
12. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (English, value for the Stefan-Boltzmann constant).  
13. ↑ Peter J. Mohr: CODATA recommended values ​​of the fundamental physical constants: 2006 . In: Reviews of Modern Physics . tape 80 , no. 2 , January 1, 2008, p. 633-730 , doi : 10.1103 / RevModPhys.80.633 ( aps.org [accessed November 7, 2016]).
14. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for Avogadro's constant).  
15. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for Faraday's constant).  
16. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for the universal gas constant).  
17. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (English, value for Loschmidt's constant under standard conditions (273.15 Kelvin , 101.325 kPa )).  
18. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (English, value for the molar volume under standard conditions (273.15 Kelvin , 101.325 kPa )).  
19. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for Rydberg's constant).  
20. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for the Rydberg energy).  
21. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for the Rydberg frequency).  
22. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for Hartree energy).  
23. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (English, value for Planck's quantum of action in the unit Js).  
24. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (English, value for Planck's quantum of action in the unit eVs).  
25. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (value for the reduced Planckian quantum of action in the unit Js).  
26. ↑ CODATA Recommended Values. NIST , accessed on August 4, 2019 (value for the first radiation constant). Since c and h are precisely defined with a finite number of decimal places, the spectral radiation constant can also be represented exactly with a finite number of decimal places: c _1L = 1  .191 042 972 397 188 414 079 489 2e^-16 Wm ² sr ⁻¹
27. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for the first radiation constant).  
28. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for the second radiation constant).  
29. ↑ CODATA Recommended Values. NIST , accessed on April 20, 2020 (English, value for the fine structure constant).  
30. ↑ CODATA Recommended Values. NIST , accessed on April 20, 2020 (English, reciprocal of the fine structure constant).  
31. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for the nuclear magneton).  
32. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (value for the magnetic flux quantum).  
33. ↑ CODATA Recommended Values. NIST , accessed August 4, 2019 (value for Josephson's constant).  
34. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (value for Von Klitzing's constant).  
35. ↑ CODATA Recommended Values. NIST , accessed on August 4, 2019 (English, value for the conductance quantum).  
36. ↑ CODATA Recommended Values. NIST , accessed on March 11, 2020 (English, value for the Fermi constant in GeV ⁻² ).  
37. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (English, value for the electron mass in kilograms).  
38. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (English, value for the electron mass in the atomic mass unit).  
39. ↑ CODATA Recommended Values. NIST , accessed August 4, 2019 (value for the Compton wavelength of the electron).  
40. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for the Bohr radius).  
41. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (value for the classical electron radius).  
42. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for Bohr's magneton).  
43. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for the magnetic moment).  
44. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for the Landé factor of the free electron).  
45. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for the gyromagnetic ratio).  
46. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for the specific charge of the electron).  
47. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (English, value for the neutron mass in kilograms).  
48. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (English, value for the mass of the neutron in the atomic mass unit u).  
49. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (value for the gyromagnetic ratio of the neutron).  
50. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for the neutron's magnetic moment).  
51. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (English, value for those in kilograms).  
52. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (English, value for the mass of the proton in the atomic mass unit u).  
53. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (value for the gyromagnetic ratio of the proton).  
54. ↑ CODATA Recommended Values. NIST , accessed June 3, 2019 (value for the magnetic moment of the proton).  
55. ↑ CODATA Recommended Values. NIST , accessed on June 3, 2019 (English, value for the ratio of proton mass and electron mass).