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A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant, which has a fixed numerical value, but does not directly involve any physical measurement.

There are many physical constants in science, some of the most widely recognized being the speed of light in vacuum c, the gravitational constant G, the Planck constant h, the electric constant ε0, and the elementary charge e. Physical constants can take many dimensional forms: the speed of light signifies a maximum speed for any object and its dimension is length divided by time; while the fine-structure constant α, which characterizes the strength of the electromagnetic interaction, is dimensionless.

The term fundamental physical constant is sometimes used to refer to universal but dimensioned physical constants such as those mentioned above.[1] Increasingly, however, physicists reserve the use of the term fundamental physical constant for dimensionless physical constants, such as the fine-structure constant α.

Physical constant in the sense under discussion in this article should not be confused with other quantities called "constants" that are assumed to be constant in a given context without the implication that they are fundamental, such as the "time constant" characteristic of a given system, or material constants, such as the Madelung constant, electrical resistivity, and heat capacity.

The International Bureau of Weights and Measures decided to redefine several SI base units as from 20 May 2019 by fixing the SI value of several physical constants, including the Planck constant, h, the elementary charge, e, the Boltzmann constant, kB, and the Avogadro constant, NA. The new fixed values are based on the best measurements of the constants based on the earlier definitions, including the kilogram, to ensure minimal impact. As a consequence, the uncertainty in the value of many physical constants when expressed in SI units will reduce substantially.

Contents

Choice of unitsEdit

Whereas the physical quantity indicated by a physical constant does not depend on the unit system used to express the quantity, the numerical values of dimensional physical constants do depend on choice of unit system. The term "physical constant" refers to the physical quantity, and not to the numerical value within any given system of units. For example, the speed of light is defined as having the numerical value of 299,792,458 in SI units, and as having the numerical value of 1 in natural units. While its numerical value can be defined at will by the choice of units, the speed of light itself is a single physical constant.

Any ratio between physical constants of the same dimensions results in a dimensionless physical constant, for example, the proton-to-electron mass ratio. Any relation between physical quantities can be expressed as a relation between dimensionless ratios via a process known as nondimensionalisation.

The term of "fundamental physical constant" is reserved for physical quantities which, according to the current state of knowledge, are regarded as immutable and as non-derivable from more fundamental principles. Notable examples are the speed of light c, and the gravitational constant G.

The fine-structure constant α is the best known dimensionless fundamental physical constant. It is the value of the elementary charge squared expressed in Planck units. This value has become a standard example when discussing the derivability or non-derivability of physical constants. Introduced by Arnold Sommerfeld, its value as determined at the time was consistent with 1/137. This motivated Arthur Eddington (1929) to construct an argument why its value might be 1/137 precisely, which related to the Eddington number, his estimate of the number of protons in the Universe.[2] By the 1940s, it became clear that the value of the fine-structure constant deviates significantly from the precise value of 1/137, refuting Eddington's argument.[3]

With the development of quantum chemistry in the 20th century, however, a vast number of previously inexplicable dimensionless physical constants were successfully computed from theory. In light of that, some theoretical physicists still hope for continued progress in explaining the values of other dimensionless physical constants.

It is known that the Universe would be very different if these constants took values significantly different from those we observe. For example, a few percent change in the value of the fine structure constant would be enough to eliminate stars like our Sun. This has prompted attempts at anthropic explanations of the values of some of the dimensionless fundamental physical constants.

Natural unitsEdit

Using dimensional analysis, it is possible to combine dimensional universal physical constants to define a system of units of measurement that has no reference to any human construct. Depending on the choice and arrangement of constants used, the resulting natural units may have useful physical meaning. For example, Planck units, shown in the table below, use c, G, ħ, ε0, and kB in such a manner to derive units relevant to unified theories such as quantum gravity.

Name Quantity Expression Value (SI units)
Planck length Length (L)   1.616229(38)×10−35 m[4]
Planck mass Mass (M)   2.176470(51)×10−8 kg[5]
Planck time Time (T)   5.39116(13)×10−44 s[6]
Planck charge Electric charge (Q)   1.875 545 956(41) × 10−18 C[7][8][9]
Planck temperature Temperature (Θ)   1.416808(33)×1032 K[10]

Number of fundamental constantsEdit

The number of fundamental physical constants depends on the physical theory accepted as "fundamental". Currently, this is the theory of general relativity for gravitation and the Standard Model for electromagnetic, weak and strong nuclear interactions and the matter fields. Between them, these theories account for a total of 19 independent fundamental constants. There is, however, no single "correct" way of enumerating them, as it is a matter of arbitrary choice which quantities are considered "fundamental" and which as "derived". Uzan (2011) lists 22 "unknown constants" in the fundamental theories, which give rise to 19 "unknown dimensionless parameters", as follows:

The number of 19 independent fundamental physical constants is subject to change under possible extensions of the Standard Model, notably by the introduction of neutrino mass (equivalent to seven additional constants, i.e. 3 Yukawa couplings and 4 lepton mixing parameters).[11]

The discovery of variability in any of these constants would be equivalent to the discovery of "new physics".[12]

The question as to which constants are "fundamental" is neither straightforward nor meaningless, but a question of interpretation of the physical theory regarded as fundamental; as pointed out by Lévy-Leblond 1979, not all physical constants are of the same importance, with some having a deeper role than others. Lévy-Leblond 1979 proposed a classification schemes of three types of fundamental constant:

  • A: characteristic of a particular system
  • B: characteristic of a class of physical phenomena
  • C: universal constants

The same physical constant may move from one category to another as the understanding of its role deepens; this has notably happened to the speed of light, which was a class A constant (characteristic of light) when it was first measured, but became a class B constant (characteristic of electromagnetic phenomena) with the development of classical electromagnetism, and finally a class C constant with the discovery of special relativity.[13]

Tests on time-independenceEdit

By definition, fundamental physical constants are subject to measurement, so that their being constant (independent on both the time and position of the performance of the measurement) is necessarily an experimental result and subject to verification.

Paul Dirac in 1937 speculated that physical constants such as the gravitational constant or the fine-structure constant might be subject to change over time in proportion of the age of the universe. Experiments can in principle only put an upper bound on the relative change per year. For the fine-structure constant, this upper bound is comparatively low, at roughly 10−17 per year (as of 2008).[14]

The gravitational constant is much more difficult to measure with precision, and conflicting measurements in the 2000s have inspired the controversial suggestions of a periodic variation of its value in a 2015 paper.[15] However, while its value is not known to great precision, the possibility of observing type Ia supernovae which happened in the universe's remote past, paired with the assumption that the physics involved in these events is universal, allows for an upper bound of less than 10−10 per year for the gravitational constant over the last nine billion years.[16]

Similarly, an upper bound of the change in the proton-to-electron mass ratio has been placed at 10−7 over a period of 7 billion years (or 10−16 per year) in a 2012 study based on the observation of methanol in a distant galaxy.[17][18]

It is problematic to discuss the proposed rate of change (or lack thereof) of a single dimensional physical constant in isolation. The reason for this is that the choice of a system of units may arbitrarily select as its basis, making the question of which constant is undergoing change an artefact of the choice of units.[19][20][21]

For example, in SI units, the speed of light was given a defined value in 1983. Thus, it was meaningful to experimentally measure the speed of light in SI units prior to 1983, but it is not so now. Similarly, with effect from May 2019, the Planck constant has a defined value, such that all SI base units are now defined in terms of fundamental physical constants. With this change, the kilogram is being retired as the last physical object used in the definition of any SI unit.

Tests on the immutability of physical constants look at dimensionless quantities, i.e. ratios between quantities of like dimensions, in order to escape this problem. Changes in physical constants are not meaningful if they result in an observationally indistinguishable universe. For example, a "change" in the speed of light c would be meaningless if accompanied by a corresponding change in the elementary charge e so that the ratio e2/(4πε0ħc) (the fine-structure constant) remained unchanged.[22]

Fine-tuned UniverseEdit

Some physicists have explored the notion that if the dimensionless physical constants had sufficiently different values, our Universe would be so radically different that intelligent life would probably not have emerged, and that our Universe therefore seems to be fine-tuned for intelligent life. The anthropic principle states a logical truism: the fact of our existence as intelligent beings who can measure physical constants requires those constants to be such that beings like us can exist. There are a variety of interpretations of the constants' values, including that of a divine creator (the apparent fine-tuning is actual and intentional), or that ours is one universe of many in a multiverse (e.g. the Many-worlds interpretation of quantum mechanics), or even that, if information is an innate property of the universe and logically inseparable from consciousness, a universe without the capacity for conscious beings cannot exist.

Table of physical constantsEdit

Universal constantsEdit

Quantity Symbol Value[23] Relative standard uncertainty
characteristic impedance of vacuum   376.730313461... Ω[24] 0
electric constant (vacuum permittivity)   8.854187817...×10−12 F⋅m−1[7] 0
magnetic constant (vacuum permeability)   12.566370614...×10−7 N⋅A−2[25] 0
Newtonian constant of gravitation   6.67408(31)×10−11 m3⋅kg−1⋅s−2[26] 4.7×10−5
Planck constant   6.626070040(81)×10−34 J⋅s[27] 1.2×10−8
reduced Planck constant   1.054571800(13)×10−34 J⋅s[8] 1.2×10−8
speed of light in vacuum   299792458 m/s[9] 0

Electromagnetic constantsEdit

Quantity Symbol Value[23] (SI units) Relative standard uncertainty
Bohr magneton   927.4009994(57)×10−26 J⋅T−1[28] 6.2×10−9
conductance quantum   7.7480917310(18)×10−5 S[29] 2.3×10−10
elementary charge   1.6021766208(98)×10−19 C[30] 6.1×10−9
inverse conductance quantum   12906.4037278(29) Ω[31] 2.3×10−10
Josephson constant   483597.8525(30)×109 Hz⋅V−1[32] 6.1×10−9
magnetic flux quantum   2.067833831(13)×10−15 Wb[33] 6.1×10−9
nuclear magneton   5.050783699(31)×10−27 J⋅T−1[34] 6.2×10−9
von Klitzing constant   25812.8074555(59) Ω[35] 2.3×10−10

Atomic and nuclear constantsEdit

Quantity Symbol Value (SI units) Relative standard uncertainty
Bohr radius   0.52917721067(12)×10−10 m[36] 2.3×10−10
classical electron radius   2.8179403227(19)×10−15 m[37] 6.8×10−10
electron g-factor   −2.00231930436182(52)[38] 2.6×10−13
electron mass   9.10938356(11)×10−31 kg[39] 1.2×10−8
Fermi coupling constant   1.1663787(6)×10−5 GeV−2[40] 5.1×10−7
fine-structure constant   7.2973525664(17)×10−3[41] 2.3×10−10
Hartree energy   4.359744650(54)×10−18 J[42] 1.2×10−8
inverse fine-structure constant   137.035999139(31)[43] 2.3×10−10
proton mass   1.672621898(21)×10−27 kg[44] 1.2×10−8
quantum of circulation   3.6369475486(17)×10−4 m2⋅s−1[45] 4.5×10−10
Rydberg constant   10973731.568508(65) m−1[46] 5.9×10−12
Thomson cross section   0.66524587158(91)×10−28 m2[47] 1.4×10−9
weak mixing angle   0.2223(21) 9.5 × 10−3

Physico-chemical constantsEdit

Quantity Symbol Value[23] (SI units) Relative standard uncertainty
Atomic mass constant   1.660539040(20)×10−27 kg[48] 1.2×10−8
Avogadro constant   6.022140857(74)×1023 mol−1[49] 1.2×10−8
Boltzmann constant   1.38064852(79)×10−23 J⋅K−1[50] 5.7×10−7
Faraday constant   96485.33289(59) C⋅mol−1[51] 6.2×10−9
first radiation constant   3.741771790(46)×10−16 W⋅m2[52] 1.2×10−8
first radiation constant for spectral radiance   1.191042953(15)×10−16 W⋅m2⋅sr−1[53] 1.2×10−8
Loschmidt constant   = 273.15 K,   = 100 kPa   2.651 6467(15) × 1025 m−3 5.7 × 10−7
  = 273.15 K,   = 101.325 kPa 2.686 7811(15) × 1025 m−3 5.7 × 10−7
gas constant   8.3144598(48) J⋅mol−1⋅K−1[54] 5.7×10−7
molar Planck constant   3.9903127110(18)×10−10 J⋅s⋅mol−1[55] 4.5×10−10
molar volume of an ideal gas   = 273.15 K,   = 100 kPa   2.271 0947(13) × 10−2 m3⋅mol−1 5.7 × 10−7
  = 273.15 K,   = 101.325 kPa 2.241 3962(13) × 10−2 m3⋅mol−1 5.7 × 10−7
Sackur–Tetrode constant   = 1 K,   = 100 kPa  
 
−1.151 7084(14) 1.2 × 10−6
  = 1 K,   = 101.325 kPa −1.164 8714(14) 1.2 × 10−6
second radiation constant   1.43877736(83)×10−2 m⋅K[56] 5.7×10−7
Stefan–Boltzmann constant   5.670367(13)×10−8 W⋅m−2⋅K−4[57] 2.3×10−6
Wien wavelength displacement law constant   2.8977729(17)×10−3 m⋅K[58] 5.7×10−7
Wien frequency displacement law constant   5.8789238(34)×1010 Hz⋅K−1[59] 5.7×10−7
Wien-Bonal entropy displacement law constant[60]   3.002 9152(05) × 10−3 m⋅K 5.7 × 10−7

Adopted valuesEdit

Quantity Symbol Value (SI units) Relative standard uncertainty
conventional value of Josephson constant[61]   483597.9×109 Hz⋅V−1[62] 0
conventional value of von Klitzing constant[63]   25812.807 Ω[64] 0
molar mass constant   0.001 kg⋅mol−1[65] 0
molar mass of carbon-12   12×10−3 kg⋅mol−1[66] 0
standard acceleration of gravity   9.80665 m⋅s−2[67] 0
standard atmosphere   101325 Pa[68] 0
Caesium standard (defines the SI second)   9192631770 Hz[69] 0

See alsoEdit

ReferencesEdit

  1. ^ "Archived copy". Archived from the original on 2016-01-13. Retrieved 2016-01-14.CS1 maint: Archived copy as title (link) NIST
  2. ^ A.S Eddington (1956). "The Constants of Nature". In J.R. Newman (ed.). The World of Mathematics. 2. Simon & Schuster. pp. 1074–1093.
  3. ^ H. Kragh (2003). "Magic Number: A Partial History of the Fine-Structure Constant". Archive for History of Exact Sciences. 57 (5): 395. doi:10.1007/s00407-002-0065-7.
  4. ^ "CODATA Value: Planck length". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2017-06-22. 2014 CODATA recommended values
  5. ^ "CODATA Value: Planck mass". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2017-06-22. 2014 CODATA recommended values
  6. ^ "CODATA Value: Planck time". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2017-06-22. 2014 CODATA recommended values
  7. ^ a b "CODATA Value: electric constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  8. ^ a b "CODATA Value: Planck constant over 2 pi". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  9. ^ a b "CODATA Value: speed of light in vacuum". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  10. ^ "CODATA Value: Planck temperature". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2017-06-22. 2014 CODATA recommended values
  11. ^ Uzan, Jean-Philippe (2011). "Varying Constants, Gravitation and Cosmology" (PDF). Living Reviews in Relativity. 14. doi:10.12942/lrr-2011-2. Any constant varying in space and/or time would reflect the existence of an almost massless field that couples to matter. This will induce a violation of the universality of free fall. Thus, it is of utmost importance for our understanding of gravity and of the domain of validity of general relativity to test for their constancy.
  12. ^ Uzan, Jean-Philippe (2011). "Varying Constants, Gravitation and Cosmology" (PDF). Living Reviews in Relativity. 14 (1): 2. doi:10.12942/lrr-2011-2. PMC 5256069. PMID 28179829.
  13. ^ Lévy-Leblond, J.-M. (1979). "The importance of being (a) Constant". In Toraldo di Francia, G. (ed.). Problems in the Foundations of Physics, Proceedings of the International School of Physics 'Enrico Fermi' Course LXXII, Varenna, Italy, July 25 – August 6, 1977. New York: NorthHolland. pp. 237–263.
  14. ^ T. Rosenband; et al. (2008). "Frequency Ratio of Al+ and Hg+ Single-Ion Optical Clocks; Metrology at the 17th Decimal Place". Science. 319 (5871): 1808–12. Bibcode:2008Sci...319.1808R. doi:10.1126/science.1154622. PMID 18323415.
  15. ^ J.D. Anderson; G. Schubert; V. Trimble; M.R. Feldman (April 2015), "Measurements of Newton's gravitational constant and the length of day" (PDF), EPL, 110 (1): 10002, arXiv:1504.06604, Bibcode:2015EL....11010002A, doi:10.1209/0295-5075/110/10002
  16. ^ J. Mould; S. A. Uddin (2014-04-10), "Constraining a Possible Variation of G with Type Ia Supernovae", Publications of the Astronomical Society of Australia, 31: e015, arXiv:1402.1534, Bibcode:2014PASA...31...15M, doi:10.1017/pasa.2014.9, archived from the original on 2014-04-08
  17. ^ Bagdonaite, Julija; Jansen, Paul; Henkel, Christian; Bethlem, Hendrick L.; Menten, Karl M.; Ubachs, Wim (December 13, 2012). "A Stringent Limit on a Drifting Proton-to-Electron Mass Ratio from Alcohol in the Early Universe". Science. 339 (6115): 46–48. Bibcode:2013Sci...339...46B. doi:10.1126/science.1224898. hdl:1871/39591. PMID 23239626.
  18. ^ Moskowitz, Clara (December 13, 2012). "Phew! Universe's Constant Has Stayed Constant". Space.com. Archived from the original on December 14, 2012. Retrieved December 14, 2012.
  19. ^ Michael Duff (2014). "How fundamental are fundamental constants?". arXiv:1412.2040. doi:10.1080/00107514.2014.980093 (inactive 2019-02-17).
  20. ^ Duff, M. J. (13 August 2002). "Comment on time-variation of fundamental constants". arXiv:hep-th/0208093.
  21. ^ Duff, M. J.; Okun, L. B.; Veneziano, G. (2002). "Trialogue on the number of fundamental constants". Journal of High Energy Physics. 3 (3): 023. arXiv:physics/0110060. Bibcode:2002JHEP...03..023D. doi:10.1088/1126-6708/2002/03/023.
  22. ^ Barrow, John D. (2002), The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe, Pantheon Books, ISBN 978-0-375-42221-8 "[An] important lesson we learn from the way that pure numbers like α define the World is what it really means for worlds to be different. The pure number we call the fine structure constant and denote by α is a combination of the electron charge, e, the speed of light, c, and Planck's constant, h. At first we might be tempted to think that a world in which the speed of light was slower would be a different world. But this would be a mistake. If c, h, and e were all changed so that the values they have in metric (or any other) units were different when we looked them up in our tables of physical constants, but the value of α remained the same, this new world would be observationally indistinguishable from our World. The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature. If all masses were doubled in value you cannot tell, because all the pure numbers defined by the ratios of any pair of masses are unchanged."
  23. ^ a b c The values are given in the so-called concise form; the number in parentheses the standard uncertainty, which is the value multiplied by the relative standard uncertainty, and indicates the amount by which the least significant digits of the value are uncertain.
  24. ^ "CODATA Value: characteristic impedance of vacuum". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2017-01-18. 2014 CODATA recommended values
  25. ^ "CODATA Value: magnetic constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  26. ^ "CODATA Value: Newtonian constant of gravitation". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  27. ^ "CODATA Value: Planck constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  28. ^ "CODATA Value: Bohr magneton". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-02-05. 2014 CODATA recommended values
  29. ^ "CODATA Value: conductance quantum". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  30. ^ "CODATA Value: elementary charge". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-22. 2014 CODATA recommended values
  31. ^ "CODATA Value: inverse of conductance quantum". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-02-02. 2014 CODATA recommended values
  32. ^ "CODATA Value: Josephson constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  33. ^ "CODATA Value: magnetic flux quantum". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  34. ^ "CODATA Value: nuclear magneton". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-02-06. 2014 CODATA recommended values
  35. ^ "CODATA Value: von Klitzing constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2017-01-18. 2014 CODATA recommended values
  36. ^ "CODATA Value: Bohr radius". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-02-09. 2014 CODATA recommended values
  37. ^ "CODATA Value: classical electron radius". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-02-09. 2014 CODATA recommended values
  38. ^ "CODATA Value: electron g factor". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  39. ^ "CODATA Value: electron mass". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  40. ^ "CODATA Value: Fermi coupling constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-04-19. 2014 CODATA recommended values
  41. ^ "CODATA Value: fine-structure constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  42. ^ "CODATA Value: Hartree energy". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-02-09. 2014 CODATA recommended values
  43. ^ "CODATA Value: inverse fine-structure constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  44. ^ "CODATA Value: proton mass". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  45. ^ "CODATA Value: quantum of circulation". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-03-05. 2014 CODATA recommended values
  46. ^ "CODATA Value: Rydberg constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  47. ^ "CODATA Value: Thomson cross section". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-02-09. 2014 CODATA recommended values
  48. ^ "CODATA Value: atomic mass constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  49. ^ "CODATA Value: Avogadro constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  50. ^ "CODATA Value: Boltzmann constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  51. ^ "CODATA Value: Faraday constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  52. ^ "CODATA Value: first radiation constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-02-17. 2014 CODATA recommended values
  53. ^ "CODATA Value: first radiation constant for spectral radiance". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-02-17. 2014 CODATA recommended values
  54. ^ "CODATA Value: molar gas constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  55. ^ "CODATA Value: molar Planck constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-02-17. 2014 CODATA recommended values
  56. ^ "CODATA Value: second radiation constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-02-17. 2014 CODATA recommended values
  57. ^ "CODATA Value: Stefan–Boltzmann constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  58. ^ "CODATA Value: Wien wavelength displacement law constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  59. ^ "CODATA Value: Wien frequency displacement law constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-03-02. 2014 CODATA recommended values
  60. ^ Delgado-Bonal, Alfonso (10 May 2017). "Entropy of radiation: the unseen side of light". Scientific Reports. 7 (1642). Bibcode:2017NatSR...7.1642D. doi:10.1038/s41598-017-01622-6.
  61. ^ This is the value adopted internationally for realizing representations of the volt using the Josephson effect.
  62. ^ "CODATA Value: conventional value of Josephson constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-02-20. 2014 CODATA recommended values
  63. ^ This is the value adopted internationally for realizing representations of the ohm using the quantum Hall effect.
  64. ^ "CODATA Value: conventional value of von Klitzing constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-02-20. 2014 CODATA recommended values
  65. ^ "CODATA Value: molar mass constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-02-09. 2014 CODATA recommended values
  66. ^ "CODATA Value: molar mass of carbon-12". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-02-09. 2014 CODATA recommended values
  67. ^ "CODATA Value: standard acceleration of gravity". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-02-17. 2014 CODATA recommended values
  68. ^ "CODATA Value: standard atmosphere". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2019-02-17. 2014 CODATA recommended values
  69. ^ "Resolutions of the CGPM: 13th meeting (10-16 October 1967)". Outcomes of the General Conference. BIPM. Retrieved 2019-03-07. SI unit of time (second)

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