Natural units

Short description: Units of measurement based on universal physical constants

In physics, natural unit systems are measurement systems for which selected physical constants have been set to 1 through nondimensionalization of physical units. For example, the speed of light c may be set to 1, and it may then be omitted, equating mass and energy directly E = m rather than using c as a conversion factor in the typical mass–energy equivalence equation E = mc2. A purely natural system of units has all of its dimensions collapsed, such that the physical constants completely define the system of units and the relevant physical laws contain no conversion constants.

While natural unit systems simplify the form of each equation, it is still necessary to keep track of the non-collapsed dimensions of each quantity or expression in order to reinsert physical constants (such dimensions uniquely determine the full formula).

## Contents

* 1 Systems of natural units
* 1.1 Summary table
* 1.2 Stoney units
* 1.3 Planck units
* 1.4 Schrödinger units
* 1.5 Geometrized units
* 1.6 Atomic units
* 1.7 Natural units (particle and atomic physics)
* 1.8 Strong units
* 2 See also
* 3 Notes and references
* 4 External links

## Systems of natural units

### Summary table

Quantity  | Planck | Stoney | Atomic | Particle and atomic physics  | Strong  | Schrödinger

|
|
|
|
|
|

Defining constants  | c, G, ℏ, kB | c, G, ℏ, ke, kB | c, G, ℏ, ε0, kB | c, G, e, ke | e, me, ℏ, ke | c, me, ℏ, ε0 | c, mp, ℏ | ℏ, G, e, ke
c | 1 | 1 | 1 | 1  | ⁠1/α⁠ | 1 | 1  | ⁠1/α⁠
ℏ | 1 | 1 | 1  | ⁠1/α⁠ | 1 | 1 | 1 | 1
e | -- | α | 2πα | 1 | 1  | 2πα | -- | 1
ε0 | -- | ⁠1/4π⁠ | 1  | ⁠1/4π⁠ | ⁠1/4π⁠ | 1  | -- | 14π
G | 1 | 1 | 1 | 1  | ⁠ηe/α⁠ | ηe | ηp | 1

where:

* α is the fine-structure constant, e2 / 4πε0ℏc = 7.2973525693(11)×10−3
* ke is the Coulomb constant, 1 / 4πε0 ≈ 8987551786, so assigning it a value also assigns ε0 a value.
* ηe = Gme2 / ℏc = 10068233135085418604437102.57πGme2 ≈ 1.7518×10−45
* me is the mass of an electron, 9.1093837015(28)×10−31
* ηp = Gmp2 / ℏc = 10068233135085418604437102.57πGmp2 ≈ 5.9061×10−39
* mp is the mass of a proton, 1.67262192369(51)×10−27
* -- indicates where the system is not sufficient to express the quantity.
* kB, the Boltzmann constant, has no interactions with the other constants - it is used only to redefine temperature.

### Stoney units

Main article: Stoney units

Stoney system dimensions in SI units  Quantity  | Expression  | Approx.
metric value

|
|

Length | eGkec2 | 1.380×10−36 m[1]
Mass | ekeG | 1.859×10−9 kg[1]
Time | eGkec3 | 4.605×10−45 s[1]
Electric charge | e | 1.602×10−19 C

The Stoney unit system uses the following defining constants:

c, G, ke, e,

where c is the speed of light, G is the gravitational constant, ke is the Coulomb constant, and e is the elementary charge.

George Johnstone Stoney's unit system preceded that of Planck by 30 years. He presented the idea in a lecture entitled "On the Physical Units of Nature" delivered to the British Association in 1874.[2] Stoney units did not consider the Planck constant, which was discovered only after Stoney's proposal.

### Planck units

Main article: Planck units

Planck dimensions in SI units  Quantity  | Expression  | Approx.
metric value

|
|

Length | ℏGc3 | 1.616×10−35 m[3]
Mass | ℏcG | 2.176×10−8 kg[4]
Time | ℏGc5 | 5.391×10−44 s[5]
Temperature | ℏc5kBG | 1.417×1032 K[6]
Electric charge
ε0 is also chosen  | ε0ℏc | 5.291×10−19 C
Electric charge
ke is also chosen  | ℏcke | 1.876×10−18 C

The Planck unit system uses the following defining constants:

c, ℏ, G, kB,

where c is the speed of light, ℏ is the reduced Planck constant, G is the gravitational constant, and kB is the Boltzmann constant.

Planck units form a system of natural units that is not defined in terms of properties of any prototype, physical object, or even elementary particle. They only refer to the basic structure of the laws of physics: c and G are part of the structure of spacetime in general relativity, and ℏ is at the foundation of quantum mechanics. This makes Planck units particularly convenient and common in theories of quantum gravity, including string theory. Planck considered only the units based on the universal constants G, h, c, and kB to arrive at natural units for length, time, mass, and temperature, but no electromagnetic units.[7] The Planck system of units is now understood to use the reduced Planck constant, ℏ, in place of the Planck constant, h.[8]

### Schrödinger units

Schrödinger system dimensions in SI units  Quantity  | Expression  | Approx. metric value

|
|

Length | ℏ2e3Gke3 | 2.593×10−32 m
Mass | ekeG | 1.859×10−9 kg
Time | ℏ3e5Gke5 | 1.185×10−38 s
Electric charge | e | 1.602×10−19 C[9]

The Schrödinger system of units (named after Austrian physicist Erwin Schrödinger) is seldom mentioned in literature. Its defining constants are:[10][11]

e, ħ, G, ke.

### Geometrized units

Main article: Geometrized unit system

Defining constants
c, G.

The geometrized unit system,[12]: 36 used in general relativity; the base physical units are chosen so that the speed of light, c, and the gravitational constant, G, are set to one.

### Atomic units

Main article: Physics:Atomic units

Atomic-unit dimensions in SI units  Quantity  | Expression  | Metric value

|
|

Length | 4πϵ0ℏ2mee2 | 5.292×10−11 m[13]
Mass | me | 9.109×10−31 kg[14]
Time | 16π2ϵ02ℏ3mee4 | 2.419×10−17 s[15]
Electric charge | e | 1.602×10−19 C[16]

The atomic unit system[17] uses the following defining constants:[18]: 349 [19]

me, e, ħ, 4πε0 (this is exactly the same as using ke, except in which constant you use when expressing the conversion).

The atomic units were first proposed by Douglas Hartree and are designed to simplify atomic and molecular physics and chemistry, especially the hydrogen atom.[18]: 349 For example, in atomic units, in the Bohr model of the hydrogen atom an electron in the ground state has orbital radius, orbital velocity and so on with particularly simple numeric values.

### Natural units (particle and atomic physics)

Particle and Atomic Physics system dimensions in SI units  Quantity  | Expression  | Metric value

|
|

Length | ℏmec | 3.862×10−13 m[20]
Mass | me | 9.109×10−31 kg[21]
Time | ℏmec2 | 1.288×10−21 s[22]
Electric charge | ε0ℏc | 5.291×10−19 C

This natural unit system, used only in the fields of particle and atomic physics, uses the following defining constants:[23]: 509

c, me, ħ, ε0,

where c is the speed of light, me is the electron mass, ħ is the reduced Planck constant, and ε0 is the vacuum permittivity.

The vacuum permittivity ε0 is implicitly used as a nondimensionalization constant, as is evident from the physicists' expression for the fine-structure constant, written α = e2/(4π),[24][25] which may be compared to the corresponding expression in SI: α = e2/(4πε0ħc).[26]: 128

### Strong units

Strong-unit dimensions in SI units  Quantity  | Expression  | Metric value

|
|

Length | ℏmpc | 2.103×10−16 m
Mass | mp | 1.673×10−27 kg
Time | ℏmpc2 | 7.015×10−25 s

Defining constants:

c, mp, ħ.

Here, mp is the proton rest mass. Strong units are "convenient for work in QCD and nuclear physics, where quantum mechanics and relativity are omnipresent and the proton is an object of central interest".[27]

## See also

* Anthropic units
* Astronomical system of units
* Dimensionless physical constant
* International System of Units
* N-body units
* Outline of metrology and measurement
* Unit of measurement

## Notes and references

1. ↑ 1.0 1.1 1.2 Barrow, John D. (1983), "Natural units before Planck", Quarterly Journal of the Royal Astronomical Society 24: 24–26, Bibcode: 1983QJRAS..24...24B, https://articles.adsabs.harvard.edu/full/1983QJRAS..24...24B
2. ↑ Ray, T.P. (1981). "Stoney's Fundamental Units". Irish Astronomical Journal 15: 152. Bibcode: 1981IrAJ...15..152R.
3. ↑ "2018 CODATA Value: Planck length". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. http://physics.nist.gov/cgi-bin/cuu/Value?plkl. Retrieved 2019-05-20.
4. ↑ "2018 CODATA Value: Planck mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. http://physics.nist.gov/cgi-bin/cuu/Value?plkm. Retrieved 2019-05-20.
5. ↑ "2018 CODATA Value: Planck time". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. http://physics.nist.gov/cgi-bin/cuu/Value?plkt. Retrieved 2019-05-20.
6. ↑ "2018 CODATA Value: Planck temperature". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. http://physics.nist.gov/cgi-bin/cuu/Value?plktmp. Retrieved 2019-05-20.
7. ↑ However, if it is assumed that at the time the Gaussian definition of electric charge was used and hence not regarded as an independent quantity, 4πε0 would be implicit in the list of defining constants, giving a charge unit √4πε0ℏc.
8. ↑ Tomilin, K. A., 1999, "Natural Systems of Units: To the Centenary Anniversary of the Planck System ", 287–296.
9. ↑ "2018 CODATA Value: elementary charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. http://physics.nist.gov/cgi-bin/cuu/Value?e. Retrieved 2019-05-20.
10. ↑ Stohner, Jürgen; Quack, Martin (2011). "Conventions, Symbols, Quantities, Units and Constants for High-Resolution Molecular Spectroscopy". Handbook of High-resolution Spectroscopy. p. 304. doi:10.1002/9780470749593.hrs005. ISBN 9780470749593. https://www.ir.ethz.ch/handbook/MQ333_Handbook_Stohner_Quack_bearbeitet.pdf. Retrieved 19 March 2023.
11. ↑ A bot will complete this citation soon. Click here to jump the queue arXiv:hep-th/0208093.
12. ↑ Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (2008). Gravitation (27. printing ed.). New York, NY: Freeman. ISBN 978-0-7167-0344-0.
13. ↑ "2018 CODATA Value: atomic unit of length". The NIST Reference on Constants, Units, and Uncertainty. NIST. http://physics.nist.gov/cgi-bin/cuu/Value?Abohrrada0.
14. ↑ "2018 CODATA Value: atomic unit of mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. http://physics.nist.gov/cgi-bin/cuu/Value?Ame.
15. ↑ "2018 CODATA Value: atomic unit of time". The NIST Reference on Constants, Units, and Uncertainty. NIST. http://physics.nist.gov/cgi-bin/cuu/Value?aut.
16. ↑ "2018 CODATA Value: atomic unit of charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. http://physics.nist.gov/cgi-bin/cuu/Value?Ae.
17. ↑ Shull, H.; Hall, G. G. (1959). "Atomic Units". Nature 184 (4698): 1559. doi:10.1038/1841559a0. Bibcode: 1959Natur.184.1559S.
18. ↑ 18.0 18.1 Levine, Ira N. (1991). Quantum chemistry. Pearson advanced chemistry series (4 ed.). Englewood Cliffs, NJ: Prentice-Hall International. ISBN 978-0-205-12770-2.
19. ↑ McWeeny, R. (May 1973). "Natural Units in Atomic and Molecular Physics" (in en). Nature 243 (5404): 196–198. doi:10.1038/243196a0. ISSN 0028-0836. Bibcode: 1973Natur.243..196M. https://www.nature.com/articles/243196a0.
20. ↑ "2018 CODATA Value: natural unit of length". The NIST Reference on Constants, Units, and Uncertainty. NIST. http://physics.nist.gov/cgi-bin/cuu/Value?eqNecomwlbar.
21. ↑ "2018 CODATA Value: natural unit of mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. http://physics.nist.gov/cgi-bin/cuu/Value?Nme.
22. ↑ "2018 CODATA Value: natural unit of time". The NIST Reference on Constants, Units, and Uncertainty. NIST. http://physics.nist.gov/cgi-bin/cuu/Value?nut.
23. ↑ Guidry, Mike (1991). "Appendix A: Natural Units". Gauge Field Theories. Weinheim, Germany: Wiley-VCH Verlag. pp. 509–514. doi:10.1002/9783527617357.app1. ISBN 978-0-471-63117-0.
24. ↑ Frank Wilczek (2005), "On Absolute Units, I: Choices", Physics Today 58 (10): 12, doi:10.1063/1.2138392, Bibcode: 2005PhT....58j..12W, http://ctpweb.lns.mit.edu/physics_today/phystoday/Abs_limits388.pdf, retrieved 2020-05-31
25. ↑ Frank Wilczek (2006), "On Absolute Units, II: Challenges and Responses", Physics Today 59 (1): 10, doi:10.1063/1.2180151, Bibcode: 2006PhT....59a..10W, http://ctpweb.lns.mit.edu/physics_today/phystoday/Abs_limits393.pdf, retrieved 2020-05-31
26. ↑ International Bureau of Weights and Measures (2019-05-20), SI Brochure: The International System of Units (SI) (9th ed.), ISBN 978-92-822-2272-0, https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9.pdf
27. ↑ Wilczek, Frank (2007). "Fundamental Constants". arXiv:0708.4361 [hep-ph].. Further see.

## External links

* The NIST website (National Institute of Standards and Technology) is a convenient source of data on the commonly recognized constants.
* K.A. Tomilin: NATURAL SYSTEMS OF UNITS; To the Centenary Anniversary of the Planck System A comparative overview/tutorial of various systems of natural units having historical use.
* Pedagogic Aides to Quantum Field Theory Click on the link for Chap. 2 to find an extensive, simplified introduction to natural units.
* Natural System Of Units In General Relativity (PDF), by Alan L. Myers (University of Pennsylvania). Equations for conversions from natural to SI units.

* v
* t
* e

SI units

Base units|

* ampere
* candela
* kelvin
* kilogram
* {{{sp}}}}}|us|er|re}}
* mole
* second

Derived units
with special names|

* becquerel
* coulomb
* degree Celsius
* farad
* gray
* henry
* hertz
* joule
* katal
* lumen
* lux
* newton
* ohm
* pascal
* radian
* siemens
* sievert
* steradian
* tesla
* volt
* watt
* weber

Other accepted units|

* astronomical unit
* dalton
* day
* decibel
* degree of arc
* electronvolt
* hectare
* hour
* {{{sp}}}}}|us|er|re}}
* minute
* minute and second of arc
* neper
* tonne

See also|

* Conversion of units
* Metric prefixes
* 2005–2019 definition
* 2019 redefinition
* Systems of measurement

* Book
* Category

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