# Natural units

In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge e may be used as a natural unit of electric charge, and the speed of light c may be used as a natural unit of speed. A purely natural system of units has all of its units defined such that each of these can be expressed as a product of powers of defining physical constants.

Through nondimensionalization, physical quantities may then redefined so that the defining constants can be omitted from mathematical expressions of physical laws, and while this has the apparent advantage of simplicity, it may entail a loss of clarity due to the loss of information for dimensional analysis. It precludes the interpretation of an expression in terms of constants, such as e and c, unless it is known which units (in dimensionful units) the expression is supposed to have. In this case, the reinsertion of the correct powers of e, c, etc., can be uniquely determined.

## Systems of natural units

### Planck units

Quantity Expression Metric value
Length (L) ${\displaystyle l_{\text{P}}={\sqrt {\hbar G \over c^{3}}}}$ 1.616×10−35 m[1]
Mass (M) ${\displaystyle m_{\text{P}}={\sqrt {\hbar c \over G}}}$ 2.176×10−8 kg[2]
Time (T) ${\displaystyle t_{\text{P}}={\sqrt {\hbar G \over c^{5}}}}$ 5.391×10−44 s[3]
Temperature (Θ) ${\displaystyle T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{G{k_{\text{B}}}^{2}}}}}$ 1.417×1032 K[4]

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.[citation needed]

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.[5] The Planck system of units is now understood to use the reduced Planck constant, ħ, in place of the Planck constant, h.[6]

### Stoney units

Quantity Expression Metric value
Length (L) ${\displaystyle l_{\text{S}}={\sqrt {\frac {Gk_{\text{e}}e^{2}}{c^{4}}}}}$ 1.38068×10−36 m
Mass (M) ${\displaystyle m_{\text{S}}={\sqrt {\frac {k_{\text{e}}e^{2}}{G}}}}$ 1.85921×10−9 kg
Time (T) ${\displaystyle t_{\text{S}}={\sqrt {\frac {Gk_{\text{e}}e^{2}}{c^{6}}}}}$ 4.60544×10−45 s
Electric charge (Q) ${\displaystyle q_{\text{S}}=e\ }$ 1.60218×10−19 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. He presented the idea in a lecture entitled "On the Physical Units of Nature" delivered to the British Association in 1874.[7] Stoney units did not consider the Planck constant, which was discovered only after Stoney's proposal.

Stoney units are rarely used in modern physics for calculations, but they are of historical interest.

### Atomic units

Quantity Expression Metric value
Length (L) ${\displaystyle l_{\text{A}}={\frac {(4\pi \epsilon _{0})\hbar ^{2}}{m_{\text{e}}e^{2}}}}$ 5.292×10−11 m
Mass (M) ${\displaystyle m_{\text{A}}=m_{\text{e}}}$ 9.109×10−31 kg
Time (T) ${\displaystyle t_{\text{A}}={\frac {(4\pi \epsilon _{0})^{2}\hbar ^{3}}{m_{\text{e}}e^{4}}}}$ 2.419×10−17 s
Electric charge (Q) ${\displaystyle q_{\text{A}}=e}$ 1.602×10−19 C

The Hartree atomic unit system uses the following defining constants:

e, me, ħ, ke.

The Coulomb constant, ke, is generally expressed as 1/4πε0 when working with this system.

These units are designed to simplify atomic and molecular physics and chemistry, especially the hydrogen atom, and are widely used in these fields. The Hartree units were first proposed by Douglas Hartree.

The units are designed especially to characterize the behavior of an electron in the ground state of a hydrogen atom. For example, in Hartree atomic units, in the Bohr model of the hydrogen atom an electron in the ground state has orbital radius (the Bohr radius) a0 = 1 lA, orbital velocity = 1 lAtA−1, angular momentum = 1 mAlAtA−1, ionization energy = 1/2 mAlA2tA−2, etc.

The unit of energy is called the Hartree energy in the Hartree system. The speed of light is relatively large in Hartree atomic units (c = 1/α lAtA−1 ≈ 137 lAtA−1) since an electron in hydrogen tends to move much more slowly than the speed of light. The gravitational constant is extremely small in atomic units (G ≈ 10−45 mA−1lA3tA−2), which is due to the gravitational force between two electrons being far weaker than the Coulomb force between them.

A less commonly used closely related system is the system of Rydberg atomic units, in which e2/2, 2me, ħ, ke are used as the defining constants, with resulting units lR = a0 = (4πε0)ħ2/mee2, tR = 2(4πε0)2ħ3/mee4, mR = 2me, qR = e2.[8]

### Natural units (particle and atomic physics)

Quantity Expression Metric value
Length (L) ${\displaystyle {\frac {\hbar }{m_{\text{e}}c}}}$ 3.862×10−13 m[9]
Mass (M) ${\displaystyle m_{\text{e}}\ }$ 9.109×10−31 kg[10]
Time (T) ${\displaystyle {\frac {\hbar }{m_{\text{e}}c^{2}}}}$ 1.288×10−21 s[11]
Electric charge (Q) ${\displaystyle {\sqrt {\varepsilon _{0}\hbar c}}}$ 5.291×10−19 C

This natural unit system, used only in the fields of particle and atomic physics, uses the following defining constants:[12]: 126

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),[13][14] which may be compared to the same expression in SI: α = e2/(4πε0ħc).[15]: 128

### Quantum chromodynamics units

Quantity Expression Metric value
Length (L) ${\displaystyle l_{\mathrm {QCD} }={\frac {\hbar }{m_{\text{p}}c}}}$ 2.103×10−16 m
Mass (M) ${\displaystyle m_{\mathrm {QCD} }=m_{\text{p}}\ }$ 1.673×10−27 kg
Time (T) ${\displaystyle t_{\mathrm {QCD} }={\frac {\hbar }{m_{\text{p}}c^{2}}}}$ 7.015×10−25 s
Electric charge (Q) ${\displaystyle q_{\mathrm {QCD} }={\sqrt {\varepsilon _{0}\hbar c}}}$ 5.291×10−19 C

Defining constants:

c, mp, ħ, ε0.

Here, mp is the proton rest mass. Strong units, also called quantum chromodynamics (QCD) 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".[16]

### Geometrized units

Defining constants:

c, G.

The geometrized unit system, used in general relativity, is an incompletely defined system. In this system, the base physical units are chosen so that the speed of light and the gravitational constant are coherent units and often used for nondimensionalization. Other units may be treated however desired. Planck units and Stoney units are examples of geometrized unit systems.

### Summary table

Quantity / Symbol Planck Stoney Hartree particle and atomic physics QCD
Defining constants ${\displaystyle c}$, ${\displaystyle G}$, ${\displaystyle \hbar }$, ${\displaystyle k_{\text{B}}}$ ${\displaystyle c}$, ${\displaystyle G}$, ${\displaystyle e}$, ${\displaystyle k_{\text{e}}}$ ${\displaystyle e}$, ${\displaystyle m_{\text{e}}}$, ${\displaystyle \hbar }$, ${\displaystyle k_{\text{e}}}$ ${\displaystyle c}$, ${\displaystyle m_{\text{e}}}$, ${\displaystyle \hbar }$, ${\displaystyle \varepsilon _{0}}$ ${\displaystyle c}$, ${\displaystyle m_{\text{p}}}$, ${\displaystyle \hbar }$, ${\displaystyle \varepsilon _{0}}$
Speed of light
${\displaystyle c}$
${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle {\frac {1}{\alpha }}}$ ${\displaystyle 1}$ ${\displaystyle 1}$
Reduced Planck constant
${\displaystyle \hbar ={\frac {h}{2\pi }}}$
${\displaystyle 1}$ ${\displaystyle {\frac {1}{\alpha }}}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$
Elementary charge
${\displaystyle e}$
${\displaystyle -}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle {\sqrt {4\pi \alpha }}}$ ${\displaystyle {\sqrt {4\pi \alpha }}}$
Vacuum permittivity
${\displaystyle \varepsilon _{0}}$
${\displaystyle -}$ ${\displaystyle {\frac {1}{4\pi }}}$ ${\displaystyle {\frac {1}{4\pi }}}$ ${\displaystyle 1}$ ${\displaystyle 1}$
Gravitational constant
${\displaystyle G}$
${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle {\frac {\eta _{\mathrm {e} }}{\alpha }}}$ ${\displaystyle \eta _{\mathrm {e} }}$ ${\displaystyle \eta _{\mathrm {p} }}$

where:

• α is the fine-structure constant (α = e2/4πε0ħc ≈ 0.007297)
• ηe = G${\displaystyle m_{\text{e}}}$2/ħc1.7518×10−45
• ηp = G${\displaystyle m_{\text{p}}}$2/ħc5.9061×10−39
• A dash (–) indicates where the system is not sufficient to express the quantity.

## Notes and references

1. ^ "2018 CODATA Value: Planck length". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
2. ^ "2018 CODATA Value: Planck mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
3. ^ "2018 CODATA Value: Planck time". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
4. ^ "2018 CODATA Value: Planck temperature". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
5. ^ 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, the Coulomb constant ke = 1/4πε0 would be implicitly added to the list of defining constants, this would yield a charge unit ħc/ke.
6. ^ Tomilin, K. A., 1999, "Natural Systems of Units: To the Centenary Anniversary of the Planck System Archived 2020-12-12 at the Wayback Machine", 287–296.
7. ^ Ray, T.P. (1981). "Stoney's Fundamental Units". Irish Astronomical Journal. 15: 152. Bibcode:1981IrAJ...15..152R.
8. ^ "Atomic Rydberg Units" (PDF).
9. ^ "2018 CODATA Value: natural unit of length". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2020-05-31.
10. ^ "2018 CODATA Value: natural unit of mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2020-05-31.
11. ^ "2018 CODATA Value: natural unit of time". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2020-05-31.
12. ^ International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), ISBN 92-822-2213-6, archived (PDF) from the original on 2021-06-04, retrieved 2021-12-16
13. ^ Frank Wilczek (2005), "On Absolute Units, I: Choices" (PDF), Physics Today, 58 (10): 12, Bibcode:2005PhT....58j..12W, doi:10.1063/1.2138392, retrieved 2020-05-31
14. ^ Frank Wilczek (2006), "On Absolute Units, II: Challenges and Responses" (PDF), Physics Today, 59 (1): 10, Bibcode:2006PhT....59a..10W, doi:10.1063/1.2180151, retrieved 2020-05-31
15. ^ Le Système international d’unités [The International System of Units] (PDF) (in French and English) (9th ed.), International Bureau of Weights and Measures, 2019, ISBN 978-92-822-2272-0
16. ^ Wilczek, Frank (2007). "Fundamental Constants". arXiv:0708.4361 [hep-ph].