Natural units

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 $E = mc 2$ . 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). Dimensional analysis in the collapsed system is uninformative as most quantities have the same dimensions.

Systems of natural units

Summary table

Quantity	Planck	Stoney	Atomic	Particle and atomic physics	Strong	Schrödinger
Defining constants	$c$ , $G$ , $\hbar$ , $k_{\text{B}}$	$c$ , $G$ , $e$ , $k_{\text{e}}$	$e$ , $m_{\text{e}}$ , $\hbar$ , $k_{\text{e}}$	$c$ , $m_{\text{e}}$ , $\hbar$ , $\varepsilon _{0}$	$c$ , $m_{\text{p}}$ , $\hbar$	$\hbar$ , $G$ , $e$ , $k_{\text{e}}$
Speed of light $c$	$1$	$1$	${1}/{\alpha }$	$1$	$1$	${1}/{\alpha }$
Reduced Planck constant $\hbar$	$1$	${1}/{\alpha }$	$1$	$1$	$1$	$1$
Elementary charge $e$	—	$1$	$1$	${\sqrt {4\pi \alpha }}$	—	$1$
Vacuum permittivity $\varepsilon _{0}$	—	${1}/{4\pi }$	${1}/{4\pi }$	$1$	—	${1}/{4\pi }$
Gravitational constant $G$	$1$	$1$	${\eta _{\mathrm {e} }}/{\alpha }$	$\eta _{\mathrm {e} }$	$\eta _{\mathrm {p} }$	$1$

where:

$α$ is the fine-structure constant ( $α = e 2 / 4 πε 0 ħc$ ≈ 0.007297)
$η e = Gm e 2 / ħc$ ≈ 1.7518×10⁻⁴⁵
$η p = Gm p 2 / ħc$ ≈ 5.9061×10⁻³⁹
A dash (—) indicates where the system is not sufficient to express the quantity.

Stoney units

Stoney system dimensions in SI units
Quantity	Expression	Approx. metric value
Length	${\sqrt {{Gk_{\text{e}}e^{2}}/{c^{4}}}}$	1.380×10⁻³⁶ m^[1]
Mass	${\sqrt {{k_{\text{e}}e^{2}}/{G}}}$	1.859×10⁻⁹ kg^[1]
Time	${\sqrt {{Gk_{\text{e}}e^{2}}/{c^{6}}}}$	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

,

k e

,

e

,

where $c$ is the speed of light, $G$ is the gravitational constant, $k e$ 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

Planck dimensions in SI units
Quantity	Expression	Approx. metric value
Length	${\sqrt {{\hbar G}/{c^{3}}}}$	1.616×10⁻³⁵ m^[3]
Mass	${\sqrt {{\hbar c}/{G}}}$	2.176×10⁻⁸ kg^[4]
Time	${\sqrt {{\hbar G}/{c^{5}}}}$	5.391×10⁻⁴⁴ s^[5]
Temperature	${\sqrt {{\hbar c^{5}}/{G{k_{\text{B}}}^{2}}}}$	1.417×10³² K^[6]

The Planck unit system uses the following defining constants:

c

,

ħ

,

G

,

k B

,

where $c$ is the speed of light, $ħ$ is the reduced Planck constant, $G$ is the gravitational constant, and $k B$ 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 $k$ _B 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	${\sqrt {{\hbar ^{4}G(4\pi \varepsilon _{0})^{3}}/{e^{6}}}}$	2.593×10⁻³² m
Mass	${\sqrt {{e^{2}}/{4\pi \varepsilon _{0}G}}}$	1.859×10⁻⁹ kg
Time	${\sqrt {{\hbar ^{6}G(4\pi \varepsilon _{0})^{5}}/{e^{10}}}}$	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

,

k e

.

Geometrized units

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

Atomic-unit dimensions in SI units
Quantity	Expression	Metric value
Length	${(4\pi \epsilon _{0})\hbar ^{2}}/{m_{\text{e}}e^{2}}$	5.292×10⁻¹¹ m^[13]
Mass	$m_{\text{e}}$	9.109×10⁻³¹ kg^[14]
Time	${(4\pi \epsilon _{0})^{2}\hbar ^{3}}/{m_{\text{e}}e^{4}}$	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]

m e

,

e

,

ħ

,

4 πε 0

.

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)

Quantity	Expression	Metric value
Length	${\hbar }/{m_{\text{e}}c}$	3.862×10⁻¹³ m^[20]
Mass	$m_{\text{e}}$	9.109×10⁻³¹ kg^[21]
Time	${\hbar }/{m_{\text{e}}c^{2}}$	1.288×10⁻²¹ s^[22]
Electric charge	${\sqrt {\varepsilon _{0}\hbar c}}$	5.291×10⁻¹⁹ C

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

c

,

m e

,

ħ

,

ε 0

,

where $c$ is the speed of light, $m$ _e is the electron mass, $ħ$ is the reduced Planck constant, and $ε$ ₀ is the vacuum permittivity.

The vacuum permittivity $ε$ ₀ is implicitly used as a nondimensionalization constant, as is evident from the physicists' expression for the fine-structure constant, written $α = e 2 /(4 π)$ ,^[24]^[25] which may be compared to the correspoding expression in SI: $α = e 2 /(4 πε 0 ħc)$ .^[26]^: 128

Strong units

Strong-unit dimensions in SI units
Quantity	Expression	Metric value
Length	${\hbar }/{m_{\text{p}}c}$	2.103×10⁻¹⁶ m
Mass	$m_{\text{p}}$	1.673×10⁻²⁷ kg
Time	${\hbar }/{m_{\text{p}}c^{2}}$	7.015×10⁻²⁵ s