Natural units

Quantity	Expression	Metric value	Name
Length (L)	${\displaystyle l_{\text{P}}={\sqrt {\hbar G \over c^{3}}}}$	1.616×10−35 m[3]	Planck length
Mass (M)	${\displaystyle m_{\text{P}}={\sqrt {\hbar c \over G}}}$	2.176×10−8 kg[4]	Planck mass
Time (T)	${\displaystyle t_{\text{P}}={\sqrt {\hbar G \over c^{5}}}}$	5.391×10−44 s[5]	Planck time
Temperature (Θ)	${\displaystyle T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{G{k_{\text{B}}}^{2}}}}}$	1.417×1032 K[6]	Planck temperature

The Planck unit system uses the following constants to have numeric value 1 in terms of the resulting units:

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 are 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 captures the relationship between energy and frequency which is at the foundation of quantum mechanics. This makes Planck units particularly useful and common in theories of quantum gravity, including string theory.

Planck units may be considered "more natural" even than other natural unit systems discussed below, as Planck units are not based on any arbitrarily chosen prototype object or particle. For example, some other systems use the mass of an electron as a parameter to be normalized. But the electron is just one of 16 known massive elementary particles, all with different masses, and there is no compelling reason, within fundamental physics, to emphasize the electron mass over some other elementary particle's mass.

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]

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 constants to have numeric value 1 in terms of the resulting units:

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. He presented the idea in a lecture entitled "On the Physical Units of Nature" delivered to the British Association in 1874.[9] 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.

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 constants to have numeric value 1 in terms of the resulting units:

e, m_e, ℏ, k_e.

Coulomb's constant, k_e, 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 l_A, orbital velocity = 1 l_A⋅t_A⁻¹, angular momentum = 1 m_A⋅l_A⋅t_A⁻¹, ionization energy = 1/2 m_A⋅l_A²⋅t_A⁻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 l_A⋅t_A⁻¹ ≈ 137 l_A⋅t_A⁻¹) since an electron in hydrogen tends to move much slower than the speed of light. The gravitational constant is extremely small in atomic units (G ≈ 10⁻⁴⁵ m_A⁻¹⋅l_A³⋅t_A⁻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, 2m_e, ℏ, k_e are used as the normalized constants, with resulting units l_R = a₀ = (4πε₀)ℏ²/m_ee², t_R = 2(4πε₀)²ℏ³/m_ee⁴, m_R = 2m_e, q_R = ​^e⁄_√2.[10]

Quantity	Expression	Metric value
Length (L)	${\displaystyle {\frac {\hbar }{m_{\text{e}}c}}}$	3.862×10−13 m[11]
Mass (M)	${\displaystyle m_{\text{e}}\ }$	9.109×10−31 kg[12]
Time (T)	${\displaystyle {\frac {\hbar }{m_{\text{e}}c^{2}}}}$	1.288×10−21 s[13]
Electric charge (Q)	${\displaystyle {\sqrt {\varepsilon _{0}\hbar c}}}$	5.291×10−19 C

The natural unit system used only in the fields of particle and atomic physics uses the following constants to have numeric value 1 in terms of the resulting units:[14]^:126

c, m_e, ℏ, ε₀,

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

The vacuum permittivity ε₀ is implicitly normalized, as is evident from the physicists' expression for the fine-structure constant, written α = e²/(4πℏc),[15][16] which may be compared to the same expression in SI: α = e²/(4πε₀ℏc).[17]^:128

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)	${\displaystyle q_{\mathrm {QCD} }=e}$ (original)	1.602×10−19 C
	${\displaystyle q_{\mathrm {QCD} }={\frac {e}{\sqrt {4\pi \alpha }}}}$ (rat.)	5.291×10−19 C
	${\displaystyle q_{\mathrm {QCD} }={\frac {e}{\sqrt {\alpha }}}}$ (non-rat.)	1.876×10−18 C

c = m_p = ℏ = 1; if rationalized, then ${\displaystyle \epsilon _{0}}$ is 1, if not, ${\displaystyle 4\pi \epsilon _{0}}$ is 1 (in the original QCD units, e is 1 instead.)

The electron rest mass is replaced with that of the proton. 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".[18]

c = G = 1

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 set equal to unity. Other units may be treated however desired. Planck units and Stoney units are examples of geometrized unit systems.

Quantity / Symbol	Planck	Stoney	Hartree	Rydberg
Defining constants	${\displaystyle c}$, ${\displaystyle G}$, ${\displaystyle \hbar }$, ${\displaystyle k_{\text{B}}}$	${\displaystyle c}$, ${\displaystyle G}$, ${\displaystyle e}$, ${\displaystyle k_{\text{e}}}$	${\displaystyle e}$, ${\displaystyle m_{\text{e}}}$, ${\displaystyle \hbar }$, ${\displaystyle k_{\text{e}}}$	${\displaystyle e^{2}/2}$, ${\displaystyle 2m_{\text{e}}}$, ${\displaystyle \hbar }$, ${\displaystyle k_{\text{e}}}$
Speed of light ${\displaystyle c}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle {\frac {1}{\alpha }}}$	${\displaystyle {\frac {2}{\alpha }}}$
Reduced Planck constant ${\displaystyle \hbar ={\frac {h}{2\pi }}}$	${\displaystyle 1}$	${\displaystyle {\frac {1}{\alpha }}}$	${\displaystyle 1}$	${\displaystyle 1}$
Elementary charge ${\displaystyle e}$	${\displaystyle -}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle {\sqrt {2}}}$
Gravitational constant ${\displaystyle G}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle G}$	${\displaystyle G}$
Boltzmann constant ${\displaystyle k_{\text{B}}}$	${\displaystyle 1}$	${\displaystyle -}$	${\displaystyle -}$	${\displaystyle -}$
Electron rest mass ${\displaystyle m_{\text{e}}}$	${\displaystyle m_{\text{e}}}$	${\displaystyle m_{\text{e}}}$	${\displaystyle 1}$	${\displaystyle {\frac {1}{2}}}$

where:

• α is the fine-structure constant, α = e²/4πε₀ħc ≈ 0.007297,
• A dash (–) indicates where the system is not sufficient to express the quantity.