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 be 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

Main article: Planck units

Quantity	Expression	Metric value	Name
Length (L)	$l_{\text{P}}={\sqrt {4\pi \hbar G \over c^{3}}}$ (L–H)	5.729×10−35 m	Planck length
	$l_{\text{P}}={\sqrt {\hbar G \over c^{3}}}$ (G and original)	1.616×10−35 m
Mass (M)	$m_{\text{P}}={\sqrt {\hbar c \over 4\pi G}}$ (L–H)	6.140×10−9 kg	Planck mass
	$m_{\text{P}}={\sqrt {\hbar c \over G}}$ (G and original)	2.176×10−8 kg
Time (T)	$t_{\text{P}}={\sqrt {4\pi \hbar G \over c^{5}}}$ (L–H)	1.911×10−43 s	Planck time
	$t_{\text{P}}={\sqrt {\hbar G \over c^{5}}}$ (G and original)	5.391×10−44 s
Electric charge (Q)	$q_{\text{P}}=e$ (original)	1.602×10−19 C	Planck charge
	$q_{\text{P}}={\sqrt {\hbar c\epsilon _{0}}}$ (L–H)	5.291×10−19 C
	$q_{\text{P}}={\sqrt {\hbar c(4\pi \epsilon _{0})}}$ (G)	1.876×10−18 C
Temperature (Θ)	$T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{4\pi G{k_{\text{B}}}^{2}}}}$ (L–H)	3.997×1031 K	Planck temperature
	$T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{G{k_{\text{B}}}^{2}}}}$ (G and original)	1.417×1032 K

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.

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. The Planck system of units is now understood to use 4πG and the reduced Planck constant ħ, in place of the gravitational constant G and the Planck constant h.

Stoney units

Main article: Stoney units

Quantity	Expression	Metric value
Length (L)	$l_{\text{S}}={\sqrt {\frac {Gk_{\text{e}}e^{2}}{c^{4}}}}$	1.38068×10−36 m
Mass (M)	$m_{\text{S}}={\sqrt {\frac {k_{\text{e}}e^{2}}{G}}}$	1.85921×10−9 kg
Time (T)	$t_{\text{S}}={\sqrt {\frac {Gk_{\text{e}}e^{2}}{c^{6}}}}$	4.60544×10−45 s
Electric charge (Q)	$q_{\text{S}}=e\$	1.60218×10−19 C
Temperature (Θ)	$T_{\text{S}}={\sqrt {\frac {c^{4}k_{\text{e}}e^{2}}{G{k_{\text{B}}}^{2}}}}$	1.21028×1031 K

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. Stoney units did not consider the Planck constant, which was discovered only after Stoney's proposal. The Stoney units and the Planck units differ by the constant factor ${\sqrt {e^{2}/hc}}$

Atomic units

Main article: Hartree atomic units

Quantity	Expression	Metric value
Length (L)	$l_{\text{A}}={\frac {\hbar ^{2}(4\pi \epsilon _{0})}{m_{\text{e}}e^{2}}}$ (both Hartree and Rydberg)	5.292×10−11 m
Mass (M)	$m_{\text{A}}=m_{\text{e}}\$ (Hartree)	9.109×10−31 kg
	$m_{\text{A}}=2m_{\text{e}}\$ (Rydberg)	1.822×10−30 kg
Time (T)	$t_{\text{A}}={\frac {\hbar ^{3}(4\pi \epsilon _{0})^{2}}{m_{\text{e}}e^{4}}}$ (Hartree)	2.419×10−17 s
	$t_{\text{A}}={\frac {2\hbar ^{3}(4\pi \epsilon _{0})^{2}}{m_{\text{e}}e^{4}}}$ (Rydberg)	4.838×10−17 s
Electric charge (Q)	$q_{\text{A}}=e\$ (Hartree)	1.602×10−19 C
	$q_{\text{A}}={\frac {e}{\sqrt {2}}}\$ (Rydberg)	1.133×10−19 C
Temperature (Θ)	$T_{\text{A}}={\frac {m_{\text{e}}e^{4}}{\hbar ^{2}(4\pi \epsilon _{0})^{2}k_{\text{B}}}}$ (Hartree)	3.158×105 K
	$T_{\text{A}}={\frac {m_{\text{e}}e^{4}}{2\hbar ^{2}(4\pi \epsilon _{0})^{2}k_{\text{B}}}}$ (Rydberg)	1.579×105 K

The Hartree atomic unit system uses the following defining constants:

e, m_e, ħ, k_e.

The Coulomb 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 more slowly 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 defining 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.

Natural units (particle and atomic physics)

Unit	Metric value	Derivation
1 eV−1 of length (L)	1.97×10−7 m	$={\frac {\hbar c}{1\,{\text{eV}}}}$
1 eV of mass (M)	1.78×10−36 kg	$={\frac {1\,{\text{eV}}}{c^{2}}}$
1 eV−1 of time (T)	6.58×10−16 s	$={\frac {\hbar }{1\,{\text{eV}}}}$
1 unit of electric charge (Q)	5.29×10−19 C (L–H)	$={\frac {e}{\sqrt {4\pi \alpha }}}$
	1.88×10−18 C (G)	$={\frac {e}{\sqrt {\alpha }}}$
1 eV of temperature (Θ)	1.16×104 K	$={\frac {1\,{\text{eV}}}{k_{\text{B}}}}$

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

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 used as a nondimensionalization constant, as is evident from the physicists' expression for the fine-structure constant, written α = e²/(4π), which may be compared to the same expression in SI: α = e²/(4πε₀ħc).

Quantum chromodynamics units

Quantity	Expression	Metric value
Length (L)	$l_{\mathrm {QCD} }={\frac {\hbar }{m_{\text{p}}c}}$	2.103×10−16 m
Mass (M)	$m_{\mathrm {QCD} }=m_{\text{p}}\$	1.673×10−27 kg
Time (T)	$t_{\mathrm {QCD} }={\frac {\hbar }{m_{\text{p}}c^{2}}}$	7.015×10−25 s
Electric charge (Q)	$q_{\mathrm {QCD} }=e$ (original)	1.602×10−19 C
	$q_{\mathrm {QCD} }={\frac {e}{\sqrt {4\pi \alpha }}}$ (L–H)	5.291×10−19 C
	$q_{\mathrm {QCD} }={\frac {e}{\sqrt {\alpha }}}$ (G)	1.876×10−18 C
Temperature (Θ)	$T_{\mathrm {QCD} }={\frac {m_{\text{p}}c^{2}}{k_{\text{B}}}}$	1.089×1013 K

Defining constants:

c, m_p, ħ, ε₀.

Here, m_p 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".

Schrödinger units

Quantity	Expression	Metric value
Length (L)	$l_{\text{Sch}}={\sqrt {\hbar ^{4}G(4\pi \epsilon _{0})^{3} \over e^{6}}}$	2.593×10−32 m
Mass (M)	$m_{\text{Sch}}={\sqrt {e^{2} \over 4\pi \epsilon _{0}G}}$	1.859×10−9 kg
Time (T)	$t_{\text{Sch}}={\sqrt {\hbar ^{6}G(4\pi \epsilon _{0})^{5} \over e^{10}}}$	1.185×10−38 s
Electric charge (Q)	$q_{\text{Sch}}=e$	1.602×10−19 C‍
Temperature (Θ)	$T_{\text{Sch}}={\sqrt {e^{10} \over \hbar ^{4}G(4\pi \epsilon _{0})^{5}k_{\text{B}}^{2}}}$	6.445×1026 K

The Schrödinger unit system uses the following defining constants:

e, G, ħ, k_e.

This system is seldom mentioned in literature as Schrödinger's system of units (after Austrian physicist Erwin Schrödinger).

In this system of units the speed of light changes in inverse proportion to the fine structure constant, therefore it has gained some interest recent years in the niche hypothesis of time-variation of fundamental constants.

Geometrized units

Main article: Geometrized unit system

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

System Quantity	Planck			Stoney	Schrödinger	Atomic			"Natural"		Quantum chromodynamics
	original	with L–H	with Gauss			Hartree	Rydberg	new	with L–H	with Gauss	original	with L–H	with Gauss
Speed of light $c\,$	$1\,$	$1\,$	$1\,$	$1\,$	${\frac {1}{\alpha }}\$	${\frac {1}{\alpha }}\$	${\frac {2}{\alpha }}\$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$
Reduced Planck constant $\hbar ={\frac {h}{2\pi }}$	$1\,$	$1\,$	$1\,$	${\frac {1}{\alpha }}\$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$
Vacuum permittivity $\varepsilon _{0}\,$	${\frac {1}{4\pi \alpha }}$	$1\,$	${\frac {1}{4\pi }}$	${\frac {1}{4\pi }}$	${\frac {1}{4\pi }}$	${\frac {1}{4\pi }}$	${\frac {1}{4\pi }}$	${\frac {1}{4\pi \alpha }}$	$1\,$	${\frac {1}{4\pi }}$	${\frac {1}{4\pi \alpha }}$	$1\,$	${\frac {1}{4\pi }}$
Coulomb constant $k_{e}={\frac {1}{4\pi \epsilon _{0}}}\,$	$\alpha$	${\frac {1}{4\pi }}$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$\alpha$	${\frac {1}{4\pi }}$	$1\,$	$\alpha$	${\frac {1}{4\pi }}$	$1\,$
Gravitational constant $G\,$	$1\,$	${\frac {1}{4\pi }}$	$1\,$	$1\,$	$1\,$	${\frac {\alpha _{G}}{\alpha }}\,$	${\frac {8\alpha _{G}}{\alpha }}\,$	$\alpha _{G}$	${\frac {\alpha _{G}}{{m_{\text{e}}}^{2}}}\approx 6.71\times 10^{-57}{\text{eV}}^{-2}$	${\frac {\alpha _{G}}{{m_{\text{e}}}^{2}}}\approx 6.71\times 10^{-57}{\text{eV}}^{-2}$	$\mu ^{2}\alpha _{G}$	$\mu ^{2}\alpha _{G}$	$\mu ^{2}\alpha _{G}$
Boltzmann constant $k_{\text{B}}\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$
Elementary charge $e\,$	$1\,$	${\sqrt {4\pi \alpha }}\,$	${\sqrt {\alpha }}\,$	$1\,$	$1\,$	$1\,$	${\sqrt {2}}\,$	$1\,$	${\sqrt {4\pi \alpha }}$	${\sqrt {\alpha }}$	$1\,$	${\sqrt {4\pi \alpha }}\,$	${\sqrt {\alpha }}\,$
Electron rest mass $m_{\text{e}}\,$	${\sqrt {\alpha _{G}}}\,$	${\sqrt {4\pi \alpha _{G}}}\,$	${\sqrt {\alpha _{G}}}\,$	${\sqrt {\frac {\alpha _{G}}{\alpha }}}\,$	${\sqrt {\frac {\alpha _{G}}{\alpha }}}\,$	$1\,$	${\frac {1}{2}}\,$	$1\,$	$m_{e}\approx 511{\text{ keV}}$	$m_{e}\approx 511{\text{ keV}}$	${\frac {1}{\mu }}$	${\frac {1}{\mu }}$	${\frac {1}{\mu }}$
Proton rest mass $m_{\text{p}}\,$	$\mu {\sqrt {\alpha _{G}}}\,$	$\mu {\sqrt {4\pi \alpha _{G}}}\,$	$\mu {\sqrt {\alpha _{G}}}\,$	$\mu {\sqrt {\frac {\alpha _{G}}{\alpha }}}\,$	$\mu {\sqrt {\frac {\alpha _{G}}{\alpha }}}\,$	$\mu \,$	${\frac {\mu }{2}}\,$	$\mu \,$	$\mu m_{\text{e}}\approx 938{\text{ MeV}}$	$\mu m_{\text{e}}\approx 938{\text{ MeV}}$	$1\,$	$1\,$	$1\,$
Vacuum permeability $\mu _{0}={\frac {1}{\epsilon _{0}c^{2}}}\,$	$4\pi \alpha$	$1\,$	$4\pi$	$4\pi$	$4\pi \alpha ^{2}$	$4\pi \alpha ^{2}$	$\pi \alpha ^{2}$	$4\pi \alpha$	$1\,$	$4\pi$	$4\pi \alpha$	$1\,$	$4\pi$
Impedance of free space $Z_{0}={\frac {1}{\epsilon _{0}c}}=\mu _{0}c\,$	$4\pi \alpha$	$1\,$	$4\pi$	$4\pi$	$4\pi \alpha$	$4\pi \alpha$	$2\pi \alpha$	$4\pi \alpha$	$1\,$	$4\pi$	$4\pi \alpha$	$1\,$	$4\pi$
Bohr radius $a_{0}={\frac {4\pi \epsilon _{0}\hbar ^{2}}{m_{\text{e}}e^{2}}}={\frac {\hbar }{m_{\text{e}}c\alpha }}$	${\frac {1}{\alpha {\sqrt {\alpha _{G}}}}}$	${\frac {1}{\alpha {\sqrt {4\pi \alpha _{G}}}}}$	${\frac {1}{\alpha {\sqrt {\alpha _{G}}}}}$	${\frac {1}{\sqrt {\alpha ^{3}\alpha _{G}}}}$	${\sqrt {\frac {\alpha }{\alpha _{G}}}}\,$	$1\,$	$1\,$	${\frac {1}{\alpha }}$	${\frac {1}{m_{e}\alpha }}\approx 2.68\times 10^{-4}{\text{eV}}^{-1}$	${\frac {1}{m_{e}\alpha }}\approx 2.68\times 10^{-4}{\text{eV}}^{-1}$	${\frac {\mu }{\alpha }}$	${\frac {\mu }{\alpha }}$	${\frac {\mu }{\alpha }}$
Bohr magneton $\mu _{B}={\frac {e\hbar }{2m_{e}}}$	${\frac {1}{2{\sqrt {\alpha _{G}}}}}$	${\frac {\alpha }{4\alpha _{G}}}$	${\frac {\alpha }{4\alpha _{G}}}$	${\frac {1}{\sqrt {4\alpha \alpha _{G}}}}$	${\sqrt {\frac {\alpha }{4\alpha _{G}}}}\,$	${\frac {1}{2}}$	${\sqrt {2}}$	${\frac {1}{2}}$	${\frac {\sqrt {4\pi \alpha }}{2m_{e}}}\approx 2.96\times 10^{-7}{\text{eV}}^{-1}$	${\frac {\sqrt {\alpha }}{2m_{e}}}\approx 8.36\times 10^{-8}{\text{eV}}^{-1}$	${\frac {\mu }{2}}$	${\frac {\mu {\sqrt {4\pi \alpha }}}{2}}$	${\frac {\mu {\sqrt {\alpha }}}{2}}$
Josephson constant $K_{\text{J}}={\frac {e}{\pi \hbar }}\,$	${\frac {1}{\pi }}\,$	${\sqrt {\frac {4\alpha }{\pi }}}\,$	${\frac {\sqrt {\alpha }}{\pi }}\,$	${\frac {\alpha }{\pi }}\,$	${\frac {1}{\pi }}\,$	${\frac {1}{\pi }}\,$	${\frac {\sqrt {2}}{\pi }}\,$	${\frac {1}{\pi }}\,$	${\sqrt {\frac {4\alpha }{\pi }}}\,$	${\frac {\sqrt {\alpha }}{\pi }}\,$	${\frac {1}{\pi }}\,$	${\sqrt {\frac {4\alpha }{\pi }}}\,$	${\frac {\sqrt {\alpha }}{\pi }}\,$
von Klitzing constant $R_{\text{K}}={\frac {2\pi \hbar }{e^{2}}}\,$	$2\pi \,$	${\frac {1}{2\alpha }}$	${\frac {2\pi }{\alpha }}\,$	${\frac {2\pi }{\alpha }}\,$	$2\pi \,$	$2\pi \,$	$\pi \,$	$2\pi \,$	${\frac {1}{2\alpha }}$	${\frac {2\pi }{\alpha }}$	$2\pi \,$	${\frac {1}{2\alpha }}$	${\frac {2\pi }{\alpha }}\,$
Rydberg constant $R_{\infty }={\frac {m_{\text{e}}e^{4}}{8\epsilon _{0}^{2}h^{3}c}}={\frac {\alpha ^{2}m_{\text{e}}c}{4\pi \hbar }}$	${\frac {\sqrt {\alpha ^{4}\alpha _{G}}}{4\pi }}$	${\sqrt {\frac {\alpha ^{4}\alpha _{G}}{4\pi }}}$	${\frac {\sqrt {\alpha ^{4}\alpha _{G}}}{4\pi }}$	${\frac {\sqrt {\alpha ^{5}\alpha _{G}}}{4\pi }}$	${\frac {\sqrt {\alpha \alpha _{G}}}{4\pi }}$	${\frac {\alpha }{4\pi }}$	${\frac {\alpha }{4\pi }}$	${\frac {\alpha ^{2}}{4\pi }}$	${\frac {\alpha ^{2}m_{\text{e}}}{4\pi }}\approx 2.17{\text{eV}}$	${\frac {\alpha ^{2}m_{\text{e}}}{4\pi }}\approx 2.17{\text{eV}}$	${\frac {\alpha ^{2}}{4\pi \mu }}$	${\frac {\alpha ^{2}}{4\pi \mu }}$	${\frac {\alpha ^{2}}{4\pi \mu }}$
Stefan–Boltzmann constant $\sigma ={\frac {\pi ^{2}k_{\text{B}}^{4}}{60\hbar ^{3}c^{2}}}\,$	${\frac {\pi ^{2}}{60}}\,$	${\frac {\pi ^{2}}{60}}\,$	${\frac {\pi ^{2}}{60}}\,$	${\frac {\pi ^{2}\alpha ^{3}}{60}}\,$	${\frac {\pi ^{2}\alpha ^{2}}{60}}\,$	${\frac {\pi ^{2}\alpha ^{2}}{60}}\,$	${\frac {\pi ^{2}\alpha ^{2}}{240}}\,$	${\frac {\pi ^{2}}{60}}\,$	${\frac {\pi ^{2}}{60}}\,$	${\frac {\pi ^{2}}{60}}\,$	${\frac {\pi ^{2}}{60}}\,$	${\frac {\pi ^{2}}{60}}\,$	${\frac {\pi ^{2}}{60}}\,$

where:

• α is the fine-structure constant, ⁠k_ee²/ħc⁠ = (⁠e/q_Planck⁠)²
  ≈ 0.0072973525693
• α_G is the gravitational coupling constant, ⁠Gm_e²/ħc⁠ = (⁠m_e/m_Planck⁠)²
  ≈ 1.7518×10⁻⁴⁵
• µ is the proton-to-electron mass ratio, ⁠m_p/m_e⁠ ≈ 1836.15267247

See also

• Anthropic units
• Dimensional analysis
• Dimensionless physical constant
• International System of Units (SI units)
• N-body units
• Physical constant
• Astronomical system of units
• Planck units
• Unit of measurement

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 Archived 2016-05-12 at the Wayback Machine 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.