# Natural units

In physics, natural units are physical units of measurement based only on universal physical constants. For example, the elementary charge e is a natural unit of electric charge, and the speed of light c is a natural unit of speed. A purely natural system of units has all of its units defined in this way, and usually such that the numerical values of the selected physical constants in terms of these units are exactly 1. These constants are then typically 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 fundamental physical 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.[1][2]

## Introduction

Natural units are intended to simplify particular equations appearing in the laws of physics or to normalize some chosen physical quantities that are properties of universal elementary particles and are reasonably believed to be constant. However, there is a choice of which quantities to set to unity in a natural system of units, and quantities which are set to unity in one system may take a different value or even be assumed to vary in another natural unit system.

Natural units are "natural" because the origin of their definition comes only from properties of nature and not from any human construct. Planck units are often, without qualification, called "natural units", although they constitute only one of several systems of natural units, albeit the best known such system. Planck units (up to a simple multiplier for each unit) might be considered one of the most "natural" systems in that the set of units is not based on properties of any prototype, object, or particle but are solely derived from the properties of free space.

Dimensioned physical constants may be thought of as conversion factors between inconsistent, rather arbitrary "human-constructed" units of measure. For example, spatial distance may be measured in terms of the rather arbitrary unit of the meter and time may be measured in terms of the rather arbitrary unit of the second. In relativity theory, no distinction is drawn between space-like distances and time-like distances, and so the speed of light is a conversion factor between the meter and the second. Likewise, the Avogadro number is a conversion factor between quantities measured in particle number and in moles (which are in turn defined by the rather arbitrary mass unit of the gram). The universal gas constant is simply Boltzmann's constant when quantity is measured in moles, and Boltzmann's constant may be thought of as a conversion factor between energy measured in, say, Joules and in Kelvin, and also (controversially) as a conversion factor between thermodynamic entropy and information entropy.

As with other systems of units, the base units of a set of natural units will include definitions and values for length, mass, time, temperature, and electric charge (in lieu of electric current). It is possible to disregard temperature as a fundamental physical quantity, since it states the energy per degree of freedom of a particle, which can be expressed in terms of energy (or mass, length, and time). Virtually every system of natural units normalizes Boltzmann's constant kB to 1, which can be thought of as simply a way of defining the unit of temperature.

In SI, electric charge is a separate fundamental dimension of physical quantity, but in natural unit systems charge is expressed in terms of the mechanical units of mass, length, and time, similarly to cgs. There are two common ways to relate charge to mass, length, and time: In Lorentz–Heaviside units (also called "rationalized"), Coulomb's law is F = q1q2/r2, and in Gaussian units (also called "non-rationalized"), Coulomb's law is F = q1q2/r2.[3] Both possibilities are incorporated into different natural unit systems.

### Summary table

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

where:

## Notation and use

Natural units are most commonly used by setting the units to one. For example, many natural unit systems include the equation c = 1 in the unit-system definition, where c is the speed of light. If a velocity v is half the speed of light, then as v = c/2 and c = 1, hence v = 1/2. The equation v = 1/2 means "the velocity v has the value one-half when measured in Planck units", or "the velocity v is one-half the Planck unit of velocity".

The equation c = 1 can be plugged in anywhere else. For example, Einstein's equation E = mc2 can be rewritten in Planck units as E = m. This equation means "The energy of a particle, measured in Planck units of energy, equals the mass of the particle, measured in Planck units of mass."

Compared to SI or other unit systems, natural units have both advantages and disadvantages:

• Simplified equations: By setting constants to 1, equations containing those constants appear more compact and in some cases may be simpler to understand. For example, the special relativity equation E2 = p2c2 + m2c4 appears somewhat complicated, but the natural units version, E2 = p2 + m2, appears simpler.
• Physical interpretation: Space and time are put on equal footing and are both measured in the same units. Natural unit systems automatically subsume dimensional analysis. For example, in Planck units, the units are defined by properties of quantum mechanics and gravity. Not coincidentally, the Planck unit of length is approximately the distance at which quantum gravity effects become important. Likewise, atomic units are based on the mass and charge of an electron, and not coincidentally the atomic unit of length is the Bohr radius describing the "orbit" of the electron in a hydrogen atom.
• No prototypes: A prototype is a physical object that defines a unit, such as the International Prototype of the Kilogram, a physical cylinder of metal whose mass used to be by definition exactly one kilogram (as of 2019 one kilogram is instead defined in terms of fundamental constants). A prototype definition always has imperfect reproducibility between different places and between different times, and it is an advantage of natural unit systems that they use no prototypes. (They share this advantage with some non-natural unit systems, such as conventional electrical units.)
• Less precise measurements: SI units are designed to be used in precision measurements. For example, the second is defined by an atomic transition frequency in cesium atoms, because this transition frequency can be precisely reproduced with atomic clock technology. Natural unit systems are generally not based on quantities that can be precisely reproduced in a lab. Therefore, in order to retain the same degree of precision, the fundamental constants used still have to be measured in a laboratory in terms of physical objects that can be directly observed. If this is not possible, then a quantity expressed in natural units can be less precise than the same quantity expressed in SI units. For example, Planck units use the gravitational constant G, which is measurable in a laboratory only to four significant digits.

## Choosing constants to normalize

Out of the many physical constants, the designer of a system of natural unit systems must choose a few of these constants to normalize (set equal to 1). It is not possible to normalize just any set of constants. For example, the mass of a proton and the mass of an electron cannot both be normalized: if the mass of an electron is defined to be 1, then the mass of a proton has to be approximately 1836. In a less trivial example, the fine-structure constant, α1/137, cannot be set to 1 because it is a dimensionless number defined in terms of other quantities. The fine-structure constant is related to other physical constants through α = kee2/ħc, where ke is the Coulomb constant, e is the elementary charge, ħ is the reduced Planck constant, and c is the speed of light. Thus, it is impossible to set all of ke, e, ħ, and c to 1; at most three of this set can be normalized to 1.

## Electromagnetism units

In SI units, electric charge is expressed in coulombs, a separate unit which is additional to the "mechanical" units (mass, length, time), even though the traditional definition of the ampere refers to some of these other units.

In order to build natural units in electromagnetism one can use:

Of these, Lorentz–Heaviside is somewhat more common,[4] mainly because Maxwell's equations are simpler in Lorentz–Heaviside units than they are in Gaussian units.

In the two unit systems, the Planck unit charge qP is:

• qP = αħc (Lorentz–Heaviside),
• qP = αħc (Gaussian)

where ħ is the reduced Planck constant, c is the speed of light, and α1/137.036 is the fine-structure constant.

In a natural unit system where c = 1, Lorentz–Heaviside units can be derived from SI units by setting ε0 = μ0 = 1. Gaussian units can be derived from SI units by a more complicated set of transformations, such as multiplying all electric fields by (4πε0)−​12, multiplying all magnetic susceptibilities by , and so on.[5]

## Systems of natural units

### Planck units

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

Planck units are defined by

c = ħ = G = ke = kB = 1,

where c is the speed of light, ħ is the reduced Planck constant, G is the gravitational constant, ke is the Coulomb constant, and kB 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, ħ, c, and kB to arrive at natural units for length, time, mass, and temperature.[6] Planck did not adopt any electromagnetic units. However, since the non-rationalized gravitational constant, G, is set to 1, a natural extension of Planck units to a unit of electric charge is to also set the non-rationalized Coulomb constant, ke, to 1 as well (as well as the Stoney units).[7] This is the non-rationalized Planck units (Planck units with the Gaussian version), which is more convenient but not rationalized, there is also a Planck system which is rationalized (Planck units with the Lorentz-Heaviside version), set 4πG and ε0 (instead of G and ke) to 1, which may be less convenient but is rationalized. Another convention is to use the elementary charge as the basic unit of electric charge in the Planck system.[8] Such a system is convenient for black hole physics. The two conventions for unit charge differ by a factor of the square root of the fine-structure constant. Planck's paper also gave numerical values for the base units that were close to modern values.

The original Planck units are based on Gaussian units, thus ${\displaystyle G=k_{e}=1}$ and thus ${\displaystyle \epsilon _{0}={\frac {1}{4\pi }}}$ and ${\displaystyle \mu _{0}=4\pi }$. However, the Planck units can also be based on Lorentz–Heaviside units, thus ${\displaystyle G=k_{e}={\frac {1}{4\pi }}}$ and ${\displaystyle \epsilon _{0}=\mu _{0}=Z_{0}=1}$ (this is often called rationalized Planck units, e.g. the rationalized Planck energy). Both conventions of Planck units set ${\displaystyle c=\hbar =k_{B}=1}$.

### 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
Temperature (Θ) ${\displaystyle T_{\text{S}}={\sqrt {\frac {c^{4}k_{\text{e}}e^{2}}{G{k_{\text{B}}}^{2}}}}}$ 1.21028×1031 K
Electric charge (Q) ${\displaystyle q_{\text{S}}=e\ }$ 1.60218×10−19 C

Stoney units are defined by:

c = G = ke = e = kB = 1,

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

George Johnstone Stoney was the first physicist to introduce the concept of natural units. He presented the idea in a lecture entitled "On the Physical Units of Nature" delivered to the British Association in 1874.[9] Stoney units differ from Planck units by fixing the elementary charge at 1, instead of the Planck constant (only discovered 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) ${\displaystyle l_{\text{A}}={\frac {\hbar ^{2}(4\pi \epsilon _{0})}{m_{\text{e}}e^{2}}}}$ (both Hartree and Rydberg) 5.292×10−11 m
Mass (M) ${\displaystyle m_{\text{A}}=m_{\text{e}}\ }$ (Hartree) 9.109×10−31 kg
${\displaystyle m_{\text{A}}=2m_{\text{e}}\ }$ (Rydberg) 1.822×10−30 kg
Time (T) ${\displaystyle t_{\text{A}}={\frac {\hbar ^{3}(4\pi \epsilon _{0})^{2}}{m_{\text{e}}e^{4}}}}$ (Hartree) 2.419×10−17 s
${\displaystyle t_{\text{A}}={\frac {2\hbar ^{3}(4\pi \epsilon _{0})^{2}}{m_{\text{e}}e^{4}}}}$ (Rydberg) 4.838×10−17 s
Temperature (Θ) ${\displaystyle T_{\text{A}}={\frac {m_{\text{e}}e^{4}}{\hbar ^{2}(4\pi \epsilon _{0})^{2}k_{\text{B}}}}}$ (Hartree) 3.158×105 K
${\displaystyle T_{\text{A}}={\frac {m_{\text{e}}e^{4}}{2\hbar ^{2}(4\pi \epsilon _{0})^{2}k_{\text{B}}}}}$ (Rydberg) 1.579×105 K
Electric charge (Q) ${\displaystyle q_{\text{A}}=e\ }$ (Hartree) 1.602×10−19 C
${\displaystyle q_{\text{A}}={\frac {e}{\sqrt {2}}}\ }$ (Rydberg) 1.133×10−19 C

There are two types of atomic units, closely related.

Hartree atomic units:

e = me = ħ = ke = kB = 1
c = 1/α

Rydberg atomic units:[10]

e/2 = 2me = ħ = ke = kB = 1
c = 2/α

Coulomb's constant is generally expressed as

ke = 1/ε0.

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, and are more common than the Rydberg units.

The units are designed especially to characterize the behavior of an electron in the ground state of a hydrogen atom. For example, using the Hartree convention, in the Bohr model of the hydrogen atom, an electron in the ground state has orbital velocity = 1, orbital radius = 1, angular momentum = 1, ionization energy = 1/2, etc.

The unit of energy is called the Hartree energy in the Hartree system and the Rydberg energy in the Rydberg system. They differ by a factor of 2. The speed of light is relatively large in atomic units (137 in Hartree or 274 in Rydberg), which comes from the fact that an electron in hydrogen tends to move much slower than the speed of light. The gravitational constant is extremely small in atomic units (around 10−45), which comes from the fact that the gravitational force between two electrons is far weaker than the Coulomb force. The unit length, lA, is the Bohr radius, a0.

The values of c and e shown above imply that e = αħc, as in Gaussian units, not Lorentz–Heaviside units.[11] However, hybrids of the Gaussian and Lorentz–Heaviside units are sometimes used, leading to inconsistent conventions for magnetism-related units.[12]

### Quantum chromodynamics (QCD) 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
Temperature (Θ) ${\displaystyle T_{\mathrm {QCD} }={\frac {m_{\text{p}}c^{2}}{k_{\text{B}}}}}$ 1.089×1013 K
Electric charge (Q) ${\displaystyle q_{\mathrm {QCD} }=e}$ (original) 1.602×10−19 C
${\displaystyle q_{\mathrm {QCD} }={\frac {e}{\sqrt {4\pi \alpha }}}}$ (L–H) 5.291×10−19 C
${\displaystyle q_{\mathrm {QCD} }={\frac {e}{\sqrt {\alpha }}}}$ (G) 1.876×10−18 C
c = mp = ħ = kB = 1 (in the original QCD units, e is also 1, if the QCD units are based on Lorentz–Heaviside units, then ${\displaystyle \epsilon _{0}}$ is 1, and if the QCD units are based on Gaussian units, then ${\displaystyle k_{e}={\frac {1}{4\pi \epsilon _{0}}}}$ is 1)

The electron rest mass is replaced with that of the proton. 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".[13]

### "Natural units" (particle physics and cosmology)

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

In particle physics and cosmology, the phrase "natural units" generally means:[14][15]

ħ = c = kB = 1.

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

Both Planck units and QCD units are this type of Natural units. Like the other systems, the electromagnetism units can be based on either Lorentz–Heaviside units or Gaussian units. The unit of charge is different in each.

Finally, one more unit is needed to construct a usable system of units that includes energy and mass. Most commonly, electronvolt (eV) is used, despite the fact that this is not a "natural" unit in the sense discussed above – it is defined by a natural property, the elementary charge, and the anthropogenic unit of electric potential, the volt. (The SI prefixed multiples of eV are used as well: keV, MeV, GeV, etc.)

With the addition of eV (or any other auxiliary unit with the proper dimension), any quantity can be expressed. For example, a distance of 1.0 cm can be expressed in terms of eV, in natural units, as:[15]

1.0 cm = 1.0 cm/ħc ≈ 51000 eV−1

If the addition is 4πG (for Lorentz–Heaviside units) or G (for Gaussian units) instead of eV, then the two kinds of Natural units are the same as the two kinds of Planck units.

### Geometrized units

c = G = 1

The geometrized unit system, used in general relativity, is not a completely 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.

## Notes and references

1. ^ What are natural units?, Sabine Hossenfelder, 2011-11-07.
2. ^ Planck Units – Part 1 of 3, DrPhysicistA, 2012-02-14.
3. ^ Kowalski, Ludwik, 1986, "A Short History of the SI Units in Electricity, Archived 2009-04-29 at the Wayback Machine" The Physics Teacher 24(2): 97–99. Alternate web link (subscription required)
4. ^ Walter Greiner; Ludwig Neise; Horst Stöcker (1995). Thermodynamics and Statistical Mechanics. Springer-Verlag. p. 385. ISBN 978-0-387-94299-5.
5. ^ See Gaussian units#General rules to translate a formula and references therein.
6. ^ *Tomilin, K. A., 1999, "Natural Systems of Units: To the Centenary Anniversary of the Planck System", 287–296.
7. ^ Pavšic, Matej (2001). The Landscape of Theoretical Physics: A Global View. Fundamental Theories of Physics. 119. Dordrecht: Kluwer Academic. pp. 347–352. arXiv:gr-qc/0610061. doi:10.1007/0-306-47136-1. ISBN 978-0-7923-7006-2.
8. ^ Tomilin, K. (1999). "Fine-structure constant and dimension analysis". Eur. J. Phys. 20 (5): L39–L40. Bibcode:1999EJPh...20L..39T. doi:10.1088/0143-0807/20/5/404.
9. ^ Ray, T.P. (1981). "Stoney's Fundamental Units". Irish Astronomical Journal. 15: 152. Bibcode:1981IrAJ...15..152R.
10. ^ Turek, Ilja (1997). Electronic structure of disordered alloys, surfaces and interfaces (illustrated ed.). Springer. p. 3. ISBN 978-0-7923-9798-4.
11. ^ Reiher, Markus; Wolf, Alexander (2009-05-13). Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science. p. 7. ISBN 9783527627493.
12. ^ A note on units lecture notes. See the atomic units article for further discussion.
13. ^ Wilczek, Frank, 2007, "Fundamental Constants," Frank Wilczek web site.
14. ^ Gauge field theories: an introduction with applications, by Guidry, Appendix A
15. ^ a b An introduction to cosmology and particle physics, by Domínguez-Tenreiro and Quirós, p422