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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 dimensionless $1$ . 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.

Introduction

Natural units are intended to elegantly simplify particular algebraic expressions 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.

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 $k B$ to 1, which can be thought of as simply a way of defining the unit 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 $F = q 1 q 2 / 4π r 2$ , and in Gaussian units (also called "non-rationalized"), Coulomb's law is $F = q 1 q 2 / r 2$ .^[1] Both possibilities are incorporated into different natural unit systems.

Summary table

Quantity / Symbol	Planck (with L-H)	Planck (with Gauss)	Stoney	Hartree	Rydberg	"Natural" (with L-H)	"Natural" (with Gauss)	Quantum chromodynamics (original)	Quantum chromodynamics (with L-H)	Quantum chromodynamics (with Gauss)
Speed of light $c,$	$1,$	$1,$	$1,$	$frac1alpha$	$frac2alpha$	$1,$	$1,$	$1,$	$1,$	$1,$
Reduced Planck constant $hbar=frach2 pi$	$1,$	$1,$	$frac1alpha$	$1,$	$1,$	$1,$	$1,$	$1,$	$1,$	$1,$
Elementary charge $e,$	$displaystyle sqrt 4pi alpha ,$	$sqrtalpha ,$	$1,$	$1,$	$sqrt2 ,$	$sqrt4pialpha$	$sqrtalpha$	$1,$	$displaystyle sqrt 4pi alpha ,$	$sqrtalpha ,$
Vacuum permittivity $displaystyle varepsilon _0,$	$1,$	$displaystyle frac 14pi$	$displaystyle frac 14pi$	$displaystyle frac 14pi$	$displaystyle frac 14pi$	$1,$	$displaystyle frac 14pi$	$displaystyle frac 14pi alpha$	$1,$	$displaystyle frac 14pi$
Vacuum permeability $displaystyle mu _0=frac 1epsilon _0c^2,$	$1,$	$4pi$	$4pi$	$displaystyle 4pi alpha ^2$	$displaystyle pi alpha ^2$	$1,$	$4pi$	$displaystyle 4pi alpha$	$1,$	$4pi$
Impedance of free space $displaystyle Z_0=frac 1epsilon _0c=mu _0c,$	$1,$	$4pi$	$4pi$	$displaystyle 4pi alpha$	$displaystyle 2pi alpha$	$1,$	$4pi$	$displaystyle 4pi alpha$	$1,$	$4pi$
Josephson constant $K_textJ =fracepi hbar ,$	$sqrtfrac4alphapi ,$	$fracsqrtalphapi ,$	$fracalphapi ,$	$frac1pi ,$	$fracsqrt2pi ,$	$sqrtfrac4alphapi ,$	$fracsqrtalphapi ,$	$frac1pi ,$	$sqrtfrac4alphapi ,$	$fracsqrtalphapi ,$
von Klitzing constant $R_textK =frac2 pi hbare^2 ,$	$frac12alpha$	$frac2pialpha ,$	$frac2pialpha ,$	$2pi ,$	$pi ,$	$frac12alpha$	$frac2 pialpha$	$2pi ,$	$frac12alpha$	$frac2pialpha ,$
Coulomb's constant $displaystyle k_e=frac 14pi epsilon _0$	$displaystyle frac 14pi$	$1,$	$1,$	$1,$	$1,$	$displaystyle frac 14pi$	$1,$	$alpha$	$displaystyle frac 14pi$	$1,$
Gravitational constant $G,$	$displaystyle frac 14pi$	$1,$	$1,$	$fracalpha_textGalpha ,$	$frac8 alpha_textGalpha ,$	$fracalpha_textGm_texte^2 ,$	$fracalpha_textGm_texte^2 ,$	$displaystyle mu ^2alpha _textG$	$displaystyle mu ^2alpha _textG$	$displaystyle mu ^2alpha _textG$
Boltzmann constant $k_textB ,$	$1,$	$1,$	$1,$	$1,$	$1,$	$1,$	$1,$	$1,$	$1,$	$1,$
Proton rest mass $displaystyle m_textP,$	$displaystyle mu sqrt 4pi alpha _textG,$	$displaystyle mu sqrt alpha _textG,$	$displaystyle mu sqrt frac alpha _textGalpha ,$	$mu ,$	$displaystyle frac mu 2,$	$displaystyle 938text MeV$	$displaystyle 938text MeV$	$1,$	$1,$	$1,$
Electron rest mass $m_texte ,$	$displaystyle sqrt 4pi alpha _textG,$	$sqrtalpha_textG ,$	$sqrtfracalpha_textGalpha ,$	$1,$	$frac12 ,$	$511 text keV$	$511 text keV$	$frac 1mu$	$frac 1mu$	$frac 1mu$

where:

$α$ is the fine-structure constant, $(e / q Planck) 2 \approx 0.007297$ ,

$α G$ is the gravitational coupling constant, $(m e / m Planck) 2 \approx 6955175200000000000♠ 1.752 \times 10 -45$ .

$µ$ is proton-to-electron mass ratio, $(m P / m e) \approx 1836.15267247$

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 $E = mc 2$ 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."

Advantages and disadvantages

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 $E 2 = p 2 c 2 + m 2 c 4$ appears somewhat complicated, but the natural units version, $E 2 = p 2 + m 2$ , 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 Kilogram, a physical cylinder of metal whose mass is by definition exactly one kilogram. 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 other 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, $α \approx 1 / 137$ , 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 $α = k e e 2 / ħc$ , where $k e$ is the Coulomb constant, $e$ is the elementary charge, $ħ$ is the reduced Planck constant, and $c$ is the speed of light. Thus, we cannot set all of $k e$ , $e$ , $ħ$ , and $c$ to 1, we can normalize at most three of this set 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 natural unit systems, however, electric charge has units of $[mass] 1 ⁄ 2 [length] 3 ⁄ 2 [time] -1$ .

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

Lorentz–Heaviside units (classified as a rationalized system of electromagnetism units).

Gaussian units (classified as a non-rationalized system of electromagnetism units).

Of these, Lorentz–Heaviside is somewhat more common,^[2] mainly because Maxwell's equations are simpler in Lorentz–Heaviside units than they are in Gaussian units.

In the two unit systems, the elementary charge $e$ satisfies:

$e = \sqrt 4π αħc$ (Lorentz–Heaviside),

$e = \sqrt αħc$ (Gaussian)

where $ħ$ is the reduced Planck constant, $c$ is the speed of light, and $α \approx 1 / 137$ is the fine-structure constant.

In a natural unit system where $c = 1$ , Lorentz–Heaviside units can be derived from SI units by setting $ε 0 = μ 0 = 1$ . Gaussian units can be derived from SI units by a more complicated set of transformations, such as multiplying all electric fields by $(4π ε 0) - 1 ⁄ 2$ , multiplying all magnetic susceptibilities by $4π$ , and so on.^[3]

Systems of natural units

Planck units

Quantity	Expression	Metric value	Name
Length (L)	$displaystyle l_textP=sqrt 4pi hbar G over c^3$ (L–H)	6965572900000000000♠5.729×10⁻³⁵ m	Planck length
Length (L)	$l_textP=sqrt hbar G over c^3$ (G)	6965161600000000000♠1.616×10⁻³⁵ m	Planck length
Mass (M)	$displaystyle m_textP=sqrt hbar c over 4pi G$ (L–H)	6991614000000000000♠6.140×10⁻⁹ kg	Planck mass
Mass (M)	$m_textP=sqrt hbar c over G$ (G)	6992217600000000000♠2.176×10⁻⁸ kg	Planck mass
Time (T)	$displaystyle t_textP=sqrt 4pi hbar G over c^5$ (L–H)	6957191100000000000♠1.911×10⁻⁴³ s	Planck time
Time (T)	$t_textP=sqrt hbar G over c^5$ (G)	6956539099999999999♠5.391×10⁻⁴⁴ s	Planck time
Temperature (Θ)	$displaystyle T_textP=sqrt frac hbar c^54pi Gk_textB^2$ (L–H)	7031399700000000000♠3.997×10³¹ K	Planck temperature
Temperature (Θ)	$T_textP = sqrtfrachbar c^5G k_textB^2$ (G)	7032141700000000000♠1.417×10³² K	Planck temperature
Electric charge (Q)	$displaystyle q_textP=sqrt hbar cepsilon _0$ (L–H)	6981529100000000000♠5.291×10⁻¹⁹ C	Planck charge
Electric charge (Q)	$displaystyle q_textP=sqrt hbar c(4pi epsilon _0)$ (G)	6982187600000000000♠1.876×10⁻¹⁸ C	Planck charge

Planck units are defined by

c = ħ = G = k e = k B = 1

,

where $c$ is the speed of light, $ħ$ is the reduced Planck constant, $G$ is the gravitational constant, $k e$ is the Coulomb 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.

The original Planck units are based on Gaussian units, thus $displaystyle G=k_e=1$ and thus $displaystyle epsilon _0=frac 14pi$ and $displaystyle mu _0=4pi$ . However, the Planck units can also based on Lorentz–Heaviside units, thus $displaystyle G=k_e=frac 14pi$ and $displaystyle epsilon _0=mu _0=Z_0=1$ (this is often called reduced Planck units, e.g. reduced Planck energy). Both kinds of Planck units have $displaystyle c=hbar =k_B=1$ .

Stoney units

Quantity	Expression	Metric value
Length (L)	$l_textS=sqrt frac Gk_textee^2c^4$	6964138100000000000♠1.381×10⁻³⁶ m
Mass (M)	$m_textS=sqrt frac k_textee^2G$	6991185900000000000♠1.859×10⁻⁹ kg
Time (T)	$t_textS=sqrt frac Gk_textee^2c^6$	6955460500000000000♠4.605×10⁻⁴⁵ s
Temperature (Θ)	$T_textS=sqrt frac c^4k_textee^2Gk_textB^2$	7031121000000000000♠1.210×10³¹ K
Electric charge (Q)	$q_textS = e$	6981160200000000000♠1.602×10⁻¹⁹ C

Stoney units are defined by:

c = G = k e = e = k B = 1

,

where $c$ is the speed of light, $G$ is the gravitational constant, $k e$ is the Coulomb constant, $e$ is the elementary charge, and $k B$ 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.^[4] 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)	$l_textA = frachbar^2 (4 pi epsilon_0)m_texte e^2$ (both Hartree and Rydberg)	6989529200000000000♠5.292×10⁻¹¹ m
Mass (M)	$m_textA = m_texte$ (Hartree)	6969910900000000000♠9.109×10⁻³¹ kg
Mass (M)	$displaystyle m_textA=2m_texte$ (Rydberg)	6970182200000000000♠1.822×10⁻³⁰ kg
Time (T)	$t_textA = frachbar^3 (4 pi epsilon_0)^2m_texte e^4$ (Hartree)	6983241900000000000♠2.419×10⁻¹⁷ s
Time (T)	$displaystyle t_textA=frac 2hbar ^3(4pi epsilon _0)^2m_textee^4$ (Rydberg)	6983483800000000000♠4.838×10⁻¹⁷ s
Temperature (Θ)	$T_textA = fracm_texte e^4hbar^2 (4 pi epsilon_0)^2 k_textB$ (Hartree)	7005315800000000000♠3.158×10⁵ K
Temperature (Θ)	$displaystyle T_textA=frac m_textee^42hbar ^2(4pi epsilon _0)^2k_textB$ (Rydberg)	7005157900000000000♠1.579×10⁵ K
Electric charge (Q)	$q_textA = e$ (Hartree)	6981160200000000000♠1.602×10⁻¹⁹ C
Electric charge (Q)	$displaystyle q_textA=frac esqrt 2$ (Rydberg)	6981113300000000000♠1.133×10⁻¹⁹ C

There are two types of atomic units, closely related.

Hartree atomic units:

e = m e = ħ = k e = k B = 1

c = 1 / α

Rydberg atomic units:^[5]

e / \sqrt 2 = 2 m e = ħ = k e = k B = 1

c = 2 / α

Coulomb's constant is generally expressed as

k e = 1 / 4π ε 0

.

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, $l A$ , is the Bohr radius, $a 0$ .

The values of $c$ and $e$ shown above imply that $e = \sqrt αħc$ , as in Gaussian units, not Lorentz–Heaviside units.^[6] However, hybrids of the Gaussian and Lorentz–Heaviside units are sometimes used, leading to inconsistent conventions for magnetism-related units.^[7]

Quantum chromodynamics (QCD) units

Quantity	Expression	Metric value
Length (L)	$l_mathrmQCD = frachbarm_textp c$	6984210300000000000♠2.103×10⁻¹⁶ m
Mass (M)	$m_mathrmQCD = m_textp$	6973167300000000000♠1.673×10⁻²⁷ kg
Time (T)	$t_mathrmQCD = frachbarm_textp c^2$	6975701500000000000♠7.015×10⁻²⁵ s
Temperature (Θ)	$T_mathrmQCD = fracm_textp c^2k_textB$	7013108900000000000♠1.089×10¹³ K
Electric charge (Q)	$displaystyle q_mathrm QCD =e$ (original)	6981160200000000000♠1.602×10⁻¹⁹ C
	$displaystyle q_mathrm QCD =frac esqrt 4pi alpha$ (L–H)	6981529100000000000♠5.291×10⁻¹⁹ C
	$displaystyle q_mathrm QCD =frac esqrt alpha$ (G)	6982187600000000000♠1.876×10⁻¹⁸ C

c = m p = ħ = k B = 1

(in the original QCD units,

e

is also 1, if the QCD units are based on Lorentz–Heaviside units, then

epsilon _0

is 1, and if the QCD units are based on Gaussian units, then

displaystyle k_e=frac 14pi 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".^[8]

"Natural units" (particle physics and cosmology)

Unit	Metric value	Derivation
1 eV⁻¹ of length	6993196999999999999♠1.97×10⁻⁷ m	$displaystyle =frac hbar c1,texteV$
1 eV of mass	6964178000000000000♠1.78×10⁻³⁶ kg	$displaystyle =frac 1,texteVc^2$
1 eV⁻¹ of time	6984658000000000000♠6.58×10⁻¹⁶ s	$displaystyle =frac hbar 1,texteV$
1 eV of temperature	7004116000000000000♠1.16×10⁴ K	$displaystyle =frac 1,texteVk_textBcdot frac 2f$ with $displaystyle f=2$
1 unit of electric charge (L–H)	6981529000000000000♠5.29×10⁻¹⁹ C	$displaystyle =frac esqrt 4pi alpha$
1 unit of electric charge (G)	6982188000000000000♠1.88×10⁻¹⁸ C	$displaystyle =frac esqrt alpha$

In particle physics and cosmology, the phrase "natural units" generally means:^[9]^[10]

ħ = c = k B = 1

.

where $ħ$ is the reduced Planck constant, $c$ is the speed of light, and $k B$ 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:^[10]

1.0 cm = 1.0 cm / ħc \approx 51000 eV -1

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

^ 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)

^ Walter Greiner; Ludwig Neise; Horst Stöcker (1995). Thermodynamics and Statistical Mechanics. Springer-Verlag. p. 385. ISBN 978-0-387-94299-5..mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em

^ See Gaussian units#General rules to translate a formula and references therein.

^ Ray, T.P. (1981). "Stoney's Fundamental Units". Irish Astronomical Journal. 15: 152. Bibcode:1981IrAJ...15..152R.

^ Turek, Ilja (1997). Electronic structure of disordered alloys, surfaces and interfaces (illustrated ed.). Springer. p. 3. ISBN 978-0-7923-9798-4.

^ Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science, by Markus Reiher, Alexander Wolf, p7 [books.google.com/books?id=YwSpxCfsNsEC&pg=PA7 link]

^ A note on units lecture notes. See the atomic units article for further discussion.

^ Wilczek, Frank, 2007, "Fundamental Constants," Frank Wilczek web site.

^ Gauge field theories: an introduction with applications, by Guidry, Appendix A

^ ^a^b An introduction to cosmology and particle physics, by Domínguez-Tenreiro and Quirós, p422

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 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.

[1] 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)

[GreinerNeise1995-2] Walter Greiner; Ludwig Neise; Horst Stöcker (1995). Thermodynamics and Statistical Mechanics. Springer-Verlag. p. 385. ISBN 978-0-387-94299-5..mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em

[3] See Gaussian units#General rules to translate a formula and references therein.

[4] Ray, T.P. (1981). "Stoney's Fundamental Units". Irish Astronomical Journal. 15: 152. Bibcode:1981IrAJ...15..152R.

[5] Turek, Ilja (1997). Electronic structure of disordered alloys, surfaces and interfaces (illustrated ed.). Springer. p. 3. ISBN 978-0-7923-9798-4.

[6] Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science, by Markus Reiher, Alexander Wolf, p7 [books.google.com/books?id=YwSpxCfsNsEC&pg=PA7 link]

[7] A note on units lecture notes. See the atomic units article for further discussion.

[8] Wilczek, Frank, 2007, "Fundamental Constants," Frank Wilczek web site.

[9] Gauge field theories: an introduction with applications, by Guidry, Appendix A

[DT-10] An introduction to cosmology and particle physics, by Domínguez-Tenreiro and Quirós, p422

v t e Planck's natural units
*Planck_units"> Planck constant Planck units
Base Planck units	Planck time Planck length Planck mass Planck charge Planck temperature
Derived Planck units	Planck energy Planck force Planck power Planck density Planck angular frequency Planck pressure Planck current Planck voltage Planck impedance Planck momentum Planck area Planck volume Planck acceleration

v t e SI units
Base units	ampere candela kelvin kilogram metre mole second
Derived units with special names	becquerel coulomb degree Celsius farad gray henry hertz joule katal lumen lux newton ohm pascal radian siemens sievert steradian tesla volt watt weber
Other accepted units	astronomical unit bar dalton day decibel degree of arc electronvolt hectare hour litre minute minute and second of arc neper tonne atomic units natural units
See also	Conversion of units Metric prefixes 2019 redefinition Systems of measurement
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