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]}
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 "humanconstructed" 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 spacelike distances and timelike 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 k_{B} 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 LorentzHeaviside units (also called "rationalized"), Coulomb's law is F = , and in Gaussian units (also called "nonrationalized"), Coulomb's law is F = .^{[3]} Both possibilities are incorporated into different natural unit systems.
System Quantity

Planck  Stoney  Atomic  "Natural"  Quantum chromodynamics  

original  with LH  with Gauss  Hartree  Rydberg  new  with LH  with Gauss  original  with LH  with Gauss  
Speed of light 

Reduced Planck constant 

Elementary charge 

Vacuum permittivity 

Vacuum permeability 

Impedance of free space 

Coulomb constant 

Gravitational constant 

Boltzmann constant 

Proton rest mass 

Electron rest mass 

Josephson constant 

von Klitzing constant 

Bohr radius 

Bohr magneton 

Rydberg constant 

StefanBoltzmann constant 
where:
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 unitsystem definition, where c is the speed of light. If a velocity v is half the speed of light, then as v = and c = 1, hence v = . The equation v = means "the velocity v has the value onehalf when measured in Planck units", or "the velocity v is onehalf 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."
Compared to SI or other unit systems, natural units have both advantages and disadvantages:
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 finestructure constant, ? ? , cannot be set to 1 because it is a dimensionless number defined in terms of other quantities. The finestructure constant is related to other physical constants through ? = , where k_{e} is the Coulomb constant, e is the elementary charge, ? is the reduced Planck constant, and c is the speed of light. Thus, it is impossible to set all of k_{e}, e, ?, and c to 1; at most three of this set can be normalized to 1.
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, LorentzHeaviside is somewhat more common,^{[4]} mainly because Maxwell's equations are simpler in LorentzHeaviside units than they are in Gaussian units.
In the two unit systems, the Planck unit charge q_{P} is:
where ? is the reduced Planck constant, c is the speed of light, and ? ? is the finestructure constant.
In a natural unit system where c = 1, LorentzHeaviside units can be derived from SI units by setting ?_{0} = ?_{0} = 1. Gaussian units can be derived from SI units by a more complicated set of transformations, such as multiplying all electric fields by (4??_{0})^{}, multiplying all magnetic susceptibilities by 4?, and so on.^{[5]}
Quantity  Expression  Metric value  Name 

Length (L)  (LH)  Planck length  
(G and original)  
Mass (M)  (LH)  Planck mass  
(G and original)  
Time (T)  (LH)  Planck time  
(G and original)  
Temperature (?)  (LH)  Planck temperature  
(G and original)  
Electric charge (Q)  (original)  Planck charge  
(LH)  
(G) 
Planck units are defined by
where c is the speed of light, ? 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 ? 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, ?, c, and k_{B} to arrive at natural units for length, time, mass, and temperature.^{[6]} Planck did not adopt any electromagnetic units. However, since the nonrationalized gravitational constant, G, is set to 1, a natural extension of Planck units to a unit of electric charge is to also set the nonrationalized Coulomb constant, k_{e}, to 1 as well (as well as the Stoney units).^{[7]} This is the nonrationalized 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 LorentzHeaviside version), set 4?G and ?_{0} (instead of G and k_{e}) 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 finestructure 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 and thus and . However, the Planck units can also be based on LorentzHeaviside units, thus and (this is often called rationalized Planck units, e.g. the rationalized Planck energy). Both conventions of Planck units set .
Quantity  Expression  Metric value 

Length (L)  
Mass (M)  
Time (T)  
Temperature (?)  
Electric charge (Q) 
Stoney units are defined by:
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.^{[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.
Quantity  Expression  Metric value 

Length (L)  (both Hartree and Rydberg)  
Mass (M)  (Hartree)  
(Rydberg)  
Time (T)  (Hartree)  
(Rydberg)  
Temperature (?)  (Hartree)  
(Rydberg)  
Electric charge (Q)  (Hartree)  
(Rydberg) 
There are two types of atomic units, closely related.
Hartree atomic units:
Rydberg atomic units:^{[10]}
Coulomb's constant is generally expressed as
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 = , 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 = , as in Gaussian units, not LorentzHeaviside units.^{[11]} However, hybrids of the Gaussian and LorentzHeaviside units are sometimes used, leading to inconsistent conventions for magnetismrelated units.^{[12]}
Quantity  Expression  Metric value 

Length (L)  
Mass (M)  
Time (T)  
Temperature (?)  
Electric charge (Q)  (original)  
(LH)  
(G) 
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]}
Unit  Metric value  Derivation 

1 eV^{1} of length  
1 eV of mass  
1 eV^{1} of time  
1 eV of temperature  
1 unit of electric charge (LH) 

1 unit of electric charge (G) 
In particle physics and cosmology, the phrase "natural units" generally means:^{[14]}^{[15]}
where ? 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 LorentzHeaviside 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]}
If the addition is 4?G (for LorentzHeaviside 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.
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.