Natural units

In physics, natural units are physical units of measurement defined in terms of universal physical constants in such a manner that some chosen physical constants take on the numerical value of one when expressed in terms of a particular set of natural units. Natural units are intended to elegantly simplify particular algebraic expressions appearing in physical law or to normalize some chosen physical quantities that are properties of a universal elementary particles and that may be reasonably believed to be constant. However, what may be believed and forced to be constant in one system of natural units can very well be allowed or even assumed to vary in another natural unit system. Naural 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" but are only one system of natural units among other systems. Planck units might be considered unique in that the set of units are not based on properties of any prototype, object, or particle but are based only on properties of free space.

As with any set of base units or fundamental units the base units of a set of natural units will include definitions and values for length, mass, time, temperature, and electric charge. Some physicists have not recognized temperature as a fundamental dimension of physical quantity since it simply expresses 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 the Boltzmann constant, k=1, which can be thought of as simply another expression of the definition of the unit temperature. In addition, some physicists recognize electric charge as a separate fundamental dimension of physical quantity, even if it has been expressed in terms of mass, length, and time in unit systems such as the electrostatic cgs system. Virtually every system of natural units normalize the permittivity of free space, ε₀=4π.

Candidate physical constants used in natural unit systems

The candidate physical constants to be normalized are chosen from those in the following table. Note that only a smaller subset of the following can be normalized in any one system of units without contradiction in definition (i.e. m_e and m_p cannot both be defined as the unit mass in a single system).

Constant	Symbol	Dimension
speed of light in vacuum	${c}\$	L T-1
Gravitational constant	${G}\$	M-1L3T-2
Dirac's constant or "reduced Planck's constant"	$\hbar ={\frac {h}{2\pi }}$ where ${h}\$ is Planck's constant	ML2T-1
Coulomb force constant	${\frac {1}{4\pi \epsilon _{0}}}$ where ${\epsilon _{0}}\$ is the permittivity of free space	Q-2 M L3 T-2
Elementary charge	$e\$	Q
Electron mass	$m_{e}\$	M
Proton mass	$m_{p}\$	M
Bohr radius	$a_{0}={\frac {4\pi \epsilon _{0}\hbar ^{2}}{m_{e}e^{2}}}\$	L
Boltzmann constant	${k}\$	ML2T-2Θ-1

Dimensionless fundamental physical constants such as the fine-structure constant

$\alpha \equiv {\frac {e^{2}}{\hbar c4\pi \epsilon _{0}}}={\frac {1}{137.03599911}}$

cannot take on a different numerical value no matter what system of units are used. Judiciously choosing units can only normalize physical constants that have dimension. In some natural unit systems, the numerical value of some important constants take on a value that is a sole function of α when expressed in terms of the particular natural units, alluding to the fundamental nature of the fine-structure constant.

Geometrized units

Main article: Geometrized unit system

$c=1\$

$G=1\$

The Geometrized unit system is not a completely defined or unique 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 leaving latitude to also set some other constant such as the Boltzmann constant and Coulomb force constant equal to unity:

$k=1\$

${\frac {1}{4\pi \epsilon _{0}}}=1$

If Dirac's constant (also called the "reduced Planck's constant") is also set equal to unity,

$\hbar =1\$

then geometrized units are identical to Planck units.

Planck units

Main article: Planck units

$c=1\$

$G=1\$

$\hbar =1\$

${\frac {1}{4\pi \epsilon _{0}}}=1$

$k=1\$

$e^{2}=\alpha \$

The physical constants that Planck units normalize are properties of free space and not properties (such as charge, mass, radius) of any object or elementary particle (that would have to be arbitrarily chosen).

Quantity	Expression
Length (L)	$l_{P}={\sqrt {\frac {\hbar G}{c^{3}}}}$
Mass (M)	$m_{P}={\sqrt {\frac {\hbar c}{G}}}$
Time (T)	$t_{P}={\sqrt {\frac {\hbar G}{c^{5}}}}$
Electric charge (Q)	$q_{P}={\sqrt {\hbar c4\pi \epsilon _{0}}}$
Temperature (Θ)	$T_{P}={\sqrt {\frac {\hbar c^{5}}{Gk^{2}}}}$

Stoney units

$c=1\$

$G=1\$

$e=1\$

${\frac {1}{4\pi \epsilon _{0}}}=1$

$k=1\$

$\hbar ={\frac {1}{\alpha }}\$

Proposed by George Stoney. Stoney units fix the elementary charge and allow Planck's constant to float. They can be obtained from Planck units with the substitution:

$\hbar \leftarrow \alpha \hbar ={\frac {e^{2}}{c4\pi \epsilon _{0}}}$.

Quantity	Expression
Length (L)	$l_{S}={\sqrt {\frac {Ge^{2}}{4\pi \epsilon _{0}c^{4}}}}$
Mass (M)	$m_{S}={\sqrt {\frac {e^{2}}{4\pi \epsilon _{0}G}}}$
Time (T)	$t_{S}={\sqrt {\frac {Ge^{2}}{4\pi \epsilon _{0}c^{6}}}}$
Electric charge (Q)	$q_{S}=e\$
Temperature (Θ)	$T_{S}={\sqrt {\frac {c^{4}e^{2}}{4\pi \epsilon _{0}Gk^{2}}}}$

"Schrödinger" units

$e=1\$

$G=1\$

$\hbar =1\$

${\frac {1}{4\pi \epsilon _{0}}}=1$

$k=1\$

$c={\frac {1}{\alpha }}\$

The name coined by Michael Duff[1]. They can be obtained from Planck units with the substitution:

$c\leftarrow \alpha c={\frac {e^{2}}{\hbar 4\pi \epsilon _{0}}}$.

Quantity	Expression
Length (L)	$l_{\psi }={\sqrt {\frac {\hbar ^{4}G(4\pi \epsilon _{0})^{3}}{e^{6}}}}$
Mass (M)	$m_{\psi }={\sqrt {\frac {e^{2}}{G4\pi \epsilon _{0}}}}$
Time (T)	$t_{\psi }={\sqrt {\frac {\hbar ^{6}G(4\pi \epsilon _{0})^{5}}{e^{10}}}}$
Electric charge (Q)	$q_{\psi }=e\$
Temperature (Θ)	$T_{\psi }={\sqrt {\frac {e^{10}}{\hbar ^{4}(4\pi \epsilon _{0})^{5}Gk^{2}}}}$

Atomic units (Hartree)

Main article: Atomic units

$e=1\$

$m_{e}=1\$

$\hbar =1\$

${\frac {1}{4\pi \epsilon _{0}}}=1$

$a_{0}=1\$

$k=1\$

$c={\frac {1}{\alpha }}\$

First proposed by Douglas Hartree to simplify the physics of Hydrogen atom. Michael Duff[2] calls these "Bohr units". The unit energy in this system is the total energy of the electron in the Bohr atom and called the Hartree energy. They can be obtained from “Schrödinger” units with the substitution:

$G\leftarrow \alpha G\left({\frac {m_{P}}{m_{e}}}\right)^{2}={\frac {e^{2}}{4\pi \epsilon _{0}}}\$.

The unit length is the Bohr radius.

Quantity	Expression
Length (L)	$l_{A}={\frac {\hbar ^{2}(4\pi \epsilon _{0})}{m_{e}e^{2}}}$
Mass (M)	$m_{A}=m_{e}\$
Time (T)	$t_{A}={\frac {\hbar ^{3}(4\pi \epsilon _{0})^{2}}{m_{e}e^{4}}}$
Electric charge (Q)	$q_{A}=e\$
Temperature (Θ)	$T_{A}={\frac {m_{e}e^{4}}{\hbar ^{2}(4\pi \epsilon _{0})^{2}k}}$

Electronic system of units

$c=1\$

$e=1\$

$m_{e}=1\$

${\frac {1}{4\pi \epsilon _{0}}}=1$

$k=1\$

$\hbar ={\frac {1}{\alpha }}\$

Michael Duff[3] calls these "Dirac units". They can be obtained from Stoney units with the substitution:

$G\leftarrow \alpha G\left({\frac {m_{P}}{m_{e}}}\right)^{2}={\frac {e^{2}}{4\pi \epsilon _{0}}}\$.

They can be also obtained from Atomic units with the substitution:

$\hbar \leftarrow \alpha \hbar ={\frac {e^{2}}{c4\pi \epsilon _{0}}}$.

Quantity	Expression
Length (L)	$l_{e}={\frac {e^{2}}{c^{2}m_{e}4\pi \epsilon _{0}}}$
Mass (M)	$m_{e}=m_{e}\$
Time (T)	$t_{e}={\frac {e^{2}}{c^{3}m_{e}4\pi \epsilon _{0}}}$
Electric charge (Q)	$q_{e}=e\$
Temperature (Θ)	$T_{e}={\frac {m_{e}c^{2}}{k}}$

Quantum electrodynamical system of units (Stille)

$c=1\$

$e=1\$

$m_{p}=1\$

${\frac {1}{4\pi \epsilon _{0}}}=1$

$k=1\$

$\hbar ={\frac {1}{\alpha }}\$

Similar to the electronic system of units except that the proton mass is normalized rather that the electron mass.

Quantity	Expression
Length (L)	$l_{\mathrm {QED} }={\frac {e^{2}}{c^{2}m_{p}4\pi \epsilon _{0}}}$
Mass (M)	$m_{\mathrm {QED} }=m_{p}\$
Time (T)	$t_{\mathrm {QED} }={\frac {e^{2}}{c^{3}m_{p}4\pi \epsilon _{0}}}$
Electric charge (Q)	$q_{\mathrm {QED} }=e\$
Temperature (Θ)	$T_{\mathrm {QED} }={\frac {m_{p}c^{2}}{k}}$

See also

• dimensional analysis
• physical constants

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.