Natural units: Information from Answers.com

In physics, natural units are physical units of measurement defined in such a way that certain selected universal physical constants are normalized to unity; that is, their numerical value becomes exactly 1.

1 Introduction
2 Candidate physical constants used in natural unit systems
3 Systems of natural units
4 See also
5 References
6 External links

Introduction

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 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.

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", when in fact they are only one of several systems of natural units, albeit the best known such system. 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 solely derived from the 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 do not recognize temperature as a fundamental 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 Boltzmann's constant k_B to 1, which can be thought of as simply a way of defining 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 normalizes the permittivity of free space to ε₀=(4π)^-1, an expression which can be thought of as defining the unit charge. This suggests that the preference in SI units for expressing Coulomb's law in the "rationalized" form F= (4πε₀)^-1q₁q₂r^-2, rather than as F=kq₁q₂r^-2, as originally advocated (controversially) by the Italian electrical engineer G. Giorgi,^[1] may not have been the most natural choice after all. (In this respect, natural units are more similar to Gaussian units than to SI.)

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 (e.g., 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
Planck's constant (reduced)	$\hbar=\frac{h}{2 \pi}$	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
Boltzmann constant	${ k_B } \$	ML²T^-2Θ^-1

Judiciously choosing units can only normalize physical constants that have dimension. Dimensionless physical constants cannot take on a different numerical value no matter what system of units is used. One such constant important to physics is the fine-structure constant, α:

$\alpha \ \equiv \frac{e^2}{\hbar c (4 \pi \epsilon_0)} = \frac{1}{137.03599911}$

Since α is a fixed dimensionless number not equal to 1, it is not possible to define a system of natural units that will normalize all of the physical constants that comprise α. Any three of the four constants: c, ℏ, e, or 4πε₀, can be normalized (leaving the remaining physical constant to take on a value that is a simple function of α, attesting to the fundamental nature of the fine-structure constant) but not all four.

Systems of natural units

Planck units

Main article: Planck units

Quantity	Expression	Metric value
Length (L)	$l_P = \sqrt{\frac{\hbar G}{c^3}}$	1.61609735×10^-35 m
Mass (M)	$m_P = \sqrt{\frac{\hbar c}{G}}$	2.17644(11)×10^-8 kg
Time (T)	$t_P = \sqrt{\frac{\hbar G}{c^5}}$	5.3907205×10^-44 s
Electric charge (Q)	$q_P = \sqrt{\hbar c (4 \pi \epsilon_0)}$	1.87554573×10^-18 C
Temperature (Θ)	$T_P = \sqrt{\frac{\hbar c^5}{G {k_B}^2}}$	1.4169206×10³² K

$c = G = \hbar = \frac{1}{4 \pi \epsilon_0} = k_B = 1 \$

$e = \sqrt{\alpha} \$

Planck units hold place of pride among systems of natural units, because they do not invoke any properties (specifically, the charge, mass, spin, or orbital radius) of any elementary particle. For example, the proton and electron are equally sensible choices, and thus are equally arbitrary. But their masses differ considerably, a fact having nontrivial implications for all other systems of natural units, because these all invoke one or more properties of protons or electrons.

By contrast, the physical constants that Planck units normalize are all properties of free space. In particular, the definition of Planck units does not invoke the elementary charge, whose numerical value, when measured in units of Planck charge, is the square root of the fine-structure constant α. Hence any observed variation over space or time in the value of α is attributed to variation in the elementary charge.

Stoney units

Main article: Stoney scale units

Quantity	Expression	Metric Value
Length (L)	$l_S = \sqrt{\frac{G e^2}{c^4 (4 \pi \epsilon_0)}}$	1.38068×10^-36 m
Mass (M)	$m_S = \sqrt{\frac{e^2}{G (4 \pi \epsilon_0)}}$	1.85921×10^-9 kg
Time (T)	$t_S = \sqrt{\frac{G e^2}{c^6 (4 \pi \epsilon_0)}}$	4.60544×10^-45 s
Electric charge (Q)	$q_S = e \$	1.60218×10^-19 C
Temperature (Θ)	$T_S = \sqrt{\frac{c^4 e^2}{G (4 \pi \epsilon_0) {k_B}^2}}$	1.21028×10³¹ K

$c = G = e = \frac{1}{4 \pi \epsilon_0} = k_B = 1 \$

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

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.^[2] Stoney units fix the elementary charge and allow Planck's constant (only discovered after Stoney's proposal) to float. They can be obtained from Planck units with the substitution:

$\hbar \leftarrow \alpha \hbar = \frac{e^2}{c (4 \pi \epsilon_0)}$ .

This removes Planck's constant from the definitions and the value it takes on in Stoney units is the reciprocal of the fine-structure constant, 1/α. Hence any observed variation over space or time in the value of α is attributed to variation in Planck's constant.

Heaviside-Lorentz units

Main article: Lorentz–Heaviside units

Heaviside-Lorentz units with

$\hbar={c}={\epsilon_0}={\mu_{0}}={{Z_0}}={1}$

are often used in relativistic calculations, and in particle and nuclear physics. The units are particularly convenient for calculations in spatial dimensions greater than three, as in string theory. Besides the speed of propagation of the electromagnetic interaction being normalized, so also is the characteristic impedance of electromagnetic waves.

This has the consequence that $\scriptstyle \alpha = \frac{e^{2}}{4\pi}$ and so can be used to define a unit of electric charge so that the elementary charge is

$e = \sqrt{4 \pi \alpha} = 0.30282212..$ .

Similarly to geometrized units, without an additional independent constraint, Heaviside-Lorentz units do not fully define a complete set for length, time, and mass. Only the unit charge is fully defined. If the gravitational constant G is also constrained, Heaviside-Lorentz units would be identical to Planck units save for constant factors of $\scriptstyle \sqrt{4 \pi}$ or the reciprocal. If G is defined similarly to the Coulomb constant $\scriptstyle \frac{1}{4 \pi \epsilon_0}$ , then G would be fixed to $\scriptstyle G = \frac{1}{4 \pi}$ , rather than normalized to 1 and, likewise to electromagnetic waves, the gravitational interaction propagates with normalized speed and normalized characteristic impedance. The latter does not occur with Planck units.

"Schrödinger" units

Quantity	Expression	Metric Value
Length (L)	$l_{\psi} = \sqrt{\frac{\hbar^4 G (4 \pi \epsilon_0)^3}{e^6}}$	2.59276×10^-32 m
Mass (M)	$m_{\psi} = \sqrt{\frac{e^2}{G (4 \pi \epsilon_0)}}$	1.85921×10^-9 kg
Time (T)	$t_{\psi} = \sqrt{\frac{\hbar^6 G (4 \pi \epsilon_0)^5}{e^{10}}}$	1.18516×10^-38 s
Electric charge (Q)	$q_{\psi} = e \$	1.602176487×10^-19 C
Temperature (Θ)	$T_{\psi} = \sqrt{\frac{e^{10}}{\hbar^4 (4 \pi \epsilon_0)^5 G {k_B}^2}}$	6.44490×10²⁶ K

$e = G = \hbar = \frac{1}{4 \pi \epsilon_0} = k_B = 1 \$

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

The name was coined by Michael Duff^[3]. They can be obtained from Planck units with the substitution:

$c \leftarrow \alpha c = \frac{e^2}{\hbar (4 \pi \epsilon_0)}$ .

This removes the speed of light from the definitions and the value it takes on in Schrödinger units is the reciprocal of the fine-structure constant, 1/α. Hence any observed variation over space or time in the value of α is attributed to variation in the speed of light.

Atomic units (Hartree)

Main article: Atomic units

Quantity	Expression	Metric Value
Length (L)	$l_A = \frac{\hbar^2 (4 \pi \epsilon_0)}{m_e e^2}$	5.29177×10^-11 m
Mass (M)	$m_A = m_e \$	9.10938×10^-31 kg
Time (T)	$t_A = \frac{\hbar^3 (4 \pi \epsilon_0)^2}{m_e e^4}$	2.41889×10^-17 s
Electric charge (Q)	$q_A = e \$	1.60218×10^-19 C
Temperature (Θ)	$T_A = \frac{m_e e^4}{\hbar^2 (4 \pi \epsilon_0)^2 k_B}$	3.15774×10⁵ K

$e = m_e = \hbar = \frac{1}{4 \pi \epsilon_0} = k_B = 1 \$

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

First proposed by Douglas Hartree to simplify the physics of the Hydrogen atom. Duff^[3] calls these "Bohr units". The unit energy in this system is the total energy of the electron in the first circular orbit of the Bohr atom and called the Hartree energy, E_h. The unit velocity is the velocity of that electron, the unit mass is the electron mass, m_e, and the unit length is the Bohr radius, $a_0 = 4 \pi \epsilon_0\hbar^2/m_e e^2 \$ . 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 m_e^2} \$ .

This removes the speed of light (as well as the gravitational constant) from the definitions and its numerical value in atomic units is the reciprocal of the fine-structure constant, 1/α. Hence any observed variation over space or time in the value of α is attributed to variation in the speed of light.

Natural Units (Particle Physics)

Quantity	Expression	Metric Value
Length (L)	$l_{nu} = \frac{\hbar c}{m_e c^2}$	3.86159×10^-13 m
Mass (M)	$m_{nu} = m_e \$	9.10938×10^-31 kg
Time (T)	$t_{nu} = \frac{\hbar}{m_e c^2}$	1.28809×10^-21 s
Electric charge (Q)	$q_{nu} = e \$	1.60218×10^-19 C
Temperature (Θ)	$T_{nu} = \frac{m_e c^2}{k_B}$	5.92989×10⁹ K

$c = e = m_e = \frac{\hbar}{m_e c^2} = 1$

These are used in high-energy particle physics and referred to in the SI handbook; however, they are not officially endorsed.^[4]

Also known by the abbreviation "n.u.".

Electronic system of units

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)	$T_e = \frac{m_e c^2}{k_B}$

$c = e = m_e = \frac{1}{4 \pi \epsilon_0} = k_B = 1 \$

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

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

$G \leftarrow \alpha G \left( \frac{m_P}{m_e} \right)^2 = \frac{e^2}{4 \pi \epsilon_0 m_e^2} \$ .

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

$\hbar \leftarrow \alpha \hbar = \frac{e^2}{c (4 \pi \epsilon_0)}$ .

As is the case with Stoney units, any observed variation over space or time in the value of α is attributed to variation in Planck's constant.

Quantum chromodynamical system of units (Strong)

Quantity	Expression
Length (L)	$l_{\mathrm{QCD}} = \frac{\hbar}{m_p c}$
Mass (M)	$m_{\mathrm{QCD}} = m_p \$
Time (T)	$t_{\mathrm{QCD}} = \frac{\hbar}{m_p c^2}$
Electric charge (Q)	$q_{\mathrm{QCD}} = e \$
Temperature (Θ)	$T_{\mathrm{QCD}} = \frac{m_p c^2}{k_B}$

$c = e = m_p = \hbar = k_B = 1 \$

$\frac{1}{4 \pi \epsilon_0} = \alpha$

The electron mass is replaced with that of the proton and the permittivity of free space is not fixed by definition. 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^[5]. In QCD, any observed variation over space or time in the value of α is attributed to variation in $\epsilon_0 \$ .

Geometrized units

Main article: Geometrized unit system

$c = 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_B = 1 \$

$\frac{1}{4 \pi \epsilon_0} = 1$

If the reduced Planck constant is also set equal to unity,

$\hbar = 1 \$

then geometrized units are identical to Planck units.

N-body units

Quantity	Expression
Length (R)	$\frac{1}{R} = \frac{1}{N(N-1)} \sum_{i=1}^{N} \sum_{j=1}^{N} \frac{1}{r_j-r_i}$
Mass (M)	$M = \sum_{i=1}^{N} m_i$

$M = G = R = 1 \$

N-body units are a completely self-contained system of units used for N-body simulations of self gravitating systems in astrophysics. In this system, the base physical units are chosen so that the total mass (M), the gravitational constant (G) and the virial radius (R) are set equal to unity. The underlying assumption is that the system of N objects (stars) satisfies the virial theorem. The consequence of standard N-body units is that the velocity dispersion of the system is $\scriptstyle v = 1/\sqrt{2}$ and that the dynamical -crossing- time scales as $\scriptstyle t = 2\sqrt{2}$ . The first mention of standard N-body units was by Michel Hénon (1971) [1]. They were taken up by Haldan Cohn (1979) [2] and later widely advertised and generalized by Douglas Heggie and Robert Mathieu (1986). [3]

References

^ Kowalski, Ludwik, 1986, "A Short History of the SI Units in Electricity," The Physics Teacher 24(2): 97-99. Alternate web link (subscription required)
^ Ray, T.P. (1981). "Stoney's Fundamental Units". Irish Astronomical Journal 15: 152.
^ ^a ^b ^c Duff, M. J., "Comment on time-variation of fundamental constants"
^ SI brochure, Table 7 (Section 4.1)
^ Wilczek, Frank, 2007, "Fundamental Constants," Frank Wilczek web site.

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.

v • d • e Systems of measurement

Metric systems	International System of Units · metre-kilogram-second · centimetre-gram-second · metre-tonne-second · gravitational system

Natural units	Geometric · Planck · Stoney · Lorentz–Heaviside · "Schrödinger" · Atomic · Electronic · Quantum chromodynamical · N-body

Conventional systems	Astronomical · Electrical · Temperature

Customary systems	Avoirdupois · Apothecaries' · Canadian · Chinese · Danish · Dutch · English · Finnish · French · German · Hindu · Hong Kong · Imperial · Irish · Japanese · Maltese · Norwegian · Pegu · Polish · Romanian · Russian · Scottish · Spanish and Portuguese · Swedish · Taiwanese Tatar · Troy · US ·

Ancient systems	Greek · Roman · Egyptian · Hebrew · Arabic · Mesopotamian · Persian · Indian

Other systems	Non-standard · Mesures usuelles

v • d • e Planck's natural 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

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