Natural units: Wikis

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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.

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 kB 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., me and mp cannot both be defined as the unit mass in a single system).

Constant Symbol Dimension
Speed of light in vacuum $\displaystyle { c } = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \$ L T-1
Gravitational constant ${ G } \$ M-1 L3 T-2
Planck constant (reduced) $\hbar = \frac{h}{2 \pi}$ M L2 T-1
Coulomb constant $\displaystyle k_\mathrm{e} = \frac{1}{4 \pi \epsilon_0} = \frac{\mu_0 c^2}{4 \pi} \$ Q-2 M L3 T-2
Electric constant $\displaystyle \epsilon_0 = \frac{1}{\mu_0 c^2} \$ Q2 M-1 L-3 T2
Magnetic constant $\displaystyle \mu_0 = \frac{1}{\epsilon_0 c^2} \$ Q-2 M L
Characteristic impedance $\displaystyle Z_0 = \mu_0 c = \frac{1}{\epsilon_0 c} \$ Q-2 M L2 T-1
Elementary charge ${ e } \$ Q
Electron mass ${ m_e } \$ M
Proton mass ${ m_p } \$ M
Boltzmann constant ${ k_B } \$ M L2 T-2 Θ-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:

$\alpha \ \equiv \frac{e^2}{\hbar c (4 \pi \epsilon_0)} = \frac{1}{137.035999679} = 7.2973525376 \cdot 10^{-3}$

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

Quantity Expression Metric value
Length (L) $l_P = \sqrt{\frac{\hbar G}{c^3}}$ 1.616252×10-35 m
Mass (M) $m_P = \sqrt{\frac{\hbar c}{G}}$ 2.17644(11)×10-8 kg
Time (T) $t_P = \sqrt{\frac{\hbar G}{c^5}}$ 5.39124 ×10-44 s
Electric charge (Q) $q_P = \sqrt{\hbar c (4 \pi \epsilon_0)}$ 1.87554573×10-18 C
Temperature (Θ) $T_P = \sqrt{\frac{\hbar c^5}{G {k_B}^2}}$ 1.416785×1032 K
$c = G = \hbar = \frac{1}{4 \pi \epsilon_0} = k_B = 1 \$
$e = \sqrt{\alpha} \$

Planck units are unique among systems of natural units, because they are not defined in terms of properties of any prototype, physical object, or elementary particle (specifically, the charge, mass, spin, or mean orbital radius). For example, the proton and electron may be considered equally sensible choices for a prototype object, 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 refer to 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 when using Planck units.

Heaviside-Lorentz 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

$\alpha = \frac{e^{2}}{4\pi}$

and defines a unit 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.

Stoney 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.

"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)

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, Eh. The unit velocity is the velocity of that electron, the unit mass is the electron mass, me, 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.

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.

"Natural Units" (Particle Physics)

Quantity Expression Metric Value
Length (L) $l_{nu} = \frac{\hbar}{m_e c}$ 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 = \hbar = k_B = 1 \$
$\frac{1}{4 \pi \epsilon_0} = \alpha$

These are used in high-energy particle physics and referred to in the SI handbook; however, they are not officially endorsed.[4] Unlike the previous systems (or SI), the permittivity of free space is not fixed by definition.

These are also known by the abbreviation "n.u.". In n.u., any observed variation over space or time in the value of α is attributed to variation in the vacuum permittivity $\epsilon_0 \$.

Quantum chromodynamics (QCD) system of units

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 the vacuum permittivity $\epsilon_0 \$.

Geometrized units

$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.

Summary table

Quantity / Symbol Planck Heaviside-Lorentz** Stoney Schrödinger Atomic Electronic "Natural Units" QCD
Speed of light in vacuum
$c \,$
$1 \,$ $1 \,$ $1 \,$ $\frac{1}{\alpha} \$ $\frac{1}{\alpha} \$ $1 \,$ $1 \,$ $1 \,$
Electric constant (vacuum permittivity)
$\epsilon_0 \,$
$\frac{1}{4 \pi} \,$ $1 \,$ $\frac{1}{4 \pi} \,$ $\frac{1}{4 \pi} \,$ $\frac{1}{4 \pi} \,$ $\frac{1}{4 \pi} \,$ $\frac{1}{4 \pi \alpha} \,$ $\frac{1}{4 \pi \alpha} \,$
Magnetic constant (vacuum permeability)
$\mu_0 = \frac{1}{\epsilon_0 c^2} \,$
$4 \pi \,$ $1 \,$ $4 \pi \,$ $4 \pi \alpha^2 \,$ $4 \pi \alpha^2 \,$ $4 \pi \,$ $4 \pi \alpha \,$ $4 \pi \alpha \,$
Characteristic impedance of vacuum
$Z_0 = \frac{1}{\epsilon_0 c} \,$
$4 \pi \,$ $1 \,$ $4 \pi \,$ $4 \pi \alpha \,$ $4 \pi \alpha \,$ $4 \pi \,$ $4 \pi \alpha \,$ $4 \pi \alpha \,$
Planck's constant (reduced)
$\hbar=\frac{h}{2 \pi}$
$1 \,$ $1 \,$ $\frac{1}{\alpha} \$ $1 \,$ $1 \,$ $\frac{1}{\alpha} \$ $1 \,$ $1 \,$
Elementary charge
$e \,$
$\sqrt{\alpha} = \frac{e}{q_P} \,$ $\sqrt{4 \pi \alpha \ } \,$ $1 \,$ $1 \,$ $1 \,$ $1 \,$ $1 \,$ $1 \,$
Josephson constant
$K_J =\frac{e}{\pi \hbar} \,$
$\frac{\sqrt{\alpha}}{\pi} \,$ $2\sqrt{\frac{\alpha}{\pi}} \,$ $\frac{\alpha}{\pi} \,$ $\frac{1}{\pi} \,$ $\frac{1}{\pi} \,$ $\frac{\alpha}{\pi} \,$ $\frac{1}{\pi} \,$ $\frac{1}{\pi} \,$
von Klitzing constant
$R_K =\frac{2 \pi \hbar}{e^2} \,$
$\frac{2\pi}{\alpha} \,$ $\frac{1}{2 \alpha} \,$ $\frac{2\pi}{\alpha} \,$ $2\pi \,$ $2\pi \,$ $\frac{2\pi}{\alpha} \,$ $2\pi \,$ $2\pi \,$
Gravitational constant
$G \,$
$1 \,$ $\frac{1}{4 \pi} \,$** $1 \,$ $1 \,$ $\alpha_G = \left(\frac{m_e}{m_P}\right)^2 \,$ $\alpha_G \,$ $\alpha_G \,$ $\mu^2 \alpha_G \,$
Electron mass
$m_e \,$
$\sqrt{\alpha_G} = \frac{m_e}{m_P} \,$ $\sqrt{4 \pi \alpha_G \ } \,$** $\sqrt{\frac{\alpha_G}{\alpha}} \,$ $\sqrt{\frac{\alpha_G}{\alpha}} \,$ $1 \,$ $1 \,$ $1 \,$ $\frac{1}{\mu} = \frac{m_e}{m_p} \,$
α is the fine-structure constant, about 7.2973525376 × 10−3 = (137.035999679)-1.
αG is the gravitational coupling constant, about 1.7518 × 10−45.
μ is the proton-to-electron mass ratio, about 1836.15267247.
**The Heaviside-Lorentz natural units do not commit to a unit mass. The values shown are such that the gravitational constant is rationalized in the same manner as the coulomb constant in Heaviside-Lorentz units.

Other "prototype"-less unit systems

In SI units, the kilogram has a particularly non-"natural" definition: The kilogram is defined as the mass of a certain block of metal stored in a vault in France. This type of definition, where a unit is defined by a prototype, is considered undesirable, because the prototype may change over time, and because an extremely precise measurement would require traveling to France. Better is a definition based on something in nature, which can be reproduced at any time and in any laboratory.

Natural unit systems fulfill this requirement: The units are based purely on constants of nature. (Of course, not all are suitable for precision measurement, for other reasons.)

Other unit systems besides natural units also fulfill this requirement. One is conventional electrical units, an offshoot of SI units which replaced some SI definitions with defined values for the Josephson constant and von Klitzing constant.

The SI unit system itself may fulfill this requirement someday, but does not yet, because of the kilogram. In fact, there used to be a prototype metre, but it was replaced in 1960 by a definition based on the wavelength of an atomic transition. Likewise, there is active research working towards finding a replacement definition of the kilogram not based on a prototype.

References

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

Simple English

Natural units are ways of measuring things that depend on some basic characteristics of nature that do not change. Which of these basic quantities to choose can depend on the physics problems being investigated, and sometimes choosing one thing as a natural unit means that the size of something else does not become used as a natural unit in that system.

The old system of English measurements such as the pound are based on convenient objects in the natural world. The "grain" is the smallest of these objects, and originally it meant the weight of a grain of wheat or barley. Each individual grain might be slightly larger or smaller than the next, but the more grains were added together to make a larger measure, the more these little differences would even out. Even so, these measure were not entirely precise and did not relate to other measures such as the inch or the foot.

In order to make calculations simpler and units more precise, it turned out that the mass of an electron or the mass of a proton were more useful standards to use for weight. All electrons are believed to have the same mass, and all protons are believed to have their own standard mass. But there is no simple mathematical relationship between the two masses.

The speed of light, c, is a constant. So c is a very natural choice to use as a standard for measurement.

One choice for a standard of length is the Bohr radius. The simplest atom, hydrogen, only has one electron, and its smallest possible orbit, that with the lowest energy, is at a distance from the nucleus called the Bohr radius.

With a standard for measuring distances and a standard for velocity, it would be possible to derive one standard unit of time. In practice, there are several ways of defining units of time. One of the most widely known ways to measure time by natural cycles is by using atomic clocks.