Natural units

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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, or the speed of light c is a natural unit of speed. A purely natural system of units is defined in such a way that some set of selected universal physical constants are normalized to unity; that is, their numerical values in terms of these units become 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 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). 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 the SI unit system, 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=q1q2/(4πr2), and in Gaussian units (also called "non-rationalized"), Coulomb's law is F=q1q2/r2.[1] Both possibilities are incorporated into different natural unit systems.

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 from the equations v = 12c and c = 1, the consequence is v = 12. The equation v = 12 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 = mc2 can be rewritten in Planck units as E = m. This equation means "The rest-energy of a particle, measured in Planck units of energy, equals the rest-mass of a 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 E2 = p2c2 + m2c4 appears somewhat complicated, but the natural units version, E2 = p2 + m2, appears simpler.
  • Physical interpretation: Natural unit systems automatically incorporate 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 length where 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 certain cylinder 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, a quantity measured in natural units can have fewer digits of precision than the same quantity measured in SI. For example, Planck units use the gravitational constant G, which is measurable in a laboratory only to four significant digits.
  • Greater ambiguity: Consider the equation a = 1010 in Planck units. If a represents a length, then the equation means a = 1.6×10−25 m. If a represents a mass, then the equation means a = 220 kg. Therefore, if the variable a was not clearly defined, then the equation a = 1010 might be misinterpreted. By contrast, in SI units, the equation would be a = 220 kg, and it would be automatically clear that a represents a mass, not a length or anything else. In fact, natural units are especially useful when this ambiguity is deliberate: For example, in special relativity space and time are so closely related that it can be useful to not specify whether a variable represents a distance or a time.

Choosing constants to normalize

Out of the many physical constants, natural unit systems choose a few 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 ≈1836. In a less trivial example, the fine-structure constant, α≈1/137, cannot be set to 1, because it is a dimensionless number. The fine-structure constant is related to other fundamental constants

\alpha = \frac{k_\mathrm{e} e^2}{\hbar c},

where ke is the Coulomb constant, e is the elementary charge, ℏ is the reduced Planck constant, and c is the speed of light. Therefore it is not possible to simultaneously normalize all four of the constants c, ℏ, e, and ke.

Electromagnetism units

For any natural unit system, electromagnetism units are treated in one of two ways:

In Lorentz–Heaviside units, there are factors of 4\pi in Coulomb's law and the Biot–Savart law but not in Maxwell's equations; In Gaussian units, it is the reverse. Both systems are used, although Heaviside-Lorentz is more common.[2] In either unit system, electric charge is expressible in terms of the "mechanical" units (mass, length, time). In fact, the elementary charge e satisfies:

  • e = \sqrt{4 \pi \alpha \hbar c} (Lorentz–Heaviside),
  • e = \sqrt{\alpha \hbar c} (Gaussian)

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

Electromagnetism units are more complicated than mechanical units because there are different forms of the electromagnetic equations themselves. For example, Newton's law is F = ma in any system of units. However Coulomb's law is F = q1q2/4πr2 in Lorentz–Heaviside units, but F = q1q2/r2 in Gaussian units. Additionally, Maxwell's equations in CGS units (Gaussian or Lorentz-Heaviside) cannot be derived from the equivalent SI equations merely by normalizing some constants: the constant c appears explicitly despite normalizing ε0 and μ0.

Systems of natural units

Planck units

Quantity Expression Metric value Name
Length (L) l_P = \sqrt{\frac{\hbar G}{c^3}} 1.616×10−35 m Planck length
Mass (M) m_P = \sqrt{\frac{\hbar c}{G}} 2.176×10−8 kg Planck mass
Time (T) t_P = \sqrt{\frac{\hbar G}{c^5}} 5.3912×10−44 s Planck time
Temperature (Θ) T_P = \sqrt{\frac{\hbar c^5}{G {k_B}^2}} 1.417×1032 K Planck temperature
Electric charge (Q) q_P = e/\sqrt{4\pi\alpha} (L–H) 5.291×10−18 C
q_P = e/\sqrt{\alpha} (G) 1.876×10−18 C

Planck units are defined by

 c =  G =  \hbar =  k_B = 1 \

where c is the speed of light, G is the gravitational constant, \hbar is the reduced Planck constant, and kB 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.

Some may consider Planck units to be "more natural" even than other natural unit systems discussed below. For example, some other systems use the mass of an electron as a parameter to be normalized. But the electron is just one of 15 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.

Like the other systems (see above), the electromagnetism units in Planck units can be based on either Lorentz–Heaviside units or Gaussian units. The unit of charge is different in each.

"Natural units" (particle physics)

Unit Metric value Derivation
1 eV−1 of length 1.97×10−7 m =(1\text{eV}^{-1})\hbar c
1 eV of mass 1.78×10−36 kg = (1 \text{eV})/c^2
1 eV−1 of time 6.58×10−16 s =(1\text{eV}^{-1})\hbar
1 eV of temperature 1.16×104 K = 1 \text{eV}/k_B
1 unit of electric charge
(L–H)
5.29×10−19 C =e/\sqrt{4\pi\alpha}
1 unit of electric charge
(G)
1.88×10−19 C =e/\sqrt{\alpha}

In particle physics, the phrase "natural units" generally means:[3][4]

 \hbar = c = k_B = 1.

where \hbar is the reduced Planck constant, c is the speed of light, and kB is the Boltzmann constant.

Like the other systems (see above), the electromagnetism units in Planck 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. Most commonly, electron-volt (eV) is used, despite the fact that this is not a "natural" unit in the sense discussed above. (The SI prefixed multiples of eV are used as well: keV, MeV, GeV, etc.)

With the addition of eV (or any other auxiliary unit), any quantity can be expressed. For example, a distance of 1 cm can be expressed in terms of eV, in natural units, as:[4]

1\, \text{cm} = \frac{1\, \text{cm}}{\hbar c} \approx 51000\, \text{eV}^{-1}

Stoney units

Quantity Expression Metric value
Length (L) l_S = \sqrt{\frac{G e^2}{c^4 (4 \pi \epsilon_0)}} 1.381×10−36 m
Mass (M) m_S = \sqrt{\frac{e^2}{G (4 \pi \epsilon_0)}} 1.859×10−9 kg
Time (T) t_S = \sqrt{\frac{G e^2}{c^6 (4 \pi \epsilon_0)}} 4.605×10−45 s
Temperature (Θ) T_S = \sqrt{\frac{c^4 e^2}{G (4 \pi \epsilon_0) {k_B}^2}} 1.210×1031 K
Electric charge (Q) q_S = e \ 1.602×10−19 C

Stoney units are defined by:

 c =  G = e = k_B = 1 \
 \hbar = \frac{1}{\alpha} \

where c is the speed of light, G is the gravitational constant, e is the elementary charge, kB is the Boltzmann constant, \hbar is the reduced Planck constant, and α is the fine-structure 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.[5] Stoney units differ from Planck units by fixing the elementary charge at 1, instead of Planck's 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
(Hartree atomic units)
Metric value
(Hartree atomic units)
Length (L) l_A = \frac{\hbar^2 (4 \pi \epsilon_0)}{m_e e^2} 5.292×10−11 m
Mass (M) m_A = m_e \ 9.109×10−31 kg
Time (T) t_A = \frac{\hbar^3 (4 \pi \epsilon_0)^2}{m_e e^4} 2.419×10−17 s
Electric charge (Q) q_A = e \ 1.602×10−19 C
Temperature (Θ) T_A = \frac{m_e e^4}{\hbar^2 (4 \pi \epsilon_0)^2 k_B} 3.158×105 K

There are two types of atomic units, closely related: Hartree atomic units:

 e = m_e = \hbar = k_B = 1 \
 c = \frac{1}{\alpha} \

Rydberg atomic units:[6]

 \frac{e}{\sqrt{2}} = 2m_e = \hbar = k_B = 1 \
 c = \frac{2}{\alpha} \

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 = ½, 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, mA, is the Bohr radius, a0.

The values of c and e shown above imply that e = \sqrt{\alpha \hbar c}, as in Gaussian units, not Lorentz–Heaviside units.[7] However, hybrids of the Gaussian and Lorentz–Heaviside units are sometimes used, leading to inconsistent conventions for magnetism-related units.[8]

Quantum chromodynamics (QCD) system of units

Quantity Expression Metric value
Length (L) l_{\mathrm{QCD}} = \frac{\hbar}{m_p c} 2.103 × 10−16 m
Mass (M) m_{\mathrm{QCD}} = m_p \ 1.673 × 10−27 kg
Time (T) t_{\mathrm{QCD}} = \frac{\hbar}{m_p c^2} 7.015 × 10−25 s
Temperature (Θ) T_{\mathrm{QCD}} = \frac{m_p c^2}{k_B} 1.089 × 1013 K
Electric charge (Q) q_P = e/\sqrt{4\pi\alpha} (L–H) 5.291×10−18 C
q_P = e/\sqrt{\alpha} (G) 1.876×10−18 C
 c = m_p =  \hbar =  k_B = 1 \

The electron 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".[9]

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. By normalizing appropriate other units, geometrized units become identical to Planck units.

Summary table

Quantity / Symbol Planck
(with Gaussian)
Stoney Hartree Rydberg "Natural"
(with L-H)
"Natural"
(with Gaussian)
Speed of light in vacuum
c \,
1 \, 1 \, \frac{1}{\alpha} \ \frac{2}{\alpha} \ 1 \, 1 \,
Planck's constant (reduced)
\hbar=\frac{h}{2 \pi}
1 \, \frac{1}{\alpha} \ 1 \, 1 \, 1 \, 1 \,
Elementary charge
e \,
\sqrt{\alpha} \, 1 \, 1 \, \sqrt{2} \, \sqrt{4\pi\alpha} \sqrt{\alpha}
Josephson constant
K_J =\frac{e}{\pi \hbar} \,
\frac{\sqrt{\alpha}}{\pi} \, \frac{\alpha}{\pi} \, \frac{1}{\pi} \, \frac{\sqrt{2}}{\pi} \, \sqrt{\frac{4\alpha}{\pi}} \, \frac{\sqrt{\alpha}}{\pi} \,
von Klitzing constant
R_K =\frac{2 \pi \hbar}{e^2} \,
\frac{2\pi}{\alpha} \, \frac{2\pi}{\alpha} \, 2\pi \, \pi \, \frac{1}{2\alpha} \frac{2 \pi}{\alpha}
Gravitational constant
G \,
1 \, 1 \, \frac{\alpha_G}{\alpha} \, \frac{8 \alpha_G}{\alpha} \, \frac{\alpha_G}{{m_e}^2} \, \frac{\alpha_G}{{m_e}^2} \,
Boltzmann constant
k_B \,
1 \, 1 \, 1 \, 1 \, 1 \, 1 \,
Electron mass
m_e \,
\sqrt{\alpha_G} \, \sqrt{\frac{\alpha_G}{\alpha}} \, 1 \, \frac{1}{2} \, 511\, \text{keV} 511\, \text{keV}

where:

See also

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. http://books.google.com/books?id=12DKsFtFTgYC&pg=PA385 Thermodynamics and statistical mechanics, by Greiner, Neise, Stöcker
  3. Gauge field theories: an introduction with applications, by Guidry, Appendix A
  4. 4.0 4.1 An introduction to cosmology and particle physics, by Domínguez-Tenreiro and Quirós, p422
  5. Ray, T.P. (1981). "Stoney's Fundamental Units". Irish Astronomical Journal. 15: 152. 
  6. Turek, Ilja (1997). Electronic structure of disordered alloys, surfaces and interfaces (illustrated ed.). Springer. p. 3. ISBN 978-0-7923-9798-4 
  7. Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science, by Markus Reiher, Alexander Wolf, p7 [books.google.com/books?id=YwSpxCfsNsEC&pg=PA7 link]
  8. A note on units lecture notes. See the atomic units article for further discussion.
  9. Wilczek, Frank, 2007, "Fundamental Constants," Frank Wilczek web site.

External links