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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, 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 dimensionless 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 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.

Contents

1 Introduction

1.1 Summary table

2 Notation and use

2.1 Advantages and disadvantages

3 Choosing constants to normalize 4 Electromagnetism units 5 Systems of natural units

5.1 Planck units 5.2 Stoney units 5.3 Atomic units 5.4 Quantum chromodynamics
Quantum chromodynamics
(QCD) units 5.5 "Natural units" (particle physics and cosmology) 5.6 Geometrized units

6 See also 7 Notes and references 8 External links

Introduction[edit] Natural units
Natural units
are intended to elegantly simplify particular algebraic expressions 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
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. 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 kB to 1, which can be thought of as simply a way of defining the unit 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 Lorentz–Heaviside units (also called "rationalized"), Coulomb's law is F = q1q2/4πr2, and in Gaussian units
Gaussian units
(also called "non-rationalized"), Coulomb's law
Coulomb's law
is F = q1q2/r2.[1] Both possibilities are incorporated into different natural unit systems. Summary table[edit]

Quantity / Symbol Planck (with Gauss) Stoney Hartree Rydberg "Natural" (with L-H) "Natural" (with Gauss)

Speed
Speed
of light

c

displaystyle c,

1

displaystyle 1,

1

displaystyle 1,

1 α

 

displaystyle frac 1 alpha

2 α

 

displaystyle frac 2 alpha

1

displaystyle 1,

1

displaystyle 1,

Planck constant
Planck constant
(reduced)

ℏ =

h

2 π

displaystyle hbar = frac h 2pi

1

displaystyle 1,

1 α

 

displaystyle frac 1 alpha

1

displaystyle 1,

1

displaystyle 1,

1

displaystyle 1,

1

displaystyle 1,

Elementary charge

e

displaystyle e,

α

displaystyle sqrt alpha ,

1

displaystyle 1,

1

displaystyle 1,

2

displaystyle sqrt 2 ,

4 π α

displaystyle sqrt 4pi alpha

α

displaystyle sqrt alpha

Josephson constant

K

J

=

e

π ℏ

displaystyle K_ text J = frac e pi hbar ,

α

π

displaystyle frac sqrt alpha pi ,

α π

displaystyle frac alpha pi ,

1 π

displaystyle frac 1 pi ,

2

π

displaystyle frac sqrt 2 pi ,

4 α

π

displaystyle sqrt frac 4alpha pi ,

α

π

displaystyle frac sqrt alpha pi ,

von Klitzing constant

R

K

=

h

e

2

displaystyle R_ text K = frac h e^ 2 ,

2 π

α

displaystyle frac 2pi alpha ,

2 π

α

displaystyle frac 2pi alpha ,

2 π

displaystyle 2pi ,

π

displaystyle pi ,

1

2 α

displaystyle frac 1 2alpha

2 π

α

displaystyle frac 2pi alpha

Gravitational constant

G

displaystyle G,

1

displaystyle 1,

1

displaystyle 1,

α

G

α

displaystyle frac alpha _ text G alpha ,

8

α

G

α

displaystyle frac 8alpha _ text G alpha ,

α

G

m

e

2

displaystyle frac alpha _ text G m_ text e ^ 2 ,

α

G

m

e

2

displaystyle frac alpha _ text G m_ text e ^ 2 ,

Boltzmann constant

k

B

displaystyle k_ text B ,

1

displaystyle 1,

1

displaystyle 1,

1

displaystyle 1,

1

displaystyle 1,

1

displaystyle 1,

1

displaystyle 1,

Electron rest mass

m

e

displaystyle m_ text e ,

α

G

displaystyle sqrt alpha _ text G ,

α

G

α

displaystyle sqrt frac alpha _ text G alpha ,

1

displaystyle 1,

1 2

displaystyle frac 1 2 ,

511

 keV

displaystyle 511 text keV

511

 keV

displaystyle 511 text keV

where:

α is the fine-structure constant, (e/qPlanck)2 ≈ 0.007297, αG is the gravitational coupling constant, (me/mPlanck)2 ≈ 6955175200000000000♠1.752×10−45.

Notation and use[edit] Natural units
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 as v = c/2 and c = 1, hence v = 1/2. The equation v = 1/2 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 energy of a particle, measured in Planck units of energy, equals the mass of the particle, measured in Planck units of mass." Advantages and disadvantages[edit] 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 subsume 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 distance at which 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
Prototype
Kilogram, a physical cylinder of metal 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
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, in order to retain the same degree of precision, the fundamental constants used still have to be measured in a laboratory in terms of physical objects that can be directly observed. If this is not possible, then a quantity expressed in natural units can be less precise than the same quantity expressed in SI units. For example, Planck units use the gravitational constant G, which is measurable in a laboratory only to four significant digits.

Choosing constants to normalize[edit] 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 fine-structure constant, α ≈ 1/137, cannot be set to 1, at least not independently, because it is a dimensionless number defined in terms of other quantities, some of which one may want to set to unity as well. The fine-structure constant is related to other fundamental constants through α = kee2/ħc, where ke is the Coulomb
Coulomb
constant, e is the elementary charge, ħ is the reduced Planck constant, and c is the speed of light. Electromagnetism units[edit] Main articles: Lorentz–Heaviside units and Gaussian units 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 natural unit systems, however, electric charge has units of [mass]​1⁄2 [length]​3⁄2 [time]−1. In order to build natural units in electromagnetism one can use:

Lorentz–Heaviside units (classified as a rationalized system of electromagnetism units). Gaussian units
Gaussian units
(classified as a non-rationalized system of electromagnetism units).

Of these, Lorentz–Heaviside is somewhat more common,[2] mainly because Maxwell's equations
Maxwell's equations
are simpler in Lorentz–Heaviside units than they are in Gaussian units. In the two unit systems, the elementary charge e satisfies:

e = √4παħc (Lorentz–Heaviside), e = √αħc (Gaussian)

where ħ is the reduced Planck constant, c is the speed of light, and α ≈ 1/137 is the fine-structure constant. In a natural unit system where c = 1, Lorentz–Heaviside units can be derived from SI units
SI units
by setting ε0 = μ0 = 1. Gaussian units
Gaussian units
can be derived from SI units
SI units
by a more complicated set of transformations, such as multiplying all electric fields by (4πε0)−​1⁄2, multiplying all magnetic susceptibilities by 4π, and so on.[3] Systems of natural units[edit] Planck units[edit] Main article: Planck units

Quantity Expression Metric value Name

Length
Length
(L)

l

P

=

ℏ G

c

3

displaystyle l_ text P = sqrt hbar G over c^ 3

6965161600000000000♠1.616×10−35 m Planck length

Mass
Mass
(M)

m

P

=

ℏ c

G

displaystyle m_ text P = sqrt hbar c over G

6992217600000000000♠2.176×10−8 kg Planck mass

Time
Time
(T)

t

P

=

ℏ G

c

5

displaystyle t_ text P = sqrt hbar G over c^ 5

6956539120000000000♠5.3912×10−44 s Planck time

Temperature
Temperature
(Θ)

T

P

=

c

5

G

k

B

2

displaystyle T_ text P = sqrt frac hbar c^ 5 G k_ text B ^ 2

7032141700000000000♠1.417×1032 K Planck temperature

Electric charge
Electric charge
(Q)

q

P

=

ℏ c

k

e

displaystyle q_ text P = sqrt hbar c over k_ text e

6982187600000000000♠1.876×10−18 C Planck charge

Planck units are defined by

c = ħ = G = ke = kB = 1,

where c is the speed of light, ħ is the reduced Planck constant, G is the gravitational constant, ke is the Coulomb
Coulomb
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. 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.

Stoney units[edit] Main article: Stoney units

Quantity Expression Metric value

Length
Length
(L)

l

S

=

G

k

e

e

2

c

4

displaystyle l_ text S = sqrt frac Gk_ text e e^ 2 c^ 4

6964138100000000000♠1.381×10−36 m

Mass
Mass
(M)

m

S

=

k

e

e

2

G

displaystyle m_ text S = sqrt frac k_ text e e^ 2 G

6991185900000000000♠1.859×10−9 kg

Time
Time
(T)

t

S

=

G

k

e

e

2

c

6

displaystyle t_ text S = sqrt frac Gk_ text e e^ 2 c^ 6

6955460500000000000♠4.605×10−45 s

Temperature
Temperature
(Θ)

T

S

=

c

4

k

e

e

2

G

k

B

2

displaystyle T_ text S = sqrt frac c^ 4 k_ text e e^ 2 G k_ text B ^ 2

7031121000000000000♠1.210×1031 K

Electric charge
Electric charge
(Q)

q

S

= e  

displaystyle q_ text S =e

6981160200000000000♠1.602×10−19 C

Stoney units are defined by:

c = G = ke = e = kB = 1,

where c is the speed of light, G is the gravitational constant, ke is the Coulomb
Coulomb
constant, e is the elementary charge, and kB is the Boltzmann constant. George Johnstone Stoney
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.[4] Stoney units differ from Planck units by fixing the elementary charge at 1, instead of the Planck constant
Planck 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[edit] Main article: Atomic units

Quantity Expression (Hartree atomic units) Metric value (Hartree atomic units)

Length
Length
(L)

l

A

=

2

( 4 π

ϵ

0

)

m

e

e

2

displaystyle l_ text A = frac hbar ^ 2 (4pi epsilon _ 0 ) m_ text e e^ 2

6989529200000000000♠5.292×10−11 m

Mass
Mass
(M)

m

A

=

m

e

 

displaystyle m_ text A =m_ text e

6969910900000000000♠9.109×10−31 kg

Time
Time
(T)

t

A

=

3

( 4 π

ϵ

0

)

2

m

e

e

4

displaystyle t_ text A = frac hbar ^ 3 (4pi epsilon _ 0 )^ 2 m_ text e e^ 4

6983241900000000000♠2.419×10−17 s

Temperature
Temperature
(Θ)

T

A

=

m

e

e

4

2

( 4 π

ϵ

0

)

2

k

B

displaystyle T_ text A = frac m_ text e e^ 4 hbar ^ 2 (4pi epsilon _ 0 )^ 2 k_ text B

7005315800000000000♠3.158×105 K

Electric charge
Electric charge
(Q)

q

A

= e  

displaystyle q_ text A =e

6981160200000000000♠1.602×10−19 C

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

e = me = ħ = ke = kB = 1 c = 1/α

Rydberg atomic units:[5]

e/√2 = 2me = ħ = ke = kB = 1 c = 2/α

Coulomb's constant is generally expressed as

ke = 1/4πε0.

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
Bohr model
of the hydrogen atom, an electron in the ground state has orbital velocity = 1, orbital radius = 1, angular momentum = 1, ionization energy = 1/2, 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
Coulomb
force. The unit length, lA, is the Bohr radius, a0. The values of c and e shown above imply that e = √αħc, as in Gaussian units, not Lorentz–Heaviside units.[6] However, hybrids of the Gaussian and Lorentz–Heaviside units are sometimes used, leading to inconsistent conventions for magnetism-related units.[7]

Quantum chromodynamics
Quantum chromodynamics
(QCD) units[edit]

Quantity Expression Metric value

Length
Length
(L)

l

Q C D

=

m

p

c

displaystyle l_ mathrm QCD = frac hbar m_ text p c

6984210300000000000♠2.103×10−16 m

Mass
Mass
(M)

m

Q C D

=

m

p

 

displaystyle m_ mathrm QCD =m_ text p

6973167300000000000♠1.673×10−27 kg

Time
Time
(T)

t

Q C D

=

m

p

c

2

displaystyle t_ mathrm QCD = frac hbar m_ text p c^ 2

6975701500000000000♠7.015×10−25 s

Temperature
Temperature
(Θ)

T

Q C D

=

m

p

c

2

k

B

displaystyle T_ mathrm QCD = frac m_ text p c^ 2 k_ text B

7013108900000000000♠1.089×1013 K

Electric charge
Electric charge
(Q)

q

Q C D

=

e

4 π α

displaystyle q_ mathrm QCD = frac e sqrt 4pi alpha

(L–H) 6981529100000000000♠5.291×10−19 C

q

Q C D

=

e

α

displaystyle q_ mathrm QCD = frac e sqrt alpha

(G) 6982187600000000000♠1.876×10−18 C

c = mp = ħ = kB = 1

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".[8]

"Natural units" (particle physics and cosmology)[edit]

Unit Metric value Derivation

1 eV−1 of length 6993196999999999999♠1.97×10−7 m

=

ℏ c

1

eV

displaystyle = frac hbar c 1, text eV

1 eV of mass 6964178000000000000♠1.78×10−36 kg

=

1

eV

c

2

displaystyle = frac 1, text eV c^ 2

1 eV−1 of time 6984658000000000000♠6.58×10−16 s

=

1

eV

displaystyle = frac hbar 1, text eV

1 eV of temperature 7004116000000000000♠1.16×104 K

=

1

eV

k

B

2 f

displaystyle = frac 1, text eV k_ text B cdot frac 2 f

with

f = 2

displaystyle f=2

1 unit of electric charge (L–H) 6981529000000000000♠5.29×10−19 C

=

e

4 π α

displaystyle = frac e sqrt 4pi alpha

1 unit of electric charge (G) 6982188000000000000♠1.88×10−18 C

=

e

α

displaystyle = frac e sqrt alpha

In particle physics and cosmology, the phrase "natural units" generally means:[9][10]

ħ = c = kB = 1.

where ħ is the reduced Planck constant, c is the speed of light, and kB 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 Lorentz–Heaviside 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:[10]

1.0 cm = 1.0 cm/ħc ≈ 51000 eV−1

Geometrized units[edit] Main article: Geometrized unit system

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. Planck units and Stoney units are examples of geometrized unit systems. See also[edit]

Anthropic units Dimensional analysis Dimensionless physical constant SI base unit N-body units Physical constant Units of measurement

Notes and references[edit]

^ Kowalski, Ludwik, 1986, "A Short History of the SI Units in Electricity, Archived 2009-04-29 at the Wayback Machine." The Physics Teacher 24(2): 97–99. Alternate web link (subscription required) ^ Walter Greiner; Ludwig Neise; Horst Stöcker (1995). Thermodynamics and Statistical Mechanics. Springer-Verlag. p. 385. ISBN 978-0-387-94299-5.  ^ See Gaussian units#General rules to translate a formula
Gaussian units#General rules to translate a formula
and references therein. ^ Ray, T.P. (1981). "Stoney's Fundamental Units". Irish Astronomical Journal. 15: 152. Bibcode:1981IrAJ...15..152R.  ^ Turek, Ilja (1997). Electronic structure of disordered alloys, surfaces and interfaces (illustrated ed.). Springer. p. 3. ISBN 978-0-7923-9798-4.  ^ Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science, by Markus Reiher, Alexander Wolf, p7 [books.google.com/books?id=YwSpxCfsNsEC&pg=PA7 link] ^ A note on units lecture notes. See the atomic units article for further discussion. ^ Wilczek, Frank, 2007, "Fundamental Constants," Frank Wilczek web site. ^ Gauge field theories: an introduction with applications, by Guidry, Appendix A ^ a b An introduction to cosmology and particle physics, by Domínguez-Tenreiro and Quirós, p422

External links[edit]

The NIST
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. Pedagogic Aides to Quantum Field Theory Click on the link for Chap. 2 to find an extensive, simplified introduction to natural units.

v t e

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Metric

Overview Introduction Outline History Metrication

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Overview Comparison Foot–pound–second (FPS)

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Metric

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v t e

Planck's natural units

Planck constant Planck 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 Planck area Planck volume Planck acceleration

v t e

SI units

Authority: International System of Units
International System of Units
(BIPM)

Base units

ampere candela kelvin kilogram metre mole second

Derived units with special names

becquerel coulomb degree Celsius farad gray henry hertz joule katal lumen lux newton ohm pascal radian siemens sievert steradian tesla volt watt weber

Other accepted units

astronomical unit bar dalton day decibel degree of arc electronvolt hectare hour litre minute minute of arc neper second of arc tonne atomic units natural units

See also

Conversion of units Metric prefixes Proposed redefinitions Systems of measurement

Bo

.