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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]] is defined in such a way that some set of selected universal physical constants are each normalized to unity; that is, their numerical values in terms of these units are exactly 1. While this has the advantage of simplicity, there is a potential disadvantage in terms of loss of clarity and understanding, as these constants are then omitted from mathematical expressions of physical laws.
| | Shares in LVMH, the world's biggest luxury goods group, fell sharply on Wednesday after an unexpected slowdown in sales growth at its fashion and leather business, which includes the Louis Vuitton, [http://www.pcs-systems.co.uk/Images/celinebag.aspx Celine Bag Online] and [http://photo.net/gallery/tag-search/search?query_string=Dior+brands Dior brands].<br>The share price was down 6.4 percent at 135.50 euros by 1140 GMT, a six-week low and wiping around 4.8 billion euros ($6.5 billion)off the market value of France's fourth-biggest listed company.<br>The group, which also owns Ruinart champagne and Hennessy cognac, saw sales growth at the fashion and leather division slide to 3 percent in the third quarter, against expectations of 7 to 8 percent.<br>In a conference call, Chief Financial Officer Jean-Jacques Guiony blamed price increases in Japan for the slowdown, as well as softer demand for some brands.<br>However, one London-based analyst noted that Japan accounted for only around 15 percent of LVMH's fashion and leather sales, "so we did not get a full explanation".<br>LVMH has been trying to stem a decline in Louis Vuitton's sales growth by introducing new and pricier leather bags, which analysts expected would lead to short-term losses in sales.<br>"I understand that the repositioning of Louis Vuitton takes time and may be a bumpy ride," said Exane BNP Paribas analyst Luca Solca.<br>Before the results were announced, LVMH shares were up 4.4 percent since Jan. 1, underperforming the overall luxury sector, which was up more than 20 percent.<br>Analysts said concerns about the future growth of Louis Vuitton had been exacerbated by the announcement this month that it was parting company with its star designer, Marc Jacobs.<br><br> |
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| ==Introduction==
| | SUPPLY CONSTRAINTS<br>Guiony said the recent launch of new leather collections had not come in time to have a meaningful impact on sales, and acknowledged that production was constrained by a lack of high-quality leather.<br>"Without these supply constraints, we would produce more than what we do," Guiony said.<br>Louis Vuitton shop assistants polled by Reuters last month said they had been provided with only a small number of new handbags, such as the Capucines model, priced at 3,500 euros, which had flown off the shelves.<br>LVMH has been buying tanneries to secure supplies but experts say the market is under pressure partly because the number of calves raised and slaughtered is driven more by demand for meat -- which has been in decline -- than by demand for quality hides.<br>In addition, China, the luxury industry's main driver since the late 2000s, has started to run out of steam in the last year due to an economic slowdown and a government crackdown on gift-giving.<br>Guiony said Vuitton's sales in mainland China were "flattish" but, thanks to sales to Chinese tourists, overall sales growth to Chinese customers was in the "mid-single digits plus".<br>Guiony said trends in watches and jewellery had slightly improved in China, but not in fashion and leather.<br>He said trading remained difficult in Europe, particularly for perfume and cosmetics, where sales were "flattish".<br>LVMH's overall sales grew 2 percent in the third quarter. Organic growth was 8 percent, of which 6 percentage points were accounted for by the relative weakness of the U.S. dollar and the Japanese yen against the euro. ($1=0.7406 euros) (Additional reporting by James Regan; Editing by James Jukwey and Kevin Liffey) |
| Natural units are intended to [[Nondimensionalization|elegantly simplify]] particular [[algebraic expression]]s appearing in the laws of physics or to [[Normalizing constant|normalize]] some chosen physical quantities that are properties of universal [[elementary particle]]s and are reasonably believed to be constant. However there is a choice of the set of natural units chosen, and quantities which are set to unity in one system may take a different value or even assumed to vary in another natural unit system.
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| 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 [[subatomic particle|particle]] but are solely derived from the properties of [[free space]].
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| 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 units|fundamental physical quantity]]{{citation needed|date=July 2013}}, since it 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''<sub>B</sub> to 1, which can be thought of as simply a way of defining the unit temperature.
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| 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 {{nowrap|1=''F'' = ''q''<sub>1</sub>''q''<sub>2</sub>/(4π''r''<sup>2</sup>)}}, and in [[Gaussian units]] (also called "non-rationalized"), Coulomb's law is {{nowrap|1=''F'' = ''q''<sub>1</sub>''q''<sub>2</sub>/''r''<sup>2</sup>}}.<ref>Kowalski, Ludwik, 1986, "[http://alpha.montclair.edu/~kowalskiL/SI/SI_PAGE.HTML A Short History of the SI Units in Electricity,]" ''The Physics Teacher'' 24(2): 97–99. [http://dx.doi.org/10.1119/1.2341955 Alternate web link (subscription required)]</ref> Both possibilities are incorporated into different natural unit systems.
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| ==Notation and use==
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| Natural units are most commonly used by ''setting the units to one''. For example, many natural unit systems include the equation {{nowrap|1=''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 {{nowrap|1=''v'' = {{sfrac|1|2}}''c''}} and {{nowrap|1=''c'' = 1}}, hence {{nowrap|1=''v'' = {{sfrac|1|2}}}}. The equation {{nowrap|1=''v'' = {{sfrac|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".
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| The equation ''{{nowrap|1=c'' = 1}} can be plugged in anywhere else. For example, Einstein's equation {{nowrap|1=''E'' = ''mc''<sup>2</sup>}} can be rewritten in Planck units as {{nowrap|1=''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."
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| === Advantages and disadvantages ===
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| {{procon|date=November 2012}}
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| Compared to [[SI]] or other unit systems, natural units have both advantages and disadvantages:
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| *'''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 [[invariant mass|special relativity]] equation {{nowrap|1=''E''<sup>2</sup> = ''p''<sup>2</sup>''c''<sup>2</sup> + ''m''<sup>2</sup>''c''<sup>4</sup>}} appears somewhat complicated, but the natural units version, {{nowrap|1=''E''<sup>2</sup> = ''p''<sup>2</sup> + ''m''<sup>2</sup>}}, appears simpler.
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| *'''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]].
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| *'''No prototypes:''' A [[Prototype#Metrology|prototype]] is a physical object that defines a unit, such as the [[kilogram|International 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]].)
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| *'''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, 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|gravitational constant G]], which is measurable in a laboratory only to four significant digits.
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| *'''Greater ambiguity:''' Consider the equation {{nowrap|1=''a'' = 10<sup>10</sup>}} in Planck units. If ''a'' represents a length, then the equation means {{nowrap|1=''a'' = {{val|1.6|e=-25|u=m}}}}. If ''a'' represents a mass, then the equation means {{nowrap|1=''a'' = {{val|220|u=kg}}}}. Therefore, if the variable ''a'' was not clearly defined, then the equation {{nowrap|1=''a'' = 10<sup>10</sup>}} might be misinterpreted. By contrast, in [[SI]] units, the equation would be (for example) ''a'' = 220 kg, and it would be 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 not to have to specify whether a variable represents a distance or a time.
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| ==Choosing constants to normalize==
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| Out of the many [[physical constant]]s, 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 ≈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
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| :<math>\alpha = \frac{k_\mathrm{e} e^2}{\hbar c},</math>
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| where ''k''<sub>e</sub> 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 ''k''<sub>e</sub>.
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| ==Electromagnetism units==
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| {{main|Lorentz–Heaviside units|Gaussian units}}
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| 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 {{nowrap|[mass]<sup>1/2</sup> [length]<sup>3/2</sup> [time]<sup>−1</sup>}}.
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| There are two main natural unit systems for electromagnetism:
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| *'''[[Lorentz–Heaviside units]]''' (classified as a '''rationalized''' system of electromagnetism units).
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| *'''[[Gaussian units]]''' (classified as a '''non-rationalized''' system of electromagnetism units).
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| Of these, Heaviside-Lorentz is somewhat more common,<ref name="GreinerNeise1995">{{cite book|author1=Walter Greiner|author2=Ludwig Neise|author3=Horst Stöcker|title=Thermodynamics and Statistical Mechanics|url=http://books.google.com/books?id=12DKsFtFTgYC&pg=PA385|year=1995|publisher=Springer-Verlag|isbn=978-0-387-94299-5|page=385}}</ref> mainly because [[Maxwell's equations]] are simpler in Lorentz-Heaviside units than they are in Gaussian units.
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| In the two unit systems, the [[elementary charge]] ''e'' satisfies:
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| *<math>e = \sqrt{4 \pi \alpha \hbar c}</math> (Lorentz–Heaviside),
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| *<math>e = \sqrt{\alpha \hbar c}</math> (Gaussian)
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| where ℏ is the [[reduced Planck constant]], ''c'' is the [[speed of light]], and α≈1/137 is the [[fine-structure constant]].
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| In a natural unit system where {{nowrap|1=[[speed of light|''c'']] = 1}}, Lorentz-Heaviside units can be derived from SI units by setting {{nowrap|[[electric constant|ε<sub>0</sub>]] = [[magnetic constant|μ<sub>0</sub>]] = 1}}. Gaussian units can be derived from SI units by a more complicated set of transformations, such as multiplying all [[electric field]]s by (4πε<sub>0</sub>)<sup>-1/2</sup>, multiplying all [[magnetic susceptibility|magnetic susceptibilities]] by 4π, and so on.<ref>See [[Gaussian units#General rules to translate a formula]] and references therein.</ref>
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| ==Systems of natural units==
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| ===Planck units===
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| {{Main|Planck units}}
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| {| class="wikitable" align="right" style="margin-left: 1em; background-color: #ffffff"
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| ! Quantity
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| ! Expression
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| ! Metric value
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| ! Name
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| |- align="left"
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| | [[Length]] (L)
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| | <math>l_P = \sqrt{\frac{\hbar G}{c^3}}</math>
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| | {{val|1.616|e=-35|u=m}}
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| | [[Planck length]]
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| |-
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| | [[Mass]] (M)
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| | <math>m_\text{P} = \sqrt{\frac{\hbar c}{G}}</math>
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| | {{val|2.176|e=-8|u=kg}}
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| | [[Planck mass]]
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| |-
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| | [[Time]] (T)
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| | <math>t_\text{P} = \sqrt{\frac{\hbar G}{c^5}} </math>
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| | {{val|5.3912|e=-44|u=s}}
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| | [[Planck time]]
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| |-
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| | [[Temperature]] (Θ)
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| | <math>T_\text{P} = \sqrt{\frac{\hbar c^5}{G {k_\text{B}}^2}}</math>
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| | {{val|1.417|e=32|u=K}}
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| | [[Planck temperature]]
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| |-
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| |rowspan=2| [[Electric charge]] (Q)
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| | <math>q_\text{P} = e/\sqrt{4\pi\alpha}</math> ([[Lorentz–Heaviside units|L–H]])
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| | {{val|5.291|e=-19|u=C}}
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| |-
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| | <math>q_\text{P} = e/\sqrt{\alpha}</math> ([[Gaussian units|G]])
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| | {{val|1.876|e=-18|u=C}}
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| |}
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| Planck units are defined by
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| :<math> c = G = \hbar = k_\text{B} = 1 \ </math>
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| where ''c'' is the [[speed of light]], ''G'' is the [[gravitational constant]], ℏ is the [[reduced Planck constant]], and ''k''<sub>B</sub> is the [[Boltzmann constant]].
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| 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]].
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| 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 particle]]s, 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.
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| 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.
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| {{-}}
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| ==="Natural units" (particle physics and cosmology)===
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| {| class="wikitable" align="right" style="margin-left: 1em; background-color: #ffffff"
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| ! Unit
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| ! Metric value
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| ! Derivation
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| |- align="left"
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| | 1 eV<sup>−1</sup> of length
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| | {{val|1.97|e=-7|u=m}}
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| | <math>=(1\text{eV}^{-1})\hbar c </math>
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| |-
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| | 1 eV of mass
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| | {{val|1.78|e=-36|u=kg}}
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| | <math>= (1 \text{eV})/c^2</math>
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| |-
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| | 1 eV<sup>−1</sup> of time
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| | {{val|6.58|e=-16|u=s}}
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| | <math>=(1\text{eV}^{-1})\hbar </math>
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| |-
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| | 1 eV of temperature
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| | {{val|1.16|e=4|u=K}}
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| | <math>= 1 \text{eV}/k_\text{B}</math>
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| |-
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| | 1 unit of electric charge <br> ([[Lorentz–Heaviside units|L–H]])
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| | {{val|5.29|e=-19|u=C}}
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| | <math>=e/\sqrt{4\pi\alpha} </math>
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| |-
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| | 1 unit of electric charge <br> ([[Gaussian units|G]])
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| | {{val|1.88|e=-18|u=C}}
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| | <math>=e/\sqrt{\alpha} </math>
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| |-
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| | 1 <math>(eV)^2</math> of magnetic field times electron charge
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| | {{val|1.68|e=-2|u=e Tesla}}
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| | <math>1~\mathrm{Tesla} = \mathrm{kg C}^{-1} \mathrm{s}^{-1}</math>
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| |}
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| In [[particle physics]] and [[cosmology]], the phrase "natural units" generally means:<ref>[http://books.google.com/books?id=kLYx_ZnanW4C&pg=PA511 ''Gauge field theories: an introduction with applications'', by Guidry, Appendix A]</ref><ref name=DT>[http://books.google.com/books?id=OF4TCxwpcN0C&pg=PA422 ''An introduction to cosmology and particle physics'', by Domínguez-Tenreiro and Quirós, p422]</ref>
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| :<math> \hbar = c = k_\text{B} = 1.</math>
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| where <math>\hbar</math> is the [[reduced Planck constant]], ''c'' is the [[speed of light]], and ''k''<sub>B</sub> is the [[Boltzmann constant]].
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| 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.
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| Finally, one more unit is needed. Most commonly, [[electron-volt|electron-volt (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 prefix]]ed multiples of eV are used as well: keV, MeV, GeV, etc.)
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| With the addition of eV (or any other auxiliary unit), any quantity can be expressed. For example, a distance of 1 [[centimeter|cm]] can be expressed in terms of eV, in natural units, as:<ref name=DT/>
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| :<math>1\, \text{cm} = \frac{1\, \text{cm}}{\hbar c} \approx 51000\, \text{eV}^{-1}</math>
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| {{-}}
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| ===Stoney units===
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| {{main|Stoney units}}
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| {| class="wikitable" align="right" style="margin-left: 1em; background-color: #ffffff"
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| ! Quantity
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| ! Expression
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| ! Metric value
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| |- align="left"
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| | [[Length]] (L)
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| | <math>l_\text{S} = \sqrt{\frac{G e^2}{c^4 (4 \pi \epsilon_0)}}</math>
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| | {{val|1.381|e=-36|u=m}}
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| |-
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| | [[Mass]] (M)
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| | <math>m_\text{S} = \sqrt{\frac{e^2}{G (4 \pi \epsilon_0)}}</math>
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| | {{val|1.859|e=-9|u=kg}}
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| |-
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| | [[Time]] (T)
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| | <math>t_\text{S} = \sqrt{\frac{G e^2}{c^6 (4 \pi \epsilon_0)}} </math>
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| | {{val|4.605|e=-45|u=s}}
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| |-
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| | [[Temperature]] (Θ)
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| | <math>T_\text{S} = \sqrt{\frac{c^4 e^2}{G (4 \pi \epsilon_0) {k_\text{B}}^2}}</math>
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| | {{val|1.210|e=31|u=K}}
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| |-
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| | [[Electric charge]] (Q)
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| | <math>q_\text{S} = e \ </math>
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| | {{val|1.602|e=-19|u=C}}
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| |}
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| Stoney units are defined by:
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| :<math> c = G = e = k_\text{B} = 1 \ </math>
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| :<math> \hbar = \frac{1}{\alpha} \ </math>
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| where ''c'' is the [[speed of light]], ''G'' is the [[gravitational constant]], ''e'' is the [[elementary charge]], ''k''<sub>B</sub> is the [[Boltzmann constant]], ℏ is the [[reduced Planck constant]], and α is the [[fine-structure constant]].
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| [[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.<ref>{{cite journal | last=Ray | first = T.P. | year=1981 | title=Stoney's Fundamental Units | journal=Irish Astronomical Journal | volume=15 | page=152|bibcode = 1981IrAJ...15..152R }}</ref> Stoney units differ from Planck units by fixing the [[elementary charge]] at 1, instead of [[Planck's constant]] (only discovered after Stoney's proposal).
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| Stoney units are rarely used in modern physics for calculations, but they are of historical interest.
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| {{-}}
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| ===Atomic units===
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| {{Main|Atomic units}}
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| {| class="wikitable" align="right" style="margin-left: 1em; background-color: #ffffff"
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| ! Quantity
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| ! Expression<br>(Hartree atomic units)
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| ! Metric value<br>(Hartree atomic units)
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| |- align="left"
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| | [[Length]] (L)
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| | <math>l_\text{A} = \frac{\hbar^2 (4 \pi \epsilon_0)}{m_\text{e} e^2}</math>
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| | {{val|5.292|e=-11|u=m}}
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| |-
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| | [[Mass]] (M)
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| | <math>m_\text{A} = m_\text{e} \ </math>
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| | {{val|9.109|e=-31|u=kg}}
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| |-
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| | [[Time]] (T)
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| | <math>t_\text{A} = \frac{\hbar^3 (4 \pi \epsilon_0)^2}{m_\text{e} e^4} </math>
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| | {{val|2.419|e=-17|u=s}}
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| |-
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| | [[Electric charge]] (Q)
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| | <math>q_\text{A} = e \ </math>
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| | {{val|1.602|e=-19|u=C}}
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| |-
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| | [[Temperature]] (Θ)
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| | <math>T_\text{A} = \frac{m_\text{e} e^4}{\hbar^2 (4 \pi \epsilon_0)^2 k_\text{B}}</math>
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| | {{val|3.158|e=5|u=K}}
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| |}
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| There are two types of atomic units, closely related.
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| '''Hartree atomic units''':
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| :<math> e = m_\text{e} = \hbar = 1 \ </math>
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| :<math> c = \frac{1}{\alpha} \ </math>
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| '''Rydberg atomic units''':<ref>{{Cite book
| |
| | last = Turek
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| | first = Ilja
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| | title = Electronic structure of disordered alloys, surfaces and interfaces
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| | publisher = Springer
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| | year = 1997
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| | edition = illustrated
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| | pages =3
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| | url = http://books.google.com/books?id=15wz64DPVqAC&pg=PA3
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| | isbn =978-0-7923-9798-4
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| | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}
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| }}</ref>
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| :<math> \frac{e}{\sqrt{2}} = 2m_\text{e} = \hbar = k_\text{B} = 1 \ </math>
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| :<math> c = \frac{2}{\alpha} \ </math>
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| 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.
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| 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]] = {{sfrac|2}}, etc.
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| 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<sup>−45</sup>), which comes from the fact that the gravitational force between two electrons is far weaker than the [[Coulomb force]]. The unit length, ''l''<sub>A</sub>, is the [[Bohr radius]], ''a''<sub>0</sub>.
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| The values of ''c'' and ''e'' shown above imply that ''e''=(αℏ''c'')<sup>1/2</sup>, as in [[Gaussian units]], ''not'' [[Lorentz–Heaviside units]].<ref>''Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science'', by Markus Reiher, Alexander Wolf, p7 [books.google.com/books?id=YwSpxCfsNsEC&pg=PA7 link]</ref> However, hybrids of the Gaussian and Lorentz–Heaviside units are sometimes used, leading to inconsistent conventions for magnetism-related units.<ref>[http://info.phys.unm.edu/~ideutsch/Classes/Phys531F11/Atomic%20Units.pdf ''A note on units'' lecture notes]. See the [[atomic units]] article for further discussion.</ref>
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| {{-}}
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| ===Quantum chromodynamics (QCD) system of units===
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| {| class="wikitable" align="right" style="margin-left: 1em; background-color: #ffffff"
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| ! Quantity
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| ! Expression
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| ! Metric value
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| |- align="left"
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| | [[Length]] (L)
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| | <math>l_{\mathrm{QCD}} = \frac{\hbar}{m_\text{p} c}</math>
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| | {{val|2.103|e=-16|u=m}}
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| |-
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| | [[Mass]] (M)
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| | <math>m_{\mathrm{QCD}} = m_\text{p} \ </math>
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| | {{val|1.673|e=-27|u=kg}}
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| |-
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| | [[Time]] (T)
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| | <math>t_{\mathrm{QCD}} = \frac{\hbar}{m_\text{p} c^2}</math>
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| | {{val|7.015|e=-25|u=s}}
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| |-
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| | [[Temperature]] (Θ)
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| | <math>T_{\mathrm{QCD}} = \frac{m_\text{p} c^2}{k_\text{B}}</math>
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| | {{val|1.089|e=13|u=K}}
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| |-
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| |rowspan=2| [[Electric charge]] (Q)
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| | <math>q_\mathrm{QCD} = e/\sqrt{4\pi\alpha}</math> ([[Lorentz–Heaviside units|L–H]])
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| | {{val|5.291|e=-19|u=C}}
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| |-
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| | <math>q_\mathrm{QCD} = e/\sqrt{\alpha}</math> ([[Gaussian units|G]])
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| | {{val|1.876|e=-18|u=C}}
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| |}
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| :<math> c = m_\text{p} = \hbar = k_\text{B} = 1 \ </math>
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| The [[electron mass]] is replaced with that of the [[proton]]. ''Strong units'' are "convenient for work in [[Quantum chromodynamics|QCD]] and nuclear physics, where quantum mechanics and relativity are omnipresent and the proton is an object of central interest".<ref>Wilczek, Frank, 2007, "[http://frankwilczek.com/Wilczek_Easy_Pieces/416_Fundamental_Constants.pdf Fundamental Constants,]" ''Frank Wilczek'' web site.</ref>
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| ===Geometrized units===
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| {{Main|Geometrized unit system}}
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| :<math> c = G = 1 \ </math>
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| 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 other appropriate units, geometrized units become identical to [[Planck units]].
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| ===Summary table===
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| {| class="wikitable" style="margin: 1em auto 1em auto; background-color: #ffffff"
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| ! Quantity / Symbol
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| ! Planck<br>(with [[Gaussian units|Gauss]])
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| ! Stoney
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| ! Hartree
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| ! Rydberg
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| ! "Natural"<br>(with [[Lorentz–Heaviside units|L-H]])
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| ! "Natural"<br>(with [[Gaussian units|Gauss]])
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| |-
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| |[[Speed of light in vacuum]] <br> <math>c \,</math>
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| |<math>1 \,</math>
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| |<math>1 \,</math>
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| |<math>\frac{1}{\alpha} \ </math>
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| |<math>\frac{2}{\alpha} \ </math>
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| |<math>1 \,</math>
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| |<math>1 \,</math>
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| |-
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| |[[Planck's constant]] (reduced) <br> <math>\hbar=\frac{h}{2 \pi}</math>
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| |<math>1 \,</math>
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| |<math>\frac{1}{\alpha} \ </math>
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| |<math>1 \,</math>
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| |<math>1 \,</math>
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| |<math>1 \,</math>
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| |<math>1 \,</math>
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| |-
| |
| |[[Elementary charge]] <br> <math>e \,</math>
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| |<math>\sqrt{\alpha} \,</math>
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| |<math>1 \,</math>
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| |<math>1 \,</math>
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| |<math>\sqrt{2} \,</math>
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| |<math>\sqrt{4\pi\alpha}</math>
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| |<math>\sqrt{\alpha}</math>
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| |-
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| |[[Josephson constant]] <br> <math>K_\text{J} =\frac{e}{\pi \hbar} \,</math>
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| |<math>\frac{\sqrt{\alpha}}{\pi} \,</math>
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| |<math>\frac{\alpha}{\pi} \,</math>
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| |<math>\frac{1}{\pi} \,</math>
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| |<math>\frac{\sqrt{2}}{\pi} \,</math>
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| |<math>\sqrt{\frac{4\alpha}{\pi}} \,</math>
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| |<math>\frac{\sqrt{\alpha}}{\pi} \,</math>
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| |-
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| |[[von Klitzing constant]] <br> <math>R_\text{K} =\frac{2 \pi \hbar}{e^2} \,</math>
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| |<math>\frac{2\pi}{\alpha} \,</math>
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| |<math>\frac{2\pi}{\alpha} \,</math>
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| |<math>2\pi \,</math>
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| |<math>\pi \,</math>
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| |<math>\frac{1}{2\alpha} </math>
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| |<math>\frac{2 \pi}{\alpha} </math>
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| |-
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| |[[Gravitational constant]] <br> <math>G \,</math>
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| |<math>1 \,</math>
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| |<math>1 \,</math>
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| |<math>\frac{\alpha_\text{G}}{\alpha} \,</math>
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| |<math>\frac{8 \alpha_\text{G}}{\alpha} \,</math>
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| |<math>\frac{\alpha_\text{G}}{{m_\text{e}}^2} \,</math>
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| |<math>\frac{\alpha_\text{G}}{{m_\text{e}}^2} \,</math>
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| |-
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| |[[Boltzmann constant]] <br> <math>k_\text{B} \,</math>
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| |<math>1 \,</math>
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| |<math>1 \,</math>
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| |<math>1 \,</math>
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| |<math>1 \,</math>
| |
| |<math>1 \,</math>
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| |<math>1 \,</math>
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| |-
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| |[[Electron mass]] <br> <math>m_\text{e} \,</math>
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| |<math>\sqrt{\alpha_\text{G}} \,</math>
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| |<math>\sqrt{\frac{\alpha_\text{G}}{\alpha}} \,</math>
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| |<math>1 \,</math>
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| |<math>\frac{1}{2} \,</math>
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| |<math>511 \text{ keV}</math>
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| |<math>511 \text{ keV}</math>
| |
| |}
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| where:
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| *α is the [[fine-structure constant]], approximately 0.007297,
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| *α<sub>G</sub> is the [[gravitational coupling constant]], {{nowrap|(''m''<sub>e</sub>/''m''<sub>Planck</sub>)<sup>2</sup> ≈ {{val|1.752|e=-45}}}},
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| ==See also==
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| {{multicol}}
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| * [[Anthropic units]]
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| * [[Dimensional analysis]]
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| * [[Fundamental unit]]
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| {{multicol-break}}
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| * [[N-body units]]
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| * [[Physical constant]]
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| * [[Units of measurement]]
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| {{multicol-end}}
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| == Notes and references ==
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| {{Reflist|2}}
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| ==External links==
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| *[http://physics.nist.gov/cuu/ The NIST website] ([[NIST|National Institute of Standards and Technology]]) is a convenient source of data on the commonly recognized constants.
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| *[http://www.ihst.ru/personal/tomilin/papers/tomil.pdf 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.
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| *[http://www.quantumfieldtheory.info Pedagogic Aides to Quantum Field Theory] Click on the link for Chap. 2 to find an extensive, simplified introduction to natural units.
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| {{Systems of measurement}}
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| {{Planck's natural units}}
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| {{SI units}}
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| {{DEFAULTSORT:Natural Units}}
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| [[Category:Natural units| ]]
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| [[Category:Metrology]]
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