Atomic units: Difference between revisions
imported>John R. Brews (quote) |
imported>John R. Brews (some points about c0) |
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|[[a.u. velocity]] | |[[a.u. velocity]] | ||
|'''v<sub>B</sub> | |'''v<sub>B</sub> = αc<sub>0</sub> = a<sub>0</sub>E<sub>h</sub>/ℏ''' | ||
|[[velocity]] | |[[velocity]] | ||
|2.187 691 2633(73) × 10<sup>6</sup> m/s | |2.187 691 2633(73) × 10<sup>6</sup> m/s | ||
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|[[a.u. permittivity]] | |[[a.u. permittivity]] | ||
|'''e<sup>2</sup>/a<sub>0</sub>E<sub>h</sub>''' = 10<sup>7</sup>/'''c<sup>2</sup>''' | |'''e<sup>2</sup>/a<sub>0</sub>E<sub>h</sub>''' = 10<sup>7</sup>/'''c<sub>0</sub><sup>2</sup>''' | ||
|[[permittivity]] | |[[permittivity]] | ||
|1.112 650 056... × 10<sup>−10</sup> F m<sup>−1</sup> (exact) | |1.112 650 056... × 10<sup>−10</sup> F m<sup>−1</sup> (exact) | ||
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Here, ''c'' = | Here, ''c<sub>0</sub>'' = [[SI units]] defined value for the [[speed of light]] in [[classical vacuum]], ''α'' = [[fine structure constant]] and ''μ<sub>B</sub>'' is the Bohr magneton.<ref name=fine_structure>An overview of the importance and determination of the fine structure constant is found in {{cite book |title=Lepton dipole moments |author=G. Gabrielse |chapter=Determining the fine structure constant |editor=B. Lee Roberts, William J. Marciano, eds |isbn=9814271837 |year=2010 |publisher=World Scientific |url=http://books.google.com/books?id=1TNXUSxJs6IC&pg=PA195 |pages=pp. 195 ''ff''}} | ||
</ref> | </ref> The a.u. unit of velocity is ''v<sub>B</sub>'' = ''αc<sub>0</sub>'' and changes size with refinement in measurement of the fine structure constant. However, obviously the defined value of ''c<sub>0</sub>'' = ''v<sub>B</sub>/α'' is unaffected by such refinements. | ||
==Notes== | ==Notes== | ||
<references/> | <references/> | ||
Revision as of 12:04, 2 April 2011
The atomic units, abbreviated a.u. is a set of units used in atomic calculations.[1][2] In the a.u. system any four of the five quantities charge e, mass me, action ℏ, length a0, and energy Eh may be taken as base quantities, and other quantities are derived.[3] In particular, time is a derived quantity, ℏ/Eh, with the interpretation as the period of an electron circling in the first Bohr orbit divided by 2π.[4] Because length is a basic unit, the speed of light is a measured quantity in a.u., c=1/α a.u. of velocity where the (dimensionless) fine structure constant is given by (in SI units):
and has the value:[5]
- α =7.297 352 5376(50) × 10-3 = 1/137.035 999 679(94).
Here, e is the elementary charge, ε0 is the electric constant, ℏ is the reduced Planck's constant h/(2π), and c0 is the SI units defined speed of light in vacuum. In SI units the a.u. unit of length, the Bohr radius a0, is:
where R∞ is the Rydberg constant.
Units
| Basic atomic units [2] | |||
|---|---|---|---|
| Name | Symbol | Quantity | Value in SI units |
| elementary charge | e | charge | 1.602 176 53(14) × 10−19 C |
| Bohr radius (bohr) | a0 | length | 0.529 177 2108(18) × 10−10 m |
| electron mass | me | mass | 9.109 3826(16) × 10−31 kg |
| reduced Planck constant | ℏ | action | 1.054 571 68(18) × 10−34 Js |
| Hartree energy (hartree) | Eh | energy | 4.359 744 17(75) × 10−18 J |
Evidently in atomic units, if we choose e, ℏ, me, and a0 as the four basic units, then these entities all become 1 in algebraic expressions, and because a0 = (4πε0)(ℏ/e)2/me, it follows that 4πε0 = 1 as well. The hartree, Eh = (4πε0)−1 e2/a0 also is automatically 1.
| Derived atomic units [1][6] | |||
|---|---|---|---|
| Name | Formula | Quantity | Value in SI units |
| a.u. velocity | vB = αc0 = a0Eh/ℏ | velocity | 2.187 691 2633(73) × 106 m/s |
| a.u. time | ℏ/Eh | time | 2.418 884 326 505(16) × 10−17 s |
| a.u. current | eEh/ℏ | current | 6.623 617 63(17) × 10−3 A |
| a.u. electric potential | Eh/e | electric potential | 27.211 383 86(68) V |
| a.u. magnetic flux density | ℏ/ea02 | magnetic flux density | 2.350 517 382(59) × 105 T |
| a.u. magnetic dipole moment | ℏe/me = 2μB | magnetic dipole moment | 1.854 801 830(46) × 10−23 J T−1 |
| a.u. permittivity | e2/a0Eh = 107/c02 | permittivity | 1.112 650 056... × 10−10 F m−1 (exact) |
Here, c0 = SI units defined value for the speed of light in classical vacuum, α = fine structure constant and μB is the Bohr magneton.[7] The a.u. unit of velocity is vB = αc0 and changes size with refinement in measurement of the fine structure constant. However, obviously the defined value of c0 = vB/α is unaffected by such refinements.
Notes
- ↑ 1.0 1.1 For an introduction, see Gordon W. F. Drake (2006). “§1.2 Atomic units”, Springer handbook of atomic, molecular, and optical physics, Volume 1, 2nd ed. Springer, p. 6. ISBN 038720802X.
- ↑ 2.0 2.1 Tabulated values from (2008) Barry N. Taylor, Ambler Thompson: International System of Units (SI), NIST special publication 330 • 2008 ed. DIANE Publishing, Table 7, p.34. ISBN 1437915582.
- ↑ According to Taylor, as cited above, page 33: "any four of the five quantities charge, mass, action, length and energy are taken as base quantities."
- ↑ Volker Schmidt (1997). “§6.1 Atomic units”, Electron spectrometry of atoms using synchrotron radiation. Cambridge University Press, pp. 273 ff. ISBN 052155053X.
- ↑ fine-structure constant. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-03-14.
- ↑ PJ Mohr, BN Taylor, and DB Newell (2008). "CODATA recommended values of the fundamental physical constants: 2006; Table LIII". Rev. Mod. Phys. vol. 80 (No. 2): p. 717.
- ↑ An overview of the importance and determination of the fine structure constant is found in G. Gabrielse (2010). “Determining the fine structure constant”, B. Lee Roberts, William J. Marciano, eds: Lepton dipole moments. World Scientific, pp. 195 ff. ISBN 9814271837.