Atomic units: Difference between revisions
imported>John R. Brews (rearrange; split out a few examples) |
imported>John R. Brews (move def of fine structure constant to first occurrence) |
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:<math> a_0 = \alpha/4{\rm\pi} R_\infty = 4{\rm\pi} \epsilon_0 \hbar^2/m_{\rm e} e^2 =\mathrm {0.529 177 208 59(36) \ \times \ 10^{-10}\ m } \ , </math> | :<math> a_0 = \alpha/4{\rm\pi} R_\infty = 4{\rm\pi} \epsilon_0 \hbar^2/m_{\rm e} e^2 =\mathrm {0.529 177 208 59(36) \ \times \ 10^{-10}\ m } \ , </math> | ||
where ''R<sub>∞</sub>'' is the [[Rydberg constant]]. | where ''R<sub>∞</sub>'' is the [[Rydberg constant]], and where the (dimensionless) [[fine structure constant]] ''α'' is given by (in [[SI units]]): | ||
:<math> \alpha = \frac{e^2}{4 \pi \varepsilon_0 \hbar c_0} \ , </math> | |||
and has the value:<ref name=fsc> | |||
{{cite web |title=Fine-structure constant |url=http://physics.nist.gov/cgi-bin/cuu/Value?eqalph|search_for=fine+structure+constant |publisher=NIST |work=The NIST reference on constants, units, and uncertainty|accessdate=2011-03-14}} | |||
</ref> | |||
:''α'' =7.297 352 5376(50) × 10<sup>-3</sup> = 1/137.035 999 679(94). | |||
===Energy=== | ===Energy=== | ||
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===Velocity=== | ===Velocity=== | ||
Using the unit of time, and the expression for the hartree, the a.u. unit of velocity is: | Using the unit of time, and the expression for the hartree, the a.u. unit of velocity is one bohr per a.u. unit of time: | ||
:<math>v_B =a_0/(\hbar/E_h) = \frac{e^2}{4 \pi \varepsilon_0 \hbar} = \alpha c_0 \ | :<math>v_B =a_0/(\hbar/E_h) = \frac{e^2}{4 \pi \varepsilon_0 \hbar} = \alpha c_0 \ .</math> | ||
Here, ''e'' is the [[elementary charge]], ''ε<sub>0</sub>'' is the [[electric constant]], ℏ is the reduced [[Planck's constant]] ''h/(2π)'', and ''c<sub>0</sub>'' is the SI units defined [[speed of light]] in [[classical vacuum]]. | Here, ''e'' is the [[elementary charge]], ''ε<sub>0</sub>'' is the [[electric constant]], ℏ is the reduced [[Planck's constant]] ''h/(2π)'', and ''c<sub>0</sub>'' is the SI units defined [[speed of light]] in [[classical vacuum]]. | ||
Revision as of 14:27, 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]
Example units
The expressions for a few example atomic units in terms of formulas in SI units are discussed next.
Length
In SI units the a.u. unit of length, the Bohr radius a0 or bohr, is:[4]
where R∞ is the Rydberg constant, and where the (dimensionless) fine structure constant α 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).
Energy
The unit of energy, the hartree, is the energy of two a.u. charges separated by one bohr in a medium of permittivity given by 1 a.u. of permittivity, 4πε0:
- Failed to parse (syntax error): {\displaystyle E_h = \frac{1}{4\pi \varepsilon_0} \frac{e^2}{a_0} \ =\ \mathrm{ 4.359\ 744\ 17(75)\ \times \ 10^{−18}\ J}\ . }
Time
Somewhat unusually, 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.[6]
- ℏ/Eh = 2.418 884 326 505(16) × 10−17 s
Velocity
Using the unit of time, and the expression for the hartree, the a.u. unit of velocity is one bohr per a.u. unit of time:
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 classical vacuum.
The a.u. unit of velocity changes size with refinement in measurement of the fine structure constant. However, the defined value of c0 = vB/α is unaffected by such refinements.
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, defining the a.u. unit of permittivity. The hartree, Eh = (4πε0)−1 e2/a0 also is automatically 1.
| Derived atomic units [1][7] | |||
|---|---|---|---|
| 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=4πε0 | 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.[8]
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."
- ↑ Bohr radius. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-04-02.
- ↑ Fine-structure constant. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-03-14.
- ↑ Volker Schmidt (1997). “§6.1 Atomic units”, Electron spectrometry of atoms using synchrotron radiation. Cambridge University Press, pp. 273 ff. ISBN 052155053X.
- ↑ 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.