Atomic units: Difference between revisions

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Tabulated values from {{cite book |title=International System of Units (SI) |editor=Barry N. Taylor, Ambler Thompson |isbn=1437915582 |publisher=DIANE Publishing |year=2008 |edition=NIST special publication 330 • 2008 ed |url=http://books.google.com/books?id=I-BlErBBeL8C&pg=PA34 |pages=Table 7, p.34}}
Tabulated values from {{cite book |title=International System of Units (SI) |editor=Barry N. Taylor, Ambler Thompson |isbn=1437915582 |publisher=DIANE Publishing |year=2008 |edition=NIST special publication 330 • 2008 ed |url=http://books.google.com/books?id=I-BlErBBeL8C&pg=PA34 |pages=Table 7, p.34}}


</ref> In the a.u. system any four of the five quantities charge ''e'', mass ''m<sub>e</sub>'', action ℏ, length ''a<sub>0</sub>'', and energy ''E<sub>h</sub>'' may be taken as base quantities, and other quantities are derived. In particular, ''time'' is a derived quantity, ℏ''/E<sub>h</sub>'', with the interpretation as the period of an electron circling in the first Bohr orbit divided by 2&pi;.<ref name=Schmidt>
</ref> In the a.u. system any four of the five quantities charge ''e'', mass ''m<sub>e</sub>'', action ℏ, length ''a<sub>0</sub>'', and energy ''E<sub>h</sub>'' may be taken as base quantities, and other quantities are derived.<ref name=Taylor2>
 
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."
 
</ref> A listing of numerical values in terms of [[SI units]] can be found on the [[NIST]] website.<ref name=listing>
{{cite web |title=Fundamental physical constants: atomic units |url=http://physics.nist.gov/cgi-bin/cuu/Results?search_for=atomic+unit |publisher=NIST |work=The NIST reference on constants, units, and uncertainty |accessdate=2011-09-04}}
</ref>
 
==Example units==
 
The expressions for a few example atomic units in terms of formulas in [[SI units]] are discussed next.
 
===Mass===
 
The atomic unit of mass is the electron mass, ''m<sub>e</sub>''.<ref name=NISTe>
 
{{cite web |title=Fundamental physical constants: atomic unit of mass, ''m<sub>e</sub>'' |url=http://physics.nist.gov/cgi-bin/cuu/Value?ttme |publisher=NIST |url=http://physics.nist.gov/cgi-bin/cuu/Value?ttme|search_for=atomic+unit|work=The NIST reference on constants, units, and uncertainty |accessdate=2011-09-08}}
 
</ref> However, confusingly, the terminology "atomic unit of mass" often is taken instead to refer to the [[Unified atomic mass unit|''unified atomic mass unit'']], the ''Dalton'', symbol ''u''.
The Dalton is defined as 1/12 the mass of a free carbon 12 atom,<ref name=Thompson>
{{cite book |title=Guide for the Use of the International System of Units (SI) (rev. ): The Metric System |author=Ambler Thompson |chapter=Table 7: Non-SI units accepted for use with the SI |pages=p. 9 |url=http://books.google.com/books?id=pTw-SCI7EkoC&pg=PA9 |isbn=1437915590 |year=2009 |publisher=DIANE Publishing}}
</ref> with a value of 1.660538921(73) × 10<sup>-27</sup> kg.,<ref name=NIST>
 
{{cite web |title=Fundamental physical constants: atomic mass unit-kilogram relationship 1 ''u''|url=http://physics.nist.gov/cgi-bin/cuu/Value?ukg |publisher=NIST |work=The NIST reference on constants, units, and uncertainty |accessdate=2011-09-08}}
 
</ref>  or 931.494 061(21) MeV.<ref name=NIST2>
 
{{cite web |title=Fundamental physical constants: atomic mass unit-electron volt relationship 1 ''u'' (''c<sub>0</sub>''<sup>2</sup>)|url=http://physics.nist.gov/cgi-bin/cuu/Value?uev |publisher=NIST |work=The NIST reference on constants, units, and uncertainty |accessdate=2011-09-04}}
 
</ref> A proton has a mass of approximately 1.007 276 466 812 ''u''.<ref name=mpu>
{{cite web |title=Fundamental physical constants: proton mass in ''u'': ''m<sub>p</sub>'' |url=http://physics.nist.gov/cgi-bin/cuu/Value?mpu|search_for=proton+mass |publisher=NIST |work=The NIST reference on constants, units, and uncertainty |accessdate=2011-09-04}}
 
</ref>  The electron mass is about 5.485 799 0946 × 10<sup>-4</sup> ''u''.<ref name=meu>
{{cite web |title=Fundamental physical constants: electron mass in ''u'': ''m<sub>e</sub>'' |url=http://physics.nist.gov/cgi-bin/cuu/Value?meu|search_for=electron+mass |publisher=NIST |work=The NIST reference on constants, units, and uncertainty |accessdate=2011-09-04}}
 
</ref>
 
===Length===
 
In [[SI units]] the a.u. unit of length, the Bohr radius ''a<sub>0</sub>'' or ''bohr'', is:<ref name=bohr>
 
{{cite web |title=Fundamental physical constants: atomic unit of length ''a<sub>0</sub>'' |url=http://physics.nist.gov/cgi-bin/cuu/Value?tbohrrada0|search_for=atomic+unit |publisher=NIST |work=The NIST reference on constants, units, and uncertainty|accessdate=2011-09-04}}
</ref>
 
:<math> a_0 = \alpha/4{\rm\pi} R_\infty = 4{\rm\pi} \epsilon_0 \hbar^2/m_{\rm e} e^2 =\mathrm {0.529 177 210 92(17) \ \times \ 10^{-10}\ m }  \ , </math>
 
where ''R<sub>&infin;</sub>'' is the [[Rydberg constant]], ''e'' is the [[elementary charge]], ''&epsilon;<sub>0</sub>'' is the [[electric constant]], ℏ is the reduced [[Planck's constant]] ''h/(2&pi;)'', ''m<sub>e</sub>'' is the electron mass, and where the (dimensionless) [[fine structure constant]] ''&alpha;'' 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=Fundamental physical constants: fine-structure constant ''&alpha; |url=http://physics.nist.gov/cgi-bin/cuu/Value?alph|search_for=fine+structure+constants |publisher=NIST |work=The NIST reference on constants, units, and uncertainty|accessdate=2011-09-04}}
 
</ref> 
:<math>\alpha =\mathrm {7.297 352 5698(24) \times 10^{-3}} = \mathrm {1/137.035 999 074(44)} \ .</math>
 
The Bohr radius was the distance of an electron from the nucleus of a hydrogen atom predicted by the Bohr theory of the atom, which required an integer number of wavelengths around the electron orbit.<ref name=Bohr_atom>
 
A discussion can be found in {{cite book |title=Atoms, molecules and photons: an introduction to atomic-, molecular-, and quantum-physics |author=W. Demtröder |pages=pp. 112''ff'' |chapter=Chapter 3: Development of quantum physics |url=http://www.amazon.com/Introduction-Atomic-Molecular-Physics/dp/3540206310/ref=sr_1_1?s=books&ie=UTF8&qid=1302198584&sr=1-1#reader_3540206310 |isbn=3540206310 |publisher=Birkhäuser |year=2006}}
 
</ref> In modern quantum mechanics the Bohr radius is the distance of maximum likelihood for finding the electron in the hydrogen atom in its ground state. <ref name=Demtröder>
 
See previously cited work {{cite book |title=Atoms, molecules and photons: an introduction to atomic-, molecular-, and quantum-physics |author=W. Demtröder |pages=p. 164 |chapter=§5.1.4 Spatial distributions and expectation values of the electron in different quantum states |url=http://books.google.com/books?id=Eehsvr8jVpAC&pg=PA164 |publisher=Birkhäuser |year=2006}}
 
</ref>
 
===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, 4''&pi;&epsilon;<sub>0</sub>'':<ref name=hartree>
{{cite web |title=Fundamental physical constants: atomic unit of energy ''E<sub>h</sub>'' |url=http://physics.nist.gov/cgi-bin/cuu/Value?thr|search_for=atomic+unit |publisher=NIST |work=The NIST reference on constants, units, and uncertainty|accessdate=2011-09-04}}
</ref>
 
:<math>E_h = \frac{1}{4\pi \varepsilon_0} \frac{e^2}{a_0}  </math> <math> =\mathrm {4.359 744 34(19)  \ \times \ 10^{-18}\ J }  \ . </math>
 
===Time===
Somewhat unusually, ''time'' is a derived quantity, ℏ''/E<sub>h</sub>'', with the interpretation as the period of an electron circling in the first Bohr orbit divided by 2&pi;.<ref name=Schmidt>


{{cite book |title=Electron spectrometry of atoms using synchrotron radiation |author=Volker Schmidt |chapter=§6.1 Atomic units |url=http://books.google.com/books?id=1oXrE5YV19IC&pg=PA273 |pages=pp. 273 ''ff'' |isbn=052155053X |publisher=Cambridge University Press |year=1997}}  
{{cite book |title=Electron spectrometry of atoms using synchrotron radiation |author=Volker Schmidt |chapter=§6.1 Atomic units |url=http://books.google.com/books?id=1oXrE5YV19IC&pg=PA273 |pages=pp. 273 ''ff'' |isbn=052155053X |publisher=Cambridge University Press |year=1997}}  




</ref> Because ''length'' is a basic unit, the [[speed of light]] is a measured quantity in a.u., ''c=1/&alpha;'' a.u. of velocity where the (dimensionless) [[fine structure constant]] ''&alpha;'' = 1/137.035 999 11(46).
</ref><ref name=time>
{{cite web |title=Fundamental physical constants: atomic unit of time ''ℏ/E<sub>h</sub>'' |url=http://physics.nist.gov/cgi-bin/cuu/Value?aut|search_for=atomic+unit |publisher=NIST |work=The NIST reference on constants, units, and uncertainty|accessdate=2011-09-04}}
</ref>
 
:<math>\hbar/E_h = \mathrm{2.418 884 326 502(12)} \ \times \ 10^{-17}\ \mathrm{ s} </math>
 
===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:
 
:<math>v_B =a_0/(\hbar/E_h) = \frac{e^2}{4 \pi \varepsilon_0 \hbar} = \alpha c_0 \ .</math>
 
Here ''c<sub>0</sub>'' is the SI units defined [[speed of light]] in [[classical vacuum]] and ''&alpha;'' is the fine structure constant. Its value is:<ref name=velocity>
{{cite web |title=Fundamental physical constants: atomic unit of time ''a<sub>0</sub>E<sub>h</sub>/''ℏ |url=http://physics.nist.gov/cgi-bin/cuu/Value?auvel|search_for=atomic+unit |publisher=NIST |work=The NIST reference on constants, units, and uncertainty|accessdate=2011-09-04}} </ref>
 
:<math>a_0/(\hbar/E_h) = \mathrm{2.187 691 263 79(71)} \times 10^6 \mathrm{m/s} \ . </math>
 
The a.u. unit of velocity changes size with refinement in measurement of the fine structure constant. However, the defined value of ''c<sub>0</sub>'' = ''v<sub>B</sub>/&alpha;'' is unaffected by such refinements. See the articles [[Speed_of_light#Relation_to_the_metre|speed of light]] and [[Metre (unit)|metre]] for more detail about the defined value ''c<sub>0</sub>'' = 299 792 458 m/s (exactly).


==Units==
==Tabulation==


{| class="wikitable" style="margin:1em auto 1em auto"
{| class="wikitable" style="margin:1em auto 1em auto"
! colspan="4" |Basic atomic units <ref name=Taylor/>
! colspan="4" |Basic atomic units <ref name=listing/>


|-
|-
Line 31: Line 123:
|'''e'''
|'''e'''
|[[charge]]
|[[charge]]
|1.602 176 53(14) × 10<sup>−19</sup> C
|1.602 176 565(35) × 10<sup>−19</sup> C
|-
|-
|-
|-
Line 37: Line 129:
|'''a<sub>0</sub>'''
|'''a<sub>0</sub>'''
|[[length]]
|[[length]]
|0.529 177 2108(18) × 10<sup>−10</sup> m
|0.529 177 210 92(17) × 10<sup>−10</sup> m
|-
|-
|[[electron mass]]
|[[electron mass]]
|'''m<sub>e</sub>'''
|'''m<sub>e</sub>'''
|[[mass]]
|[[mass]]
|9.109 3826(16) × 10<sup>−31</sup> kg
|9.109 382 91(40) × 10<sup>−31</sup> kg
|-
|-
|Reduced [[Planck constant]]
|reduced [[Planck constant]]
|'''ℏ'''
|'''ℏ'''
|[[action]]
|[[action]]
|1.054 571 68(18) × 10<sup>−34</sup> Js
|1.054 571 726(47) × 10<sup>−34</sup> Js
|-
|-
|[[Hartree energy]] (hartree)
|[[Hartree energy]] (hartree)
|'''E<sub>h</sub>'''
|'''E<sub>h</sub>'''
|[[energy]]
|[[energy]]
|4.359 744 17(75)  × 10<sup>−18</sup> J
|4.359 744 34(19)  × 10<sup>−18</sup> J
|-
|-
|}
|}
Evidently in atomic units, if we choose ''e'', ℏ, ''m<sub>e</sub>'', and ''a<sub>0</sub>'' as the four basic units, then these entities all become 1 in algebraic expressions, and because ''a<sub>0</sub> = (4&pi;&epsilon;<sub>0</sub>)''(ℏ/''e'')<sup>2</sup>/''m<sub>e</sub>'', it follows that ''4&pi;&epsilon;<sub>0</sub>'' = 1 as well, defining the a.u. unit of permittivity. The hartree, ''E<sub>h</sub>'' = ''(4&pi;&epsilon;<sub>0</sub>)<sup>−1</sup>'' ''e<sup>2</sup>/a<sub>0</sub>'' also is automatically 1.
{| class="wikitable" style="margin:1em auto 1em auto"
{| class="wikitable" style="margin:1em auto 1em auto"
! colspan="4" |Derived atomic units <ref name=Drake/>
! colspan="4" |Derived atomic units <ref name=listing/><ref name=Drake/><ref name=Mohr>
 
{{cite journal |title=CODATA recommended values of the fundamental physical constants: 2006; Table LIII |url=http://physics.nist.gov/cuu/Constants/RevModPhys_80_000633acc.pdf |author=PJ Mohr, BN Taylor, and DB Newell |journal=Rev. Mod. Phys. |volume=vol. 80 |issue=No. 2 |year=2008 |pages=p. 717}}
 
 
</ref>
|-
|-
|'''Name'''
|'''Name'''
|'''Symbol'''
|'''Formula'''
|'''Quantity'''
|'''Quantity'''
|'''Value in [[International System of Units|SI units]]'''
|'''Value in [[International System of Units|SI units]]'''
|-
|-
|[[a.u. velocity]]  
|[[a.u. velocity]]  
|'''v<sub>B</sub>≡αc'''
|'''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
|2.187 691 263 79(71) × 10<sup>6</sup> m/s
|-
|-
|[[a.u. time]]
|[[a.u. time]]
|'''ℏ/E<sub>h</sub>'''
|'''ℏ/E<sub>h</sub>'''
|[[time]]
|[[time]]
|2.418 884 326 505(16) × 10<sup>−17</sup> s
|2.418 884 326 502(12) × 10<sup>−17</sup> s
|-
|[[a.u. current]]
|'''eE<sub>h</sub>/ℏ'''
|[[Electric current|current]]
|6.623 617 95(15) × 10<sup>−3</sup> A
|-
|[[a.u. electric potential]]
|'''E<sub>h</sub>/e'''
|[[electric potential]]
|27.211 385 05(60) V
|-
|[[a.u. magnetic flux density]]
|'''ℏ/ea<sub>0</sub><sup>2</sup>'''
|[[magnetic flux density]]
|2.350 517 464(52) × 10<sup>5</sup> T
|-
|[[a.u. magnetic dipole moment]]
|'''ℏe/m<sub>e</sub>''' = 2'''&mu;<sub>B</sub>'''
|[[magnetic dipole moment]]
|1.854 801 936(41)  × 10<sup>−23</sup> J T<sup>−1</sup>
|-
|[[a.u. permittivity]]
|'''e<sup>2</sup>/a<sub>0</sub>E<sub>h</sub>''' = 10<sup>7</sup>/'''c<sub>0</sub><sup>2</sup>'''='''4&pi;&epsilon;<sub>0</sub>'''
|[[permittivity]]
|1.112 650 056...  × 10<sup>−10</sup> F m<sup>−1</sup> (exact)
|-
|-
|}
|}
Here, ''c'' = [[speed of light]] and ''&alpha;'' = [[fine structure constant]].<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''}}
Here, ''c<sub>0</sub>'' = [[SI units]] defined value for the [[speed of light]] in [[classical vacuum]], ''&epsilon;<sub>0</sub>'' is the [[electric constant]], ''&alpha;'' = [[fine structure constant]] and ''&mu;<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>


==Notes==
==Notes==
<references/>
<references/>[[Category:Suggestion Bot Tag]]

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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] A listing of numerical values in terms of SI units can be found on the NIST website.[4]

Example units

The expressions for a few example atomic units in terms of formulas in SI units are discussed next.

Mass

The atomic unit of mass is the electron mass, me.[5] However, confusingly, the terminology "atomic unit of mass" often is taken instead to refer to the unified atomic mass unit, the Dalton, symbol u. The Dalton is defined as 1/12 the mass of a free carbon 12 atom,[6] with a value of 1.660538921(73) × 10-27 kg.,[7] or 931.494 061(21) MeV.[8] A proton has a mass of approximately 1.007 276 466 812 u.[9] The electron mass is about 5.485 799 0946 × 10-4 u.[10]

Length

In SI units the a.u. unit of length, the Bohr radius a0 or bohr, is:[11]

where R is the Rydberg constant, e is the elementary charge, ε0 is the electric constant, ℏ is the reduced Planck's constant h/(2π), me is the electron mass, and where the (dimensionless) fine structure constant α is given by (in SI units):

and has the value:[12]

The Bohr radius was the distance of an electron from the nucleus of a hydrogen atom predicted by the Bohr theory of the atom, which required an integer number of wavelengths around the electron orbit.[13] In modern quantum mechanics the Bohr radius is the distance of maximum likelihood for finding the electron in the hydrogen atom in its ground state. [14]

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:[15]

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.[16][17]

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 c0 is the SI units defined speed of light in classical vacuum and α is the fine structure constant. Its value is:[18]

The a.u. unit of velocity changes size with refinement in measurement of the fine structure constant. However, the defined value of c0 = vB is unaffected by such refinements. See the articles speed of light and metre for more detail about the defined value c0 = 299 792 458 m/s (exactly).

Tabulation

Basic atomic units [4]
Name Symbol Quantity Value in SI units
elementary charge e charge 1.602 176 565(35) × 10−19 C
Bohr radius (bohr) a0 length 0.529 177 210 92(17) × 10−10 m
electron mass me mass 9.109 382 91(40) × 10−31 kg
reduced Planck constant action 1.054 571 726(47) × 10−34 Js
Hartree energy (hartree) Eh energy 4.359 744 34(19) × 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 [4][1][19]
Name Formula Quantity Value in SI units
a.u. velocity vB = αc0 = a0Eh/ℏ velocity 2.187 691 263 79(71) × 106 m/s
a.u. time ℏ/Eh time 2.418 884 326 502(12) × 10−17 s
a.u. current eEh/ℏ current 6.623 617 95(15) × 10−3 A
a.u. electric potential Eh/e electric potential 27.211 385 05(60) V
a.u. magnetic flux density ℏ/ea02 magnetic flux density 2.350 517 464(52) × 105 T
a.u. magnetic dipole moment ℏe/me = 2μB magnetic dipole moment 1.854 801 936(41) × 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, ε0 is the electric constant, α = fine structure constant and μB is the Bohr magneton.[20]

Notes

  1. Jump up to: 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. 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. 
  3. 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."
  4. Jump up to: 4.0 4.1 4.2 Fundamental physical constants: atomic units. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
  5. Fundamental physical constants: atomic unit of mass, me. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-08.
  6. Ambler Thompson (2009). “Table 7: Non-SI units accepted for use with the SI”, Guide for the Use of the International System of Units (SI) (rev. ): The Metric System. DIANE Publishing, p. 9. ISBN 1437915590. 
  7. Fundamental physical constants: atomic mass unit-kilogram relationship 1 u. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-08.
  8. Fundamental physical constants: atomic mass unit-electron volt relationship 1 u (c02). The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
  9. Fundamental physical constants: proton mass in u: mp. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
  10. Fundamental physical constants: electron mass in u: me. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
  11. Fundamental physical constants: atomic unit of length a0. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
  12. Fundamental physical constants: fine-structure constant α. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
  13. A discussion can be found in W. Demtröder (2006). “Chapter 3: Development of quantum physics”, Atoms, molecules and photons: an introduction to atomic-, molecular-, and quantum-physics. Birkhäuser, pp. 112ff. ISBN 3540206310. 
  14. See previously cited work W. Demtröder (2006). “§5.1.4 Spatial distributions and expectation values of the electron in different quantum states”, Atoms, molecules and photons: an introduction to atomic-, molecular-, and quantum-physics. Birkhäuser, p. 164. 
  15. Fundamental physical constants: atomic unit of energy Eh. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
  16. Volker Schmidt (1997). “§6.1 Atomic units”, Electron spectrometry of atoms using synchrotron radiation. Cambridge University Press, pp. 273 ff. ISBN 052155053X. 
  17. Fundamental physical constants: atomic unit of time ℏ/Eh. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
  18. Fundamental physical constants: atomic unit of time a0Eh/. The NIST reference on constants, units, and uncertainty. NIST. Retrieved on 2011-09-04.
  19. 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.
  20. 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.