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| The '''Rydberg constant''', symbol ''R''<sub>∞</sub> or ''R''<sub>H</sub>, named after the Swedish [[physicist]] [[Johannes Rydberg]], is a [[physical constant]] relating to atomic [[spectrum|spectra]], in the science of [[spectroscopy]]. The constant first arose as an empirical fitting parameter in the [[Rydberg formula]] for the [[hydrogen spectral series]], but [[Niels Bohr]] later showed that its value could be calculated from more fundamental constants, explaining the relationship via his "[[Bohr model]]". As of 2012, ''R''<sub>∞</sub> and electron spin [[Anomalous magnetic dipole moment|g-factor]] are the most accurately measured [[physical constant|fundamental physical constants]].<ref name="pohl">{{cite journal |title=The size of the proton |journal=Nature |volume=466 |issue=7303 |pages=213–216|year=2010 |pmid=20613837|doi=10.1038/nature09250|bibcode = 2010Natur.466..213P |last2=Antognini |last3=Nez |last4=Amaro |last5=Biraben |last6=Cardoso |last7=Covita |last8=Dax |last10=Fernandes |first10=Luis M. P. |last11=Giesen |last12=Graf |last13=Hänsch |last14=Indelicato |last15=Julien |last16=Kao |last17=Knowles |last18=Le Bigot |last19=Liu |first19=Yi-Wei |last20=Lopes |first20=José A. M. |last21=Ludhova |last22=Monteiro |last23=Mulhauser |last24=Nebel |last25=Rabinowitz |last26=Dos Santos |last27=Schaller |last28=Schuhmann |last29=Schwob |first29=Catherine |last30=Taqqu |first30=David |last1=Pohl |first1=Randolf |first2=Aldo |first3=François |first4=Fernando D. |first5=François |first6=João M. R. |first7=Daniel S. |first8=Andreas |last9=Dhawan |first9=Satish |first11=Adolf |first12=Thomas |first13=Theodor W. |first14=Paul |first15=Lucile |first16=Cheng-Yang |first17=Paul |first18=Eric-Olivier |first21=Livia |first22=Cristina M. B. |first23=Françoise |first24=Tobias |first25=Paul |first26=Joaquim M. F. |first27=Lukas A. |first28=Karsten }}</ref>
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| The Rydberg constant represents the limiting value of the highest [[wavenumber]] (the inverse wavelength) of any photon that can be emitted from the hydrogen atom, or, alternatively, the wavenumber of the lowest-energy photon capable of ionizing the hydrogen atom from its ground state. The spectrum of hydrogen can be expressed simply in terms of the Rydberg constant, using the [[Rydberg formula]].
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| The '''Rydberg unit of energy''', symbol Ry, is closely related to the Rydberg constant. It corresponds to the energy of the photon whose wavenumber is the Rydberg constant, i.e. the ionization energy of the hydrogen atom.
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| ==Value of the Rydberg constant and Rydberg unit of energy== | | '''MathML''' |
| According to the 2010 [[CODATA]], the constant is:
| | :<math forcemathmode="mathml">E=mc^2</math> |
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| :<math>R_\infty = \frac{m_\text{e} e^4}{8 {\varepsilon_0}^2 h^3 c} = 1.097\;373\;156\;8539(55) \times 10^7 \,\text{m}^{-1},</math><ref name="codata"/>
| | <!--'''PNG''' (currently default in production) |
| where <math>m_\text{e}</math> is the [[rest mass]] of the [[electron]], <math>e</math> is the [[elementary charge]], <math>\varepsilon_0</math> is the [[permittivity of free space]], <math>h</math> is the [[Planck constant]], and <math>c</math> is the [[speed of light]] in a vacuum.
| | :<math forcemathmode="png">E=mc^2</math> |
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| This constant is often used in [[atomic physics]] in the form of the Rydberg unit of energy:
| | '''source''' |
| :<math>1 \ \text{Ry} \equiv h c R_\infty = 13.605\;692\;53(30) \,\text{eV}.</math><ref name="codata"/> | | :<math forcemathmode="source">E=mc^2</math> --> |
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| ==Occurrence in Bohr model== | | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
| {{main|Bohr model}}
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| The [[Bohr model]] explains the atomic [[spectrum]] of hydrogen (see [[hydrogen spectral series]]) as well as various other atoms and ions. It is not perfectly accurate, but is a remarkably good approximation in many cases, and historically played an important role in the development of [[quantum mechanics]]. The Bohr model posits that electrons revolve around the atomic nucleus in a manner analogous to planets revolving around the sun.
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| In the simplest version of the Bohr model, the mass of the atomic nucleus is considered to be infinite compared to the mass of the electron,<ref name="coffman">{{cite journal |title=Correction to the Rydberg Constant for Finite Nuclear Mass |journal=American Journal of Physics|volume=33 |issue=10 |pages=820–823 |year=1965 |doi=10.1119/1.1970992|bibcode = 1965AmJPh..33..820C |last1=Coffman |first1=Moody L. }}</ref> so that the center of mass of the system lies at the [[center of mass|barycenter]] of the nucleus. This infinite mass approximation is what is alluded to with the <math>\infty</math> subscript. The Bohr model then predicts that the wavelengths of hydrogen atomic transitions are (see [[Rydberg formula]]):
| | ==Demos== |
| :<math>\frac{1}{\lambda} = R_\infty\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)=\frac{m_\text{e} e^4}{8 \varepsilon_0^2 h^3 c} \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right) </math>
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| where ''n''<sub>1</sub> and ''n''<sub>2</sub> are any two different positive integers (1, 2, 3, ...), and <math>\lambda</math> is the wavelength (in vacuum) of the emitted or absorbed light.
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| A refinement of the Bohr model takes into account the fact that the mass of the atomic nucleus is not actually infinite compared to the mass of the electron. Then the formula is:<ref name="coffman"/>
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
| :<math>\frac{1}{\lambda} = R_M\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)</math>
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| where <math>R_M = R_\infty/(1+m_{\text{e}}/M),</math> and ''M'' is the total mass of the atom. This formula comes from substituting the [[reduced mass]] for the mass of the electron.
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| A generalization of the Bohr model describes a hydrogen-like ion; that is, an atom with atomic number ''Z'' that has only one electron, such as C<sup>5+</sup>. In this case, the wavenumbers and photon energies are scaled up by a factor of ''Z''<sup>2</sup> in the model.
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| ==Precision measurement==
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| The Rydberg constant is one of the most well-determined physical constants, with a relative experimental uncertainty of fewer than 7 parts in 10<sup>12</sup>. The ability to measure it to such a high precision constrains the proportions of the values of the other physical constants that define it.<ref name="codata">P.J. Mohr, B.N. Taylor, and D.B. Newell (2011), "The 2010 CODATA Recommended Values of the Fundamental Physical Constants" (Web Version 6.0). This database was developed by J. Baker, M. Douma, and S. Kotochigova. Available: http://physics.nist.gov/constants. National Institute of Standards and Technology, Gaithersburg, MD 20899. [http://physics.nist.gov/cgi-bin/cuu/Value?ryd Link to R<sub>∞</sub>], [http://physics.nist.gov/cgi-bin/cuu/Value?rydhcev Link to hcR<sub>∞</sub>]. Published in {{cite journal|doi=10.1103/RevModPhys.84.1527|postscript=""|title=CODATA recommended values of the fundamental physical constants: 2010|year=2012|last1=Mohr|first1=Peter J.|last2=Taylor|first2=Barry N.|last3=Newell|first3=David B.|journal=Reviews of Modern Physics|volume=84|issue=4|pages=1527|arxiv = 1203.5425 |bibcode = 2012RvMP...84.1527M }} and {{Cite journal|doi=10.1063/1.4724320|postscript=""|title=CODATA Recommended Values of the Fundamental Physical Constants: 2010|year=2012|last1=Mohr|first1=Peter J.|last2=Taylor|first2=Barry N.|last3=Newell|first3=David B.|journal=Journal of Physical and Chemical Reference Data|volume=41|issue=4|pages=043109|bibcode = 2012JPCRD..41d3109M }}.</ref> ''See'' [[precision tests of QED]].
| | ==Test pages == |
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| Since the Bohr model is not perfectly accurate, due to [[fine structure]], [[hyperfine splitting]], and other such effects, the Rydberg constant <math>R_{\infty}</math> cannot be ''directly'' measured at very high accuracy from the [[atomic spectral line|atomic transition frequencies]] of hydrogen alone. Instead, the Rydberg constant is inferred from measurements of atomic transition frequencies in three different atoms ([[hydrogen]], [[deuterium]], and [[antiprotonic helium]]). Detailed theoretical calculations in the framework of [[quantum electrodynamics]] are used to account for the effects of finite nuclear mass, fine structure, hyperfine splitting, and so on. Finally, the value of <math>R_{\infty}</math> comes from the [[best fit]] of the measurements to the theory.<ref name=codata2006paper>{{cite journal |doi=10.1103/RevModPhys.80.633 |title=CODATA recommended values of the fundamental physical constants: 2006 |journal=Reviews of Modern Physics |volume=80 |pages=633–730 |year=2008|bibcode=2008RvMP...80..633M |issue=2|arxiv = 0801.0028 |last2=Taylor |last3=Newell |last1=Mohr |first1=Peter J. |first2=Barry N. |first3=David B. }}</ref>
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| ==Alternative expressions==
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| The Rydberg constant can also be expressed as in the following equations.
| | *[[Url2Image|Url2Image (private Wikis only)]] |
| | | ==Bug reporting== |
| :<math>R_\infty = \frac{\alpha^2 m_\text{e} c}{4 \pi \hbar} = \frac{\alpha^2}{2 \lambda_{\text{e}}} = \frac{\alpha}{4\pi a_0}</math>
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| :<math>h c R_\infty = m_{\text{e}} c^2 \frac{\alpha^2}{2} = \frac{m_{\text{e}} e^4}{32 \pi^2 \varepsilon_0^2 \hbar^2} = \frac{m_{\text{e}} c^2 r_e}{2 a_0} =\frac{h c \alpha^2}{2 \lambda_{\text{e}}} = \frac{h f_{\text{C}} \alpha^2}{2} = \frac{\hbar \omega_{\text{C}}}{2} \alpha^2 = \dfrac{\hbar^2}{2m_{\text{e}}a_0^2}=\frac{e^2}{(4\pi\varepsilon_0)2a_0}.</math>
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| where
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| :<math>m_\text{e}</math> is the [[electron rest mass]]
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| :<math>e</math> is the [[electric charge]] of the electron,
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| :<math>h</math> is the [[Planck constant]]
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| :<math>\hbar= h/2\pi</math> is the [[reduced Planck constant]],
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| :<math>c</math> is the [[speed of light]] in a vacuum,
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| :<math>\varepsilon_0</math> is the [[permittivity]] of free space,
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| :<math>\alpha</math> is the [[fine-structure constant]],
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| :<math>\lambda_{\text{e}} = h/m_\text{e} c</math> is the [[Compton wavelength]] of the electron,
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| :<math>f_{\text{C}}=m_{\text{e}} c^2/h</math> is the Compton frequency of the electron,
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| :<math>\omega_{\text{C}}=2\pi f_{\text{C}}</math> is the Compton angular frequency of the electron,
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| :<math>a_0=\frac{4\pi\varepsilon_0\hbar^2}{e^2m_{\text{e}}}</math> is the [[Bohr radius]], | |
| :<math>r_\mathrm{e} = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{m_{\mathrm{e}} c^2} </math> is the [[Classical electron radius]].
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| The last expression in the first equation shows that the wavelength of light needed to ionize a hydrogen atom is 4π/α times the Bohr radius of the atom.
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| The second equation is relevant because its value is the coefficient for the energy of the atomic orbitals of a hydrogen atom: <math>E_n = -h c R_\infty / n^2 </math>.
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| ==See also== | |
| * [[Rydberg formula]], includes a discussion of Rydberg's original discovery.
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| ==References==
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| {{Reflist}}
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| [[Category:Emission spectroscopy]]
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| [[Category:Physical constants]]
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| [[Category:Units of energy]]
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