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Rydberg constant
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<h2 id="value">Value</h2> <h3>Rydberg constant</h3> <p>The <a href="/facts/CODATA/ANfTN84l">CODATA</a> value is </p> R ∞ = m e e 4 8 ε 0 2 h 3 c = {\displaystyle R_{\infty }={\frac {m_{\text{e}}e^{4}}{8\varepsilon _{0}^{2}h^{3}c}}=} 10973731.568157(12) m−1,<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> <p>where </p> <ul><li> m e {\displaystyle m_{\text{e}}} is the <a href="/facts/Rest_mass/aRloR0nW">rest mass</a> of the <a href="/facts/Electron/L0KBo5cW">electron</a> (i.e. the <a href="/facts/Electron_mass/xlG83aQl">electron mass</a>),</li> <li> e {\displaystyle e} is the <a href="/facts/Elementary_charge/lPcj6CoQ">elementary charge</a>,</li> <li> ε 0 {\displaystyle \varepsilon _{0}} is the <a href="/facts/Permittivity_of_free_space/gWRUhPGY">permittivity of free space</a>,</li> <li> h {\displaystyle h} is the <a href="/facts/Planck_constant/qGHAe3qq">Planck constant</a>, and</li> <li> c {\displaystyle c} is the <a href="/facts/Speed_of_light/sKSQwdNx">speed of light</a> in vacuum.</li></ul> <p>The symbol ∞ {\displaystyle \infty } means that the nucleus is assumed to be infinitely heavy, an improvement of the value can be made using the <a href="/facts/Reduced_mass/GQgUNp8D">reduced mass</a> of the atom: </p> μ = 1 1 m e + 1 M {\displaystyle \mu ={\frac {1}{{\frac {1}{m_{\text{e}}}}+{\frac {1}{M}}}}} <p>with M {\displaystyle M} the mass of the nucleus. The corrected Rydberg constant is: </p> R M = μ m e R ∞ {\displaystyle R_{\text{M}}={\frac {\mu }{m_{\text{e}}}}R_{\infty }} <p>that for hydrogen, where M {\displaystyle M} is the mass m p {\displaystyle m_{\text{p}}} of the <a href="/facts/Proton/clCjepXP">proton</a>, becomes: </p> R H = m p m e + m p R ∞ ≈ 1.09678 × 10 7 m − 1 , {\displaystyle R_{\text{H}}={\frac {m_{\text{p}}}{m_{\text{e}}+m_{\text{p}}}}R_{\infty }\approx 1.09678\times 10^{7}{\text{ m}}^{-1},} <p>Since the Rydberg constant is related to the spectrum lines of the atom, this correction leads to an <a href="/facts/Isotopic_shift/5KqGqyLg">isotopic shift</a> between different isotopes. For example, deuterium, an isotope of hydrogen with a nucleus formed by a proton and a <a href="/facts/Neutron/oRCM1geb">neutron</a> ( M = m p + m n ≈ 2 m p {\displaystyle M=m_{\text{p}}+m_{\text{n}}\approx 2m_{\text{p}}} ), was discovered thanks to its slightly shifted spectrum.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h3>Rydberg unit of energy</h3> <p>The Rydberg unit of energy is </p> 1 Ry ≡ h c R ∞ = α 2 m e c 2 / 2 {\displaystyle 1\ {\text{Ry}}~~\equiv hc\,R_{\infty }=\alpha ^{2}m_{\text{e}}c^{2}/2} = 2.1798723611030(24)×10−18 J<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> = 13.605693122990(15) eV<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> <h3>Rydberg frequency</h3> c R ∞ {\displaystyle cR_{\infty }} = 3.2898419602500(36)×1015 Hz.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> <h3>Rydberg wavelength</h3> 1 R ∞ = 9.112 670 505 826 ( 10 ) × 10 − 8 m {\displaystyle {\frac {1}{R_{\infty }}}=9.112\;670\;505\;826(10)\times 10^{-8}\ {\text{m}}} . <p>The corresponding <a href="/facts/Angular_wavelength/kcJU0ymz">angular wavelength</a> is </p> 1 2 π R ∞ = 1.450 326 555 77 ( 16 ) × 10 − 8 m {\displaystyle {\frac {1}{2\pi R_{\infty }}}=1.450\;326\;555\;77(16)\times 10^{-8}\ {\text{m}}} . <h2 id="bohr-model">Bohr model</h2> <p class="note">Main article: <a href="/facts/Bohr_model/FmTHjaxy">Bohr model</a></p> <p>The <a href="/facts/Bohr_model/FmTHjaxy">Bohr model</a> explains the atomic <a href="/facts/Spectrum/wPUjkpqI">spectrum</a> of hydrogen (see <i><a href="/facts/Hydrogen_spectral_series/BhTeHzBt">Hydrogen spectral series</a></i>) 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 <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>. The Bohr model posits that electrons revolve around the atomic nucleus in a manner analogous to planets revolving around the Sun. </p><p>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,<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> so that the center of mass of the system, the <a href="/facts/Center_of_mass/n4Lr9GAz">barycenter</a>, lies at the center of the nucleus. This infinite mass approximation is what is alluded to with the ∞ {\displaystyle \infty } subscript. The Bohr model then predicts that the wavelengths of hydrogen atomic transitions are (see <i><a href="/facts/Rydberg_formula/xKRGYj5V">Rydberg formula</a></i>): </p> 1 λ = R y ⋅ 1 h c ( 1 n 1 2 − 1 n 2 2 ) = m e e 4 8 ε 0 2 h 3 c ( 1 n 1 2 − 1 n 2 2 ) {\displaystyle {\frac {1}{\lambda }}=\mathrm {Ry} \cdot {1 \over hc}\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)} <p>where <i>n</i>1 and <i>n</i>2 are any two different positive integers (1, 2, 3, ...), and λ {\displaystyle \lambda } is the wavelength (in vacuum) of the emitted or absorbed light, giving </p> 1 λ = R M ( 1 n 1 2 − 1 n 2 2 ) {\displaystyle {\frac {1}{\lambda }}=R_{M}\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)} <p>where R M = R ∞ 1 + m e M , {\displaystyle R_{M}={\frac {R_{\infty }}{1+{\frac {m_{\text{e}}}{M}}}},} and <i>M</i> is the total mass of the nucleus. This formula comes from substituting the <a href="/facts/Reduced_mass/GQgUNp8D">reduced mass</a> of the electron. </p> <h2 id="precision-measurement">Precision measurement</h2> <p class="note">See also: <a href="/facts/Precision_tests_of_QED/ddojHb67">Precision tests of QED</a></p> <p>The Rydberg constant was one of the most precisely determined physical constants, with a relative standard uncertainty of 1.1×10−12.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> This precision constrains the values of the other physical constants that define it.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p><p>Since the Bohr model is not perfectly accurate, due to <a href="/facts/Fine_structure/6lejlmN2">fine structure</a>, <a href="/facts/Hyperfine_splitting/aqj0dMut">hyperfine splitting</a>, and other such effects, the Rydberg constant R ∞ {\displaystyle R_{\infty }} cannot be <i>directly</i> measured at very high accuracy from the <a href="/facts/Atomic_spectral_line/mj71FuNK">atomic transition frequencies</a> of hydrogen alone. Instead, the Rydberg constant is inferred from measurements of atomic transition frequencies in three different atoms (<a href="/facts/Hydrogen/BoYHjl0J">hydrogen</a>, <a href="/facts/Deuterium/Bv4wfCkw">deuterium</a>, and <a href="/facts/Antiprotonic_helium/0nnuFFpn">antiprotonic helium</a>). Detailed theoretical calculations in the framework of <a href="/facts/Quantum_electrodynamics/zPteAr6G">quantum electrodynamics</a> are used to account for the effects of finite nuclear mass, fine structure, hyperfine splitting, and so on. Finally, the value of R ∞ {\displaystyle R_{\infty }} is determined from the <a href="/facts/Best_fit/zrLKw6Xj">best fit</a> of the measurements to the theory.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> <h2 id="alternative-expressions">Alternative expressions</h2> <p>The Rydberg constant can also be expressed as in the following equations. </p> R ∞ = α 2 m e c 2 h = α 2 2 λ e = α 4 π a 0 {\displaystyle R_{\infty }={\frac {\alpha ^{2}m_{\text{e}}c}{2h}}={\frac {\alpha ^{2}}{2\lambda _{\text{e}}}}={\frac {\alpha }{4\pi a_{0}}}} <p>and in energy units </p> Ry = h c R ∞ = 1 2 m e c 2 α 2 = 1 2 e 4 m e ( 4 π ε 0 ) 2 ℏ 2 = 1 2 m e c 2 r e a 0 = 1 2 h c α 2 λ e = 1 2 h f C α 2 = 1 2 ℏ ω C α 2 = 1 2 m e ( ℏ a 0 ) 2 = 1 2 e 2 ( 4 π ε 0 ) a 0 , {\displaystyle {\text{Ry}}=hcR_{\infty }={\frac {1}{2}}m_{\text{e}}c^{2}\alpha ^{2}={\frac {1}{2}}{\frac {e^{4}m_{\text{e}}}{(4\pi \varepsilon _{0})^{2}\hbar ^{2}}}={\frac {1}{2}}{\frac {m_{\text{e}}c^{2}r_{\text{e}}}{a_{0}}}={\frac {1}{2}}{\frac {hc\alpha ^{2}}{\lambda _{\text{e}}}}={\frac {1}{2}}hf_{\text{C}}\alpha ^{2}={\frac {1}{2}}\hbar \omega _{\text{C}}\alpha ^{2}={\frac {1}{2m_{\text{e}}}}\left({\dfrac {\hbar }{a_{0}}}\right)^{2}={\frac {1}{2}}{\frac {e^{2}}{(4\pi \varepsilon _{0})a_{0}}},} <p>where </p> <ul><li> m e {\displaystyle m_{\text{e}}} is the <a href="/facts/Electron_rest_mass/xlG83aQl">electron rest mass</a>,</li> <li> e {\displaystyle e} is the <a href="/facts/Electric_charge/xxsBfmlB">electric charge</a> of the electron,</li> <li> h {\displaystyle h} is the <a href="/facts/Planck_constant/qGHAe3qq">Planck constant</a>,</li> <li> ℏ = h / 2 π {\displaystyle \hbar =h/2\pi } is the <a href="/facts/Reduced_Planck_constant/qGHAe3qq">reduced Planck constant</a>,</li> <li> c {\displaystyle c} is the <a href="/facts/Speed_of_light/sKSQwdNx">speed of light</a> in vacuum,</li> <li> ε 0 {\displaystyle \varepsilon _{0}} is the <a href="/facts/Electric_constant/gWRUhPGY">electric constant</a> (vacuum permittivity),</li> <li> α = 1 4 π ε 0 e 2 ℏ c {\displaystyle \alpha ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{\hbar c}}} is the <a href="/facts/Fine-structure_constant/zyzpmTJV">fine-structure constant</a>,</li> <li> λ e = h / m e c {\displaystyle \lambda _{\text{e}}=h/m_{\text{e}}c} is the <a href="/facts/Compton_wavelength/og2h1J2q">Compton wavelength</a> of the electron,</li> <li> f C = m e c 2 / h {\displaystyle f_{\text{C}}=m_{\text{e}}c^{2}/h} is the Compton frequency of the electron,</li> <li> ω C = 2 π f C {\displaystyle \omega _{\text{C}}=2\pi f_{\text{C}}} is the Compton angular frequency of the electron,</li> <li> a 0 = 4 π ε 0 ℏ 2 / e 2 m e {\displaystyle a_{0}={4\pi \varepsilon _{0}\hbar ^{2}}/{e^{2}m_{\text{e}}}} is the <a href="/facts/Bohr_radius/vvP1lghl">Bohr radius</a>,</li> <li> r e = 1 4 π ε 0 e 2 m e c 2 {\displaystyle r_{\mathrm {e} }={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{m_{\mathrm {e} }c^{2}}}} is the <a href="/facts/Classical_electron_radius/dU9jFS10">classical electron radius</a>.</li></ul> <p>The last expression in the first equation shows that the wavelength of light needed to ionize a hydrogen atom is 4<i>π</i>/<i>α</i> times the Bohr radius of the atom. </p><p>The second equation is relevant because its value is the coefficient for the energy of the atomic orbitals of a hydrogen atom: E n = − h c R ∞ / n 2 {\displaystyle E_{n}=-hcR_{\infty }/n^{2}} . </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Lyman_limit/aq5PglAX">Lyman limit</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Pohl, Randolf; Antognini, Aldo; Nez, François; Amaro, Fernando D.; Biraben, François; Cardoso, João M. R.; Covita, Daniel S.; Dax, Andreas; Dhawan, Satish; Fernandes, Luis M. P.; Giesen, Adolf; Graf, Thomas; Hänsch, Theodor W.; Indelicato, Paul; Julien, Lucile; Kao, Cheng-Yang; Knowles, Paul; Le Bigot, Eric-Olivier; Liu, Yi-Wei; Lopes, José A. M.; Ludhova, Livia; Monteiro, Cristina M. B.; Mulhauser, Françoise; Nebel, Tobias; Rabinowitz, Paul; Dos Santos, Joaquim M. F.; Schaller, Lukas A.; Schuhmann, Karsten; Schwob, Catherine; Taqqu, David (2010). "The size of the proton". Nature. 466 (7303): 213–216. Bibcode:2010Natur.466..213P. doi:10.1038/nature09250. PMID 20613837. S2CID 4424731. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"2022 CODATA Value: Rydberg constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18. <a href="https://physics.nist.gov/cgi-bin/cuu/Value?ryd" target="_blank">https://physics.nist.gov/cgi-bin/cuu/Value?ryd</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Quantum Mechanics (2nd Edition), B.H. Bransden, C.J. Joachain, Prentice Hall publishers, 2000, ISBN 0-582-35691-1 <a href="/wiki/Charles_J._Joachain" target="_blank">/wiki/Charles_J._Joachain</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>"2022 CODATA Value: Rydberg constant times hc in J". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18. <a href="https://physics.nist.gov/cgi-bin/cuu/Value?rydhcj" target="_blank">https://physics.nist.gov/cgi-bin/cuu/Value?rydhcj</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>"2022 CODATA Value: Rydberg constant times hc in eV". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18. <a href="https://physics.nist.gov/cgi-bin/cuu/Value?rydhcev" target="_blank">https://physics.nist.gov/cgi-bin/cuu/Value?rydhcev</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>"2022 CODATA Value: Rydberg constant times c in Hz". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18. <a href="https://physics.nist.gov/cgi-bin/cuu/Value?rydchz" target="_blank">https://physics.nist.gov/cgi-bin/cuu/Value?rydchz</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Coffman, Moody L. (1965). "Correction to the Rydberg Constant for Finite Nuclear Mass". American Journal of Physics. 33 (10): 820–823. Bibcode:1965AmJPh..33..820C. doi:10.1119/1.1970992. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>"2022 CODATA Value: Rydberg constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18. <a href="https://physics.nist.gov/cgi-bin/cuu/Value?ryd" target="_blank">https://physics.nist.gov/cgi-bin/cuu/Value?ryd</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>P.J. Mohr, B.N. Taylor, and D.B. Newell (2015), "The 2014 CODATA Recommended Values of the Fundamental Physical Constants" (Web Version 7.0). 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. Link to R∞, Link to hcR∞. Published in Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2012). "CODATA recommended values of the fundamental physical constants: 2010". Reviews of Modern Physics. 84 (4): 1527–1605. arXiv:1203.5425. Bibcode:2012RvMP...84.1527M. doi:10.1103/RevModPhys.84.1527. S2CID 103378639""{{cite journal}}: CS1 maint: postscript (link) and Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2012). "CODATA Recommended Values of the Fundamental Physical Constants: 2010". Journal of Physical and Chemical Reference Data. 41 (4): 043109. arXiv:1507.07956. Bibcode:2012JPCRD..41d3109M. doi:10.1063/1.4724320""{{cite journal}}: CS1 maint: postscript (link). <a href="/wiki/Svetlana_Kotochigova" target="_blank">/wiki/Svetlana_Kotochigova</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA recommended values of the fundamental physical constants: 2006". Reviews of Modern Physics. 80 (2): 633–730. arXiv:0801.0028. Bibcode:2008RvMP...80..633M. doi:10.1103/RevModPhys.80.633. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> </ol>