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Rydberg constant

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Physical constants of energy and wavenumber

Inspectroscopy, theRydberg constant, symbol $R_{\infty }$ for heavy atoms or $R_{\text{H}}$ for hydrogen, named after the Swedishphysicist Johannes Rydberg, is aphysical constant relating to the electromagneticspectra of an atom. The constant first arose as an empirical fitting parameter in theRydberg formula for thehydrogen spectral series, butNiels Bohr later showed that its value could be calculated from more fundamental constants according to hismodel of the atom.

Before the2019 revision of the SI, $R_{\infty }$ and the electron sping-factor were the most accurately measuredphysical constants.^[1]

The constant is expressed for either hydrogen as $R_{\text{H}}$ , or at the limit of infinite nuclear mass as $R_{\infty }$ . In either case, the constant is used to express the limiting value of the highestwavenumber (inverse wavelength) of any photon that can be emitted from a hydrogen atom, or, alternatively, the wavenumber of the lowest-energy photon capable of ionizing a hydrogen atom from itsground state. Thehydrogen spectral series can be expressed simply in terms of the Rydberg constant for hydrogen $R_{\text{H}}$ and theRydberg formula.

Inatomic physics,Rydberg unit of energy, symbol Ry, corresponds to the energy of the photon whose wavenumber is the Rydberg constant, i.e. the ionization energy of the hydrogen atom in a simplified Bohr model.^{[citation needed]}

Value

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Rydberg constant

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TheCODATA value is

R_{\infty }={\frac {m_{\text{e}}e^{4}}{8\varepsilon _{0}^{2}h^{3}c}}=

10973731.568157(12) m⁻¹,^[2]

where

$m_{\text{e}}$ is therest mass of theelectron (i.e. theelectron mass),
$e {\displaystyle e}$ is theelementary charge,
$\varepsilon _{0}$ is thepermittivity of free space,
$h {\displaystyle h}$ is thePlanck constant, and
$c {\displaystyle c}$ is thespeed of light in vacuum.

The symbol $\infty$ means that the nucleus is assumed to be infinitely heavy, an improvement of the value can be made using thereduced mass of the atom:

\mu ={\frac {1}{{\frac {1}{m_{\text{e}}}}+{\frac {1}{M}}}}

with $M {\displaystyle M}$ the mass of the nucleus. The corrected Rydberg constant is:

R_{\text{M}}={\frac {\mu }{m_{\text{e}}}}R_{\infty }

that for hydrogen, where $M {\displaystyle M}$ is the mass $m_{\text{p}}$ of theproton, becomes:

R_{\text{H}}={\frac {m_{\text{p}}}{m_{\text{e}}+m_{\text{p}}}}R_{\infty }\approx 1.09678\times 10^{7}{\text{ m}}^{-1},

Since the Rydberg constant is related to the spectrum lines of the atom, this correction leads to anisotopic shift between different isotopes. For example, deuterium, an isotope of hydrogen with a nucleus formed by a proton and aneutron ( $M=m_{\text{p}}+m_{\text{n}}\approx 2m_{\text{p}}$ ), was discovered thanks to its slightly shifted spectrum.^[3]

Rydberg unit of energy

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The Rydberg unit of energy is

1\ {\text{Ry}}~~\equiv hc\,R_{\infty }=\alpha ^{2}m_{\text{e}}c^{2}/2

=2.1798723611030(24)×10⁻¹⁸ J‍^[4]

=13.605693122990(15) eV‍^[5]

Rydberg frequency

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cR_{\infty }

=3.2898419602500(36)×10¹⁵ Hz.^[6]

Rydberg wavelength

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{\frac {1}{R_{\infty }}}=9.112\;670\;505\;826(10)\times 10^{-8}\ {\text{m}}

The correspondingangular wavelength is

{\frac {1}{2\pi R_{\infty }}}=1.450\;326\;555\;77(16)\times 10^{-8}\ {\text{m}}

Bohr model

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Main article:Bohr model

TheBohr model explains the atomicspectrum of hydrogen (seeHydrogen 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 ofquantum mechanics. The Bohr model posits that electrons revolve around the atomic nucleus in a manner analogous to planets revolving around the Sun.

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,^[7] so that the center of mass of the system, thebarycenter, lies at the center of the nucleus. This infinite mass approximation is what is alluded to with the $\infty$ subscript. The Bohr model then predicts that the wavelengths of hydrogen atomic transitions are (seeRydberg formula):

{\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)

wheren₁ andn₂ are any two different positive integers (1, 2, 3, ...), and $\lambda$ is the wavelength (in vacuum) of the emitted or absorbed light, giving

{\frac {1}{\lambda }}=R_{M}\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)

where $R_{M}={\frac {R_{\infty }}{1+{\frac {m_{\text{e}}}{M}}}},$ andM is the total mass of the nucleus. This formula comes from substituting thereduced mass of the electron.

Precision measurement

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Alternative expressions

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The Rydberg constant can also be expressed as in the following equations.

R_{\infty }={\frac {\alpha ^{2}m_{\text{e}}c}{2h}}={\frac {\alpha ^{2}}{2\lambda _{\text{e}}}}={\frac {\alpha }{4\pi a_{0}}}

and in energy units

{\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}}},

where

$m_{\text{e}}$ is theelectron rest mass,
$e {\displaystyle e}$ is theelectric charge of the electron,
$h {\displaystyle h}$ is thePlanck constant,
$\hbar =h/2\pi$ is thereduced Planck constant,
$c {\displaystyle c}$ is thespeed of light in vacuum,
$\varepsilon _{0}$ is theelectric constant (vacuum permittivity),
$\alpha ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{\hbar c}}$ is thefine-structure constant,
$\lambda _{\text{e}}=h/m_{\text{e}}c$ is theCompton wavelength of the electron,
$f_{\text{C}}=m_{\text{e}}c^{2}/h$ is the Compton frequency of the electron,
$\omega _{\text{C}}=2\pi f_{\text{C}}$ is the Compton angular frequency of the electron,
$a_{0}={4\pi \varepsilon _{0}\hbar ^{2}}/{e^{2}m_{\text{e}}}$ is theBohr radius,
$r_{\mathrm {e} }={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{m_{\mathrm {e} }c^{2}}}$ is theclassical electron radius.

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.

The second equation is relevant because its value is the coefficient for the energy of the atomic orbitals of a hydrogen atom: $E_{n}=-hcR_{\infty }/n^{2}$ .

References

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^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 ^b"2022 CODATA Value: Rydberg constant".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
^Quantum Mechanics (2nd Edition), B.H. Bransden,C.J. Joachain, Prentice Hall publishers, 2000,ISBN 0-582-35691-1
^"2022 CODATA Value: Rydberg constant times hc in J".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
^"2022 CODATA Value: Rydberg constant times hc in eV".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
^"2022 CODATA Value: Rydberg constant times c in Hz".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
^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.
^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, andS. Kotochigova. Available:http://physics.nist.gov/constants. National Institute of Standards and Technology, Gaithersburg, MD 20899.Link to R_∞,Link to hcR_∞. Published inMohr, 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) andMohr, 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).
^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.

v t e Scientists whose names are used in physical constants
Physical constants	Isaac Newton (Newtonian constant of gravitation) Amedeo Avogadro (Avogadro constant) Michael Faraday (Faraday constant) Johann Josef Loschmidt (Loschmidt constant) Johann Jakob Balmer Josef Stefan (Stefan–Boltzmann constant) Ludwig Boltzmann (Boltzmann constant,Stefan–Boltzmann constant) Johannes Rydberg (Rydberg constant) J. J. Thomson Max Planck (Planck constant) Wilhelm Wien Otto Sackur Niels Bohr (Bohr radius) Edwin Hubble (Hubble constant) Hugo Tetrode Douglas Hartree Brian Josephson (Josephson constant) Klaus von Klitzing (von Klitzing constant)
List of scientists whose names are used as SI units andnon SI units

Scientists whose names are used in physical constants

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