In spectroscopy, the Rydberg constant, symbol \( {\displaystyle R_{\infty }} \) for heavy atoms or \( R_{\text{H}} \) for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectra of an atom. 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 via his Bohr model. As of 2018, \( {\displaystyle R_{\infty }} \) and electron spin g-factor are the most accurately measured physical constants.[1]
The constant is expressed for either hydrogen as \( R_{\text{H}} \), or at the limit of infinite nuclear mass as \( {\displaystyle R_{\infty }} \). In either case, the constant is used to express the limiting value of the highest wavenumber (inverse wavelength) of any photon that can be emitted from an atom, or, alternatively, the wavenumber of the lowest-energy photon capable of ionizing an atom from its ground state. The hydrogen spectral series can be expressed simply in terms of the Rydberg constant for hydrogen \( R_{\text{H}} \) and the Rydberg formula.
In atomic 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.
Value
Rydberg constant
The CODATA value is[2]
\( {\displaystyle R_{\infty }={\frac {m_{\text{e}}e^{4}}{8\varepsilon _{0}^{2}h^{3}c}}} = 10973731.568160(21) \) m−1,
where
\( m_{\text{e}} \) is the rest mass of the electron,
e is the elementary charge,
\( \varepsilon _{0} \) is the permittivity of free space,
h is the Planck constant, and
c is the speed of light in vacuum.
The Rydberg constant for hydrogen may be calculated from the reduced mass of the electron:
\( {\displaystyle R_{\text{H}}=R_{\infty }{\frac {m_{\text{p}}}{m_{\text{e}}+m_{\text{p}}}}\approx 1.09678\times 10^{7}{\text{ m}}^{-1},}\)
where
\( m_{\text{e}} \) is the mass of the electron,
\( m_\text{p} \) is the mass of the nucleus (a proton).
Rydberg unit of energy
\( {\displaystyle 1\ {\text{Ry}}\equiv hcR_{\infty }={\frac {m_{\text{e}}e^{4}}{8\varepsilon _{0}^{2}h^{2}}}=2.179\;872\;361\;1035(42)\times 10^{-18}\ {\text{J}}}[3] = 13.605 693 122 994 ( 26 ) eV . {\displaystyle \ =13.605\;693\;122\;994(26)\ {\text{eV}}.} {\displaystyle \ =13.605\;693\;122\;994(26)\ {\text{eV}}.\) }[4]
Rydberg frequency
\( {\displaystyle cR_{\infty }=3.289\;841\;960\;2508(64)\times 10^{15}\ {\text{Hz}}.}\) [5]
Rydberg wavelength
\( {\displaystyle {\frac {1}{R_{\infty }}}=9.112\;670\;505\;824(17)\times 10^{-8}\ {\text{m}}}.\)
The angular wavelength is
\({\displaystyle {\frac {1}{2\pi R_{\infty }}}=1.450\;326\;555\;7696(28)\times 10^{-8}\ {\text{m}}}.\)
Occurrence in Bohr model
Main article: Bohr model
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.
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,[6] so that the center of mass of the system, the barycenter, lies at the center of the nucleus. This infinite mass approximation is what is alluded to with the ∞ {\displaystyle \infty } \infty subscript. The Bohr model then predicts that the wavelengths of hydrogen atomic transitions are (see Rydberg formula):
\( {\displaystyle {\frac {1}{\lambda }}=\mathrm {Ry} \cdot {1 \over hc}\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)={\frac {m_{\text{e}}e^{4}}{8\varepsilon _{0}^{2}h^{3}c}}\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)}\)
where n1 and n2 are any two different positive integers (1, 2, 3, ...), and \( \lambda \) is the wavelength (in vacuum) of the emitted or absorbed light.
\( {\displaystyle {\frac {1}{\lambda }}=R_{M}\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)} \)
where \( R_{M}=R_{\infty }/(1+m_{{{\text{e}}}}/M) \), and M is the total mass of the nucleus. This formula comes from substituting the reduced mass of the electron.
Precision measurement
See also: Precision tests of QED
The Rydberg constant is one of the most precisely determined physical constants, with a relative standard uncertainty of under 2 parts in 1012.[2] This precision constrains the values of the other physical constants that define it.[7]
Since the Bohr model is not perfectly accurate, due to fine structure, hyperfine splitting, and other such effects, the Rydberg constant R ∞ {\displaystyle R_{\infty }} R_{{\infty }} cannot be directly measured at very high accuracy from the 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 R ∞ {\displaystyle R_{\infty }} R_{{\infty }} is determined from the best fit of the measurements to the theory.[8]
Alternative expressions
The Rydberg constant can also be expressed as in the following equations.
\( R_{\infty }={\frac {\alpha ^{2}m_{{\text{e}}}c}{4\pi \hbar }}={\frac {\alpha ^{2}}{2\lambda _{{{\text{e}}}}}}={\frac {\alpha }{4\pi a_{0}}}\)
and
\( {\displaystyle hcR_{\infty }={\frac {1}{2}}m_{\text{e}}c^{2}\alpha ^{2}={\frac {1}{2}}{\frac {e^{4}m_{\text{e}}}{(4\pi \varepsilon _{0})^{2}\hbar ^{2}}}={\frac {1}{2}}{\frac {m_{\text{e}}c^{2}r_{\text{e}}}{a_{0}}}={\frac {1}{2}}{\frac {hc\alpha ^{2}}{\lambda _{\text{e}}}}={\frac {1}{2}}hf_{\text{C}}\alpha ^{2}={\frac {1}{2}}\hbar \omega _{\text{C}}\alpha ^{2}={\frac {1}{2m_{\text{e}}}}\left({\dfrac {\hbar }{a_{0}}}\right)^{2}={\frac {1}{2}}{\frac {e^{2}}{(4\pi \varepsilon _{0})a_{0}}}.}\)
where
\( m_{\text{e}}\) is the electron rest mass,
e is the electric charge of the electron,
h is the Planck constant,
\( \hbar =h/2\pi \) is the reduced Planck constant,
c is the speed of light in a vacuum,
\( \varepsilon _{0} \) is the electrical field constant (permittivity) of free space,
\( {\displaystyle \alpha ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{\hbar c}}} \) is the fine-structure constant,
\( \lambda _{{{\text{e}}}}=h/m_{{\text{e}}}c \) is the Compton 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,
\( a_{0}={\frac {4\pi \varepsilon _{0}\hbar ^{2}}{e^{2}m_{{{\text{e}}}}}} \) is the Bohr radius,
\( r_{{\mathrm {e}}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{m_{{{\mathrm {e}}}}c^{2}}}\) is the classical 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: \( E_{n}=-hcR_{\infty }/n^{2}.\)
See also
Rydberg formula, includes a discussion of Rydberg's original discovery
Related physical constants:
Bohr radius
Planck constant
Fine-structure constant
Electron rest mass
References
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.
"2018 CODATA Value: Rydberg constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
"2018 CODATA Value: Rydberg constant times hc in J". NIST. The NIST Reference on Constants, Units, and Uncertainty. Retrieved 2020-02-06.
"2018 CODATA Value: Rydberg constant times hc in eV". NIST. The NIST Reference on Constants, Units, and Uncertainty. Retrieved 2020-02-06.
"2018 CODATA Value: Rydberg constant times c in Hz". NIST. The NIST Reference on Constants, Units, and Uncertainty. Retrieved 2020-02-05.
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, 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.arXiv:1203.5425. Bibcode:2012RvMP...84.1527M. doi:10.1103/RevModPhys.84.1527"" 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"".
Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA recommended values of the fundamental physical constants: 2006".arXiv: 80 (2): 633–730. arXiv:0801.0028. Bibcode:2008RvMP...80..633M. doi:10.1103/RevModPhys.80.633.
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Scientists whose names are used in physical constants
Physical constants
Isaac Newton (gravitational constant) Charles-Augustin de Coulomb (Coulomb constant) Amedeo Avogadro (Avogadro constant) Michael Faraday (Faraday constant) Johann Josef Loschmidt 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 Klaus von Klitzing
Hellenica World - Scientific Library
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