Editing Rydberg constant

{{Use American English|date = March 2019}}
{{Short description|Physical constants of energy and wavenumber}}
In [[spectroscopy]], the '''Rydberg constant''', symbol <math>R_\infty</math> for 
heavy atoms or <math>R_\text{H}</math> for hydrogen, named after the Swedish [[physicist]] [[Johannes Rydberg]], is a [[physical constant]] relating to the electromagnetic [[spectrum|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 according to his [[Bohr model|model of the atom]]. 

Before the [[2019 revision of the SI]], <math>R_\infty</math> and the electron spin [[g-factor (physics)|''g''-factor]] were the most accurately measured [[physical constant]]s.<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 |s2cid=4424731 }}</ref>

The constant is expressed for either hydrogen as <math>R_\text{H}</math>, or at the limit of infinite nuclear mass as <math>R_\infty</math>. 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 a hydrogen atom, or, alternatively, the wavenumber of the lowest-energy photon capable of ionizing a hydrogen atom from its [[ground state]]. The [[hydrogen spectral series]] can be expressed simply in terms of the Rydberg constant for hydrogen <math>R_\text{H}</math> 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.{{cn|date=April 2019}}

== Value ==
=== Rydberg constant === 
The [[CODATA]] value is
: <math> R_\infty = \frac{m_\text{e} e^4}{8 \varepsilon_{0}^{2} h^3 c} = </math> {{physconst|Rinf|after=,}}
where
* <math>m_\text{e}</math> is the [[rest mass]] of the [[electron]] (i.e. the [[electron mass]]),
* <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 vacuum.

The symbol <math>\infty</math> means that the nucleus is assumed to be infinitely heavy, an improvement of the value can be made using the [[reduced mass]] of the atom:
: <math>\mu = \frac{ 1 }{ \frac{1}{m_\text{e}} + \frac{1}{M} }</math>
with <math>M</math> the mass of the nucleus. The corrected Rydberg constant is:
: <math>R_\text{M} = \frac{ \mu }{ m_\text{e} }R_\infty </math>
that for hydrogen, where <math>M</math> is the mass <math>m_\text{p}</math> of the [[proton]], becomes:
: <math>R_\text{H} = \frac{ m_\text{p} }{ m_\text{e}+m_\text{p} }R_\infty  \approx 1.09678 \times 10^7 \text{ m}^{-1} ,</math>

Since the Rydberg constant is related to the spectrum lines of the atom, this correction leads to an [[isotopic shift]] between different isotopes. For example, deuterium, an isotope of hydrogen with a nucleus formed by a proton and a [[neutron]] (<math>M = m_\text{p} + m_\text{n}\approx 2m_\text{p}</math>), was discovered thanks to its slightly shifted spectrum.<ref>''Quantum Mechanics'' (2nd Edition), B.H. Bransden, [[Charles J. Joachain|C.J. Joachain]], Prentice Hall publishers, 2000, {{ISBN|0-582-35691-1}}</ref>

=== Rydberg unit of energy ===
The Rydberg unit of energy is
: <math>1 \ \text{Ry} ~~ \equiv h c\, R_\infty = \alpha^2 m_\text{e} c^2 /2</math>
::: = {{physconst|Rinfhc}}
::: = {{physconst|Rinfhc_eV}}

=== Rydberg frequency ===
: <math>c R_\infty</math> = {{physconst|Rinfc|after=.}}

=== Rydberg wavelength ===
: <math>\frac 1 {R_\infty} = 9.112\;670\;505\;826(10) \times 10^{-8}\ \text{m}</math>.

The corresponding [[angular wavelength]] is
: <math>\frac 1 {2\pi R_\infty} = 1.450\;326\;555\;77(16) \times 10^{-8}\ \text{m}</math>.

== Bohr model ==
{{main|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,<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, the [[center of mass|barycenter]], lies at the center 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]]''):
: <math>\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) </math>
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, giving
: <math>\frac{1}{\lambda} =  R_M\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)</math>
where <math>R_M = \frac{R_\infty}{1+\frac{m_{\text{e}}}{M}},</math> 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 was one of the most precisely determined physical constants, with a relative standard uncertainty of {{physconst|Rinf|runc=yes|after=.}} This precision constrains the values of the other physical constants that define it.<ref name="codata">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 [[Svetlana Kotochigova|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=1527–1605|arxiv = 1203.5425 |bibcode = 2012RvMP...84.1527M |s2cid=103378639}} 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 |arxiv=1507.07956}}.</ref>

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> is determined 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>

== Alternative expressions ==
The Rydberg constant can also be expressed as in the following equations.
: <math>R_\infty = \frac{\alpha^2 m_\text{e} c}{2h} = \frac{\alpha^2}{2 \lambda_{\text{e}}} = \frac{\alpha}{4\pi a_0}</math>
and in energy units
: <math>\text{Ry} = h c R_\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{h c \alpha^2}{\lambda_{\text{e}}} = \frac{1}{2} h f_{\text{C}} \alpha^2 = \frac{1}{2} \hbar \omega_{\text{C}} \alpha^2 = \frac{1}{2 m_{\text{e}}}\left(\dfrac{\hbar}{a_0}\right)^2 = \frac{1}{2}\frac{e^2}{(4\pi\varepsilon_0)a_0} ,</math>
where
* <math>m_\text{e}</math> is the [[electron rest mass]],
* <math>e</math> is the [[electric charge]] of the electron,
* <math>h</math> is the [[Planck constant]],
* <math>\hbar= h/2\pi</math> is the [[reduced Planck constant]],
* <math>c</math> is the [[speed of light]] in vacuum,
* <math>\varepsilon_0</math> is the [[electric constant]] (vacuum permittivity),
* <math>\alpha = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{\hbar c}</math> is the [[fine-structure constant]],
* <math>\lambda_{\text{e}} = h/m_\text{e} c</math> is the [[Compton wavelength]] of the electron,
* <math>f_{\text{C}}=m_{\text{e}} c^2/h</math> is the Compton frequency of the electron,
* <math>\omega_{\text{C}}=2\pi f_{\text{C}}</math> is the Compton angular frequency of the electron,
* <math>a_0={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]].

The last expression in the first equation shows that the wavelength of light needed to ionize a hydrogen atom is 4''π''/''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: <math>E_n = -h c R_\infty / n^2 </math>.

== See also ==
* [[Lyman limit]]

== References ==
{{reflist}}

{{Scientists whose names are used in physical constants}}

[[Category:Emission spectroscopy]]
[[Category:Physical constants]]
[[Category:Units of energy]]