Rydberg constant: Difference between revisions
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The '''Rydberg constant''', often denoted as ''R<sub>∞</sub>'', originally defined empirically in terms of the spectrum of hydrogen, is given a theoretical value by the Bohr theory of the atom as:<ref name= > | The '''Rydberg constant''', often denoted as ''R<sub>∞</sub>'', originally defined empirically in terms of the spectrum of hydrogen, is given a theoretical value by the Bohr theory of the atom as (in [[SI units]]):<ref name= > | ||
{{cite book |title=The Spectrum of atomic hydrogen--advances: a collection of progress reports by experts |url=http://books.google.com/books?id=ELV3YdX16ooC&pg=PA485 |pages=p. 485 |isbn=9971502615 |year=1988 |publisher=World Scientific |author=GW Series |chapter=Chapter 10: Hydrogen and the fundamental atomic constants}} | {{cite book |title=The Spectrum of atomic hydrogen--advances: a collection of progress reports by experts |url=http://books.google.com/books?id=ELV3YdX16ooC&pg=PA485 |pages=p. 485 |isbn=9971502615 |year=1988 |publisher=World Scientific |author=GW Series |chapter=Chapter 10: Hydrogen and the fundamental atomic constants}} | ||
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:<math>R_{\infty} = \frac{m_ee^4}{4\pi \hbar^3 | :<math>R_{\infty} = \frac{m_ee^4}{4\pi \hbar^3 c_0}\ \left( \frac{\mu_0 c_0^2}{4 \pi}\right)^2 \ . </math> | ||
The best value (in 2005) was:<ref name= Grynberg> | The best value (in 2005) was:<ref name= Grynberg> | ||
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:R<sub>∞</sub>/(''hc'') = 10 973 731.568 525 (8) ''m''<sup>−1</sup>, | :R<sub>∞</sub>/(''hc<sub>0</sub>'') = 10 973 731.568 525 (8) ''m''<sup>−1</sup>, | ||
where ''h'' = [[Planck's constant]] and ''c'' = [[speed of light]] in vacuum. | where ''h'' = [[Planck's constant]] and ''c<sub>0</sub>'' = [[SI units]] defined value for the [[speed of light]] in vacuum. | ||
==Background== | |||
The Rydberg constant is the common scaling factor for all hydrogen transitions.<ref name=Cagnac> | |||
This discussion is based upon the review by {{cite journal |title=The hydrogen atom, a tool for metrology |author=B Cagnac, MD Plimmer, L Julien and F Biraben |journal=Rep. Prog. Phys. |volume=vol. 57 |year=1994 |pages=pp. 853-893 |publisher=IOP Publishing Ltd.}} | |||
</ref> Its measurement has proven often to be a testing ground for theoretical results. The original introduction of this constant by [[JR Rydberg]] in 1889 was through a formula for the wavelengths associated with alkali metal transitions, which can be formulated for the hydrogen atom as: | |||
:<math>\frac{1}{\lambda} = R\ \left(\frac{1}{p^2} - \frac{1}{n^2} \right ) \ , </math> | |||
where ''R'' is a constant and ''n'' and ''p'' are any integers, with ''n''>''p''. The case for ''p''=2 is called the ''Balmer series'' and for ''p''=1 the ''Lyman series''. The first value for the Rydberg constant was found using this formula, and was totally empirical. | |||
In 1913 [[Neils Bohr]] developed a theory of the atom predicting the hydrogen atom to have energy levels (in [[SI units]]): | |||
:<math>E_n = -\frac{me^4}{8\varepsilon _0^2 h^2 n^2} \ , </math> | |||
with ''e'' the electron charge, ''m'' the electron mass, ε<sub>0</sub> the [[electric constant]], ''h'' [[Planck's constant]], and ''n'' the so-called ''principal quantum number''. According to this model, the wavelength ''λ'' of a transition of an electron moving from state ''n'' to state ''p'' is then: | |||
:<math> \frac{1}{\lambda} = \frac{me^4}{8\varepsilon_0^2h^3c_0}\left ( \frac{1}{p^2} - \frac{1}{n^2} \right) \ , </math> | |||
with ''c<sub>0</sub>'' the SI defined valued for the [[speed of light]] in vacuum. Thus, a theoretical expression for the Rydberg constant is obtained. | |||
==Notes== | ==Notes== | ||
<references/> | <references/> | ||
Revision as of 08:57, 14 March 2011
The Rydberg constant, often denoted as R∞, originally defined empirically in terms of the spectrum of hydrogen, is given a theoretical value by the Bohr theory of the atom as (in SI units):[1]
The best value (in 2005) was:[2]
- R∞/(hc0) = 10 973 731.568 525 (8) m−1,
where h = Planck's constant and c0 = SI units defined value for the speed of light in vacuum.
Background
The Rydberg constant is the common scaling factor for all hydrogen transitions.[3] Its measurement has proven often to be a testing ground for theoretical results. The original introduction of this constant by JR Rydberg in 1889 was through a formula for the wavelengths associated with alkali metal transitions, which can be formulated for the hydrogen atom as:
where R is a constant and n and p are any integers, with n>p. The case for p=2 is called the Balmer series and for p=1 the Lyman series. The first value for the Rydberg constant was found using this formula, and was totally empirical.
In 1913 Neils Bohr developed a theory of the atom predicting the hydrogen atom to have energy levels (in SI units):
with e the electron charge, m the electron mass, ε0 the electric constant, h Planck's constant, and n the so-called principal quantum number. According to this model, the wavelength λ of a transition of an electron moving from state n to state p is then:
with c0 the SI defined valued for the speed of light in vacuum. Thus, a theoretical expression for the Rydberg constant is obtained.
Notes
- ↑ GW Series (1988). “Chapter 10: Hydrogen and the fundamental atomic constants”, The Spectrum of atomic hydrogen--advances: a collection of progress reports by experts. World Scientific, p. 485. ISBN 9971502615.
- ↑ Gilbert Grynberg, Alain Aspect, Claude Fabre (2010). Introduction to Quantum Optics: From the Semi-classical Approach to Quantized Light. Cambridge University Press, p. 297. ISBN 0521551129.
- ↑ This discussion is based upon the review by B Cagnac, MD Plimmer, L Julien and F Biraben (1994). "The hydrogen atom, a tool for metrology". Rep. Prog. Phys. vol. 57: pp. 853-893.