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| [[Image:Rydbergformula.jpg|thumb|Rydberg's formula as it appears in a November 1888 record]]
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| The '''Rydberg formula''' is used in [[atomic physics]] to describe the wavelengths of [[spectral lines]] of many [[chemical element]]s. It was formulated by the Swedish [[physicist]] [[Johannes Rydberg]], and presented on November 5, 1888.
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| ==History==
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| In the 1880s, Rydberg worked on a formula describing the relation between the wavelengths in spectral lines of alkali metals. He noticed that lines came in series and he found that he could simplify his calculations by using the [[wavenumber]] (the number of waves occupying the [[unit length]], equal to 1/''λ'', the inverse of the [[wavelength]]) as his unit of measurement. He plotted the wavenumbers (''n'') of successive lines in each series against consecutive integers which represented the order of the lines in that particular series. Finding that the resulting curves were similarly shaped, he sought a single function which could generate all of them, when appropriate constants were inserted.
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| First he tried the formula: <math>\scriptstyle n=n_0 - \frac{C_0}{m+m'}</math>, where ''n'' is the line's wavenumber, ''n''<sub>0</sub> is the series limit, ''m'' is the line's ordinal number in the series, ''m''' is a constant different for different series and ''C''<sub>0</sub> is a universal constant. This did not work very well.
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| Rydberg was trying: <math>\scriptstyle n=n_0 - \frac{C_0}{\left(m+m'\right)^2}</math> when he became aware of [[Balmer's formula]] for the [[hydrogen spectrum]] <math>\scriptstyle \lambda={hm^2 \over m^2-4}</math> In this equation, ''m'' is an integer and ''h'' is a constant (not to be confused with the later [[Planck's constant]], a completely different number now usually given the symbol ''ħ'', which refers to Planck's Constant divided by 2π).
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| Rydberg therefore rewrote Balmer's formula in terms of wavenumbers, as <math>\scriptstyle n=n_0 - {4n_0 \over m^2}</math>.
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| This suggested that the Balmer formula for hydrogen might be a special case with <math>\scriptstyle m'=0\!</math> and <math>\scriptstyle C_0=4n_0\!</math>, where <math>\scriptstyle n_0=\frac{1}{h}</math>, the reciprocal of Balmer's constant (this constant '''h''' is written '''B''' in the [[Balmer equation]] article, again to avoid confusion with Planck's constant).
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| The term ''C''<sub>o</sub> was found to be a universal constant common to all elements, equal to 4/''h''. This constant is now known as the [[Rydberg constant]], and ''m''' is known as the [[quantum defect]].
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| As stressed by [[Niels Bohr]],<ref>{{Cite book |first=N. |last=Bohr |chapter=Rydberg's discovery of the spectral laws |title=Collected works |editor-first=J. |editor-last=Kalckar |publisher=North-Holland Publ. Cy. |location=Amsterdam |year=1985 |volume=10 |pages=373–379 }}</ref> expressing results in terms of wavenumber, not wavelength, was the key to Rydberg's discovery. The fundamental role of wavenumbers was also emphasized by the [[Rydberg-Ritz combination principle]] of 1908. The fundamental reason for this lies in [[quantum mechanics]]. Light's wavenumber is proportional to frequency <math>\scriptstyle \frac{1}{\lambda}=\frac{f}{c}</math>, and therefore also proportional to light's quantum energy ''E''. Thus, <math>\scriptstyle \frac{1}{\lambda}=\frac{E}{hc}</math>. Modern understanding is that Rydberg's findings were a reflection of the underlying simplicity of the behavior of spectral lines, in terms of fixed (quantized) ''energy'' differences between [[electron]] orbitals in atoms. Rydberg's 1888 classical expression for the form of the spectral series was not accompanied by a physical explanation. [[Walther Ritz|Ritz]]'s ''pre-quantum'' 1908 explanation for the ''mechanism'' underlying the spectral series was that atomic electrons behaved like magnets and that the magnets could vibrate with respect to the atomic nucleus (at least temporarily) to produce electromagnetic radiation,<ref name=Ritz>{{Cite journal |first=W. |last=Ritz |title={{lang|de|Magnetische Atomfelder und Serienspektren}} |journal=[[Annalen der Physik]] |volume=330 |issue=4 |year=1908 |pages=660–696 |doi=10.1002/andp.19083300403 |bibcode = 1908AnP...330..660R }}</ref> but this theory was superseded in 1913 by [[Niels Bohr]]'s [[Bohr model|model of the atom]].
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| In Bohr's conception of the atom, the integer Rydberg (and Balmer) ''n'' numbers represent electron orbitals at different integral distances from the atom. A frequency (or spectral energy) emitted in a transition from ''n''<sub>1</sub> to ''n''<sub>2</sub> therefore represents the photon energy emitted or absorbed when an electron makes a jump from orbital 1 to orbital 2.
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| ==Rydberg formula for hydrogen==
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| :<math>\frac{1}{\lambda_{\mathrm{vac}}} = R\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)</math>
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| Where
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| :<math>\lambda_{\mathrm{vac}} \!</math> is the [[wavelength]] of electromagnetic radiation emitted in [[vacuum]],
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| :<math>R\!</math> is the [[Rydberg constant]], approximately 1.097 x 10<sup>7</s> m<sup>-1</sup>,
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| :<math>n_1\!</math> and <math>n_2\!</math> are integers greater than or equal to 1 such that <math>n_1 < n_2\!</math>.
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| By setting <math>n_1</math> to 1 and letting <math>n_2</math> run from 2 to infinity, the spectral lines known as the [[Lyman series]] converging to 91 nm are obtained, in the same manner:
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| {|class=wikitable
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| ! n<sub>1</sub> !! n<sub>2</sub> !! Name !! Converge toward
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| | 1 || 2 → ∞ || [[Lyman series]] || 91.13 nm ([[Ultraviolet|UV]])
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| | 2 || 3 → ∞ || [[Balmer series]] || 364.51 nm ([[Visible spectrum|Visible]])
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| | 3 || 4 → ∞ || [[Paschen series]] || 820.14 nm ([[Infrared|IR]])
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| | 4 || 5 → ∞ || [[Brackett series]] || 1458.03 nm (Far IR)
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| | 5 || 6 → ∞ || [[Pfund series]] || 2278.17 nm (Far IR)
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| | 6 || 7 → ∞ || [[Humphreys series]] || 3280.56 nm (Far IR)
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| |}
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| [[Image:HydrogenSpectrum.PNG|thumb|A visual comparison of the Hydrogen spectral series for n1 = 1 to n1 = 6 on a Log scale.]]
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| ==Rydberg formula for any hydrogen-like element==
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| The formula above can be extended for use with any [[Hydrogen-like atom|hydrogen-like]] [[chemical element]]s with
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| :<math>\frac{1}{\lambda_{\mathrm{vac}}} = RZ^2 \left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)</math>
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| where
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| :<math>\lambda_{\mathrm{vac}}\!</math> is the [[wavelength]] of the light emitted in [[vacuum]]; | |
| :<math>R\!</math> is the [[Rydberg constant]] for this element;
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| :<math>Z\!</math> is the [[atomic number]], i.e. the number of [[proton]]s in the [[atomic nucleus]] of this element;
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| :<math>n_1\!</math> and <math>n_2\!</math> are integers such that <math>n_2 < n_1\!</math>.
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| It's important to notice that this formula can be directly applied only to hydrogen-like, also called ''hydrogenic'' atoms of [[chemical element]]s, i.e. atoms with only one electron being affected by an effective nuclear charge (which is easily estimated). Examples would include He<sup>+</sup>, Li<sup>2+</sup>, Be<sup>3+</sup> etc., where no other electrons exist in the atom.
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| But the Rydberg formula also provides correct wavelengths for distant electrons, where the effective nuclear charge can be estimated as the same as that for hydrogen, since all but one of the nuclear charges have been screened by other electrons, and the core of the atom has an effective positive charge of +1.
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| Finally, with certain modifications (replacement of '''''Z''''' by '''''Z''−1''', and use of the integers 1 and 2 for the ''n''s to give a numerical value of {{frac|3|4}} for the difference of their inverse squares), the Rydberg formula provides correct values in the special case of [[K-alpha]] lines, since the transition in question is the K-alpha transition of the electron from the 1s orbital to the 2p orbital. This is analogous to the [[Lyman-alpha line]] transition for hydrogen, and has the same frequency factor. Because the 2p electron is not screened by any other electrons in the atom from the nucleus, the nuclear charge is diminished only by the single remaining 1s electron, causing the system to be effectively a hydrogenic atom, but with a diminished nuclear charge ''Z''−1. Its frequency is thus the Lyman-alpha hydrogen frequency, increased by a factor of (''Z''−1)<sup>2</sup>. This formula of ''f'' = ''c''/''λ'' = (Lyman-alpha frequency)⋅(''Z''−1)<sup>2</sup> is historically known as [[Moseley's law]] (having added a factor '''''c''''' to convert wavelength to frequency), and can be used to predict wavelengths of the K<sub>α</sub> (K-alpha) X-ray spectral emission lines of chemical elements from aluminum to gold. See the biography of [[Henry Moseley]] for the historical importance of this law, which was derived empirically at about the same time it was explained by the [[Bohr model]] of the atom.
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| For other spectral transitions in multi-electron atoms, the Rydberg formula generally provides ''incorrect'' results, since the magnitude of the screening of inner electrons for outer-electron transitions is variable and not possible to compensate for in the simple manner above.
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| ==See also==
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| * [[Rydberg–Ritz combination principle]]
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| * [[Balmer series]]
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| * [[Hydrogen line]]
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| ==References==
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| {{reflist}}
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| * {{cite journal |first=Mike |last=Sutton |pages=38–41|title=Getting the numbers right: The lonely struggle of the 19th century physicist/chemist Johannes Rydberg |journal=Chemistry World |volume=1 |issue=7 |date=July 2004|issn=1473-7604}}
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| * {{cite journal | year = 2005 | title = Janne Rydberg – his life and work | journal = NIM B | volume = 235 | pages = 17–22 | url = http://www.physics.utoledo.edu/~ljc/rydberg_sist.PDF | doi = 10.1016/j.nimb.2005.03.137 |bibcode = 2005NIMPB.235...17M | last1 = Martinson | first1 = I. | last2 = Curtis | first2 = L.J. }}
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| [[Category:Atomic physics]]
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| [[Category:Foundational quantum physics]]
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| [[Category:Hydrogen physics]]
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| [[Category:History of physics]]
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