Editing Bohr model (section)

== Rydberg formula ==
{{Main|Rydberg formula}}
Beginning in late 1860s, [[Johann Balmer]] and later [[Johannes Rydberg]] and [[Walther Ritz]] developed increasingly accurate empirical formula matching measured atomic spectral lines. 
Critical for Bohr's later work, Rydberg expressed his formula in terms of wave-number, equivalent to frequency.<ref name=Bohr1985>{{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> These formula contained a constant, <math>R</math>, now known the [[Rydberg constant]] and a pair of integers indexing the lines:<ref name=PaisInwardBound>
{{Cite book 
|last=Pais 
|first=Abraham 
|title=Inward bound: of matter and forces in the physical world 
|date=2002 
|publisher=Clarendon Press [u.a.] 
|isbn=978-0-19-851997-3 
|edition=Reprint 
|location=Oxford}}
</ref>{{rp|247}} 
<math display="block">\nu = R \left( \frac{1}{m^2} - \frac{1}{n^2} \right).</math>
Despite many attempts, no theory of the atom could reproduce these relatively simple formula.<ref name=PaisInwardBound/>{{rp|169}}

In Bohr's theory describing the energies of transitions or [[Atomic electron transition|quantum jumps]] between orbital energy levels is able to explain these formula. For the hydrogen atom Bohr starts with his derived formula for the energy released as a free electron moves into a stable circular orbit indexed by <math>\tau</math>:<ref name=Bohr1913_I>{{Cite journal |last=Bohr |first=N. |date=July 1913 |title=I. On the constitution of atoms and molecules |url=https://www.tandfonline.com/doi/full/10.1080/14786441308634955 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |language=en |volume=26 |issue=151 |pages=1–25 |doi=10.1080/14786441308634955 |bibcode=1913PMag...26....1B |issn=1941-5982}}</ref>
<math display="block">W_\tau = \frac{2\pi^2me^4}{h^2\tau^2}</math>
The energy difference between two such levels is then:
<math display="block">h\nu = W_{\tau_2} - W_{\tau_1}  = \frac{2\pi^2me^4}{h^2}\left( \frac{1}{\tau_2^2} - \frac{1}{\tau_1^2} \right)</math>
Therefore, Bohr's theory gives the Rydberg formula and moreover the numerical value the Rydberg constant for hydrogen in terms of more fundamental constants of nature, including the electron's charge, the electron's mass, and the [[Planck constant]]:<ref name=Baggott2013>{{Cite book |last=Baggott |first=J. E. |title=The quantum story: a history in 40 moments |date=2013 |publisher=Oxford Univ. Press |isbn=978-0-19-965597-7 |edition=Impression: 3 |location=Oxford}}</ref>{{rp|31}}<ref name=Baily2013>{{Cite journal |last=Baily |first=C. |date=2013-01-01 |title=Early atomic models – from mechanical to quantum (1904–1913) |url=https://link.springer.com/article/10.1140/epjh/e2012-30009-7 |journal=The European Physical Journal H |language=en |volume=38 |issue=1 |pages=1–38 |doi=10.1140/epjh/e2012-30009-7 |issn=2102-6467|arxiv=1208.5262 |bibcode=2013EPJH...38....1B }}</ref>
<math display="block">cR_H = \frac{2\pi^2me^4}{h^3}.</math>

Since the energy of a photon is
: <math>E = \frac{hc}{\lambda},</math>
these results can be expressed in terms of the wavelength of the photon given off:
: <math>\frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right).</math>
Bohr's derivation of the Rydberg constant, as well as the concomitant agreement of Bohr's formula with experimentally observed spectral lines of the [[Lyman series|Lyman]] ({{math|''n<sub>f</sub>''}} =1), [[Balmer series|Balmer]] ({{math|''n<sub>f</sub>''}} =2), and [[Paschen series|Paschen]] ({{math|''n<sub>f</sub>''}} =3) series, and successful theoretical prediction of other lines not yet observed, was one reason that his model was immediately accepted.<ref name=Baily2013/>{{rp|34}}

To apply to atoms with more than one electron, the Rydberg formula can be modified by replacing {{math|''Z''}} with {{math|''Z''&nbsp;−&nbsp;''b''}} or {{math|''n''}} with {{math|''n''&nbsp;−&nbsp;''b''}} where {{math|''b''}} is constant representing a screening effect due to the inner-shell and other electrons (see [[Electron shell]] and the later discussion of the "Shell Model of the Atom" below). This was established empirically before Bohr presented his model.