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The **Rydberg formula** is used in "atomic physics to describe the wavelengths of "spectral lines of many "chemical elements. It was formulated by the Swedish "physicist "Johannes Rydberg, and presented on 5 November 1888.

In 1880, 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.

First he tried the formula: , where *n* is the line's wavenumber, *n*_{0} is the series limit, *m* is the line's ordinal number in the series, *m'* is a constant different for different series and *C*_{0} is a universal constant. This did not work very well.

Rydberg was trying: when he became aware of "Balmer's formula for the "hydrogen spectrum In this equation, *m* is an integer and *h* is a constant (not to be confused with the later "Planck's constant).

Rydberg therefore rewrote Balmer's formula in terms of wavenumbers, as .

This suggested that the Balmer formula for hydrogen might be a special case with and , where , 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).

The term *C*_{o} 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.

As stressed by "Niels Bohr,^{[1]} 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 , and therefore also proportional to light's quantum energy *E*. Thus, . 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. "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,^{[2]} but this theory was superseded in 1913 by Niels Bohr's "model of the atom.

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*_{1} to *n*_{2} therefore represents the photon energy emitted or absorbed when an electron makes a jump from orbital 1 to orbital 2.

Later models found that the values for *n*_{1} and *n*_{2} corresponded to the "principal quantum numbers of the two orbitals.

Where

- is the "wavelength of electromagnetic radiation emitted in "vacuum,
- is the "Rydberg constant, approximately 1.097 x 10
^{7}m^{−1}, - and are integers greater than or equal to 1 such that , corresponding to the "principal quantum numbers of the orbitals occupied before and after the "'quantum leap'.

By setting to 1 and letting run from 2 to infinity, the spectral lines known as the "Lyman series converging to 91 nm are obtained, in the same manner:

n_{1} |
n_{2} |
Name | Converge toward |
---|---|---|---|

1 | 2 → ∞ | "Lyman series | 91.13 nm ("UV) |

2 | 3 → ∞ | "Balmer series | 364.51 nm ("Visible) |

3 | 4 → ∞ | "Paschen series | 820.14 nm ("IR) |

4 | 5 → ∞ | "Brackett series | 1458.03 nm (Far IR) |

5 | 6 → ∞ | "Pfund series | 2278.17 nm (Far IR) |

6 | 7 → ∞ | "Humphreys series | 3280.56 nm (Far IR) |

The formula above can be extended for use with any "hydrogen-like "chemical elements with

where

- is the "wavelength of the light emitted in "vacuum;
- is the "Rydberg constant for this element;
- is the "atomic number, i.e. the number of "protons in the "atomic nucleus of this element;
- and are integers such that , corresponding to the "principal quantum numbers of the orbitals occupied before and after.

It's important to notice that this formula can be directly applied only to "hydrogen-like, also called *hydrogenic* atoms of "chemical elements, i.e. atoms with only one electron being affected by an effective nuclear charge (which is easily estimated). Examples would include He^{+}, Li^{2+}, Be^{3+} etc., where no other electrons exist in the atom.

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.

Finally, with certain modifications (replacement of * Z* by

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.

**^**Bohr, N. (1985). "Rydberg's discovery of the spectral laws". In Kalckar, J.*Collected works*.**10**. Amsterdam: North-Holland Publ. Cy. pp. 373–379.**^**Ritz, W. (1908). "Magnetische Atomfelder und Serienspektren".*"Annalen der Physik*(in German).**330**(4): 660–696. "Bibcode:1908AnP...330..660R. "doi:10.1002/andp.19083300403.

- Sutton, Mike (July 2004). "Getting the numbers right: The lonely struggle of the 19th century physicist/chemist Johannes Rydberg".
*Chemistry World*.**1**(7): 38–41. "ISSN 1473-7604. - Martinson, I.; Curtis, L.J. (2005). "Janne Rydberg – his life and work" (PDF).
*NIM B*.**235**: 17–22. "Bibcode:2005NIMPB.235...17M. "doi:10.1016/j.nimb.2005.03.137.