Rydberg formula

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: ${\displaystyle \scriptstyle n=n_{0}-{\frac {C_{0}}{m+m'}}}$, where n is the line's wavenumber, n₀ is the series limit, m is the line's ordinal number in the series, m' is a constant different for different series and C₀ is a universal constant. This did not work very well.

Rydberg was trying: ${\displaystyle \scriptstyle n=n_{0}-{\frac {C_{0}}{\left(m+m'\right)^{2}}}}$ when he became aware of Balmer's formula for the hydrogen spectrum ${\displaystyle \scriptstyle \lambda ={hm^{2} \over m^{2}-4}}$ In this equation, m is an integer and h is a constant (not to be confused with the later Planck's constant).

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