Rydberg–Ritz combination principle

The Rydberg–Ritz combination principle is an empirical rule proposed by Walther Ritz in 1908 to describe the relationship of the spectral lines for all atoms, as a generalization of an earlier rule by Johannes Rydberg for the hydrogen atom and the alkali metals. The principle states that the spectral lines of any element include frequencies that are either the sum or the difference of the frequencies of two other lines. Lines of the spectra of elements could be predicted from existing lines.[1][2] Since the frequency of light is proportional to the wavenumber or reciprocal wavelength, the principle can also be expressed in terms of wavenumbers which are the sum or difference of wavenumbers of two other lines.

Another related version is that the wavenumber or reciprocal wavelength of each spectral line can be written as the difference of two terms.[3][4] The simplest example is the hydrogen atom, described by the Rydberg formula

${\displaystyle {\frac {1}{\lambda }}=R\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)}$

where ${\displaystyle \lambda }$ is the wavelength, ${\displaystyle R}$ is the Rydberg constant, and ${\displaystyle n_{1}}$ and ${\displaystyle n_{2}}$ are positive integers such that ${\displaystyle n_{1}<n_{2}}$. This is the difference of two terms of form ${\displaystyle T_{n}={\frac {R_{H}}{n^{2}}}}$.[3]

The exact Ritz Combination formula was mathematically derived from this where:

${\displaystyle {\tilde {\nu }}\;=\;{\frac {1}{\lambda }},}$

${\displaystyle {\tilde {\nu }}=A-{\frac {N}{(m+{\alpha }+{\beta }({\alpha }-{\tilde {\nu }}))^{2}}}}$

Where:

${\displaystyle {\tilde {\nu }}}$ is the wavenumber,

${\displaystyle A}$ is the limit of the series,

${\displaystyle N}$ is a universal constant, (now known as R)[5]

${\displaystyle m}$ is the numeral, (now known as n)[6]

${\displaystyle {\alpha }}$ and ${\displaystyle {\beta }}$ are constants.[7][8]