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{{Short description|Principle of spectral lines}}
{{Use American English|date = March 2019}}
{{More citations needed|date=March 2015}}
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 metal]]s. The principle states that the [[Spectroscopy|spectral lines]] of any element include [[Frequency|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.<ref name= FR2>{{cite journal | last = Jastrow | first = Robert | date =1948 | title = On the Rydberg-Ritz Formula in Quantum Mechanics | journal = Phys. Rev. | volume = 73 | pages = 60 | url =http://journals.aps.org/pr/abstract/10.1103/PhysRev.73.60 | doi =10.1103/PhysRev.73.60 |bibcode = 1948PhRv...73...60J }}</ref><ref>{{cite web|url=http://gallica.bnf.fr/ark:/12148/bpt6k33824|title=Gesammelte Werke / Walther Ritz,...; [préface de Pierre Weiss]; oeuvres publiées par la Société suisse de physique|first=Walther (1878-1909)|last=Ritz|date=1 January 1911|publisher=Gauthier-Villars|via=gallica.bnf.fr}}</ref> 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.<ref name=Atkins>{{cite book |last1=Atkins |first1=Peter |last2=de Paula |first2=Julio |date=2006 |title=Physical Chemistry |edition=8th |publisher=W.H.Freeman |isbn=0-7167-8759-8 |pages=[https://archive.org/details/atkinsphysicalch00pwat/page/320 320-1] |url-access=registration |url=https://archive.org/details/atkinsphysicalch00pwat/page/320 }}</ref><ref>{{cite book |last1=Tralli |first1=Nunzio |last2=Pomilla |first2=Frank R. |date=1969 |title=Atomic Theory. An Introduction to Wave Mechanics |publisher=McGraw-Hill |isbn= 0070651329 |page=5}}</ref> The simplest example is the [[hydrogen atom]], described by the [[Rydberg formula]]

:<math>\frac{1}{\lambda} = R\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)</math>

where <math>\lambda</math> is the wavelength, <math>R</math> is the [[Rydberg constant]], and <math>n_1</math> and <math>n_2</math> are positive integers such that <math>n_1 < n_2</math>. This is the difference of two terms of form <math>T_n = \frac{R_H}{n^2}</math>.<ref name=Atkins/>

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

:<math>\tilde{\nu} \;=\; \frac{1}{\lambda},</math>

:<math>\tilde{\nu} = A- \frac{N}{(m+{\alpha}+{\beta}({\alpha}-\tilde{\nu}))^2}</math>

Where:

<math>\tilde{\nu}</math> is the wavenumber,

<math>A</math> is the limit of the series,

<math>N</math> is a universal constant, (now known as ''R'')<ref>See [[Rydberg formula]] original of Rydberg’s writings showing this constant as ''N''.</ref>

<math>m</math> is the numeral, (now known as ''n'')<ref>See [[Rydberg formula]] original of Rydberg’s writings showing this constant as ''m''.</ref>

<math>{\alpha}</math> and <math>{\beta}</math> are constants.<ref>{{Cite journal| doi=10.1086/141591| author=Walther Ritz |url=http://gallica.bnf.fr/ark:/12148/bpt6k33824.image.f185 | title=On a new law of series spectra |date=1908 | journal=[[Astrophysical Journal]] |volume=28 |pages=237–243 | bibcode=1908ApJ....28..237R| doi-access=free }}</ref><ref>“Rydberg and the development of atomic spectroscopy (Centennial of J. R. Rydberg's paper on the laws governing atomic spectra),” 
El'yashevich, M A et al.,
Soviet Physics Uspekhi (1990),33(12): 1047.
https://doi.org/10.1070/PU1990v033n12ABEH002666</ref>

==Relation to quantum theory==
The combination principle is explained using quantum theory. Light consists of [[photon]]s whose energy ''E'' is proportional to the frequency {{math|''&nu;''}} and wavenumber of the light: {{math|''E {{=}} h&nu; {{=}} hc/&lambda;''}} (where ''h'' is the Planck constant, ''c'' is the speed of light, and {{math|''&lambda;''}} is the wavelength). A combination of frequencies or wavenumbers is then equivalent to a combination of energies.

According to the [[Bohr model|quantum theory of the hydrogen atom]] proposed by [[Niels Bohr]] in 1913, an atom can have only certain [[energy levels]]. [[Absorption (electromagnetic radiation)|Absorption]] or [[Spontaneous emission|emission]] of a particle of light or [[photon]] corresponds to a [[Atomic electron transition|transition]] between two possible energy levels, and the photon energy equals the difference between their two energies. On dividing by ''hc'', the photon wavenumber equals the difference between two ''terms'', each equal to an energy divided by ''hc'' or an ''energy in wavenumber units'' (cm<sup>−1</sup>). Energy levels of atoms and molecules are today described by [[term symbol]]s which indicate their [[quantum number]]s.

Also, a transition from an initial to a final energy level involves the same energy change whether it occurs in a single step or in two steps via an intermediate state. The energy of transition in a single step is the ''sum'' of the energies of transition in two steps: {{math|(''E''<sub>3</sub> – ''E''<sub>1</sub>) {{=}} (''E''<sub>2</sub> – ''E''<sub>1</sub>) + (''E''<sub>3</sub> – ''E''<sub>2</sub>)}}.

The NIST database tables of lines of spectra contains observed lines and the lines calculated by use of the Ritz combination principle.<ref>{{cite web|url=http://physics.nist.gov/cgi-bin/ASD/lines1.pl?spectra=H&low_wl=&upp_wn=&upp_wl=&low_wn=&unit=1&submit=Retrieve+Data&de=0&java_window=3&java_mult=&format=0&line_out=0&en_unit=0&output=0&bibrefs=1&page_size=15&show_obs_wl=1&show_calc_wl=1&order_out=0&max_low_enrg=&show_av=2&max_upp_enrg=&tsb_value=0&min_str=&A_out=0&intens_out=on&max_str=&allowed_out=1&forbid_out=1&min_accur=&min_intens=&conf_out=on&term_out=on&enrg_out=on&J_out=on|title=NIST ASD Output: Lines|website=physics.nist.gov}}</ref>

==History==
The spectral lines of [[hydrogen]] had been analyzed and found to have a mathematical relationship in the [[Balmer series]].  This was later extended to a general formula called the [[Rydberg formula]].  This could only be applied to hydrogen-like atoms.  In 1908 Ritz derived a relationship that could be applied to all atoms which he calculated prior to the first 1913 quantum atom and his ideas are based on classical mechanics.<ref>M A El'yashevich, N G Kembrovskaya, L M Tomil'chik, "Walter Ritz as a theoretical physicist and his research on the theory of atomic spectra", PHYS-USP, 1995, 38 (4), 435–455</ref>  This principle, the Rydberg–Ritz combination principle, is used today in identifying the transition lines of atoms.

==References==
{{Reflist}}

==External links==
* {{Cite journal| doi=10.1086/141591| author=Walther Ritz |url=http://gallica.bnf.fr/ark:/12148/bpt6k33824.image.f185 | title=On a new law of series spectra |date=1908 | journal=[[Astrophysical Journal]] |volume=28 |pages=237–243 | bibcode=1908ApJ....28..237R| doi-access=free }}

{{DEFAULTSORT:Rydberg-Ritz Combination Principle}}
[[Category:Emission spectroscopy]]
[[Category:Old quantum theory]]