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In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. The formula was primarily presented as a generalization of the Balmer series for all atomic electron transitions of hydrogen. It was first empirically stated in 1888 by the Swedish physicist Johannes Rydberg, then theoretically by Niels Bohr in 1913, who used a primitive form of quantum mechanics. The formula directly generalizes the equations used to calculate the wavelengths of the hydrogen spectral series. 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 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: n = n 0 − C 0 m + m ′ {displaystyle extstyle 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: n = n 0 − C 0 ( m + m ′ ) 2 {displaystyle extstyle n=n_{0}-{frac {C_{0}}{left(m+m' ight)^{2}}}} when he became aware of Balmer's formula for the hydrogen spectrum λ = h m 2 m 2 − 4 {displaystyle extstyle 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). Rydberg therefore rewrote Balmer's formula in terms of wavenumbers, as n = n 0 − 4 n 0 m 2 {displaystyle extstyle n=n_{0}-{4n_{0} over m^{2}}} . This suggested that the Balmer formula for hydrogen might be a special case with m ′ = 0 {displaystyle extstyle m'=0} and C 0 = 4 n 0 {displaystyle { ext{C}}_{0}=4n_{0}} , where n 0 = 1 h {displaystyle extstyle n_{0}={frac {1}{h}}} , 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 0 {displaystyle { ext{C}}_{0}} 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, 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 1 λ = f c {displaystyle extstyle {frac {1}{lambda }}={frac {f}{c}}} , and therefore also proportional to light's quantum energy E. Thus, 1 λ = E h c {displaystyle extstyle {frac {1}{lambda }}={frac {E}{hc}}} . 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, but this theory was superseded in 1913 by Niels Bohr's model of the atom.

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