Rydberg formula

History

In 1890, Rydberg proposed 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 specifying the lines in terms of their wavenumber (the number of waves occupying the unit length, equal to 1/λ, the inverse of the wavelength) rather than their wavelength. 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: ${\displaystyle n=n_0-\frac{C_0}{m+m^{\prime }}}$, 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: ${\displaystyle n=n_0-\frac{C_0}{{\left(m+m^{\prime }\right)}^2}}$ when he became aware of Balmer's formula for the hydrogen spectrum ${\displaystyle \lambda =\frac{h{m}^2}{m^2-4}}$ In this equation, m is an integer and h is a constant (not to be confused with the later Planck constant).

Rydberg therefore rewrote Balmer's formula in terms of wavenumbers, as ${\displaystyle n=n_0-\frac{4n_0}{m^2}}$.

This suggested that the Balmer formula for hydrogen might be a special case with ${\displaystyle m^{\prime }=0}$ and ${\displaystyle {\text{C}}_0=4n_0}$, where ${\displaystyle n_0=\frac{1}{h}}$, the reciprocal of Balmer's constant (this constant h is written B in the Balmer equation article, again to avoid confusion with the Planck constant).

The term ${\displaystyle {\text{C}}_0}$ 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 ${\displaystyle \frac{1}{\lambda }=\frac{f}{c}}$, and therefore also proportional to light's quantum energy E. Thus, ${\displaystyle \frac{1}{\lambda }=\frac{E}{hc}}$ (in this formula the h represents the Planck constant). Modern and legitimate 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. Walther 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.

Bohr's interpretation and derivation of the constant

Rydberg's published formula was $${\displaystyle \pm \frac{n}{N_0}=\frac{1}{(m_1+{\mu }_1{)}^2}-\frac{1}{(m_2+{\mu }_2{)}^2}}$$ where ${\displaystyle n}$ is the observed wavenumber, ${\displaystyle N_0}$ is a constant for all spectral series and elements, and the remaining values, ${\displaystyle m_1,{\mu }_1,m_2,{\mu }_2}$ are integers indexing the various lines. When Bohr analyzes his model for the atom he writes $${\displaystyle \nu =\frac{2{\pi }^{2}m{e}^4}{h^3}\left(\frac{1}{{\tau }_2^2}-\frac{1}{{\tau }_1^2}\right)}$$ where he uses frequency ${\displaystyle \nu }$ (proportional to wavenumber). Thus he has been able to compute the value of Rydberg's heuristic constant ${\displaystyle N_0}$ from his atom theory and set the integers ${\displaystyle {\mu }_1}$ and ${\displaystyle {\mu }_2}$ to zero. The effect is to predict new series corresponding to ${\displaystyle {\tau }_2=1}$ in the extreme ultraviolet unknown to Rydberg.

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 n₁ to n₂ 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 n₁ and n₂ corresponded to the principal quantum numbers of the two orbitals.

The Rydberg formula can be interpreted both through the semi-classical Bohr model and through fully quantum-mechanical treatments of the hydrogen atom. In the Bohr model, electrons occupy quantized orbits whose energies vary. When an electron transitions from a higher level to a lower level a photon is emitted with a wavelength matching the Rydberg expression. Modern quantum mechanics arrives at the same result from the Schrödinger equation for an electron bound by a Coulomb potential. Differences in the energy eigenvalues of the hydrogen atom reproduce the observed Rydberg dependence, while relativistic and spin corrections appear when the Dirac equation, fine-structure interactions, and quantum electrodynamics (QED) effects are included. These refinements explain small deviations from the simple formula, such as the Lamb shift and hyperfine splittings in hydrogen-like systems.