The spectrum of He⁺ as a proving ground for Bohr’s model of the atom

The mysterious lines that Williamina Fleming discovered in the spectrum of ζ Puppis attracted significant attention from the international scientific community, who sought to determine their precise cause. Originally it was thought they might be an unusual “cosmic” form of hydrogen, or “proto-hydrogen,” as labeled by Norman Lockyer.¹ In 1912, Alfred Fowler reported finding the same lines in the laboratory using a discharge tube filled with helium and hydrogen, but not in one with only hydrogen. The lines had not, however, been observed to that point in any experiment with pure helium.²

Then, in the first paper of his 1913 trilogy on the atomic model, a young Niels Bohr noted:³

According to Bohr’s early version of his atomic model, the wavelength, λ, of an emission line from a state $n_2$ to a state $n_1$ where $\left(n_2 > n_1\right)$ of a hydrogenic (one-electron) atom was given by (in modern notation but in the cgs units used by Bohr):

$${\lambda }^{-1} = R_Z(\frac{1}{n_1^2} - \frac{1}{n_2^2})$$

where $R_Z$ is the Rydberg constant pertaining to a one-electron atom with atomic number $Z$

$$R_Z = \frac{2{\mathrm{\pi }}^2{\mathrm{m}}_{\mathrm{e}}{\mathrm{Z}}^2{\mathrm{e}}^4}{h^{3}c}$$

$m_e$ is the electron mass, $e$ is the electron charge, $h$ is Planck’s constant, and $c$ is the speed of light. Upon reading Bohr’s paper, Fowler pointed out to Bohr a discrepancy between Bohr’s treatment of the Pickering series and the positions of the spectral lines of the series as measured in his own laboratory and confirmed by Evan Jenkins Evans at Rutherford’s laboratory.⁴ Indeed, the ratio of the Rydberg constants given by Bohr’s formula for He⁺ and for atomic hydrogen would be

$$\frac{R_{H{e}^{+}}}{R_H} = \frac{Z_{H{e}^{+}}^2}{Z_H^2} = 4$$

whereas for the Pickering series and the Balmer series,⁵ this ratio came out experimentally as

$$\frac{R_{H{e}^{+}}}{R_H} \approx 4.0016$$

Whereupon Bohr went back to the drawing board and replaced in his expression for the Rydberg constant the electron mass with the reduced mass, $m_e\mapsto {\mu }_{Z }\equiv m_e{M}_Z/(m_e+M_Z)$ , of the nucleus–electron system, as he should have done to begin with:⁶

$$R_Z = \frac{2{\mathrm{\pi }}^2{\mathrm{\mu }}_{\mathrm{Z}}{\mathrm{Z}}^2{\mathrm{e}}^4}{h^{3}c}$$

With this amendment, the ratio of the Rydberg constants for He⁺ and H became

$$\frac{R_{H{e}^{+}}}{R_H} = \frac{Z_{H{e}^{+}}^2{\mu }_{H{e}^{+}}}{Z_H^2{\mu }_H} = 4.00163$$

in a five-digit agreement with the accurate laboratory data, as subsequently acknowledged by Fowler.⁷

Had the wavelengths of the Pickering series published by Pickering in 1897 been used to determine $R_{H{e}^{+}}$ , the ratio $\frac{R_{H{e}^{+}}}{R_H}$ would have come out as 4.00101 using the data from the left ζ-labeled column in the table that Edward Pickering published in his 1897 article, “The spectrum of ζ Puppis,” and 4.00011 using the data from the right ζ-labeled column.⁸

Interestingly enough, the blue $\lambda =4686$ Å line corresponding to the transition between the $n_1=3$ and $n_2=4$ levels was not reported in communications from the Harvard College Observatory in the 1890s. This line typifies the Pickering series as observed in laboratory discharge-tube spectra.⁹

In 1915, Bohr provided further comments on the differences between the Balmer series and the Pickering series and strengthened his case by referring to 1914 experimental data on the ionization potentials of hydrogen and helium.¹⁰

The explanation of the Pickering series was a major triumph for Bohr’s theory. Thereby, Bohr’s theory proved capable of not just reproducing what was well known and well established, such as the Balmer series or the Rydberg constant for hydrogen, but it also demonstrated its predictive and explanatory power.

In particular, the accurate rendition of the ratio of the Rydberg constants for He⁺ and H had made a deep impression. When Albert Einstein heard about it at a 1913 conference, he exclaimed, “Then the frequency of the light does not depend at all on the frequency of the electron … this is an enormous achievement. The theory of Bohr must then be right.”¹¹

In his 1951 reflections on the advent of quantum theory, Einstein took the opportunity to pay tribute to Bohr’s achievement once more:¹²

Bohr’s model of the hydrogenic atom was amended in 1916 by Arnold Sommerfeld, who made use of relativistic old quantum theory to explain the fine structure of the H and He⁺ spectra and to provide the means to meet the challenge of unriddling the anomalous Zeeman effect.¹³ According to Walther Gerlach, it was the fine structure of the spectrum of He⁺ that convinced his teacher, Friedrich Paschen—a skeptical experimentalist—that the theory of relativity was correct. Paschen is said to have remarked, “As of today, relativity exists”¹⁴

Sommerfeld and, independently, Peter Debye concluded that not just the magnitudes of the electronic orbital angular momenta but also the spatial orientations of the electronic orbits with respect to an external magnetic field are quantized.¹⁵ In 1922, the Bohr–Sommerfeld–Debye model was subjected to a nonspectroscopic test, the Stern–Gerlach experiment, which confirmed the existence of space quantization and thus ruled unequivocally in favor of quantum theory as epitomized by the Bohr–Sommerfeld–Debye model.¹⁶

Bohr’s model of the atom thus continued playing the role of a touchstone on the path to quantum mechanics until its advent between 1925 and 1927. We add that a definitive treatment of the fine structure of hydrogenic atoms as due to spin–orbit coupling and relativistic effects was provided by Sommerfeld in 1940 and reviewed by numerous authors since, including by Condon and Shortley.¹⁷

References

1. “New hydrogen spectra,” Nature, 90, 1912, 466–467, https://doi.org/10.1038/090466a0.
2. A. Fowler, “Observations of the principal and other series of lines in the spectrum of hydrogen,” Monthly Notices of the Royal Astronomical Society, 73, 1912, 62–72. doi:10.1093/mnras/73.2.62.
3. N. Bohr, “I. On the constitution of atoms and molecules,” London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 26, 1913, 1–25, doi:10.1080/14786441308634955.
4. A. Fowler, “The spectra of helium and hydrogen,” Nature, 92, 1913, 95–96, doi:10.1038/092095b0; A. Fowler, letter to the editor, Nature, 92, 1913, 232–233, doi.org:10.1038/0922232a0; and Evan J. Evans, “The spectra of helium and hydrogen. Nature, 92, 1913, 5. doi:10.1038/092005a0.
5. Joseph Sweetman Ames, “V. On some gaseous spectra: Hydrogen, nitrogen.” London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 30, 1890, 48–58. doi:10.1080/14786449008619987.
6. N. Bohr, “The spectra of hydrogen and helium,” Nature, 92, 231–232, doi:10.1038/092231d0.
7. Fowler, letter, op. cit. (4).
8. Edward C. Pickering, “The spectrum of ζ Puppis,” Astronomical Journal, 5, 1897, 92–94, doi:10.1086/140312.
9. Fowler, “Observations,” op. cit. (2).
10. N. Bohr, “The spectra of hydrogen and helium,” Nature, 95, 1915, 6–7, doi:10.1038/095006a0.
11. John Stachel, Einstein from ‘B’ to ‘Z’ (Birkhäuser, 2002), pp. 369–370.
12. Albert Einstein, “The advent of the quantum theory,” Science, 113, 1951, 82–84. doi:10.1126/science.113.2926.82.
13. A. Sommerfeld, “Zur Quantentheorie der Spektrallinien,” Annalen der Physik, 51, 1916, 125–167, doi:10.1002/andp.19163561702; F. Paschen, “Bohrs Heliumlinien,” Annalen der Physik, 355, no. 16, 1916, 901–940, doi:10.1002/andp.19163551603; Horst Schmidt-Böcking, Gernot Gruber, and Bretislav Friedrich, “One hundred years ago Alfred Landé unriddled the anomalous Zeeman effect and presaged electron spin,” Physica Scripta, 98, 014005, 2023, doi:10.1088/1402-4896/ac9c9b.
14. Walther Gerlach, interviewed by Thomas S. Kuhn, Febraury 18 and 23, 1963, AIP Niels Bohr Library & Archives, p. 9, https://repository.aip.org/node/130451: “Und dann kam er und sagte: ‘Dr. Gerlach, die Relativitätstheorie ist richtig. Seit heute gibt es Relativitätstheorie’” [emphasis in original German transcript].
15. P. Debye, “Quantenhypothese und Zeeman-effekt,” Physikalische Zeitschrift, 17, 1916, 507–512.
16. W. Gerlach and O. Stern, “Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld,” Zeitschrift für Physik, 9, 1922, 349–352, doi:10.1007/BF01326983; Bretislav Friedrich, “A century ago the Stern–Gerlach experiment ruled unequivocally in favor of quantum mechanics,” Israel Journal of Chemistry, 63, e2023000, 2023, doi:10.1002/ijch.202300047.
17. Arnold Sommerfeld, “Zur Feinstruktur der Wasserstofflinien. Geschichte und gegenwärtiger Stand der Theorie,” Naturwissenschaften, 28, 1940, 417–423, doi:10.1007/BF01490583; Edward U. Condon and George H. Shortley, The Theory of Atomic Spectra (Cambridge University Press, 1951).

Cite this resource

Bretislav Friedrich and Maria McEachern, “Williamina Fleming,” Women in the History of Quantum Physics collection, American Institute of Physics, 2026, https://www.aip.org/history/williamina-fleming.