Barlow Bob’s Corner – The Balmer Series

The following article has been provided by Barlow Bob, founder & organizer of the NEAF Solar Star Party and regional event host & lecturer on all things involving solar spectroscopy. You can read more about Barlow Bob and see some of his other articles at www.neafsolar.com/barlowbob.html.

The February 2014 issue of Astronomy magazine contained an article about the fate of the Sun. There was an illustration showing the differences between the various types of dark Fraunhofer absorption lines in the spectrum of the Sun, a hot blue star and a white dwarf star.

The solar spectrum consisted of many thin dark lines of different elements. The hot blue star spectrum consisted of only thin dark lines of the Balmer Series of hydrogen. The white dwarf spectrum also contained only the Balmer Series lines. In the white dwarf spectrum, however, these lines were very thick.

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The spectrum of the Sun, a white dwarf, and blue giant. Image taken from: pasthorizonspr.com.

Reference books and articles about spectroscopy state that the Fraunhofer lines in the spectrum of hot stars with a high-pressure atmosphere are thin. The lines of cool stars with a low-pressure atmosphere are thick. Why does a white dwarf with an extremely high-pressure atmosphere have wide Fraunhofer lines in its spectrum?

Sue French provided the explanation below, which is reprinted here with permission.

“It’s a question of density and pressure differences between the different luminosity classes of stars. Hydrogen lines broaden from luminosity class I (luminous supergiant) to luminosity class V (main sequence). The lines are generated by collisions in a star’s photosphere. Close-passing atoms can slightly disturb an electron’s energy level such that the electron can absorb at a wavelength that is a bit offset from the center of the line. Whole bunches of these interactions put together broaden the line, and higher photospheric density (class V) promotes more interactions. For example, a B5V star and a B5I star would have about the same photospheric temperature, but the lines would be broader in the former because of its higher photospheric density. Thus for the white dwarf, where the photospheric density is very high, the lines are broadened with respect to stars of similar photospheric temperature.”

From 1859 until his death at age 73, Johann Jakob Balmer (1825-1898) was a high school teacher at a girl’s school in Basel, Switzerland. His primary academic interest was geometry, but in the middle 1880’s he became fascinated with four numbers: 6,562.10, 4,860.74, 4,340.1, and 4,101.2. These are not pretty numbers, but for the mathematician Balmer, they became an intriguing puzzle. Was there a pattern to the four numbers that could be represented mathematically? The four numbers Balmer chose were special because these numbers pertained to the spectrum of the hydrogen atom. By the time Balmer became interested in the problem, the spectra of many chemical elements had been studied and it was clear that each element gave rise to a unique set of spectral lines. Balmer was a devoted Pythagorean: he believed that simple numbers lay behind the mysteries of the universe. His interest was not directed toward spectra, which he knew little about, nor was it directed toward the discovery of some hidden physical mechanism inside the atom that would explain the observed spectra. Balmer was intrigued by the numbers themselves.

In the mid-1880’s, Balmer began his examination of the four numbers associated with the hydrogen spectrum. At his disposal were the four numbers measured by Anders Jonas Angström (1814-1874): 6,562.10, 4,860.74, 4,340.1, and 4,101.2. These numbers represented the wavelengths, in units of Angströms, of the four visible spectral lines in the hydrogen atom spectrum.

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The Balmer Series for hydrogen. Image taken from en.wikipedia.org/wiki/Balmer_series

In 1885, Balmer published a paper in which his successful formulation was communicated to the scientific world. Balmer showed that the four wavelengths could be obtained with the formula that bears his name: wavelength = B x (m^2)/(m^2-n^2), with B = 3645.6 Angströms. He had found a simple mathematical formula that expressed a law by which the hydrogen wavelengths could be represented with striking precision. He further suggested that there might be additional lines in the hydrogen spectrum. Other spectral lines with their own wavelengths were predicted by Balmer and later found by other scientists. Angström measured the wavelengths of the spectral lines of hydrogen, but Balmer showed that the wavelengths of the spectral lines are not arbitrary. The values of the wavelengths are the expression of a single mathematical formula – and this Balmer Series equation altered how scientists thought about spectral lines. Before Balmer published his results, scientists drew an analogy between spectral lines and musical harmonies. They assumed that there were simple harmonic ratios between the frequencies of spectral lines. After Balmer’s work, all scientists recognized that spectral wavelengths could be represented by simple numerical relationships.

Balmer disappeared from the ranks of working scientists and continued his classroom work teaching young ladies mathematics. Neither he nor his students recognized that his paper on the spectrum of hydrogen would bring him scientific immortality. The spectral lines of hydrogen that were the focus of Balmer’s attention are now known as the Balmer Series.

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