## Johann Jakob Balmer

Historically, the simple and regular Balmer spectrum has inspired . . . pathbreaking discoveries.

From 1859 until his death at age seventy-three, Johann Jakob Balmer (1825-1898) was a high-school teacher at a girls' school in Basel, Switzerland. His primary academic interest was geometry, but in the mid-1880s 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 specific numbers that commanded Balmer's attention were four of many, many such numbers Balmer could have examined. But the four numbers Balmer chose were special because these numbers pertained to the atom of hydrogen. We shall return to these numbers shortly.

The significance of an everyday object often reaches far beyond its own apparent simplicity. A little toy compass whose pivoting pointer mysteriously orients itself along a north-south direction was a source of inspiration to the young Albert Einstein—the sense of awe it inspired in him never waned. A glass prism captures the bright light of the Sun or the feeble glimmer of a candle and sparkles with surprising brilliance. With such a simple glass prism, Isaac Newton demonstrated that the Sun's white light was not what it seemed: it was, instead, a mixture of many pure colors.

Most of what we know about the material makeup of the universe, from the Sun that commands our solar system to the minerals that make up the Earth's crust, has come by examining in detail how atoms either absorb or emit light. In order to learn about the properties of atoms, however, a way must be found to examine individual wavelengths of light. Since the light from most sources is, like sunlight, a composite of many wavelengths, the challenge is to separate the composite into its individual wavelength parts. This is what the glass prism achieves for sunlight. Take a glass prism from a chandelier and you hold in your hands the means to probe into the atomic nature of matter.

When a narrow beam of sunlight enters one side of a prism, the beam bends slightly and then emerges from the prism as a broadened beam displaying the colors of the rainbow: red, orange, yellow, green, blue, indigo, and violet. The different wavelengths associated with these colors—from the longer wavelength of red light to the shorter wavelength of violet light—make up the visible spectrum. However, these colors constitute only a small part of the radiant energy coming from the Sun. In 1800, William Herschel (1738-1822), who discovered the planet Uranus, used a thermometer to determine the heating effects of light with different colors. He found that the temperature increased as he moved the thermometer away from the violet toward the red light, but, more interestingly, that the heating effect continued to increase as he moved the thermometer out of the red light into the darkened region beyond the red. From this, he correctly inferred the presence of invisible light, which we now call the infra-red region of the spectrum. In 1801, the German physicist Johann Wilhelm

Ritter (1776-1810) discovered the presence of another invisible radiation at the other end of the visible spectrum, beyond the violet or, as we now call it, the ultraviolet.

From the beginning of the nineteenth century, scientists relied on the glass prism as an active element in optical experimentation. During the year following the discoveries of Herschel and Ritter, the British scientist William Wollaston (1766-1828) made a seminal discovery in a way that established the terminology scientists still use today. Up to this time, scientists had followed Newton's example, allowing sunlight to pass through a small circular hole in an opaque shield. Through the hole came a beam of sunlight with a cross section like the hole itself: circular. Wollaston changed this. He cut a slit in the barrier, and from this slit, a ribbon of light fell on his glass prism. When he examined the Sun's visible spectrum, he noticed several dark images of the slit. Wollaston concluded that the dark images represented certain wavelengths in the visible light coming from the Sun that were missing, and these missing wavelengths revealed themselves as missing light, or dark lines in the spectrum. The dark images in the solar spectrum came to be called spectral lines.

The dark lines discovered by Wollaston quickly attracted the attention of other scientists. Joseph Fraunhofer (1787-1826) observed 574 dark lines in the solar spectrum and he labeled and mapped the more prominent ones. Further, and most significantly, Fraunhofer found that the two dark lines in the solar spectrum, which he labeled "D," coincided in position with the two bright lines from the sodium lamp he had in his laboratory. Fraunhofer did not explicitly link these two observations, but this coincidence between the light from the Sun and that from a light source on Earth was a coincidence that awaited further explanation. Fraunhofer did more: he examined the light from the planets and found patterns of spectral lines similar to those he had observed in the Sun's light. He also examined the light from Sirius and other bright stars and he found both consistencies and differences in the spectral line patterns from one star to another.

By this time, scientists were studying light from as many sources as they could conjure. In 1822, the Scotsman David Brewster (1781-1868) invented a device that, by means of a flame, vaporized small amounts of material. The light from this vaporized material could then be studied. He added 1,600 new spectral lines to those discovered by Fraunhofer and other investigators. During the same year, 1822, John Herschel (1792-1871), William Herschel's son, vaporized various metallic salts and established that the light from the flames could be used to detect the presence of these metals in very small samples. A few years later, William Talbot (1800-1877) showed that the spectrum of each of the chemical elements was unique and that it was possible to identify the chemical elements from their spectra.

It often takes time for the implications of experimental data to be understood and to be acted upon. Fraunhofer's earlier observation that the solar D-lines coincided with the spectral lines of a sodium lamp eventually prompted further important experiments. In 1849, Jean Bernard Léon Foucault (1819-1868), a Parisian physicist, made an unexpected discovery. He passed sunlight through a vapor of sodium and he found that the solar D-lines were darker. His conclusion was that the sodium vapor "presents us with a medium which emits the rays D on its own account, and which absorbs them when they come from another quarter."1 The consequences of Foucault's experiment, however, were expressed more cogently by Sir William Thomson (later Lord Kelvin). He drew the following explicit conclusion: "That the double line D, whether bright or dark, is due to the vapor of sodium . . . That Fraunhofer's double dark line D, of solar and stellar spectra, is due to the presence of vapor of sodium in atmospheres surrounding the Sun and those stars in whose spectra it has been observed."2

Thomson's recognition that the dark D-lines of the Sun's light were somehow connected with the bright lines of sodium light and that both were due to the element sodium can be cited as the beginning of astrophysics. But the foundation of spectroscopy was put in place in 1859 by Gustav Robert Kirchhoff (1824-1887) and Robert Bunsen (1811-1899). Kirchhoff repeated Fraun-hofer's earlier experiment (without knowing that Fraunhofer had already done it) of passing sunlight through sodium vapor. Like Fraunhofer, he saw that the dark lines of the solar spectrum got darker when the Sun's light was passed through a vapor of sodium. Kirchhoff and Bunsen, however, articulated the general principle on which spectroscopy rests; namely, that under the same physical conditions, the emission of light by an element (which gives rise to the bright lines) and the absorption oflight by the same element (which gives rise to the dark lines) produce spectral lines with identical wavelengths.

The vast array of numbers, thousands of numbers, representing the wavelengths of these spectral lines required an explanation. Was there an underlying pattern? If so, what was happening inside the atom to cause the observed pattern of spectral lines? George Johnstone Stoney (1826-1911) proposed in a 1868 paper that spectral lines were caused by some kind of periodic motion inside the atom. Arthur Schuster (1851-1934) refuted Stoney's idea in 1881, but concluded, "Most probably some law hitherto undiscovered exists."3

This brings us back to Balmer, the high-school mathematics teacher. 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 nature. Thus, his interest was not directed toward spectra per se, 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 spec-

Figure 3.1 The visible spectrum of hydrogen, called the Balmer series. The wavelengths of these spectral lines are, from left to right, 4,101.2 A, 4,340.1 A, 4,860.74 A, and 6,562.10 A.

Figure 3.1 The visible spectrum of hydrogen, called the Balmer series. The wavelengths of these spectral lines are, from left to right, 4,101.2 A, 4,340.1 A, 4,860.74 A, and 6,562.10 A.

tra; Balmer was intrigued by the numbers themselves. Was there a pattern to the numbers? In the mid-1880s, Balmer began his examination of the four numbers associated with the hydrogen spectrum. At his disposal were four numbers measured by Anders Jonas Angstrom (1814-1874): 6,562.10, 4,860.74, 4,340.1, and 4,101.2. These numbers represent the wavelengths, in units of angstroms, of the four visible spectral lines in the hydrogen spectrum (Figure 3.1).4

No one knows how many unsuccessful formulations Balmer attempted. What we do know is that in 1885 Balmer published a paper in which his successful formulation was communicated to the scientific world. In this paper, Balmer showed that the four wavelengths could be obtained with the formula

In this formula, the wavelength 2 is given in angstroms (A). The symbol b, which Balmer called "the fundamental number of hydrogen," has the numerical value of 3,645.6 A; the symbol n is an integer, which Balmer gave the value 2. The symbol m is another integer, to which Balmer assigned the values starting with m = 3 and continuing with m = 4, 5, and 6. With m = 3, Balmer calculated one wavelength. With m = 4, another wavelength, and so on. The result of Balmer's calculation was stunning:

Balmer's calculated Angstrom's measured

Value of m wavelengths wavelengths m = 3 6,562.08 A 6,562.10 A

A comparison of the wavelengths calculated by Balmer's formula with those measured by Angstrom reveals their close agreement. Balmer had achieved his objective. He had found a mathematical formula that "expresses a law by which their wavelengths [hydrogen's] can be represented with striking precision."5 But Balmer did more for science than simply develop a formula that reproduced the numbers representing the wavelengths of the four visible spectral lines of hydrogen. He suggested that there might be additional lines in the hydrogen spectrum. Specifically, Balmer extended his calculation by using the next integer, m = 7, and calculated a wavelength equal to 3,969.65 A. As far as Balmer knew, this spectral line did not exist; so he was essentially making a prediction. What Balmer did not know was that Angstrom had in fact already measured the wavelength of another spectral line with the value of 3,968.10 A. Still other spectral lines with their own wavelengths were predicted by Balmer and later found by other scientists.

Angstrom measured the wavelengths of the spectral lines of hydrogen, but Balmer showed that the wavelengths of these spectral lines are not arbitrary; rather, the value of the wavelengths are the expression of one particular mathematical formula. Balmer's work illustrates the hierarchy of values for physicists: discovering an underlying order in measured numbers often counts for more than the measurements themselves.

Balmer's formula had a striking effect on the scientific investi gations of atomic spectra. To begin, it altered scientists' thinking about spectral lines. Before Balmer published his results, scientists drew an analogy between spectral lines and musical harmonics. They assumed that there were simple harmonic ratios between the frequencies of spectral lines. After Balmer's work, all scientists came to recognize that spectral wavelengths could be represented by simple numerical relationships. Even more, Balmer's success inspired scientists to believe that order lay beneath the confusing profusion of spectral lines.

In the closing paragraph of his paper, Balmer noted the "great difficulties" in finding the "fundamental number" of other chemical elements. He specifically mentioned the elements oxygen and carbon. Had Balmer chosen to apply his effort to any chemical element other than hydrogen, we would never have heard of the high-school teacher from Basel. He owed his success to a judicious choice: to study the spectral lines of hydrogen, the simplest chemical element. Through Balmer's success, the hydrogen atom prepared the way not only for an eventual understanding of atomic spectra, but also to an understanding of how spectral lines originate within the unseen atom.

Balmer unwittingly introduced a ticking bomb into the literature of physics—a bomb that would remain undisturbed for twenty-eight years. After he discovered his mathematical expression, 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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