*In this installment on the history of atom theory, physics professor (and my dad) Dean Zollman explains how 19th century physicists turned to mathematics as they sought patterns among those dark lines in the spectra, hoping to better understand how light was emitted. – Kim*

**By Dean Zollman**

After the seminal work of Gustav Kirchhoff and Robert Bunsen, the collection of information about spectra moved somewhat rapidly. Many researchers were able to obtain spectra of elements and molecules. Advances in photography enabled these physicists and chemists to increase the precision of their measurements and to improve their knowledge of visible, ultraviolet, and infrared light. However, progress on understanding how the light was emitted from matter moved much more slowly.

When deep understanding is lacking, a common technique in science is to look for patterns. If a pattern can be seen in the data (particularly a mathematical relation), then maybe that pattern can be used to discover some important underlying feature or (better) a cause for the data. This process did occur in understanding how light was emitted by atoms. However, it would unfold over about 40 years and involve several important discoveries. So, we need some patience to tell the story.

Physicists and chemists began looking for mathematical relations among the wavelengths of the various spectra by about 1870. The chemists tried to connect the spectra to the relatively new periodic table. That line of reasoning did not lead to much progress, so we will not follow it here. Physicists tried to discover mathematical connections among the wavelengths of light. As we shall see shortly, it took a mathematics teacher in a high school for girls to make the real breakthrough.

**Good Vibrations?**

One of the physicists’ dead ends is somewhat interesting in that it gives some understanding about how science works. In the 1870s, George Johnston Stoney (1826-1911) pursued the idea that the emission of light was related to vibrations of the atom or molecule. At that time, he had no idea what an atom was. However, lots of things vibrate, so it was reasonable to assume that atoms did also.

He attempted to show that the spectral lines were components in a harmonic series related to these vibrations. If that were true, he should be able to find a simple relationship similar to the octaves on the piano, where concert A is a vibration of 440 Hertz, the next highest A is 880 Hertz and so forth. (For a short discussion of harmonic series see “Physics of Music – Notes.”)

Stoney was able to come up with some possibilities, but it was a stretch. For example he concluded that some of the lines in the hydrogen spectrum were the 284th, 288th, 291st, 293rd, 296th, and 297th harmonics of a vibration which had a period of vibration of T/9.0572, where T is the time it takes light to travel one millimeter. To obtain harmonics for the entire hydrogen spectrum, Stoney needed to have four different periods of vibration, each of which resulted in a few very high harmonics being emitted.

This theory had a couple of serious issues. First, why are some of the harmonics missing? Second, what is the meaning of periods of vibrations? So, the theory was not on good grounds. The final blow came from Arthur Shuster (1851-1934). Shuster was able to show that he could obtain a similar type of “fit” to the data using a set of random numbers. Thus, the connection to the data was not much better than a mathematical connection among results of rolling dice. While the results were not positive, the idea was worth pursuing and helped eliminate one option.

In spite of this ill-fated idea, Stoney did have a successful career as a physicist. For example, he is given credit for conceiving the idea that there is a fundament unit of electricity for which he “ventured to suggest the name *electron.*” The electron was not discovered for more than 20 years after Stoney coined the term, but he was right about there being a fundamental until of electricity and the name stuck.

**Playing with Fractions**

Meanwhile, at the urging of a colleague, Johann Jakob Balmer (1825-1898) took on the task to find a mathematical relationship among four visible spectral lines in hydrogen. Balmer taught mathematics at a secondary school for girls in Basel, Switzerland, and was a part-time math faculty member at the University of Basil. The challenge was to find some common mathematical way to express the wavelengths that had been carefully measured by Anders Ångström (1814-1874).

These wavelengths were: 656.210 nanometers (nm), 486.074 nm, 434.01 nm, and 410.12 nm. (I will use modern units rather than the ones that Balmer and Ångström used. A nanometer is 1/10,000,000th of a centimeter.) I could probably stare at these numbers for many years and not see any simple relation. But Balmer had talents that I can only dream about. In a short paper in 1885, he described a remarkable conclusion: “The wavelengths of the first four hydrogen lines are obtained by multiplying the fundamental number h = 364.56 nm in succession by the coefficients 9/5; 4/3; 25/21; and 9/8.”

We might say, “So what?” We take a seemingly random number (364.56 nm) and multiply it by four fractions that also seem to be picked out of a hat and get the correct wavelengths. But Balmer had no doubt taught his pupils that we can multiply the top and bottom of a fraction by the same number and get a new fraction with the same value.

When he multiplied the numerators and denominators of the second and fourth fractions by 4, he obtained an interesting result. The new fractions were 9/5, 16/12, 25/21, and 36/32. Now the numerators are the squares of 3, 4, 5, and 6 while the denominators are (9-4), (16-4), (25-4) and (36-4). To obtain the wavelengths of each of the hydrogen lines he multiplied

9/5 x 364.56 nm = 656.208 nm

16/12 x 364.56 nm = 486.08 nm

25/21 x 364.56 nm = 434.01 nm

36/32 x 364.56 nm = 410.01 nm.

These numbers are in remarkable agreement with Ångström’s measurements. In fact, Ballmer states that, “The deviations of the formula from Ångström’s measurements amount in the most unfavorable case to not more than 1/40,000 of a wavelength.” That kind of match is extremely rare in science.

Balmer wrote his result as a mathematical equation: 364.56 x m^{2}/(m^{2}-n^{2}) where n = 2 and m = 3, 4, 5, or 6. He calculated the value of the wavelength for m = 7 and found a wavelength that should be visible. But he knew of no such observation. It turned out he was just not up-to-date on the experiments. Indeed, someone had discovered such a spectral line. So, Balmer’s formula fit the existing data and predicted another data point correctly.

We know very little about how Balmer came up with his result. He was not an academic. The paper that he wrote reads more like a blog post that a scientific document. So, he does not talk about his reasoning process or how long the process took. Some of his notes indicate that he was aware of some of Stoney’s work. We do know that Balmer liked to play with numbers, and the playing paid off in a big way.

**A New Equation**

About three years later, Johannes Rydberg (1854-1919) was able to generalize Balmer’s result. He proposed the equation:

In this equation, λ (lambda) is the wavelength of the spectral lines in hydrogen, R is a constant number, n is an integer starting with 1, and m is an integer which is greater than n. Rydberg found that when n = 1 and m= 1, 2, 3, …, he could get the wavelengths of the ultraviolet lines in hydrogen. For n = 2 and m = 3, 4, …, his equation matched Balmer’s. When n = 3 and m= 4, 5, …, he got the wavelengths for some infrared lines in hydrogen.

The agreement between Rydberg’s and Balmer’s equations and the experimental results were extremely good. But the equations did not explain how the light was emitted or what the atoms did to create the light. In fact, many people still did not agree that atoms even existed.

However, the equations created a new challenge for anyone who would try to explain how matter emitted light. Any theory that was to explain this emission needed to be able to derive these equations from the theory. This was a big challenge, but the first attempt at this theory would come. However, several important discoveries would need to be made before it could happen. More on all of this next time.

*Public domain images via Wikimedia Commons.*

**Previously**

What Are Things Made of? Depends on When You Ask.

Ancient Greeks Were the First to Hypothesize Atoms

The Poetry of Atoms

Atom Theory in Ancient India

Religion, Science Clashed over Atoms

Medieval Arabic Scholarship Might Have Preserved Scientific Knowledge

Rediscovering a Roman Poet – and Atom Theory – Centuries Later

Reconciling Atom Theory with Religion

Did Atom Theory Play a Role in Galileo’s Trouble with the Inquisition?

Did Gifted Scientist’s Belief in Atoms Led to His Obscurity?

Does Atom Theory Apply to the Earthly and the Divine?

A Duchess Inspired by Atoms

Separating Atoms from Atheism

Isaac Newton: 300 Years Ahead of His Time

Issac Newton and the Philosopher’s Stone

When Chemistry and Physics Split

Redefining Elements

Mme Lavoisier: Partner in Science, Partner in Life

With Atoms, Proportionality and Simplicity Rule

Despite Evidence of Atoms, 19th Century Skeptics Didn’t Budge

Mission of the First International Scientific Conference: Clear up Confusion

Rivalry over the First Periodic Table

The Puzzle of Dark Lines amid Rainbow Colors

The Colorful Signature of Each Element

**Dean Zollman** is university distinguished professor of physics at Kansas State University where he has been a faculty member for more than 40 years. During his career he has received four major awards — the American Association of Physics Teachers’ Oersted Medal (2014), the National Science Foundation Director’s Award for Distinguished Teacher Scholars (2004), the Carnegie Foundation for the Advancement of Teaching Doctoral University Professor of the Year (1996), and AAPT’s Robert A. Millikan Medal (1995). His present research concentrates on the teaching and learning of physics and on science teacher preparation.

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