Wednesday, July 5, 2017

The Balmer Series

  It's customary on the 4th of July for Americans view local fireworks displays. What the experts can do with patterns, colors and sound is quite amazing. History tells us that fireworks originated in ancient China and the ancient alchemists experimented with colors. Knowledge of the colors that can be produced can be acquired through the use of flame tests. This tool is still used in modern chemistry.

  The scientific study of colored light was advanced in the late 17th Century by Newton's work on the prism. At the beginning of the 19th Century the introduction of the spectroscope allowed scientists to study the Sun's spectrum and discover the dark lines known as Fraunhofer lines. At the same time flame tests of various elements allowed scientists to connect these lines to chemical elements present. In 1868 Ångström published accurate values for the lines of the solar spectrum with the elements associated with them.

In 1885 Balmer published a notice giving the formula for a series of hydrogen lines. How might he have accomplished this? His data came from a notice by Huggins on the hydrogen lines present in the spectra of certain stars. The data is included in a footnote referring to a note he received from Johnstone Stoney, a fellow of the Royal Society, who states that the lines might belong to a series.

What happens if we try to do an empirical fit for the data? Notice that Stoney also includes the wave numbers, ν=1/λ, and we can try to fit these. The lines appear to converge in one direction so we might first try to fit a formula that is quadratic in 1/n (fit1). The results are quite good with an rms err of 0.4. Using n=3 for the first line gives the best fit. The value for B is relatively quite small when compared with the others and the ratio of C to A is very close to 4. Redoing the fit for just two terms makes the ratio even closer to 4 so we can just try to get a value for A by computing a value for each line and taking the average.

Although the rms error is greater, we still get a fairly good fit to the data.

Notice that Stoney includes a curve passing through the data points. Did he know the formula for the series of lines? His table suggests he used a difference formula to fit the data.

Supplemental (Jul 5): Fraunhofer's lines:

Huggins Plate 33 showing the line spectra of a number of stars. The second row appears to be α Lyræ (Vega) containing the first twelve lines of the data above:

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