Balmer numbers

Comparison with the empirical Equation (1.4) shows that = /re /S/z eg and that n" = 2 for the Balmer series. Similarly n" = 1, 3, 4, and 5 for the Lyman, Paschen, Brackett and Pfimd series, although it is important to realize that there is an infinite number of series. Many series with high n" have been observed, by techniques of radioastronomy, in the interstellar medium, where there is a large amount of atomic hydrogen. For example, the (n = 167) — ( " = 166) transition has been observed with V = 1.425 GFIz (1 = 21.04 cm). 

Calculate the frequency, wavelength, and wave number for the series limit of the Balmer series of the hydrogen-atom spectral lines. 

This review outlines aspects of the various phases of Swiss chemistry from the introduction of alchemy in Western Europe, its transition to chemistry as a science and profession and the more recent practice. Attention is drawn to the large number of Swiss Nobel Laureates in chemistry and the contributions of Swiss physicists to chemical spectroscopy, from the days of Balmer. In all periods, Swiss chemistry has had European dimensions, and links can be traced to most host countries of previous Euroanalysis conferences and indeed to the next, at Lisbon, in AD 2000... 

There are in principle an infinite number of series beginning at higher quantum numbers with i = 6, 7, 8, 9... but they become increasingly difficult to observe. For the higher n series to be seen in the spectrum the levels have to be populated, so some hydrogen atoms must be in the n = 5 level to see the Pfund series. We shall see that the presence of the Balmer series in the spectrum of a star is indicative of the stellar temperature, which is a direct consequence of the population of the energy levels. More of this in Chapter 4. 

The Balmer series is seen in many but not all stars because the first energy level in the series is an excited state with quantum number n = 2. There has to be a mechanism by which the excited stated is populated and this is the local temperature of the star. Hence only if the star is sufficiently hot will the spectrum contain the Balmer series. The strength of the Balmer series within the stellar spectrum can be used to derive a temperature for the surface of the star to compare with black body temperature and the B/V ratio. 

It is noted without further comment that Moseley was able to characterize his observed x-ray spectra in such a way that the individual elements are identified purely by their respective atomic numbers without any use of an atomic model. Note again, however, that the denominators in the Qk and Ql expressions are exactly the frequencies of the hydrogenic Lymann and Balmer alpha lines (Moseley s nomenclature) and that therefore Qk and Ql exactly equal the nuclear charge of a one-electron hydrogenic atom which would be deduced from the frequency v of its observed Lymann and Balmer alpha lines2. 

Balmer lines spect Lines in the hydrogen spectrum, produced by transitions between = 2 and > 2 levels either in emission or in absorption here is the principal quantum number. bobmor, lTnz ... 

The lines are not spaced at random. In the spectrum of hydrogen, for example, there is a prominent red line wavelength interval a blue-violet line (Hyf and after a still shorter interval another violet line (Hi), etc. One has only to plot the frequencies of these lines as a function of their ordinal number in the sequence to get a smooth curve, which shows that they are spaced in accordance with some law. In 1885, Balmer studied these lines, now called the Balmer series, and arrived at an empirical formula which in modem notation reads... 

Most puzzling was the discovery by Balmer of a simple relationship between the wavelength distribution in the light emitted by hydrogen gas in an incandescent state, and the natural numbers. Rather than having a continuous variation of wavelength (A) this observed emission spectrum is found to consist of a series of sharp maxima, called lines, at specific wavelengths that obey the simple formula ... 

Re-examination of the first quantitative model of the atom, proposed by Bohr, reveals that this theory was abandoned before it had received the attention it deserved. It provided a natural explanation of the Balmer formula that firmly established number as a fundamental parameter in science, rationalized the interaction between radiation and matter, defined the unit of electronic magnetism and produced the fine-structure constant. These are not accidental achievements and in reworking the model it is shown, after all, to be compatible with the theory of angular momentum, on the basis of which it was first rejected with unbecoming haste. 

How, then, did Balmer arrive at the formula we know so well He first observes that the wavelengths of the first four lines bear the relations 9/5 4/3 25/21 9/8 to one another, and that the common factor h = 3645.6 mm/107 converts these numbers to the wavelengths. Multiplying then the second and fourth fractions by 4, one arrives at 9/5, 16/12, 25/21, 36/32, of which the numerators are 32, 42, 52, 62. Representing these numbers by m2, the fractions are given by m2/(m2-n2), where n, for this set, is 2. 

But experiments to resolve the fine structure of the Balmer lines were difficult as you all know, resolution was impeded by the Doppler broadening of components. So ionized helium comes into the picture, because, as Sommerfeld s formula predicted, fine structure intervals are a function of (aZ)2, so in helium they are of order four times as wide as in hydrogen and one has more chance of resolving the Doppler-broadened lines. So PASCHEN [40], in 1916. undertook an extensive study of the He+ lines and in particular, 4686 A (n = 4->3). Fine structure, indeed, was found and matched against Sommerfeld s formula. The measurements were used to determine a value of a. But the structure did not really match the theory in that the quantum numbers bore no imprint of electron spin, so even the orbital properties - which dominated the intensity rules based on a correspondence with classical radiation theory - were wrongly associated with components, and the value of a derived from this first study was later abandoned. 

The factor r] accounts for the number of emitted Balmer-a photons per molecule, which involves the type of dissociation process, t] = 1 holds for dissociative excitation with the products D°(n = 3)+D° (Is), i.e., only one atom will directly emit Balmer-a radiation whereas the other is set free already as a proton. In the worst case of a pure molecular flux the total deuterium flux would be underestimated by a factor of 2 by determining it from the Balmer-line emissions solely. More details concerning the energies of the dissociated atoms and accompanying heating mechanisms can be found in [46] and below. 

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. 

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-... 

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