Bohr-Sommerfeld theory

With these new, accurate data, Smekal,4 Coster,8 Dauvillier8 and Wentzel6 have constructed atomic systems to explain the source of each known emission line. The systems are based on the Bohr-Sommerfeld theory of the origin of characteristic X-radiation. In developing and discussing their systems, these investigators have used the values of the L absorption limits measured by us. Certain criticisms have been made of a few of our values. It has therefore seemed desirable to measure some of the limits again. 

We shall frequently encounter cases with angular momentum values of / = and / = 1. For J =, such as for electron spin, there are only two possible orientations of the angular momentum they are depicted in Figure 9. Commemorative of the old Bohr-Sommerfeld theory, we say that the angular momentum is oriented parallel to the z axis in the case of M = + and antiparallel for M = - j. 

The theory of Bohr and Sommerfeld has been applied extensively to atomic spectra with considerable success but it has proved of little use when applied to molecules and the problem of the chemical bond attempts by Pauli and Niessen to apply the theory to the simplest molecular system i.e. the hydrogen molecule ion were unsuccessful. This stable molecule, which has a bond enei of 61 kcal/gmmol, was shown on the basis of the Bohr-Sommerfeld theory to be unstable. More recently, defects have become apparent in the application of this theory to atomic spectra. 

In the Bohr-Sommerfeld theories of the atom, the electrons are moving in orbits which are precisely specified, and the velocities are given exactly. Those theories are therefore concerned with properties which cannot be measured precisely. This difficulty is avoided, however, if one develops theories based on the wave properties of electrons we have already seen, with reference to Figure 1.1, that such theories remove some of the arbitrariness inherent in the Bohr-Sommerfeld approach. Modern theories of atoms and molecules are, therefore, wave theories, which have led to a very considerable increase in our understanding. In the remainder of this chapter we will describe aspects of wave mechanics, or quantum mechanics, that will be of help to biologists in appreciating the nature of the molecular structures with which they are concerned. 

The Bohr-Sommerfeld theory provided a reasonably satisfactory explanation of the spectra of atoms having only one valence electron. With two or more electrons discrepancies occur and certain arbitrary selection rules for atomic transitions were required. In an attempt to solve this problem, several persons, including de Broglie, Schrodinger, Heisenberg, and Born, combined quantum mechanical and wave mechanical concepts. A detailed account of these efforts is beyond the scope of this book. However, some qualitative understanding of these concepts will be helpful and thus a brief account is included. 

The possible combinations of n, /, and m are shown in Table 2-1. What were assumptions of the Bohr-Sommerfeld theory for n, /c, and m become, through the solution of the Schrodinger wave equation, n, /, and m. Thus all three quantum numbers arise from the solution of the Schrodinger equation. 

More serious was the inability of even the Bohr-Sommerfeld theory to account for the spectral details of the atoms that have several electrons. But these were the 1920s and theoretical physics was enjoying its greatest period. Soon the ideas of de Broglie, Schro-dinger, and Heisenberg would put atomic theory on a sound foundation. 

I should also mention Sommerfeld, who extended Bohr s theory to try and account for the extra quantum numbers observed experimentally. Sommerfeld allowed the electrons to have an elliptic orbit rather than a circular one. 

Lewis appropriated Bohr s new atom to try to unify the physical and chemical atom. If the Bohr-Sommerfeld orbits are in fixed positions and orientations, "they may be used as the building stones of an atom which has an essentially static character." 17 Bohr s dynamic theory works for the chemist, Lewis wrote, if the "average" position of an electron in a Bohr-Sommerfeld orbit is taken to correspond to the fixed position of the electron in Lewis s static chemical model. The outermost shell of electrons constitutes the "valence" electrons, and the remaining electrons constitute the "kernel." 18... 

Figure 8. A Bohr-Sommerfeld model of the xenon atom. (From H. A. Kramers and H. Horst, The Atom and the Bohr Theory of its Structure, 1924).

The periodicity of structural regularities in spectra as related to the periodic law of the chemical elements is thus revealed in the verification of the displacement and alternation laws. Practically all the structures here referred to are in beautiful accord with the quantum theory of spectral line emission as developed by Bohr, Sommerfeld, Land and others. During the past three years more than 5000 spectral lines have been classified by Catalan, Gieseler, Kiess, Laporte, Meggers, Russell and Walters. With the further development of methods of attack, it seems probable that practically all complex spectra will be fully interpreted in a period of time not greater than that already expended in the classification of simpler spectra—especially if additional workers can be brought into this fascinating field. 

Bonino brought forward a further contribution to the theory of infrared spectra of organic liquids by incorporating the Bohr-Sommerfeld quantum conditions, including the correspondence principle of Bohr as well. This paved the way toward establishing a correlation between the physical and chemical image of molecules in the study of infrared spectra. From this series of papers on infrared spectroscopy, one can already observe the interdisciplinary character of Bonino s thought. In a lecture delivered some years later, Bonino offered these reflections on his chosen field of research ... 

The models of Bohr, Sommerfeld and de Broglie provide a firm basis for the further development of a quantum theory of chemistry by re-assessment of the more advanced theories of quantum physics. However, there is little support for such a pursuit, not if we find statements like the following [14], put out by one of the world s leading publishers of academic science ... 

It is well known that Bohr s theory of radiation explains very simply the above relation between emission and absorption wave-numbers. The relation, however, should not be absolutely exact, if, as Sommerfeld supposes, some of the atoms contain M orbits that are elliptic, and other atoms of the same chemical element contain M orbits that are circular. In this case the L absorption drops should have a complex structure. We have not been able to observe such a structure in the drops on our curves, and the effect must be exceedingly small, if it exits at all. 

Bohr s theory was in close agreement with many experimental facts regarding one-electron atoms (the hydrogen atom and hydrogen-fike atoms, snch as He and LF ), bnt it conld not explain the fine stmctnre of the spectral lines that is, the fact that certain lines were actnally a set of closely spaced lines. In 1915 and 1916 respectively, W. Wilson and A. Sommerfeld refined Bohr s theory by admitting elliptical orbits. Flowever, it became evident to many physicists, including Bohr himself, that is was time for a scientific revolution. 

Optical spectroscopy of the simple hydrogen atom has played a central role in the development of atomic physics and quantum mechanics.3 The visible Balmer spectrum was the Rosetta stone which inspired the pathbreaking discoveries of Bohr, Sommerfeld, De Broglie, Schrbdinger, Dirac, and Lamb. More than once, seemingly minute discrepancies between theory and experiment led to important breakthroughs in our understanding of quantum physics. 

Despite its shortcomings, however, the Bohr-Sommerfeld model is of enormous significance in the development of the quantum theory and, in a broader sense, as an illustration of the evolution of scientific ideas. 

The de Broglie relation and the Heisenberg uncertainty principle successfully danon-strated the major flaw in the Bohr-Sommerfeld model. Although Bohr went beyond classical physics in postulating the quantization of energy levels, his theory still relied... 

Below, we will give the quantum mechanical proof. In classical mechanics, oscillating dipoles behave in the same way. In the Bohr-Sommerfeld picture ( old quantum theory ), the transition moments are also oscillating dipoles, but contrary to the classical case, only two quantized states are possible at the end. 

Actually, Bohr s theory was an inspired (lucky) guess and, since it is valid only for the hydrogen atom, it is not particularly useful as a general theory. Attempts were made by Sommerfeld to extend the theory to two-electron systems, but without much success. To move on, we must understand more about the electron. 

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