Selection rules hydrogenlike atoms

The operation of the selection rule for l for hydrogen and hydrogenlike ions can be seen by the study of the fine structure of the lines. The phenomena are complicated, however, by the influence of electron spin.1 In alkali atoms the levels with given n and varying l are widely separated, and the selection rule for l plays an important part in determining the nature of their spectra. Theoretical calculations have also been made of the intensities of lines in these spectra with the use of wave functions such as those described in Chapter IX, leading to results in approximate agreement with experiment. 

In this chapter, we review electronic structure in hydrogenlike atoms and develop the pertinent selection rules for spectroscopic transitions. The theory of spin-orbit coupling is introduced, and the electronic structure and spectroscopy of many-electron atoms is greated. These discussions enable us to explain details of the spectra in Fig. 2.2. Finally, we deal with atomic perturbations in static external magnetic fields, which lead to the normal and anomalous Zeeman effects. The latter furnishes a useful tool for the assignment of atomic spectral lines. 

It can be shown that the angular part of this integral vanishes [1] unless A/ = / — / = 1 not zero), and unless Am = m — m = 0 or 1. Evaluation of the radial part is difficult this factor is nonzero regardless of An = n — n, provided A/ = 1. Hence we can summarize the El selection rules for one-photon transitions in hydrogenlike atoms. 

Other H atom series arise from transitions terminating in different values of n (Fig. 2.4). Since the hydrogenlike energy levels E are independent of I and m, the selection rules (2.12) do not preclude the appearance of a spectral line for any energy separation — E ), and lines appear for all combinations ( , ). These facts quantitatively account for the wavelengths of all of the H atom spectral lines in the low-resolution spectrum of Fig. 2.2. 

Another consequence of the selection rules (2.13) is that a hydrogenlike atom... 

Since each of the hydrogenlike levels with / 9 0 is now split into doublets with = / + 5, it becomes necessary to augment the El selection rules on An, A/, and Aw with El selection rules on Aj. It turns out that these are Aj = 0, 1. We will demonstrate this selection rule in the case of rYSi/2 - n Pj transitions, where j can be i or. The coupled (total) angular momentum states lsjmy = jm in atoms can be expressed as a superposition of uncoupled states /w,sWs> weighted by Qebsch-Gordan coefficients [1,3],... 

We now briefly consider the magnetic dipole (Ml) selection rules for transitions in hydrogenlike and alkali atoms. The relevant matrix element is (not as has sometimes been implied, because the... 

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