The Hydrogen Atom and X-ray Terms

We shall now investigate the question of how the statements of the atomic theory with regard to the values of the terms are to be corrected in the light of the recognition of the existence of spin. We begin with the simple example of the hydrogen atom, and with the schemes of terms which may be called hydrogen-like (alkali terms, X-ray terms). In Chapter V (pp. 112, 126) we discussed the values of 

The existence of electronic spin leads to a further correction of 

The case of hydrogen is peculiar in one respect. Experiment gives distinctly fewer terms than are specified in the term scheme of fig. 8 for = 2 only two terms are found, for n = 3 only three, and so on. The theoretical calculation shows that here (by a mathematical coincidence, so to speak) two terms sometimes coincide, the reason being that the relativity and spin corrections partly compensate each other. It is found that terms with the same inner quantum number j but different azimuthal quantum numbers I always strictly coincide, for instance, the ns and the np, term, the p. , and the d, term, and so on such pairs of terms are drawn close together in fig. 8. For the value of the terms a formula was given by Sommerfeld (1916), even before the introduction of wave mechanics the same formula is also obtained when the hydrogen atom is calculated by Dirac s relativistic (E908) 11 

Here E denotes the energy of the bound electron after deducting the rest energy, and Eq is the rest energy mc is the radial quantum number (Sommerfeld) is identical with Bohr s azimuthal quantum number /c, and corresponds therefore to the H + 1 of wave mechanics. Since, however, as we have just seen, two terms with dilierent I but the same j always coincide when we take the spin into account, discrimination between the terms by means of the quantum number is identical with discrimination by means of j we therefore have = i + I- The principal quantum number is then found as the sum n=- n r + The constant a is given by 

It should be particularly emphasized that the X-ray terms also are well represented for all elements by Sommerfeld s formula. 

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