Russell-Saunders coupling approximation

In the Russell-Saunders approximation introduced earher, which is appropriate for the 4-d, 5d, and fighter rare-earth elements, the coupling of L and S give a total angular momentum quantum number J that takes on integer values from (L + S) through L — 5 ... 

Electronic-state configurations are described by the standard quantum numbers or principal quantum number n, angular momentum quantum number L, spin quantum number S and total angular momentum quantum number J. In the so-called Russel-Saunders approximation of electron coupling, electronic states are commonly described symbolically by n For... 

The intermediate coupling, i.e., deviations from the Russell-Saunders approximation, was also treated by Bethe. We may here distinguish between systems with an even or an odd number of electrons. In spherical symmetry, J and 5 are integers, as is L, for an even number, while J and S are half integers 1/2, 3/2, 5/2, for odd electron numbers. The same intrinsic difference occurs in lower symmetries, e.g., for Oj the new quantum number r 7 has the same value for a given J as the function of L in Eq. (10). On the other hand, three values of viz., Fj e = 2), F e = 2), and Fg (e = 4), occur for octahedral systems with an odd number of electrons. Griffith proposes the names E, E, and U. ... 

A second approximation neglects coupling between the spin of an electron and its orbital momentum but assumes that coupling between orbital momenta is strong and that between spin momenta relatively weak but appreciable. This represents the opposite extreme to the 77-coupling approximation. It is known as the Russell-Saunders coupling approximation and serves as a useful basis for describing most states of most atoms and is the only one we shall consider in detail. 

However complex the atom, we can use the Russell-Saunders coupling approximation (or jj coupling, if necessary) to derive the states that arise from any configuration. The four general selection mles that apply to transitions between these states are as follows. 

Assume that the Russell-Saunders coupling approximation applies to both configurations. Answer. The ground electron configuration of zirconium (Z = 40) is (see Table 7.1)... 

In general the value of the 0-factor is neither 1 nor 2, but has some other value. In case that the electronic state of the atom is such as to approximate closely to Russell-Saunders coupling the value of the 0-factor can be, calculated in a simple way. The total angular momentum vector of the atom is the resultant of the vector corresponding to... 

To a first approximation each of several electrons in such a partly filled shell may be assigned its own private set of one-electron quantum numbers, n, /, m, and s. However, there are always fairly strong interactions among these electrons, which make this approximation unrealistic. In general the nature of these interactions is not easy to describe, but the behavior of real atoms often approximates closely to a limiting situation called the L-S or Russell-Saunders coupling scheme. 

If the approximation is made that HL > //ER, then we have the reverse of the Russell-Saunders coupling scheme. The individual electron total angular momenta, specified by /, couple together to give the total angular momentum for the set of electrons. The equivalent of equation (33) becomes... 

First, we consider the case of two tlu electrons, later we will extend the results to more particles. Since the one electron spin-orbit coupling is negligible [13], we will work in the LS (Russel-Saunders) molecular approximation. Two electron basis ket vectors are denoted by a single index I ... 

Fig. 3. Energy levels of eleven trivalent lanthanides below 43000 era-. The quantum number J is given to the right of each level, whereas the multiplicity (2S + 1) and L-value are only given to the left for such levels, where Russell-Saunders coupling is a good approximation. The black triangles give the levels most frequently found to fluoresce. The squares indicate levels to which transitions from the groundstate have pronounced pseudoquadrupolar hypersensitive character, and hence intensity in certain materials. The stippled lines indicate calculated positions of J-levels not yet identified. The diagram is taken from Ref.231...

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