Russell Saunders quantum number

In the case of atoms, deriving states from configurations, in the Russell-Saunders approximation (Section 7.1.2.3), simply involved juggling with the available quantum numbers. In diatomic molecules we have seen already that some symmetry properties must be included, in addition to the available quantum numbers, in a discussion of selection rules. 

Finally, the magnetic moments resulting from the spin and the orbital motion interact. This spin-orbit coupling is taken into account by the total angular momentum quantum number J (Russel-Saunders coupling) ... 

In the presence of Coulomb correlation only, the wave function is characterized by the total spin S = SSj and the total angular momentum L = 2,1 of the 5 f electrons, and the total momentum J is given by Hund s rule (J = L S). Important spin orbit coupling will mix LS multiplets and only J remains a good quantum number. The Russell-Saunders coupling scheme is no longer valid and an intermediate coupling scheme is more appropriate. 

We now consider many-electron atoms. We will assume Russell-Saunders coupling, so that an atomic state can be characterized by total electronic orbital and spin angular-momentum quantum numbers L and S, and total electronic angular-momentum quantum numbers J and Mj. (See Section 1.17.) The electric-dipole selection rules for L, J, and Mj can be shown to be (Bethe and Jackiw, p. 224)... 

The method used in the calculations follows that explained in detail in Ref. [43]. The input quantum defects pa defined with respect to the Russell Saunders coupling scheme, which is the appropriate short-range basis, are given in Ref. 43. At energies corresponding to v = 100 the total number of open and closed channels in the final KF matrix is 414. 

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. 

We then applied this formula to various types of single-electron wave functions, for example s,p,d, /, g, and to wave functions for various Russell-Saunders terms characterized by integral values of the quantum number L. 

The selection rules are not rigorously obeyed. In atoms that do not exhibit Russell-Saunders coupling, the quantum numbers L and S are not defined. Even in atoms that do have this type of coupling, forbidden transitions are merely of lower probability than allowed ones, and they may occur from a state from which no transitions are allowed by the rules, if conditions are such that collisions of the second kind do not remove the atom from the initial state before it radiates (e.g., at extremely low pressures). 

The spin-orbit coupling term in the Hamiltonian induces the coupling of the orbital and spin angular momenta to give a total angular momentum J = L + S. This results in a splitting of the Russell-Saunders multiplets into their components, each of which is labeled by the appropriate value of the total angular momentum quantum number J. The character of the matrix representative (MR) of the operator R(0 n) in the coupled representation is... 

Silver,71 we generally do not employ the WET for the spatial part of a spin-orbit matrix element. In the Russell-Saunders (LS) coupling scheme, we still assume, however, that S is a (fairly) good quantum number 1 and that we... 

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