Electron Russell-Saunders coupling

For a monatomic entity containing only one electron, Russell-Saunders coupling is insured by a special situation. Using lower-case letters for one electron, 1 is... 

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. 

Figure 7.4 Russell-Saunders coupling of (a) orbital angular momenta li and I2, (b) spin angular momenta Sj and 2 and (c) total orbital and total spin angular momenta, L and S, of sip and a d electron...

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)... 

The bond diagrams provide an obvious simple method of determining the allowed spectral terms for equivalent electrons with Russell-Saunders coupling, which may be convenient for the reason that it separates states of different multiplicity at the start. 

A partly filled shel > exhibits a number of states of different energies which arise as a result of the interactions or couplings of the electrons in the shell. These states can be determined using the Russell-Saunders coupling scheme (Hund s rules) (Figgis, 1966). A characteristic property of a state is the spin multiplicity which is related to the number of unpaired electrons in a shell. A singlet state has a spin multiplicity of one (two electrons of opposite spin), a doublet state has a multiplicity of two and... 

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 The Russell-Saunders coupling scheme is no longer valid and an intermediate coupling scheme is more appropriate. 

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... 

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)... 

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... 

The LSJ-coupling scheme introduced above is called the Russell-Saunders coupling scheme [RSa25], It is based on the validity of equ. (1.9). The other extreme coupling case follows if the spin-orbit interaction dominates the Coulomb interaction between the electrons. This is called the jjJ-coupling scheme and requires that... 

We are now in position to derive the electronic states arising from a given electronic configuration. These states have many names spectroscopic terms (or states), term symbols, and Russell-Saunders terms, in honor of spectroscopists H. N. Russell and F. A. Saunders. Hence, the scheme we use to derive these states is called Russell-Saunders coupling. It is also simply referred to as L-S coupling. 

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