Russell-Saunders coupling,

However, when two electrons are in equivalent orbitals (same value of n), the Pauli principle must be taken into account so that the permitted states are restricted to D, 

3P and S. Note that states with different values of /. or. S have different energies, partly because of the electron-electron interaction term in the electronic Hamiltonian. A further symmetry classification that should be mentioned is the parity of an atomic state which depends on the behaviour of the total wave function under space-fixed inversion. This is either even (g) or odd (u) and is determined by, summed over all the electrons in the atom. 

The electron spin and orbital angular momenta give rise to intrinsic magnetic moments which interact with other this interaction is known as the spin orbit coupling 

The spin-orbit coupling therefore removes the degeneracy of the so-called fine-structure states listed above. For example, simply by substituting the appropriate values of J, L and S, the three fine-structure components arising from the L = 1, S= 1 configuration are calculated to have the following first-order spin orbit energies  

In practice, for many atomic states, the coupling will be intermediate between LSand jj. 

Figvire 6.5. Russell-Savinders vector coupling scheme for two electrons. 

Values of the spin-orbit coupling constants for different atomic states have been determined fi om atomic spectra, or may be calculated from relativistic quantum mechanics. A more detailed examination of the matrix elements reveals that die spin-orbit coupling operator mixes states of different L and 5 values spin-orbit matrices for many of the most common electron configurations have been given by Condon and Shortley [2]. 

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LS coupling Russell-Saunders coupling. LSD Lysergic acid diethylamide. An hallucinogen. 

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...

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)... 

Hund37 in 1926 and of oxygen and nitric oxide by Van Vleck38 in 1928 were triumphs of the theory of spectra. The magnetic moment of an atom or monatomic ion with Russell-Saunders coupling of the quantum vectors is... 

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. 

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) ... 

Pj and Po = statistical weights of the respective states and are obtainable from Russell-Saunders coupling,... 

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. 

The interactions of type (1) are known as L-S coupling or Russell-Saunders coupling. From the vector addition rule and the constraint that the values must differ by one in the unit of h/2n, the possible values of L and S are ... 

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... 

Table IV-3.—The Lani> (/-Factor for Russell-Saunders Coupling...

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