Russell—Saunders multiplets

In Equation 1.5, A is the spin-orbit coupling within a given Russell-Saunders multiplet, which is related to the spin-orbit coupling constant of the ion, C, by the relation A = f/2.S, the + sign applying for n <7 and the - sign for n > 7 [ 15]. The effect of spin-orbit coupling is to split the terms in multiplets with same L,... 

Figure 10.8 Room temperature electronic absorption spectrum recorded on a THF solution of 6. Bands are marked by the nominal Russell-Saunders multiplet to which the excitation occurs. (Adapted from Ref. [34], Copyright (2011) Nature Publishing Group.)...

A Ce3+ ion has the configuration 4fl so that its ground state is the Russell-Saunders multiplet 4F. In a crystal of CaF2 containing dissolved CeF3, the Ce3+ ions substitute for Ca2+ ions at lattice sites so that each Ce3+ ion is at the center of a cube of F ions (see Exercise 7.4-1). This cubic field is of a strength intermediate between the... 

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... 

Since the matrix elements of L - S are given by j[/(/ + 1) — L(L + 1) - S(5 + 1)], the matrix elements of diagonal in L and S can be expressed z kL- S where A is a constant for a given (empirical) term. This is the mathematical expression of the Latide interval rule which states that within a Russell-Saunders multiplet the interval between two levels having consecutive J values is proportional to the higher J value of the pair. 

The general magnetic properties of ionic and molecular actinide compounds have been reviewed recently [15], and only the briefest of overviews is given here. The 5f3 ions U5+ and Np6+ have 2F5/2 ground multiplets in the Russell-Saunders... 

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 The Russell-Saunders coupling scheme is no longer valid and an intermediate coupling scheme is more appropriate. 

This spectrum has been compared with the BIS measurement on Nd metaP i.e. of the homologous lanthanide. Trivalent Nd has a localized 4f initial state configuration. For U, a 5 f or a 5 f initial state configuration are usually assumed, with a tetravalent or a trivalent core respectively. While in Nd the 4f and 4f multiplet states, as evaluated in an atomic-like Russell-Saunders scheme, can be well recognized in the XPS/BIS combined results, and are well separated from a (weak) d-emission at the Fermi edge, in U the occupied states and the empty states spectra join in a continuous band at Ep. Therefore, only the symmetry of 5 f states, given by the position of the main peaks in the joint spectrum, can be recognized with certainty. 

Example 8.2-1 Examine the effect of spin-orbit coupling on the states that result from an intermediate field of O symmetry on the Russell-Saunders term 4F. Correlate these states with those produced by the effect of a weak crystal field of the same symmetry on the components produced by spin-orbit coupling on the 4F multiplet. 

Multiplet structure is also observed in the spectra of the open 4/ shells of the rare earths. Cox has provided an explanation (47). Consider that the various eigenfunctions are constructed, in Russell-Saunders J, L, S, Mj coupling, for the 4/ -1 configuration. Racah showed that a 4fn [/, L, S, Mj] state may be expressed as a sum involving the 4/ 1 eigenfunctions to which a 4/ electron is added, namely, neglecting antisymmetrization,... 

It is well-known that the electron repulsion perturbation gives rise to LS terms or multiplets (also known as Russell-Saunders terms) which in turn are split into LSJ spin-orbital levels by spin-orbit interaction. These spin-orbital levels are further split into what are known as Stark levels by the crystalline field. The energies of the terms, the spin-orbital levels and the crystalline field levels can be calculated by one of two methods, (1) the Slater determinantal method [310-313], (2) the Racah tensor operator method [314-316]. 

The breakdown of Russell-Saunders coupling is obvious for the 5D multiplets (Fig. 22) and strong admixture with 50 multiplets is expected. The departure from Russell-Saunders coupling is exhibited by the majority of the Eu3+ excited levels. 

For most of the crystals mentioned in section 5.3, the periodicity about the c-axis is greater than two-fold (i.e., n > 2). For such so-called uniaxial crystals it is not too difficult to distinguish different types of radiation. Three different spectra need to be compared the axial (for which k c, and hence necessarily Elc and H I c) the transverse n (for which fe L c, c, and hence necessarily H 1 c) and the transverse a (for which k c, Elc, and hence necessarily H c). As Runciman (1958) pointed out, a line coincident in both the axial and a spectra is electric-dipole, while one coincident in both the axial and n spectra is magnetic-dipole or electric-quadrupole. The reader who attempts to verify these assertions soon discovers that the condition n > 2 is crucially important, since the components of D, L -I- 2S and for which q = l must connect a particular lower sublevel to an upper one that is of a different type from that reached by the components for which q = 0. The identifications of D2 of 4f and the levels of the multiplet of 4f referred to in section 5.2 were made by studying the polarizations of the various allowed transitions from sublevel to sublevel. The opportunity for several consistency checks made the J assignments very secure. It turned out that the transitions H4 -> D2 and Fq - D2 are electric-dipole, while Fi-> Dq is magnetic-dipole. Deviations from perfect Russell-Saunders coupling are responsible for the apparent violations of the selection rule AS = 0, which holds for D, L + 2S and... 

The XPS spectrum of 4f excitation often exhibits a series of peaks corresponding to the multiplet structure of the final 4f shell. The spin-orbit splitting in Yb has been discussed in section 3.3. The splitting of final states with different L quantum numbers in the Russell-Saunders scheme is evident in Sm and heavier members. The spin multiplet structure is seen in rare earths with more than half-filled shells (Tb-Tm). For example, the final state of Tb(4f ) may be in the state 4f (5 = 2) or 4ff4fi (5 = ). The splitting of these groups of states measures the exchange interaction between the 4f electrons. 

The multiplet structure in Ne arises from a pair-coupling (not Russell-Saunders) angular momentum coupling scheme, which is obeyed in rare-gas atom excited states. The ground and excited states of Ne exhibit the electron... 

Intensities (Intens.) of the final states (S L J ) of the configuration 4f" arising from electron addition to the initial state (SU) of 4f", normalized to (14 - n). The intensities of the multiplets J are calculated within the Russel-Saunders (LS) coupling scheme. Only multiplet components with normalized intensity >0.1 are listed. The relative energies E are obtained from UV-absorption data (Carnall et al. 1968, Crosswhite et al. 1969). The states are listed in the order of increasing energy. 

In the Russell-Saunders approximation, considering electrostatic but neglecting electrodynamic effects, the energy levels of a given configuration can be assembled in (multiplet) terms characterized by the quantum numbers L of orbital angular momentum and 5 of the total spin. With Mulliken, we use capital letters for the quantum numbers of more-electron systems and lower case letters for the individual orbitals. If the gaseous atom or ion contains only one partly filled shell with only one electron (or 4/ 1 electrons, i.e., a hole in the filled n,l shell which may... 

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