LS-coupling scheme

RQDO oscillator strengths, these generally satisfy a feature that is characteristic of the LS coupling scheme The sum of the f-values of the lines that start from the same level is approximately constant. Not all the remaining f-values collected in Tables 1 to 6 comply with this feature. 

In Chapter 9 we discussed the classification of the terms and energy levels of a shell of equivalent electrons using the LS coupling scheme. Here we shall consider the case of two non-equivalent electrons. As we shall see later on, generalization of the results for two non-equivalent electrons to the case of two or more shells of equivalent electrons is straightforward. 

For a pure LS coupling scheme, both the electric and magnetic multipole transitions are diagonal with regard to S and S. The multiplet strength is also symmetric with respect to the transposition of the initial and final terms... 

Many interesting features have been revealed during such studies the necessity to use various (differing from the usual LS) coupling schemes, the role of forbidden transitions, occurrence of satellite lines, unresolved transition arrays, splitting of electronic shells into subshells, etc. 

Particular attention is paid to recently worked out methods of theoretical description of highly ionized atoms. They allow us to take into account relativistic effects. Accounting for them both as corrections and in relativistic approximation, as well as the use of various (differing from traditional LS) coupling schemes is considered in detail. 

Within the two-step model, one can say that the intermediate photoionized state is the initial state for the Auger transition. For the K-LL spectrum of neon this initial state is described by ls2s22p6 2Sj/2. For the final state the possible electron configurations of the ion were shown in Fig. 2.5. Within the LS-coupling scheme which applies well to neon, these electron configurations yield the following final... 

After this excursion one returns to the neon K-LL Auger transitions and their final ionic states, given in equ. (3.1). Within the LS-coupling scheme one expects the following six transitions which are classified according to X-ray nomenclature (see Fig. 2.2) ... 

Using the basis functions which follow from the approximate Hamiltonian H° of equ. (1.3), it is the residual interaction H — H° which causes the Auger transitions. This operator, however, reduces to the Coulomb interaction if more than one electron changes its orbital.) Within the LS-coupling scheme this transition operator requires the following selection rules... 

In all cases the same Wannier exponent occurs, and for a selected photon energy the excess energies differ due to the state-dependent ionization energies Ef+. However, of importance in the present context are the state-dependent values for the constants of proportionality a0. Within the LS-coupling scheme the complete final state built from the ion core and the electron-pair wavefunction must have Sf = 0 and L = 1. Therefore one gets the following possibilities ... 

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... 

The tensorial structure of the spin-orbit operators can be exploited to reduce the number of matrix elements that have to be evaluated explicitly. According to the Wigner-Eckart theorem, it is sufficient to determine a single (nonzero) matrix element for each pair of multiplet wave functions the matrix element for any other pair of multiplet components can then be obtained by multiplying the reduced matrix element with a constant. These vector coupling coefficients, products of 3j symbols and a phase factor, depend solely on the symmetry of the problem, not on the particular molecule. Furthermore, selection rules can be derived from the tensorial structure for example, within an LS coupling scheme, electronic states may interact via spin-orbit coupling only if their spin quantum numbers S and S are equal or differ by 1, i.e., S = S or S = S 1. 

The least ambiguous and most appropriate description of the atom after the collision is in terms of the density matrix (Blum, 1981), whose elements are bilinear combinations of scattering amplitudes for different magnetic substates. For the sake of simplicity we restrict ourselves to the most common case, in which the target is initially in an S state and the excitation involves the transfer of one electron from an s orbital to a p orbital in the independent-particle approximation. In atoms with one active electron the transition is — P. If there are two active electrons it is — P. We use the LS-coupling scheme. 

We now determine the exchange ion-atom interaction that allows us to evaluate the cross section of resonant charge exchange. On this way we represent the wave function of the atom having n valence electrons of momentum within the framework of the LS—coupling scheme in the form [2, 12, 23] ... 

For practice in using the LS coupling scheme, the reader may verify, by the method used for np2, that an nd2 configuration gives rise to the statesJF, 3P, lG, lD and lS, and that an np3 configuration gives the states 4S, 2P, 2D. 

In this work we interpret the two-body matrix elements of in the representation of the LS-coupling scheme as done by Wildenthal, Brown and co-workers [48,63] and Hosaka and Toki [64]. This representation allows for a more direct comparison with the NN interaction, which is expressed in terms of partial waves. The LS-coupled matrix element of a given component (afi)LSJ Tj f J(yS)LSJ T), with a = Ha/ji, is related to the corresponding matrix elements of the total interaction in the jf/-scheme by... 

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