Line and multiplet strengths

A number of ideas of the theory of electronic transitions were discussed in Chapter 4. In Part 6 we are going to consider this issue in more detail. Let us start with the definition of the main characteristics of electronic transitions, common for both electric and magnetic multipole radiation. 

As we have seen while considering energy spectra, the energy levels of free atoms are always degenerate relative to the projections M of the total angular momentum J. Further we shall learn that the characteristics of spontaneous electronic transitions do not depend on them. Let us define the line strength of the electronic transition of any multipolarity k as the modulus of the relevant matrix element squared, i.e. 

The total line strength does not depend on the coupling scheme, i.e. 

Equation (24.3) is also valid for intermediate coupling. Applying the Wigner-Eckart theorem (5.15) to (24.1) and performing the summation, we arrive at the result 

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 

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Line and multiplet strengths are useful theoretical characteristics of electronic transitions, because they are symmetric, additive and do not depend on the energy parameters. However, they are far from the experimentally measured quantities. In this respect it is much more convenient to utilize the concepts of oscillator strengths and transition probabilities, already directly connected with the quantities measured experimentally (e.g. line intensities). Oscillator strength fk of electric or magnetic electronic transition aJ — a J of multipolarity k is defined as follows ... 

The presence of an outer open shell in an atom, even if this shell does not participate in the transitions under consideration, influences the X-ray radiation spectrum. Interaction of the vacancy with the open shell, particularly in the final state when the vacancy is not in a deep shell, splits the levels of the core. Depending on level widths and relative strength of various intra-atomic interactions, this multiplet splitting leads to broadening of diagram lines, their asymmetry, the occurrence of satellites, or splitting of the spectrum into large numbers of lines. 

The RQDO /-values conform with the recommended values by Wiese et al. [18] for the 3p ( P) 4s-3p ( P) 4p transitions in C//better than those of the much more complex theoretical procedure of Ojha and Hibbert [17], who used large multiconfiguration expansions in the atomic structure code CIVS. On the other hand, the discrepancies between the length and velocity CIVS oscillator strengths are not negligible. Wiese et al. [18] remark that, for the case of the 4s " P-4p multiplet, the stronger lines measured by Bengtson et al. [16] seem to be affected by self-absorption (Table 2). 

The quantity g(J)/g(LS) stands for the relative statistical weight of the J level in the LS term. Therefore, formulas (24.12) and (24.13) show that the sum of relative line strengths of the multiplet with given initial or final level is equal to the relative statistical weight of this level in the term. 

Figure 2.3 shows the spectra lined up on a ppm scale rather than a hertz scale (the vertical scale is increased and the tall methyl and OH peaks are clipped off). This is the universally accepted format for presenting NMR data. Although all the peaks (resonances) appear at the same place in the spectrum (same chemical shift in ppm), the multiplet patterns appear to shrink horizontally as we go to higher field strength because the J couplings in hertz get smaller and smaller on the ppm scale. For example, on a 200-MHz spectrometer, a typical... 

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