Electric dipole radiation selection rules

Selection rules also arise on considering the point-group symmetry of tfo(Qeq). In the case of electric dipole radiation the perturbation Y, which describes the interaction with the radiation field, may be expressed in terms of the x, y, z components of the dipole moment operator r. The operators (t) transform as the x, y, or z components of r. 

In the particular case of electric dipole radiation A/ = 1, i.e. El-transitions are permitted between configurations of opposite parity. For 2-transitions Al = 0, 2 (excluding transitions ns — n s), i.e. they are allowed between levels of one and the same configuration or between configurations of the same parity. M 1-transitions may take place only between levels of one and the same configuration. There are no restrictions on An for /c-transitions. Selection rules for J and M follow from the Clebsch-Gordan coefficient... 

The quality of the SOC calculation in O2 can be checked by estimation of the fc Sj" — A3E transition probability. The transition is forbidden by selection rules for electric dipole radiation with account of SOC, and occurs as magnetic dipole spin-current borrowing intensity from microwave transitions between spin-sublevels of the ground state [41]. 

The actual site symmetry of the Pr3+ ion in the lattice is C3[,. However, the doubly degenerate levels of D3j, still retain their degeneracy in C3h and the additional crystal-field parameter that arises in C3[, is not significant. The selection rules for electric dipole radiation in C3j, symmetry, however, are different. 

The laser can be used to exdte particles by one-photon or multiphoton absorption. A sublevel a of the upper state under study can be populated selectively by making use of the selection rules for electric dipole radiation and/or—if the spectral bandwidth of the laser is sufficiently narrow in relation to the experimental linewidth—by driving only the transition, which connects the lower state and the excited state sublevel a. This rather simple process is usually followed by an rf transition within the excited state. 

A schematic representation of the valence k, n, n and 7t molecular orbitals is given in Fig. 3. The selection rules for electric dipole radiation do not permit the X Aj and A Aj states to combine. As a result, the 0° transition is formally electric dipole forbidden. (M denotes a vibronic transition involving a quanta in the upper state and b quanta in the lower state, while a level is designated M. In the case of the 0—0 origin transition, the mode is labelled 0 and the transition Oq.)... 

Fig.5.1. Selection rules for magnetic quantum number m and polarization of electric dipole radiation observed in the direction 6=0.

We now derive the selection rules for electric dipole radiation using atomic wavefunctions in the L-S coupling approximation (equations (5.24) and (5.25)). We shall find that certain of the selection rules are rigorous but that others are only applicable if L-S coupling is a good description of the atomic system. 

Magnetic dipole and electric quadrupole transitions, which we have ignored so far, will be discussed in detail in Chapter 7. However, it should be noted that the selection rules governing these forbidden types of radiation can be derived by the application of the techniques already discussed here. The results are summarized in Table 7.1 together with the rules for electric dipole radiation previously derived in this chapter. 

We have seen in Chapter 5 that the transition probability for electric dipole radiation, equation (4.23), is only non-zero if certain selection rules are satisfied. In particular the initial and final states of the system must have opposite parity, where the parity of an electron configuration is... 

The selection rules for electric dipole radiation have already been considered in detail in Chapter 5 and are summarized in Table 7.1. Those for magnetic dipole and electric quadrupole transitions may be derived from equations (7.5) and (7.10) by the application of similar techniques. The task is therefore left as an exercise for the student (Problem 7.4) however, the comments which follow indicate the line of reasoning used and may prove helpful. 

Electric dipole n polarized emissions are those in which the electric field of the emitted light is parallel to z. Thus the selection rule for n emissions from level A1 are those defined by the direct product A1 x A2. By an inspection of the character table of group 1)3 (Table 7.6), we can easily prove that Ai x A2 = A2, so that only the A A2 emission is allowed by n polarized radiation (as shown in Figures 7.7 and 7.8). 

What are the electric-dipole selection rules for a particle in a three-dimensional box exposed to isotropic radiation ... 

Lines, corresponding to different transitions from initial states with vacancy in the shells with the same n, compose a series of spectra, e.g. K-, L-, M-series etc. Main diagram lines correspond to electric dipole ( 1) transitions between shells with different n. The lines of 2-transitions also belong to diagram lines. Selection rules of 1-radiation as well as the one-particle character of the energy levels of atoms with closed shells and one inner vacancy cause, as a rule, a doublet nature of the spectra, similar to optical spectra of alkaline elements. X-ray spectra are even simpler than optical spectra because their series consist of small numbers of lines, smaller than the number of shells in an atom. The main lines of the X-ray radiation spectrum, corresponding to transitions in inner shells, preserve their character also for the case of an atom with open outer shells, because the outer shells hardly influence the properties of inner shells. 

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