MULTIPOLAR selection rules

We find that the intensity of emission is a function of the oscillator strength (for spontaneous emission). This leads us to multipolar selection rules, which are a useful way to classify electronic transitions and interactions. We can classify such transitions as dipole - dipole (dd) and dipole-quadrupole (dq) interactions (see 5.6.6. given above). 

A. Jorio, A.G.S. Filho, V.W. Brar, A.K. Swan, M.S. Unlu, B.B. Goldberg, A. Righi, J.H. Hafner, C.M. Lieber, R. Saito, G. Dresselhaus, and M.S. Dresselhaus, Polarized resonant Raman study of isolated single-wall carbon nanotubes symmetry selection rules, dipolar and multipolar antenna effects. Phys. Rev. B 65, 121402.1-121402.4 (2002). 

All selection rules are contained in the expressions for matrix elements of the electron transition operators. Therefore, not knowing them cannot cause the occurrence of false transitions, because they will be automatically equal to zero. However, studies of the selection rules allow us in many cases to determine the transition type and multipolarity (Ek or Mk), and to estimate their relative intensities without carrying out accurate... 

As we have seen from the selection rules for non-relativistic magnetic dipole (M1) radiation, these transitions are permitted only between levels of one and the same configuration. However, this is not so for higher multipolarities (k > 1). Therefore we present here the appropriate formulas to cover the general cases needed in practice. So, it may be of use to have the following expression for the submatrix element of the operator of Mfc-transitions between the levels of two different two-shell configurations ... 

As has been emphasized in Chapter 24 the probabilities of Ek- and Mfc-transitions are rapidly decreasing functions of k (see also Chapter 30). Therefore, we can usually restrict ourselves to examination of the radiation of the lowest multipolarity permitted by the appropriate selection rules. Between levels of one and the same configuration both E2- and Ml-transitions are allowed, that is why we must consider them together. 

The situation is more complex in NGR spectra since in general both ground and excited nuclear levels have magnetic moments so that (18.2) and (18.3) apply to both. In addition, the selection rule for the transitions is governed by the multipolarity of the gamma radiation (magnetic dipole, electric quadrupole, etc., or some mixture thereof). 

The number of multipole parameters is reduced by the requirements of symmetry. As discussed in chapter 3, the only allowed multipolar functions are those having the symmetry of the site, which are invariant under the local symmetry operations. For example, only / = even multipoles can have nonzero populations on a centrosymmetric site, while for sites with axial symmetry the dipoles must be oriented along the symmetry axis. For a highly symmetric site having 6 mm symmetry, the lowest allowed / 0 is d66+ all lower multipoles being forbidden by the symmetry. The index-picking rules listed in appendix D give the information required for selection of the allowed parameters. 

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