Electric and Magnetic Multipole Transitions

This operator, too, is a scalar in the space of total angular momentum for an electron. Tensors in this space are, for example, the operators of electric and magnetic multipole transitions (4.12), (4.13), (4.16). So, the operator of electric multipole transition (4.12) in the second-quantization representation is... 

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... 

The general definition of the electron transition probability is given by (4.1). More concrete expressions for the probabilities of electric and magnetic multipole transitions with regard to non-relativistic operators and wave functions are presented by formulas (4.10), (4.11) and (4.15). Their relativistic counterparts are defined by (4.3), (4.4) and (4.8). They all are expressed in terms of the squared matrix elements of the respective electron transition operators. There are also presented in Chapter 4 the expressions for electric dipole transition probabilities, when the corresponding operator accounts for the relativistic corrections of order a2. If the wave functions are characterized by the quantum numbers LJ, L J, then the right sides of the formulas for transition probabilities must be divided by the multiplier 2J + 1. 

Half-lives Ty2 = n2)/Ti for electric and magnetic multipole transitions are shown as functions of the y-ray energy in Fig. 2.13. The half-lives embrace more than 32 orders of magnitude from 10 to 10 s. For comparison the mean lives (r = Ti/2/0.6931) of atomic levels shall be given here ... 

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 seen above, the general treatment of the photon-molecule interaction expands the transition operator as a series with electric and magnetic multipole contributions ... 

The electric and magnetic dipole, quadrupole, octupole, etc., transitions are denoted by El, E2, E3,. .. and Ml, M2, M3,. .., respectively. The selection rules deduced from angular momentum and parity conservation laws for electric and magnetic multipole radiations are summarized in Table 2.6. 

Fig. 1-11. Calculated conversion coefficients for the emission of K electrons for the light elements of the Pt group (Ru, Rh, Pd) versus the transition energy E —EJ [37]. The electric and magnetic multipole orders are denoted as usual by E and (x = 1 to 4), respectively.

The angular distribution of the intensity of electromagnetic radiation is given by specific analytic functions written in terms of an angle, W(Q,mi), relative to the quantization axis, Z, and the magnetic quantum number, mi. The patterns depend on the order of the multipole, dipole, quadra pole, and so forth, but they are the same for electric and magnetic transitions with the same order. For example, the angular distributions for dipole radiation are... 

As was pointed out in Chapter 4, division of the radiation into electric and magnetic is connected with the existence of two types of multipoles, characterized by the parities (—l)fc and (—l)fc+1, respectively. The first ones we have studied quite thoroughly in Chapters 24-26. Here let us consider in a similar way the M/c-transitions. Again, as we have seen in Chapter 4, the potential of the electromagnetic field in this case does not depend on gauge. Therefore only one relativistic expression (4.8) was established for the probability of M/c-radiation, described by the appropriate operator (4.9). The probability of non-relativistic M/c-transitions (in atomic units) is given by formula (4.15), whereas the corresponding non-relativistic operator has the form (4.16). 

Some arguments that imply limitations on the concept of molecular multipole moments due to the requirements of gauge invariance are presented. The discussion is based on a pair of polarization field vectors which are natural generalizations of multipole series. A class of transformations that mix some of the components of the polarization field vectors, so spoiling a separation into electric and magnetic types, is identified. The results are related to the gauge-invariant transition amplitudes. 

The parity of the various contributions to the cross-section merit attention K = 1,2,.. ., 21 + 1 for the magnetic multipole transition and K = 2,4,..., 21 for electric multipole transitions. The latter involves only B(K, K), i.e., the electric multipole is purely spin. An orbital contribution to the electric multipole arises in scattering events which engage electrons in different /-states that are non-degenerate and possess different radial wavefunctions. The quantity B(K, K)... 

Endt (1981) evaluated 1,340 y-ray transitions in A = 91-150 nuclek The transition rates were classified according to electric or magnetic multipole character and were expressed in Weisskopf units. Transitions, which can be mixed in principle, are only listed if the mixing ratio has been measured. The strengths S = Ti Ti) usually scatter appreciably, nevertheless, upper limits could be established fi-om the data. These are 0.01,300,100, and 30 W. u. for El, E2, E3, and E4 and 1, 1, 10, and 30 W. u. for Ml, M2, M3, and M4 radiations, respectively. For upper limits in other regions see (Firestone et al. 1996). The strengths are usually enhanced for E2, and E3, while the El, M2 transitions are usually hindered. 

In its broadest sense, spectroscopy is concerned with interactions between light and matter. Since light consists of electromagnetic waves, this chapter begins with classical and quantum mechanical treatments of molecules subjected to static (time-independent) electric fields. Our discussion identifies the molecular properties that control interactions with electric fields the electric multipole moments and the electric polarizability. Time-dependent electromagnetic waves are then described classically using vector and scalar potentials for the associated electric and magnetic fields E and B, and the classical Hamiltonian is obtained for a molecule in the presence of these potentials. Quantum mechanical time-dependent perturbation theory is finally used to extract probabilities of transitions between molecular states. This powerful formalism not only covers the full array of multipole interactions that can cause spectroscopic transitions, but also reveals the hierarchies of multiphoton transitions that can occur. This chapter thus establishes a framework for multiphoton spectroscopies (e.g., Raman spectroscopy and coherent anti-Stokes Raman spectroscopy, which are discussed in Chapters 10 and 11) as well as for the one-photon spectroscopies that are described in most of this book. 

While this discussion based on electrostatics ignores the time dependence in PF(t) and omits the effects of magnetic fields associated with the light wave, it does anticipate some of our final results in this section regarding electric dipole and electric quadrupole contributions to the matrix elements . It yields no insight into magnetic multipole transitions or into the nature of the time-ordered integrals in the Dyson series expansion of Eq. 1.96. 

Determine which types of multipole transitions are embodied in this second-order term. What kinds of electric and magnetic field gradients are generally required to effect these types of transitions By what factor do these transition probabilities differ from those of Ml and E2 transitions ... 

For the sake of simplicity and a more instructive description, we shah restrict ourselves to the case of unpolarized single line sources of 7 = 3/2v / = 1/2 magnetic dipole transitions (Ml) as for example in Fe, which has only a negligible electric quadrupole (E2) admixture. It will be easy to extend the relations to arbitrary nuclear spins and multipole transitions. A more rigorous treatment has been given in [76, 78] and [14] in Chap. 1. The probability P for a nuclear transihon of multipolarity Ml (L=l) from a state I, m ) to a state h, m2) is equal to... 

As stated in an earlier paragraph, the sharp emission and absorption lines observed in the trivalent rare earths correspond to/->/transitions, that is, between free ion states of the same parity. Since the electric-dipole operator has odd parity,/->/matrix elements of it are identically zero in the free ion. On the other hand, however, because the magnetic-dipole operator has even parity, its matrix elements may connect states of the same parity. It is also easily shown that electric quadrupole, and other higher multipole transitions are possible. 

Figure 9.3 Weisskopf single-particle estimates of the transition rates for (a) electric multipoles and (b) magnetic multipoles. From Condon and Odishaw, Handbook on Physics, 2nd Edition. Copyright 1967 hy McGraw-Hill Company, Inc. Reprinted by permission of McGraw-Hill Book Company, Inc.

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