Multipole 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... 

Pk in the form of the magnetic multipole transition operator in the longwave approximation. The T-even operators Qk are then obtained from the... 

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. 

By evaluating the commutator of Q q with Hnr the generalization of the acceleration form of the transition operator to cover the case of electric multipole transitions of any multipolarity was obtained [77]. Unfortunately, the resultant operator has a much more complex and cumbersome form than Q q and Q q, since it contains both one-electron and two-electron parts. 

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. 

For the expressions of matrix elements of multipole transitions, containing the CFP, the selection rules of the kind Av = +1 may be established, however they are rather approximate. 

The highest probabilities are for transitions between configurations with i = n2 and h = h + 1. In the final state the coupling, close to LS, holds for neighbouring shells then the matrix element of electric multipole transition is defined by formulas (25.28), (25.30). Similar expressions for other coupling schemes may be easily found starting with the data of Part 6 and Chapter 12. 

Such expressions can be easily generalized to cover the case of the electric multipole transition operator with an unspecified value of the gauge condition K of electromagnetic field potential (4.10) or (4.11). 

In this section we first present a set of general transformation formulae for tensor operators associated with SRMs. These then serve as a mathematical tool for the formulation of Wigner-Eckart theorems and selection rules for irreducible tensor operators associated with multipole transitions of SRMs. The concept of isometric groups will allow a formulation of selection rules in strict analogy to the group theoretical treatment of quasirigid molecules first presented by Wigner5. ... 

The dipole polarizability can be used in place of the dipole moment function, and this will lead to Raman intensities. Likewise, one can compute electrical quadrupole and higher multipole transition moments if these are of interest. 

Generalizations of the Jaynes-Cummings model (34) in the case of quad-mpole and other high-order multipole transitions can be constructed in the same way. 

Similar results can be obtained for an arbitrary atomic multipole transition in much the same way as above. For example, in the case of the excited atomic... 

For any atomic multipole transition, the excited state can be described in terms of the dual representation of corresponding SU(2) algebra, describing the azimuthal quantum phase of the angular momentum. In particular, the exponential of the phase operator and phase states can be constructed. The quantum phase variable has a discrete spectrum with (2j + 1) different eigenvalues. 

Based on the single-particle model, Blatt and Weisskopf have calculated probable lifetimes for excited states assuming a model nucleus with a radius of 6 fm. For 2 -multipole transitions of electric (E) or magnetic (M) type they derived the following equations... 

As seen, for decay involving electric multipole transitions the average lifetime r is proportional to r, and for decay involving magnetic multipoles, to in the... 

Multipol-Strahlungsiibergange, s. a. Gber-gangswahrscheinlichkeiten, multipole transition probabilities 43, 46. 

Multipole transitions, cf. transition probabilities, Multipol-Strahlungsubergdnge, s. auch Vbergangswahrscheinlichkeiten 43, 46. 

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)... 

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