Multipole radiation

For quadrupole radiation, they estimate P 2x 10-9, whereas for magnetic-dipole radiation their result was P 2 x 10-8. The experimental values lie in the range of 10 7 to 10 5. From these estimates, one concludes that the probability of significant electric-quadrupole and higher-order-multipole radiation is very small indeed. The magnetic-dipole radiation is weak but probably is of some importance, particularly in cases where the electric-dipole emission is strictly prohibited. 

The single-particle limit for magnetic multipole radiation obtained by assuming that the change in current is due to a single nucleon is... 

Now, by use of formulas (2.23)-(2.26) we are in a position to present in. /-representation all the operators needed. For example, the non-relativistic operator of electric multipole radiation will have the form... 

Let us consider the non-relativistic limit of the relativistic operators describing radiation. Expressing the small components of the four-component wave functions (bispinors) in terms of the large ones and expanding the spherical Bessel functions in a power series in cor/c, we obtain, in the non-relativistic limit, the following two alternative expressions for the probability of electric multipole radiation ... 

The transformation of the relativistic expression for the operator of magnetic multipole radiation (4.8) may be done similarly to the case of electric transitions. As has already been mentioned, in this case the corresponding potential of electromagnetic field does not depend on the gauge condition, therefore, there is only the following expression for the non-relativistic operator of Mk-transitions (in a.u.) ... 

Relativistic corrections of order v2/c2 to the non-relativistic transition operators may be found either by expanding the relativistic expression of the electron multipole radiation probability in powers of v/c, or semiclas-sically, by replacing p in the Dirac-Breit Hamiltonian by p — (l/c)A (here A is the vector-potential of the radiation field) and retaining the terms linear in A. Calculations show that in the general case the corresponding corrections have very complicated expressions, therefore we shall restrict ourselves to the particular case of electric dipole radiation and to the main corrections to the length and velocity forms of this operator. 

The operator of the hyperfine structure, caused by electric multipole radiation, may be presented in the form... 

General expressions for electric (Ek) and magnetic (Mk) multipole radiation quantities... 

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 an example of these methods, consider the B cyclic theorem for multipole radiation, which can be developed for the multipole expansion of plane-wave radiation to show that the B<3) field is irrotational, divergentless, and fundamental for each multipole component. The magnetic components of the plane wave are defined, using Silver s notation [112] as... 

This is the phaseless magnetic field of multipole radiation on the 0(3) level. The solution (777) reduces to the simple... 

The probability of y-ray emission is given by the sum of the probabilities for the emission of the individual multipole radiations, which decrease drastically with increasing L. Furthermore, for a certain multipole, the probability of the emission of electric multipole radiation is about two orders of magnitude higher than that of the emission of magnetic multipole radiation. 

On the basis of the shell model of the nuclei, Weisskopf derived the following equations for the probabilities of y-ray emission, given by the decay constants Xe for electric multipole radiation and Am for magnetic multipole radiation ... 

Table 5.3. Half-lives of y transitions calculated by application of the model of multipole radiation.

In turn, the monochromatic multipole photons are described by the scalar wavenumber k (energy), parity (type of radiation either electric or magnetic), angular momentum j 1,2,..., and projection m = —j,..., / [2,26,27]. This means that even in the simplest case of monochromatic dipole (j = 1) photons of either type, there are three independent creation or annihilation operators labeled by the index m = 0, 1. Thus, the representation of multipole photons has much physical properties in comparison with the plane waves of photons. For example, the third spin state is allowed in this case and therefore the quantum multipole radiation is specified by three different polarizations, two transversal and one longitudinal (with respect to the radial direction from the source) [27,28], In contrast to the plane waves of photons, the projection of spin is not a quantum number in the case of multipole photons. Therefore, the polarization is not a global characteristic of the multipole radiation but changes with distance from the source [22],... 

Then, the positive-frequency part of the operator vector potential of the multipole radiation of a given type X takes the form [2,27]... 

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