Electric multipole radiation

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

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 ... 

In contrast to the plane waves, the field strengths of the multipole radiation can have any direction. In fact, the electric multipole radiation obey the condition -r = 0, while it can have nonzero longitudinal component (S r 0) of the electric field strength [25], In other words, this is the transversal magnetic radiation. In turn, the magnetic multipole field is characterized by the relations... 

Here we again consider a monochromatic radiation field. Let us stress that this expression describes the spatial anisotrophy of the electric field and therefore specifies the polarization of the electric multipole radiation. In the case of magnetic multipole radiation, the spatial anisotrophy of magnetic induction can be described by the following polarization matrix [89] ... 

Much of our knowledge of molecules is obtained from experimental studies of the way they interact with electromagnetic radiation, and the recent growth in non-linear spectroscopies and molecular electronics has focused attention on our ability (or otherwise) to predict and rationalize the electric properties of molecules. The idea of an electric multipole is an important one, so let s begin the discussion there. 

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 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. 

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. 

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],... 

As can be seen from the equations (21)-(22) and (23)-(24), there is an essential difference between the representations of plane and multipole waves of photons. In particular, a monochromatic plane wave of photons is specihed by only two different quantum numbers a = x, y, describing the linear polarization in Cartesian coordinates. In turn, the monochromatic multipole photons are described by much more quantum numbers. Even in the simplest case of the electric dipole radiation when X = E and j = 1, we have three different states of multipole photons in (23) with m = 0, 1. Besides that, the plane waves of photons have the same polarization a everywhere, while the states of multipole photons have given m. It is seen from (24) that, in this case, the polarization described by the spin index p can have different values at different distances from the singular point. In Section V we discuss the polarization properties of the multipole radiation in greater detail. 

We now turn to the problem of the SU(2) quantum phase of multipole radiation. As a particular example of some considerable interest, we investigate the electric dipole field. All other types of the multipole radiation can be considered in the same way. 

In Section III.B, we introduced the atomic quantum phase states through the use of the representation of the SU(2) algebra (37) and dual representation (48), corresponding to the angular momentum of the excited atomic state. The multipole radiation emitted by atoms carries the angular momentum of the excited atomic state and can also be specified by the angular momentum [2,26,27], The bare operators of the angular momentum of the electric dipole... 

Taking into account the physical meanings of the atomic operators (41) and (66) and the integrals of motion (64), we can consider the field operators (63) and (69) as the nonnormalized exponential operators of the radiation phase, which, by construction, is the SU(2) phase of the multipole (electric dipole) radiation. By performing a similar analysis to that described in Section III.B, we can define the cosine and sine operators of the radiation phase as follows [36]... 

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