Magnetic multipoles

A magnetic multipole expansion is obtained by can be obtained by inserting the vector potential proposed by Bloch (146) into the interaction energy of Eq.(120) 

Note that, contrary to the electric multipole expansion (148), there is no zeroth order term corresponding to a magnetic charge interacting with the value of the vector potential at the expansion point, this reflecting the absence of magnetic monopoles. 

The relativistic magnetic multipole Hamiltonian is obtained by inserting the relativistic expression of the current density (121) into the magnetic multipole expansion (161) 

These operators are symmetric in all indices j. .. jn), but not including the index denoting the component of m. A magnetic multipole of order n accordingly constitute 3(n -t- 2) (n - -1 )/2 linearly independent quantities. 

The corresponding expression for the non-relativistic magnetic moments are not at all trivial, in particular since the spin-dependent part entails a triple vector product, for which the associative law does not hold. After quite a bit of manipulations we obtain the operator 

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

The symbol p is a transition moment operator, of which there are various kinds, namely, those corresponding to changes in electric or magnetic dipoles, higher electric or magnetic multipoles, or polarizability tensors. 

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

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.

The corresponding expression for the relativistic transition probability of the magnetic multipole (Mk) radiation has the form... 

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.) ... 

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... 

The hyperfine structure (splitting) of energy levels is mainly caused by electric and magnetic multipole interactions between the atomic nucleus and electronic shells. From the known data on hyperfine structure we can determine the electric and magnetic multipole momenta of the nuclei, their spins and other parameters. 

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. 

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. 

The analogous selection rules for magnetic multipole (Mfc) radiation (operator (4.16)) have the form... 

We firstly define the electric and magnetic multipoles, then show how the interaction energy with external fields can be expressed in terms of the induced moments and go on to see how these are related to the field-free moments via response functions. Most of our treatment will concern the electric case, since this is the simpler, and since magnetic effects can be treated by analogy. 

There are unfortunately various definitions of magnetic multipole moments in the literature - a recent paper by Raab16 discusses the various definitions and properties, and creates some order from an apparent chaos. 

Mixed-Symmetry Interpretation of Some Low-Lying Bands in Deformed Nuclei and the Distribution of Collective Magnetic Multipole Strength... 

Proceeding as above, the general expression (58) can be used to calculate other particular cases involving higher multipoles, on resorting to Tables 4—7, where the multipole elements are listed. Interactions between magnetic multipoles are also the subject of discussion. The theory of electrostatic interactions for electric multipoles has been dealt with in various approaches by Frenkel, Pople, Jansen, and others, as well as by Gray. The above presentation follows the concise uniform treatment of Kielich. > > ... 

Similarly to (54), one can define the potential energy of interaction between magnetic multipoles (43) and an external magnetic induction... 

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 ... 

For a Gaussian beam, the fields of the radiating electric and magnetic multipoles satisfy the same boundary conditions (vanishing faster than 1/p as p oo) so that the fields in the plane(s) defined by the transverse E (H) field and the optical axis are symmetric. It is difficult to generate a balanced hybrid mode in conventional smooth-walled metallic waveguide instead, one may use a component called a scalar horn. 

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