Electric/magnetic multipolarity

Nonradiative energy transfer is due to electric or magnetic multipolar interactions or to exchange interactions. ... 

Dexter, following the classic work by Forster, considered energy transfer between a donor (or a sensitizer) S and an acceptor (or activator) A in a solid. This process occurs if the energy difference between the ground and excited states of S and A are equal (resonance condition) and if a suitable interaction between both systems exists. The interaction may be either an exchange interaction (if we have wave function overlap) or an electric or magnetic multipolar interaction. In practice the resonance condition can be tested by considering the spectral overlap of the S emission and the A absorption spectra. The Dexter result looks as follows ... 

The coupling of adjacent ions in such a case can arise via exchange interaction if their wave-functions overlap, via super exchange interactions involving intervening ions, or via various electric or magnetic multipolar interactions. 

An important class of properties arise from multipolar expansions of the interaction of nuclear moments with the electric and magnetic fields set up by surrounding electrons and nuclei. Restrictions apply to the possible nuclear moments 2 [93]. In general I < 21, where I is the nuclear spin. Furthermore, electric (magnetic) moments are restricted to even(odd) values of /. The lowest nuclear electric multipole is accordingly the electric quadmpole moment... 

Here B[E/(M)X is the reduced nuclear probability, atomic radial matrix elements of electric (magnetic) [E/M] multipolarity A ji f and 7i,f are the angular momenta of the electronic and nuclear states correspondingly. The atomic radial matrix elements P (ft)jv) of electric (magnetic) [E/M] multipolarity A are expressed by means of the integral ... 

In the Hamiltonian conventionally used for derivations of molecular magnetic properties, the applied fields are represented by electromagnetic vector and scalar potentials [1,20] and if desired, canonical transformations are invoked to change the magnetic gauge origin and/or to introduce electric and magnetic fields explicitly into the Hamiltonian, see e.g. refs. [1,20,21]. Here we take as our point of departure the multipolar Hamiltonian derived in ref. [22] without recourse to vector and scalar potentials. 

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... 

The interaction Hamiltonian that appears in Equation (5.37) can involve different types of interactions namely, multipolar (electric and/or magnetic) interactions and/or a quantum mechanical exchange interaction. The dominant interaction is strongly dependent on the separation between the donor and acceptor ions and on the nature of their wavefunctions. 

Energy transfer probabilities due to multipolar magnetic interactions also behave in a similar way to that previously discussed for multipolar electric interactions. Thus, the transfer probability for a magnetic dipole-dipole interaction also varies with 1 / 7 , and higher order magnetic interactions are only influential at short distances. In any case, the multipolar magnetic interactions are always much less important than the electric ones. 

There exists another more consistent way of obtaining the electron transition operators. We can start with the quantum-electrodynamical description of the interaction of the electromagnetic field with an atom. In this case we find the relativistic operators of electronic transitions with respect to the relativistic wave functions. After that they may be transformed to the well-known non-relativistic ones, accounting for the relativistic effects, if necessary, as corrections to the usual non-relativistic operators. Here we shall consider the latter in more detail. It gives us a closed system of universal expressions for the operators of electronic transitions, suitable to describe practically the radiation in any atom or ion, including very highly ionized atoms as well as the transitions of any multipolarity and any type of radiation (electric or magnetic). 

Line and multiplet strengths are useful theoretical characteristics of electronic transitions, because they are symmetric, additive and do not depend on the energy parameters. However, they are far from the experimentally measured quantities. In this respect it is much more convenient to utilize the concepts of oscillator strengths and transition probabilities, already directly connected with the quantities measured experimentally (e.g. line intensities). Oscillator strength fk of electric or magnetic electronic transition aJ — a J of multipolarity k is defined as follows ... 

Even though the nonlinearity of BN is predominantly electric-dipolar, recent experiments show that nonlocal higher-order multipolar (magnetic and electric quadrupolar) contributions to SFG from BN are measurable [30]. However, the polarization of all three fields in bulk SFG experiments may always be chosen such that only the chiral electric-dipolar signals are observed. 

Fourier series mode expansions for the multipolar electric displacement and magnetic field operators may be written in terms of fhe creation and destruction operators as... 

The particular merit of multipolar gauge is that is allows one to express the scalar and vector potentials directly in terms of the fields E and B, thus facilitating the identification of electric and magnetic multipoles for generally time-dependent fields. We will follow the three-vector derivation given by Bloch [68]. We will furthermore in this section make extensive use of the Einstein summation convention for coordinate indices. Consider a Taylor expansion of the scalar potential... 

---