Multipole fields

For polarizable charge distributions, additional classical-type interactions arise from the induced dipole, quadrupole, and higher moments on each monomer, which are proportional to the fields created by the asymmetric charge distribution on the other monomer. The proportionality constants for each multipole field are the monomer polarizabilities aa and ah (a111 for dipole fields, a(Q) for quadrupole fields, etc.). The leading two induction interactions are ... 

So our choices of the two antennas is not unique for separately emphasizing the Lorenz vector and scalar potentials. All that is required is for the two to have the same exterior fields (say, electric dipole fields, or more general multipole fields) with different potentials (related by the gauge condition). In a classical electromagnetic sense, these antennas cannot be distinguished by exterior measurements. This is a classical nonuniqueness of sources. In a QED sense, the same is the case due to gauge invariance in its currently accepted form. 

If at least one of the interacting particles is a molecule, further induction mechanisms arise. Molecules are surrounded by an electric field which may be viewed as a superposition of multipole fields. A collisional partner will be polarized in the multipole field and thus give rise to induced dipole components. In the case of symmetric diatoms like H2 or N2, the lowest-order multipole is a quadrupole and asymptotically, for R - 00, the quadrupole-induced dipole may be written as [288, 289]... 

On a scale of the order of atomic size, molecular multipole fields vary strongly with orientation and separation. As a consequence, one will generally find induced dipole components arising from field gradients of first and higher order which interact with the so-called dipole-multipole polarizability tensor components, such as the A and E tensors. 

While exchange- and dispersion-induced dipole components are of a quantum nature, the multipole-induced dipole components can be modeled by classical relationships, if the quantum effects are small. For many systems of practical interest, multipolar induction generates the dominant dipole components. The classical multipole induction approximation has been very successful, except for the weakly polarizable partners (e.g., He atoms) [193]. It models the dipole induced in the collisional partner by polarization in the molecular multipole fields. 

Both photon-assisted collisions and collision-induced absorption deal with transitions which occur because a dipole moment is induced in a collisional pair. The induction proceeds, for example, via the polarization of B in the electric multipole field of A. A variety of photon-assisted collisions exist for example, the above mentioned LICET or pair absorption process, or the induction of a transition which is forbidden in the isolated atom [427], All of these photon-assisted collision processes are characterized by long-range transition dipoles which vary with separation, R, as R n with n — 3 or 4, depending on the symmetry of the states involved. Collision-induced spectra, on the other hand, frequently arise from quadrupole (n = 4), octopole (n = 5) and hexadecapole (n = 6) induction, as we have seen. At near range, a modification of the inverse power law due to electron exchange is often quite noticeable. The importance of such overlap terms has been demonstrated for the forbidden oxygen —> lD emission induced by collision with rare gases [206] and... 

The coupling operator of an irreducible operator (A, ( )) of rank s for a semirigid model in a multipole field (Esa) may be taken in the form... 

Rose, M. E. Multipole Fields. New York John Wiley 1955. 

S. Kielich. Time-correlation functions for new cross-multipole field fluctuations in binary light scattering by unlike polar molecules. J. Phys. (Paris), 45 L.389-L.394 (1982). 

The potential function V still must be made explicit in order to complete the description of the system. A general multipole expansion in terms of first, second rank, etcetera interactions depending only on the relative orientation between each pair of bodies can be taken, as well as a multipole-field term (e.g., a dipole-field) for the pairwise interaction between each body and field. Finally each stochastic field is subjected to a harmonic potential, to parametrize in the most economical way the amplitude of the stochastic fluctuations. The complete potential is then... 

Unlike the case of plane waves of photons, the multipole field (18) propagates as a uniformly expanding spherical shell rather than propagates along a given direction of k. Instead of the symmetry relations (12), for the spherical waves of photons we get the following reciprocity relations [2,27] ... 

The case of outgoing and incoming spherical waves can be examined under the standard assumption that the atom located at the origin has a finite size that permits us to avoid the divergence at kr — 0. It is seen that, in some small vicinity of the atom, the zero-point oscillations, corresponding to the multipole field in an infinite space, strongly exceed those in the ideal spherical cavity (see Fig. 2). 

To illustrate the exchange of the phase information between the atomic transition and the multipole field, consider the electric dipole Jaynes-Cummings model (34). Assume that the field consists of two circularly polarized components in a coherent state each. The atom is supposed to be initially in the ground state. Then, the time-dependent wave function of the system has the form [53]... 

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... 

Hence, the polarization of either multipole radiation should be specified by the spatial anisotrophy of the field strengths rather than the transversal anisotrophy as in the case of plane waves [28,46,54,88]. Thus, the polarization of the classical multipole field should be described by bilinear forms in all three components of the field strengths which leads to the Hermitian (3 x 3) matrix with the elements [28,46]... 

Now consider the monochromatic multipole field with given X and j. Exactly this field is emitted by an atomic transition. Employing (21) and (17) then gives... 

It is seen from (143) that the vacuum polarization matrix is independent of index X, describing the type of the multipole field. This seems to be natural, because the vacuum properties are affected by the presence of the singular point (atom) without respect to the type of radiation that might be emitted. 

The Stark effect and the Zeeman effect of molecules in an inhomogeneous external field serve to select molecules due to their different rotational quantum states and are thus used to produce molecules with well defined preferential orientations in a beam molecules with different orientation are differently deflected in these fields. With regard to state selectors one can distinguish between simple deflection devices like Stern-Gerlach magnets (and their electrical analogues) and multipole fields where molecules with a well-defined Stark or Zeeman-effect are focused into the detector. In both cases the state selector works as a filter enhancing the relative number of molecules in a certain quantum state. 

Franzen, J. Simulation study of an ion cage with superimposed multipole fields. Int. J. Mass Spectrom. Ion Processes 1991, 106, 63-78. 

To investigate the dependence on N, Samec has evaluated Pg averaged over the linear multipole fields given by Eq. (8). 

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