Magnetic Multipole Expansion

Though the ESR Hamiltonian is typically expressed in terms of effective electronic and nuclear spins, it can, of course, also be derived from the more fundamental Breit-Pauli Hamiltonian, when the magnetic fields produced by the moving nuclei are explicitly taken into account. In order to see this, we shall recall that in classical electrodynamics the magnetic dipole equation can be derived in a multipole expansion of the current density. For the lowest order term the expansion yields (59)... 

In the usual texts a multipole expansion involving spherical Bessel functions and spherical vector harmonics is also introduced [16,23,23,26]. The fields from electric and magnetic dipoles correspond to the lowest-order terms ( =1) in the expansion. If we define dipole by this expansion then our toroidal antenna is an electric dipole. In any event, the fields away from the source are the same. This is perhaps a matter of consistency in definitions. 

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... 

Starting from a multipole expansion of intramolecular Coulomb interactions, we present an efficient configuration interaction calculation for the electron terms = 2,3, 4, and the hole terms (hf)", n = 2-5. We have studied magnetic moments for the electron and hole terms. The coupling of spin and orbital momenta differs from the Lande g-factor scheme of atoms. The magnetic moments do not depend on the orientation of the molecule with respect to an external magnetic field. 

Finally, performing multipole expansion of the terms representing the magnetic field effect, as shown in the deduction of Eq. (4.9), we obtain the terms iqu>j aP% and iQwj bpQ, which also have to be added to the righthand sides of (5.94) and (5.93). 

In a multipole expansion of the interaction of a molecule with a radiation field, the contribution of the magnetic dipole is in general much smaller than that of the electric dipole. The prefactor for a magnetic dipole transition probability differs from the one for an electric dipole by a2/4 1.3 x 1 () 5. Magnetic dipoles may play an important role, however, when electric dipole transitions are symmetry-forbidden as, e.g., in homonuclear diatomics. 

Terms of higher order in the field amplitudes or in the multipole expansion are indicated by. . . The other two tensors in (1) are the electric polarizability ax and the magnetizability The linear response tensors in (1) are molecular properties, amenable to ab initio computations, and the tensor elements are functions of the frequency m of the applied fields. Because of the time derivatives of the fields involved with the mixed electric-magnetic polarizabilities, chiroptical effects vanish as a> goes to zero (however, f has a nonzero static limit). Away from resonances, the OR parameter is given by [32]... 

To some purposes, one can define new molecular tensors that are independent of the origin [40]. At any rate, it can be easily proven that the induced moments (69)-(72) and fields (73)-(74) are, order by order, independent of the origin chosen for the multipole expansion, provided that all the terms of the same order of magnitude are retained. Thus, within the quadrupole approximation, both the magnetic field and the electric field gradient must be taken... 

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)... 

For particles not small compared with the wavelength, the interaction properties must be found by solving a complicated boundary-value problem for the electric and magnetic fields. For spherical particles, this theory is well established and the scattering and extinction cross sections are calculated from the multipole expansions. 

In contrast to magnetic properties, the theory of electric-field-like properties is much easier to cast into a set of working equations. One of them has attracted particular interest, and that is the electric field gradient (EFG). This property is of decisive importance to Mossbauer spectroscopy, i.e., to the spectroscopy of excited nuclear states whose energies are modulated by the molecular structure (the chemical environment ). In order to see how this property arises, we study the electrostatic electron-nucleus interaction of extended, not spherically symmetric charge distributions. For this we apply a multipole expansion in order to generate the properties term by term. 

To carry out the calculations it is customary to use a multipole expansion and to separate electric and magnetic couplings. A nucleus can only possess even electric moments, i.e. a monopole moment (/ = 0), namely its protonic charge and a quadrupole moment (/ = 2) if the charge distribution is not spherical. Higher moments are neglected. Magnetically only odd moments exist. It suffices to consider the dipole moment (/ = 1). 

Magnetic Multipole Expansion and Interaction with Magnetic Fields... 

The system of equations obtained, (5.22) and (5.23), in broad line approximation in many cases allows us to carry out the analysis of non-linear optical pumping of both atoms and molecules in an external magnetic field. Some examples will be considered in Section 5.5, among them the comparatively unexplored problem of transition from alignment to orientation under the influence of the dynamic Stark effect. But before that we will return to the weak excitation and present, as examples, some cases of the simultaneous application of density matrix equations (5.7) and expansion over state multipoles (5.20). 

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... 

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... 

The representation of hyperfine interactions in the form of multipoles follows from expansion of the potentials of the electric and magnetic fields of a nucleus, conditioned by the distribution of nuclear charges and currents, in a series of the corresponding multipole momenta. It follows from the properties of the operators obtained with respect to the inversion operation that the nucleus can possess non-zero electric multipole momenta of the order k = 0,2,4,..., and magnetic ones with... 

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