Spherical vector

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. 

The dipole operator is expressed in terms of spherical vector components, Eqs. 4.6 and 4.14, according to... 

This matrix I 1 (A) differs from that in eq. (11.6.19) which describes the transformation of the basis x y z). The first term in the symmetrized basis in eq. (25) is the spherical vector... 

The irreducible tensor product between two (spherical) vectors is defined in Eq. (37). An important feature of this Hamiltonian is that it explicitly describes the dependence of the coupling constants J, Am, and T, on the distance vectors rPP between the molecules and on the orientations phenomenological Hamiltonian (139). Another important difference with the latter is that the ad hoc single-particle spin anisotropy term BS2y, which probably stands implicitly for the magnetic dipole-dipole interactions, has been replaced by a two-body operator that correctly represents these interactions. The distance and orientational dependence of the coupling parameters J, A, , and Tm has been obtained as follows. 

Irreducible tensor operators of the so9(3) subalgebra and construction of spherical vector operators... 

The compound irreducible tensor operators of the second rank are constructed from the first-rank tensors (spherical vectors) as follows... 

Let us begin with the orbital, or current, matrix element. Write p = -iftV and work with spherical vector components labelled by = 0, 1. Atomic states are labelled by 6, J, M, where 6 contains all the quantum numbers required to define a state other than J, M. The magnetic neutron operator contains the quantity K X V, and the many-electron matrix element is found to be... 

The concept of GMM and the details of computation were presented by Yu-Lin Xu in a series of papers (e.g. Xu 1995,1997,1998a). The main idea is to express the scattering field of the particle j as an infinite series of spherical vector wave functions (SVWF) (i.e. spherical harmonics for vector fields) that are defined for... 

O. Cruzan, Translational addition theorems for spherical vector wave functions. Q. Appl. Math. 20 (1), 33 0 (1962)... 

The spherical vector wave functions (SVWF) are the general solution of the vectorial Helmholtz differential equation in spherical coordinates (Xu 1995) ... 

The polarization unit vector of a linearly polarized vector plane wave is given by (1.18). If the vector plane wave propagates along the z-axis we have / = a = 0 and for / = 0, the spherical vector harmonics are zero unless m = 1. Using the special values of the angular functions and when (3 = 0,... 

For a system of Af particles with o = f3i = 7 = 0, I = 1,2,. ..,AA, the transformations of the vector spherical vector wave functions involve only the addition theorem under coordinate translations, i.e.,... 

The spherical vector wave expansion of the dyad gl is of basic importance in our analysis and is given by [175,228,229]... 

The regular and radiating spherical vector wave functions can be expressed as integrals over vector spherical harmonics [26]... 

The following theorems state the completeness and linear independence of the system of regular and radiating spherical vector wave functions on two enclosing surfaces. 

K.A. Aydin, A. Hizal, On the completeness of the spherical vector wave functions, J. Math. Anal. Appl. 117, 428 (1986)... 

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