Bipolar harmonics

In the parametrization of equ. (4.68) the terms associated with the Legendre polynomials Pk(cos ab) represent that part of the angular correlation which is independent of the light beam, while the terms associated with the bipolar harmonics are due to the multipole expansion of the interactions of the electrons with the electric field vector. The link between geometrical angular functions and dynamical parameters is made by the summation indices ku k2 and k. These quantities are related to the orbital angular momenta of the two individual emitted electrons, and they are subject to the following conditions ... 

We will now dwell on relaxation processes. Since we intend to discuss only relaxation which is isotropic with respect to orientations of J, its probability depends only on the angle 0 between the initial 0, p and the final 0,

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Several papers have been written on the subject of analyzing the anisotropic distribution of v and j vectors of molecules produced by a photodissociation reaction [17,36-42]. The most satisfying scheme is the renormalized bipolar moment analysis proposed by Dixon [18]. The basic premise is to use bipolar harmonics as the basis set to describe the angular distribution of a single recoil velocity v and fragment angular momentum j ... 

Because of the conservation of the total angular momentum (J = j -f-1), we have to search for a proper combination of spherical harmonics. Introducing then the coupling coefficients of quantum mechanical angular momenta (the Clebsch-Gordan coefficients), we obtain a bipolar harmonic... 

After the expansion of the wave function in terms of bipolar harmonics, we obtain ... 

Starting with the parametrization for two-electron emission given in equ. (4.68), and using equ. (4.67) which relates the bipolar spherical harmonics to spherical harmonics attached to the actual directions ka = a, electron emission, the triple differential cross section for two-electron emission can be rewritten as... 

Reference 2 also describes the on-off characteristics of the power semiconductors used in the bridges e.g. diodes, thyristors, triads, gate turn-off thyristors and bipolar power transistors. Only the steady state operations of bridges are described herein. For such operations it is assumed that the load is well matched to the rating of the bridge. The remainder of this section is an introduction to the subject of harmonic voltages and currents that are caused by variable speed systems for DC and AC motors. It emphasises the main aspects that affect the supply power systems. 

The value of the first harmonic appeared to be relatively low compared to that of the dc component (Figs. 4(a) and 4(b)] indicating the piezoelectric coefficient depended on the driving frequency. The amount of enhancement in piezoelectric coefficient by a reverse field also depended on the frequency. And that can be seen from a larger increase in the dc component than in the first harmonic in the measurement of a semi-bipolar mode. Variation patterns of the dc and the first harmonic are quite similar to that of the... 

The normalized dc and first harmonic components [Figs. 5(a) and 5(b)] exhibit similar variation patterns to that of the normalized strain [Fig. 3(a)]. The shifted piezoelectric loss tangent [Fig. 5(c)] shows a decreasing tendency followed by some fluctuation in a semi-bipolar measurement, and an increase trend in a unipolar measurement. 

Here, v, unlike n, Z m, is not a quantum number but a dummy non-negative integer. We will now derive a two-range ADT for the two types of orbitals defined in Eq. 4.5 by further breaking them up into convenient components. Two-range ADTs are more naturally expressed in terms of bipolar spherical harmonics, X, defined as [3] ... 

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