Optical potential formal

Probably the most accurate positron-hydrogen s-wave phase shifts are those obtained by Bhatia et al. (4974), who avoided the possibility of Schwartz singularities by using a bounded variational method based on the optical potential formalism described previously. These authors chose their basis functions spanning the closed-channel Q-space, see equation (3.44), to be of essentially the same Hylleraas form as those used in the Kohn trial function, equation (3.42), and their most accurate results were obtained with 84 such terms. By extrapolating to infinite u in a somewhat similar way to that described in equation (3.54), they obtained phase shifts which are believed to be accurate to within 0.0002 rad. They also established that there are no Feshbach resonances below the positronium formation threshold. 

The multiple scattering optical potential formalism of Watson [Wa 53] provides a formal solution of the nonrelativistic many-body Scluodinger equation for the projectile-nucleus system in terms of a systematic expansion in elemental, many-body operators defined in ref. [Wa53]. The many-body Schrodinger equation is... 

We have provided a pedagogical derivation of the traditional, nonrelativistic form of multiple scattering theory based on the optical potential formalism. We have also discussed in detail each of the important advances made over the past ten years in the numerical application of the NR formalism. These include the full-folding calculation of the first-order optical potential, off-shell NN t-matrix contributions, relativistic kinematics and Lorentz boost of the NN t-matrix, electromagnetic effects, medium corrections arising from Pauli blocking and binding potentials in intermediate states, nucleon... 

In the projection operator formalism, which leads to a rigorous basis for the optical potential, the absorptive imaginary part is associated with transitions out of the elastic channel from which no return occurs. Whereas Pgl transitions are in this category, excitation transfer (ET) transitions are not, since return ( virtual excitation ) can occur during the ET collision. In the event that a localized avoided curve crossing with one other state dominates the inelastic process (expected for many endoergic transfers), the total absorption probability (opacity) can still be defined ... 

The coupled-channels-optical equations are formally analogous to the Lippmann—Schwinger equivalent of (7.29) in which the coupling potential includes the potential V (7.40) and a polarisation potential that describes the real (on-shell) and virtual (off-shell) excitation of the complementary channel space, called Q space. The total coupling potential is the optical potential... 

These are the coupled-channels-optical equations, which are formally identical to (6.73) except that the channels are restricted to P space and the potential V is replaced by the optical potential (7.118). The extension of (7.123) to the distorted-wave representation is analogous to the extension of (6.73) to (6.87). 

Because of these difficulties we turn to inversion procedures which are valid in the semiclassical limit since this approximation has proved to be applicable for most of the atomic and molecular collisions. Solutions of the second step, the determination of the potential, are treated in Section IV.B.2. In general, the input information will be the phase shifts or the deflection function. Only in the high energy approximation can the potential be derived directly from the cross section. For a detailed discussion of these procedures see Buck (1974). The possibilities of determining the phase shifts or the deflection function from the cross section are treated in Section IV.B.3. The advantage of such procedures and the general requirements on the data are discussed in Section IV.B.4. The emphasis will be on procedures which have been applied to real data. Extensions to non-central or optical interaction potentials are available. Most of them, however, are still in a formal state, so that a direct application to molecular physics is not obvious. Two exceptions should be mentioned. One is a special inversion procedure for optical potentials derived by a perturbation formalism (Roberts and Ross,... 

We also want to mention that a Dyson-equation approach for propagators like the polarisation and the particle-pcurticle propagator has been formulated and used to derive a self-consistent extension of the RPA, also called cluster-Hartree-Fock approximation, that has been applied in the fields of plasma and nuclear physics [12-14], This formalism, however, has similar problems like Feshbach s theory and does not yield a universal, well-behaved optical potential because the two-particle space has to be restricted in order to make the approach well-defined [14]. 

Nonrelativistic multiple scattering formalism and the optical potential 232... 

A pedagogical discussion of nonrelativistic multiple scattering formalisms is presented, followed by a description of the approximation schemes used in numerical applications of the theory. Recent theoretical developments in the nonrelativistic approach, including medium corrections to the effective projectile-taiget nucleon interaction, off-shell contributions, and full integration ( full-folding ) of the nucleon-nucleus optical potential are discussed in detail. 

In the Watson formalism the Coulomb contribution to the proton-nucleus optical potential is simply... 

Isobar excitation in the two-body NN system is incorporated in the NN r-matrix, whether it is empirical or calculated from a model. Direct three-body and higher order intermediate isobar excitation effects are formally included in the first portion of the second-order optical potential. For example, the elastic channel to isobar channel transition t-matrix can be obtained from the nucleon-isobar coupled channels model. Estimates indicate that this effect is significant for p + He scattering predictions but that it quickly diminishes in importance as target mass increases and is negligible for the heavier nuclei under consideration here [Wa77, Wa81]. 

The formalism of ref. [Lu 87] and the IA2 invariant amplitudes were used by Kaki [Ka 90] to extend the study of correlation effects in relativistic models to lower energies. As in the preceding case [Lu 87], the calculations required just that portion of the IA2 invariant amplitude which contributes to the first-order optical potential for even-even nuclei. Again, small dianges from the first-order, IA2 predictions for the scattering observables were found, even at the lowest energy of 100 MeV. 

As a starting point for further theoretical consideration, we briefly describe tbe procedure, known as Parrat formalism, of calculations of tbe reflection amplitude from tbe mean optical potential ... 

One is purely formal, it concerns the departure from symmetry of an approximate solution of the Schrodinger equation for the electrons (ie within the Bom-Oppenheimer approximation). The most famous case is the symmetry-breaking of the solutions of the Hartree-Fock equations [1-4]. The other symmetry-breaking concerns the appearance of non symmetrical conformations of minimum potential energy. This phenomenon of deviation of the molecular structure from symmetry is so familiar, confirmed by a huge amount of physical evidences, of which chirality (i.e. the existence of optical isomers) was the oldest one, that it is well accepted. However, there are many problems where the Hartree-Fock symmetry breaking of the wave function for a symmetrical nuclear conformation and the deformation of the nuclear skeleton are internally related, obeying the same laws. And it is one purpose of the present review to stress on that internal link. 

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