The optical potential

The use of a finite-basis expansion to represent the continuum is reminiscent of the use of quadratures to represent an integration. Heller, Reinhardt and Yamani (1973) showed that use of the Laguerre basis (5.56) is equivalent to a Gaussian-type quadrature rule. The underlying orthogonal polynomials were shown by Yamani and Reinhardt (1975) to be of the Pollaczek (1950) class. 

The Lippmann—Schwinger equations (6.73) are written formally in terms of a discrete notation i) for the complete set of target states, which includes the ionisation continuum. For a numerical solution it is necessary to have a finite set of coupled integral equations. We formulate the coupled-channels-optical equations that describe reactions in a channel subspace, called P space. This is projected from the chaimel space by an operator P that includes only a finite set of target states. The entrance channel 0ko) is included in P space. The method was first discussed by Feshbach (1962). Its application to the momentum-space formulation of electron—atom scattering was introduced by McCarthy and Stelbovics 

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 

The polarisation potential is complex and nonlocal. The imaginary part is due to on-shell amplitudes for the excitation of Q space from P space. At long range the potential is real. We will show its relationship for large r, where it is due to virtual dipole excitations, to the classical dipole potential where a is the polarisability. 

The channels to be included in P space are the entrance channel, those for which we want to describe experimental observations and others that are so strongly coupled that numerical investigation shows their inclusion to be necessary. 

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Jolicard G, Leforestier C and Austin E J 1988 Resonance states using the optical potential model. Study of Feshbach resonances and broad shape resonances J. Chem. Phys. 88 1026... 

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 ... 

It has been shown by Gailitis (1965) that the optical potential defined by equation (3.22) is less attractive than the exact optical potential for energies below the lowest eigenvalue of QHQ, and the resulting phase shifts are therefore lower bounds on the exact values. Furthermore, as the number of basis functions used in the matrix representation of the operator QHQ is increased, the optical potential becomes more attractive and the resulting phase shift therefore also increases, becoming closer to the exact value. 

When elastic scattering is the only open channel, k is positive but all other values of kf, and all values of nj, are negative. Consequently, all the functions F)(ri) and Gj(p), except for Fi(ri), decay exponentially for large values of r and p. The resulting equation for Fj(ri) is similar in form to equation (3.20), in which the optical potential Vopt was introduced indeed a truncated coupled-state expansion essentially defines an approximation to the optical potential which satisfies the conditions for the phase shifts to be lower bounds on the exact values. 

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. 

G. Jolicard, J. Humbert, Study of the one-channel resonance states. Method without a stabilization procedure in the framework of the optical potential model, Chem. Physics, 118 (3) (1987) 397. 

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). 

The optical potential, defined by (7.111,7.115) can only be calculated exactly if we can solve the whole collision problem to find the spectral representation of H. We must approximate it as closely as possible with the rationale that, since strongly-coupled channels are treated explicitly in P space, a reasonable approximation for the remaining channels should not cause significant errors in the amplitudes for the excitation of P-space channels from the entrance channel. 

In choosing the partition xq, xq of the set x of coordinate-spin variables we have broken the symmetry of the problem. It will be restored by explicit symmetrisation of the expression for the optical potential. 

In order to implement the approximation to the optical potential we must choose a form for the potential Vopul (7.132) in which the Green s function is calculated (7.135). We choose different types of potential for the discrete and continuum channels of Q space, projected respectively by Q and Q+. For Q space we choose the average potential for the target state i). Its coordinate representation is obtained from (7.48,7.63,7.66)... 

In applying the distorted-wave second Born approximation we have the same difficulty as in calculating the optical potential. We must calculate the spectrum of the Green s function of (6.87). The first iteration of (6.87) is written as... 

The dimerization is easily understood considering the optical potential created by the trapping laser. Figure 18.2b shows the calculated optical potential experienced by a silver nanoparticle that is fi ee to move in a Gaussian laser focus at a wavelength of 830 nm. The particle is also affected by the optical interparticle force from an immobilized silver particle located at different separations from the laser focus. It is clear that a deep potential minimum is induced when the trapped particle approaches the immobilized one, giving rise to spontaneous optical dimerization and a SERS hot spot in the optical trap. Note that the two particles are expected to ahgn parallel to the laser polarization in this case, as has been demonstrated experimentally recently [88]. 

So far no approximation for a form of the optical potential U has been made. At this point we assume static-exchange approximation defined by the eq. (4). First two terms of eq. (4) describe electrostatic interaction consisting of nuclear attraction and static repulsion with the bound electrons. Corresponding matrix elements for the static part Us come out as a Fourier transform of the charge density ... 

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