Coupled channel scattering methods

I. Bray, D.A. Konovalov, I.E. McCarthy, Convergence of an L2 approach in the coupled-channel optical-potential method for e-H scattering, Phys. Rev. A 43 (1991) 1301. 

The lowess normalization method is typically coupled with scatter-plots (e.g., M vs. A) illustrating the relationship between the two dye channels, as a function of the overall intensity (i.e., geometric mean of the intensities) (47). The lowess normalization smooths the microarray data in... 

A powerful way of achieving this goal uses the coupled-channels expansion, a method widely used in calculations of scattering cross sections [6]. In the context of quantized matter-radiation problems, the coupled-channels method amounts to expanding E, n, N ) in number states. Concentrating on the expansion in the /th mode, we write E, n, N ) as... 

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... 

Fig. 8.3. Differential cross section for electron scattering to the Is, 2s and 2p states of hydrogen at 54.4 eV. Experimental data for Is are interpolated (Williams, 1975), for 2s and 2p they are taken from Williams (1981). Calculations are solid curve, convergent close coupling (Bray and Stelbovics, 1992h) long-dashed curve, coupled channels optical (Bray et al, 1991c) short-dashed curve, distorted-wave second Born (Madison et al, 1991) chain curve, intermediate-energy R matrix (Scholz et al, 1991) dotted curve, pseudostate method (van Wyngaarden and Walters, 1986).

Fig. 8.9. Integrated cross section for electron scattering to the 2s (above) and 2p (below) states of hydrogen below the n=3 threshold. The positions and quantum numbers of resonances are shown on the upper scale. Experiment, Williams (1988) solid curve, coupled channels optical (equivalent local) (McCarthy and Shang, 1992) long-dashed curve, pseudostate method (Callaway, 1982) short-dashed curve, 9-state coupled channels.

Table 8.8. Total cross sections for electron-helium scattering. CCO, coupled-channels-optical (equivalent local) method (McCarthy et al.,1991) experiment. Nickel et al. (1985). Units are KT cmi ...

A direct measurement of the electronic energy loss as a function of the impact parameter is a hard task to be performed from the experimental point of view and only a few experiments have been performed for fast light ions. Experiments in gas targets under single collision condition provide a more direct and precise comparison of the theoretical results with the experimental data. Here we compare the results of the coupled-channel method for collisions of protons with He as a function of the projectile scattering angle. 

The coupled-channels method may be developed within the language of wave-mechanics, or more formally (and more compactly) by means of operator equations. The common feature of both approaches is that the total scattering state is expanded in internal states of reactants and products. The nature of the colliding particles and the quantum numbers of the interna] states define the reaction channel index c = a, b,. We begin with the wave-mechanical approach, some of whose features have been presented in the section on statistical theories. For the total wavefunction [pa of reactants in channel a, with relative wave vector ka, we can write... 

We would like to complete this section by briefly describing some of the recent developments on electronically non-adiabatic reactions. From the standpoint of the coupled-channels method, there is in principle no added difficulty in treating more than one electronic state of the reactive system. This may be done, for example, by keeping electronic degrees of freedom in the Hamiltonian and expanding the total scattering wavefunction in the electronic states of reactants and products. In practice, however, some new difficulties may arise, such as non-orthogonality of vibrational states on different electronic potential surfaces. There is at present a lack of quantum mechanical results on this problem. 

Another approach for developing approximations to CC and CS reactive scattering calculations is to use distorted wave theory. In this approach, one considers that reaction is only a small perturbation on the nonreactive collision dynamics. As a result, the reactive scattering matrix can be approximated by the matrix element of a perturbative Hamiltonian operator using reagent and product nonreactive wavefunctions. Variations on this idea can be developed by using different approximations to the nonreactive wavefunctions. At the top of the hierachy of these methods is the coupled channel distorted wave (CCDW) method, followed by coupled states distorted wave (CSDW). 

Coupled-channel equations arise in scattering dynamics when all but one of the degrees of freedom of the system are expanded in a square integral basis (of "channels"). The coupled channel equations are then solved numerically and describe motion in the unbound, or scattering coordinate. The principal difficulty of any reactive scattering calculation is that the coordinate system which best describes the asymptotic motions of reactants differs from the coordinate system best suited for products. Consequently, computational methods commonly use different coordinate systems in different parts of configuration space. Boundary conditions are expressed in terms of Jacobi coordinates (sometimes referred to as "cartesian coordinates"), where in the A -BC arrangement... 

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