Scattering coupled channels

Although the Sclirodinger equation associated witii the A + BC reactive collision has the same fonn as for the nonreactive scattering problem that we considered previously, it cannot he. solved by the coupled-channel expansion used then, as the reagent vibrational basis functions caimot directly describe the product region (for an expansion in a finite number of tenns). So instead we need to use alternative schemes of which there are many. 

For a structureless continuum (i.e., in the absence of resonances), assuming that the scattering projection of the potential can only induce elastic scattering, the channel phase vanishes. The simplest model of this scenario is depicted schematically in Fig. 5a. Here we consider direct dissociation of a diatomic molecule, assuming that there are no nonadiabatic couplings, hence no inelastic scattering. This limit was observed experimentally (e.g., in ionization of H2S). 

Mizutani T. and Koltun D. S. Coupled channel theory of pion-deuteron reaction applied to threshold scattering, Annals Phys. 109, 1, (1977)... 

Most of the recent literature on molecular collisions in external fields [1-3, 9, 10, 15, 16, 18, 19, 21, 23, 25, 27-84, 92] is focused on collisions of molecules at low and ultralow temperatures. As mentioned in the introduction, it is at temperatures < 10 K that strong electromagnetic fields are expected to have a noticeable effect on the scattering properties of molecules. The coupled-channel calculations... 

Takatsuka and Gordon (21a) have developed a "full collision" formulation of photodissociation which describes a multichannel process on the repulsive surface for both direct and indirect events. The scattering wavefunctions that are used to generate the T-matrix and the FC overlaps are not zeroth-order uncoupled functions, but solutions of the coupled-channel problem. 

Archer, B.J., Parker, G.A. and Pack, R.T. (1990). Positron-hydrogen-atom S-wave coupled-channel scattering at low energies. Phys. Rev. A 41 1303-1310. 

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. 

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... 

The internal solutions are matched at r = a to solutions of the scattering problem in the external region. Here it is a simple coupled-channels problem in which exchange and target-correlation terms are negligible. The matching matrix is the R matrix. 

Fig. 8.2. Differential cross section for the elastic scattering of electrons on hydrogen. Circles, Williams (1975) solid curve, coupled-channels-optical calculation long-dashed curve, one channel with discrete polarisation potential only short-dashed curve, one channel without polarisation potential. Adapted from Bray et al. (1991h).

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. 8.8. Differential cross section at 30 and 90 for electron—hydrogen elastic scattering below the n = 2 threshold. Experiment, Williams (1976f>) solid curve, coupled channels optical (equivalent local) (McCarthy and Shang, 1992). From McCarthy and Shang (1992).

Fig. 8.10. Differential cross section for elastic electron scattering on sodium. Open circles, Lorentz and Miller (1991) closed circles, Srivastava and Vuskovic (1980) crosses, 54.4 eV, Allen et al. (1987), and 100 eV, Teubner, Buckman and Noble (1978) solid curve, coupled channels optical broken curve, 3-state coupled channels (Bray et al., I99ld). From Bray et al. (I99ld).

Table 8.7. The integrated cross section a p for the 3p channel of electron—sodium scattering and the total cross section ot (ICr cm ). EXP (a-ip), interpolation in the data of Enemark and Gallagher (1972) EXP (oj), Kwan et al. (1991) CCO, coupled-channels-optical calculation (Bray et al, 199 Id)...

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 ...

Fig. 8.13. Differential cross section for electron—magnesium scattering at Eq = 40 eV. Open circles, Williams and Trajmar (1978) closed circles, Brunger et al. (1988) full curves, coupled channels optical (Zhou, 1992).

Fig. 9.2. Spin asymmetries for electron scattering to the 2s and 2p states of lithium plotted against incident energy for the indicated scattering angles (Bray et al, 1993). 2s experiment, Baum et al. (1986) 2p experiment, Baumet al. (1989) full curve, coupled channels optical broken curve, 13-state coupled channels.

Fig. 9.4. Elastic asymmetry for electron—sodium scattering at 1.0 and 1.6 eV (Bray and McCarthy, 1992). Circles, Lorentz et al. (1991) full curves, 15-state coupled channels with core-polarisation in the bound states broken curve, the same reaction calculation omitting core polarisation.

Essentially-complete agreement with experiment is achieved by the coupled-channels-optical calculation. We can therefore ask if scattering is so sensitive to the structure details in the calculation that it constitutes a sensitive probe for structure. The coupled-channels calculations in fig. 9.3 included the polarisation potential (5.82) in addition to the frozen-core Hartree—Fock potential. Fig. 9.4 shows that addition of the polarisation potential has a large effect on the elastic asymmetry at 1.6 eV, bringing it into agreement with experiment. However, in general the probe is not very sensitive to this level of detail. 

It is useful to test approximations for the total ionisation cross section of helium, since it is a common target for the scattering and ionisation reactions treated in chapters 8, 10 and 11. Fig. 10.15 compares the data reported as the experimental average by de Heer and Jansen (1977) with the distorted-wave Born approximation and the coupled-channels-optical calculation using the equivalent-local polarisation potential. Cross sections... 

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. 

These results confirm and complement the earlier work of Beswick and Jortner who compared DWBA widths with those from collinear coupled-channel scattering calculations(33) where, as here, good agreement was observed. In the next section the DWBA widths are calculated for the chemically bonded HCO radical to give H+CO. Based on the present comparisons with exact results we are optimistic that the DWBA will provide realistic widths for this system. 

We have presented a sample of resonance phenomena and calculations in reactive and non-reactive three-body systems. In all cases a two-mathematical dimensional dynamical space was considered> leading to a great simplification in the computational effort. For the H-K 0 system, low-energy coupled-channel calculations are planned in the future to test the reliablity of the approximations used here, i.e., the scattering path hamiltonian as well as the distorted wave Born approximation. Hopefully these approximations will prove useful in larger systems where coupled-channel calculations would be prohibitively difficult to do. Such approximations will be necessary as resonance phenomena will continue to attract the attention of experimentalists and theorists for many years. 

---