Scattering theory differential cross section

D. Partial Cross Sections, Product State Distributions, and Differential Cross Sections III. Reactive Scattering Theory... 

The first part of the review deals with aspects of photodissociation theory and the second, with reactive scattering theory. Three appendix sections are devoted to important technical details of photodissociation theory, namely, the detailed form of the parity-adapted body-fixed scattering wavefunction needed to analyze the asymptotic wavefunction in photodissociation theory, the definition of the initial wavepacket in photodissociation theory and its relationship to the initial bound-state wavepacket, and finally the theory of differential state-specific photo-fragmentation cross sections. Many of the details developed in these appendix sections are also relevant to the theory of reactive scattering. 

If the interaction operator V can be regarded as a perturbation to the Hamiltonian H0 (this is the case for fast particles the velocity of which is much greater than those of atomic electrons), the function if>+ can be found using the perturbation theory. Such an approach was named the Born approximation. In the first Born approximation we replace the function ip+ by that of the initial state of the scattering system,48 that is, put i//+ = 0, and thereby do not have to solve Eq. (4.2). In this way, for the differential cross section of direct scattering, we get... 

Fig. 3.14. Differential cross sections for positron-argon elastic scattering at the following energies (a) , 2.2 eV (b) , 3.4 eV (a) o, 6.7 eV (b) o, 8.7 eV. (The corresponding fc-values are respectively 0.4, 0.5, 0.7 and 0.8 a.u.) The solid curves give the theory of Schrader (1979) whilst the broken curves are the scaled results of McEachran, Ryman and Stauffer (1979).

The main difference between the new potential and that of Buck et al., i.e. the depth of potential wells, cannot be verified by this high energy scattering calculation in order to provide major insights, work is in progress to compute the MO-VB PES at a higher level of theory and to calculate differential cross-sections at the lower collision energy of 275 cm 1 [69], where total inelastic and some state resolved differential cross sections are available [65]. 

Note that the entire theory needs to be modified only slightly to accommodate control of scattering into different angles, that is, the differential cross section, into channel q. A specific example of this type of control is discussed in Section 3.4 where we apply this approach to manipulating electric currents in semiconductors. 

Fig. 9.6. Angular distribution of S, T and U and the derived moduli and relative phases of the scattering parameters for elastic scattering from Xe at 60 eV. Experiment o, Berger and Kessler (1986) Mollenkamp et al. (1984) and Wiibker, Mollenkamp and Kessler (1982). Absolute measured differential cross sections used in the derivation of the moduli / and g , Register et al. (1986) g, Williams and Crowe (1975). Theory —, McEachran and Stauffer (1986) ., Haberland et al. (1986) -----, Awe et al. (1983).

Clary, D.C. (1994) Four-atom reaction dynamics, J. Phys. Chem. 98, 10678-10688. Pack, R.T. and Parker, G.A. (1987) Quantum reactive scattering in three dimensions using hyperspherical (APH) coordinates. Theory, J. Chem. Phys. 87, 3888-3921. Truhlar, D.G., Mead, C.A. and Brandt, 5I.A. (1975) Time-Reversal Invariance, Representations for Scattering Wavcfunctions, Symmetry of the Scattering Matrix, and Differential Cross-Sections, Adv. Chem. Phys. 33, 295-344. 

Differential cross section Deflection function. First we describe methods which take advantage of the close relationship between semi-classical cross sections and deflection function as outlined in Section III. A procedure which uses nearly all measurable quantities has been proposed and applied by Buck (1971). In order to unfold the multivalued character of 6(9), the deflection function is separated into monotonic functions g/[b) such that 0(6) = . gj(6)and6 = g (9). The g are represented by the usual functional approximations made in the semiclassical scattering theory ... 

More information on excitation processes can be gained from measurements of the differential cross section of the elastically or inelastically scattered particles. The phenomena expected in the differential cross sections will be the same as found in the corresponding ion-atom scattering experiments. They have been demonstrated most beautifully in the standard case He+ + Ne which has been studied thoroughly both experimentally and theoretically.97-99100101 102 A fully developed theory exists for the interpretation of differential cross sections in cases where excited states play an important role.10 96 9798... 

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