Total cross section potential scattering

In the frame of the theoretical formulation, in which the Penning process is described by the local quantities V+ R), T(/ ), and V+(R), the total cross section can be calculated as either (1) total absorption cross section atotaI from the complex phase shift for scattering by the complex potential V(R)= V (R)- r(R) or (2) as the sum of the partial cross sections a(Pgl), a(AI), and a(QI), into whose calculation also V+(R) enters in the form of matrix elements involving nuclear wave functions in this potential. 

In terms of spectroscopic observables, the potential energy function is that function V(r) which, when inserted into the quantum mechanical formulation of the vibration problem, gives the observed vibrational levels. In beam experiments, it is the potential which gives the observed scattering. In chemical excitation processes, it is the surface which predicts the observed total cross section and the observed distribution of products over internal energy states. Potential energy functions may be calculated from first principles14, or they may be constructed... 

That is, the scattering is isotropic, i.e., it is independent of the deflection angle (which is, in general, not the case for more realistic potentials). The total cross-section takes the form... 

Here E, q, n ) denotes the incoming scattering solutions associated with the product in state E, q1, n 0), and Vq is the component of the total potential that vanishes as the A to B distance becomes arbitrarily large. The cross section for scattering into arrangement q, independent of the product internal state n, is then... 

Figure 2. Comparison of theoretical and experimental total cross sections for e-H, scattering. Curves are as follows (1) represents the effective optical potential (2) represents the experiment by Golden and (3) represents the experiment by...

In both cases, la and lb, the total photodissociation cross section is completely determined by the short-time dynamics in the Franck-Condon region. In contrast, the partial cross sections, which determine the vibrational, rotational, and electronic-state distributions of the products, involves longer time dynamics. To obtain all of the relevant information about the reaction, the wavepacket evolution must be followed out into the product region of the potential energy surface and projected onto the various different vibrational and rotational states of the fragments. The partial cross section for scattering... 

The influence of chemical reactions on elastic scattering has been extensively studied in the past. Nearly all treatments are based on the optical model (for a review see Ross and Green, 1970). Both the imaginary part of the potential (here assumed to be local) and the opacity function have been parameterized (Mariott and Micha, 1969 Harris and Wilson, 1971 and references cited therein). For a study of the total cross section see Diiren et al. (1972), A semiclassical study of a bimolecular exchange reaction where the three atoms are constrained to move on a straight line but the whole system is free to rotate in three dimensions, predicts a new kind of rainbow (Connor and Child, 1970),... 

The number of parameters which can be determined from such a minimization procedure is restricted by the information content of the measurement. The easiest way to estimate this information content is to calculate the number of partial waves which contribute significantly to the scattering amplitude. For very low energies this information content reduces to one parameter so that such a measurement does not contain very much information on the potential (Farrar and Lee, 1972). Another method has been proposed by Ury and Wharton (1972) for the total cross section. An upper limit of the number of parameters is estimated from the order of a power series necessary to describe the experimental curve to that degree of experimental accuracy established when the fractional x2 of the polynomial with respect to the data equals the fractional variance of the experiment. In spite of the merits of the x2-procedure the disadvantages of a fit procedure may be summarized as follows. 

Only a few elastic scattering experiments of molecules which contain an alkali atom have been performed. Recently published results on the scattering of alkali dimers on molecules show that the rainbow structure in the differential cross section is only slightly less resolved than the structure for the monomers, thus indicating a weak anisotropy of the long range potential (Hardin and Grice, 1973). For a discussion of total cross section measurements of alkali halides on rare gas atoms with a Boltzmann rotational state distribution and state selected beams see David et al. (1973). 

Fig. 24. Total cross section multiplied by v° as a function of the primary beam velocity for the scattering of D2 by various molecules. The solid line is calculated from a LJ 12-6 potential. Deviations only occur for the system D2-C02.

When the analysis of the neon data had been completed, the modification of the potential was so drastic that we expected some difficulty in the calculation of macroscopic properties. Considering the still primitive state of scattering measurements, the agreement with experiment seems reasonable for both argon and neon. The neon potential can and should be subject to further tests. Measurements of X-ray scattering and measurements of the total cross section at low velocities should be sensitive to the intermediate-range forces. 

The total cross section for interaction of particle a with the nucleus X is greater than a) because of elastic scattering of a by the potential fields surrounding X. Some elastic scattering is also obtained by re-emission of a particle a from the compound nucleus, with partial width which is included in F, It is therefore also useful to write... 

The scattering length R may also be determined from direct measurements of the total cross section in thick material. In this case, the neutrons with energies corresponding to the resonance energies are removed by absorption and the measured transmission depends mainly on those neutrons whose energies lies between resonances. These neutrons are attenuated effectively only by potential scattering. 

The aforementioned optical model calculations assumed square well potentials in order to interpret total cross-section data. In attempting to fit the observed angular distribution of elastically scattered protons, Woods and Saxon introduced a tapered edge to the complex potential of Feshbach, Porter, and Weisskopf. The form Woods and Saxon employ is... 

The conclusion is that also in the higher energies level the Bom approximation finely combines with the Yukawa potential to correctly model the photonic scatterings on Coulombic manifested interaction. Worth therefore to further compute, from the scattering amplitude, the dififerential and total cross section induced by the Yukawa (and then Coulombic) potentials - a matter in the next approached. 

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