Total collision cross-section

The total collision cross sections for excitation transfer, - ipa/a) and... 

IV. Measurements of the Anisotropy of the Total Collision Cross Section. 397... 

Bennewitz et al. (1964) were the first to get this information on the AIP from total collision cross section measurements. In the last three sections we shall confine ourselves to this type of measurements and discuss a more recent experimental set up (Section IV) and the results obtained so far (Section VI). In Section V we shall discuss the theory which connects the AIP with the orientation dependent part of the total collision cross section. Our main technique will be the first order distorted wave approximation (d.w.a.). 

The velocity dependence of the total collision cross section is a restricted source of information regarding the intermolecular potential (IP), Bernstein (1973). Much more information is contained in differential cross section measurements which therefore form a great challenge for future scattering experiments with oriented molecules. 

Measurements on this system were done by Gengenbach et al. (1972) who determined the velocity dependence and the absolute value of the total collision cross section. From these measurements the isotropic part of the intermolecular potential (IIP) was obtained, see Fig. 1. Recently Riehl et al. 

Much less is known about the AIP (and IIP) for all other anisotropic systems. In order to extract information from measurements of the orientation dependent part of the total collision cross section under these unfavourable circumstances one proceeds as follows for the IIP one uses an empirical two parameter potential, the Lennard-JOnes 12-6 potential, with the parameters determined from independent scattering experiments. These potentials are implemented by an angle dependent part so that one has, for instance for the case of atom-non-polar diatom collisions (3) with... 

If collisions are considered between a polar diatom and an atom of radial symmetry, angular dependent long range induction terms are present. Their influence on the total collision cross section is discussed by Stolte (1972) and found either to be negligible or to change only the effective value of p2.6. 

The fact that the centre of rotation of the polar diatomic does not coincide with the centre of charge of the electron cloud gives rise to extra terms in the AIP, Ree (1971) Legendre polynomials of odd order appear. Their contribution to the orientation dependent part of the total cross section has been shown to vanish in first order, Stolte (1972). Again, therefore, the angular dependence of (3) appears to be satisfactory for the interpretation of the total collision cross section measurements. 

Moerkerken et al. (1970) applied an RF field to H 2-molecules of a molecular beam passing through a fairly conventional arrangement of A-, B- and C-fields. Molecules in a well-defined rotational state undergo a transition into a state with different Zeeman-effect when they pass through the C-field where the RF field is applied (see Fig. 2). This combination of deflecting fields and spectroscopic techniques permits the production of a beam of preferentially oriented non-polar molecules. The scattering chamber is also shown in Fig. 2 where the beam of selected molecules can be attenuated for determination of the total collision cross section. 

IV. MEASUREMENTS OF THE ANISOTROPY OF THE TOTAL COLLISION CROSS SECTION... 

Concerning the total collision cross section measurements one may be hopeful that the full apparatus of many coupled equations to be solved is not needed, but that perturbation theory suffices to cope with the influence of the AIP this perturbation theory must treat the IIP rigorously. The vehicle for such an approach is the d.w.a. Concerning the orientation dependent part of the total collision cross section, the AIP shows up in first order, i.e. by terms proportional to q2.6 and q2.12, in contrast to the cross section for rotational excitation and to the orientationally averaged total cross section where only terms in q. 6 and q .12 and higher orders occur. This fact may be taken as the first indication that d.w.a. is a suitable means for the calculation of the orientation dependent part of the total collision cross section. 

Within the framework of the linear d.w.a. one finds for the potential of (4) and (5) that the orientation dependent part Aer M of the total collision cross section can split into two contributions... 

Fig. 5. Isotropic and anisotropic glory structure. For a system resembling NO-Ar in the experimental set up of Stolte (1972), the total collision cross section a and its orientation dependent part Act /cr are calculated in d.w.a. as functions of the primary beam velocity vN0. Extremes of a coincide with zeros of Acrn/cr. The orientation dependent part oscillates around the average value (Aff /<7,0,) nB.

The combined effect of Acrg and Ameasured orientational dependency of the total collision cross section is an expression where the difference is formed between <7 and aL according to Stolte (1972) and Kuijpers (1973)... 

The H 2-noble gas systems were investigated by Moerkerken et al. (1970 and 1973). Their choice was dictated by the consideration that in the velocity range of the experiment (1000-2000 m/s), inelastic processes could be ruled out and strict quantum mechanical calculations could be compared with the measurements, e.g. Reuss et al. (1969). Moreover, a large body of experimental results exists on the total collision cross section, Helbing (1968), and on the differential cross section, Kuppermann (1973). A relatively easy and secure intepretation was therefore hoped for. In Fig. 7 we present the latest results of the Nijmegen group. 

Ar and Oz-Kr the other one based on combination rules and the potential parameters found in Hirschfelder (1965). Although the first choice is the only one based upon experiments on related systems one has to keep in mind that differential cross section measurements explore a different region of the IP than do the total collision cross section measurements. In other cases (Butz, 1971 and Kupperman, 1973) differences of about 30% were... 

At the end of this review it seems proper to ask what has been learned from anisotropy measurements of the total collision cross section with respect to the AIP. In Fig. 8 the Vm and the FAIP are sketched as functions of R. We assume that the Kip is known from other experiments. The anisotropy measurements produce two pieces of information, i.e. Ang and Ag. With this information, the whole curve AIP cannot be determined, but some features can be unambiguously established. From (23) we see that measurements of Ag determine the position of the minimum of VMr. Experience has shown that the value of q2.l2/q2.6 is rather independent of the individual values of e and Rm employed in the analysis, as long as the product Rm remains constant (Moerkerken, 1974). Therefore, one needs an independent determination of Rm of the VllP and a measurement of A g to establish the value of Km a/Rm- Naturally, the absolute value of Rma can only be found, then,... 

Equation (4.10) defines the total collision cross-section, cr(E). between an energetic particle of energy E and the target atoms. The total cross-section gives a measure of the probability for any type of collision to occur where energy-transfers are possible, for energies up to and including the maximum value... 

Fig. 14. The total collision cross section of neon as a function of the velocity (k = 2-niwlh). The solid line was calculated from the Lennard-Jones 6-12 potential, the dashed curve from the 6-8 potential, and the dot-dashed curve from the scattering potential of Fig. 3.

To obtain a final expression for k, recall that the total collision cross section is defined in terms of the impact parameter, b, as and that the orbital angular momentum of the system, f, is given by p.ub = Vc( + )h, where u is the relative velocity of the two fragments. Thus, the total cross section is... 

Collisions which occur with impact parameter b have their loci on a cylinder of radius b. Thus, the collision cross section for an impact parameter between b and b + db % the annular area Inb db. The fraction of collisions which occur with this range of impact parameters is therefore Inb db divided by the total collision cross section na. Since reaction only occurs if 6 < bfe), the fraction of reactive collisions is... 

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