Impact parameter rainbow

This RE is radially unstable if j / 2mr ) + V r) is a maximum, radially stable if it is a minimum. If an unstable RE occurs, the deflection function 0/ =/(h,), [41,76], displays rainbows (0/ is the final angle of exit of the particle in the inertial frame, h,- is the initial impact parameter). The structure of these rainbows is well known in the classical or quantum cases [77]. For such an integrable Hamiltonian like equation (45), there are as many singularities (rainbows) of the deflection function as integer numbers each singularity is characterized by an increase by 1 of k = mod(0/, 2ti). There is one impact parameter b such that... 

We shall discuss the rainbow cross section as an example in detail since this is the most important feature of the differential cross section at intermediate energies. The result for the cross section is when the contribution of the third impact parameter, of Fig. 2, is included (Boyle, 1971 Mullen and Thomas, 1973) ... 

Figure 2.13 Trajectories for collisions at different impact parameters showing the deflection / at different values of b = b/Rm- For large initial f) the trajectory is a shade pulled in by the long-range force. The deflection is maximal at the rainbow, which is at far Ffm- For closer-in approach the trajectory begins to sample the repulsive potential. At what is called the glorythe net deflection is zero because the initial attraction is fully counterbalanced by the repulsion closer in. Below the glory, fa< fag, repulsion dominates and the scattering is backwards. Rq isthe distance of closest approach.

The presence of the well in the potential means that impact parameters somewhat below and somewhat above the rainbow impact parameter lead to the same angle of deflection. Approximating the potential about its minimum as a harmonic well, the approximation x (b) oc V(b)/E suggests that near the rainbow x(b) = Xr + c(b - brf where c depends on the collision energy. There are then two different trajectories, with impact parameters just above and just below br, that scatter into the same angle. 

If we could sample the -dependence of the deflection function we would have a very sensitive probe of the intermolecular potential. In fact we shall see below that, at least for higher is, the functional dependence of / (i) on b is that of V b), the potential evaluated at = i. in particular, the existence of a minimum in x (i) is an indication of a well in the potential. Therefore the rainbow impact parameter should serve as a good indicator of Tim, the equilibrium distance of the potential. This expectation is used in drawing Figure 4.5. 

At the high level of final state resolution provided by such experiments we can discern quantal interference effects. The more prominent feature for inelastic excitation is a rotational rainbow that arises by a mechanism similar to the intense scattering of the final velocity into certain directions (Section 2.2.5). Here too, the rainbow arises from different trajectories scattered into the same final state except that the state is specified not only by the direction of v but also by the rotational state of the molecule, NO in the case of Figure 10.9. This is a stereodynamic effect because the final state is determined not only by the impact parameter but also by the angle of approach, as shown for scattering by a hard ellipsoid in Figure 10.10. 

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