Asymptotic scattering region

FIGURE 3.24 Comparison between the resonance (upper) and Gaussian (lower) curves wile the last is fast decreasing to zero outside of the resonance (central) region, the first one presents the long (asymptotical) tail specific for (asymptotic) scatterings. 

Scattering from the drawn samples in the higher- region was greater than for identical undrawn samples and was nearer to that expected for an adjacent re-entry model (all isotropic polypropylene samples, regardless of their thermal histories, have approximately the same asymptotic scattering level). A tentative interpretation of this result is that the stems come closer together as the polymer is drawn. 

Since TD-DFT is applied to scattering problems in its QFD version, two important consequences of the nonlocal nature of the quantum potential are worth stressing in this regard. First, relevant quantum effects can be observed in regions where the classical interaction potential V becomes negligible, and more important, where p(r, t) 0. This happens because quantum particles respond to the shape of K, but not to its intensity, p(r, t). Notice that Q is scale-invariant under the multiplication of p(r, t) by a real constant. Second, quantum-mechanically the concept of asymptotic or free motion only holds locally. Following the classical definition for this motional regime,... 

At sufficiently large distances from tlje origin (kr 1), in the far-field region, the scattered electric field E5 is approximately transverse (er E5 — 0) and has the asymptotic form (see, e.g., Jackson, 1975, p. 748)... 

In the region outside the sphere jn and yn are well behaved therefore, the expansion of the scattered field involves both of these functions. However, it is convenient if we now switch our allegiance to the spherical Hankel functions h[]) and h% We can show that only one of these functions is required by considering the asymptotic expansions of the Hankel functions of order v for large values of p (Watson, 1958, p. 198) ... 

All the information on the measurable scattering quantities is concentrated into the asymptotic part of the continuum wavefunction, regardless of the dynamics of the physical system described by the inner-region part of the wavefunction. Significant effects of the inner-region physics should be reflected on the asymptotic quantities. This section is concerned with such effects of QBSs, or resonance effects, manifesting themselves in these asymptotic quantities. 

In the absence of any interaction, the outgoing wave packet comes out to the asymptotic region without delay. When scattering occurs, the wave packet spends some time in the potential region before coming out from there if At is positive. If a resonance occurs, the time delay is calculable from Eqs. (15) and (19) as... 

This asymptotic decrease in intensity in the high q tail is described in the Porod law region and arises when qR > 4, where R refers to the dimension of the scattering heterogeneity. 

---