Scattering projection

The first term of Eq. (34) describes a direct transition from the bound state to the scattering projection of the structured continuum. The second term describes a resonance-mediated transition. 

Consider first the case where the scattering projection of the Hamiltonian, PHMP, can induce only elastic scattering (referred to as an uncoupled or elastic continuum). In this situation the partial wave in Eq. (37) reduces to... 

For a structureless continuum (i.e., in the absence of resonances), assuming that the scattering projection of the potential can only induce elastic scattering, the channel phase vanishes. The simplest model of this scenario is depicted schematically in Fig. 5a. Here we consider direct dissociation of a diatomic molecule, assuming that there are no nonadiabatic couplings, hence no inelastic scattering. This limit was observed experimentally (e.g., in ionization of H2S). 

Figure 8.41. Liquid scattering projected on the meridian, / j ( 3), originating from random placement of lamellae along a shish. The form factor of the dilute system is displayed for reference. Structure parameters as in Fig. 8.40. Uniform crystallite thickness leads to characteristic minima in the scattering curves. Note the varying strength of the oscillations...

Another swept approach is shown in Figure 16.21 [85]. In the center, a semi-transparent bidirectional Lambertian scattering projection screen rotates around its center axis at... 

If we are only interested in elastic scattering, a useful procedure [Wa53] is to absorb the contributions from virtual excitation of the target nucleus (and internal excitation of the projectile) into an effective operator (the optical potential operator) and to solve a scattering equation in which only elastic channel intermediate states appear. (Analogous procedures can be used for inelastic scattering.) Projecting onto the elastic channel in eq. (2.18) yields... 

Colijn A P, Beekman F J (2004) Accelerated simulation of cone beam x-ray scatter projections IEEE Trans. Med.Imaging 23 584-590. 

In oriented systems (fibres or stretched films), the scattered image often appears as a two-bar or a four-point pattern with the scattering maximum at or near the meridian (fibre axis). The one-dimensional scattered intensity along the meridian must be calculated by the projection method using the following fonnalism... 

The energy and state resolved tiansition probabilities are the ratio of two quantities obtained by projecting the initial wave function on incoming plane waves (/) and the scattered wave function on outgoing plane waves [F)... 

Like e, t is the product of two contributions the concentration N/V of the centers responsible for the effect and the contribution per particle to the attenuation. It may help us to become oriented with the latter to think of the scattering centers as opaque spheres of radius R. These project opaque cross sections of area ttR in the light path. The actual cross section is then multiplied by the scattering efficiency factor optical cross... 

It is awkward to use two different angles to describe the intensity of light scattered along a particular line of sight, but this situation is easily remedied by referring back to Fig. 10.5. It is apparent from Fig. 10.5 that r cos 0 is the projection of r along either the x, y, or z axis, depending on the choice of 0. We therefore see that... 

The easiest way to proceed is to use vectors to describe this part of the problem. We represent the distance between the pair of scattering sites by the vector OP the length of which is simply r. To express di and d2 in terms of OP we construct the unit vectors a and b which are parallel to the incident and scattered directions, respectively. The projection of OP into direction a, given by the dot product of these two vectors, equals dj. Likewise, the projection of OP into direction b gives d2. Therefore we can write... 

Laser light-scattering 1-1800 mean projected area... 

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