chapter9 - Chapter 9 SCATTERING THEORY 9.1 General...

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Chapter 9 SCATTERING THEORY 9.1 General Considerations In this chapter we consider a situation of considerable experimental and theoretical inter- est, namely, the scattering of particles o¤ of a medium containing some type of scattering centers, such as atoms, molecules, or nuclei. The basic experimental situation of interest is indicated in the …gure below. An incident beam of particles impinges upon a target, which maybe a cell con- taining atoms or molecules in a gas, a thin metallic foil, or a beam of particles moving at right angles to the incident beam. As a result of interactions between the particles in the initial beam and those in the target, some of the particles in the beam are de‡ected and emerge from the target traveling along a direction ( μ; Á ) with respect to the original beam direction, while some are left unscattered and emerge out the other side having undergone no de‡ection (or undergo “forward scattering”). The number of particles de‡ected along a given direction are then counted in a detector of some sort. The kinds of interactions and the analyis of general scattering situations of this type can be quite complicated. We will focus in the following discussion on the scattering of incident particles by scattering centers in the target uner the following conditions: 1. The incident beam is composed of idealized spinless, structureless, point particles. 2. The interaction of the particles with the scattering centers is assumed to be elastic so that the energy of the scattered particle is …xed, the internal structure of the scatterer (if any) and, thus, the potential seen by the scattered particle does not change during the scattering event. 3. There is no multiple scattering, so that each incident particle interacts with at most one scattering center, a condition that can be obtained with su¢ciently thin or dilute targets.
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264 Scattering Theory 4. The scattering potential V ( ~ r 1 ;~ r 2 ) = V ( j ~ r 1 ¡ ~ r 2 j ) between the incident particle and the scattering center is a central potential, so we can work in the relative coordinate and reduced mass of the system. Under these conditions, the picture of interest reduces to that depicted below, in with an incident particle characterized by a plane wave of wavevector ~ k = k ^ z along a direction that we take parallel to the z -axis, and scattered particles emerging through a in…nitesimal solid angle d - along some direction ( μ; Á ) . We may characterize the incident beam by its (assumed uniform) current density ~ J i along the z axis. Classically ~ J i = n~v i where n = dN=dV is the particle number density characterizing the beam. The incident particle current through a speci…ed surface S is then the surface integral I = dN S dt = Z S ~ J i ¢ d ~ S: The scattered particle current dI S into a far away detector subtending solid angle d - along ( μ; Á ) is found to be proportional to (i) the magnitude of the incident ‡ux density J i , and (ii) the magnitude of the solid angle d - subtended by the detector. We write dI S = ( μ; Á ) d - J i d -
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