Elastic scattering amplitudes

Shapiro, M., and Gerber, R. B. (1976), Extraction of Interaction Potentials from the Elastic Scattering Amplitudes An Accurate Quantum Mechanical Procedure, Chem. Phys. 13, 227. 

Thus, the nonresonant elastic scattering amplitude may be represented in the form... 

At intermediate and higher energies it is appropriate to expand the forward elastic scattering amplitude in a Born series ... 

The real part of the forward elastic scattering amplitude can be related to the total scattering cross section by means of the following dispersion relation ... 

Since large-angle scattering seldom occurs in heavy-particle collisions, the scattering amplitude /+(0) is concentrated around 0 = 0 and 0 = 77. Therefore, we can write the square of the total elastic scattering amplitude f (0) as... 

The problem has therefore been reduced to potential scattering by the interactions U = (U associated with elastic scattering amplitudes/. Hence the elastic (/ = /) and inelastic (/ j) amplitudes are... 

Thus, the formation of this state does not change the elastic scattering amplitude Add shown in Figure 10.8. 

The projectile-nucleus elastic scattering amplitude may also be obtained by solving a one-body Schrodinger equation for the elastic channel using the optical potential where from eq. (2.21)... 

To summarize, the p + NR elastic scattering amplitude is conveniently calculated using the relativistic DWBA formalism, where the following dianges from the procedure desoibed in ref. [Ra 88b] are carried out (1) NR distorted waves and bound state wave functions are used for the upper components (2) the lower components are set to the NR or free partide limit [i.e., no potential terms included in (a k)/ E + m)] (3) ) corresponds to the dioice of t according to eq. (6.5) and may include density dependence (4) the pA recoil factor, EJ E + ), is included. 

In line with approximation vii"), only the asymptotic forms of the nuclear wave functions are relevant. (Note that these forms contain the exact elastic scattering amplitudes.) The insertion of (2.8) and (2.9) into Eq. (2.7) gives four terms. We now make a peaking approximation ... 

In the conventional theory of elastic image fomiation, it is now assumed that the elastic atomic amplitude scattering factor is proportional to the elastic atomic phase scattering factor, i.e. 

V..(R) is the static interaction for elastic scattering. The Bom scattering amplitude is a pure fiinction only of... 

B-e collisions, then the Bom approximation for atom-atom collisions is also recovered for general scattering amplitudes. For slow atoms B, is dominated by s-wave elastic scattermg so thaty g = -a and cr g = 4ti... 

Here the distortion (diagonal) and back coupling matrix elements in the two-level equations (section B2.2.8.4) are ignored so that = exp(ik.-R) remains an imdistorted plane wave. The asymptotic solution for ij-when compared with the asymptotic boundary condition then provides the Bom elastic ( =f) or inelastic scattering amplitudes... 

The amplitude of the elastic scattering, Ao(Q), is called the elastic incoherent structure factor (EISF) and is determined experimentally as the ratio of the elastic intensity to the total integrated intensity. The EISF provides information on the geometry of the motions, and the linewidths are related to the time scales (broader lines correspond to shorter times). The Q and ft) dependences of these spectral parameters are commonly fitted to dynamic models for which analytical expressions for Sf (Q, ft)) have been derived, affording diffusion constants, jump lengths, residence times, and so on that characterize the motion described by the models [62]. 

The structure factor S(q as defined in Eq. (54) in terms of the Ising pseudospins Si, in the framework of the first Bom approximation describes elastic scattering of X-rays, neutrons, or electrons, from the adsorbed layer. SCq) is particularly interesting, since in the thermodynamic limit it allows to estimate both the order parameter amplitude tj/, the order parameter susceptibility X4, and correlati length since for q near the superstructure Bragg reflection q we have (k = q— q%)... 

According to Eq. (1.16), the elastic coherent X-ray scattering amplitude is the Fourier transform of the electron density in the crystal. The crystal is a three-dimensional periodic function described by the convolution of the unit cell density and the periodic translation lattice. For an infinitely extended lattice,... 

The form displayed in eq. (2-40) implies that the ratios of the amplitudes for scattering into different exit channels are independent of the entrance channel. This, of course, will only be true if the resonance is long lived, so that memory of the initial state can be lost. Note that Aga is a symmetric function, which is a consequence of time-reversal invariance. Note also that, within the approximations used, the phase shift associated with a given channel is just the elastic scattering phase shift for that channel. Finally, the partial widths are proportional to the probability of decay from channel fi. Equation (2-41) is, then, merely a statement that the total probability of decay from channel is the sum of the probabilities of decay into individual channels. 

Here fj(k) is the field amplitude of the light scattered by particle j with its center of mass at the position ry, Nis the number of particles in the scattering volume, K = K/ — K is the transferred momentum, and k = K. For elastic scattering K, = K/ = 2tuis/2q, where is the incident light wavelength in vacuum. Thus,... 

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