Scattering amplitudes

The cross section for scattering into the differential solid angle dD centred in the direction (9,(l)), is proportional to the square of the scattering amplitude ... 

Neutron scattering depends upon nuclear properties, which are related to fluctuations in the neutron scattering cross section a between the scatterer and the surroundings. The scattered amplitude from a collection of scatterers can thus be written as (similar to (B 1.9.29)) ... 

This equation describes the Fourier transfonn of the scattering potential V r). It should be noted that, in the Bom approximation the scattering amplitude/(0) is a real quantity and the additional phase shift q(9) is zero. For atoms with high atomic number this is no longer tme. For a rigorous discussion on the effects of the different approximations see [2] or [5]. 

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... 

Solution of the scattering amplitude may then be detemrined from the asymptotic fomi of 4 (r jdirectly or from the integral representation... 

The optical theorem relates the integral cross section to the unaginary part of the forward scattering amplitude by... 

As well as obtaining tlie scattering amplitude from the above asymptotic boundary conditions, can also be obtained from the integral representation for the scattering amplitude is... 

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 close-coupling equations are also applicable to electron-molecule collision but severe computational difficulties arise due to the large number of rotational and vibrational channels that must be retained in the expansion for the system wavefiinction. In the fixed nuclei approximation, the Bom-Oppenlieimer separation of electronic and nuclear motion pennits electronic motion and scattering amplitudes f, (R) to be detemiined at fixed intemuclear separations R. Then in the adiabatic nuclear approximation the scattering amplitude for ... 

The END trajectories for the simultaneous dynamics of classical nuclei and quantum electrons will yield deflection functions. For collision processes with nonspherical targets and projectiles, one obtains one deflection function per orientation, which in turn yields the semiclassical phase shift and thus the scattering amplitude and the semiclassical differential cross-section... 

Figure 2 Variations in the neutron scattering amplitude or scattering length as a function of the atomic weight. The irregularities arise from the superposition of resonance scattering on a slowly increasing potential scattering. For comparison the scattering amplitudes for X rays under two different conditions are shown. Unlike neutrons, the X-ray case exhibits a monotonic increase as a function of atomic weight.

Where, /(k) is the sum over N back-scattering atoms i, where fi is the scattering amplitude term characteristic of the atom, cT is the Debye-Waller factor associated with the vibration of the atoms, r is the distance from the absorbing atom, X is the mean free path of the photoelectron, and is the phase shift of the spherical wave as it scatters from the back-scattering atoms. By talcing the Fourier transform of the amplitude of the fine structure (that is, X( )> real-space radial distribution function of the back-scattering atoms around the absorbing atom is produced. 

Anti-phase boundaries are interfaces leading a phase shift in the scattering amplitudes between two domains. They may be distinguished from those where the two domains differ in orientation. 

The above discussion, thus, has made somewhat more precise the sense of the limiting procedure involved in writing down the representation (10-216) for the scattering amplitude. This representation also makes clear that a knowledge of Green s function... 

To second order in the external field, the scattering amplitude is given by... 

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