Spherical-wave outgoing, amplitude

The second term in (6-9) expresses that nearest and next nearest neighbors dominate scattering contributions to the EXAFS signal, while contributions from distant shells are weak. The dependence of the amplitude on 1/r2 reflects that the outgoing electron is a spherical wave, the intensity of which decreases with the distance squared. The term exp(-2r/X) represents the exponential attenuation of the electron when it travels through the solid, as in the electron spectroscopies of Chapter 3. The factor 2 is there because the electron has to make a round trip between the emitting and the scattering atom in order to cause interference. 

It is easily verified that any function of the form P(r,t) - P0 = (l/r)f(t-r/c0) is a solution, the negative sign corresponding to an outgoing wave about the center. Thus the form of an infinitesimal spherical wave does not change as it spreads out at the same velocity, but its amplitude decreases proportional to 1/r, the inverse of the radial distance... 

The formalism of electron-atom scattering has been extensively dealt with elsewhere./27,28,29/ We shall only recall its main features here. Because of the assumed spherical symmetry, the partial-wave scattering approach is convenient. Namely, an incoming spherical wave h[2 kr) y/ (r), (—can scatter only into the outgoing spherical wave hf kr) Y/"(r) (here hf and hf2 are Hankel functions of the first and second kinds, k = 2n/h(2mE) i, E is the kinetic energy and r — r ). This occurs with amplitude t( (f is an element of the diagonal atomic <-matrix), which is related to the phase shifts 6l through... 

In the case of inelastic processes, the asymptotic form of Ri(kr) given in Eq. (5) no longer holds. This is because the amplitude of the outgoing spherical wave must necessarily be less than that of the incoming wave. These amplitudes however refer only to the elastic component of the scattering amplitude. The breakdown of the radial standing wave pattern in Ri(kr) results in a nett inward flux towards the origin. In fact these are the waves which go into various inelastic channels. This is taken into accmmt by... 

Equations (12.4)-(12.9) describe an outgoing transverse spherical wave propagating radially with the phase velocity v = cojk and having mutually perpendicular complex electric and magnetic field vectors. The wave is homogeneous in that the real and imaginary parts of the complex wave vector kx are parallel. The surfaces of constant phase coincide with the surfaces of constant amplitude and are spherical. Obviously,... 

The S (scattering) matrix is defined by considering the amplitudes of the incoming spherical waves in the incident channel and the outgoing spherical waves in all channels. Let correspond to r y) for this case for (if no... 

The scattering amplitude is similarly determined by equating the coefficients of the outgoing spherical waves, after first expanding... 

A schematic illustration of elastic neutron scattering from an isolated point scatterer (nuclei) is given in Figure la. An incidental plane wave with wave vector, Jcq is incident from the left and produces an outgoing spherical wave of wave vector k ki = ko) and amplitude... 

The first term is a plane wave of incoming reactants in channel Nr, and the second term is a sum of outgoing spherical waves in all open channels. The asymptotic scattering amplitude /n t. Nti which has units of length, can also be expressed as... 

---