Continuum scattering phase shifts

The continuum electron-phase shifts induced by the short-range scattering off the chiral molecular potential are most conveniently introduced by a third choice of continuum function, obtained by diagonalizing the K-matrix by a transformation U, resulting in a set of real eigenchannel functions (apart from normalization) [41] ... 

When experimental results are later introduced, it will be seen that the significance of the final-state scattering in PECO measurements is confirmed by the observation that for C li core ionizations, which must therefore proceed from an initial orbital that is achiral by virtue of its localized spherical symmetry, there is no suggestion that the dichroism is attenuated. The sense of the chirality of the molecular frame in these cases can only come from final-state continuum electron scattering off the chiral potential. Generally then, the induced continuum phase shifts are expected to be of paramount importance in quantifying the observed dichroism. 

Johnson and Rice used an LCAO continuum orbital constructed of atomic phase-shifted coulomb functions. Such an orbital displays all of the aforementioned properties, and has only one obvious deficiency— because of large interatomic overlap, the wavefunction does not vanish at each of the nuclei of the molecule. Use of the LCAO representation of the wavefunction is equivalent to picturing the molecule as composed of individual atoms which act as independent scattering centers. However, all the overall molecular symmetry properties are accounted for, and interference effects are explicitly treated. Correlation effects appear through an assigned effective nuclear charge and corresponding quantum defects of the atomic functions. 

To find the values of the eigen phase shift rp we simply replace jtv by jit for all the open channels in the determinant of Eq. (20.12) and solve Eq. (20.12) for the two possible values of r, which are rp, p = 1,2. If there are P open channels there are P values of rp. While it is not transparent from the discussion up to this point that this procedure is reasonable, as we shall see, these values of rp lead to scattering and reactance matrices which are diagonal, and continuum wavefunc-tions which are the normal scattering modes. 

This phase shift is a direct consequence of the higher frequency within the range of the potential. As the potential is made either wider or deeper, states are sucked in (fi-om the box) and are localized in the potential well. In the case of a one-dimensional square well of depth Vand width L, the number of bound states for a particle of mass misN= 1 + [ 2mV) L/(nh)] where the square brackets stand for the integer part. Every time the phase shift passes Jt, another one of the continuum (particle-in-a-box) states is sucked into the well. This phase shift, which plays a central role in scattering problems, records the asymptotic compression of the wave function, but the number of (1/2) oscillations (each producing a bound state) is lost. However, the number of bound states for each , N, can be recovered as it is encoded in the zero energy (i.e., threshold) phase shift, 5 (0) = N( n. This is known as Levinson s Theorem. 

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