Geometrical interpretation of the quantum defect surface

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. 

When there are multiple values of r, the dimensionality of the quantum defect surface is reduced. In the region below the second limit the quantum defect surface defined by Eq. (20.12) is a two dimensional surface. Above the second limit, for each value of v3 there are two values of rp, and the quantum defect surface is two lines. 

Finally, if we consider the energy region above the third limit of Fig. 20.2, all three channels are open for all values of r. In this case we can put a box of any r rc about the ionic core and inquire as to the normal modes for electron scattering from the contents of the box. We have in fact already solved this problem. The normal modes are the Ha wavefunctions. Since the energy is above all the ionization limits we no longer need to ignore the r — °° boundary conditions they play no role. 

In any orbit of the Rydberg electron, most of the time is spent when the electron is far from the nucleus, where the wavefunction is most reasonably characterized in terms of Ah the coefficients of the 0, wavefunctions. Correspondingly, many of the properties depend upon the values of At in a very direct way. As shown by Cooke and Cromer,3 a particularly attractive feature of QDT is that the values of A,2, i.e. the composition of the wavefunction in terms of the collision channels, can be determined by inspecting the quantum defect surface. If we define the cofactor matrix Cia of the matrix of Eq. (20.12) by 

If the gradient defined by Eq. (20.18) is represented by a vector perpendicular to the quantum defect surface, its projections on the axes are proportional to the values of Aj. In a two limit problem Eq. (20.18) reduces to the simple form8 

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