Quantum defect surface

Let us for a moment consider the three channel problem depicted in Fig. 20.1. We have just discussed the case in which Wt < 0 for all i, i.e. the region below the lowest limit. The quantum defect surface defined by Eq. (20.12) is a two dimensional surface inscribed in a cube of length Av, = 1 on a side. Now let us consider the region between the first and second ionization limits, where channel 1 is open. Since is a continuum wave the r — boundary condition does not... 

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... 

Fig. 21.2 Quantum defect surface for the two channel problem showing r, or equivalently r, the phase shift in the open channel, channel 1, divided by — n. The Vj axis is reversed to...

All the energy dependence in the cross section arises from the factors A2 and (vb B v2 ) 2. These are, respectively, the spectral density of the channel 2 autoionizing states, and the overlap integral between the bound and continuum ni states with effective quantum numbers vb and v2. We have already seen that A is simply given by -dvl/dv2, the derivative of the quantum defect surface, and repeats modulo 1 in v2. The overlap integral is given is closed form by5... 

In the last expression of Eq. (21.20) the squared overlap integral is approximately constant and equal to v3 for v3 vb, and (6s r 6p) and fl3 are constant as well, so the cross section is directly proportional to A2. Knowing that the structure of Fig. 21.11 has no contribution from interference in the excitation amplitudes vastly simplifies the task of interpreting the spectrum. In fact, knowing that all the information appearing in the spectrum is in A2, we know that it must be found in the quantum defect surface. 

In Fig. 21.12 we show the quantum defect surface for the three channel 6pnd J = 3 problem.16 This surface, inscribed in a cube, is valid for all energies below the 6pI/2 limit. If there were no interchannel interactions the surface of Fig. 21.12 would consist of three intersecting planes at = 0.2, v2 = 0.2, and v3 = 0.3. The... 

Fig. 21.12 The three dimensional, quantum defect surface for the 6pnd states below the 6p1/2 limit. Here v2 and v3 are the effective quantum numbers relative to the 6p1/2 and 6p3/2 limits, respectively. is the continuum phase. The direction of the normal to the surface at any point indicates the amounts of 6p1/2nd5/2,6p3/2ndJ-, and continuum character in the...

Now we need to determine which points of the surface of Fig. 21.12 correspond to the energy scan of the spectrum of Fig. 21.11. The path along the quantum defect surface is determined by the energy constraint... 

Before we see how the spectrum of Fig. 21.11 emerges from Fig. 21.12, let us try a simpler case, the 6sl6d 1D2 —> 6p1/216d5/2 ICE transition, which exhibits a Lorentzian peak at v2 =13.3.16 Since the peak falls at the center of the (vbdB v2d) overlap integral the cross section is proportional to A2, which should thus have a maximum at v2 = 13.3. Inspecting Fig. 21.14 we can see that the 6p1/216d5/2 state lies on the last full branch of the v2, v3 curve, from v3 = 6.85 to v3 = 6.98 and v2 = 13 to v2 = 14. This branch of the curve of Fig. 21.14 is almost parallel to the v2 axis and lies at v3 6.9 where the normal of the quantum defect surface points in the direction except at v2 13.3 where it points predominantly in the v2 direction. In other words the normal to the quantum defect surface also tells us that A2 is... 

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