Electron to the continuum

Using the finite-size scaling method, study of the analytical behavior of the energy near the critical point shows that the open-shell system, such as the lithium-like atoms, is completely different from that of a closed-shell system, such as the helium-like atoms. The transition in the closed-shell systems from a bound state to a continuum resemble a first-order phase transition, while for the open-shell system the transition of the valence electron to the continuum is a continuous phase transition [9]. 

The time-dependence of the energy levels in a supercritical heavy-ion collision is depicted in Figure 8.33. An electron (or hole) which was in a certain molecular eigenstate at the beginning of the collision can be transfered with a certain probability into different states by the dynamics of the collision. This can lead to the hole production in an inner shell by excitation of an electron to a higher state and/or hole production by ionization of an electron to the continuum. Further possibilities are induced positron production by excitation of an electron from the lower continuum to an empty bound level and direct pair production [59]. 

Bang and Hansteen (1959) and later Hansteen and Mosebekk (1973) treated the ion-atom collision process in a semiclassical approximation, considering the projectile motion classically and the transition of the inner-shell electron to the continuum quantum mechanically. 

Inner shell ionization of electrons to the continuum in ion-atom collisions can occur by two different processes. For low Z (projectile) particles on high Z2 (target) atoms the only available process is Coulomb excitation which is variously treated by plane wave Born approximation (PWBA), the binary encounter approximation (BEA), and the semiclassical approximation (SCA). When Z becomes comparable to 7/1 and the ion velocity v is lower than the velocity of the bound electron in question, v, the electrons adjust adiabati-cally to the approach of the two nuclei and enter molecular orbitals (MO) which in the limit of fused nuclei approach the atomic orbitals of the united atom Z = Z + Z2. This stacking of electrons can lead to a promotion of an innershell electron to the continuum or to a vacant outer orbital by direct curve crossing, rotational coupling, or radial coupling between molecular levels when such channels are available. 

Compared to atomic auto-ionizing states, where lifetimes below 10"12 s are common, the molecular levels exhibit a much weaker coupling to the ionization continuum. The reason is that the molecular ionization process takes place through a coupling between nuclear motion and electronic configurations, whereas in atoms a direct electronic coupling of the excited electron to the continuum makes the auto-ionization probability by far larger. 

Figs. 10 and 11 for helium. This feature is closely analogous to the one observed for ve = vp for electron capture to the continuum mechanism or to electron loss to the continuum which is often referred to as the cusp. The electrons with v, vr 0 in the low-lying continuum states of the target ion also form a sharp peak known as the target cusp. ... 

In Figs. 24 and 25 we show the measured double differential cross sections for electron emission at zero degrees in collisions of 100-keV protons with He and H2 [39] compared to CDW-EIS predictions [39]. Uncertainties associated with the experimental results vary from 1% near the electron capture to the continuum peak to about 15% near the extreme wings of the distribution. These results have been scaled to provide a best fit with CDW-EIS calculations. In both cases there is satisfactory agreement between the CDW-EIS calculations and experiment, particularly with excellent agreement for electrons with velocities greater than v, where v is the velocity of the projectile. For lower-energy electrons the eikonal description of the initial state may have its limitations, especially for lower-impact parameters. 

There can be a difference between the dissociation of polyatomic molecules and delayed ionization in the nature of the initial excitation. In ZEKE spectroscopy the state that is optically accessed (typically via an intermediate resonantly excited state) is a high Rydberg state, that is a state where most of the available energy is electronic excitation. Such a state is typically directly coupled to the continuum and can promptly ionize, unlike the typical preparation process in a unimolecular dissociation where the state initially accessed does not have much of its energy already along the reaction coordinate. It is quite possible however to observe delayed ionization in molecules that have acquired their energy by other means so that the difference, while certainly important is not one of principle. 

Figure 7. A schematic outline of the available bound phase space. The shaded area is the region that is optically accessed. It is near the exit to the continuum. The ordinate refers to the angular quantum numbers of the electron. These can be changed primarily (but not only) by the external perturbations. The bottleneck for such changes is shown as a dashed line. The abscissa refers to the principal quantum number of the electron or to the rotational quantum number of the core. These two change in opposite directions due to the coupling to the cote.

Here, Q is the projector on the bound subspace and P projects onto the open, continuum channels. The intramolecular coupling is written as V+ U so that, as before, U is any additional coupling brough about by external perturbations. The equation H = Hq + V+U, where Ho is the zero-order Hamiltonian of the Rydberg electron and so includes only the central part of the potential due to the core plus the motion (vibration, rotation) of the core, uncoupled to the electron. The perturbations V + U can act within the bound subspace, as the operator Q(V+l/)Q is not necessarily diagonal and is the cause of any intramolecular dynamics even in the absence of coupling to the continuum. The intramolecular terms can also couple the bound and dissociative states. 

It should be noted that these expressions hold only for excitation to the continuum and that comparison of electron scattering and optical absorption for discrete transitions is only possible if the resolution of both types of experiment is known. 

The utilization of NRO is also essential when calculating the electronic transitions between the bound states and to the continuum, if the radial orbitals of the initial and final states differ significantly [205, 206]. Accounting for the non-orthogonality terms is of particular importance when studying many-electron transitions. 

The first reported study of the behaviour of double differential cross sections for positron impact ionization was that of Moxom et al. (1992) these workers conducted a search for electron capture to the continuum (ECC) in positron-argon collisions. In this experiment electrons ejected over a restricted angular range around 0° were energy-analysed to search for evidence of a cusp similar to that found in heavy-particle collisions (e.g. Rodbro and Andersen, 1979 Briggs, 1989, and references therein), which would be the signature of the ECC process. 

Schultz, D.R. and Reinhold, C.O. (1990). Electron capture to the continuum and binary ridge structures in positron-hydrogen collisions. J. Phys. B At. Mol. Opt. Phys. 23 L9-L14. 

---