3. Characterizing the kinematic regions in cross sections

Due to the generality of our approach we are able to address from a unified viewpoint a number of issues which have appeared and been discussed separately in previous treatments of double and single photoionization processes. These issues include characterization of the kinematic regions of the matrix element and cross sections (differential and integrated):

1) The dominant contributions to the total cross section for double ionization of He. These contributions are usually called the shake-off (SO) (where one electron takes almost all photon energy and has high momentum while the other shakes-off with low momentum) and the quasi-free (QF) (electrons have nearly equal and nearly opposite momenta, i.e. the relative momentum is large while the total momentum is small) contribution. The shake-off contribution has been considered extensively and often considered the only dominant contribution to the total cross section for double ionization of the ground state of He. It was obtained using V-form [4]and A-form [5] in coordinate space or V-form in a momentum space approach [6]. All these approaches agree providing the cusp condition at the nucleus is satisfied by the initial state wave function, as required by the Schrödinger-equation. The derivations of the shake-off contribution rely on the assumption that one electron is fast and the other slow, and therefore the whole contribution (at high energies) comes from the point at the nucleus. In our asymptotic FT approach this contribution is understood to represent a single FT in the large momentum ${\bf p}_{1}$ (of the fast electron) associated with the singular point ${\bf r}_{1}=0$ which, according to FT theory, gives the dominant asymptotic behavior for large momenta. Naturally, in the case of high energy single ionization (including ionization both with and without excitation) the same singularity is necessarily associated with the single large electron momentum, and so the same general discussion follows.

In the case of double ionization the point at the nucleus is not the only singular point of a two electron wave function which can be associated with a large momentum. The other point is at the electron-electron coalescence. In the case of the ground state of He (s-state and spin singlet) this singularity has the same behavior as the electron-nucleus singularity. In the case of double ionization there is a kinematic region (large relative momentum p of the two electrons and small total momentum P) for which the matrix element is a single FT in the large momentum p associated with the coalescence at the relative coordinate ${\bf r}=0$. The contribution to the cross section from this singularity has the same momentum dependence as the contribution from the electron-nucleus coalescence. It is, however, forbidden in dipole approximation, and it only appears in the first retardation correction. Our analysis confirms that the contribution from this coalescence is the quasi-free contribution predicted in [7] and [8,9]. We discuss this contribution in V-form, A-form, and in momentum space in order to compare with the existing analysis of the shake-off contribution. We discuss the energy dependence of the SO and QF contributions to the total cross section and the consequences of this energy dependence for the ratio of double to single ionization by photoabsorption. Our analysis confirm the result [8,9] that for high but nonrelativistic energies the double to single ionization ratio for photoeffect from the ground state of He is a linearly rising function in which the linear term (due to the first retardation correction) is a consequence of the existence of the cusp in the e-e coalescence of the ground state of He.

2) The dominant contributions to the spectrum. We may identify three regions in the spectrum, and we may identify the leading contribution to the matrix element in each. Two of these leading contributions (SO and QF) which are associated with specific kinematics, as already discussed, are determined by a corresponding singularity of the interaction potential energy. In a suitable form (A or V) they are independent of the final state interaction between electrons. The third such contribution to the spectrum (the final-state-interaction contribution, dominant in the intermediate energy region between the SO and the QF regions) is also determined by a coalescence point (e-N singularity). However, in all three forms (A,V,L) it requires the e-e interaction in the final state, described by the spherical waves of the relative e-e motion, which means that relatively large (but not too large) distances between the two electrons are involved. Again, there is specific kinematics (the two electrons leaving in nearly perpendicular directions) for which the matrix element is a single FT in the large momentum P associated with the e-N coalescence. This third contribution (final-state-interaction) can be obtained in a simple way using V-form, neglecting the interaction of the electrons with the nucleus in the final state. Thus in the contributions we have discussed, and with our choice of form, we have neglected either e-e or e-N interactions in the final state. Can additional contributions of the same order appear if we simultaneously include the interaction of all three particles in the final state? To get some sense of the possibilities we have used C3 functions for the final state; in this case we find no additional contributions. However, we do not know whether the exact final state wave function may in fact contain other terms (similar to the one described by the e-e spherical wave in C3) which would give such contributions. Our analysis does show that all known dominant contributions to all points of the spectrum, in addition to the SO and QF contributions which dominate the total cross section, are also determined by singularities at the two singular points of the wave functions.

3) Behavior of cross sections in other regions (the triple coalescence). Our approach also allows considerations of more differential cross sections for double ionization (i.e. differential both in energy and angles) at high energies. The discussion in points 1) and 2) is concerned with the regions in which the matrix element is a FT in one large momentum connected with a singularity. In kinematics in which both momenta are large the asymptotic FT is connected with the two singularities ${\bf r}_{1}=0$ and ${\bf r}=0$, i.e. with the triple singularity. The leading power in an inverse momentum expansion (the two large momenta are now of the same order, say p) is 1/p⁷ in the case of the He ground state. This means that, generally, in any kinematics there is at least a contribution from the triple coalescence point (which, in the case of the ground state, gives a 1/p⁷ contribution). Unequal energy sharing (away from the SO region) with electrons moving in the same or opposite directions would be examples where matrix element is determined by the triple coalescence point. Recently, the dipole matrix element for double ionization has been evaluated in the kinematics of electrons moving in the same direction with the same energies, in order to evaluate the inverse process, i.e. radiative double electron capture [10]. At high energies the result shows an energy dependence characteristic of triple coalescence, as follows from our approach.