1. Introduction

In this paper we discuss the high energy limit of double ionization of helium in photoabsorption as an example of the insights which can be achieved in the study of the high energy limit of photon atom interactions.

In general we can identify the dominant high energy processes which result when a photon is incident on a composite structure of charged particles, such as an atom, and how they differ from the corresponding processes (scattering, pair production, and photon splitting) of photons incident on free charges. Additional processes, such as excitation and photoionization of the structure, also occur. Appropriate description depends on the detail of observation; implicit sum rules relate exclusive and more inclusive characterizations of processes. We may identify the aspects of composite structure which are relevant to the description of these high energy situations. We may summarize what is known about the high energy limit of photoelectric effect, elastic (Rayleigh and Delbrück) and inelastic (Raman and Compton) scattering, and pair production (bound and free). We may discuss the relationship of high energy processes in which single or double ionization is specified. We may discuss the modifications and further features which result if the composite structure is in an excited state, rather than a ground state. We can note the further considerations if the composite structure is not isolated, but in an environment, such as a plasma, a strong field, or a lattice.

The study of atomic processes at high energies is of both fundamental and practical interest. Basic mechanisms are often revealed and isolated in their most transparent ways at high energies. We think here particularly of the study of multi-particle correlated interactions. This provides insight for the development of approaches to the description of these mechanisms, which can then be utilized more generally, as in the example which we discuss below. For example, at high energy there are situations in which one can neglect final state interactions, and this leads to the use of high energies in studying composite particles (as in photoeffect, Rayleigh and Compton scattering, and double ionization). High energy studies are also now much more firmly motivated, since the processes are now observable and theoretical views are subject to experimental test. We now have both modern high energy and intensity synchrotron sources and sources of fast highly charged ions.

Here we focus on a limited, but fundamental, aspect of these issues. We study the double ionization of the ground state of He in photoabsorption at high photon energies $\omega$ (but still $\omega \ll m$), to lowest order in the electron-photon interaction Hamiltonian, including the first order retardation term in that interaction for the kinematics in which it gives a dominant contribution to the total ionization cross section. At these energies the portions of the matrix element which dominate the spectrum are asymptotic regions of single (rather than double) Fourier transforms (FT) in a single large momentum, of a function with coordinate-space singularities (i.e. points at which the function is not differentiable). These singularities originate in the Coulombic (electron-nucleus and electron-electron) interaction potential energy; in the matrix element they enter through wave functions (initial and final) and/or the electron-photon interaction, depending on the form used. We may use the properties of the asymptotic behavior of a Fourier transform (FT) of such singular functions in order to study the dominant contributions to the spectrum in double photoionization of He.

The matrix element for photoionization is an integral over a product of initial state wave function, final state wave function and electron-photon interaction operator. These wave functions, being solutions of the Schrödinger equation with Coulombic interactions, have singularities at the singularities of the potential. These singularities are located at the points where the particles coalesce and we will use the term coalescence points for the locations of the singularities. These are electron-nucleus (e-N) coalescences, when one electron is at the nucleus, and an electron-electron (e-e) coalescence when the two electrons coincide. There is also a triple coalescence point when all three particles coincide. The properties of wave functions in the near vicinity of these singularities which, for bound states are well understood [1,2], can be extracted from the Schrödinger equation. They are known as coalescence properties and in the case of s-states are often called Kato cusp conditions. We will here also discuss these general properties for continuum states, as well as related properties for high energy electrons.

Our discussion is general and does not depend on the choice of the form (length, velocity, acceleration etc.) of the photoionization matrix element. It is based on a general theorem of Fourier transforms, which states that a slow asymptotic decrease for large p, such as 1/pⁿ in a 1/p asymptotic power expansion, of the Fourier transform of a well localized function, comes only from singularities of that function. Namely, according to the Fourier transform theorem [3], a function f(x,y,z), infinitely differentiable at all points (x,y,z), which decreases for large r=(x²+y²+z²)^1/2 faster than any power of r, has a Fourier transform F(p_x,p_y,p_z) which decreases faster than any power of p=(p_x²+p_y²+p_z²)^1/2 for large p. Functions which appear in our photoionization matrix elements are well localized (due to the fact that the bound state is localized) and they are differentiable everywhere except at coalescence points, where they are singular [1], i.e. non-differentiable. The asymptotic form of the FT of such a function, for singularities resulting from point Coulomb potentials, is expandable in powers of 1/p. It can be obtained from expansion of the function in powers of r around the singular points, since for such potentials solutions of the Schrödinger equation can be expanded in powers of r in the vicinity of the singular points.