Cross section for double photoionization at threshold

Six coordinates are necessary in order to describe the positions rj and r2 of the two helium electrons in a coordinate frame fixed at the nucleus. One possible choice is three hyperspherical coordinates and three Euler angles. The hyperspherical coordinates determine the triangle given by the positions of the three charged particles. They are defined by 

If the hyperspherical coordinates are inserted into the Schrodinger equation (see equ. (1.1)) one obtains as a differential equation for the wavefunction (/ a, l2) (see [Fan83]) 

Here the quantity U is an effective potential that contains three contributions the kinetic energy for the radial movement of the electrons (in the coordinate a), a centrifugal potential energy, and the Coulomb potential energy — C(a, 12)/R of the system. In the present context of double photoionization it is this Coulomb energy which determines the features of two-electron emission (in atomic units)  

In order to elucidate how the total cross section for double photoionization, equ. (5.76), can be derived from the triple-differential cross section, equ. (4.84b), the necessary integration steps will be listed (for details see [HSW91]). Assuming for simplicity completely linearly polarized incident light with the electric field vector defining the reference axis, the triple-differential cross section from equ. (4.84b) including also a constant of proportionality can be reproduced here  

integration of equ. (5.79) gives for the total cross section 

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