Photoionization differential cross-section

A detailed derivation of the photoionization differential cross-section expression, leading ultimately to the angular distribution in Eq. (4), is provided in Appendix A. This will help provide a detailed understanding of the photoelectron dynamics that determine the angular distribution parameters, as will be discussed in a subsequent section, but for now it may help develop the reader s appreciation of this phenomenon to provide a simple, if necessarily inexact, mechanical analogy. 

J.-Z. Tang, I. Shimamura, Differential cross sections for photoionization of He, J. Electron Spectrosc. Related Phenomena 79 (1996) 259. 

Starting in a manner similar to the treatment of single photoionization described in Section 2.1, double photoionization in helium caused by linearly polarized light will be treated first with uncorrelated wavefunctions. A calculation of the differential cross section for double photoionization then requires the evaluation... 

Figure 4.43 Energy- and angle-resolved triple-differential cross section for direct double photoionization in helium at 99 eV photon energy. The diagram shows the polar plot of relative intensity values for one electron (ea) kept at a fixed position while the angle of the coincident electron (eb) is varied. The data refer to electron emission in a plane perpendicular to the photon beam direction for partially linearly polarized light (Stokes parameter = 0.554) and for equal energy sharing of the excess energy, i.e., a = b = 10 eV. Experimental data are given by points with error bars, theoretical data by the solid curve.

The matrix element Mfi derived so far for the differential cross section of double photoionization in helium is based on uncorrelated wavefunctions in the initial and final states. For simplicity the initial state will be left uncorrelated, but electron correlations in the final state will now be included. The significance of final state correlations can be inferred from Fig. 4.43 without these correlations an intensity... 

Collecting together the information contained in equs. (4.78) and (4.81), the triple-differential cross section for double photoionization can be represented as... 

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 ... 

Finally, the differential cross section for photoionization, dff/dfl, will be given explicitly for the dipole approximation and the length form of the matrix element by collecting all the individual steps. This cross section is related to the transition rate w by... 

In order finally to derive the differential cross section of photoionization one inserts equ. (8.26) in equ. (8.24) and replaces the number nph of incident photons by nPh = ce0Alo)/2ti (see equs. (8.4b) and (8.8a) and (8.8b)) and the interaction operator by equ. (8.21). Then one removes the factor h2/m0 resulting from the normalization of the continuum function from the matrix element and incorporates it in the final prefactor (see footnote concerning equ. (7.28d)), and one introduces the fine structure constant a using a = el/4ne0hc. This leads to (for the summations over magnetic quantum numbers see below)... 

Parametrization of the differential cross section for two-electron emission As with single photoionization, the observables for two-electron emission depend on the properties of the incident light (energy, intensity and polarization), the... 

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