Theoretical Shapes of Angular Electron Distributions

The angle-dependent electron intensity 7(i, ) is in principle given by the square of the matrix element (1). For the description of an actual measurement one has to sum coherently over contributions from different intermediate states (p,a) and incoherently over those from different initial and final states (ju.j) and (p,/). This yields 

Expanding the ejected electron wave function k) into partial waves and summing over the magnetic quantum numbers m, of the ejected electron s spin— which is generally not observed—one can write the electron intensity in terms of multipole components of the density matrix  

In this way the angular distribution is determined by three types of quantities pfc, R, and F. The tensor components characterize the excited atom, each yielding a contribution to the electron intensity that is proportional to p)- The quantities R are products of reduced matrix 

In an actual process spin-dependent interactions are often unimportant and therefore the involved quantities depend only on the orbital angular momenta L rather than on J. For a comparison with experimental data one therefore has to consider what the quantities R, p/, and Fj are in such cases. First of all the autoionization process itself can often be regarded as spin independent, at least in those cases in which the normal autoionization due to Coulomb interaction between the electrons is allowed. In that case the reduced matrix elements and thus the coefiicients R can be expressed in terms of L-dependent matrix elements by  

If the autoionization process only depends on L rather than on / the excited atom has to be characterized by the L-dependent tensor components PkqiL) in place of the /-dependent ones. This makes sense anyway, since the excitation processes in heavy particle collisions are mostly spin independent. Therefore the spin can be regarded as spectator during the excitation process, and at time t = 0, when the excitation takes place, the density matrix factorizes into an L-dependent part and a spin-dependent one which is diagonal. The pkq L,T = 0) therefore characterize the excitation process and are the quantities that one would like to know in order to obtain information about excitation mechanisms. 

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