Efficiency tensors

The interpretation of the density (and efficiency) matrices is rather obvious the diagonal elements represent the probabilities of finding a certain magnetic quantum number of the ensemble, and the non-diagonal elements contain the phase information of the system for the corresponding different magnetic quantum numbers. However, in calculations with these matrices one still has to work out the cumbersome summations over these quantum numbers. It is more convenient to replace the density (and efficiency) matrices by statistical and efficiency tensors, also called state multipoles. 

The transition rate P then follows from the statistical and efficiency tensors of the three-particle system in the final state - ion (Jf), Auger electron (J2), and photoelectron (j j) - as... 

In this expression the well-defined angular momenta are indicated in the respective tensors, and only for the photoelectron described by jk do different partial waves exist. The three efficiency tensors of the final system are easy to write down ... 

With these efficiency tensors and the corresponding statistical tensors the transition rate P then follows as... 

Taking into account the conditions fcf = 0 and k = 0 from the efficiency tensor one can first couple p00(Jt) with pk2K2(j2,j 2) to form pilKI(J,). The attached Clebsch-Gordan coefficient brings in the condition k, = k2 and k, = k2, and allows the replacement of the coefficients fc, and k, which link the first and second steps via the intermediate state. (Only the coefficients ku k1 and k2, k2 which refer to the first and second steps, respectively, are retained.)... 

Note the difference in the transformation properties of statistical (efficiency) tensors as compared to angular functions (compare equ. (8.82) with equs. (8.80)). 

---