Auger electron, wavefunction

The selection rules have to be fulfilled for the transition from the ls2s22p6 2Se initial state to the possible final states. Thus, the final state contains one of the final ionic states listed in Table 3.2 and the wavefunction for the emitted Auger electron in its partial wave expansion (see equ. (7.28b)). Due to the selection rules, only a few t values from the partial wave expansion contribute. In the present case there is only one possibility which will be characterized by si. Therefore, one... 

Because the Coulomb operator is a two-particle operator, the transition matrix element Mn is non-zero only for cases in which at most two orbitals differ in the initial- and final-state wavefunctions. For normal Auger transitions it will turn out that these are just the electron orbitals used to characterize the Auger transition, including the Auger electron itself. To show this for the K-LfLf 0 transition one starts with the matrix element... 

This result shows that the original matrix element containing the orbitals of all electrons factorizes into a two-electron Coulomb matrix element for the active electrons and an overlap matrix element for the passive electrons. Within the frozen atomic structure approximation, the overlap factors yield unity because the same orbitals are used for the passive electrons in the initial and final states. Considering now the Coulomb matrix element, one uses the fact that the Coulomb operator does not act on the spin. Therefore, the ms value in the wavefunction of the Auger electron is fixed, and one treats the matrix element Mn as... 

The wavefunctions hydrogenic wavefunctions. In the present context such wavefunctions for discrete orbitals and for the continuum describing the emission of a photo- or Auger electron are of interest. 

The summations over Mf and ms are needed because no observation is made with respect to these final-state quantum numbers. ka and Kph are the wavenumber vectors of the Auger electron and the photoelectron, the minus sign indicates the correct asymptotic boundary condition for the wavefunctions, Vc is the Coulomb interaction between the electrons causing the Auger transition, and is the dipole operator causing the photoionization process. 

NCs contain approximately 100-10,000 atoms. Because of the strong spatial confinement of electronic wavefunctions and reduced electronic screening, the effects of carrier-carrier Coulomb interactions are greatly enhanced in NCs compared with those in bulk materials. These interactions open a highly efficient decay channel via Auger recombination and just such a strong carrier-carrier interaction in NCs is responsible for carrier multiplication (61)-(63). 

Nevertheless, the one-electron approach does have its deHciencies, and we believe that a major theoretical effort must now be devoted to improving on it. This is not only in order to obtain better quantitative results but, perhaps more importantly, to develop a framework which can encompass all types of charge-transfer processes, including Auger and quasi-resonant ones. To do so is likely to require the use of many-electron multi-configurational wavefunctions. There have been some attempts along these lines and we have indicated, in detail, how such a theory might be developed. The few many-electron calculations which have been made do differ qualitatively from the one-electron results for the same systems and, clearly, further calculations on other systems are required. 

The Is and 2s orbitals which are affected by neither the photoionization nor the Auger process are omitted for simplicity.) If these wavefunctions are constructed from single-electron orbitals of a common basis set (the frozen atomic structure approximation), the photon operator as a one-particle operator allows a change of only one orbital. Hence, the photon operator induces the change 2p to r in these matrix elements ... 

Correlation, or many-body effects, can be classified according to the many-body factor Xx- If Xx is close to 1, the MO picture, the aufbau principle, a Koopmans theorem and the quasi-particle picture hold. The analysis of the Auger spectrum can then be conducted solely in terms of MO theory. When more than one Xx enters in the wavefunction, we have hole-mixing effects and electronic interference in the transition cross sections, in analogy to the case of photoelectron spectra. When only one Xx is large, but this Xx is present in more than one state, one can then not associate a one-to-one correspondence between MOs (or MO factors in Eq. 3.39) and spectral bands (states). The states in question are thus associated with a breakdown of the MO picture. It could, finally, also be that no Xx is large, in which case we talk about a correlation-state satellite. 

As previously discussed, according to the Fermi golden rule, the intensity of processes like photoemission and Auger decay is expressed by a transition matrix element between initial and final states of the dipole and, respectively, the Coulomb operator. In both cases the final state belongs to the electronic continuum and we already observed that an representation lacks a number of relevant properties of a continuum wavefunction. Nevertheless, it was also observed that the transition moment, due to the presence of the initial bound wavefunction, implies an integration essentially over the molecular space and then even an l representation of the final state may provide information on the transition process. We consider now a numerical technique that allows us to compute the intensity for a transition to the electronic continuum from the results of I calculations that have the advantage, in comparison with the simple atomic one-center model, to supply a correct multicenter description of the continuum orbital. 

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