Free-electron final-state approximation

The two 3-functions in Eq. (3.2.2.19), providing a Idnematical description of the photoemission process, represent a very stringent condition under which photoemission intensity can be observed, especially when many-body interactions are neglected, and + E - Ei = eB(kj) represents the single-particle band structure of the solid according to Koopman s theorem. The convenient picture of direct transitions arises [19] in a band structure plotted in the reduced zone scheme, the conservation of wave vectors up to a reciprocal lattice vector means that an electron is excited vertically, that is, at constant crystal momentum, from an initial-state band to a final-state band eB(k/) —> C (kj) with energy levels separated by the photon energy hv. In the free-electron final-state approximation, the upper level is... 

The free-electron final state may seem like a crude approximation for a photoelectron propagating through a periodic potential, but it has nevertheless proven to be quite successful, in the traditional three-step model [7] of photoemission, the process is described as a sequence of the following three steps (i) photoexcitation of an electron into an unoccupied state within the sample band structure, (ii) prop-... 

Application of the formalism of the impulse approximation to the double differential cross section in terms of the dielectric response (Equation 12), that is, using free-electron-like final states E = p+q 2/2m in the calculation ofU(p+q, E +7ko)... 

For X-rays well above the absorption edge (AE > 30 eV), the final electron is unbound, that is, has a continuous distribution of allowed energies, and the density of allowed states p(E ) at the final-state energy Ef is then a smooth function which may be approximated by the density of states of a free electron of energy... 

In the case that both initial and final states are unbound, an electron can absorb a photon of frequency i/g to move to a state with energy given by hvs = l/2me(f2 — v3). The free-free absorption coefficient for one atom and one free electron per unit volume can be written in the hydrogenic approximation as... 

The free-electron approximation described in Chapter 15 is so successful that it is natural to expect that any effects of the pseudopotential can be treated as small perturbations, and this turns out to be true for the simple metals. This is only possible, however, if it is the pseudopotential, not the true potential, which is treated as the perturbation. If we were to start with a free-electron gas and slowly introduce the true potential, states of negative energy would occur, becoming finally the tightly bound core states these are drastic modifications of the electron gas. If, however, we start with the valence-electron gas and introduce the pseudopotential, the core states are already there, and full, and the effects of the pseudopotential are small, as would be suggested by the small magnitude of the empty-core pseudopotential shown in Fig. 15-3. 

In the electric dipole approximation, Hx reduces to the electric dipole radiation operator, in a tensorial form, a pair of photon creation and annihilation operators (their order depends on the type of the radiative process), and a pair of creation and annihilation operators that change one electron initial state of a matter into the final state due absorp-tion/annihilation of a photon. The atomic part of the first-order term, when the free ionic system approximation and the electric dipole approximation are adopted, is defined by the matrix element. 

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