Wannier theory

Ashley, Moxom and Laricchia (1996) measured the positron impact-ionization cross section in helium and found that its energy dependence up to 10 eV beyond the threshold was quite accurately represented by a power law, as in equation (5.8), but with the exponent having the value 2.27 rather than Klar s value of 2.651. This discrepancy prompted Ihra et al. (1997) to extend the Wannier theory to energies slightly above the ionization threshold using hidden crossing theory. They derived a modified threshold law of the form... 

Wannier (1953) who treated the problem classically. The Wannier theory was confirmed by Peterkop (1971) and Ran (1971) using semiclassical methods. This earlier work, as well as more recent work on near-threshold ionisation, have all emphasised the role of radial and angular correlations in the final two-electron state. In the Wannier theory, and its semiclassical extensions, the details of the collision process and the structure of the target play no role, since only the asymptotic region is considered. 

The Wannier theory for threshold ionization has been extended to positron impact by Klar [4.15, 4.16] and Grujic [4.17]. Within their model, the electron and positron escape along classical orbits, both in the same direction and with the electron trailing the positron. For ionization of neutral particles (Z = 1), they find —2.15 and, in marked contrast to the electron-impact case, that... 

Wannier, G. H. (1959). Solid State Theory . Cambridge University Press, Cambridge... 

In the classical treatment of near-threshold electron impact ionization developed by Wannier (1953), the repulsion between the two electrons causes them to emerge with very similar energies but in opposite directions along the so-called Wannier ridge. This effect is depicted in Figure 5.6, where it is contrasted with the case for positron impact described below. According to this theory the energy dependence of the ionization cross section for electron impact is predicted to be... 

STATISTICAL PHYSICS, Gregory H. Wannier. Classic text combines thermodynamics, statistical mechanics and kinetic theory in one unified presentation of thermal physics. Problems with solutions. Bibliography. 532pp. 55 x 85. 

The spectra may also be described in the language of solid state theory. The atomic excited states are the same as the excitons that were described, for semiconductors, at the close of Chapter 6. They are electrons in the conduction band that are bound to the valence-band hole thus they form an excitation that cannot carry current. The difference between atomic excited states and excitons is merely that of different extremes the weakly bound exciton found in the semiconductor is frequently called a Mott-Wannier exciton-, the tightly bound cxciton found in the inert-gas solid is called a Frenkel exciton. The important point is that thecxcitonic absorption that is so prominent in the spectra for inert-gas solids does not produce free carriers and therefore it docs not give a measure of the band gap but of a smaller energy. Values for the exciton energy are given in Table 12-4. 

Besides the mentioned aperiodicity problem the treatment of correlation in the ground state of a polymer presents the most formidable problem. If one has a polymer with completely filled valence and conduction bands, one can Fourier transform the delocalized Bloch orbitals into localized Wannier functions and use these (instead of the MO-s of the polymer units) for a quantum chemical treatment of the short range correlation in a subunit taking only excitations in the subunit or between the reference unit and a few neighbouring units. With the aid of the Wannier functions then one can perform a Moeller-Plesset perturbation theory (PX), or for instance, a coupled electron pair approximation (CEPA) (1 ), or a coupled cluster expansion (19) calculation. The long range correlation then can be approximated with the help of the already mentioned electronic polaron model (11). 

Wannier, G. H. Elements of solid state theory. Cambridge Cambridge Uni-... 

The discussion of co-operative phenomena given here is based on the simple Bragg-Williams model. The modern theories of order-disorder changes have undergone rapid development recently. The situation in 1938 is admirably reviewed by Nix and Shockley 1 more recent summaries of both theoretical and experimental developments will be found in papers by Lipson and Wannier. See also Guggenheim,Rush-brooke, and footnote p. 305. 

The foundation of excitons theory was formulated by Frenkel, Peierls and Wannier (l)-(5) more than 70 years ago. After that time the theory has been enriched by many new aspects. The theory has also been exposed to continuous experimental verification, which has confirmed the role of excitons in such processes as absorption of light, luminescence and energy transfer, photochemical processes, etc. Before we examine the experiments, which illustrate the presence and the role of excitons in crystals, we will briefly describe the basic models of excitons, which are mostly used in the interpretation of experimental results. 

The theory of Wannier-Mott excitons and, in particular, the limits of validity of eqn (1.3), can be found, for instance, in the textbook by Knox (8) and the review article by Haken (19). 

A similar situation appears in the theory of Wannier-Mott excitons (see, for example, (17), 4), where the interaction between the electron and the hole is given by —e2/er only at large enough distances. 

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