Potential scattering model

In potential scattering models the basic "mathematical" reason for exponential decay is a complex pole in the fourth quadrant of the momentum complex plane (second Riemann sheet of the energy plane), which, through its exponentially decaying residue, dominates the dynamics for some time. A simple analytical example of the deviation from exponentiality follows from the integral expression for the survival amplitude. 

In the cellular multiple scattering model , finite clusters of atoms are subjected to condensed matter boundary conditions in such a manner that a continuous spectrum is allowed. They are therefore not molecular calculations. An X type of exchange was used to create a local potential and different potentials for up and down spin-states could be constructed. For uranium pnictides and chalcogenides compounds the clusters were of 8 atoms (4 metal, 4 non-metal). The local density of states was calculated directly from the imaginary part of the Green function. The major features of the results are ... 

I would like to ask Prof. J. Troe whether he could discuss some typical situations where the SAC approximation may fail. For example, consider the F + HBr — FHBr — HF(u) + Br reaction with energy E just above the potential barrier V41. In this situation, the adiabatic channels in the transition state ( ) should be populated only in the vibrational ground state, and they should, therefore, yield products HF(u = 0) + Br, according to the assumption of adiabatic channels. This is in contrast with population inversion in the experimental results that is, the preferred product channels are HF(i/) + Br, where v = 3, 4 [1] see also the quantum scattering model simulations in Ref. [2]. The fact that dynamics cannot be rigorously adiabatic (as in the most literal interpretation of SAC) has been discussed by Green et al. [3], and the most recent results (for the case of ketene) are in Ref. 4. 

Here, we shall assume that we are dealing with d-wave superconductors. Since for unconventional superconductors there is no qualitative difference between these two types of scattering, we shall confine ourselves to the study of potential scattering. Even with this limitation there is a wide range of theoretical predictions as regards rc-suppression, density of states, transport properties etc, depending on the way disorder is modelled and depending on the analytical and numerical approximations employed to derive experimentally verifiable conclusions. [Atkinson et al., 2000]... 

It has been emphasized by Bard et al. that there may be exceptions to the model derived above, insofar as Fermi level pinning by surface states may occur in a similar fashion to that at semiconductor-metal junctions [33]. Such an effect would lead to an unpinning of bands at the interface. There are some examples in the literature, such as FeS2 in aqueous solutions [34, 35] and Si in methanol [36] for which an unpinning of bands has been reported. In some cases, such as TiO2, experimental values of flatband potentials scatter considerably. This is mostly due to changes in surface chemistry and doping profiles. 

The large branching ratios predicted by the multiple scattering model is certainly an Indication of the poor representation of the R dependence of the molecular ion potential in this model. It is important to note that substantial non-Franck-Condon behavior of this branching ratio occurs at photon energies well above the peak position of the resonance (27). 

Natoli CR (1983) Near-edge absorption stracture in the framework of the multiple scattering model. Potential resonance on barrier effects. In A Bianconi, L Incoccia, S Stipcich (eds) EXAFS and Nearedge Stracture. Springer Ser Chem Phys 27 43-47... 

Recently the same problem has been reanalyzed by Dicus et al. [86], and indeed they confirmed that the survival probability deviates from exponential at long times. This model and its variants have been applied to study the effect of a distant detector (by adding an absorptive potential) [87], anomalous decay from a flat initial state [44], resonant state expansions [3], initial state reconstruction (ISR) [58], or the relevance of the non-Hermitian Hamiltonian concept (associated with a projector formalism for internal and external regions of space) in potential scattering [88]. In Ref. [88] the model was extended to a chain of delta functions to study overlapping resonances. 

Edwards (1971a, b) has discussed the phenomenon of the localization of electrons in disordered systems for the case of a simple model which is conveniently expressed in terms of Feynman-like path integrals (Feynman (1965)). The model is that of an electron in a system of very dense, random, weak scatterers. If p is the density of the scatterers and v (r) is the scattering potential, the model is employed in the limit... 

For the non-symmetrical Anderson model with C/ + 2ef 0, one has in the Coqblin Schrieffer and Kondo Hamiltonians in addition to the exchange a potential scattering term. The Kondo resonance is no longer at the Fermi energy ep = 0 but is shifted. This shift leads to a smaller resistivity p(T), the maximum of which, however, is still at T = 0. The potential scattering has a dramatic effect on the thermoelectric power, which vanishes in the symmetric case C/ + 2ef = 0, but has a huge peak near 7k for C/ 4- 2ef 0. 

An alternative development in the theory rather concentrated on the Austin-Heine-Sham type of pseudopotential, to construct a model pseudopotential, which is rather flat, but which gives the same electron scattering as the true potential . This model potential has been parametrized, and tables of the parameters have been published for many elements of the... 

To calculate the amplitude connected with T violation in the process of potential scattering, we will use the square-well model approximation. Then, according to eq. (4.35), the difference of matrix... 

In spite of the apparent consistency of the experimental results with calculations based on the t-matrix theory, the question of the validity of the application of this theory to strong-scattering liquid metals and alloys remains. Harris et al. (1978) have approached this question theoretically by means of a coherent-potential approximation displaying a Debye-Waller factor type of temperature dependence. They succeed by means of this strong-scattering model to reproduce, in semiquantitative fashion, the negative dp, /dr for liquid SnCe. 

Another statistical mechanical approach makes use of the radial distribution function g(r), which gives the probability of finding a molecule at a distance r from a given one. This function may be obtained experimentally from x-ray or neutron scattering on a liquid or from computer simulation or statistical mechanical theories for model potential energies [56]. Kirkwood and Buff [38] showed that for a given potential function, U(r)... 

There are many large molecules whose mteractions we have little hope of detemiining in detail. In these cases we turn to models based on simple mathematical representations of the interaction potential with empirically detemiined parameters. Even for smaller molecules where a detailed interaction potential has been obtained by an ab initio calculation or by a numerical inversion of experimental data, it is usefid to fit the calculated points to a functional fomi which then serves as a computationally inexpensive interpolation and extrapolation tool for use in fiirtlier work such as molecular simulation studies or predictive scattering computations. There are a very large number of such models in use, and only a small sample is considered here. The most frequently used simple spherical models are described in section Al.5.5.1 and some of the more common elaborate models are discussed in section A 1.5.5.2. section Al.5.5.3 and section Al.5.5.4. 

Infomiation about interatomic potentials comes from scattering experiments as well as from model potentials fitted to the themiodynamic and transport properties of the system. We will confine our discussion mainly to... 

---