Phenomenological potential

Experimental data on the deuterium-hydrogen isotope shift (see Table 12.4 below) have an accuracy of about 0.1 kHz and, hence, a more accurate theoretical result for the polarizability contribution is required. In order to obtain such a result it is necessary to go beyond the zero range approximation, and take the deuteron structure into account in more detail. Fortunately, there exist a number of phenomenological potentials which describe the properties of the deuteron in all details. Some calculations with realistic proton-neutron potentials were performed [40, 41, 42, 43]. The most precise results were obtained in [43]... 

Let us conclude with a short remark on the concentration dependence of the phenomenological potentials p, and, in particular, when point defects are involved. It is common and convenient to split the chemical potentials into two parts 1) r (P, T), which does not depend on the composition variables TV,-, and 2) the composition dependent term R T- In , which for ideal solutions (a, = N,) is simply R T- In TV,. For non-ideal solutions, one introduces the excess term R T- In ft = R T-In a —R T-Iri TV,-. Let us write In f, as a power series of the form... 

For a phenomenological description of the magnetic properties of the crystal under consideration, it is necessary at first to carry out a symmetry analysis to define the magnetic modes of the irreducible representations, and to generate from them invariants which allow us to obtain a phenomenological potential. 

The combination of the discrete variable with the finite element method allows not only to compute atomic data for the hydrogen atom. Atomic data for alkali-metal atoms and alkaU-like ions can be obtained by a suitable phenomenological potential, which mimics the multielectron core. The basic idea of model potentials is to represent the influence between the non-hydrogenic multielectron core and the valence electron by a semi-empirical extension to the Coulomb term, which results in an analytical potential function. The influence of the non-hydrogenic core on the outer electron is represented by an exponential extension to the Coulomb term [18] ... 

Thus, by comparing the experimental NN scattering data with the predictions by the Paris potential, meson-theory is not put to a test. This comparison tests the phenomenological potential which is, indeed, consistent with the data. The satisfactory x of the fit of the data by the Paris potential (cf. Table 1) cannot be used as a proof that meson-theory is correct for the low-energy NN... 

In the 1960s, the rapidly growing experimental data made it possible to construct phenomenological potentials that could describe various effects of NN interaction. Such are the Hamada-Johnston potential with hard core (Hamada and Johnston 1962), the Yale potential with hard core (Lassila et al. 1962), the nonlocal Tabakin potential (Tabakin 1964), and the Reid potential with soft core (Reid 1968). 

Meson-field theoretical NN potentials give a reasonable guidance for the construction of phenomenological potentials, too. At the same time, both the phenomenological and the realistic, meson-field theoretical potentials are very complicated, and their application in nuclear structure calculations is rather difficult. 

The McMillan theory has been further refined by several authors [62-64] to improve the quantitative agreement with experiment. However, the basic structure of the theory remains the same. This theory presents another example of a successful application of a simple mean-field approach. On the other hand, there are several limitations of the McMillan theory that cannot be ignored. Firstly, the theory is based on the semi phenomenological potential that does not allow determination of the smectic period in a self-consistent way. Secondly, the model poten-... 

The adiabatic approximation makes it possible to work with an effective potential R) given by (1.8) if we are dealing with the motion of the ions. The evaluation of Eg(R) on the basis of (1.5), however, is a very difficult problem in general. A more practical approach is to assume some phenomenological potential for 4>(R) which contains only a few parameters. For the diatomic molecule, for example, a good choice is the Morse potential given by (5.1). Phenomenological potentials for atoms in solids are discussed in Chap.4. 

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

Here the nucleation barrier AO is the excess thermodynamic potential needed to form the critical embryo within the uniform metastable state, while the prefactor Jq is determined by the kinetic characteristics for the embryo diffusion in the space of its size a. Expressions for both AO and Jo given by Zeldovich include a number of phenomenological parameters. 

It is possible that the stationary-state situations leading to an active ion transport occur only in localized regions of the membrane, i.e., at ATPase molecule units with diameters of about 50 A and a length of 80 A. The vectorial ion currents at locations with a mixed potential and special equipotential lines would appear phenomenologically like ionic channels. If the membrane area where the passive diffusion occurs is large, it may determine the rest potential of the whole cell. 

According to our analytical results on the solid-state redox reaction of LiNi02 based on the phenomenological expression for solid-state redox potentials of insertion electrodes [23], the reaction consists of three redox systems characterized by potentials of 4.23, 3.93, and 3.63V with re-... 

Within the broad spectrum of systems described, only few can be considered well understood. Unfortunately, many past investigations were superficial and more involved studies have been avoided because of the inherent complex phenomenology. This has been harmful, and misleading conclusions have been reached about the real potential of furan monomers. It is hoped that this review has shown the fallacy of this approach and will therefore stimulate further skilled work in this field. 

Electrochemical promotion or NEMCA is the main concept discussed in this book whereby application of a small current (1-104 pA/cm2) or potential ( 2 V) to a catalyst, also serving as an electrode (electrocatalyst) in a solid electrolyte cell, enhances its catalytic performance. The phenomenology, origin and potential practical applications of electrochemical promotion, as well as its similarities and differences with classical promotion and metal-support interactions, is the main subject of this book. 

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