Phenomenological operators

We shall consider only the leading part of the spin-orbit operator assumed in the phenomenological fonn... 

The first theoretical handling of the weak R-T combined with the spin-orbit coupling was carried out by Pople [71]. It represents a generalization of the perturbative approaches by Renner and PL-H. The basis functions are assumed as products of (42) with the eigenfunctions of the spin operator conesponding to values E = 1/2. The spin-orbit contribution to the model Hamiltonian was taken in the phenomenological form (16). It was assumed that both interactions are small compared to the bending vibrational frequency and that both the... 

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... 

When polymer melts are deformed, polymer molecules not only slide past each other, but they also tend to uncoil—or at least they are deformed from their random coiled-up configuration. On release of the deforming stresses these molecules tend to revert to random coiled-up forms. Since molecular entanglements cause the molecules to act in a co-operative manner some recovery of shape corresponding to the re-coiling occurs. In phenomenological terms we say that the melt shows elasticity. 

The model is able to predict the influence of mixing on particle properties and kinetic rates on different scales for a continuously operated reactor and a semibatch reactor with different types of impellers and under a wide range of operational conditions. From laboratory-scale experiments, the precipitation kinetics for nucleation, growth, agglomeration and disruption have to be determined (Zauner and Jones, 2000a). The fluid dynamic parameters, i.e. the local specific energy dissipation around the feed point, can be obtained either from CFD or from FDA measurements. In the compartmental SFM, the population balance is solved and the particle properties of the final product are predicted. As the model contains only physical and no phenomenological parameters, it can be used for scale-up. 

The stochastic problem is to describe properly the time evolution of the Heisenberg operator d(t) averaged over all the realizations of collisional process in the interval (0,t). The averaging, performed in the impact theory, results in the phenomenological kinetic equation [170, 158]... 

Previous theoretical kinetic treatments of the formation of secondary, tertiary and higher order ions in the ionization chamber of a conventional mass spectrometer operating at high pressure, have used either a steady state treatment (2, 24) or an ion-beam approach (43). These theories are essentially phenomenological, and they make no clear assumptions about the nature of the reactive collision. The model outlined below is a microscopic one, making definite assumptions about the kinematics of the reactive collision. If the rate constants of the reactions are fixed, the nature of these assumptions definitely affects the amount of reaction occurring. 

Whatever mechanisms operate to cause deformation-hardening, it is phenomenologically the most general determinant of hardness for metals. 

---