We have used the dielectric-continuum model to study the surface and interface optical-phonon modes in a four-layer heterostructure with finite thickness. The eige-nvectors, the dispersion relation and the interface charge densities are obtained. All the previous results for a dielectric slab, bilayer, trilayer and infinite four-layer he-terostructures can be reded-iced from our general relation for a finite four-layer he-terostructure. The dispersion relations and their numerical illustration for some inte-resting and practical structures such as a finite step-quantum-well are also given. This work can be regarded as a generalization of the formalism in reference [8].

Using a diagrammatic computation method of the Wigner distribution function for geometric-optical systems, a 2-dimensional coordinate transform system is designed theoretically. The precision of this system may be better than that of the Bryngdahl co-ordnate transformer.

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We demonstrate that quantum size effects play a crucial role in the row structure of metal-induced Si KEYWORDS: models of surface kinetics; quantum effects; surface relaxation and reconstruction; metalsemiconductor interfacesThe reconstruction of the Si(111) surface is a well-known process that has been the subject of numerous experimental and theoretical studies. However, metal-induced reconstruction of Si(111) continues to present theoretical challenges, particularly concerning the coverage-dependent structural arrangements. In such systems a submonolayer of metal atoms is deposited onto the Si(111) surface, resulting in a reconstruction of the Si(111) surface and arrangement of the metal adsorbate into rows of monolayer height.1 -17 Using scanning tunnelling microscopy, these rows are visible and both the height and width of the rows are evident. We introduce here a theoretical model that demonstrates that the quantum size effect plays a crucial role in determining the widths of these rows, and we show that our model provides complete agreement with all recent metal-induced M/Si(111)-3 ð 1 reconstructions (where M D Li, Na, K, Rb, Ag) and Si(111)-4 ð 1-In reconstructions. The quantum size effect has proved to be a crucial element in understanding and modelling the work function, 18 resistivity 19 and stability 20 of metallic thin films, and in the pattern formation of metal-on-metal surfaces. 21 The quantum size effect has proved to be a valuable tool for studying small, confined free-electron systems, and our application to metal-induced reconstruction of Si(111) surfaces represents another success of this approach.

By employing the method of ligand field theory, we calculated the additional potential energies of d electron states of V3X(X = Si, Ga, Ge, Sb, Sn) superconducting compounds, these energies are induced by the D2d) crystal field. We obtained the analytic relation between the. lattice constant and the relative displacement of d sublevels △E⊥, which is the characteristic quantity for the model of d band relative displacement. The numerical values of △E⊥ and the magnetic susceptibilities calculated by using them are also presented.