Mössbauer study of a novel series of ternary rare-earth iron-rich intermetallics: ND(Fe1-xTi x )6+y(x ∼ 0.1 and 0≤y≤5)

A study of the Nd-Fe-Ti phases at the Fe-rich end of the phase diagram has been carried out using⁵⁷Fe Mössbauer spectroscopy, scanning electron microscopy (SEM) and energy dispersive X-ray analysis (EDAX). In particular,⁵⁷Fe Mössbauer spectroscopy has been used to characterize the phases present in these samples. In addition to the two well-known 217 and 112 phases, twonovel phases have been identified, with compositions denoted by3–29 and9–94, which are closely linked to the orthorhombic ScFe₆Ga₆-type and the hexagonal TbCu₇-type structures, respectively. The possible crystal structures of these new phases will be discussed in terms of the dumbbell substitution sequence in the CaCu₅ structure.     

Chaotic behavior of channeling particles

Channeling describes the collimated motion of energetic charged particles along the lattice plane or axis in a crystal. The energetic particles are steered through the channels formed by strings of atomic constituents in the lattice. In the case of planar channeling, the motion of a charged particle between the atomic planes can be periodic or quasiperiodic, such as a simple oscillatory motion in the transverse direction. In practice, however, the periodic motion of the channeling particles can be accompanied by an irregular, chaotic behavior. In this paper, the Moliere potential, which is considered as a good analytical approximation for the interaction of channeling particles with the rows of atoms in the lattice, is used to simulate the channeling behavior of positively charged particles in a tungsten (100) crystal plane. By appropriate selection of channeling parameters, such as the projectile energy E(0) and incident angle psi(0), the transition of channeling particles from regular to chaotic motion is demonstrated. It is argued that the fine structures that appear in the angular scan channeling experiments are due to the particles' chaotic motion.     

Origin and manifestation of the anharmonic potential felt by an ion-cloud in an actual Paul trap

The anharmonic potential felt by a single-species ions confined in a rf quadrupole trap which results from a non-ideal trap configuration and the charge distribution of the ion cloud is studied. The rf resonance-absorption spectra are explained by a Duffing oscillator and a representation of the line-shape parameter is derived. For > 0.77, the electric signals will exhibit hysteresis. The relation with the anharmonic potential is discussed.     

Finite-element analysis of planar dielectric waveguide discontinuities

The finite element method is combined with the mode-matching method and the multi-mode network method to analyze the scattering and radiation characteristics of a class of planar dielectric waveguide discontinuities. Unlike the conventional method to treat the radiation as a source-field problem, in the present approach, the dispersion characteristics of dielectric guided-wave structures are calculated first, and then the radiation problem is transferred to the propagation problem of a series of surface-waves and space waves from the viewpoint of scattering, thus the analysis is tremendously simplified.     

Foxby duality and Gorenstein injective and projective modules

In 1966, Auslander introduced the notion of the -dimension of a finitely generated module over a Cohen-Macaulay noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander and Bridger (Mem. Amer. Math. Soc., vol. 94, 1969)). Enochs and Jenda attempted to dualize the notion of -dimensions. It seemed appropriate to call the modules with -dimension 0 Gorenstein projective, so the basic problem was to define Gorenstein injective modules. These were defined in Math. Z. 220 (1995), 611--633 and were shown to have properties predicted by Auslander's results. The way we define Gorenstein injective modules can be dualized, and so we can define Gorenstein projective modules (i.e. modules of -dimension 0) whether the modules are finitely generated or not. The investigation of these modules and also Gorenstein flat modules was continued by Enochs, Jenda, Xu and Torrecillas. However, to get good results it was necessary to take the base ring Gorenstein. H.-B. Foxby introduced a duality between two full subcategories in the category of modules over a local Cohen-Macaulay ring admitting a dualizing module. He proved that the finitely generated modules in one category are precisely those of finite -dimension. We extend this result to modules which are not necessarily finitely generated and also prove the dual result, i.e. we characterize the modules in the other class defined by Foxby. The basic result of this paper is that the two classes involved in Foxby's duality coincide with the classes of those modules having finite Gorenstein projective and those having finite Gorenstein injective dimensions. We note that this duality then allows us to extend many of our results to the original Auslander setting.

     

Divisors on Generic Complete Intersections in Projective Space

Let be a generic complete intersection of hypersurfaces of degree in -dimensional projective space. We study the question when a divisor on is nonrational or of general type, and give an alternative proof of a result of Ein. We also give some improvement of Ein's result in the case .