Dynamical Invariant and Exact Mechanical Analyses for the Caldirola–Kanai Model of Dissipative Three Coupled Oscillators

We study the dynamical invariant for dissipative three coupled oscillators mainly from the quantum mechanical point of view. It is known that there are many advantages of the invariant quantity in elucidating mechanical properties of the system. We use such a property of the invariant operator in quantizing the system in this work. To this end, we first transform the invariant operator to a simple one by using a unitary operator in order that we can easily manage it. The invariant operator is further simplified through its diagonalization via three-dimensional rotations parameterized by three Euler angles. The coupling terms in the quantum invariant are eventually eliminated thanks to such a diagonalization. As a consequence, transformed quantum invariant is represented in terms of three independent simple harmonic oscillators which have unit masses. Starting from the wave functions in the transformed system, we have derived the full wave functions in the original system with the help of the unitary operators.     

Propagation of sound waves through a spatially homogeneous but smoothly time-dependent medium

The propagation of sound through a spatially homogeneous but non-stationary medium is investigated within the framework of fluid dynamics. For a non-vortical fluid, especially, a generalized wave equation is derived for the (scalar) potential of the fluid velocity distribution in dependence of the equilibrium mass density of the fluid and the sound wave velocity. A solution of this equation for a finite   transition period τ$\tau$ is determined in terms of the hypergeometric function for a phenomenologically realistic, sigmoidal change of the mass density and sound wave velocity. Using this solution, it is shown that the energy flux of the sound wave is not conserved but increases always   for the propagation through a non-stationary medium, independent of whether the equilibrium mass density is increased or decreased. It is found, moreover, that this amplification of the transmitted wave arises from an energy exchange with the medium and that its flux is equal to the (total) flux of the incident and the reflected wave. An interpretation of the reflected wave as a propagation of sound backward in time is given in close analogy to Feynman and Stueckelberg for the propagation of anti-particles. The reflection and transmission coefficients of sound propagating through a non-stationary medium is analyzed in more detail for hypersonic waves with transition periods τ$\tau$ between 15 and 200 ps as well as the transformation of infrasound waves in non-stationary oceans.     

High resolution measurements of energy spectra of protons scattered from silicon crystals in the case of planar channelling

The energy spectrum of backscattered protons in the case of incidence along several planar directions shows a fine structure near the high-energy edge. This structure, an oscillatory dependence of the probability of backscattering vs. depth in the crystal, offers a possibility to study the proton trajectory in the lattice and also to obtain the stopping power of protons near planes in silicon.

Application of a simple model for the proton trajectory yields a stopping power near the planes 4 to 5 times higher than for random incidence. These effects have been observed using primary energies in the range 40–140 keV and for incidence along (110). (111), (100) and (112) planes.     

Transformation optics and cloaking

Invisibility, a long sought-for speculation in science fiction, has been turned into reality in the laboratory through the use of a theoretical technique called Transformation Optics. The principles of transformation optics show that any desired smooth deformation of the electromagnetic field can be implemented exactly by an appropriately engineered metamaterial. All demonstrations of cloaking to date have had limitations, however, reflecting our technological inability to implement the transformation optics algorithm exactly. However, the scientific principles leading to perfect invisibility are now established, and practical improvements on the initial designs are now occurring very rapidly. Most recently, researchers have re-examined transformation optics to include time as well as space, describing and then implementing the concept of a cloak that hides events, a conceptual breakout that promises many new applications. This review describes the general ideas underlying transformation optics, and how the various types of cloak based on these ideas have been implemented practically to date.     

Dynamic mechanical and morphological studies of homopolymer/block copolymer blends

A triblock copolymer of styrene-butadiene-styrene (SBS) was blended with a high and a low molecular weight polybutadiene [designated as PB(H) and PB(L), respectively]. Electron micrographs of these blends show that PB(L) was solubilized into the butadiene domains of the SBS, while PB(H) was present in a separate phase. Dynamic mechanical data of the SBS&PB(L) blends indicate the presence of an intermediate loss peak between those of the glass transitions of the styrene and butadiene blocks in SBS, which can be attributed to the slippage of untrapped entanglements of PB(L) chains. Similar data for blends containing PB(H) also show an intermediate loss peak, which is, however, due to crystallization and melting of the polycisbutadiene chains. The peak due to the primary glass transition of the butadiene phase was shifted to lower temperatures upon incorporation of PB(L) as a result of plasticization effect of PB(L). The same transition for PB(H) blends is split into a doublet because of the presence of the homopolymer in a separate phase. Dynamic mechanical data for cross-linked blends were also taken for comparison.     

Phase transformation behaviour in continuously cooled Zr65Al7.5Ni10Cu17.5− x Pd x (x = 0 − 17.5) glass-forming alloys and consequences for structure and property control

Continuous cooling transformation (CCT) diagrams were successfully constructed for Zr₆₅Al_7.5Ni₁₀Cu_{17.5? x} Pd _x (x?=?0???17.5) glass-forming alloys, comparing phase-transformation features in the alloy system to composition. While a low-Pd alloy (x?=?5) showed a single transformation curve, corresponding to the formation of a crystalline phase on the high-temperature side of the undercooled-liquid region, for a given time-scale, a high-Pd alloy (x?=?17.5) revealed an additional curve, corresponding to quasicrystalline phase formation on the lower temperature side. The result provides a clue to the structural and property control on the alloy system. Glassy specimens of the same size but with different intrinsic structure, evaluated by structural relaxation during continuous heating, could be fabricated for the low-Pd alloy (x?=?5). Plasticity was found to increase proportionally with the relaxation enthalpy. On the other hand, the critical size for glass formation could be improved considerably from 5 to 7 mm in diameter for the high-Pd alloy (x?=?17.5).     

Layer-stacking irregularities in C36-type Nb–Cr and Ti–Cr Laves phases and their relation with polytypic phase transformations

Specific layer-stacking irregularities have been identified in C36 (4H) Nb–Cr and Ti–Cr Laves phases on the basis of X-ray diffraction line-profile analysis and high-resolution transmission electron microscopy. Domain boundaries and transformation errors within domains could be distinguished. The layer-stacking irregularities in both C36-NbCr₂ and C36-TiCr₂ can be associated with a preceding C14 (2H) → C36 (4H) phase transformation carried out by glide of mobile synchro-Shockley partial dislocation dipoles in an ordered fashion. The stacking irregularities observed can be interpreted as deviations from such perfect “ordered glide”. The interpretation is supported by the observation that, in the case of C36-NbCo₂, where no preceding C14 → C36 transformation occurs, different layer-stacking irregularities are observed.     

BINARY DARBOUX TRANSFORMATION AND QUASIDETERMINANT SOLUTIONS OF THE CHIRAL FIELD

The standard binary Darboux transformation is composed and is used to obtain exact multisoliton solutions of the chiral field model in two dimensions. The solutions are expressed in terms of quasideterminants. It has been shown that the standard binary Darboux transformation is equivalent to the elementary binary Darboux transformation.     

THE TRANSFORMATION BETWEEN THE AKNS HIERARCHY AND THE KN HIERARCHY WITH SELF-CONSISTENT SOURCES

It is shown that the AKNS hierarchy with self-consistent sources can transform to KN hierarchy with self-consistent sources through a transformation operator and gauge transformation. Besides, there exists transformation in their conservation laws and Hamiltonian structures.     

Dielectric and structural relaxation phenomena in Na0.5Bi0.5TiO3 single crystal

Abstract

Dielectric permittivity studies of Na_0.5Bi_0.5TiO₃ single crystals in a broad range of frequency up to 10 MHz and temperature 300—823 K are reported. In this temperature range dielectric dispersion below 1 MHz has been found. The obtained data were fitted to the Cole-Cole relation. The mean relaxation time τ is strongly temperature dependent (0.04 ? 2.6 × 10^?5 s). A remarkable hysteresis effect in the values of τ on cooling and heating took place. The Δε(T) dependence (the maximal value of Δε ～ 400) is similar to the global ε′(T) response at low frequency. An isothermal structural transformation in Na_0.5Bi_0.5TiO₃ was observed by X-ray measurements. The order of the time in which the transformation takes place (～300 minutes) corresponds to the time in which the strongest time evolution of electric permittivity and time changes of dielectric dispersion were detected.