Abstract

In this talk I will review some recent articles in Kitaev model. Kitaev model was introduced by Alexei Kitaev in 2008. It is a quantum spin model with anisotropic couplings, which can be solved exactly in two dimensions.

Abstract

Abstract

Abstract

Nano-research requires the ability to form and characterize nanostructures. The precision and accuracy required to produce structures in the nanometer scale and to characterize them demand commensurate tools. Although several techniques (e.g. wet and dry etching techniques, electron-beam lithography, etc.) have been employed for the fabrication of nanostructures, ion irradiation has also been demonstrated to be promising tool for the fabrication of uniform and highly ordered quantum dot arrays. Ion irradiation is a promising tool for nanopatterning of materials’ surfaces because of its compatibility and reproducibility with easily controllable growth conditions. It has very high spatial selectivity and is not governed by chemical compatibility. Thus, one can tune physical properties of a material by varying the ion species and energy, the charge state, and the ion fluence. The present talk will deal with wide range of ion energy, starting from keV ions (at IOP, Krakow, Poland) to a few MeV ions (at IOP, Bhubaneswar) to hundreds of MeV ions (at IUAC, New Delhi) to demonstrate irradiation-induced nanoscale patterning of semiconductors (Ge, Si, GaAs, InP, GaSb, InSb, InAs) and alkali halide system (KBr).

Abstract

The number of potential that admit simple analytical expressions for the wave functions as well as the reflection and transmission amplitudes is limited. However, various physical problems in different disciplines of science need to be addressed by the interaction potentials which may not allow simple solution in the format of Schrödinger equation. In the general case, the Schrödinger equation is solved numerically or by approximating the studied potential by solvable ones. But the question arises whether it is possible to obtain simple and easy-to-use expressions for reflection and transmission coefficients for a smoothly varying potential which is not amenable to exact analytical solutions and evaluate the correct eigenvalues which represent bound, quasi-bound, and resonances states etc., generated by the potential. It may be mentioned that the applicability of the popular Runge-kutta method of numerical integration of Schrödinger equation is exposed in the cases of long ranged potentials having nonzero values at large distances, nevertheless, it is possible to obtain an exact solution by imposing constraints on the potentials. We derive the general expressions for reflection and transmission coefficients for a potential constituting n number of rectangular wells and barriers. These expressions are readily used for the estimation of eigenvalues of a smooth potential which is simulated by a multi-step potential. The applicability of this method is demonstrated in this work with success in potentials with different forms including the most versatile Ginocchio potential where the widely used numerical method like Runge-kutta integration algorithm fails to yield the result. Accurate evaluation of eigenvalues free from numerical problem for any form of potentials, whether analytically solvable or not, is the highlight of our multi-step approximation method in the theory of potential scattering. Here we give a brief review of heavy ion scattering and fusion. We see that it is quiet difficult to find a unique potential that can take care of both scattering and fusion processes simultaneously. For the description of elastic scattering and fusion cross section simultaneously, we use the Woods-Saxon form for the internuclear potential with a very small value for the diffuseness parameter ′a′ and the imaginary part used for absorption of flux is taken to be weak. Further, this potential is not made dependent on incident energy. To analyze the data of angular variations of elastic scattering cross sections an analytical recursive formula of the partial wave scattering matrix for the total effective complex potential of nucleus-nucleus collisions is derived. Further, another expression of cross sections for the absorption from arbitrary small intervals is derived. This leads to the explanation of the fusion cross section data at various center-of-mass energies Ec.m. by collecting the absorption contributions in the interior region of the effective potential. By virtue of the weakly absorbing character, nucleus-nucleus potential supports resonance states in different partial wave trajectories. Consequently, occurrence of these resonances is shown to be the physical origin of the observed oscillatory structure in the variation respect to energy of the quantity D(Ec.m.) = d2(Ec.m.σfus)/dEc.m.2, the second derivative of the product Ec.m.σfus with respect to Ec.m.. We investigate two well known systems namely 12C+208Pb and 16O+208Pb, and obtain simultaneous and very successful explanations of cross sections for elastic scattering, fusion, and the result of D(Ec.m.). These results obtained by using our convenient multi-step approximation method demonstrate the unified description of scattering and fusion for interacting heavy ion systems.