Dispersion and energy relations in periodic chains and grids

In this study wave propagation, dispersion relations, and energy relations for linear elastic periodic systems are analyzed. In particular, the dispersion relations for monoatomic chain of infinite dimension are obtained analytically by writing the Block-type wave equation for a unit cell in order to capture the dynamic behavior for chains under prescribed vibration. By comparing the discretized model (mass-spring chain) with the solid bar system, the nonlinearity of the dispersion relation for chain indicates that the periodic lattice is dispersive in contrast to the continuous rod, which is non dispersive. Further investigations have been performed considering one-dimensional diatomic linear elastic mass-spring chain. The dispersion relations, energy velocity, and group velocity have been derived. At certain range of frequencies harmonic plane waves do not propagate in contrast with monoatomic chain. Also, since the diatomic chain considered is a linear elastic chain, both of the energy velocity and the group velocity are identical. As long as the linear elastic condition is considered the results show zero flux condition without residual energy. In addition, this paper shows that the diatomic chain dispersion relations are independent on the unit cell scheme. Finally, an extension for the study covers the dispersion and energy relations for 2D- grid system. The 2x2 grid system show a periodicity of the dispersion surface in the wavenumber domain. In addition, the symmetry of the surface can be exploited to identify an Irreducible Brillouin Zone (IBZ). Compact representations of the dispersion properties of multidimensional periodic systems are obtained by plotting frequency as the wave vector’s components vary along the boundary of the IBZ, which leads to a widely accepted and effective visualization of bandgaps and overall dispersion properties.

Cadmium Telluride Magic-Sized Clusters and Superstructures

The aim of the present work is to gain new insights into the formation mechanism of CdTe magic-sized clusters (MSCs) at low temperatures, as well as on their evolution towards 1D and 2D nanostructures and assemblies thereof, under mild reaction conditions. The reaction system included toluene as solvent, octylamine as primary alkylamine, trioctylphosphine-Te as chalcogenide precursor and Cd(oleate)2 as metal precursor. UV-Vis absorption spectroscopy and transmission electron microscopy (TEM) were used to analyze samples containing concentrations of octylamine of 0.2, 0.8 and 2 M: well-defined, sharp absorption peaks were observed, with peaks maxima at 449, 417 and 373 nm respectively, and 1D structures with a string-like appearance were displayed in the TEM images. Investigating peaks growth, step-wise peaks shift to lower energies and reverse, step-wise peak shift to higher energies allowed to propose a model to describe the system, based on interconnected [CdTe]x cluster units originating an amine-capped, 1-dimensional, polymer-like structure, in which different degrees of electronic coupling between the clusters are held responsible for the different absorption transitions. The many parameters involved in the synthesis procedure were then investigated, starting from the Cd:Te ratio, the role of the amine, the use of different phosphine-Te and Cd precursors. The results allowed to gain important information of the reaction mechanism, as well as on the different behavior of the species featuring the sharp absorption peaks in each case. Using Cd(acetate)2 as metal precursor, 2D structures were found to evolve from the MSCs solutions over time, and their tendency to self-assemble was then analyzed employing two amines of different alkyl chain length, octylamine (C-8) and oleylamine (C-18). Their co-presence led to the formation of free-floating triangular nanosheets, which tend to readily aggregate if only octylamine is present in solution.

Diffusion model of the hollow electron lens for HL-LHC

In the upcoming years, various upgrades and improvements are planned for the CERN Large Hadron Collider (LHC) and represent the mandate of the High-Luminosity project. The upgrade will allow for a total stored beam energy of about 700 MJ, which will need, among others, an extremely efficient collimation system. This will be achieved with the addition of a hollow electron lens (HEL) system to help control the beam-halo depletion and mitigate the effects of fast beam losses. In this master thesis, we present a diffusion model of the HEL for HL-LHC. In particular, we explore several scenarios to use such a device, focusing on the halo depletion efficiency given by different noise regimes.