Structure and effects of annealing in colloidal matrix‐free Ge quantum dots

The structure of small (2–5 nm) Ge quantum dots prepared by the colloidal synthesis route is examined. Samples were synthesized using either GeO₂ or GeCl₄ as precursor. As‐prepared samples were further annealed under Ar or H₂/Ar atmosphere at different temperatures in order to understand the effect of annealing on their structure. It was found that as‐prepared samples possess distinctly different structures depending on their synthesis route as indicated by their long‐range ordering. An appreciable amount of oxygen was found to be bound to Ge in samples prepared with GeO₂ as a precursor; however, not for GeCl₄. Based on combined transmission electron microscope, Raman, X‐ray diffraction and X‐ray absorption measurements, it is suggested that as‐prepared samples are best described by the core‐shell model with a small nano‐crystalline core and an amorphous outer layer terminated either with oxygen or hydrogen depending on the synthesis route. Annealing in an H₂Ar atmosphere leads to sample crystallization and further nanoparticle growth, while at the same time reducing the Ge—O bonding. X‐ray diffraction measurements for as‐prepared and annealed samples indicate that diamond‐type and metastable phases are present.     

Ab initio study of the mechanism of forming a spiro‐heterocyclic ring compound involving Si and Ge from dichlorosilylene germylidene(Cl2SiGe:) and formaldehyde

The mechanism of the cycloaddition reaction between singlet dichlorosilylene germylidene (Cl₂Si?Ge:) and formaldehyde has been investigated with the CCSD(T)//MP2/6‐31G* method. From the potential energy profile, it could be predicted that the reaction has one dominant reaction pathway. The reaction rule presented is that the two reactants first form a four‐membered Si‐heterocyclic ring germylene through the [2 + 2] cycloaddition reaction. Because of the 4p unoccupied orbital of Ge atom in the four‐membered Si‐heterocyclic ring germylene and the π orbital of formaldehyde forming a π→p donor–acceptor bond, the four‐membered Si‐heterocyclic ring germylene further combines with formaldehyde to form an intermediate. Because the Ge atom in intermediate undergoes sp3 hybridization after transition state, then, the intermediate isomerizes to a spiro‐heterocyclic ring compound involving Si and Ge via a transition state. © 2012 Wiley Periodicals, Inc.     

Pressure-induced phase transitions in the ZrXY (X= Si,Ge, Sn;Y= S,Se, Te) family compounds

Pressure is an effective and clean way to modify the electronic structures of materials, cause structural phase transitions and even induce the emergence of superconductivity. Here, we predicted several new phases of the ZrXY family at high pressures using the crystal structures search method together with first-principle calculations. In particular, the ZrGeS compound undergoes an isosymmetric phase transition from P4/nmm-I to P4/nmm-II at approximately 82 GPa. Electronic band structures show that all the high-pressure phases are metallic. Among these new structures, P4/nmm-II ZrGeS and P4/mmm ZrGeSe can be quenched to ambient pressure with superconducting critical temperatures of approximately 8.1 K and 8.0 K, respectively. Our study provides a way to tune the structure, electronic properties, and superconducting behavior of topological materials through pressure.     

Extension of PT-symmetric quantum mechanics to the Dirac theory with position-dependent mass

We present a new method to construct the exactly solvable PT-symmetric potentials within the framework of the position-dependent effective mass Dirac equation with the vector potential coupling scheme in 1 + 1 dimensions. In order to illustrate the procedure, we produce three PT-symmetric potentials as examples, which are PT-symmetric harmonic oscillator-like potential, PT-symmetric potential with the form of a linear potential plus an inversely linear potential, and PT-symmetric kink-like potential, respectively. The real relativistic energy levels and corresponding spinor components for the bound states are obtained by using the basic concepts of the supersymmetric quantum mechanics formalism and function analysis method.     

Nonlinear Landau--Zener Tunnelling with Two and Three-Body Interactions

We investigate the nonlinear Landau-Zener tunnelling of Bose-Einstein condensate （BEC） in an accelerating optical lattice with two- and three-body interactions between the particles. The influence of the three-body interaction on the eigenstates and the transition probability are discussed both anaJytically and numerically. The analytical eigenstates and the tunnelling probability are obtained, which are verified by numerical methods. It is shown that the eigenstates and the tunnelling probability are modified dramatically by three-body interaction.     

Exact Solution of Klein--Gordon Equation by Asymptotic Iteration Method

Using the asymptotic iteration method （AIM） we obtain the spectrum of the Klein-Gordon equation for some choices of scalar and vector potentials. In particular, it is shown that the AIM exactly reproduces the spectrum of some solvable potentials.     

Exact Solutions of the Dirac Equation for an Electron in a Magnetic Field with Shape Invariant Method

Based on the shape invariance property we obtain exact solutions oI the Dirac equation for an electron moving in the presence of a certain varying magnetic field, then we also show its non-relativistic limit.     

A New Method for Constructing Travelling Wave Solutions to the modified Benjamin--Bona--Mahoney Equation

We present a new method to find the exact travelling wave solutions of nonlinear evolution equations, with the aid of the symbolic computation. Based on this method, we successfully solve the modified BenjaminBona-Mahoney equation, and obtain some new solutions which can be expressed by trigonometric functions and hyperbolic functions, It is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.     

The improved quantization rule and the Langer modification

An improved quantization rule is used to obtain a generalized formulation of Langer modification. The relations between the improved quantization rule and the Langer modification are studied. Two typical quantum systems, hydrogen atom and harmonic oscillator, are studied to show the relations between them.