Effects of a mixed vector-scalar kink-like potential for spinless particles in two-dimensional space time

The intrinsically relativistic problem of spinless particles subject to a general mixing of vector and scalar kink- like potentials (similar to tanh gamma x) is investigated. The problem is mapped into the exactly solvable Sturm - Liouville problem with the Rosen - Morse potential and exact bounded solutions for particles and antiparticles are found. The behavior of the spectrum is discussed in some detail. An apparent paradox concerning the uncertainty principle is solved by recurring to the concept of effective Compton wavelength.

Oscilador harmônico unidimensional via transformada unilateral de Fourier e mapeamento do potencial de morse unidimensional nos potenciais pseudo-harmônico e de kratzer tridimensionais

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Oscilador harmônico unidimensional via transformada unilateral de Fourier e mapeamento do potencial de morse unidimensional nos potenciais pseudo-harmônico e de kratzer tridimensionais

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Evolution and detection of cyanobacterial hepatotoxin synthetase genes

Mass occurrences (blooms) of cyanobacteria are common in aquatic environments worldwide. These blooms are often toxic, due to the presence of hepatotoxins or neurotoxins. The most common cyanobacterial toxins are hepatotoxins: microcystins and nodularins. In freshwaters, the main producers of microcystins are Microcystis, Anabaena, and Planktothrix. Nodularins are produced by strains of Nodularia spumigena in brackish waters. Toxic and nontoxic strains of cyanobacteria co-occur and cannot be differentiated by conventional microscopy. Molecular biological methods based on microcystin and nodularin synthetase genes enable detection of potentially hepatotoxic cyanobacteria. In the present study, molecular detection methods for hepatotoxin-producing cyanobacteria were developed, based on microcystin synthetase gene E (mcyE) and the orthologous nodularin synthetase gene F (ndaF) sequences. General primers were designed to amplify the mcyE/ndaF gene region from microcystin-producing Anabaena, Microcystis, Planktothrix, and Nostoc, and nodularin-producing Nodularia strains. The sequences were used for phylogenetic analyses to study how cyanobacterial mcy genes have evolved. The results showed that mcy genes and microcystin are very old and were already present in the ancestor of many modern cyanobacterial genera. The results also suggested that the sporadic distribution of biosynthetic genes in modern cyanobacteria is caused by repeated gene losses in the more derived lineages of cyanobacteria and not by horizontal gene transfer. Phylogenetic analysis also proposed that nda genes evolved from mcy genes. The frequency and composition of the microcystin producers in 70 lakes in Finland were studied by conventional polymerase chain reaction (PCR). Potential microcystin producers were detected in 84% of the lakes, using general mcyE primers, and in 91% of the lakes with the three genus-specific mcyE primers. Potential microcystin-producing Microcystis were detected in 70%, Planktothrix in 63%, and Anabaena in 37% of the lakes. The presence and co-occurrence of potential microcystin producers were more frequent in eutrophic lakes, where the total phosphorus concentration was high. The PCR results could also be associated with various environmental factors by correlation and regression analyses. In these analyses, the total nitrogen concentration and pH were both associated with the presence of multiple microcystin-producing genera and partly explained the probability of occurrence of mcyE genes. In general, the results showed that higher nutrient concentrations increased the occurrence of potential microcystin producers and the risk for toxic bloom formation. Genus-specific probe pairs for microcystin-producing Anabaena, Microcystis, Planktothrix, and Nostoc, and nodularin-producing Nodularia were designed to be used in a DNA-chip assay. The DNA-chip can be used to simultaneously detect all these potential microcystin/nodularin producers in environmental water samples. The probe pairs detected the mcyE/ndaF genes specifically and sensitively when tested with cyanobacterial strains. In addition, potential microcystin/nodularin producers were identified in lake and Baltic Sea samples by the DNA-chip almost as sensitively as by quantitative real-time PCR (qPCR), which was used to validate the DNA-chip results. Further improvement of the DNA-chip assay was achieved by optimization of the PCR, the first step in the assay. Analysis of the mcy and nda gene clusters from various hepatotoxin-producing cyanobacteria was rewarding; it revealed that the genes were ancient. In addition, new methods detecting all the main producers of hepatotoxins could be developed. Interestingly, potential microcystin-producing cyanobacterial strains of Microcystis, Planktothrix, and Anabaena, co-occurred especially in eutrophic and hypertrophic lakes. Protecting waters from eutrophication and restoration of lakes may thus decrease the prevalence of toxic cyanobacteria and the frequency of toxic blooms.

Theory of polymer breaking under tension

We consider the breaking of a polymer molecule which is fixed at one end and is acted upon by a force at the other. The polymer is assumed to be a linear chain joined together by bonds which satisfy the Morse potential. The applied force is found to modify the Morse potential so that the minimum becomes metastable. Breaking is just the decay of this metastable bond, by causing it to go over the barrier. Increasing the force causes the potential to become more and more distorted and eventually leads to the disappearance of the barrier. The limiting force at which the barrier disappears is D(e)a/2,D-e with a the parameters characterizing the Morse potential. The rate of breaking is first calculated using multidimensional quantum transition state theory. We use the harmonic approximation to account for vibrations of all the units. It includes tunneling contributions to the rate, but is valid only above a certain critical temperature. It is possible to get an analytical expression for the rate of breaking. We have calculated the rate of breaking for a model, which mimics polyethylene. First we calculate the rate of breaking of a single bond, without worrying about the other bonds. Inclusion of other bonds under the harmonic approximation is found to lower this rate by at the most one order of magnitude. Quantum effects are found to increase the rate of breaking and are significant only at temperatures less than 150 K. At 300 K, the calculations predict a bond in polyethylene to have a lifetime of only seconds at a force which is only half the limiting force. Calculations were also done using the Lennard-Jones potential. The results for Lennard-Jones and Morse potentials were rather different, due to the different long-range behaviors of the two potentials. A calculation including friction was carried out, at the classical level, by assuming that each atom of the chain is coupled to its own collection of harmonic oscillators. Comparison of the results with the simulations of Oliveira and Taylor [J. Chem. Phys. 101, 10 118 (1994)] showed the rate to be two to three orders of magnitude higher. As a possible explanation of discrepancy, we consider the translational motion of the ends of the broken chains. Using a continuum approximation for the chain, we find that in the absence of friction, the rate of the process can be limited by the rate at which the two broken ends separate from one another and the lowering of the rate is at the most a factor of 2, for the parameters used in the simulation (for polyethylene). In the presence of friction, we find that the rate can be lowered by one to two orders of magnitude, making our results to be in reasonable agreement with the simulations.

Performance metrics in a hybrid MPI-OpenMP based molecular dynamics simulation with short-range interactions

We discuss the computational bottlenecks in molecular dynamics (MD) and describe the challenges in parallelizing the computation-intensive tasks. We present a hybrid algorithm using MPI (Message Passing Interface) with OpenMP threads for parallelizing a generalized MD computation scheme for systems with short range interatomic interactions. The algorithm is discussed in the context of nano-indentation of Chromium films with carbon indenters using the Embedded Atom Method potential for Cr-Cr interaction and the Morse potential for Cr-C interactions. We study the performance of our algorithm for a range of MPI-thread combinations and find the performance to depend strongly on the computational task and load sharing in the multi-core processor. The algorithm scaled poorly with MPI and our hybrid schemes were observed to outperform the pure message passing scheme, despite utilizing the same number of processors or cores in the cluster. Speed-up achieved by our algorithm compared favorably with that achieved by standard MD packages. (C) 2013 Elsevier Inc. All rights reserved.

Estimation of mechanical properties of single wall carbon nanotubes using molecular mechanics approach

Molecular mechanics based finite element analysis is adopted in the current work to evaluate the mechanical properties of Zigzag, Armchair and Chiral Single wall Carbon Nanotubes (SWCNT) of different diameters and chiralities. Three different types of atomic bonds, that is Carbon Carbon covalent bond and two types of Carbon Carbon van der Waals bonds are considered in the carbon nanotube system. The stiffness values of these bonds are calculated using the molecular potentials, namely Morse potential function and Lennard-Jones interaction potential function respectively and these stiffness's are assigned to spring elements in the finite element model of the CNT. The geometry of CNT is built using a macro that is developed for the finite element analysis software. The finite element model of the CNT is constructed, appropriate boundary conditions are applied and the behavior of mechanical properties of CNT is studied.

Size- And Temperature-Dependent Thermal Expansion Coefficient Of A Nanofilm

The thermal expansion coefficient (TEC) of an ideal crystal is derived by using a method of Boltzmann statistics. The Morse potential energy function is adopted to show the dependence of the TEC on the temperature. By taking the effects of the surface relaxation and the surface energy into consideration, the dimensionless TEC of a nanofilm is derived. It is shown that with decreasing thickness, the TEC can increase or decrease, depending on the surface relaxation of the nanofilm.

A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula

Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (pdf) of the random shear tensor due to point masses in the limit of an infinite number of stars. Up to this order, the pdf depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the star's mass. As a consequence, the pdf's of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic pdf of the shear magnitude in the limit of an infinite number of stars is also presented. All the results on the random microlensing shear are given for a general point in the lens plane. Extending to the general random distributions (not necessarily uniform) of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of global expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars. © 2009 American Institute of Physics.

Application of [Lambda] to the fourth perturbation theory in calculating the equation of state of rare gas solids and fcc metals

We have calculated the thermodynamic properties of monatomic fcc crystals from the high temperature limit of the Helmholtz free energy. This equation of state included the static and vibrational energy components. The latter contribution was calculated to order A4 of perturbation theory, for a range of crystal volumes, in which a nearest neighbour central force model was used. We have calculated the lattice constant, the coefficient of volume expansion, the specific heat at constant volume and at constant pressure, the adiabatic and the isothermal bulk modulus, and the Gruneisen parameter, for two of the rare gas solids, Xe and Kr, and for the fcc metals Cu, Ag, Au, Al, and Pb. The LennardJones and the Morse potential were each used to represent the atomic interactions for the rare gas solids, and only the Morse potential was used for the fcc metals. The thermodynamic properties obtained from the A4 equation of state with the Lennard-Jones potential, seem to be in reasonable agreement with experiment for temperatures up to about threequarters of the melting temperature. However, for the higher temperatures, the results are less than satisfactory. For Xe and Kr, the thermodynamic properties calculated from the A2 equation of state with the Morse potential, are qualitatively similar to the A 2 results obtained with the Lennard-Jones potential, however, the properties obtained from the A4 equation of state are in good agreement with experiment, since the contribution from the A4 terms seem to be small. The lattice contribution to the thermal properties of the fcc metals was calculated from the A4 equation of state, and these results produced a slight improvement over the properties calculated from the A2 equation of state. In order to compare the calculated specific heats and bulk moduli results with experiment~ the electronic contribution to thermal properties was taken into account~ by using the free electron model. We found that the results varied significantly with the value chosen for the number of free electrons per atom.

On the equation of state and atomic mean square displacement of crystals

We have presented a Green's function method for the calculation of the atomic mean square displacement (MSD) for an anharmonic Hamil toni an . This method effectively sums a whole class of anharmonic contributions to MSD in the perturbation expansion in the high temperature limit. Using this formalism we have calculated the MSD for a nearest neighbour fcc Lennard Jones solid. The results show an improvement over the lowest order perturbation theory results, the difference with Monte Carlo calculations at temperatures close to melting is reduced from 11% to 3%. We also calculated the MSD for the Alkali metals Nat K/ Cs where a sixth neighbour interaction potential derived from the pseudopotential theory was employed in the calculations. The MSD by this method increases by 2.5% to 3.5% over the respective perturbation theory results. The MSD was calculated for Aluminum where different pseudopotential functions and a phenomenological Morse potential were used. The results show that the pseudopotentials provide better agreement with experimental data than the Morse potential. An excellent agreement with experiment over the whole temperature range is achieved with the Harrison modified point-ion pseudopotential with Hubbard-Sham screening function. We have calculated the thermodynamic properties of solid Kr by minimizing the total energy consisting of static and vibrational components, employing different schemes: The quasiharmonic theory (QH), ).2 and).4 perturbation theory, all terms up to 0 ().4) of the improved self consistent phonon theory (ISC), the ring diagrams up to o ().4) (RING), the iteration scheme (ITER) derived from the Greens's function method and a scheme consisting of ITER plus the remaining contributions of 0 ().4) which are not included in ITER which we call E(FULL). We have calculated the lattice constant, the volume expansion, the isothermal and adiabatic bulk modulus, the specific heat at constant volume and at constant pressure, and the Gruneisen parameter from two different potential functions: Lennard-Jones and Aziz. The Aziz potential gives generally a better agreement with experimental data than the LJ potential for the QH, ).2, ).4 and E(FULL) schemes. When only a partial sum of the).4 diagrams is used in the calculations (e.g. RING and ISC) the LJ results are in better agreement with experiment. The iteration scheme brings a definitive improvement over the).2 PT for both potentials.

Phonon dispersion curves and atomic mean square displacement for several fcc and bcc materials

The atomic mean square displacement (MSD) and the phonon dispersion curves (PDC's) of a number of face-centred cubic (fcc) and body-centred cubic (bcc) materials have been calclllated from the quasiharmonic (QH) theory, the lowest order (A2 ) perturbation theory (PT) and a recently proposed Green's function (GF) method by Shukla and Hiibschle. The latter method includes certain anharmonic effects to all orders of anharmonicity. In order to determine the effect of the range of the interatomic interaction upon the anharmonic contributions to the MSD we have carried out our calculations for a Lennard-Jones (L-J) solid in the nearest-neighbour (NN) and next-nearest neighbour (NNN) approximations. These results can be presented in dimensionless units but if the NN and NNN results are to be compared with each other they must be converted to that of a real solid. When this is done for Xe, the QH MSD for the NN and NNN approximations are found to differ from each other by about 2%. For the A2 and GF results this difference amounts to 8% and 7% respectively. For the NN case we have also compared our PT results, which have been calculated exactly, with PT results calculated using a frequency-shift approximation. We conclude that this frequency-shift approximation is a poor approximation. We have calculated the MSD of five alkali metals, five bcc transition metals and seven fcc transition metals. The model potentials we have used include the Morse, modified Morse, and Rydberg potentials. In general the results obtained from the Green's function method are in the best agreement with experiment. However, this improvement is mostly qualitative and the values of MSD calculated from the Green's function method are not in much better agreement with the experimental data than those calculated from the QH theory. We have calculated the phonon dispersion curves (PDC's) of Na and Cu, using the 4 parameter modified Morse potential. In the case of Na, our results for the PDC's are in poor agreement with experiment. In the case of eu, the agreement between the tlleory and experiment is much better and in addition the results for the PDC's calclliated from the GF method are in better agreement with experiment that those obtained from the QH theory.

Thermodynamic properties and Debye-Waller factor of fee materials

We have calculated the equation of state and the various thermodynamic properties of monatomic fcc crystals by minimizing the Helmholtz free energy derived in the high temperature limit for the quasiharmonic theory, QH, and the lowest-order (cubic and quartic), 'A2, anharmonic terms of the perturbation theory, PT. The total energy in each case is obtained by adding the static energy. The calculation of the thermal properties was carried out for a nearest-neighbour central-force model of the fcc lattice by means of the appropriate thermodynamic relations. We have calculated the lattice constant, the thermal expansion, the coefficient of volume expansion, the specific heat at constant volume and at constant pressure, the isothermal and adiabatic bulk moduli, and the Griineisen parameter, for the rare-gas solids Kr and Xe, and gold. Morse potential and modified Morse potential were each used to represent the atomic interaction for the three fcc materials. For most of the calculated thermodynamic properties from the QH theory, the results for Kr and Xe with the modified Morse potential show an improvement over the results for the Morse potential when compared with the experimental data. However, the results of the 'A 2 equation of state with the modified Morse potential are in good agreement with experiment only in the case of the specific heat at constant volume and at constant pressure. For Au we have calculated the lattice contribution from the QH and 'A 2 PT and the electronic contribution to the thermal properties. The electronic contribution was taken into account by using the free electron model. The results of the thermodynamic properties calculated with the modified Morse potential were similar to those obtained with the Morse potential. U sing the minimized equation of state we also calculated the Mossbauer recoilless fraction for Kr and Xe and the Debye-Waller factor (DWF) for Pb, AI, eu, Ag, and Au. The Mossbauer recoilless fraction was obtained for the above two potentials and Lennard-Jones potential. The L-J potential gives the best agreement with experiment for Kr. No experimental data exists for Xe. At low temperature the calculated DWF results for Pb, AI, and eu show a good agreement with experimental values, but at high temperature the experimental DWF results increase very rapidly. For Ag the computed values were below the expected results at all temperatures. The DWF results of the modified Morse potential for Pb, AI, eu and Ag were slightly better than those of the Morse potential. In the case of Au the calculated values were in poor agreement with experimental results. We have calculated the quasiharmonic phonon dispersion curves for Kr, Xe, eu, Ag, and Au. The calculated and experimental results of the frequencies agree quite well for all the materials except for Au where the longitudinal modes show serious discrepancies with the experimental results. In addition, the two lowest-order anharmonic contributions to the phonon frequency were derived using the Green's function method. The A 2 phonon dispersion curves have been calculated only for eu, and the results were similar to those of the QH dispersion curves. Finally, an expression for the Griineisen parameter "( has been derived from the anharmonic frequencies, and calculated for these materials. The "( results are comparable with those obtained from the thermodynamic definition.

Exercise sheet 3

Exercises and solutions in LaTex

Exercise sheet 3 pdf

Exercises and solutions in PDF