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At least 667 records · Page 37

Practical Probabilistic Programming

Recent advances in probabilistic programming languages (PPLs) have provided the capability for exact inference: computing a closed-form probability distribution for a given probabilistic program. In particular, the new language Roulette uses a language oriented programming (LOP) approach, wherein analysts build new programming languages on top of a set of primitives provided by Roulette, which then translates these structures into a weighted model counting problem which can be solved by automated reasoning tools. However, because Roulette provides few convenience features, developing these new languages is challenging even for expert users. We developed a standard library of common probability functions for Roulette with the goal of improved usability. This included approximation of continuous probability density functions using discrete probability mass functions. We demonstrated this approach by modeling a cosmic ray striking a RAM controller. We found that Roulette provides a powerful interface for highly expressive probabilistic programs to be generated. In collaboration with the NNSA Advanced Simulation and Computing program, which resulted in development of a tool called Circulette, we were able to model complex circuits expressed in Verilog using probabilistic programs with an expressivity not previously possible. Our research question that motivated the development of a Roulette standard library was to determine whether non-experts could use a PPL to model relevant problems regarding radiation effects on microelectronics. This standard library improved the expressivity of Roulette by implementing common probability density functions, mathematical operators on distributions, and support for empirical distributions. While Roulette is a powerful modeling language, the untyped, LOP approach makes error messages difficult to understand and requires expert aid. We recommend further research on Roulette, especially with its error messages, to enable improved usability. At the same time, this project demonstrated that for users familiar with Roulette and the LOP approach, Roulette provides powerful new capabilities that can be integrated with other Sandia modeling capabilities.

97 MATHEMATICS AND COMPUTING↗

Electronic structure and topology across T c in the magnetic Weyl semimetal Co 3 Sn 2 S 2

Co 3 Sn 2 S 2 is a magnetic Weyl semimetal, in which ferromagnetic ordering at 177 K is predicted to stabilize Weyl points. We perform temperature and spatial dependent angle--resolved photoemission spectroscopy measurements through the Curie temperature (T c ), which show large band shifts and renormalization concomitant with the onset of magnetism. We argue that Co 3 Sn 2 S 2 evolves from a Mott ferromagnet below T c to a correlated metallic state above T c . To understand the magnetism, we derive a tight-binding model of Co-3d x 2 -y 2 orbitals on the kagome lattice. At the filling obtained by first-principles calculations, this model reproduces the ferromagnetic ground state, and results in the reduction of Coulomb interactions due to cluster effects. Using a disordered local moment simulation, we show how this reduced Hubbard U leads to a collapse of the bands across the magnetic transition, resulting in a correlated state which carries associated characteristic photoemission signatures that are distinct from those of a simple lifting of exchange splitting. Finally, the behavior of topology across T c is discussed in the context of this description of the magnetism.

59 BASIC BIOLOGICAL SCIENCES↗

Spin Wave Resonance Frequency in the Bilayer Ferromagnetic Film

The dependence of spin wave resonance (SWR) frequency on the surface anisotropy, the interlayer and interfacial coupling, the ferromagnetic layer thickness and the external magnetic field in the bilayer ferromagnetic film have been investigated by using the linear spin wave approximation and Green’s function technique. We find that the SWR frequency of the ferromagnetic bilayer films are shifted to higher values corresponding to those of above factors, respectively. The mode of the energetically highest can be controlled by the interfacial coupling, the thickness and the symmetry of the film. The SWR frequency of mode number m = 1 is almost flat except for applied magnetic field. Analytical details and the validity of the method are shown and discussed.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Topics in Theoretical Physics

This award supported a broadly based research effort in theoretical particle physics, including research aimed at uncovering the laws of nature at short (subatomic) and long (cosmological) distances. These theoretical developments apply to experiments in laboratories such as CERN, the facility that operates the Large Hadron Collider outside Geneva, as well as to cosmological investigations done using telescopes and satellites. The results reported here apply to physics beyond the so-called Standard Model of particle physics; physics of high energy collisions such as those observed at the Large Hadron Collider; theoretical and mathematical tools and frameworks for describing the laws of nature at short distances; cosmology and astrophysics; and analytic and computational methods to solve theories of short distance physics. Some specific research accomplishments include + Theories of the electroweak interactions, the forces that give rise to many forms of radioactive decay; + Physics of the recently discovered Higgs boson. + Models and phenomenology of dark matter, the mysterious component of the universe, that has so far been detected only by its gravitational effects. + High energy particles in astrophysics and cosmology. + Algorithmic research and Computational methods for physics of and beyond the Standard Model. + Theory and applications of relativity and its possible limitations. + Topological effects in field theory and cosmology. + Conformally invariant systems and AdS/CFT. This award also supported significant training of students and postdoctoral fellows to lead the research effort in particle theory for the coming decades. These students and fellows worked closely with other members of the group as well as theoretical and experimental colleagues throughout the physics community. Many of the research projects funded by this grant arose in response to recently obtained experimental results in the areas of particle physics and cosmology. We describe a few of these below. Relativity is founded on a symmetry property of nature called "Lorentz Invariance". Like all symmetry properties, it is essential to determine precisely how symmetric nature actually is; that is, do the laws of nature fully respect the symmetry or is there room for tiny symmetry violating effects? An important consequence of Lorentz invariance is the existence of a universal limiting velocity for all physical particles. Light travels at this limiting velocity so it is frequently referred to as simply "the speed of light", but relativity requires that ALL particles travel more slowly than this speed. Once the Higgs particle was discovered in 2012 a natural question was whether or not this particle's speed was consistent with relativity. Although the speed of the Higgs particle is not measurable directly, Cohen has shown that, if the maximal speed of the Higgs particle was not precisely the same as the speed of light, then the Higgs would have some unusual properties. In some cases the Higgs would be unstable to some unusual decay modes; in other cases the interactions of the Higgs with other particles would change the properties of these other particles in ways that could be observed in so-called cosmic rays, very energetic particles (such as photons, protons and other atomic nuclei) coming from space. Once these particles hit the upper atmosphere they produce a "shower" of particles that can be seen by ground-based instruments. If the Higgs has a maximal speed that differs even a tiny bit from the speed of light these showers would look quite different from what is observed. In this way Cohen was able to establish that the Higgs travels with a maximal speed that cannot differ from the speed of light by more than one part in a thousand-trillion. This is by far the most precisely determined property of the Higgs particle. Cohen and Schmaltz reviewed evidence from the Large Hadron Collider (LHC), a particle physics experiment operating at the CERN laboratory near Geneva, for a new particle sometimes called a W'. This evidence included certain unexpected by-products in collisions of protons at very high energy. While the evidence was not significant enough to claim a discovery, it was sufficiently intriguing that many particle theorists worked to construct explanations for this signal. Cohen and Schmaltz were able to determine that such explanations are highly constrained by previous experiments involving collisions of very energetic particles. Nevertheless they were able to construct a theory that adequately explains the LHC data and remain consistent with prior experiments. Their explanation predicts the existence of yet another new particle, called a Z', with a mass slightly greater than that of the W'. This additional particle, if it exists, should be seen as more data is collected from the LHC. Amusingly, there is one collision by-product that has already been seen by the CMS experiment at the LHC that supports the existence of this new particle; however, it is not unlikely that this single event is a so-called "background" event, that is a somewhat atypical by-product of a conventional Standard Model process. This theory for the anomalous LHC data will either be confirmed or excluded with further data-taking at the LHC. The ratio of the number of electrons produced in bottom quark decays over the number of muons produced has been measured at the LHC. This ratio is interesting because it can be predicted very precisely from a basic property of the Standard Model: lepton universality. If lepton universality is correct, the ratio of electrons to muons is predicted to be equal to 1. The first measurements of this ratio find a value different from 1 with a statistical significance of about 3 standard deviations. Schmaltz and collaborator proposed a new extension of the Standard Model which can explain the new data. In addition, Schmaltz and collaborators proposed several new measurements of ratios of decay rates which can confirm or rule out the surprising results from the earlier LHC data. The most recent and precise measurements of the cosmic microwave background from the Planck satellite, from a combination of measurements of the dark matter distribution in the universe, and from a measurement of the expansion rate of the universe today show some disagreement when interpreted in terms of the so-called LambdaCDM model. Schmaltz and collaborators proposed an alternative model to LambdaCDM in which the usual cold dark matter is replaced by a new ``dark sector". This sector consists of a cold dark matter particle which interacts with a newly postulated dark radiation component of the universe. The dark radiation can help explain the discrepancy in measurements of the expansion rate, and the dark matter interactions subtly modify the clumping of dark matter at large scales, thus potentially explaining both kinds of tensions in the data. In two publications Schmaltz described the new model and then performed a precision comparison of the predictions of the model with all currently available cosmological data. The results favor the new model at the level of three standard deviations with current data. Quantum Field Theory (QFT) is the language we use to describe quantum systems which are consistent with Einstein’s theory of Special Relativity. In particular, the requirement of Einstein’s theory that signals not travel faster than the speed of light constrains the types of interactions which particles can engage in. One consequence of relativity is that these interactions cannot preserve particle number. The stronger the interactions, the more severe the particle number violation in a given Relativistic QFT. When particle number violation is strong, it becomes very difficult to adequately parameterize the quantum wave function (which characterizes the state of a quantum system). For example, though we can formulate the QFT which describes the strong force as a set of interactions between quark and gluon particles, we have no clear idea how to express the proton state in terms of these quarks and gluons. This is because the proton, though a bound state of quarks and gluons, is not a state of a fixed number of particles due to strong interactions. Yet, understanding the proton state is very important in order to theoretically predict the reaction rates observed at the LHC in Geneva, which is a proton-proton collider. Katz has formulated a new approach to QFT, which among other things offers a way to adequately approximate the quantum wave function of a bound state at strong coupling. The approximation scheme is related to the fact that any sensible QFT (including that of the strong interactions) is at short distances approximately self-similar upon rescaling of space and time. It turns out that keeping track of the response upon this rescaling is important in efficiently parameterizing the state. Katz and collaborators have used this observation to approximate the state of the proton in toy versions of the strong force. In the late 60s Sheldon Glashow, Abdus Salam and Steven Weinberg (1979 Nobel Prize awardees) proposed a theory unifying weak and electromagnetic interaction which assumed the existence of new particles, the W and Z bosons. The W and Z bosons were eventually detected in high-energy collision in a particle accelerator at CERN, and the recent discovery of the Higgs meson at the Large Hadron Collider (LHC), always at CERN, completed the picture. However, deep theoretical considerations indicate that the theory by Glashow, Weinberg and Salam, often referred to as "the standard model" cannot be the whole story: the existence of new particles and new interactions at yet higher energies is widely anticipated. The experiments at the LHC are looking for these, while theorists, like Brower, Rebbi and collaborators, are investigating models for these new interactions. Working in a large national collaboration with access to the most powerful DOE computers Brower, Rebbi and colleagues have been using calculational techniques, similar to those successfully employed for many years to investigate the interactions among quarks in nucleons, to study theories that can describe the expected "beyond the standard model" (BSM) interactions. Their results, which include also a model for dark matter, have been published in several refereed papers in prestigious journals. Various ideas in topologically interesting field theories predict hypothetical objects such as fractional charges and Majorana excitations. However, such fascinating objects have not been seen in particle physics. Nevertheless, these objects demonstrate possible phenomena that quantum field theory can support. Pi used condensed matter physics as a laboratory to study possible realizations and observable effects of these objects predicted by quantum field theory. In recent times there has developed considerable interest among condensed matter field theorists in precisely the same geometrical and topological structures, which were first discovered in particle physics field theories. From particle physicists' point of view, this is an interesting development, since condensed matter provides an arena in which one can concretely realize particle physics ideas. Moreover, particle physicists can learn new ideas from condensed matter physics. Higgs phenomenon is precisely an important particle physics realization of condensed matter ideas. In contrast to the small distance characterizing condensed matter systems, field theory also describes large distance physics characterizing cosmology. Pi worked on various geometrical effects in the standard theory of cosmology, viz general relativity.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

An Immersed Boundary method with divergence-free velocity interpolation and force spreading

The Immersed Boundary (IB) method is a mathematical framework for constructing robust numerical methods to study fluid–structure interaction in problems involving an elastic structure immersed in a viscous fluid. The IB formulation uses an Eulerian representation of the fluid and a Lagrangian representation of the structure. The Lagrangian and Eulerian frames are coupled by integral transforms with delta function kernels. The discretized IB equations use approximations to these transforms with regularized delta function kernels to interpolate the fluid velocity to the structure, and to spread structural forces to the fluid. It is well-known that the conventional IB method can suffer from poor volume conservation since the interpolated Lagrangian velocity field is not generally divergence-free, and so this can cause spurious volume changes. In practice, the lack of volume conservation is especially pronounced for cases where there are large pressure differences across thin structural boundaries. The aim of this paper is to greatly reduce the volume error of the IB method by introducing velocity-interpolation and force-spreading schemes with the properties that the interpolated velocity field in which the structure moves is at least $\mathscr{C}$ 1 and satisfies a continuous divergence-free condition, and that the force-spreading operator is the adjoint of the velocity-interpolation operator. We confirm through numerical experiments in two and three spatial dimensions that this new IB method is able to achieve substantial improvement in volume conservation compared to other existing IB methods, at the expense of a modest increase in the computational cost. Further, the new method provides smoother Lagrangian forces (tractions) than traditional IB methods. Finally, the method presented here is restricted to periodic computational domains. Its generalization to non-periodic domains is important future work.

97 MATHEMATICS AND COMPUTING↗

Effects of phase singularity in the interference of two laser fields with a wide spectrum

The interference of laser beams with a wide spectrum is studied theoretically and numerically. It is shown that in the cross section of such intersecting beams in a small vicinity of zero of the complex mutual coherence function, the change in the argument of the slow amplitude of this function corresponds to the phase singularity. An approximate analytical representation of the mutual coherence function in the vicinity of the singularity point is given. It is found that an average ‘funnel’ diameter of the modulus of the mutual coherence function that arises in the singularity region does not depend on the intensity of the light beams and remains constant when the longitudinal coordinate changes. The emergence of phase dislocations in the interference pattern is established. (laser beams)

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Tagged spectator DIS off a polarized spin-1 target

We discuss the process of deep-inelastic electron scattering (DIS) on the polarized deuteron with detection of a nucleon in the nuclear fragmentation region ('spectator tagging'). We cover (a) the general structure of the DIS cross section; (b) the structure functions of the process in the impulse approximation using deuteron light-front wave functions; (c) the extraction of free neutron structure by extracting the residue of observables at the free nucleon pole ('pole extrapolation'). As an application we consider the extraction of the free neutron structure function g1n through polarized electron scattering on the longitudinally polarized deuteron with proton tagging and on-shell extrapolation in the recoil momentum, possible at an EIC with polarized deuteron beams and forward detectors.

Cosyn, Wim↗

Study of structural and electronic properties of ScN and ScAs in rocksalt and zincblende structure: A DFT approach

In this paper, we have studied the structural and electronic properties of ScN and ScAs in zincblende (ZB) and rocksalt (RS) phases. We have employed the full potential linearized augmented plane wave (FP-APW) method within the density functional theory (DFT). Generalized gradient approximation (GGA), due to Perdew-Burke-Ernzerhof (PBE) has been used to estimate the exchange-correlation functional. Our band structure results for ScN shows the metallic nature, whereas ScAs shows the semiconducting behavior. The obtained results are in excellent agreement with earlier reported data.

36 MATERIALS SCIENCE↗

Krylov subspace recycling for evolving structures

Krylov subspace recycling is a powerful tool when solving a long series of large, sparse linear systems that change only slowly over time. In PDE constrained shape optimization, these series appear naturally, as typically hundreds or thousands of optimization steps are needed with only small changes in the geometry. In this setting, however, applying Krylov subspace recycling can be a difficult task. As the geometry evolves, in general, so does the finite element mesh defined on or representing this geometry, including the numbers of nodes and elements and element connectivity. This is especially the case if re-meshing techniques are used. As a result, the number of algebraic degrees of freedom in the system changes, and in general the linear system matrices resulting from the finite element discretization change size from one optimization step to the next. Changes in the mesh connectivity also lead to structural changes in the matrices. In the case of re-meshing, even if the geometry changes only a little, the corresponding mesh might differ substantially from the previous one. Obviously, this prevents any straightforward mapping of the approximate invariant subspace of the linear system matrix (the focus of recycling in this work) from one optimization step to the next; similar problems arise for other selected subspaces. In this paper, we present an algorithm to map an approximate invariant subspace of the linear system matrix for the previous optimization step to an approximate invariant subspace of the linear system matrix for the current optimization step, for general meshes. This is achieved by exploiting the map from coefficient vectors to finite element functions on the mesh, combined with interpolation or approximation of functions on the finite element mesh. We demonstrate the effectiveness of our approach numerically with several proof of concept studies for a specific meshing technique.

42 ENGINEERING↗

Calculation of electron-ion temperature equilibration rates and friction coefficients in plasmas and liquid metals using quantum molecular dynamics

In this work, we discuss a method to calculate with quantum molecular dynamics simulations the rate of energy exchanges between electrons and ions in two-temperature plasmas, liquid metals, and hot solids. Promising results from this method were recently reported for various materials and physical conditions. Like other ab initio calculations, the approach offers a very useful comparison with the experimental measurements and permits an extension into conditions not covered by the experiments. The energy relaxation rate is related to the friction coefficients felt by individual ions due to their nonadiabatic interactions with electrons. Each coefficient satisfies a Kubo relation given by the time integral of the autocorrelation function of the interaction force between an ion and the electrons. These Kubo relations are evaluated using the output of quantum molecular dynamics calculations in which electrons are treated in the framework of finite-temperature density functional theory. The calculation presents difficulties that are unlike those encountered with the Kubo formulas for the electrical and thermal conductivities. In particular, the widely used Kubo-Greenwood approximation is inapplicable here. Indeed, the friction coefficients and the energy relaxation rate diverge in this approximation since it does not properly account for the electronic screening of electron-ion interactions. The inclusion of screening effects considerably complicates the calculations. We discuss the physically motivated approximations we applied to deal with these complications in order to investigate a widest range of materials and physical conditions. Unlike the standard method used for the electronic conductivities, the Kubo formulas are evaluated directly in the time domain and not in the energy domain, which spares one from needing to introduce an extraneous undetermined numerical parameter to account for the discrete character of the numerical density of states. We highlight interesting properties of the energy relaxation rate not shared by other electronic properties, in particular its self-averaging character. We then present a detailed parametric and convergence study with the numerical parameters, including the system size, the number of bands and k-points, and the physical approximations for the dielectric function and the exchange-correlation energy.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

The electronic structure of As{sub 2}S{sub 3}

The electronic band structure and density of states of crystalline As{sub 2}S{sub 3} are calculated using SIESTA code within generalized-gradient approximation (GGA) of density functional theory (DFT). Energy band gap in As{sub 2}S{sub 3} is found to be indirect with a value 1.925 eV. Density of states and partial density of states are in agreement with available experimental data.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Competing adsorption mechanisms of pyridine on Cu, Ag, Au, and Pt(110) surfaces

We explore the adsorption of pyridine on Cu, Ag, Au, and Pt(110) surfaces using density functional theory. To account for the van der Waals interaction, we use the optB86b-vdW, optB88-vdW, optPBE-vdW, revPBE-vdW, and rPW86-vdW2 functionals. For comparison, we also run calculations using the generalized gradient approximation-PBE (Perdew–Burke–Ernzerhof) functional. We find the most stable adsorption site to depend on both metal and functional, with two energetically favorable adsorption sites, namely, a vertically oriented site and a flat pyridine site. We calculate that every functional predicts pyridine to lie in the vertical configuration on the coinage metals at a low coverage. On Pt(110), by contrast, we calculate all the functionals—except rPW86-vdW2—to predict pyridine to lie flat at a low coverage. By analyzing these differences for these adsorption configurations, along with various geometric and electronic properties of the adsorbate/substrate system, we access in detail the performance of the 6 functionals we use. We also characterize the nature of the bonding of pyridine on the coinage metals from weak to strong physisorption, depending on the functional used. On Pt(110), we characterize the nature of the bonding of pyridine as ranging from strong physisorption to strong chemisorption depending again on the functional used, illustrating both the importance of the van der Waals interaction to this system and that this system can make a stringent test for computational methods.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

The effects of intrinsic noise on the behaviour of bistable cell regulatory systems under quasi-steady state conditions

We analyse the effect of intrinsic fluctuations on the properties of bistable stochastic systems with time scale separation operating under quasi-steady state conditions. We first formulate a stochastic generalisation of the quasi-steady state approximation based on the semi-classical approximation of the partial differential equation for the generating function associated with the chemical master equation. Such approximation proceeds by optimising an action functional whose associated set of Euler-Lagrange (Hamilton) equations provides the most likely fluctuation path. We show that, under appropriate conditions granting time scale separation, the Hamiltonian can be re-scaled so that the set of Hamilton equations splits up into slow and fast variables, whereby the quasi-steady state approximation can be applied. We analyse two particular examples of systems whose mean-field limit has been shown to exhibit bi-stability: an enzyme-catalysed system of two mutually inhibitory proteins and a gene regulatory circuit with self-activation. Our theory establishes that the number of molecules of the conserved species is order parameters whose variation regulates bistable behaviour in the associated systems beyond the predictions of the mean-field theory. This prediction is fully confirmed by direct numerical simulations using the stochastic simulation algorithm. This result allows us to propose strategies whereby, by varying the number of molecules of the three conserved chemical species, cell properties associated to bistable behaviour (phenotype, cell-cycle status, etc.) can be controlled.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Multipactor threshold calculation of coaxial transmission lines in microwave applications with nonstationary statistical theory

This paper presents a statistical theory for the initial onset of multipactor breakdown in coaxial transmission lines, taking both the nonuniform electric field and random electron emission velocity into account. A general numerical method is first developed to construct the joint probability density function based on the approximate equation of the electron trajectory. The nonstationary dynamics of the multipactor process on both surfaces of coaxial lines are modelled based on the probability of various impacts and their corresponding secondary emission. The resonant assumption of the classical theory on the independent double-sided and single-sided impacts is replaced by the consideration of their interaction. As a result, the time evolutions of the electron population for exponential growth and absorption on both inner and outer conductor, in response to the applied voltage above and below the multipactor breakdown level, are obtained to investigate the exact mechanism of multipactor discharge in coaxial lines. Furthermore, the multipactor threshold predictions of the presented model are compared with experimental results using measured secondary emission yield of the tested samples which shows reasonable agreement. Finally, the detailed impact scenario reveals that single-surface multipactor is more likely to occur with a higher outer to inner conductor radius ratio.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Structural phase transitions in isotropic magnetic elastomers

Magnetic elastomers represent a new type of materials that are “soft” matrices with “hard” magnetic granules embedded in them. The elastic forces of the matrix and the magnetic forces acting between granules are comparable in magnitude even under small deformations. As a result, these materials acquire a number of new properties; in particular, their mechanical and/or magnetic characteristics can depend strongly on the polymer matrix filling with magnetic particles and can change under the action of an external magnetic field, pressure, and temperature. To describe the properties of elastomers, we use a model in which the interaction of magnetic granules randomly arranged in space with one another is described in the dipole approximation by the distribution function of dipole fields, while their interaction with the matrix is described phenomenologically. A multitude of deformation, magnetic-field, and temperature effects that are described in this paper and are quite accessible to experimental observation arise within this model.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Optoelectronic properties of germanium iodide perovskites AGeI{sub 3} (A = K, Rb and Cs): first principles investigations

In this paper, we have investigated the structural, optoelectronic and elastic properties of AGeI{sub 3} (A = K, Rb and Cs) using the density functional theory with generalized gradient approximation (GGA) for potential exchange correlation. The modified Becke–Johnson (mBJ-GGA) potential approximation is also used for calculating the optoelectronic properties of the material. The results show that the band structure of the perovskites AGeI{sub 3} (have a semiconductor behavior with direct band gap at R–R direction, the gap energy values calculated with mBJ-GGA, for each compound as following: 0.58, 0.63, and 0.71 eV, respectively. The optical properties, such as real and imaginary parts of the dielectric functions, refractive index, reflectivity, conductivity and absorption coefficient are investigated. As results, these compounds are competent candidates photovoltaic application like light harvester.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗