Angle-dependent Part Research Articles

Schrödinger equation with a non-central potential: Some statistical quantities

In this paper, we study the dependence of some statistical quantities such as the free energy, the mean energy, the entropy, and the specific heat for the Schrodinger equation on the temperature, particularly in the case of a non-central potential. The basic point is to find the partition function which is obtained by a method based on the Euler-Maclaurin formula. At first, we present the analytical results supporting them with some plots for the thermal functions for one- and three-dimensional cases to find out the effect of the angular momentum. We also study then the effect of the angle-dependent part of the non-central potential. We discuss the results briefly for a phase transition for the system. We also present our results for a three-dimesional harmonic oscillator.

Exact Solutions of Schrödinger Equation with Improved Ring-Shaped Non-Spherical Harmonic Oscillator and Coulomb Potential

We propose improved ring shaped like potential of the form, V(r, θ) = V(r) + (ħ2/2Mr2)[(β sin2 θ + γ cos2 θ + λ) / sin θ cos θ]2 and its exact solutions are presented via the Nikiforov–Uvarov method. The angle dependent part V(θ) = (ħ2 / 2 Mr2)[(β sin2 θ + γ cos2 θ + λ) / sin θ cos θ]2, which is reported for the first time embodied the novel angle dependent (NAD) potential and harmonic novel angle dependent potential (HNAD) as special cases. We discuss in detail the effects of the improved ring shaped like potential on the radial parts of the spherical harmonic and Coulomb potentials.

A new Coulomb ring-shaped potential via generalized parametric Nikiforov-Uvarov method

In this paper, the Schrodinger equation is analytically solved for the Coulomb potential with a novel angledependent part. The generalized parametric Nikiforov-Uvarov method is used to obtain energy eigenvalues and corresponding eigenfunctions. We presented the effect of the angle-dependent part on radial solutions and some special cases are also discussed.

A new Morse ring-shaped potential for diatomic molecules

Using the Pekeris approximation, the Schrödinger equation is solved analytically for the Morse plus a novel angle-dependent potential. The generalized parametric Nikiforov–Uvarov method is used to obtain energy eigenvalues and corresponding eigenfunctions. We present the effect of the angle-dependent part on radial solutions, and some numerical results for diatomic molecules are also given.

Improved ASCAT Wind Retrieval Using NWP Ocean Calibration

The Advanced Scatterometer (ASCAT) wind data processor (AWDP) currently uses the so called CMOD5n geophysical model function (GMF), which was originally derived for the European Remote Sensing (ERS) scatterometers. In order to deliver a high-quality ASCAT wind product, the operational AWDP uses backscatter measurement corrections that are estimated visually (VOC) for each wind vector cell. We propose an alternative and previously established method for estimating correction tables based on numerical weather prediction ocean calibration residuals (NOC). It embodies a smooth incidence-angle dependent part that could serve as an appropriate ASCAT GMF correction, and a radar-beam-dependent residual. The incidence-angle-dependent part of these correction tables is due to differences in calibration procedure of the ERS and ASCAT scatterometers. For the high ASCAT incidence angles for which the GMF has not been assessed by ERS data, the modification is quite large, almost 1 dB. The incidence angle-dependent part is derived by fitting the OC residuals of all beams obtained over one year of data. It is subsequently used to adapt the GMF (yielding CMOD5na). The remaining radar-beam-dependent residual (NOCa) shows a wiggle pattern as function of incidence angle that is very persistent over time, apart from a seasonally varying offset. Both the effects of the GMF modification and the beam-dependent residual on the wind retrieval quality are investigated in this paper. Overall, the performance of NOC is better than that obtained with the previously used VOC calibration method, and the wind statistics show a much better symmetry of the left and right swath for NOC. The beam-dependent corrections improve the quality of the retrieved winds. NOC may thus be used for the intercalibration of the ERS and ASCAT scatterometers.

Exact Solutions of the Dirac Equation with Coulomb plus a Novel Angle-Dependent Potential

Abstract In this paper the Dirac equation is analytically solved for Coulomb plus a novel angle-dependent potential. The Nikiforov-Uvarov method is used to obtain energy eigenvalues and corresponding eigenfunctions. We also discussed the effect of the angle-dependent part on radial solutions.

Pseudospin symmetry for a new oscillatory ring-shaped noncentral potential

A new oscillatory ring-shaped noncentral potential is proposed and the pseudospin symmetry for this potential is investigated by solving the Dirac equation with equally mixed scalar and vector potentials with opposite signs. The spinor wave functions and the energy spectrum for the bound states are obtained by using the Nikiforov-Uvarov method. The effect of angle-dependent part on the radial solutions and the algebraic property of the energy equation are discussed.

Flux based modeling and simulation of dry etching for fabrication of silicon deep trench structures

Deep reactive ion etching (DRIE) process is a key growth for fabrication of micro-electromechanical system (MEMS) devices. Due to complexity of this process, including interaction of the process steps, full analytical modeling is complex. Plasma process holds deficiency of understanding because it is very easy to measure the results empirically. However, as device parameters shrink, this issue is more critical. In this paper, our process was modeled qualitatively based on "High Density Plasma Etch Model". Deep trench solutions of etch rate based on continuity equation were successfully generated first time through mathematical analysis. It was also proved that the product of fluorine and gas phase concentration in SF6 remains identical during both deposition and etching stages. The etching process was treated as a combination of isotropic, directional and angle-dependent component parts. It exploited a synergistic balance of chemical as well as physical etching for promoting silicon trenches and high aspect ratio structures. Simulations were performed for comprehensive analysis of fluxes coming towards the surface during chemical reaction of gas. It is observed that near the surface, the distribution of the arrival flux follows a cosine distribution. Our model is feasible to analyze various parameters like gas delivery, reactor volume and temperature that help to assert large scale effects and to optimize equipment design.

PSEUDOSPIN AND SPIN SYMMETRY FOR THE RING-SHAPED GENERALIZED HULTHÉN POTENTIAL

We obtain the bound state energy eigenvalues and the corresponding wave functions of the Dirac particle for the generalized Hulthén potential plus a ring-shaped potential with pseudospin and spin symmetry. The Nikiforov–Uvarov method is used in the calculations. Contribution of the angle-dependent part of the potential to the relativistic energy spectra are investigated. In addition, it is shown that the obtained results coincide with those available in the literature.

Exact solutions of the Schr?dinger equation for a ring-shaped noncentral potential

A new ring-shaped noncentral potential is proposed and the exact complete solutions of the Schr&#246,dinger equation with this potential are presented by the Nikiforov-Uvarov method.The effect of the angle-dependent part on the radial solutions and some particular cases of this potential are also discussed.

Exactly complete solutions of the Schrödinger equation with a spherically harmonic oscillatory ring-shaped potential

A spherically harmonic oscillatory ring-shaped potential is proposed and its exactly complete solutions are presented by the Nikiforov–Uvarov method. The effect of the angle-dependent part on the radial solutions is discussed.

On the exact solutions of the Dirac equation with a novel angle-dependent potential

We study the problem of the relativistic motion of a 1/2-spin particle in an exactly solvable potential, which consists of the harmonic oscillator potential plus a novel angle-dependent potential, The analytic bound state solutions of the Dirac equation for this potential are obtained by using the Nikiforov–Uvarov method. The wave functions of the radial and angle-dependent parts of the Dirac equation are derived in the form of the Laguerre and Jacobi polynomials. The contribution of the angle-dependent potential to the relativistic energy spectra is discussed under the condition that the scalar potential is equal to or minus the vector potential.

Modified ℓ-states of diatomic molecules subject to central potentials plus an angle-dependent potential

We present modified l-states of diatomic molecules by solving the radial and angle-dependent parts of the Schrodinger equation for central potentials, such as Morse and Kratzer, plus an exactly solvable angle-dependent potential V θ (θ)/r 2 within the framework of the Nikiforov–Uvarov (NU) method. We emphasize that the contribution which comes from the solution of the Schrodinger equation for the angle-dependent potential modifies the usual angular momentum quantum number l. We calculate explicitly bound state energies of a number of neutral diatomic molecules composed of a first-row transition metal and main-group elements for both Morse and Kratzer potentials plus an angle-dependent potential. We also compare the bound state energies for both potentials, taking into account spectroscopic parameters of diatomic molecules and arbitrary values of potential constants.

A novel angle-dependent potential and its exact solution

The quantum mechanics of a diatomic molecule in a noncentral potential of the type V (r) = Vθ(θ)/r2 + Vr(r) are investigated analytically. The θ-dependent part of the relevant potential is suggested for the first time as a novel angle-dependent (NAD) potential \({V_{\theta}(\theta)=\frac{\hbar^2}{2\mu}\left(\frac{\gamma +\beta \sin^2\theta +\alpha \sin^4 \theta}{\sin^2\theta \cos^2\theta}\right)}\) and the radial part is selected as the Coulomb potential or the harmonic oscillator potential, i.e., Vr(r) = − H/r or Vr(r) = Kr2, respectively. Exact solutions are obtained in the Schrodinger picture by means of a mathematical method named the Nikiforov–Uvarov (NU). The effect of the angle-dependent part on the solution of the radial part is discussed in several values of the NAD potential’s parameters as well as different values of usual quantum numbers.

Multipole decomposition of the elastic atomic photon scattering form factor and the anomalous scattering factor

The elastic atomic photon scattering form factor amplitude Af0, obtained from the high energy and low momentum transfer (K) limit of the exact (within the independent particle approximation (IPA)) Rayleigh relativistic second order S-matrix scattering amplitude Asm is decomposed into an infinite sum of its electric and magnetic field multipoles. The energy-dependent part of Af0 (the scattering angle-dependent part of Af0 is incorrect due to the low K approximation), combined with the angle-dependent part of Asm, yields a form factor amplitude Af (different versions of Af and Af0 will be described in the text) that can be used to decompose the anomalous scattering factor Aasf into its multipoles. Af0 which initially contains coherent and incoherent factors is reduced to a single sum over the multipoles of only coherent terms by assuming that the charge density is spherically symmetric. At photon energies up to about ten times the K-shell binding energy it takes up to quadrupole Aasf for light atoms of Z ≤ 20 and up to the fourth multipole for heavy atoms such as uranium for an accurate Aasf (within IPA). The multipole contributions were found to become somewhat larger as the scattering angle becomes larger (greater than 120°); however there were partial cancellations of such effects.

Kinetic theory of area-preserving maps. Application to the standard map in the diffusive regime

The evolution of the distribution function of a dynamical system governed by a general two-dimensional area-preserving iterative map is studied by the methods of nonequilibrium statistical mechanics. A closed, non-Markovian master equation determines the angle-averaged distribution function (the ‘‘density profile’’). The complementary, angle-dependent part (‘‘the fluctuations’’) is expressed as a non-Markovian functional of the density profile. Whenever there exist two widely separated intrinsic time scales, the master equation can be markovianized, yielding an asymptotic kinetic equation. The general theory is applied to the standard map in the diffusive regime, i.e., for large stochasticity parameter and large scale length. The non-Markovian master equation can be written and solved analytically in this approximation. The two characteristic time scales are exhibited. This permits the thorough study of the evolution of the density profile, its tendency toward the Markovian approximation, and eventually toward a diffusive Gaussian packet. The evolution of the fluctuations is also described in detail. The various relaxation processes are governed asymptotically by a single diffusion coefficient, which is calculated analytically. This model appears as a testing bench for the study of kinetic equations. The various previous approaches to this problem are reviewed and critically discussed.

Modified flux flow in polycrystalline YBCO: temperature and current dependence

DC resistivity of a zero-field-cooled granular YBCO sample has been studied as a function of angle between the applied field and transport current, at various fields, currents and temperatures. In addition to the large angle-independent dissipation, a small angle-dependent part has been observed which has a behaviour. This part varies with increasing fields according to a modified flux flow expression, in line with the reported microwave surface resistance data. Interestingly, the exponent, n, of the field dependence obtained from a modified flux flow expression gives values close to the one at low currents and low temperatures, whereas it reduces towards 0.5 for large values of currents and temperatures close to . With increasing current and temperature, the behaviour of the flux flow in YBCO approaches that which has been reported for BSCCO. The behaviour is considered to be due to a decrease of the coherence of vortex segments at higher currents and temperatures. Furthermore, the deviation of the voltage from at low angles has also been studied at different currents and temperatures, which further supports our previous conjecture of Lorentz-force-mediated flux redistribution between inter- and intragrain regions.

Perturbation theory of polar nonspherical molecule fluids

Perturbation theory of polar nonspherical molecule fluids was applied to systems of dipolar and polarizable dipolar Kihara molecules. Integrals corresponding to the first, second and third order perturbation terms, containing the angle-dependent part of the electrostatic pair potentials, were determined by the numerical methods for a set of parameters characterizing the nonsphericity of molecules, and expressed in the form of Padé approximants. Theoretical expressions for thermodynamic functions were applied to describe the dipolar nonspherical molecule system and compared with the simulation data. Simple models of polarizable quadrupolar and polarizable dipolar Kihara molecule fluids were considered for carbon dioxide and hydrogen chloride and theoretical expressions for the Helmholtz function, compressibility factor and the second virial coefficient were employed to describe the equilibrium behaviour of these systems. Comparison with experimental data is given.

Molecular theory of elastic constants of liquid crystals: Application to uniaxial phases.

We use density-functional theory to derive an expression for the distortion free energy of molecular ordered phases and expand it in terms of the order parameters characterizing the structure of the phase and the molecular correlation function of an effective isotropic liquid. We derive expressions for the elastic constants of the uniaxial nematic and smectic-A phases of the liquid crystals and express the results in terms of the spherical-harmonic coefficients of the direct pair-correlation function of the effective isotropic liquid. These harmonic coefficients are obtained by solving the Ornstein-Zernike equation using the Percus-Yevick closure relation for a hard-core system and using a perturbation method for a model system interacting via a pair potential that has angle-dependent attractive parts in addition to hard-core repulsion. It is shown that the repulsive part of the pair interaction makes a dominant contribution to the elastic constants of the liquid crystals.

Density-functional theory of curvature elasticity in nematic liquids. II. Effect of long-range dispersion interactions.

The theory of curvature elasticity in nematics described in earlier papers is reformulated and simplified. The extent to which the ``splay,'' twist, and ``bend'' constants (${K}_{1}$,${K}_{2}$,${K}_{3}$) of nematic liquid crystals differ from one another and the way they depend upon the degree of alignment (as characterized by the nematic order parameters P${\ifmmode\bar\else\textasciimacron\fi{}}_{n}$) are determined by the structure of the medium which in turn is characterized by the correlation functions. The direct correlation functions of the isotropic liquid in terms of which the elastic constants are expressed have broad functional similarities for systems with rather different intermolecular potentials. An approximate scheme is proposed to estimate the values of the direct pair correlation function. Numerical results for the elastic constants are presented for a range of hard-core sizes and shapes and dispersional strengths and anisotropies. It is found that the effect of the long-range dispersion interactions on the elastic constants is determined not only by the angle-dependent part of the forces, but, and more significantly, by the coupling between the isotropic part and anisotropic hard-core repulsions through the pair correlation function. The attractive interactions are found to make significant contributions to ${K}_{i}$ in ordinary nematics (nematics of rod-shaped molecules) when ${x}_{0}$ (length-to-width ratio) is small (\ensuremath{\lesssim}3.0) and in discotic nematics when 1/${x}_{0}$ is large (i.e., \ensuremath{\gtrsim}3.0). Qualitative features exhibited by our calculation for both the ordinary and the discotic nematics are in agreement with the experiment.