Variable Phase Approach Research Articles

Study of Two-Dimensional Low-Energy Scattering by the Variable Phase Approach

Study of Two-Dimensional Low-Energy Scattering by the Variable Phase Approach

Phase shift and cross section analysis of nucleon–nucleon and nucleon–nucleus scattering using second Pöschl–Teller potential

In view of nuclear structure calculations, the second Pöschl–Teller potential is parameterized to study nucleon–nucleon and nucleon–nucleus elastic scattering. By exploiting the variable phase approach to potential scattering, phase parameters, cross sections, analyzing powers, etc., are computed and compared with earlier works. The calculated cross sections agree with the experiments below 25 MeV within the error bars for the (n-p) and (n-d) systems.

Repulsive Casimir–Lifshitz pressure in closed cavities

We consider the interaction pressure acting on the surface of a dielectric sphere enclosed within a magnetodielectric cavity. We determine the sign of this quantity regardless of the geometry of the cavity for systems at thermal equilibrium, extending the Dzyaloshinskii-Lifshitz-Pitaevskii result for homogeneous slabs. As in previous theorems regarding Casimir-Lifshitz forces, the result is based on the scattering formalism. In this case the proof follows from the variable phase approach of electromagnetic scattering. With this, we present configurations in which both the interaction and the self-energy contribution to the pressure tend to expand the sphere.

VPA: Computer program for the computation of the phase shift in atom–atom potential scattering using the Variable Phase Approach

A computer code for the computation of the phase shift in atom–atom and electron–atom potential scattering is presented. The phase shift is the central quantity required for the calculation of all types of scattering cross sections. The program uses the Variable Phase Approach (VPA). This is the only exact method for the direct calculation of the scattering phase shift. All other methods are based on examining the large distance behavior of the exact solution of the Schrödinger equation. Such methods yield the phase shift only modulo π. The absolute value of the phase shift and its variation with scattering energy is, however, needed for a full understanding of the scattering process, such as for instance in the study of shape resonances and Glory oscillations. The VPA has been sparingly used owing to the instability of the underlying equations and the consequent difficulty of writing computer code to solve them. We present a computer code for the efficient implementation of the VPA method for atom–atom scattering problems over a wide range of scattering energies. The code works for potentials which are singular and for those that are non-singular at the origin. An example of the implementation of the code is given for both an interaction potential with an attractive well and for a purely repulsive potential. Program summaryProgram title: VPA:Variable Phase ApproachCPC Library link to program files: https://dx.doi.org/10.17632/zh248d2362.1Code Ocean capsule: https://dx.doi.org/10.24433/CO.8146850.v1Licensing provisions: GNU General Public License 3 (GPL)Programming language: Fortran 90Nature of problem: To compute the quantum mechanical phase shift in an atom–atom or an electron–atom potential scattering problem. The interaction potential, dependent on a single radial coordinate, is defined by the user.Solution method: The Variable Phase Approach is used. This is the only available method for the direct calculation of the phase shift. The initial internuclear separation for the integration is chosen with the help of the analytical solution for a hard sphere core for all orbital angular momentum quantum numbers L of interest. The first-order nonlinear equation is integrated numerically with a modified LSODA19,20 program. The analytic formula for calculation of the Jacobian of the equation is also provided.Additional comments including restrictions and unusual features: Computer codes for the atom–atom interaction potential, POTENT1, and for its derivative with respect to the inter nuclear separation, DPOTENT1, must be supplied by a user. The calculation of the spherical Bessel and Neumann functions required is performed explicitly for large l angular momentum quantum numbers where the standard upward iterative generation of the functions becomes unstable.

Trilateration‐based indoor localization engineering technique for visible light communication system

SummaryThis article is aimed at designing positioning system using visible light communication technology by incorporating three different diversity schemes, namely, transmit diversity (TD), receive diversity (RD), and angle diversity. For the TD, multiple transmitters with single receiver (MISO) is used to obtain the target location using frequency division multiplexing (FDM) and variable phase approach. To deploy RD, the proposed system comprises multiple receivers and single transmitter (SIMO). Only one transmitter's absolute coordinates are required to obtain the location coordinates of optical receivers. FDM and time division multiplexing access schemes have been utilized to transmit the unique signals by light‐emitting diodes (LEDs). To implement angle diversity, orientation of transmitter and receiver is considered for positioning purposes. Under this environment, there is no constraint for the LEDs to be installed on ceiling; they can be at any point within the room or at different heights. The information regarding spherical coordinates (representing the orientation) of LEDs (single or multiple) is used to determine the coordinates or the orientation of receiver having multiple or single photodiodes, respectively. Under multiple orientations, this system is analyzed with both scenarios, that is, MISO and SIMO.

Approximation of scattering phases for Argonne v18 potential

Purpose. This article is dedicated to the study of phase shifts for single-channel neutron-proton scattering. The phase shifts, which are obtained in the variable phase approach, are studied. For the approximation of phase shifts, a function — 2019 is used in the form of a known formula for a parabolic-type quadratic function y(x)=a+bx+cx2 , which was proposed by M.A. Dolgopolov, L.A. Minin and V.A. Rabotkin in the study of phase shifts for the Paris potential. It is necessary to find the coefficients a, b, c for y(x) when approximating the np- scattering phases for the Argonne v18 potential. Methods. Approximation is used to find the coefficients of the function y(x). Find the optimal values of the coefficients a, b, c at a minimum value χ 2 . Results. An approximation of the neutron-proton scattering phases obtained by various methods for the phenomenological nucleon-nucleon potential Argonne v18 was made. The calculated coefficients of this function obtained for phase shifts by means of the variable phase approach are compared with the results of the approximation of the scattering phases from the original paper. For illustration graphs are given for phase shifts for four spin states for the np- system ( 1S0, 3P1, 1D2, 3D2). Conclusions. The obtained coefficients for the function y(x) can be used to calculate the values of the scalar scattering amplitude, the full cross-section and the partial scattering amplitude for any phase value within the energy interval for the approximation carried out. In next studies one can obtain and compare the coefficients and parameters of approximation for scattering phases for other modern phenomenological nucleon-nucleon potentials. Keywords: variable phase approach, nucleon-nucleon scattering, approximation, phase shifts, Arg

Single-particle relaxation time in doped semiconductors beyond the Born approximation

We compare the magnitudes of the single-particle relaxation time exactly computed in the variable phase approach with those computed in the first Born approximation for doped semiconductors such as Si and GaAs, assuming that the Coulomb impurities are randomly distributed centers. We find that for typical dopant concentrations in Si, the Born approximation can overestimate the single-particle relaxation time by roughly 40% and underestimate it by roughly 30%. Finally, we show that the strong interference of phase shifts is missing in the strong scattering regime where the Born approximation fails.

A new fully quantum-mechanical method used to calculate the collisional broadening coefficients and shift coefficients of Rb D1 lines perturbed by noble gases He and Ar**Project supported by the National Natural Science Foundation of China (Grants Nos. 11405077 and 11575073).

In this work, a new full quantum method is proposed to calculate the broadening and shift coefficients of the D1 line in neutral collision. Based on the variable phase approach and Baranger theory, this method calculates the scattering phase shift instead of scattering matrix elements in order to simplify the calculation. As an illustration, this method is used to calculate the broadening and shift coefficients of the absorption lines of alkali metal atom Rb, as it collides with buffer gas He and Ar, in a temperature range from 150 K to 800 K. With a comparison with other calculations and experiment measurements, the reasonable agreements in all cases demonstrate the validity and simplicity of this method.

Scattering of a slow quantum particle on a central short-range potential

The linear version of the variable-phase approach in the theory of potential scattering is supplemented with a new asymptotic method. This method is intended for performing a quantum-mechanical analysis and for constructing explicit low-energy approximations for partial-wave phase shifts and radial components of the wave function for the scattering of a quantum particle on a central short-range potential. The procedure used to construct all low-energy approximations reduces to solving a recursion chain of energy-independent sets of equations, each such set consisting of two linear first-order differential equations.

Scattering of a slow quantum particle on an axially symmetric short-range potential

A linear version of the variable-phase approach in potential-scattering theory is supplemented with a new asymptotic method. This method is intended for analyzing quantum mechanically and for constructing explicit low-energy approximations for partial-wave phase shifts, amplitudes, cross sections, and radial components of the wave function for the scattering of a quantum particle on an axially symmetric short-range potential. The procedure used to construct all low-energy approximations reduces to solving a recursion chain of energy-independent sets of equations, each such set consisting of two linear first-order differential equations.

Proton scattering by a hydrogen atom in an effectively two-body model

It is assumed that the total potential of proton interaction with a hydrogen atom is the sum of the short-range nuclear soft-core Reid potential and the long-range Thomas-Fermi potential. A quantum mechanical analysis of low-energy features of the phase shift and cross section for elastic proton scattering on a hydrogen atom is given for the case of zero total angular momentum. The calculations performed in the present study within a nonlinear version of the variable-phase approach ultimately revealed that, because of a long-range character of the asymptotic behavior of the Thomas-Fermi potential, the respective cross section at low energies oscillates but has a finite number of zeros.

Theoretical calculation of total cross sections for e+ – NH3 molecule at low energies

A parameter-free spherical complex optical potential model is used to calculate the differential and total cross sections for the scattering of positrons by the NH3 molecules in the energy range 10 eV − 400 eV. The optical potential is constructed from a near-Hartree-Fock one-center expansion of ammonia (NH3) wave function and it is then treated exactly in a partial-wave analysis involving variable phase approach to extract complex phase shifts. The present theoretically calculated results show good agreement with the available experimental results for total cross sections at positron energy ≥ 30 eV.

Ionization cross section for a strongly coupled partially ionized hydrogen plasma: variable phase approach

In the present work an electron impact ionization cross section is considered. The electron impact ionization cross section is calculated with the help of a variable phase approach to potential scattering. The Calogero equation is numerically solved, based on a pseudopotential model of interaction between partially ionized plasma particles, which accounts for correlation effects. As a result, scattering phase shifts are obtained. On the basis of the scattering phase shifts, the ionization cross section is calculated.

Relativistic optical model on the basis of the Moscow potential and lower phase shifts for nucleon-nucleon scattering at laboratory energies of up to 3 GeV

Data of a partial-wave analysis of nucleon-nucleon scattering at energies of up to Elab = 3 GeV (lower partial waves) and the properties of the deuteron are described within the relativistic optical model based on deep attractive quasipotentials involving forbidden states (as exemplified by the Moscow potential). Partial-wave potentials are derived by the inverse-scattering-problem method based on the Marchenko equation by using present-day data from the partial-wave analysis of nucleon-nucleon scattering at energies of up to 3 GeV. Channel coupling is taken into account. The imaginary parts of the potentials are deduced from the phase equation of the variable-phase approach. The general situation around the manifestation of quark effects in nucleon-nucleon interaction is discussed.

Reconstruction of the optical potential from scattering data

We propose a method for reconstructing the optical potential (OP) from scattering data. The algorithm is a two-step, procedure. In the first step, the real part of the potential is determined analytically via solution of the Marchenko equation. At this point, we use a rational function fit of the corresponding unitary S matrix. In the second step, the imaginary part of the potential is determined via the phase equation of the variable phase approach. We assume that the real and imaginary parts of the OP are proportional (the Lax-type interaction). We use the phase equation to calculate the proportionality coefficient. A numerical algorithm is developed for a single and for coupled partial waves. The developed procedure is applied to analyze the ${}^{1}{S}_{0}\mathit{NN},{}^{3}{\mathit{SD}}_{1}\mathit{NN}$, and $P31\phantom{\rule{0.3em}{0ex}}{\ensuremath{\pi}}^{\ensuremath{-}}N$ data. For the NN states, we constructed partial potentials with forbidden states (Moscow potential). We examine the ${\ensuremath{\pi}}^{\ensuremath{-}}N$ and NN partial-wave analysis data and demonstrate that the Lax-type interaction may be valid for these systems at the considered energies.

Scattering and bound-state problems with non-local potentials: application of the variable-phase approach

Following the framework of the variable-phase approach, we derive an equation fo rd etermining the scattering amplitude of a non-relativistic quantum particle in a non-local potential. Its solution implies the integration of the Volterra integrodifferential equation of the first kind and allows determination of bound-state energies and wavefunctions. A fast numerical scheme for the solution of these equations is suggested and it is demonstrated that the proposed method requires the numerical efforts of the same order as in the local potential case.

The absolute definition of the phase-shift in potential scattering

The variable phase approach to potential scattering with regular spherically symmetric potentials satisfying Eq. (1), and studied by Calogero in his book [Variable Phase Approach to Potential Scattering (Acadamic, New York, 1967)] is revisited, and we show directly that it gives the absolute definition of the phase-shifts, i.e., the one which defines δl(k) as a continuous function of k for all k⩾0, up to infinity, where δl(∞)=0 is automatically satisfied. This removes the usual ambiguity ±nπ, n integer, attached to the definition of the phase-shifts through the partial wave scattering amplitudes obtained from the Lippmann–Schwinger integral equation, or via the phase of the Jost functions. It is then shown rigorously, and also on several examples, that this definition of the phase-shifts is very general, and applies as well to all potentials which have a strong repulsive singularity at the origin, for instance those which behave like gr−m, g>0, m⩾2, etc. We also give an example of application to the low-energy behavior of the S-wave scattering amplitude in two dimensions, which leads to an interesting result.

Numerical investigation of quasilinearization method in quantum mechanics

The general properties of the quasilinearization method (QLM), particularly its fast convergence, monotonicity and numerical stability are analyzed and verified on the example of scattering length calculations in the variable phase approach to quantum mechanics. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), approximates the solution of a nonlinear differential equation by treating the nonlinear terms as a perturbation about the linear ones, and is not based, unlike perturbation theories, on the existence of some kind of a small parameter. Each approximation of the method sums many orders of the perturbation theory. It is shown that already the first few iterations provide very accurate and numerically stable answers for small and intermediate values of the coupling constant. The number of iterations necessary to reach a given precision only moderately increases for its larger values. The method provides accurate and stable answers for any coupling strengths, including for super singular potentials for which each term of the perturbation theory diverges and the perturbation expansion does not exist even for a very small coupling.

Screened excitons in wide-gap semiconductors and quantum wells

The variable-phase approach is applied to scattering and bound states in screened Coulomb potentials in two and three dimensions. The degree of ionisation for the electron—hole plasma is calculated taking into account scattering in the continuum. A two-dimensional formulation of Levinson's theorem is used for bound-state counting and a simple relationship between the screening length and the number of bound states is found for an attractive Coulomb potential screened by a two-dimensional electron gas.

Variable-phase method and Levinson's theorem in two dimensions: Application to a screened Coulomb potential

The variable-phase approach is applied to scattering and bound states in an attractive Coulomb potential, statically screened by a two-dimensional (2D) electron gas. A 2D formulation of Levinson's theorem is used for bound-state counting and a hitherto undiscovered, simple relationship between the screening length and the number of bound states is found. As the screening length is increased, sets of bound states with differing quantum numbers appear degenerately.