Number Of Partial Waves Research Articles

Quantum dynamics of the P(2D) + H2 (X1[formula omitted]) → PH (X3Σ-) + H(2S) reaction

The title reaction is thought to be important for the astronomical PO formation, which may contribute to the generation of bioavailable H3PO3. Using quasi-classical trajectory (QCT) method, the state-specified (v = 0, j = 0) rate constant of this reaction was calculated. The comparison with previous temperature-averaged QCT rate constant shows that the effect of ro-vibrational excitation is relatively small below 2000 K. Then, the time-dependent wave packet (TDWP) method was applied to investigate the quantum reaction mechanisms, obtaining the state-specified (v = 0, j = 0) reaction probability, integral cross sections and rate constants. The number of partial waves and the integral interval of collision energy considered were rigorously tested to ensure the convergence of the TDWP rate constant below 2000 K. The TDWP rate constant agrees better to the latest experimental results at 291 to 685 K than those from the two QCT calculations. Also, the TDWP calculations can produce relatively reliable rate constants at 700 to 2000 K, where the Arrhenius formula fitted by experimental results ignores the temperature dependence of the activation energy. The calculated rate constants are helpful for modelling the phosphorus chemistry and determining the abundances of the P-bearing species in interstellar media (ISM).

QPC-TDSE: A parallel TDSE solver for atoms and small molecules in strong lasers

The QPC-TDSE program serves as a general tool to study laser-driven dynamics of electrons in ideal isolated atoms and molecules by solving the full-dimensional non-relativistic time-dependent Schrödinger equation (TDSE) within single-active-electron approximation. It expands the full-dimensional electronic wavefunction in spherical coordinates by spherical harmonics and B-spline functions and employs a set of parallel Crank-Nicolson propagators combined with split-operator techniques to evolve the wavefunction in time, which support centrifugal and multi-polar static potentials to treat atomic and molecular scenarios and accepts arbitrary combinations of linearly or elliptically polarized lasers within the dipole approximation. The program is capable of extracting the photo-electron momentum distribution via t-SURFF approach or projection onto either the exact scattering states or the planewaves. Its applications in different scenarios are given as examples, e.g., above threshold ionization, attosecond clock, higher-order harmonic generation. Program summaryProgram Title: QPC-TDSECPC Library link to program files:https://doi.org/10.17632/xjm3kfgv75.1Licensing provisions: GPLv3Programming language: C++External libraries: HDF5, GSL, MKLNature of problem: Numerical solution of TDSE and extraction of various types of electron spectrum.Solution method: The electronic wavefunction is expanded by B-spline functions and spherical harmonics whose range is chosen elaborately to reduce the total number of partial waves for non-linearly polarized lasers. The Crank-Nicolson approach combined with an operator-splitting scheme is used to propagate the wavefunction in time, either in velocity gauge or length gauge. Matrix inversions are solved via either dense or sparse linear algebra solvers according to their structures. The t-SURFF method and projections onto either the scattering states or planewaves are provided for the accurate extraction of the momentum distributions.Additional comments including restrictions and unusual features: Only lasers within dipole approximation are supported. For the multi-polar potentials, only pure Coulombic ones are supported. Routines for solving exact scattering states have only been implemented for centrifugal potentials. The codes are written in C++17 and can only be compiled on the platforms that support the avx instruction sets. An extension for the propagation algorithm using the avx-512 intrinsics is provided as optional.

A Kr*–Rb cold collision apparatus based on atom trap

A cold collision between atoms and molecules (<1 K) is one of the hot research fields in atomic and molecular physics. At low temperatures, the number of partial waves participating in the collision process decreases dramatically, and quantum phenomena start to emerge. The reaction is often dominated by quantum tunneling, and pronounced resonances can exist on collision cross sections. Here, we report on an apparatus designed for studying cold collisions between metastable noble gas atoms and alkali atoms. Our apparatus features a combined Magneto-Optical-Trap (MOT) and velocity map imaging (VMI) system. The center of a Rb MOT is overlapped with the VMI system. Cold Kr* atoms are launched toward the Rb atoms to induce Kr* + Rb reactions. The collision energy between the two species can be varied from 100 mK to 20 K. With this setup, we are planning to explore the quantum phenomena in Kr* + Rb cold collisions, including the shape resonance and stereodynamics in the reaction.

BREMS: Partial-wave calculation of spectra and angular distributions of electron-atom bremsstrahlung at electron energies less than 30 MeV (New Version Announcement)

BREMS: Partial-wave calculation of spectra and angular distributions of electron-atom bremsstrahlung at electron energies less than 30 MeV (New Version Announcement)

Calculating the S -matrix of low-energy heavy-ion collisions using quantum coupled-channels wave-packet dynamics

We investigate the fusion and scattering of a $^{16}\mathrm{O}$ projectile on $^{152,154}\mathrm{Sm}$ targets using the time-dependent coupled-channels wave-packet method. We benchmark calculations of the $S$-matrix elements, fusion cross sections, and scattering differential cross sections with those from the time-independent coupled-channels method, and compare the results to experimental data. We find that our time-dependent method and the time-independent method produce quantitatively similar results for the $S$-matrix elements and fusion cross sections, but our method cannot quantitatively explain the experimental scattering differential cross sections, mainly due to the low maximum number of partial waves produced by the time-dependent method. Nevertheless, the strong agreements between our method and the time-independent method demonstrates that the time-dependent coupled-channels wave-packet method can be used to address fusion reactions for a wide range of energies, with the advantage of being able to extend to time-dependent Hamiltonians for more advanced modeling of nuclear reactions.

Calculation of double differential cross sections of electron–atom bremsstrahlung by the relativistic partial-wave method at electron energies greater than 3 MeV

This work describes an extension of the relativistic partial-wave (PW) method for calculation of the double differential cross sections (DDCS) of electron–atom bremsstrahlung at incident electron energies (T1) of the order of 10 MeV. The PW method is extended to higher energies by replacing some of the radial matrix elements with interpolated values, and by extrapolating the truncated PW series to a sufficiently high value of the maximum total angular momentum of the electron (jmax). Although the current analysis has been mostly limited to the cases with the outgoing electron energy T2≤10MeV and all values of Z from 1 to 100, this method allows to obtain accurate values of the DDCS even when T2>20MeV, using a relatively small number of partial waves (jmax<300), as demonstrated by a case study described in the present work (Z=1, T1=30MeV, T2=24MeV).

Hyperspherical three-body model calculation for the bound and resonant states of the neutron dripline nuclei 42,44Mg using isospectral potential

In this paper, an effective theoretical scheme is presented to investigate ground and resonant levels of weakly bound nuclei near the neutron-dripline. The hyperspherical harmonics expansion method is employed for the bound state while supersymmetric quantum mechanics (SSQM) is used for the resonant state. The three-body Schrödinger equation is solved for the lowest bound state with chosen set of two-body potentials to obtain the energy and normalized wavefunction. The three-body effective potential has a very shallow well following a low-wide barrier giving a broad resonance due to far extended wavefunction in the asymptotic region. Numerical computation of such broad resonance is very challenging due to large computational errors. Hence, a one-parameter family of isospectral potential is constructed with the ground state energy and wavefunction, by the use of the algebra of SSQM. This enhanced potential facilitates a more accurate computation of the resonance energy and width. Effectiveness of the scheme has been checked for the J=π0+ bound and resonant states of neutron-rich nuclei 42,44Mg in the coreNN three-body cluster model where core=40,42Mg. In the present calculation, two-neutron separation energies obtained with very large number of partial waves for the J=π0+ bound states of neutron-rich nuclei 42,44Mg are found to be 0.41433 MeV and 0.44975 MeV respectively. Results of the calculation have been compared with those found in the literature.

Comparison of semiclassical transfer to continuum model with Ichimura-Austern-Vincent model in medium energy knockout reactions

The full quantum mechanical (QM) model of inclusive breakup of Ichimura-Austern-Vincent (IAV) is implemented in this paper to calculate breakup from heavy radioactive nuclei on a 9Be target at intermediate energies. So far it had been implemented and applied only to low energy reactions with light projectiles. The IAV model is successful in predicting absolute cross sections among other observables. In order to get insight on the content of the model in the case of the complicated heavy-ion reactions, results are compared with those of the semiclassical transfer to the continuum (TC) model. Because the TC is based on analytical formulae the dynamics of the breakup as it is contained in the rather involved IAV formalism will become more transparent. Heavy-ion reactions at high energies (>50 A.MeV) are demanding from the computational point of view because of the high number of partial waves involved, typically around 100. The TC constitutes a useful alternative to the full QM calculations whenever predictions and/or estimates are necessary. It allows also for a systematic, fast evaluation of breakup observables. In the applications of both methods we use state-of-the art optical potentials and structure information. Excellent agreement is found between the calculated results of both methods and with available experimental data which shows that the qualitative and quantitative understanding of most aspects of one nucleon breakup is well under control.

Scattering Observables from One- and Two-body Densities: Formalism and Application to $$\pmb \gamma $$ $${}^3\hbox {He}$$ Scattering

We introduce the transition-density formalism, an efficient and general method for calculating the interaction of external probes with light nuclei. One- and two-body transition densities that encode the nuclear structure of the target are evaluated once and stored. They are then convoluted with an interaction kernel to produce amplitudes, and hence observables. By choosing different kernels, the same densities can be used for any reaction in which a probe interacts perturbatively with the target. The method therefore exploits the factorisation between nuclear structure and interaction kernel that occurs in such processes. We study in detail the convergence in the number of partial waves for matrix elements relevant in elastic Compton scattering on $^3$He. The results are fully consistent with our previous calculations in Chiral Effective Field Theory. But the new approach is markedly more computationally efficient, which facilitates the inclusion of more partial-wave channels in the calculation. We also discuss the usefulness of the transition-density method for other nuclei and reactions. Calculations of elastic Compton scattering on heavier targets like $^4$He are straightforward extensions of this study, since the same interaction kernels are used. And the generality of the formalism means that our $^3$He densities can be used to evaluate any $^3$He elastic-scattering observable with contributions from one- and two-body operators. They are available at https://datapub.fz-juelich.de/anogga.

Spherical harmonics expansion of the evanescent waves in angular spectrum decomposition of shaped beams

The angular spectrum decomposition (ASD) is a powerful tool for analyzing the interaction between spherical particles and shaped beams, in which homogeneous plane waves and/or evanescent waves are expanded into series of vector spherical harmonics (VSH). The paper presents a numerical study on the VSH expansion of the evanescent wave and its behavior in the field reconstruction of the shaped beam. It is revealed that the beam shape coefficients (BSCs) of the evanescent waves increase exponentially along the increasing partial wave number, consequently extending the convergence of the series and restricting the field reconstruction into a limited space.

Nd-Scattering within MGL Approach for Configuration-Space Faddeev Equations

We study the neutron-deuteron elastic and breakup scattering on the basis of the configuration-space Faddeev equations. The Merkuriev–Gignoux–Laverne approach has been generalized for arbitrary nucleon-nucleon potentials and with an arbitrary number of partial waves. Neutron-deuteron observables at the incident nucleon energy 14.1 MeV are calculated using the realistic AV14 nucleon-nucleon potential. Results are compared with those of other authors and with experimental neutron-deuteron scattering data. The computational procedure is presented in details.

Low temperature scattering with the R-matrix method: argon-argon scattering

Results for elastic atom-atom scattering are obtained as a first practical application of RmatReact, a new code for generating high-accuracy scattering observables from potential energy curves. RmatReact has been created in response to new experimental methods which have paved the way for the routine production of ultracold (μK) atoms and molecules, and hence the experimental study of chemical reactions involving only a small number of partial waves. Elastic scattering between argon atoms is studied here. There is an unresolved discrepancy between different potential energy curves which give different numbers of vibrational bound states and different scattering lengths for the dimer. Depending on the number of bound states, the scattering length is either large and positive or large and negative. Scattering observables, specifically the scattering length, effective range, and partial and total cross-sections, are computed at low collision energies and compared to previous results. In general, good agreement is obtained, although our full scattering treatment yields resonances which are slightly lower in energy and narrower than previous determinations using the same potential energy curve.

Analysis of fusion excitation functions of reactions 6He +209Bi and 7Li +209Bi around Coulomb barrier

Here, we have studied the sensitivity of fusion excitation functions of reactions induced by weakly bound projectiles [Formula: see text] and [Formula: see text] on [Formula: see text] target on nuclear potential parameters and on number of partial waves. The Kemble version of WKB approximation and Hill–Wheeler formula has been used to predict the fusion transmission probability below and above the Coulomb barrier, respectively, and the optimum values for radius (r0) = 1.17 fm, diffuseness (a) = 0.5 fm and for partial waves (l) = up to 60 are proposed. The coupled channel calculations have also been performed and it is found that the matching between data and predictions have been enhanced on inclusion of coupling effects. Further, the breakup effects are also taken into account through the dynamic polarization potential (DPP) approach. It further improves matching between data and predictions.

Three-nucleon bound state calculations using the three dimensional formalism

The traditional method of carrying out few-nucleon calculations is based on the angular momentum decomposition of operators relevant to the calculation. Expressing operators using a finite-sized partial wave basis enables the calculations to be carried out using a small amount of numerical work. Unfortunately, certain calculations that involve higher energies or long range potentials, require including a large number of partial waves in order to get converged results. This is problematic because such an approach requires a numerical implementation of heavily oscillating functions. Modern computers made it possible to carry out few-nucleon calculations without using angular momentum decomposition and instead to work directly with the three dimensional degrees of freedom of the nucleons. In this paper we briefly describe the, so called 3D approach and present preliminary results related to the 3He bound state obtained within this formalism.

Supersonic beams of mixed gases: A method for studying cold collisions

We show that collisions in a single supersonic molecular beam reach characteristic temperatures in the range of a few Kelvin. Experiments have been carried out for mixtures of H2 and HD as well as HD and D2 in a pulsed supersonic expansion. From the measured time-of-flight spectrum, we find that the high velocity edge of the distribution for both species is nearly coincident, but the average speed of the heavier species is slightly greater than the lighter one. By working in the few Kelvin regime, this relatively simple technique reduces the number of partial waves in the collision process, which allows the observation of collision resonances. Additionally, copropagation of scattering partners in a single molecular beam precisely defines the direction of their relative velocity vector, which is essential for the study of stereodynamics.

Acceleration of diffraction calculations in cylindrically symmetrical optics.

We have significantly accelerated diffraction calculations using three independent acceleration devices. These innovations are restricted to cylindrically symmetrical systems. In the first case, we consider Wolf's formula for integrated flux in a circular region following diffraction of light from a point source by a circular aperture or a circular lens. Although the formula involves a double sum, we evaluate it with the effort of a single sum by use of fast Fourier transforms (FFTs) to perform convolutions. In the second case, we exploit properties of the Fresnel-Kirchhoff propagator in the Gaussian, paraxial optics approximation to achieve the propagation of a partial wave from one optical element to the next. Ordinarily, this would involve a double loop over the radial variables on each element, but we have reduced the computational cost by a factor approximately equal to the smaller number of radius values. In the third case, we reduce the number of partial waves, for which the propagation needs to be calculated, to determine the throughput of an optical system of interest in radiometry when at least one element is very small, such as a pinhole aperture. As a demonstration of the benefits of the second case, we analyze intricate diffraction effects that occur in a satellite-based solar radiometry instrument.

Variational treatment of electron–polyatomic-molecule scattering calculations using adaptive overset grids

The Complex Kohn variational method for electron-polyatomic molecule scattering is formulated using an overset grid representation of the scattering wave function. The overset grid consists of a central grid and multiple dense, atom-centered subgrids that allow the simultaneous spherical expansions of the wave function about multiple centers. Scattering boundary conditions are enforced by using a basis formed by the repeated application of the free particle Green's function and potential, $\hat{G}^+_0\hat{V}$ on the overset grid in a "Born-Arnoldi" solution of the working equations. The theory is shown to be equivalent to a specific Pad\'e approximant to the $T$-matrix, and has rapid convergence properties, both in the number of numerical basis functions employed and the number of partial waves employed in the spherical expansions. The method is demonstrated in calculations on methane and CF$_4$ in the static-exchange approximation, and compared in detail with calculations performed with the numerical Schwinger variational approach based on single center expansions. An efficient procedure for operating with the free-particle Green's function and exchange operators (to which no approximation is made) is also described.

Elastic differential cross sections for space radiation applications

The eikonal, partial wave (PW) Lippmann-Schwinger, and three-dimensional Lippmann-Schwinger (LS3D) methods are compared for nuclear reactions that are relevant for space radiation applications. Numerical convergence of the eikonal method is readily achieved when exact formulas of the optical potential are used for light nuclei $(A\ensuremath{\le}16)$, and the momentum-space representation of the optical potential is used for heavier nuclei. The PW solution method is known to be numerically unstable for systems that require a large number of partial waves, and, as a result, the LS3D method is employed. The effect of relativistic kinematics is studied with the PW and LS3D methods and is compared to eikonal results. It is recommended that the LS3D method be used for high-energy nucleon-nucleus reactions and nucleus-nucleus reactions at all energies because of its rapid numerical convergence and stability.

Properties of nuclear matter within the JISP16NNinteraction

Saturation properties of the JISP16 NN interaction are studied in symmetric nuclear matter calculations, with special attention paid to the convergence properties with respect to the number of partial waves. We also present results of pure neutron matter calculations with the JISP16 interaction.

Calculations of three-nucleon reactions with N3LO chiral forces: achievements and challenges

We discuss the application of the chiral N3LO forces to three-nucleon reactions and point to the challenges which will have to be addressed. Present approaches to solve three-nucleon Faddeev equations are based on a partial-wave decomposition. A rapid increase of the number of terms contributing to the chiral three-nucleon force when increasing the order of the chiral expansion from N2LO to N3LO forced us to develop a fast and effective method of automatized partial-wave decomposition. At low energies of the incoming nucleon below ≈20 MeV, where only a limited number of partial waves is required, this method allowed us to perform calculations of reactions in the three-nucleon continuum using N3LO two- and three-nucleon forces. It turns out that inclusion of consistent chiral interactions, with relativistic 1/m corrections and short-range 2π-contact term omitted in the N3LO three-nucleon force, does not explain the long standing low energy Ay-puzzle. We discuss problems arising when chiral forces are applied at higher energies, where large three-nucleon force effects are expected. It seems plausible that at higher energies, due to a rapid increase of a number of partial waves required to reach convergent results, a three-dimensional formulation of the Faddeev equations which avoids partial-wave decomposition is desirable.