Partial Wave Phase Shifts Research Articles

Contrast factor for standing-wave radiation forces on spheres: Series expansion in powers of sphere radius.

Recently researchers often normalize the radiation force on spheres in standing waves in inviscid fluids using an acoustic contrast factor (typically denoted by Φ) that is independent of kR where k is the wave number and R is the sphere radius. An alternative normalization uses a function Ys that depends on kR. Here, standard results for Φ are extended as a power series in kR using prior Ys results. Also, new terms are found for fluid spheres and applied to the kR dependence of Φ for strongly responsive and weakly responsive examples. Partial-wave phase shifts are used in the derivation.

Analytical solution for nuclear plus atomic Hulthén potential — A perturbative approach

In this work, we construct the analytical solution of the Schrödinger equation for a combined nuclear plus atomic Hulthén potential with different range parameters, using the Frobenius method. The atomic Hulthén potential acts as the screened Coulomb potential to represent the very short-range electromagnetic interaction. It is intended to emphasize how the impact of the combined potential in these situations is routinely examined within the context of nuclear physics. The Jost function is calculated from its integral representation in terms of the regular solution. From the phase of the Jost function, the scattering phase shifts for different partial waves, and further differential scattering cross-section and total cross-sections are calculated for alpha-proton and proton–proton systems. On comparing with the existing experimental data, we conclude that the results obtained are in close conformity with the previous works that exist in the literature.

Study of electron impact elastic scattering from Kr@C60 and Xe@C60 using a fully relativistic approach

In the present work, a detailed study has been reported on electron impact elastic scattering from krypton (Kr) and xenon (Xe) atoms when confined in two different types of C60 potentials viz (a) hard annular square well (ASW) and (b) diffused Gaussian annular square well (GASW). The Dirac equations are solved using these potentials for encaged Kr and Xe in C60. First, bound state Dirac–Fock wave functions of these encaged Kr and Xe atoms are found by utilizing modified general relativistic atomic structure package and thereafter, the charge densities and static potentials of the endohedral Kr@C60 and Xe@C60 are obtained. Further, using these, the Dirac equations are solved by the relativistic partial wave phase shift analysis method and the scattering amplitudes in terms of phase shifts are obtained. Thereafter, the electron elastic differential and integrated cross sections of Kr@C60 and Xe@C60 along with the C60 are calculated in the range of 0.1–15 eV incident electron energies. Presently, no experimental and theoretical results are available to compare our electron scattering cross section results from Kr@C60 and Xe@C60; thus, we have shown the cross section results obtained from ASW and GASW potential and compared them.

Partial-Wave Phase Shift Analysis of Elastic Pion-Nucleon Scattering

Partial-Wave Phase Shift Analysis of Elastic Pion-Nucleon Scattering

Nonperturbative two-pion exchange contributions to the nucleon-nucleon interaction in covariant baryon chiral perturbation theory

We calculate the nonperturbative two-pion exchange (TPE) contributions to the $NN$ interaction in covariant baryon chiral perturbation theory. We study how the nonperturbative resummation affects the $NN$ phase shifts for partial waves with $J \geq 3$ and $L \leq 6$. No significant differences are observed between the nonperturbative phase shifts and perturbative ones for most partial waves except for $^3D_3$, for which the nonperturbative resummation greatly improves the description of the phase shifts. However, a significant cutoff dependence is found for this partial wave and a reasonable description of the phase shifts can only be obtained with a particular cutoff. Furthermore, we compare the so-obtained nonperturbative phase shifts with those obtained in the heavy baryon chiral perturbation theory. We show that the contributions from relativistic nonperturbative TPE are more moderate than those from the nonrelativistic TPE obtained in the dimensional regularization scheme. A proper convergence pattern is observed for most of the partial waves studied except for $^3F_3$, $^3F_4$, and $^3H_6$, for which the subleading TPE contributions are a bit strong. We find that for $H$ and $I$ partial waves, the OPE alone can already describe the phase shifts reasonably well.

Study of Positron Impact Scattering from Methane and Silane Using an Analytically Obtained Static Potential with Correlation Polarization

A detailed study of positron impact elastic scattering from methane and silane is carried out using a model potential consisting of static and polarization potentials. The static potential for the molecular target is obtained analytically by using accurate Gaussian molecular wavefunctions. The molecular orbitals are expressed as a linear combination of Gaussian atomic orbitals. Along with the analytically obtained static potential, a correlation polarization potential is also added to construct the model potential. Utilizing the model potential, the Schrödinger equation is solved using the partial wave phase shift analysis method, and the scattering amplitude is obtained in terms of the phase shifts. Thereafter, the differential, integrated and total cross sections are calculated. These cross-section results are compared with the previously reported measurements and theoretical calculations.

Fully relativistic study on electron impact elastic scattering from Nq+ (q = 1–3), Na+, Arq+ (q = 1–3, 7–8), and Xeq+ (q = 2–6, 8)

AbstractA study is presented on the elastic scattering of electrons from Nq+ (q = 1–3), Na+, Arq+ (q = 1–3, 7–8), Xeq+ (q = 2–6, 8) to understand the available experimental differential cross section results. A model potential approach has been utilized to describe the scattering process. The model potential includes the static, exchange, polarization and absorption potentials. The static potentialis obtained through the charge density calculated by obtaining ionic wave functions using multi‐configuration Dirac‐Fock (MCDF) approximation. Thereafter, the static potential is added to the suitable exchange, polarisation and absorption potentials to construct the spherically averaged complex optical potential. Using the obtained potential in the Dirac equations,these are solved with the partial wave phase shift analysis method and the differential cross sections are calculated. Results for different ions exhibit prominent interference structures in the energy versus cross section curves and show good agreement on comparison with the experimental results available in the selected energy ranges.

The Relevance of Pion-Exchange Contributions Versus Contact Terms in the Chiral Effective Field Theory Description of Nucleon–Nucleon Scattering

The standard way to demonstrate the relevance of chiral symmetry for the NN interaction is to consider higher partial waves of NN scattering which are controlled entirely by chiral pion-exchanges (since contacts vanish). However, in applications of NN-potentials to nuclear structure and reactions, the lower partial waves are the important ones, making the largest contributions. Lower partial waves are sensitive to the short-range potential, and so, when the short-range contacts were to dominate over the chiral pion-contributions in lower partial waves, then the predictions from "chiral potentials" would have little to do with chiral symmetry. To address this issue, we investigate systematically the role of the (chiral) one- and two-pion exchanges, on the one hand, and the effect of the contacts, on the other hand, in the lower partial waves of NN scattering. We are able to clearly identify the signature of chiral symmetry in lower partial waves. Our study has also a pedagogical spin-off as it demonstrates in detail how the reproduction of the lower partial-wave phase shifts comes about from the various ingredients of the theory.

Study of Electron and Positron Elastic Scattering from Hydrogen Sulphide Using Analytically Obtained Static Potential

A detailed study of elastic scattering of electrons and positrons from a hydrogen sulphide (H2S) molecule is presented using the method of partial wave phase shift analysis with suitably chosen complex optical potentials. The important aspect of our present work is that we uniquely obtain static potential in an analytical form and use it along with exchange (only for electron), polarization and purely imaginary absorption potentials to define the complex optical potential. The static potential is evaluated by obtaining charge density from the H2S molecule using the molecular wavefunction represented through an accurate analytical form of the Gaussian orbitals. The primary aim of our study is to test our present approach, as applied to the electron and positron scattering from H2S. Therefore, the results for electron and positron impact differential, integral, momentum-transfer, absorption and total cross sections are obtained for the incident energies in the range of 10–500 eV. Comparisons of these different types of cross section results with the available measurements and other calculations show good agreement, which suggests the applicability of our present approach.

An approach to study electron and positron scattering from NH3 and PH3 using the analytic static potential

A detailed study of the elastic scattering of electrons and positrons from two C3v symmetrical molecules viz ammonia and phosphine molecules is presented. The partial wave phase shift analysis method with complex optical potential consisting of static, exchange, polarization and an imaginary absorption potentials has been used. In contrast to the previously used similar approaches, the present method is quite different in the sense that we have obtained the required static potential in an analytical form using correctly represented Gaussian molecular wave functions for NH3 and PH3 in our calculations. The differential, integrated, momentum transfer, absorption and total cross sections are calculated and reported in the incident electron and positron energy range of 10–500 eV. In order to test the applicability of our approach, the present results are compared with the available previous measurements and theoretical calculations and a good agreement is found.

New insights on low energy πN scattering amplitudes: comprehensive analyses at level * * Supported in part by National Nature Science Foundations of China (NSFC)(10925522, 11021092), and by the Spanish Ministerio de Economía y Competitividad (MINECO) and the European Regional Development Fund (ERDF)(FIS2017-84038-C2-1-P, FIS2017-84038-C2-2-P, SEV-2014-0398)

A production representation of partial-wave S matrix is utilized to construct low-energy elastic pion-nucleon scattering amplitudes from cuts and poles on complex Riemann sheets. Among them, the contribution of left-hand cuts is estimated using the results obtained in covariant baryon chiral perturbation theory within the extended-on-nass-shell scheme. By fitting to data on partial-wave phase shifts, it is indicated that the existences of hidden poles in S11 and P11 channels, as conjectured in our previous paper [Eur. Phys. J. C, 78(7): 543 (2018)], are firmly established. Specifically, the pole mass of the S11 hidden resonance is determined to be (895±81)−(164±23)i MeV, whereas, the virtual pole in the P11 channel locates at (966±18) MeV. It is found that analyses at the level improves significantly the fit quality, comparing with the previous one. Quantitative studies with cautious physical discussions are also conducted for the other S- and P-wave channels.

Phase-shift derivation of expansions for material and frequency dependence of progressive-wave radiation forces and backscattering by spheres.

When considering the scattering of sound and radiation forces for spheres, it has historically been helpful to understand situations lacking dissipation. In that case the scattering is characterized by real partial-wave phase shifts. At low frequencies expansions show the dependence of each phase shift on material properties and on frequency. Those expansions are used here to describe the frequency and material dependence of scattering and radiation forces beyond the usual Rayleigh-scattering approximation. Results for radiation forces on spheres in standing waves are extended to plane progressive waves. The expansion coefficients use algebraic functions. Results for movable and fixed rigid spheres are shown.

Phase-shift based expansions for material and frequency dependence of scattering and of radiation forces

When considering the scattering of sound and radiation forces for spheres, it has historically been helpful to gain a good understanding of situations lacking dissipation. In that case the linear acoustic scattering by spheres is characterized by real partial-wave phase shifts. At low frequencies it is possible to obtain simple expansions showing the dependence of each phase shift on material properties and on frequency [P. L. Marston and L. Zhang, J. Acoust. Soc. Am. 141, 3042–3049 (2017); P. L. Marston, J. Acoust. Soc. Am. 142, 1167–1170 (2017); P. L. Marston, J. Acoust. Soc. Am. 142, 3358–3361 (2017)]. Those expansions can then be used to describe the frequency and material dependence of scattering and radiation forces for standing and traveling waves at low frequencies beyond the usual Rayleigh-scattering approximation. While spherical Bessel and Hankel functions are used in the derivation, the final expansion coefficients are expressed using algebraic functions. The derivation of the simplified series expansion for the radiation force in the traveling wave case is subtler than the standing wave case. Well-known limiting expressions are recovered in both cases. Some applications to spheres in acoustic beams are also known. [Work supported by ONR.]

Comparative study of eV to GeV electrons and positrons scattering elastically from neutral atoms

Differential cross sections and spin asymmetries for leptons colliding with light and heavy targets are calculated at collision energies Ee ranging from 1 eV to 0.3 GeV. For , the partial-wave phase shift analysis for solving the relativistic Dirac equation is used within an optical model potential, and for within the nuclear potential. At , the phase shift analysis is supplemented with the distorted-wave Born approximation to account for magnetic scattering. As test cases, elastic collisions of e± with 23Na, 112Cd and 208Pb atoms are reported, and good agreement with available experimental data and other theoretical cross sections is obtained. For electrons, pronounced structures in the spin asymmetry occur at low energies and small scattering angles. They disappear beyond 50 keV and reappear beyond 50 MeV, being then particularly strong for spin-zero nuclei at large scattering angles. For positrons, the high-energy diffraction structures are shifted in phase with respect to those in the electron spin asymmetry. The strong reduction of the positron spin asymmetry and the quenching of the structures at low-energy is interpreted in terms of a dominating action of the nuclear potential.

Radiation force on spheres in standing waves in ideal fluids: Improved simple approximation for material dependence

When spheres are much smaller than the acoustic wavelength but sufficiently large that viscous dissipative properties are insignificant, the Yosioka-Kawasima approximation for the radiation force in standing waves is widely used. One consequence is that the force is taken to be proportional to the sphere’s volume. One approach to finding how corrections to that approximation depend on material properties is to start with Hasegawa’s full partial-wave-based series and convert it to partial-wave phase shifts using analytical relations [L. Zhang and P. L. Marston, J. Acoust. Soc. Am. 140, EL178–EL183 (2016)]. When all the relevant phase shifts are real and sufficiently small in magnitude, a simple approximation can be found for (possibly imaginary) frequencies giving vanishing forces. That was an unanticipated application. (Material dissipation and surface tension are taken to be insignificant.) Using that frequency root, a modification of the Yosioka-Kawasima radiation force is derived that allows for the liquid or solid sphere to be finite in size [P. L. Marston, J. Acoust. Soc. Am. 142, 1167–1170 (2017) and at press]. The force is no longer strictly proportional to the volume. This approach is supported by numerical comparisons with Hasegawa’s series when the phase shifts are small. [Research supported by ONR.]

Note on Resonant and Non-resonant Peaks in Electron-Atom Total Scattering Cross Sections

Certain broad low-energy peaks caused by a single partial wave in total cross sections are explained in terms of phase shifts. Such peaks have been associated with the real part of a Regge pole trajectory, having a maximum near an integer value of the angular momentum quantum number. At the peak energies, the pertinent partial-wave phase shift was shown to have a local maximum near a value modulo π. This implies no time delay in the semiclassical context. The phenomenon is a quantum effect, lacking a semiclassical interpretation.

Acoustic radiation force expansions in terms of partial-wave scattering phase shifts: Extended applications

When evaluating radiation forces on spheres in sound fields, the interpretation of analytical results is greatly simplified by retaining the use of s-function notation for partial-wave coefficients imported into acoustics from quantum scattering theory. This facilitates easy interpretation of various scattering efficiency factors [L. Zhang and P. Marston, J. Acoust. Soc. Am. 140, EL178 (2016)]. This also facilitates the correction of certain plane-wave results [H. Olsen et al., J. Acoust. Soc. Am. 30, 633 (1958)] and the force parameterization for a broader class of wavefields. For situations in which dissipation is negligible, each partial-wave s-function becomes characterized by a single parameter: a phase shift. These partial-wave phase shifts are associated with scattering by plane traveling waves; the incident wavefields of interest (progressive and standing wavefields and beams) are separately parameterized. (When considering outcomes, the method of fabricating symmetric objects having a desirable set of phase shifts becomes a separate issue.) The existence of negative radiation force “islands” for beams reported in 2006 by Marston is manifested. Elementary standing and traveling wave force expressions are also recovered. This approach also manifests the utility of conservation theorems [P. Marston and L. Zhang, J. Acoust. Soc. Am. 139, 3139 (2016)]. [Work supported by ONR.]

Relationship of scattering phase shifts to special radiation force conditions for spheres in axisymmetric wave-fields.

When investigating the radiation forces on spheres in complicated wave-fields, the interpretation of analytical results can be simplified by retaining the s-function notation and associated phase shifts imported into acoustics from quantum scattering theory. For situations in which dissipation is negligible, as taken to be the case in the present investigation, there is an additional simplification in that partial-wave phase shifts become real numbers that vanish when the partial-wave index becomes large and when the wave-number-sphere-radius product vanishes. By restricting attention to monopole and dipole phase shifts, transitions in the axial radiation force for axisymmetric wave-fields are found to be related to wave-field parameters for traveling and standing Bessel wave-fields by considering the ratio of the phase shifts. For traveling waves, the special force conditions concern negative forces while for standing waves, the special force conditions concern vanishing radiation forces. An intermediate step involves considering the functional dependence on phase shifts. An appendix gives an approximation for zero-force plane standing wave conditions. Connections with early investigations of acoustic levitation are mentioned and some complications associated with viscosity are briefly noted.

Radiation force expansions in terms of partial wave phase shifts for scattering: Examples

When evaluating radiation forces on spheres in acoustic beams, with or without helicity, the interpretation of analytical results is greatly simplified by retaining the use of s-function notation for partial-wave coefficients imported into acoustics from quantum scattering theory in the 1970s. This facilitates easy interpretation of various efficiency factors [L. Zhang and P. L. Marston, Phys. Rev. E. 84, 035601 (2011); L. Zhang and P. L. Marston, Bio. Opt. Express 4, 1610–1617 (2013); (E) 4, 2988 (2013)]. For situations in which dissipation is negligible each partial wave s-function becomes characterized by a single parameter: the partial wave phase shift that parameterizes possible degrees of freedom [P. Marston and L. Zhang, J. Acoust. Soc. Am. 131, 3534 (2012); P. L. Marston, J. Quant. Spectrosc. Radiat. Transf. 162, 8–17 (2015)]. The s-function and associated phase shifts are associated with scattering by plane traveling waves, and the incident wavefield of interest is separately parameterized. (When considering allowed outcomes, the method of fabricating symmetric objects having a desirable set of plane-wave scattering partial-wave phase shifts becomes a separate issue.) The present analysis illustrates the advantages of the formulation by extending some prior expansions associated with radiation forces. [Supported in part by ONR.]

Nucleon-nucleon scattering from dispersion relations: Next-to-next-to-leading order study

Nucleon-nucleon $(\mathit{NN})$ scattering is studied by applying an approach based on the $\mathit{N}/\mathit{D}$ method and chiral perturbation theory (ChPT), whose dynamical input per partial wave consists of the imaginary part of the $\mathit{NN}$ partial-wave amplitude along the left-hand cut. The latter is calculated in one-loop ChPT up to and including next-to-next-to-leading order (NNLO). A power counting for the subtraction constants is established, which is appropriate for those subtractions attached to both the left- and the right-hand cuts. A quite good reproduction of the Nijmegen partial-wave analysis phase shifts and mixing angles results, which implies a steady improvement in the accurateness achieved by increasing the chiral order in the calculation of the dynamical input. I discuss that it is not necessary to fine tune the chiral counterterms ${c}_{i}$ determined from pion-nucleon scattering to agree with $\mathit{NN}$ data, but instead one should perform the iteration of two-nucleon intermediate states in a well-defined way so as to keep proper unitarity and analyticity. It is also confirmed at NNLO the long-range correlations between the $\mathit{NN} S$-wave effective ranges and scattering lengths, when employing only once-subtracted dispersion relations, that hold up to around 10% when compared with experimental values.