Partial Wave Expansion Research Articles

Scattering cross-section in gravitating cosmic string spacetimes

We propose a modification of the partial wave approach to deal with the relativistic quantum scattering of bosonic and fermionic particles in a class of models concerning gravitating cosmic string spacetimes. These spacetimes are characterized by the Minkowski line element at the center of the vortex, non-vanishing curvature at a finite distance from the center but with a conical structure far from the core. We find the correction in the partial wave expansion and the phase shift. Consequently, we show the explicit form of the scattering amplitude and the correction to the differential cross-section for a massive scalar field. We also implement our formalism in a toy model mimicking this class of gravitating cosmic string spacetime. Moreover, we discuss the procedure to apply this formalism to a massive Dirac field.

Longitudinal and transverse PAFs for an absorptive magneto-dielectric circular cylinder in light-sheets of arbitrary wavefronts and polarization.

Based upon the expression of the heat source function in photophoresis, generalized mathematical expressions for the longitudinal (L) and transverse (T) photophoretic asymmetry factors (PAFs) for a light-absorptive magneto-dielectric circular cylinder of arbitrary relative permittivity and permeability, illuminated by an arbitrarily shaped polarized light-sheet, are derived and computed. The L- and T-PAFs are directly proportional to the L and T components of the photophoretic force vector, respectively, induced by light absorption inside the particle, and their sign predicts the behavior of the force (pulling/attractive or pushing/repulsive). The partial-wave series expansion method in cylindrical coordinates is used, and the obtained mathematical expressions for the L- and T-PAFs depend on the beam-shape coefficients and the internal coefficients of the cylinder. Numerical examples illustrate the theory for TE and TM polarized plane waves, and nonparaxial Airy light-sheets with particular emphasis on absorption inside the cylinder and varying the Airy light-sheet parameters. The generalized expressions presented here are applicable to any light-sheet of an arbitrary wavefront, and offer additional quantitative observables for the analysis of the photophoretic force in applications in electromagnetic scattering, optical light-sheet tweezers, particle manipulation, radiative transfer, and other research fields.

Notes on flat-space limit of AdS/CFT

Different frameworks exist to describe the flat-space limit of AdS/CFT, include momentum space, Mellin space, coordinate space, and partial-wave expansion. We explain the origin of momentum space as the smearing kernel in Poincare AdS, while the origin of latter three is the smearing kernel in global AdS. In Mellin space, we find a Mellin formula that unifies massless and massive flat-space limit, which can be transformed to coordinate space and partial-wave expansion. Furthermore, we also manage to transform momentum space to smearing kernel in global AdS, connecting all existed frameworks. Finally, we go beyond scalar and verify that leftlangle VVmathcal{O}rightrangle maps to photon-photon-massive amplitudes.

Electron backscattering coefficients of molybdenum and tungsten based on the Monte Carlo simulations

Monte Carlo simulation is employed for the calculation of electron backscattering coefficients of molybdenum (Mo) and tungsten (W) at normal incidence angle and at energies between 100 eV to 100 keV. For the modelling of the electron elastic scattering we have applied the partial-wave expansion method using the Mott's cross-section, where the scattering potential includes the electron electrostatic potential, exchange potential and correlation-polarization potential. A relativistic dielectric functional approach with full Penn's algorithm is adopted for the calculation of the electron inelastic cross-section. The effect of phonon excitation, the interband transition of the loosely bound valance electrons, and the inner-shells electrons excitations are considered in the energy loss functions of the materials via the optical constants. We found that our present simulated data agree well with the experimental data at energies above 20 keV for both elements while they are well above the experimental data at lower incident energies. The disagreement can be explained by the possible surface contamination of carbonaceous atomic layers exist in the measurements by additional simulation for the contaminated surfaces. Moreover, for the better and detailed understanding of the energy dependence of the backscattering coefficients, we also present the backscattered electron energy spectra, angular distributions, and depth distributions for both samples.

Mie particle assembly by a converging ultrasound field and acoustic interaction forces

There has been growing interest in exerting radiation forces to trap and cluster randomly distributed cells in body fluid, micro-particles in water, or microorganisms in fluid-like culture media. Acoustic standing waves are extensively utilized as a patterning tool to assemble particles at nodes or anti-nodes but, for frequencies above megahertz, the nodal distances become too small for particle separation. We study a mechanism to propel particles towards one central space by creating a converging wave field without significant reflections. This is achieved by strong decay of the main ultrasound beam by multiple scattering. When two opposing traveling waves are suitably decayed towards their meeting point, a converging wave field is created for particle assembly via radiation forces. This paper describes the theoretical prediction model based on the translation addition theorem and the partial-wave expansion method. The predicted acoustic pressure fields are compared with full numerical simulations for a limited number of particles, and the attenuation coefficient is validated by the existing experimental data. The results demonstrate that the converging wave field is formed, and the acoustic radiation force vectors in a bulk acoustic wave (BAW) device are directed towards the central space for meaningful clustering of micro-particles from their host fluid.

New positivity bounds from full crossing symmetry

Positivity bounds are powerful tools to constrain effective field theories. Utilizing the partial wave expansion in the dispersion relation and the full crossing symmetry of the scattering amplitude, we derive several sets of generically nonlinear positivity bounds for a generic scalar effective field theory: we refer to these as the P Q, Dsu, Dstu and {overline{D}}^{mathrm{stu}} bounds. While the PQ bounds and Dsu bounds only make use of the s↔u dispersion relation, the Dstu and {overline{D}}^{mathrm{stu}} bounds are obtained by further imposing the s↔t crossing symmetry. In contradistinction to the linear positivity for scalars, these inequalities can be applied to put upper and lower bounds on Wilson coefficients, and are much more constraining as shown in the lowest orders. In particular we are able to exclude theories with soft amplitude behaviour such as weakly broken Galileon theories from admitting a standard UV completion. We also apply these bounds to chiral perturbation theory and we find these bounds are stronger than the previous bounds in constraining its Wilson coefficients.

MAST-SEY: MAterial Simulation Toolkit for Secondary Electron Yield. A monte carlo approach to secondary electron emission based on complex dielectric functions

MAST-SEY is an open-source Monte Carlo code capable of calculating secondary electron emission using input data generated entirely from first principle (density functional theory) calculations. It utilizes the complex dielectric function and Penn’s theory for inelastic scattering processes, and relativistic Schrödinger theory by means of a partial-wave expansion method to govern elastic scattering. It allows the user to include explicitly calculated momentum dependence of the dielectric function, as well as to utilize first-principle density of states in secondary electron generation, which provides a more complete description of the underlying physics. In this paper we thoroughly describe the theoretical aspects of the modeling, as used in the code, and present sample results obtained for copper and aluminum.

Cross sections for 2-to-1 meson\u2013meson scattering

We study the processes K{bar{K}} rightarrow phi , pi D rightarrow D^*, pi {bar{D}} rightarrow {bar{D}}^*, and the production of psi (3770), psi (4040), psi (4160), and psi (4415) mesons in collisions of charmed mesons or charmed strange mesons. The process of 2-to-1 meson–meson scattering involves a quark and an antiquark from the two initial mesons annihilating into a gluon and subsequently the gluon being absorbed by the spectator quark or antiquark. Transition amplitudes for the scattering process derive from the transition potential in conjunction with mesonic quark–antiquark wave functions and the relative-motion wave function of the two initial mesons. We derive these transition amplitudes in the partial wave expansion of the relative-motion wave function of the two initial mesons so that parity and total-angular-momentum conservation are maintained. We calculate flavor and spin matrix elements in accordance with the transition potential and unpolarized cross sections for the reactions using the transition amplitudes. Cross sections for the production of psi (4040), psi (4160), and psi (4415) relate to nodes in their radial wave functions. We suggest the production of psi (4040), psi (4160), and psi (4415) as probes of hadronic matter that results from the quark–gluon plasma created in ultrarelativistic heavy-ion collisions.

Convergent close-coupling calculations of electrons scattering on electronically excited molecular hydrogen

We use the adiabatic-nuclei molecular convergent close-coupling method to perform calculations of 0.01--1000 eV electrons scattering on the $c^{3}\mathrm{\ensuremath{\Pi}}_{u}$, $a^{3}\mathrm{\ensuremath{\Sigma}}_{g}^{+}$, $B^{1}\mathrm{\ensuremath{\Sigma}}_{u}^{+}$, $C^{1}\mathrm{\ensuremath{\Pi}}_{u}$, and $EF^{1}\mathrm{\ensuremath{\Sigma}}_{g}^{+}$ states of ${\mathrm{H}}_{2}$ in the $v=0$ vibrational level. Elastic, superelastic, ionization, and grand-total cross sections are presented, as well as cross sections for excitation of the $n=2$--3 singlet and triplet states of ${\mathrm{H}}_{2}$ (where $n$ is the atomic-limit principle quantum number). Comparison with available theoretical results is made. Good agreement is found with the recent $R$-matrix results [J. Phys. B 53, 245203 (2020)] for most of the exchange and dipole-forbidden transitions, but not for the dipole-allowed transitions. The sources of disagreement were found to be an unconverged partial-wave expansion and the utilization of the fixed-nuclei approximation (as opposed to adiabatic-nuclei) in the $R$-matrix calculations.

Integral equations and binding energies for Λ − NN and Λ − NNN systems

A systematic treatment of the procedure used to construct and solve the Faddeev-Yakubovsky integral equations for three and four different particles with nonzero spin is introduced. The four-body T-matrices are written in terms of solutions of the three-body Faddeev integral equations in total angular momenta representation. General spin-angular structure of the four-body Faddeev-Yakubovsky equations is written down and reduced to a spin-angular structure of the three-body system by elimination of one body. Partial-wave expansion performed for the system of four-body integral equations gives an infinite system of one-dimensional coupled equations with integral kernels containing four types of partial components of the three-body Faddeev integral equation solutions. The given method is tested on Λ−NN and Λ−NNN systems with two-body potential written down in only separable form. Both separable NN potential and ΛN potential rewritten in separable form for s-wave are used to determine the Λ-hyperon binding energies in Λ−NN and Λ−NNN systems. Calculation also includes the hyperon conversion in ΛN→ΣN process. Λ-hyperon separation energies were calculated from binding energies and turned out to be 0.147 MeV in Λ−NN system and 2.04 MeV in Λ−NNN system, respectively, that is in close coincidence with experimental ones.

Longitudinal and transverse optical scattering asymmetry parameters for a dielectric cylinder in light-sheets of arbitrary wavefronts and polarization.

The asymmetry parameter is an important quantity used in radiative transfer modeling and scattering. This parameter specifies the amount of energy scattered by the particle along the direction of the incident illuminating field. However, a rigorous and complete analysis of the energetic scattering requires determining the energy scattered in the lateral direction as well. As such, the present work introduces generalized expressions for the scattering asymmetry parameters for a dielectric cylinder in arbitrary-shaped light-sheets, both along and perpendicular to the direction of the incident radiation. Both longitudinal and transverse scattering asymmetry parameters are defined, and their generalized expressions are obtained based on the (spatial) average cosine and sine of the scattering angle θ and the expression of the scattering cross section (or energy efficiency). The partial-wave series expansion method in cylindrical coordinates is used, and the resulting mathematical expressions depend on the beam-shaped coefficients and the scattering coefficients of the dielectric cylinder. Numerical results for arbitrary-shaped light-sheets illuminating a dielectric cylinder cross section located arbitrarily in space are presented and discussed. The longitudinal and transverse scattering asymmetry parameters defined here offer additional quantitative (quadratic) observables for the analysis of the energetic scattering in applications in electromagnetic scattering, optical light-sheet tweezers, radiative transfer computations, and remote sensing, to name a few examples.

One hadron exchange non-relativistic model for potential

One of the KN Interaction is K+p interaction which one we’re interested in. K+p potential is described as one hadron exchanges. The hadrons being exchanged include scalar meson (σ), vector mesons (ω, ρ ), hyperons (Λ, Σ), and their resonances ( Λ* (1600), Σ*(1385)). This interaction model is derived for non-relativistic calculations within Blankenbecler-Sugar reduction. The values of the potential parameters are fixed by fitting to experimental data of K+p spin-averaged differential cross section for kaon laboratory energies from 88.13 MeV to 3683.22 MeV. Scattering calculations are performed by using a 3D technique without partial wave expansion.

One-Hadron-Exchange K−p interaction model in nonrelativistic energy region

A nonrelativistic K−p interaction model is derived as one-hadron-exchanges. The exchanged particles are the scalar meson, the vector mesons, the hyperons, and the resonances Λ*(1600),Σ*(1385). The models parameters are determined by fitting to experimental data of spin-averaged differential cross section for kaon laboratory energy range of 51.27 MeV up to 376.87 MeV. We use a 3D technique without partial wave expansion to calculate K−p scattering.

Partial Wave Amplitude Basis and Selection Rules in Effective Field Theories.

We derive the generalized partial wave expansion for N→M scattering amplitude in terms of spinor helicity variables. The basis amplitudes of the expansion with definite angular momentum j consist of the Poincaré Clebsch-Gordan coefficients. Moreover, we obtain a series of selection rules that restrict the anomalous dimension matrix of effective operators and how effective operators contribute to some 2→N amplitudes at the loop level.

One hadron exchange non-relativistic model for K+p potential

K+p potential is modeled as one hadron exchange. The hadrons being exchanged include scalar meson (σ), vector mesons ( ω, p ), hyperons (Λ, Σ), and their resonances (Λ*(1600), Σ*(1385)). This interaction model is derived for non-relativistic calculations and for that purpose the potential model is formulated within Blankenbecler-Sugar reduction. The values of the potential parameters are fixed by fitting to experimental data of K+p spin-averaged differential cross section for kaon laboratory energies from 88.13 MeV to 380.58 MeV. Scattering calculations are performed by using a 3D technique without partial wave expansion.

Nonrelativistic one-hadron-exchange k−p interaction model

We have derived a nonrelativistic K−p interaction model as one-hadron-exchanges. The exchanged particles are the scalar meson σ, the vector mesons ω, ρ, the hyperons Λ, Σ, and the resonances Λ* (1600), Σ*(1385). Parameters in this model are determined by fitting to experimental data of spin-averaged differential cross section for kaon laboratory energy range of 51.27 MeV up to 900.64 MeV. The K-p scattering is calculated using a 3D technique without partial wave expansion.

Acoustic radiation torque on an off-axis elliptical cylinder in Gauss beams

As one of the nonlinear effects of acoustic waves, the time-averaged acoustic radiation torque expression is derived from the transfer of angular momentum from the incident beam to the object. In recent years, the acoustic radiation torque has received substantial attention since it is the underlying principle of well-controlled particle rotations and spins, which provides a new degree of freedom in particle manipulation and acousto-fluidic applications in addition to the translational displacement caused by the acoustic radiation force. Cylindrical particles, such as fibers, carbon nanotubes and other surface acoustic wave devices, are commonly encountered in various applications. The acoustic scattering coefficients for an elliptical cylinder arbitrarily located at the field of Gauss beam in two-dimensions are computed based on the partial-wave series expansion method and the Graf’s additional theorem for cylindrical functions to obtain the off-axis beam shape coefficients. It is worth mentioning that both the rigid and non-rigid cylinders are considered in this work, which requires different boundary conditions at the cylinder surface. Moreover, the closed-form expression of the acoustic radiation torque in this case is derived. On this basis, several numerical simulations are performed with particular emphasis on the off-axis distance, the incident angle and the beam waist. The simulated results show that both the positive and negative acoustic radiation torque can exist under certain conditions, which means that 1) the elliptical cylinder can be rotated in either the clockwise or the counterclockwise direction, 2) rigid elliptical cylinders are more likely to experience a strong acoustic radiation torque than non-rigid elliptical cylinders at low frequencies, 3) the incident wave field with a specific frequency can excite a different resonance scattering mode for a non-rigid elliptical cylinder, therefore the acoustic radiation torque peak is more related to the beam frequency than to the elliptical cylinder’s location in the field, and 4) increasing the beam width can enlarge the scattering cross section area, and thus enhancing the acoustic radiation torque on the elliptical cylinder. The results in this study are expected to provide a theoretical guide for the controllable rotation of a particle and the viscosity inversion of fluid by using the acoustic radiation torque. The exact formalism presented here by using the multipole expansion method, which is valid for any frequency range, can be used to validate other approaches by using purely numerical methods.

Elastic scattering of electrons by barium atoms using the distorted wave method

Differential and integral cross sections of elastic scattering of electrons by barium atoms at intermediate energies (10 – 200 eV) were calculated using the distorted wave method. Being an elastic scattering both the initial and final distortion potentials were taken as the static potential of a barium atom in the initial state. The distorted waves are determined by the partial wave expansion method by expanding them in terms of spherical harmonics while the radial equation corresponding to distorted waves is evaluated using the Numerov method. A computer program DWBA1, for scattering was modified to perform numerical calculations for scattering and the results obtained are compared with the experimental and theoretical results available. The present integral cross sections are in good qualitative agreement with the experimental results and most of the theoretical results. For energies in the range 30 – 100 eV the present differential cross sections are in satisfactory agreement with the other theoretical and experimental results.

A scattering amplitude in Conformal Field Theory

We define form factors and scattering amplitudes in Conformal Field Theory as the coefficient of the singularity of the Fourier transform of time-ordered correlation functions, as p2 → 0. In particular, we study a form factor F(s, t, u) obtained from a four-point function of identical scalar primary operators. We show that F is crossing symmetric, analytic and it has a partial wave expansion. We illustrate our findings in the 3d Ising model, perturbative fixed points and holographic CFTs.

Relativistic partial waves for celestial amplitudes

The formalism of relativistic partial wave expansion is developed for four-point celestial amplitudes of massless external particles. In particular, relativistic partial waves are found as eigenfunctions to the product representation of celestial Poincaré Casimir operators with appropriate eigenvalues. The requirement of hermiticity of Casimir operators is used to fix the corresponding integral inner product, and orthogonality of the obtained relativistic partial waves is verified explicitly. The completeness relation, as well as the relativistic partial wave expansion follow. Example celestial amplitudes of scalars, gluons, gravitons and open superstring gluons are expanded on the basis of relativistic partial waves for demonstration. A connection with the formulation of relativistic partial waves in the bulk of Minkowski space is made in appendices.