Partial Wave Expansion Research Articles

Cloaking resonant scatterers and tuning electron flow in graphene

We consider resonant scatterers with large scattering cross sections in graphene that are produced by a gated disk or a vacancy, and show that a gated ring can be engineered to produce an efficient electron cloak. We also demonstrate that this same scheme can be applied to tune the direction of electron flow. Our analysis is based on a partial-wave expansion of the electronic wave functions in the continuum approximation, described by the Dirac equation. Using a symmetrized version of the massless Dirac equation, we derive a general condition for the cloaking of a scatterer by a potential with radial symmetry. We also perform tight-binding calculations to show that our findings are robust against the presence of disorder in the gate potential.

Towards holographic higher-spin interactions: four-point functions and higher-spin exchange

Within holography, we calculate the contribution of an arbitrary spin-s gauge boson exchange in AdS d+1 to the four-point function with scalar operators on the boundary. As an important ingredient, we first compute the complete bulk-to-bulk propagators for massless bosonic higher-spin fields in the metric-like formulation, in any dimension and in various gauges. The split representation of the bulk-to-bulk propagators in terms of bulk-to-boundary propagators allows to present the higher-spin exchange diagram in the form of a conformal partial wave expansion. Our results provide a step towards the larger goal of the holographic reconstruction of bulk interactions, and of clarifying bulk locality.

Dual parametrization of generalized parton distributions in two equivalent representations

The dual parametrization and the Mellin-Barnes integral approach represent two frameworks for handling the double partial wave expansion of generalized parton distributions (GPDs) in the conformal partial waves and in the $t$-channel ${\rm SO}(3)$ partial waves. Within the dual parametrization framework, GPDs are represented as integral convolutions of forward-like functions whose Mellin moments generate the conformal moments of GPDs. The Mellin-Barnes integral approach is based on the analytic continuation of the GPD conformal moments to the complex values of the conformal spin. GPDs are then represented as the Mellin-Barnes-type integrals in the complex conformal spin plane. In this paper we explicitly show the equivalence of these two independently developed GPD representations. Furthermore, we clarify the notions of the $J=0$ fixed pole and the $D$-form factor. We also provide some insight into GPD modeling and map the phenomenologically successful Kumeri\v{c}ki-M\"uller GPD model to the dual parametrization framework by presenting the set of the corresponding forward-like functions. We also build up the reparametrization procedure allowing to recast the double distribution representation of GPDs in the Mellin-Barnes integral framework and present the explicit formula for mapping double distributions into the space of double partial wave amplitudes with complex conformal spin.

Computing the acoustic radiation force exerted on a sphere using the translational addition theorem.

In this paper, the translational addition theorem for spherical functions is employed to calculate the acoustic radiation force produced by an arbitrary shaped beam on a sphere arbitrarily suspended in an inviscid fluid. The procedure is also based on the partial-wave expansion method, which depends on the beam-shape and scattering coefficients. Given a set of beam-shape coefficients (BSCs) for an acoustic beam relative to a reference frame, the translational addition theorem can be used to obtain the BSCs relative to the sphere positioned anywhere in the medium. The scattering coefficients are obtained from the acoustic boundary conditions across the sphere's surface. The method based on the addition theorem is particularly useful to avoid quadrature schemes to obtain the BSCs. We use it to compute the acoustic radiation force exerted by a spherically focused beam (in the paraxial approximation) on a silicone-oil droplet (compressible fluid sphere). The analysis is carried out in the Rayleigh (i.e., the particle diameter is much smaller than the wavelength) and Mie (i.e., the particle diameter is of the order of the wavelength or larger) scattering regimes. The obtained results show that the paraxial focused beam can only trap particles in the Rayleigh scattering regime.

Basis convergence of range-separated density-functional theory.

Range-separated density-functional theory (DFT) is an alternative approach to Kohn-Sham density-functional theory. The strategy of range-separated density-functional theory consists in separating the Coulomb electron-electron interaction into long-range and short-range components and treating the long-range part by an explicit many-body wave-function method and the short-range part by a density-functional approximation. Among the advantages of using many-body methods for the long-range part of the electron-electron interaction is that they are much less sensitive to the one-electron atomic basis compared to the case of the standard Coulomb interaction. Here, we provide a detailed study of the basis convergence of range-separated density-functional theory. We study the convergence of the partial-wave expansion of the long-range wave function near the electron-electron coalescence. We show that the rate of convergence is exponential with respect to the maximal angular momentum L for the long-range wave function, whereas it is polynomial for the case of the Coulomb interaction. We also study the convergence of the long-range second-order Møller-Plesset correlation energy of four systems (He, Ne, N2, and H2O) with cardinal number X of the Dunning basis sets cc - p(C)V XZ and find that the error in the correlation energy is best fitted by an exponential in X. This leads us to propose a three-point complete-basis-set extrapolation scheme for range-separated density-functional theory based on an exponential formula.

Study on the dissociative recombination of HeH+by multi-channel quantum defect theory

The dissociative recombination of HeH+ is studied using multi-channel quantum defect theory. We investigated how the partial waves of incident electrons affect the DR cross section. The DR cross section depends on the position of the center of partial wave expansion for the adiabatic S-matrix of electron scattering. When the Rydberg states correlate with the Rydberg states of the hydrogen atom at large internuclear distances, the center should be on the hydrogen atom for a better convergence of the expansion.

Generalized theory of resonance scattering (GTRS) using the translational addition theorem for spherical wave functions.

The generalized theory of resonance scattering (GTRS) by an elastic spherical target in acoustics is extended to describe the arbitrary scattering of a finite beam using the addition theorem for the spherical wave functions of the first kind under a translation of the coordinate origin. The advantage of the proposed method over the standard discrete spherical harmonics transform previously used in the GTRS formalism is the computation of the off-axial beam-shape coefficients (BSCs) stemming from a closed-form partial-wave series expansion representing the axial BSCs in spherical coordinates. With this general method, the arbitrary acoustical scattering can be evaluated for any particle shape and size, whether the particle is partially or completely illuminated by the incident beam. Numerical examples for the axial and off-axial resonance scattering from an elastic sphere placed arbitrarily in the field of a finite circular piston transducer with uniform vibration are provided. Moreover, the 3-D resonance directivity patterns illustrate the theory and reveal some properties of the scattering. Numerous applications involving the scattering phenomenon in imaging, particle manipulation, and the characterization of multiphase flows can benefit from the present analysis because all physically realizable beams radiate acoustical waves from finite transducers as opposed to waves of infinite extent.

Partial-wave series expansions in spherical coordinates for the acoustic field of vortex beams generated from a finite circular aperture.

Stemming from the Rayleigh-Sommerfeld surface integral, the addition theorems for the spherical wave and Legendre functions, and a weighting function describing the behavior of the radial component vp1 of the normal velocity at the surface of a finite circular radiating source, partial-wave series expansions are derived for the incident field of acoustic spiraling (vortex) beams in a spherical coordinate system centered on the axis of wave propagation. Examples for vortex beams, comprising ρ-vortex, zeroth-order and higher order Bessel-Gauss and Bessel, truncated Neumann-Gauss and Hankel- Gauss, Laguerre-Gauss, and other Gaussian-type vortex beams are considered. The mathematical expressions are exact solutions of the Helmholtz equation. The results presented here are particularly useful to accurately evaluate analytically and compute numerically the acoustic scattering and other mechanical effects of finite vortex beams, such as the axial and 3-D acoustic radiation force and torque components on a sphere of any (isotropic, anisotropic, etc.) material (fluid, elastic, viscoelastic, etc.), either centered on the beam's axis of wave propagation, or placed off-axially. Numerical predictions allow optimal design of parameters in applications including but not limited to acoustical tweezers, acousto-fluidics, beamforming design, and imaging, to name a few.

In-medium nucleon-nucleon cross sections with nonspherical Pauli blocking

We present a formalism to solve the Bethe-Goldstone scattering equation without the use of partial wave expansion which is alternative to the one we developed in a previous work. The present approach is more suitable for the calculation of in-medium nucleon-nucleon cross sections, which are the focal point of this paper. The impact of removing the spherical approximation on the angle and energy dependence of, particularly, in-medium proton-proton and proton-neutron differential cross sections is discussed along with its potential implication.

Analysis of proton-Be9,10,11,12scattering using an energy-, density-, and isospin-dependent microscopic optical potential

The proton elastic scattering of ${}^{9,10,11,12}\mathrm{Be}$ isotopes at a wide energy range from 3 to 200 MeV/nucleon is analyzed using the optical model with the partial-wave expansion method. The microscopic optical potential (OP) is taken within the single-folded model. The density- and isospin-dependent M3Y-Paris nucleon-nucleon ($NN$) interaction is used for the real part and the $NN$-scattering amplitude of the high-energy approximation is used for the imaginary one. The surface contribution to the imaginary part is included. The analysis reveals that the partial-wave expansion with this microscopic OP reproduces well the basic scattering observables at energies up to 100 MeV/nucleon. For higher energies, the eikonal approximation with the same OP gives results better than the partial-wave expansion calculations. The volume integrals of the OP parts have systematic energy dependencies, and they are parameterized in empirical formulas. In addition, the volume integral's parametrizations determine the true energy dependence for the depths of the OP parts. The study of increasing the number of neutrons for a given isotope shows that the imaginary volume integrals and reaction cross sections depend on the matter radii of the scattered nuclei. Further, they are found to have larger values for the halo nucleus scattering ($^{11}\mathrm{Be}+p$) than those for the scattering of their isotopes.

Pseudo-Gaussian cylindrical acoustical beam – Axial scattering and radiation force on an elastic cylinder

Making use of the addition theorem for the cylindrical wave functions and the complex-source-point method in cylindrical coordinates, an exact solution to the Helmholtz equation is derived, which corresponds to a tightly focused (or collimated) cylindrical quasi-Gaussian beam with arbitrary waist. The solution is termed “quasi-Gaussian” to make a distinction from the standard Gaussian beam solution obtained in the paraxial approximation. The advantage of introducing this new solution is the efficient and fast computational modeling of tightly focused or quasi-collimated cylindrical wave-fronts depending on the dimensionless waist parameter kw0, where k is the wavenumber of the acoustical radiation. Moreover, a closed-form partial-wave series expansion is obtained for the incident field, which has the property that the axial scattering (i.e. along the direction of wave propagation) and the axial acoustic radiation force (which is a time-averaged quantity) on a cylinder, can be calculated without any approximations in the limit of linear acoustical waves in a nonviscous fluid. Examples are found where the extinction in the radiation force function plot is found to be correlated with conditions giving reduction of the backscattering from an elastic cylinder. Those results are useful in beam-forming design, particle manipulation in acoustic tweezers operating with focused cylindrical beams, and the prediction of the scattering and radiation forces on a cylindrical particle or liquid bridges.

Higher order light-cone distribution amplitudes of the Lambda baryon

The improved light-cone distribution amplitudes (LCDAs) of the $\Lambda$ baryon are examined on the basis of the QCD conformal partial wave expansion approach. The calculations are carried out to the next-to-leading order of conformal spin accuracy with consideration of twist 6. The next leading order conformal expansion coefficients are related to the nonperturbative parameters defined by the local three quark operator matrix elements with different Lorentz structures with a covariant derivative. The nonperturbative parameters are determined with the QCD sum rule method. The explicit expressions of the LCDAs are provided as the main results.

Unitarity constraints for Yukawa couplings in the two-Higgs-doublet model type III

Unitarity constraints for Yukawa couplings are considered in the Two-Higgs-Doublet Model type III, by using a general expansion in partial waves for fermionic scattering processes. Constraints over general Flavor Changing Neutral Currents are found from that systematic, wherein such bounds compete with those coming from Lagrangian perturbativity requirement but are weaker than those imposed from phenomenological processes and precision tests. Nevertheless, for bounds based on unitarity, the number of assumptions is the lowest among phenomenological and theoretical limits. Indeed, these new theoretical constraints are independent of scalar masses or mixing angles for this extended Higgs sector, making them less model dependent.

Cross-channel analysis of quark and gluon generalized parton distributions with helicity flip

Quark and gluon helicity flip generalized parton distributions (GPDs) address the transversity quark and gluon structure of the nucleon. In order to construct a theoretically consistent parametrization of these hadronic matrix elements, we work out the set of combinations of those GPDs suitable for the ${\rm SO}(3)$ partial wave (PW) expansion in the cross-channel. This universal result will help to build up a flexible parametrization of these important hadronic non-perturbative quantities, using for instance the approaches based on the conformal PW expansion of GPDs such as the Mellin-Barnes integral or the dual parametrization techniques.

The J and M integrals for a cylindrical cavity in a time-harmonic wave field

The invariant integrals are being widely used in the study of defects and fracture mechanics, mostly in elastostatics. However, the properties and the interpretation of these integrals in elastodynamics, especially in the case of time-harmonic excitation, remain unexplored. This contribution is focused on the derivation of the time average J integral for a cylindrical inhomogeneity and the M integral for a cylindrical cavity placed in a monochromatic plane elastic wave. It is shown in the context of antiplane linear elasticity that the J integral, or the material force acting on the inhomogeneity, resembles the radiation pressure force exerted on a dielectric cylinder by the normally incident electromagnetic wave. Based on the existing solution of this electrodynamic problem and the corresponding acoustic problem, the J integral is expressed as a function of the nondimensional wave number in the form of the partial wave expansion of the scattering theory. Employing the same classical method as with the J integral, the closed-form solution of the time average M integral for a traction-free cavity is also obtained as a function of the nondimensional wave number. The M integral, i.e., the expansion or scalar moment per unit length on an infinitely long circular cavity, is represented in terms of the scattering phase shifts as in the case of the J integral. Alternative expressions that are more convenient for numerical calculations for the cavity are also derived for both integrals. These calculations are carried out for the J and M integrals in a wide spectrum of frequencies. Asymptotic approximations of both integrals for low and high frequencies are presented. The long wavelength approximation, including the monopole and dipole contributions, has been provided for the J integral in the form of a simple analytical expression. The value of M integral in the vanishing frequency limit is also presented. In the opposite short wavelength limit, the corresponding asymptotic values are derived for both integrals. These solutions, which are valid for the empty cavity, are extended to the case of an inviscid fluid-filled cavity. The obtained results have a variety of engineering and geophysical implications. In particular, their applications to the flaw characterization of materials and further development of non-destructive evaluation techniques are briefly discussed in light of the frequency relationships derived for the J and M integrals.

Single Bessel tractor-beam tweezers

The tractor behavior of a zero-order Bessel acoustic beam acting on a fluid sphere, and emanating from a finite circular aperture (as opposed to waves of infinite extent) is demonstrated theoretically. Conditions for an attractive force acting in opposite direction of the radiating waves, determined by the choice of the beam’s half-cone angle, the size of the radiator, and its distance from a fluid sphere, are established and discussed. Numerical predictions for the radiation force function, which is the radiation force per unit energy density and cross-sectional surface, are provided using a partial-wave expansion method stemming from the acoustic scattering. The results suggest a simple and reliable analysis for the design of Bessel beam acoustical tweezers and tractor beam devices.

Interaction acoustic radiation force and torque on a cluster of spheres suspended in an inviscid fluid

The acoustic radiation force and torque exerted by a time-harmonic beam of arbitrary wavefront on a cluster of suspended spheres in an inviscid fluid is theoretically analyzed. In the proposed method, the effective incident wave is modelled as a coherent sum of an external beam and the contributions from the re-scattering events by other spheres present in the medium. Using the translational addition theorem for spherical functions the effective beam-shape and scattering coefficients are numerically computed [J. Acoust. Soc. Am. 98, 495 (1995)] for different external incident fields. The radiation force and torque exerted on the probe sphere can then be calculated using the farfield partial-wave expansion method [J. Acoust. Soc. Am. 130, 3541 (2011); Europhys. Phys. Lett. 97, 54003 (2012)]. The method was employed to obtain the radiation force due to an external plane and spherical waves on a cluster of three solid elastic or fluid spheres suspended in water. The results show that the radiation force deviates considerably from that exerted solely by the external incident wave and that the radiation torque arises on the spheres when a asymmetric spatial distribution of the effective incident acoustic field takes place in the medium. In addition, the proposed method may help on the study of acoustic tweezers devices and acoustofluidic systems, which involve several suspended particles.

Measure of the spatial size for the monopole excitation in proton scattering

We formulate a scattering radius, which will be demonstrated to be a good measure of the spatial size of a general exclusive reaction. The scattering radius is presented in a framework of the partial-wave expansion method in a general two-body scattering problem. A microscopic coupled-channel calculation is performed for proton scattering by $^{12}\mathrm{C}$ in the range of the proton's incident energy, ${E}_{p}=29.95--200$ MeV, and the scattering radii are evaluated for elastic scattering and inelastic scattering, going to the Hoyle 0${}_{2}^{+}$ state with a well-developed 3$\ensuremath{\alpha}$ structure. A prominent enhancement of the scattering radius is clearly confirmed in the 3$\ensuremath{\alpha}$ final channel in comparison to the elastic channel. The scattering radius is also calculated for excitation to the giant monopole resonance (GMR) in a microscopic coupled-channel framework. The scattering radius for the 3$\ensuremath{\alpha}$ excitation is much more enhanced than the scattering radius for the GMR excitation. The proton's incident-energy dependence of the scattering radius is also investigated, and the energy systematics strongly suggest that the scattering radius can characterize the spatial size of a reaction area, which is determined by the matter radius of a nucleus excited to a final state.

Near-field acoustic resonance scattering of a finite bessel beam by an elastic sphere

The near-field acoustic scattering from a sphere centered on the axis of a finite Bessel acoustic beam is derived stemming from the Rayleigh-Sommerfeld diffraction surface integral and the addition theorems for the spherical wave and Legendre functions. The beam emerges from a finite circular disk vibrating according to one of its radial modes corresponding to the fundamental solution of a Bessel beam J0. The incident pressure field's expression is derived analytically as a partial-wave series expansion taking into account the finite size and the distance from the center of the disk transducer. Initially, the scattered pressure by a rigid sphere is evaluated, and backscattering pressure moduli plots as well as 3-D directivity patterns for an elastic PMMA sphere centered on a finite Bessel beam with appropriate tuning of its half-cone angle, reveal possible resonance suppression of the sphere only in the zone near the Bessel transducer. Moreover, the analysis is extended to derive the mean spatial incident and scattered pressures at the surface of a rigid circular receiver of infinitesimal thickness. The transducer, sphere and receiver are assumed to be coaxial. Some applications can result from the present analysis since all physically realizable Bessel beam sources radiate finite sound beams as opposed to waves of infinite extent.

Higher order corrections to the light-cone distribution amplitudes of the sigma baryon

The light-cone distribution amplitudes (LCDAs) of the ${\mathrm{\ensuremath{\Sigma}}}^{\ifmmode\pm\else\textpm\fi{}}$ baryons up to twist-6 are investigated on the basis of the QCD conformal partial wave expansion approach. The calculations are carried out to the next-to-leading order of conformal spin accuracy. The nonperturbative parameters relevant to the LCDAs are determined in the framework of the QCD sum rule method. The explicit expressions of the LCDAs are given as the main results.