Partial Wave Expansion Research Articles

Generalized theory of resonance excitation by sound scattering from an elastic spherical shell in a nonviscous fluid

This work presents the general theory of resonance scattering (GTRS) by an elastic spherical shell immersed in a nonviscous fluid and placed arbitrarily in an acoustic beam. The GTRS formulation is valid for a spherical shell of any size and material regardless of its location relative to the incident beam. It is shown here that the scattering coefficients derived for a spherical shell immersed in water and placed in an arbitrary beam equal those obtained for plane wave incidence. Numerical examples for an elastic shell placed in the field of acoustical Bessel beams of different types, namely, a zero-order Bessel beam and first-order Bessel vortex and trigonometric (nonvortex) beams are provided. The scattered pressure is expressed using a generalized partial-wave series expansion involving the beam-shape coefficients (BSCs), the scattering coefficients of the spherical shell, and the half-cone angle of the beam. The BSCs are evaluated using the numerical discrete spherical harmonics transform (DSHT). The far-field acoustic resonance scattering directivity diagrams are calculated for an albuminoidal shell immersed in water and filled with perfluoropropane gas, by subtracting an appropriate background from the total far-field form function. The properties related to the arbitrary scattering are analyzed and discussed. The results are of particular importance in acoustical scattering applications involving imaging and beam-forming for transducer design. Moreover, the GTRS method can be applied to investigate the scattering of any beam of arbitrary shape that satisfies the source-free Helmholtz equation, and the method can be readily adapted to viscoelastic spherical shells or spheres.

Cooperativity in light scattering by cold atoms

AbstractA cloud of cold N two‐level atoms driven by a resonant laser beam shows cooperative effects both in the scattered radiation field and in the radiation pressure force acting on the cloud center‐of‐mass. The induced dipoles synchronize and the scattered light presents superradiant and/or subradiant features. We present a quantum description of the process in terms of a master equation for the atomic density matrix in the scalar, Born‐Markov approximations, reduced to the single‐excitation limit. From a perturbative approach for weak incident field, we derive from the master equation the effective Hamiltonian, valid in the linear regime. We discuss the validity of the driven timed Dicke ansatz and of a partial wave expansion for different optical thicknesses and we give analytical expressions for the scattered intensity and the radiation pressure force on the center of mass. We also derive an expression for collective suppression of the atomic excitation and the scattered light by these correlated dipoles.

Acoustic radiation force of a Bessel beam on a porous sphere

The possibility of using acoustic Bessel beams to produce an axial pulling force on porous particles is examined in an exact manner. The mathematical model utilizes the appropriate partial-wave expansion method in spherical coordinates, while Biot's model is used to describe the wave motion within the poroelastic medium. Of particular interest here is to examine the feasibility of using Bessel beams for (a) acoustic manipulation of fine porous particles and (b) suppression of particle resonances. To verify the viability of the technique, the radiation force and scattering form-function are calculated for aluminum and silica foams at various porosities. Inspection of the results has shown that acoustic manipulation of low porosity (<0.3) spheres is similar to that of solid elastic spheres, but this behavior significantly changes at higher porosities. Results have also shown a strong correlation between the backscattered form-function and the regions of negative radiation force. It has also been observed that the high-order resonances of the particle can be effectively suppressed by choosing the beam conical angle such that the acoustic contribution from that particular mode vanishes. This investigation may be helpful in the development of acoustic tweezers for manipulation of micro-porous drug delivery carrier and contrast agents.

A Lippmann–Schwinger approach for the determination of photoionization and photodetachment cross sections based on a partial wave Green's function expansion and configuration interaction wave functions

A computational approach is presented for determining partial differential cross sections for molecular photodetachment and photoionization. The approach, which extends a recent treatment of negative ion photodetachment, uses multireference configuration interaction wave functions to describe both the target or residual molecule and its interaction with the scattered electron and can treat states of the residual molecule which can be described adiabatically or are non-adiabatically coupled by conical intersections. Antisymmetry is taken into account in an approximate manner and a Lippmann–Schwinger equation is used to treat the interaction of the ionized/detached electron with the residual molecule. A partial wave expansion of the requisite Green's functions enables a unified treatment of photoionization and photodetachment and achieves computational efficiencies. A matrix representation of the quantities appearing in the iterated solution to the Lippmann–Schwinger equation using the radial Green's function permits the iterated solution to be summed eliminating potential convergence issues. Two methods for determining the radial Green's functions are presented. In the standard approach the Green's function is determined using the regular and irregular solutions to the corresponding homogeneous equation. The numerical issues associated with this well-studied form of the Green's function are non-trivial. An alternative approach is introduced which uses the complex scaling technique to construct the relevant matrix elements of the Green's function. The degree of simplification achieved in this way is significant.

Spherical geometric model for absorption electrons/positrons phenomenon: Application to the mean penetration depth calculation

I present a spherical geometric model for the slowing down and transport of electrons and positrons in solids. This latter is mainly based on the Continuous Slowing Down Approximation (CSDA) scheme. In advantage, I have developed an analytical expression of the mean penetration depth at normal and oblique incidence by combination between my model and the Vicanek and Urbassek theory. For this, I have used the Relativistic Partial Wave Expansion Method (RPWEM) and the optical dielectric model to calculate the elastic cross-sections and the ranges respectively. Good agreement was found with the experimental and theoretical data.

Partial-wave expansion for photoproduction of two pseudoscalars on a nucleon

The amplitudes for photoproduction of two pseudoscalars on a nucleon are expanded in the overall center-of-momentum (c.m.) frame in a model-independent way with respect to the contribution of the final-state partial wave of total angular momentum $J$ and its projection on the normal to the plane spanned by the momenta of the final particles. The expansion coefficients, which are analogs to the multipole amplitudes for single-meson photoproduction, contain the complete information about the reaction dynamics. Results of an explicit evaluation are presented for the moments ${W}_{jm}$ of the inclusive angular distribution of an incident photon beam with respect to the c.m. coordinate system defined by the final particles, taking photoproduction of ${\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ and ${\ensuremath{\pi}}^{0}\ensuremath{\eta}$ as an example.

Fine-structure splitting calculation in the Ar III ion: comparison of perturbative (Breit-Pauli) and non-perturbative (MCDF-EAL) predictions

We have calculated fine structure splitting in the Ar III ion using the R-matrix method. Two independent atomic structure calculations have been performed. Results from the Breit-Pauli – and the Dirac-Atomic – R-matrix relativistic calculations are analysed comparatively. Cross sections and collision strengths are provided for selected weak and intercombination transitions, allowing explicitly for resonance effects. Convergence of the partial wave expansion is ensured by examining the partial collision strengths at collision energies up to 20 Ry. Radiative data are also reported here. We discuss all these results and compare them, when possible. For the allowed transitions, for which the resonance structure is not dominant, the two calculations are nearly identical.

Two-body Coulomb scattering and complex scaling

The two-body Coulomb scattering problem is solved using the standard complex scaling method. The explicit enforcement of the scattering boundary condition is avoided. Splitting of the scattering wave function based on the Coulomb modified plane wave is considered. This decomposition leads a three-dimensional Schr\"odinger equation with source term. Partial wave expansion is carried out and the asymptotic form of the solution is determined. This splitting does not lead to simplification of the scattering boundary condition if complex scaling is invoked. A new splitting carried out only on partial wave level is introduced and this method is proved to be very useful. The scattered part of the wave function tends to zero at large inter-particle distance. This property permits of easy numerical solution: the scattered part of the wave function can be expanded on bound-state type basis. The new method can be applied not only for pure Coulomb potential butin the presence of short range interaction too.

Investigation of the Charge Symmetry conserving reaction dd →3Henπ0with WASA-at-COSY

The reaction dd → 3Henπ0 has been measured at a beam momentum of pd = 1.2 GeV/c using the WASA-at-COSY facility. For the first time data on the total cross section as well as differential distributions were obtained. The data are described with a phenomenological approach based on the combination of a quasi-free model and a partial wave expansion model for the three-body reaction.

Acoustic radiation torque of arbitrary shaped waves

In this work, a general expression of the acoustic radiation torque produced over an object by an arbitrary shaped beam is presented. The object is immersed in an inviscid fluid. To obtain the expression, the stress tensor of the angular momentum is integrated over a farfield virtual sphere, which encloses the object. The incident and scattered acoustic fields are represented through the partial wave expansion in the spherical coordinates. After performing the integration, the radiation torque is given in terms of the beam-shape and the scattering coefficients, which come from the incident and scattered partial wave series, respectively. The method is applied to compute the torque upon a fluid sphere by a vortex (first-order) Bessel beam in both on- and off-axis configurations. It is shown that the torque only arises on absorbing spheres. In this case, the angular acceleration and velocity are obtained from the torque. It is found that the angular acceleration may reverse its direction depending on the wave frequency. In conclusion, the presented theory might be useful for describing the particle dynamics of a sphere subjected to acoustic vortex beams.

Decoding the bispectrum of single-field inflation

Galileon fields arise naturally from the decoupling limit of massive gravities, and possess special self-interactions which are protected by a spacetime generalization of Galilean symmetry. We briefly revisit the inflationary phenomenology of Galileon theories. Working from recent computations of the fluctuation Lagrangian to cubic order in the most general model with second-order equations of motion, we show that a distinct shape is present but with suppressed amplitude. A similar shape has been found in other higher-derivative models. It may be visible in a theory tuned to suppress the leading-order shapes, or if the overall bispectrum has large amplitude. Using a partial-wave expansion of the bispectrum, we suggest a possible origin for the frequent appearance of this shape. It follows that models with very disparate microphysics can produce very similar bispectra. We argue that it may be more profitable to distinguish these models by searching for relations between the amplitudes of these common shapes. We illustrate this method using the examples of DBI and k-inflation.

Calculation of supercritical Dirac resonance parameters for heavy-ion systems from a coupled-differential-equation approach

Previous work [E. Ackad and M. Horbatsch, Phys. Rev. A 78, 062711 (2008)] on supercritical Dirac resonance parameters from extrapolated analytic continuation, obtained with a Fourier grid method, is generalized by numerically solving the coupled Dirac radial equations to a high precision. The equations, which contain the multipole decomposition of the two-center potential, are augmented by a complex absorbing potential and truncated at various orders in the partial wave expansion to demonstrate convergence of the resonance parameters in the limit of vanishing absorber. The convergence of the partial-wave spinor and of the multipole potential expansions is demonstrated in the supercritical regime. The comparison of critical distances with literature values shows that the work provides benchmark results for future two-center calculations without multipole expansion.

Potential energy surface and rotational cross sections for methyl formate colliding with helium

A potential energy surface for helium interacting with methyl formate has been computed using high-level electronic structure methods. The interaction energies obtained on a three-dimensional grid have been fitted by an analytic function of interatomic distances with correct asymptotic behavior for large intermonomer separations. This potential has then been refitted using partial wave expansion in terms of the distance between centers of mass and spherical angles. The latter potential has been used to compute cross sections for the rotational excitations of methyl formate at the full quantum close-coupling level. Collisional propensity rules and astrophysical implications are discussed.

Aeroacoustic Integrals Accelerated by Fast Multipole Method

The calculation of acoustic field solutions due to aeroacoustic sources is performed for a large number of observer locations. Sound is predicted by a hybrid method that uses direct calculation for near-field source computations and the Ffowcs Williams―Hawkings equation as the acoustic analogy formulation. The integrations of surface dipole and volume quadrupole source terms appearing in the Ffowcs Williams―Hawkings formulation are accelerated by a three-dimensional wideband multilevel adaptive fast multipole method. The three-dimensional wideband fast multipole method presented here applies a plane-wave expansion formulation with fast spherical filtering and interpolation and diagonal translation for calculations in the high-frequency regime and a partial-wave expansion formulation with rotational―coaxial translation in the low-frequency regime. The method is described for the solution of a three-dimensional Green's function that incorporates convective effects. The sound generated by the unsteady flow past a cylinder in the proximity of a NACA0012 airfoil is studied and the interaction of both geometries is analyzed for different cylinder positions. The developed numerical capability allows the analysis of each noise source individually. Therefore, the effects of dipoles and quadrupoles for each configuration are investigated. Results for acoustic field solutions obtained by the accelerated Ffowcs Williams―Hawkings formulation are up to 2 orders of magnitude faster than with the conventional computation of Ffowcs Williams―Hawkings equation.

An efficient method for computing backscattering from Born objects of arbitrary shape

A method is presented for efficiently computing the propagating pressure field backscattered by an arbitrarily shaped, weakly scattering, three-dimensional object. This is accomplished by drawing upon a previously reported relationship between the boundary condition on a two-dimensional radiating aperture and the pressure propagating along an axis normal to the aperture, and the fundamental theorem of diffraction tomography, which relates the Fourier transform of an object function to its scattered pressure field. Together, these two results are used to derive an integral formula that expresses the pressure field backscattered from an object as a one-dimensional Fourier transform of its scattering amplitude. This formula is then utilized to compute the backscattered pressure field from a uniform fluid sphere in the first Born approximation; the results of which are compared to the rigorous partial wave expansion.

On the relation between basis set convergence and electron correlation: a critical test for modern ab initio quantum chemistry on a “mindless” data set

The correlation of the only two error sources in the solution of the electronic Schrodinger equation is addressed: the basis set convergence (incompleteness) error (BSIE) and the electron correlation effect. The electron correlation effect and basis set incompleteness error are found to be correlated for all of the molecules in Grimme’s “mindless” data set (MB08-165). One can use an extrapolation to the HF or MP2 complete basis set (CBS) limit to see with which type of quantum chemical problem (“simple” and “hard”) the researcher is dealing. The origin of the slow convergence of the partial wave expansion can be the Kato cusp condition for electron–electron coalescence. Such an extrapolation is possible for many large molecular systems and would give the researcher an idea about the expected electron correlation level that would lead to the desired theoretical accuracy. In other words, it is possible to use not only the CBS energy value itself but the speed with which it is reached to get extra information about the molecular system under study.

Transverse Target Moments of SIDIS Vector Meson Production at HERMES

The on-going analysis of the transverse target (AUT) moments of the non-collinear cross section for SIDIS dihadron production at Hermes is discussed. These moments access the transversity (h1), pretzelocity (h⊥1T), and Sivers (f⊥1T) distribution functions convoluted with the unpolarized (D1) and Collins (H1) dihadron fragmentation functions. This measurement allows greater insight into the flavor decomposition and factorization of these functions. Additionally, the results will test the Lund/Artru fragmentation model, which predicts a sign change in the Collins function between pseudo-scalar mesons and certain partial waves of vector mesons. In preparation for this analysis, a new Monte Carlo generator has been written, and non-collinear fragmentation functions have been computed in a spectator model. Additionally, an alternate partial wave expansion is presented, providing a complementary interpretation of the fragmentation functions and allowing computation of the next-to-leading twist cross section.

Renormalization of chiral two-pion exchangeNNinteractions withΔexcitations: Correlations in the partial-wave expansion

In this work we consider the renormalization of the chiral two-pion exchange potential with explicit delta-excitations for nucleon-nucleon scattering at next-to-leading (NLO) and next-to-next-to-leading order (N2LO). Due to the singular nature of the chiral potentials, correlations between different partial waves are generated. In particular we show that two-body scattering by a short distance power like singular attractive interaction can be renormalized in all partial waves with a single counterterm, provided the singularities are identical. A parallel statement holds in the presence of tensor interactions when the eigenpotentials in the coupled channel problem also coincide. While this construction reduces the total number of counterterms to eleven in the case of nucleon-nucleon scattering with chiral two-pion exchange interactions with delta degrees of freedom, the differences in the scattering phases as compared to the case with the uncorrelated partial wave renormalization become smaller as the angular momentum is increased in the elastic scattering region.

Relativistic description of inclusive quasielastic proton-nucleus scattering with relativistic distorted-wave impulse approximation and random-phase approximation

We present a fully relativistic model for polarized inclusive quasielastic proton-nucleus scattering that includes relativistic distorted waves for the projectile and ejectile (RDWIA), as well as the relativistic random-phase approximation (RPA) applied to the target nucleus. Using a standard relativistic impulse approximation treatment of quasielastic scattering and a two-body Scalar, Pseudoscalar, Vector, Axial vector, Tensor (SPVAT) form of the current operator, it is shown how the behavior of the projectile/ejectile and target can be decoupled. Distortion effects are included via a full partial-wave expansion of the relativistic wave functions. Target correlations are included via the relativistic RPA applied to mean-field theory in quantum hadrodynamics. A number of novel analytical and numerical techniques are employed to aid in this highly nontrivial calculation. A baseline plane-wave calculation is performed for the reaction $^{40}\mathrm{Ca}$$(\mathrm{pP\vec},\mathrm{pP\vec} {}^{\ensuremath{'}})$ at an energy of 500 MeV and an angle ${\ensuremath{\theta}}_{\mathrm{c}.\mathrm{m}.}={40}^{\ifmmode^\circ\else\textdegree\fi{}}$. Here it is found that the effect of isoscalar correlations is a quenching of the cross section that is expected to become more pronounced at lower energies or for higher-density targets. A RDWIA calculation shows additional reduction and if isoscalar target correlations are included this effect is enhanced.

A Breit–Pauli distorted wave implementation for autostructure

We describe the Breit–Pauli distorted wave (BPDW) approach for the electron-impact excitation of atomic ions that we have implemented within the autostructure code. Program summaryProgram title:autostructureCatalogue identifier: AEIV_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEIV_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 130 987No. of bytes in distributed program, including test data, etc.: 1 031 584Distribution format: tar.gzProgramming language: Fortran 77/95Computer: GeneralOperating system: UnixHas the code been vectorized or parallelized?: Yes, a parallel version, with MPI directives, is included in the distribution.RAM: From several kbytes to several GbytesClassification: 2, 2.4Nature of problem: Collision strengths for the electron-impact excitation of atomic ions are calculated using a Breit–Pauli distorted wave approach with the optional inclusion of two-body non-fine-structure and fine-structure interactions.Solution method: General multi-configuration Breit–Pauli atomic structure. A jK-coupling partial wave expansion of the collision problem. Slater state angular algebra. Various model potential non-relativistic or kappa-averaged relativistic radial orbital solutions — the continuum distorted wave orbitals are not required to be orthogonal to the bound.Additional comments: Documentation is provided in the distribution file along with the test-case.Running time: From a few seconds to a few hours.