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This paper presents a study of the Bövik-Benveniste methodology for high order imperfect interface modeling of steady-state conduction problems involving coated spherical particles. Two types of imperfect interface models, that reduce the original three-phase configuration problem to the two-phase configuration problem, are discussed. In one model, the effect of the layer is accounted for via jumps in the field variables across the traces of its boundaries, while in the other via corresponding jumps across the trace of its mid-surface. Explicit expressions for the jumps are provided for both models up to the third order. The obtained higher order jump conditions are incorporated into the unit cell model of spherical particle composite. The multipole expansion method is used to derive the convergent series solutions to the corresponding boundary value problems. Numerical examples are presented to demonstrate that the use of higher order imperfect interface models allows for accurate evaluation of the local fields and overall conductivity of composites reinforced with coated spherical particles. [ABSTRACT FROM AUTHOR]

In recent years, the rapid development of dynamically tunable metasurfaces has provided a new avenue for flexible control of optical properties. This paper introduces a transmission-type electrically tunable metasurface, employing a series of subwavelength-scale silicon (Si) nanoring structures with an intermediate layer of Al2O3-ITO-Al2O3. This design allows the metasurface to induce strong Mie resonance when transverse electric (TE) waves are normally incident. When a bias voltage is applied, the interaction between light and matter is enhanced due to the formation of an electron accumulation layer at the ITO-Al2O3 interface, thereby altering the resonance characteristics of the metasurface. This design not only avoids the absorption loss of metal nanostructures and has a large modulation depth, but also shows compatibility with complementary metal oxide semiconductor (CMOS) technology. [ABSTRACT FROM AUTHOR]

Structural color, arising from the interaction between nanostructures and light, has experienced rapid development in recent years. However, high-order Mie resonances in dielectric materials often induce unnecessary sub-peaks, particularly at shorter wavelengths, reducing the vibrancy of colors. To address this, we have developed a multilayer dielectric metasurface based on silicon-rich silicon nitride (SRN), achieving expanded color gamut through precise refractive index matching and suppression of high-order resonances. This strategy introduces more design dimensions and can reduce the complexity of material deposition. It enables the generation of vibrant colors in a 3 × 3 array, with a resolution of approximately 25,400 dpi, demonstrating its potential applications in displays.

We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale l in an infinite heterogeneous correlated random medium, in a situation where the medium is only known in a box of diameter L ⪢ l around the support of the charge. We show that the algorithm in [J. Lu, F. Otto, and L. Wang, Optimal Artificial Boundary Conditions Based on Second-Order Correctors for Three Dimensional Random Ellilptic Media, preprint, arXiv:2109.01616, 2021], suggesting optimal Dirichlet boundary conditions motivated by the multipole expansion [P. Bella, A. Giunti, and F. Otto, Comm. Partial Differential Equations, 45 (2020), pp. 561--640], still performs well in correlated media. With overwhelming probability, we obtain a convergence rate in terms of l, L, and the size of the correlations for which optimality is supported with numerical simulations. These estimates are provided for ensembles which satisfy a multiscale logarithmic Sobolev inequality, where our main tool is an extension of the semigroup estimates in [N. Clozeau, Stoch. Partial Differ. Equ. Anal. Comput., 11 (2023), pp. 1254--1378]. As part of our strategy, we construct sublinear second-order correctors in this correlated setting, which is of independent interest. [ABSTRACT FROM AUTHOR]

Abstract Time-domain solutions of Maxwell’s equations in homogeneous and isotropic media are paramount to studying transient or broadband phenomena. However, analytical solutions are generally unavailable for practical applications, while numerical solutions are computationally intensive and require significant memory. Semi-analytical solutions (e.g., series expansion), such as those provided by the current theoretical framework of the multipole expansion, can be discouraging for practical case studies. This paper shows how sophisticated mathematical tools standard in modern physics can be leveraged to find semi-analytical solutions for arbitrary localized time-varying current distributions thanks to the novel time-domain Cartesian multipole expansion. We present the theory, apply it to a concrete application involving the imaging of an intricate current distribution, verify our results with an existing analytical approach, and compare the proposed method to a finite-difference time-domain numerical simulation. Thanks to the concept of current “pixels” introduced in this paper, we derive time-domain semi-analytical solutions of Maxwell’s equations for arbitrary planar geometries.

Time-domain solutions of Maxwell's equations in homogeneous and isotropic media are paramount to studying transient or broadband phenomena. However, analytical solutions are generally unavailable for practical applications, while numerical solutions are computationally intensive and require significant memory. Semi-analytical solutions (e.g., series expansion), such as those provided by the current theoretical framework of the multipole expansion, can be discouraging for practical case studies. This paper shows how sophisticated mathematical tools standard in modern physics can be leveraged to find semi-analytical solutions for arbitrary localized time-varying current distributions thanks to the novel time-domain Cartesian multipole expansion. We present the theory, apply it to a concrete application involving the imaging of an intricate current distribution, verify our results with an existing analytical approach, and compare the proposed method to a finite-difference time-domain numerical simulation. Thanks to the concept of current "pixels" introduced in this paper, we derive time-domain semi-analytical solutions of Maxwell's equations for arbitrary planar geometries. [ABSTRACT FROM AUTHOR]

A multipole expansion of the fundamental solution of the fractional power of the Laplace operator is constructed in terms of the Gegenbauer polynomials. Based on the decomposition constructed and the idea of the fast multipole method, we propose a numerical algorithm for solving the fractional differential generalization of the Poisson equation in the two-dimensional and three-dimensional spaces. [ABSTRACT FROM AUTHOR]

A semi-analytical approach is presented to solve three-dimensional dynamic Green's function for an infinitely extended acoustic field with multiple spheres subjected to the Robin boundary conditions. The multipole expansions of the acoustic field induced by a time-harmonic point source are expanded with spherical wave functions. As an alternative to the complex addition theorem, the multipole expansion is computed in a straightforward way. By taking the finite number of terms, an algebraic system is constructed and is used to obtain Green's function. This result of one sphere agrees with the available analytical solution. For the case of more than one sphere, the proposed results are verified by the numerical method such as the boundary element method (BEM). It indicates that the present solution is more accurate than that of the BEM and shows a fast convergence. Finally, the parameter study is performed to explore the influences of the exciting frequency of the point source, the surface admittance, the number and the separation of spheres on the dynamic Green's functions. The proposed results can be applied to solve the acoustic scattering problems and to increase the application of boundary integral equation method in the way of numerical Green's functions. [ABSTRACT FROM AUTHOR]

The paper presents the multipole expansion for the electron-nucleus scattering cross section at high energies within the framework of the unified electroweak theory. The electroweak currents of the nucleus are expanded into the simple components with definite angular momenta, called the multipole form factors. The multipole expansion of the cross section is a consequence of the above expansion. Besides the familiar electromagnetic form factors, there are also the vector and axial form factors, respectively, related to weak interactions. To determine multipole form factors, general formulas for the calculation of reduced matrix elements are established using the fractional parentage coefficient method and the multiparticle shell model. Calculation of them enables us to obtain more detailed information about the nuclear structure and elucidate the role played by the weak interaction in the high-energy reaction mechanisms.

Artificial nanostructures with large optical chiral responses have been intensively investigated recently. In this work, we propose a diffractive circular dichroism enhancement technique using stereoscopic plasmonic molecule structures. According to the multipole expansion analysis, the z-component of the electric dipole becomes the dominant chiral scattering mechanism during the interaction between an individual plasmonic molecule and the plane wave at a grazing angle. For a periodical structure with the designed plasmonic molecule, large diffractive circular dichroism can be obtained, which can be associated with the Wood–Rayleigh anomaly. Such a diffractive circular dichroism enhancement is verified by the good agreement between numerical simulations and experimental results. The proposed approach can be potentially used to develop enhanced spectroscopy techniques to measure chiral information, which is very important for fundamental physical and chemical research and bio-sensing applications. [ABSTRACT FROM AUTHOR]

For ultra-reliable high-data-rate communication, the beyond fifth generation (B5G) and the sixth generation (6G) wireless networks will heavily rely on beamforming, with mobile users often located in the radiative near-field of large antenna systems. Therefore, a novel approach to shape both the amplitude and phase of the electric near-field of any general antenna array topology is presented. Leveraging on the active element patterns generated by each antenna port, the beam synthesis capabilities of the array are exploited through Fourier analysis and spherical mode expansions. As a proof-of-concept, two different arrays are synthesized from the same active antenna element. These arrays are used to obtain 2D near-field patterns with sharp edges and a 30 dB difference between the fields' magnitudes inside and outside the target regions. Various validation and application examples demonstrate the full control of the radiation in every direction, yielding optimal performance for the users in the focal zones, while significantly improving the management of the power density outside of them. Moreover, the advocated algorithm is very efficient, allowing for a fast, real-time modification and shaping of the array's radiative near-field. [ABSTRACT FROM AUTHOR]

A conductive meta-atom of toroidal topology is studied both theoretically and experimentally, demonstrating a sharp and highly controllable resonant response. Simulations are performed both for a free-space periodic metasurface and a pair of meta-atoms inserted within a rectangular metallic waveguide. A quasi-dark state with controllable radiative coupling is supported, allowing to tune the linewidth (quality factor) and lineshape of the supported resonance via the appropriate geometric parameters. By conducting a rigorous multipole analysis, we find that despite the strong toroidal dipole moment, it is the residual electric dipole moment that dictates the electromagnetic response. Subsequently, the structure is fabricated with 3D printing and coated with silver paste. Importantly, the structure is planar, consists of a single metallization layer and does not require a substrate when neighboring meta-atoms are touching, resulting in a practical, thin and potentially low-loss system. Measurements are performed in the 5 GHz regime with a vector network analyzer and a good agreement with simulations is demonstrated. [ABSTRACT FROM AUTHOR]

Machine learning (ML) models, neural networks, and Gaussian processes have been used to predict the potential energy surface taking C2‐He (both static and dynamic scenario) and NCCN‐He collision systems. The surface is restricted to ∽125 points where traditional spline becomes inefficacious. Quantum dynamics is performed by solving close‐coupling equation to compute cross sections benchmarking the performance of the ML models. The current study forms a basis for any future investigation of larger molecules where conventional fitting fails due to sparser ab initio points and cuts down the computational time without compromising on the quality of the surface. [ABSTRACT FROM AUTHOR]

Following the first principles, the elements of the analytic theory of potential curves for diatomic molecules (diatomics) are presented. It is based on matching the perturbation theory at small internuclear distances R and multipole expansion at large distances, modified by exponentially small terms for homonuclear case, with the addition of the phenomenologically described equilibrium configuration, if it exists. It leads to a new class of (generalised) rational potentials (modified by exponentially small terms) with a difference in degrees of polynomials in numerator and denominator equal to 4 (6) for positively charged (neutral) diatomics. For example, the He 2 + and LiH diatomics in Born–Oppenheimer approximation are considered. For 4 He 2 + ( 3 He 2 + ) diatomics it is found the approximate analytic expression for the potential energy curves (analytic PEC) V (R) for the ground state X 2 Σ u + and the first excited state A 2 Σ g + . It provides 3–4 s.d. correctly for distances R ∈ [ 1 , 10 ] a.u. with some irregularities for A 2 Σ g + PEC at small distances (much are smaller than equilibrium distances) probably related to level (quasi)-crossings that may occur there. The analytic PEC for the ground state X 2 Σ u + supports 829 (626) rovibrational states with 3–4 s.d. of accuracy in energy, which is only by 1 state less (more) than 830 (625) reported in the literature. The analytic PEC for the excited state A 2 Σ g + supports all 9 reported weakly-bound rovibrational states. For 7 LiH it is found the analytic expression for the ground state X 1 Σ + PEC in the form of a rational function, which supports 906 rovibrational states with 3–4 s.d. accuracy in energy, it is only of 5 states more than reported in the literature (901). For both diatomics, the difference in a number of rovibrational states is related to the non-existence/existence of weakly-bound states close to the threshold (to dissociation limit). Entire rovibrational spectra are found in a single calculation using the code based on the Lagrange mesh method. [ABSTRACT FROM AUTHOR]

A semianalytic single-level fast direct solver (FDS) is presented for the method-of-moments (MoM) analysis of large planar antenna arrays with up to few thousands of complex elements on current desktop computers. Assuming a partition of the impedance matrix, the proposed scheme factorizes the near-field blocks via a singular value decomposition (SVD)-based low-rank approximation and the intermediate-field blocks via an analytical inhomogeneous plane-wave (IPW) expansion. The near- and intermediate-field bases are then unified to leverage the Woodbury formula for the inversion. The solutions for the excitation of each array ports are obtained at once with $O(N^{15/7} \ln ^{3} N)$ computational cost and $O(N^{11/7} \ln N)$ memory for array problems with $N$ basis functions. The complexities are traced precisely and the error is controlled by means of analytical formulas for the required number of IPWs. The FDS is also combined with a macrobasis function (MBF) approach to deal with antenna geometries containing many subwavelength details. This article includes simulations of regular arrays of about $169^{2}~3$ -D dipoles and an irregular array composed of seven stations of the low-frequency square kilometer array (SKA) telescope comprising each 256 log-periodic (LP) antennas. The proposed FDS allows the analysis of the coupling effect both within any one station but also between the stations of the telescope, a size and complexity of problem which has not been possible with existing published techniques. [ABSTRACT FROM AUTHOR]

The phenomenon of anapole has attracted considerable attention in the field of metamaterials as a possible realization of radiationless objects. We comprehensively study this phenomenon in the cluster-based systems of dielectric particles by considering conditions of anapole manifestation in both single trimers of disk-shaped particles and metamaterial composed on such trimers. Our analytical approach is based on the multipole decomposition method and the secondary multipole decomposition technique. They allow us to associate the anapole with the multipole moments of the trimer and the separate multipole moments of its constitutive particles. The manifestation of anapole in a two-dimensional metamaterial (metasurface) is confirmed by checking the resonant states in the reflected field as well as from the electromagnetic near-field patterns obtained from the full-wave numerical simulation. It is demonstrated that the anapole excitation in trimers results in the polarization-independent suppression of reflection with the resonant enhancement of local electromagnetic fields in the metasurface. Finally, experimental verification of the theoretical results is presented and discussed.

The model of an anisotropic interface in an elastic particulate composite with initial stress is developed as the first-order approximation of a transversely isotropic interphase between an isotropic matrix and spherical particles. The model involves eight independent parameters with a clear physical meaning and conventional dimensionality. This ensures its applicability at various length scales and flexibility in modeling the interfaces, characterized by the initial stress and discontinuity of the displacement and stress fields. The relevance of this model to the theory of material interfaces and its applicability in nanomechanics is discussed. The proposed imperfect interface model is incorporated in the unit cell model of a spherical particle composite with thermal stress owing to uniform temperature change. The rigorous solution to the model boundary value problem is obtained using the multipole expansion method. The reported accurate numerical data confirm the correctness of the developed theory, provide an estimate of its accuracy and applicability limits in the multiparticle environment, and reveal significant effects of the interphase or interface anisotropy and initial stress on the local fields and overall thermoelastic properties of the composite. [ABSTRACT FROM AUTHOR]

Electron–electron interactions and correlations form the basis of difficulties encountered in the theoretical solution of problems dealing with multi-electron systems. Accurate treatment of the electron–electron problem is likely to unravel some nice physical properties of matter embedded in the interaction. In an effort to tackle this many-body problem, a symmetry-dependent all-electron potential generalized for an n-electron atom is suggested in this study. The symmetry dependence in the proposed potential hinges on an empirically determined angular momentum-dependent partitioning fraction for the electron–electron interaction. With the potential, all atoms are treated in the same way regardless of whether they are open or closed shell using their system specific information. The non-relativistic ground-state ionization potentials for atoms with up to 103 electrons generated using the all-electron potential are in reasonable agreement with the existing experimental and theoretical data. The effects of higher-order non-relativistic interactions as well as the finite nuclear mass of the atoms are also analysed.

We are given a uniformly elliptic coefficient field that we regard as a realization of a stationary and finite-range ensemble of coefficient fields. Given a right-hand side supported in a ball of size ℓ ≫ 1 and of vanishing average, we are interested in an algorithm to compute the solution near the origin, just using the knowledge of the given realization of the coefficient field in some large box of size L ≫ ℓ . More precisely, we are interested in the most seamless artificial boundary condition on the boundary of the computational domain of size L. Motivated by the recently introduced multipole expansion in random media, we propose an algorithm. We rigorously establish an error estimate on the level of the gradient in terms of L ≫ ℓ ≫ 1 , using recent results in quantitative stochastic homogenization. More precisely, our error estimate has an a priori and an a posteriori aspect: with a priori overwhelming probability, the prefactor can be bounded by a constant that is computable without much further effort, on the basis of the given realization in the box of size L. We also rigorously establish that the order of the error estimate in both L and ℓ is optimal, where in this paper we focus on the case of d = 2 . This amounts to a lower bound on the variance of the quantity of interest when conditioned on the coefficients inside the computational domain, and relies on the deterministic insight that a sensitivity analysis with respect to a defect commutes with stochastic homogenization. Finally, we carry out numerical experiments that show that this optimal convergence rate already sets in at only moderately large L, and that more naive boundary conditions perform worse both in terms of rate and prefactor. [ABSTRACT FROM AUTHOR]

A conductive meta-atom of toroidal topology is studied both theoretically and experimentally, demonstrating a sharp and highly controllable resonant response. Simulations are performed both for a free-space periodic metasurface and a pair of meta-atoms inserted within a rectangular metallic waveguide. A quasi-dark state with controllable radiative coupling is supported, allowing to tune the linewidth (quality factor) and lineshape of the supported resonance via the appropriate geometric parameters. By conducting a rigorous multipole analysis, we find that despite the strong toroidal dipole moment, it is the residual electric dipole moment that dictates the electromagnetic response. Subsequently, the structure is fabricated with 3D printing and coated with silver paste. Importantly, the structure is planar, consists of a single metallization layer and does not require a substrate when neighboring meta-atoms are touching, resulting in a practical, thin and potentially low-loss system. Measurements are performed in the 5 GHz regime with a vector network analyzer and a good agreement with simulations is demonstrated.

The multipole expansion method (MEM) is a spatial discretization technique that is widely used in applications that feature scattering of waves from circular cylinders. Moreover, it also serves as a key component in several other numerical methods in which scattering computations involving arbitrarily shaped objects are accelerated by enclosing the objects in artificial cylinders. A fundamental question is that of how fast the approximation error of the MEM converges to zero as the truncation number goes to infinity. Despite the fact that the MEM was introduced in 1913, and has been in widespread usage as a numerical technique since as far back as 1955, a precise characterization of the asymptotic rate of convergence of the MEM has not been obtained. In this work, we provide a resolution to this issue. While our focus in this paper is on the Dirichlet scattering problem, this is merely for convenience and our results actually establish convergence rates that hold for all MEM formulations irrespective of the specific boundary conditions or boundary integral equation solution representation chosen. [ABSTRACT FROM AUTHOR]

A terahertz (THz) all-dielectric metasurface with crescent cylinder arrays for chiral drug sensing has been demonstrated. Through the multipole expansion method, we theoretically found that breaking the symmetry of the metasurface can excite higher-order resonance modes and provide stronger anisotropy as well as enhanced sensitivity for the surroundings, which gives a better sensing performance than lower-order resonance. Based on the frequency shift and transmittance at higher-order resonance, we carried out the sensing experiments on (R)-(−)-ibuprofen and (S)-(+)-ibuprofen solution on the surface of this metasurface sensor. We were able to monitor the concentrations of ibuprofen solutions, and the maximum sensitivity reached 60.42 GHz/mg. Furthermore, we successfully distinguished different chiral molecules such as (R)-(−)-ibuprofen and (S)-(+)-ibuprofen in the 5 μL trace amount of samples. The maximum differentiation was 18.75 GHz/mg. Our analysis confirms the applicability of this crescent all-dielectric metasurface to enhanced sensing and detection of chiral molecules, which provides new paths for the identification of biomolecules in a trace amount. [ABSTRACT FROM AUTHOR]

The previously reported exact potential and multipole moment (EP/MM) method for fast and precise evaluation of the intermolecular electrostatic interaction energies in molecular crystals using the pseudoatom representation of the electron density [Nguyen, Macchi & Volkov (2020), Acta Cryst. A76, 630–651] has been extended to the calculation of the electrostatic potential (ESP), electric field (EF) and electric field gradient (EFG) in an infinite crystal. The presented approach combines an efficient Ewald‐type summation (ES) of atomic multipoles up to the hexadecapolar level in direct and reciprocal spaces with corrections for (i) the net polarization of the sample (the 'surface term') due to a net dipole moment of the crystallographic unit cell (if present) and (ii) the short‐range electron‐density penetration effects. The rederived and reported closed‐form expressions for all terms in the ES algorithm have been augmented by the expressions for the surface term available in the literature [Stenhammar, Trulsson & Linse (2011), J. Chem. Phys.134, 224104] and the exact potential expressions reported in a previous study [Volkov, King, Coppens & Farrugia (2006), Acta Cryst. A62, 400–408]. The resulting algorithm, coded using Fortran in the XDPROP module of the software package XD, was tested on several small molecular crystal systems (formamide, benzene, l‐dopa, paracetamol, amino acids etc.) and compared with a series of EP/MM‐based direct‐space summations (DS) performed within a certain number of unit cells generated along both the positive and negative crystallographic directions. The EP/MM‐based ES technique allows for a noticeably more precise determination of the EF and EFG and significantly better precision of the evaluated ESP when compared with the DS calculations, even when the latter include contributions from an array of symmetry‐equivalent atoms generated within four additional unit cells along each crystallographic direction. In terms of computational performance, the ES/EP/MM method is significantly faster than the DS calculations performed within the extended unit‐cell limits but trails the DS calculations within the reduced summation ranges. Nonetheless, the described EP/MM‐based ES algorithm is superior to the direct‐space summations as it does not require the user to monitor continuously the convergence of the evaluated properties as a function of the summation limits and offers a better precision–performance balance. [ABSTRACT FROM AUTHOR]

Various optical antennas have been proposed to control the radiation pattern in a linear manner. Furthermore, the larger nonlinearity will play a vital role, and will change the refractive index of the antenna, thereby changing the radiation pattern. By modifying the effective mass of electrons in Drude model, it can be obtained that the refractive index of ITO film changes with the optical field intensity, and provides the nonlinear refractive index as a function of frequency and intensity. Nanoantennas made of indium tin oxide (ITO) will exhibit strong Kerr effect. The linear and nonlinear responses of ITO antennas was studied, the Kerr effect of ITO was used to control the radiation pattern of the antenna. Based on this model, a nonlinear optical antenna was designed to achieve tunable radiation pattern covering a broad near infrared band (1 000 nm to 1 650 nm). Moreover, a hybrid nonlinear antenna was demonstrated, composed of ITO and silicon (Si), which can control the radiation pattern more efficiency. This work breaks through the limitation that the strong nonlinear refractive index coefficient of nonlinear material only occurs at specific resonance frequency or the zero refractive index point. Our study provides a novel approach toward ultrafast dynamical control of metamaterials, for applications such as beam steering and optical modulation. [ABSTRACT FROM AUTHOR]

All-dielectric metasurfaces have attracted extensive attention because of their ability to control electromagnetic waves with low loss. However, compared with the research on the physical basis of a single nanoparticle controlling electromagnetic waves, foundational research on the influence of the periodicity of all-dielectric metasurfaces on their control of electromagnetic waves is still insufficient. Thus, the aim of this article is to further explore and emphasize the effect of the periodicity of metasurfaces on their optical performance and their electromagnetic response and coupling, and to describe how the above theoretical exploration can be employed for designing devices. We first explore some optical performance caused by the periodicity of cube silicon metasurfaces in the near-infrared band and reveal that these phenomena cannot be explained by the Mie resonances of the single nanoparticles composing the metasurfaces but by the lattice resonance theory and guided-mode resonance filter theory. Then, the influence of the periodicity on the electromagnetic response and coupling in the metasurfaces are analyzed via the multipole expansion method. We emphasize that the electromagnetic coupling between array nanoparticles enhances dipole and octupole radiation in the metasurfaces and analyze how the radiation is enhanced. We also study some phenomena and applications caused by the guided-mode resonances for future design. The results of this study may have potential applications in lasers, reflectors, absorbers, detectors, and other fields, and provide theoretical basis for the design of specific performance devices by adjusting the size of metasurfaces related to periodicity.

Despite its easy experimental implementation, macroscopic systems composed of compass needles array show a rich and complex variety of physical phenomena that are not yet fully understood. In this work we consider the experimental realization of a square lattice of equally spaced millimeter sized compass needles, as well as its theoretical approach taking into account a proper higher order moments expansion of the magnetic potential, considering the actual shape of a compass needle. A Monte Carlo simulation, done using interaction energy value given by the previous expansion, showed excelent agreement with experimental results for the squared array magnetization process. Rather than the unique commonly considered dipolar contribution, the results evidence the importance of multipole expansion in considering interactions among magnetic elements with finite size, attributing to the magnetic particle shape a critical importance that could raise higher order contributions. In addition to the clear pedagogical relevance, the conclusions deserve attention, and reopen the discussion of whether dipolar interaction is the only relevant contribution to be considered in theoretical models in which magnetostatic interactions are important, e.g. superparamagnetic nanoparticles models.

The previously reported exact potential and multipole moment (EP/MM) method for fast and accurate evaluation of the intermolecular electrostatic interaction energies using the pseudoatom representation of the electron density [Volkov, Koritsanszky & Coppens (2004). Chem. Phys. Lett.391, 170–175; Nguyen, Kisiel & Volkov (2018). Acta Cryst. A74, 524–536; Nguyen & Volkov (2019). Acta Cryst. A75, 448–464] is extended to the calculation of electrostatic interaction energies in molecular crystals using two newly developed implementations: (i) the Ewald summation (ES), which includes interactions up to the hexadecapolar level and the EP correction to account for short‐range electron‐density penetration effects, and (ii) the enhanced EP/MM‐based direct summation (DS), which at sufficiently large intermolecular separations replaces the atomic multipole moment approximation to the electrostatic energy with that based on the molecular multipole moments. As in the previous study [Nguyen, Kisiel & Volkov (2018). Acta Cryst. A74, 524–536], the EP electron repulsion integral is evaluated analytically using the Löwdin α‐function approach. The resulting techniques, incorporated in the XDPROP module of the software package XD2016, have been tested on several small‐molecule crystal systems (benzene, l‐dopa, paracetamol, amino acids etc.) and the crystal structure of a 181‐atom decapeptide molecule (Z = 4) using electron densities constructed via the University at Buffalo Aspherical Pseudoatom Databank [Volkov, Li, Koritsanszky & Coppens (2004). J. Phys. Chem. A, 108, 4283–4300]. Using a 2015 2.8 GHz Intel Xeon E3‐1505M v5 computer processor, a 64‐bit implementation of the Löwdin α‐function and one of the higher optimization levels in the GNU Fortran compiler, the ES method evaluates the electrostatic interaction energy with a numerical precision of at least 10−5 kJ mol−1 in under 6 s for any of the tested small‐molecule crystal structures, and in 48.5 s for the decapeptide structure. The DS approach is competitive in terms of precision and speed with the ES technique only for crystal structures of small molecules that do not carry a large molecular dipole moment. The electron‐density penetration effects, correctly accounted for by the two described methods, contribute 28–64% to the total electrostatic interaction energy in the examined systems, and thus cannot be neglected. [ABSTRACT FROM AUTHOR]

The Pure-type HTS undulator is proposed to achieve a high-intensity magnetic field and small size undulator for a compact free-electron laser (FEL). A high precision simulation is required before making the real machine since the sizes and positions are difficult to adjust after the HTSs are magnetized in the cryostat. For this purpose, authors have been developed a numerical simulation code for the magnetization process of HTS undulator of X-FEL based on the power-law macro-model. In this paper, the previously developed simulation code can be speeded up by carefully optimizing the parameters of the power-law macro-model for the 3-HTS magnets model and a large-scale simulation can be performed in an acceptable time by using a multipole expansion for the Biot–Savart law. In addition, for practical applications, the influence of the fluctuation of the magnets thickness and critical current for the electron trajectory are evaluated by using the speed-up simulation code. [ABSTRACT FROM AUTHOR]

In a homogeneous medium, the far field generated by a localized source can be expanded in terms of multipoles; the coefficients are determined by the moments of the localized charge distribution. We show that this structure survives to some extent for a random medium in the sense of quantitative stochastic homogenization: In three space dimensions, the effective dipole and quadrupole – but not the octupole – can be inferred without knowing the realization of the random medium far away from the (overall neutral) source and the point of interest. Mathematically, this is achieved by using the two-scale expansion to higher order to construct isomorphisms between the hetero- and homogeneous versions of spaces of harmonic functions that grow at a certain rate, or decay at a certain rate away from the singularity (near the origin); these isomorphisms crucially respect the natural pairing between growing and decaying harmonic functions given by the second Green's formula. This not only yields effective multipoles (the quotient of the spaces of decaying functions) but also intrinsic moments (taken with respect to the elements of the spaces of growing functions). The construction of these rigid isomorphisms relies on a good (and dimension-dependent) control on the higher-order correctors and their flux potentials. [ABSTRACT FROM AUTHOR]

The present work deals with the question of the origin choice to get the maximal accuracy of the dipole-Coulomb approximation in the theory of molecular Rydberg states with high orbital momentum l. Concerning birefringence in polar molecules, the question of adequate origin choice has first been raised in the classical work by Buckingham & Longuet-Higgins in 1968, where it has been taken into account that the dipole moment of a polar molecular core (cation) depends on the origin choice. This dependence also leads to inseparability of dipole and quadrupole corrections to the Rydberg electron spectra in polar molecules. In the present work a new option for the problem is proposed as applied to the dipole-Coulomb approximation. Considering a simplified model of a molecular core as a system of N effective fixed point charges, we show both analytically and numerically that the optimal choice of origin, at least for a linear core, is the geometric centre of the core. On the other hand, results of the present work as well as previous publications show that the problem raised by Buckingham and Longuet-Higgins does not have a universal solution. Abbreviations: RS: Rydberg state; RSs: Rydberg states; ZEKE: zero electron kinetic energy; MATI: mass-analysed threshold ionisation; QDT: quantum defect theory; DCA: dipole-Couloumb approximation; CD: centre of dipole; CC: centre of charge; GC: geometric centre; DCAGC: dipole-Couloumb approximation with origin in the geometric centre; DCACD: dipole-Couloumb approximation with origin in the centre of dipole [ABSTRACT FROM AUTHOR]

Dipole selection rules underpin much of our understanding in characterization of matter and its interaction with external radiation. However, there are several examples where these selection rules simply break down, for which a more sophisticated knowledge of matter becomes necessary. An example, which is increasingly becoming more fascinating, is macroscopic toroidization (density of toroidal dipoles), which is a direct consequence of retardation. In fact, dissimilar to the classical family of electric and magnetic multipoles, which are outcomes of the Taylor expansion of the electromagnetic potentials and sources, toroidal dipoles are obtained by the decomposition of the moment tensors. This review aims to discuss the fundamental and practical aspects of the toroidal multipolar moments in electrodynamics, from its emergence in the expansion set and the electromagnetic field associated with it, the unique characteristics of their interaction with external radiations and other moments, to the recent attempts to realize pronounced toroidal resonances in smart configurations of meta-molecules. Toroidal moments not only exhibit unique features in theory but also have promising technologically relevant applications, such as data storage, electromagnetic-induced transparency, unique magnetic responses and dichroism.

A terahertz (THz) all-dielectric metasurface with crescent cylinder arrays for chiral drug sensing has been demonstrated. Through the multipole expansion method, we theoretically found that breaking the symmetry of the metasurface can excite higher-order resonance modes and provide stronger anisotropy as well as enhanced sensitivity for the surroundings, which gives a better sensing performance than lower-order resonance. Based on the frequency shift and transmittance at higher-order resonance, we carried out the sensing experiments on (R)-(−)-ibuprofen and (S)-(+)-ibuprofen solution on the surface of this metasurface sensor. We were able to monitor the concentrations of ibuprofen solutions, and the maximum sensitivity reached 60.42 GHz/mg. Furthermore, we successfully distinguished different chiral molecules such as (R)-(−)-ibuprofen and (S)-(+)-ibuprofen in the 5 μL trace amount of samples. The maximum differentiation was 18.75 GHz/mg. Our analysis confirms the applicability of this crescent all-dielectric metasurface to enhanced sensing and detection of chiral molecules, which provides new paths for the identification of biomolecules in a trace amount.

We assess the performance of the Taruya, Nishimichi and Saito (TNS) model for the halo redshift space power spectrum, focusing on utilising mildly non-linear scales to constrain the growth rate of structure f. Using simulations with volume and number density typical of forthcoming Stage IV galaxy surveys, we determine ranges of validity for the model at redshifts z = 0.5 and z = 1. We proceed to perform a Bayesian MCMC analysis utilising the monopole, quadrupole, and hexadecapole spectra, followed by an exploratory Fisher matrix analysis. As previously noted in other forecasts as well as in real data analyses, we find that including the hexadecapole can significantly improve the constraints. However, a restricted range of scales is required for the hexadecapole in order for the growth parameter estimation to remain unbiased, limiting the improvement. We consistently quantify these effects by employing the multipole expansion formalism in both our Fisher and MCMC forecasts.

In this paper we derive a full expression for the propagation of multiplicative and additive shape measurement biases into the cosmic shear power spectrum. In doing so we identify several new terms that are associated with selection effects, as well as cross-correlation terms between the multiplicative and additive biases and the shear field. The computation of the resulting bias in the shear power spectrum scales as the fifth power of the maximum multipole considered. Consequently the calculation is unfeasible for large l-modes, and the only tractable way to assess the full impact of shape measurement biases on cosmic shear power spectrum is through forward modelling of the effects. To linear order in bias parameters the shear power spectrum is only affected by the mean of the multiplicative bias field over a survey and the cross correlation between the additive bias field and the shear field. If the mean multiplicative bias is zero then second order convolutive terms are expected to be orders of magnitude smaller.

A rigorous mathematical solution to the problem of synthesizing wire antennas using the multipole expansion method (MEM) is presented. The input current and required size of the antenna are estimated using the knowledge of the required field pattern. This solution is based on three steps. First, obtaining the expansion’s weighting coefficients that carry both the signature of the current and size of the antenna. In essence, the analysis of the direct modeling needs to be addressed first in order to define these coefficients. Second, extracting the coefficients from the required field pattern using the natural orthogonality property of the spherical harmonic functions. Finally, performing a numerical search for the argument of the spherical Bessel functions to satisfy the ratio between two successive moments. Once the size of the antenna is found, a recursive expression is used to obtain the input current. For the current distributions assumed herein, the odd order coefficients of the MEM are shown to be the only necessary coefficients to construct the radiation far-field. Also, it is found that the estimated model parameters are in excellent agreement with the analytical ones. [ABSTRACT FROM AUTHOR]

We developed a highly accurate and fully Maxwellian conformal mapping method for calculation of main fields of electrostatic particle optical elements. A remarkable advantage of this method is the possibility of rapid recalculations with geometric asymmetries and mispowered plates. We used this conformal mapping method to calculate the multipole terms of the high voltage quadrupoles in the storage ring of the Muon g − 2 Experiment (FNAL-E-0989). Next, we demonstrate that an effect where the observed tunes correspond to a voltage that is about 4 % higher compared to the voltage to which the Muon g − 2 quadrupoles are set is explained by the conceptual and quantitative differences between the beam optics quadrupole voltage and the quadrupole voltage at the plates. Completing the methodological framework for field computations, we present a method for extracting multipole strength falloffs of a particle optical element from a set of Fourier mode falloffs. We calculated the quadrupole strength falloff and its effective field boundary (EFB) for the Muon g − 2 quadrupole, which has explained the experimentally measured tunes, while simple estimates based on a linear model exhibited discrepancies up to 2 %. [ABSTRACT FROM AUTHOR]

The dynamics of the charge distribution produced at the inner surface of an insulating capillary by low energy ion impact are investigated theoretically in the case where the entrance, exit and outer capillary surface are grounded. Starting with the surface continuity equation, that describes the charge dynamics at the inner surface, we deduced an analytical solution of the continuity equation in the form of a linear combination of surface charge moments that satisfy the given boundary conditions. We determined the relaxation rate of each moment, which is given as a function of the dimensions and electrical properties of the capillary. We found an approximate expression of the time evolution of the electric potential and field generated by a charge patch during and after beam irradiation. The time evolution of the total charge is also described in detail. Our findings are illustrated with examples. [ABSTRACT FROM AUTHOR]

The deformed relativistic Hartree–Bogoliubov theory in continuum (DRHBc) has been proved as one of the best models to probe the exotic structures in deformed nuclei. In DRHBc, the potentials and densities are expressed in terms of the multipole expansion with Legendre polynomials, the dependence on which has only been touched for light nuclei so far. In this paper, taking a light nucleus 2 0 Ne and a heavy nucleus 2 4 2 U as examples, we investigated the dependence on the multipole expansion of the potentials and densities in DRHBc. It is shown that the total energy converges well with the expansion truncation both in the absence of and presence of the pairing correlation, either in the ground state or at a constrained quadrupole deformation. It is found that to reach the same accuracy of the total energy, even to the same relative accuracy by percent, a larger truncation is required by a heavy nucleus than a light one. The dependence of the total energy on the truncation increases with deformation. By decompositions of the neutron density distribution, it is shown that a higher-order component has a smaller contribution. With the increase of deformation, the high-order components get larger, while at the same deformation, the high-order components of a heavy nucleus play a more important role than that of a light one. [ABSTRACT FROM AUTHOR]

Based on linear potential theory, this study develops an analytical solution to water wave scattering by a submerged perforated reef ball. The series solutions for velocity potentials in the external and internal fluid domains of the perforated reef ball are developed using the multipole expansions and the separation of variables, respectively. The unknowns in the series solutions are determined by matching the porous boundary condition on the perforated reef ball. Also a numerical solution using three-dimensional multi-domain boundary element method (BEM) is developed. The special case of an impermeable reef ball is also examined. The calculation methods for the horizontal and vertical wave forces acting on the reef ball and the free surface elevation around the reef ball are given. The convergence of the calculation results is very rapid, and the results with five-figure accuracy are obtained using a few truncated terms in the series solution. The results by the analytical solution agree well with that by the multi-domain BEM solution. Typical calculation examples are presented to examine the effects of reef ball radius, porous effect parameter of reef ball and incident wave frequency on the wave forces and the free surface elevation. The results of analysis are valuable for engineering design and application of reef balls. The present analytical solution gives a new reliable benchmark for three-dimensional numerical models to wave interactions with perforated structures. Moreover, better understanding on wave scattering by perforated structures can be obtained during the analytically solving procedure. [ABSTRACT FROM AUTHOR]

In this paper, we study some classical aspects of Podolsky electrodynamics in the static regime. We develop the multipole expansion for the theory in both the electrostatic and the magnetostatic cases. We also address the problem of consistently truncating the infinite series associated with the several kinds of multipoles, yielding approximations for the static Podolskian electromagnetic field to any degree of precision required. Moreover, we apply the general theory of multipole expansion to some specific physical problems. In those problems, we identify the first terms of the series with the monopole, dipole, and quadrupole terms in the generalized theory. We also propose a situation in which Podolsky theory can be experimentally tested. [ABSTRACT FROM AUTHOR]

A numerical method for finding the potential of a multiatomic system in the real space is proposed. A distinctive feature of this method is the decomposition of the electron density and the potential into two parts and , where is the sum of the spherical atom densities and the potential is generated by the density . The potential is found by solving Poisson's equation. The boundary conditions are obtained by expanding the reciprocal distance between two points in a series in Legendre polynomials. To improve the accuracy of the method, the computation domain is decomposed into Voronoi polyhedra, and asymptotic estimates of iterations are used when the characteristic function is replaced by its smooth approximations. Poisson's equation is numerically solved using the two-grid method and the Fourier transform. An estimate , where is the grid size and is a fixed number greater than one, is obtained for the accuracy of the method. The error of the method is analyzed using a two-atom problem as an example. [ABSTRACT FROM AUTHOR]

We propose a consistent approach to the definition of electric, magnetic and toroidal multipole moments. Electric and magnetic fields are split into potential, vortex and radiative terms, with the latter ones dropped off in the quasistatic approximation. The potential part of the electric field, the vortex parts of the magnetic field and vector potential contain gradients of scalar functions. Formally introducing magnetic and toroidal analogs of the electric charge, we apply multipole expansions for those scalars. Closed-form expressions are derived in an arbitrary order for electric, magnetic and toroidal multipoles, which constitute a full system for expansions of the electromagnetic field. [ABSTRACT FROM AUTHOR]

It is well known that Green’s function can be expressed by multipole expansion, plane-wave expansion, and exponential expansion (spectral representation). These three expansions constitute of the foundations of the fast multipole algorithm (FMA). The plane-wave expansion has the low-frequency breakdown issue due to its failure in capturing the evanescent spectra, while the multipole expansion is inefficient at high frequencies. The spectral representation usually involves in direction-dependent issue. In this communication, the 2-D FMA is interpreted as Parseval’s theorem in Fourier transform. To achieve a stable and accurate transition between the multipole expansion and the plane-wave expansion, a novel diagonalization in the 2-D FMA is proposed with scaled special functions based on a discrete Fourier transform. A wideband fast algorithm with high accuracies can be achieved efficiently. [ABSTRACT FROM AUTHOR]

Abstract The multiple cavity model of nanoporous solid has been developed which takes into account the elastic stiffness of matrix solid, porosity, shape, size and arrangement of cavities as well as the effects of their interaction. To describe the pertinent to nano level size effect of the stress field and effective stiffness, the Gurtin–Murdoch model that includes both surface tension and surface stiffness has been applied. A complete semi-analytical solution to the model problem has been obtained by the multipole expansion method. The displacement perturbation field induced by each cavity is expanded over a set of the vector solutions of Lamé equation in spheroidal basis. Expansion of the displacement and stress fields in terms of the vector surface harmonics reduces the surface conditions to an infinite linear system for the multipole strengths. The superposition principle and re-expansion formulas for the partial vector solutions extend this theory to problems involving multiple cavities. The method provides accurate evaluation of the stress field with the interaction effects taken into account. Also, the derived solution constitutes a mathematical background of the modified Maxwell homogenization scheme. The stiffness tensor of nanoporous solid is found by equating the induced dipole moment of equivalent inclusion to the total dipole moment of individual cavities. Numerical study demonstrate an accuracy and computational efficiency of the developed approach and shows quite a significant combined effect of the cavity shape and surface tension and stiffness on the stress field and macroscopic stiffness of nanoporous solid. [ABSTRACT FROM AUTHOR]

An analytical approach to the elasticity homogenization problem for a spheroidal particle composite with general imperfect interface has been developed. The considered model takes into account the volume content and elastic moduli of constituents, shape, size and orientation of inhomogeneities and interface stiffness. The multipole expansion method has been applied to obtain an accurate solution to the problem for a single spheroidal inhomogeneity with imperfect interface. This solution constitutes a mathematical background of the Maxwell homogenization scheme. The formula for effective stiffness is derived by equating the induced dipole moment of equivalent inclusion to the total dipole moment of individual inhomogeneities. Numerical study shows a significant combined effect of interface stiffness and particle size, shape and orientation on the stress field and macroscopic stiffness of composite. [ABSTRACT FROM AUTHOR]