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All sourcesNASA NTRSDOE OSTI
NASA NTRS1987

Complex space multipole theory for scattering and diffraction problems

Classical multipole theory can be extended to multipoles located in complex space and applied in scattering and diffraction problems with the advantage that, if the point of the multipole is correctly chosen, the first term may give an order of magnitude better approximation to the source than when the multipole is in real space. The basic theory, given elsewhere, is presented here in a more straightforward manner and the improvement in radiation pattern is demonstrated for sources of constant polarization. Applications on scattering by spheroidal dielectric bodies and diffraction by a dielectric half-space are discussed.

Lindell, Ismo V.↗
DOE OSTI2019

An adaptive fast multipole method in Cartesian basis, enabled by algorithmic differentiation

A novel form of the fast multipole method is realized by the combination of adaptive data structures with implementation of the multipole expansions and their translations in Cartesian basis, and enabled by algorithmic differentiation. It may be possible to generalize it to any kernel that allows its formulation as a function of small quantities with respect to which they can be expanded. There is no hard and fast rule recipe that works for all cases. However, it is shown how to do it for the softened Coulomb kernel, which is different from the Coulomb kernel. The characteristics and performance of the method are mapped and shown to provide an attractive option for a generic fast multipole algorithm in a wide region of the parameter space. The algorithm is illustrated with a beam dynamics application.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Atom-partitioned multipole expansions for electrostatic potential boundary conditions

Applications such as grid-based real-space density functional theory (DFT) use the Poisson equation to compute electrostatics. However, the expected long tail of the electrostatic potential requires either the use of a large and costly outer domain or Dirichlet boundary conditions estimated via multipole expansion. We find that the oft-used single-center spherical multipole expansion is only appropriate for isotropic mesh domains such as spheres and cubes. In this work, we introduce a method suitable for high aspect ratio meshes whereby the charge density is partitioned into atomic domains and multipoles are computed for each domain. While this approach is moderately more expensive than a single-center expansion, it is numerically stable and still a small fraction of the overall cost of a DFT calculation. The net result is that when high aspect ratio systems are being studied, form-fitted meshes can now be used in lieu of cubic meshes to gain computational speedup.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Multipole Expansion of Atomic Electron Density Fluctuation Interactions in the Density-Functional Tight-Binding Method

The accuracy of the density-functional tight-binding (DFTB) method in describing noncovalent interactions is limited due to its reliance on monopole-based spherical charge densities. In this study, we present a multipole-extended second-order DFTB (mDFTB2) method that takes into account atomic dipole and quadrupole interactions. Furthermore, we combine the multipole expansion with the monopole-based third-order contribution, resulting in the mDFTB3 method. To assess the accuracy of mDFTB2 and mDFTB3, we evaluate their performance in describing noncovalent interactions, proton transfer barriers, and dipole moments. Here, our benchmark results show promising improvements even when using the existing electronic parameters optimized for the original DFTB3 model. Both mDFTB2 and mDFTB3 outperform their monopole-based counterparts, DFTB2 and DFTB3, in terms of accuracy. While mDFTB2 and mDFTB3 perform comparably for neutral and positively charged systems, mDFTB3 exhibits superior performance over mDFTB2 when dealing with negatively charged systems and proton transfers. Overall, the incorporation of the multipole expansion significantly enhances the accuracy of the DFTB method in describing noncovalent interactions and proton transfers.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Resonance Scattering Treatment with the Windowed Multipole Formalism

A new method for directly sampling the resonance upscattering effect is presented. Alternatives have relied on inefficient rejection sampling techniques or large tabular storage of relative velocities. None of these approaches, which require pointwise energy data, are particularly well suited to the windowed multipole cross-section representation. The new method, called multipole analytic resonance scattering, overcomes these limitations by inverse transform sampling from the target relative velocity distribution where the cross section is expressed in the multipole formalism. The closed-form relative speed distribution contains a novel special function we deem the incomplete Faddeeva function, and we present the first results on its efficient numerical evaluation.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2015

Transferable Atomic Multipole Machine Learning Models for Small Organic Molecules

Accurate representation of the molecular electrostatic potential, which is often expanded in distributed multipole moments, is crucial for an efficient evaluation of intermolecular interactions. In this paper we introduce a machine learning model for multipole coefficients of atom types H, C, O, N, S, F, and Cl in any molecular conformation. The model is trained on quantum-chemical results for atoms in varying chemical environments drawn from thousands of organic molecules. Multipoles in systems with neutral, cationic, and anionic molecular charge states are treated with individual models. The models’ predictive accuracy and applicability are illustrated by evaluating intermolecular interaction energies of nearly 1,000 dimers and the cohesive energy of the benzene crystal.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2013

Zero-multipole summation method for efficiently estimating electrostatic interactions in molecular system

The zero-multipole summation method has been developed to efficiently evaluate the electrostatic Coulombic interactions of a point charge system. This summation prevents the electrically non-neutral multipole states that may artificially be generated by a simple cutoff truncation, which often causes large amounts of energetic noise and significant artifacts. The resulting energy function is represented by a constant term plus a simple pairwise summation, using a damped or undamped Coulombic pair potential function along with a polynomial of the distance between each particle pair. Thus, the implementation is straightforward and enables facile applications to high-performance computations. Any higher-order multipole moment can be taken into account in the neutrality principle, and it only affects the degree and coefficients of the polynomial and the constant term. The lowest and second moments correspond respectively to the Wolf zero-charge scheme and the zero-dipole summation scheme, which was previously proposed. Relationships with other non-Ewald methods are discussed, to validate the current method in their contexts. Good numerical efficiencies were easily obtained in the evaluation of Madelung constants of sodium chloride and cesium chloride crystals.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2014

The zero-multipole summation method for estimating electrostatic interactions in molecular dynamics: Analysis of the accuracy and application to liquid systems

In the preceding paper [I. Fukuda, J. Chem. Phys. 139, 174107 (2013)], the zero-multipole (ZM) summation method was proposed for efficiently evaluating the electrostatic Coulombic interactions of a classical point charge system. The summation takes a simple pairwise form, but prevents the electrically non-neutral multipole states that may artificially be generated by a simple cutoff truncation, which often causes large energetic noises and significant artifacts. The purpose of this paper is to judge the ability of the ZM method by investigating the accuracy, parameter dependencies, and stability in applications to liquid systems. To conduct this, first, the energy-functional error was divided into three terms and each term was analyzed by a theoretical error-bound estimation. This estimation gave us a clear basis of the discussions on the numerical investigations. It also gave a new viewpoint between the excess energy error and the damping effect by the damping parameter. Second, with the aid of these analyses, the ZM method was evaluated based on molecular dynamics (MD) simulations of two fundamental liquid systems, a molten sodium-chlorine ion system and a pure water molecule system. In the ion system, the energy accuracy, compared with the Ewald summation, was better for a larger value of multipole moment l currently induced until l ≲ 3 on average. This accuracy improvement with increasing l is due to the enhancement of the excess-energy accuracy. However, this improvement is wholly effective in the total accuracy if the theoretical moment l is smaller than or equal to a system intrinsic moment L. The simulation results thus indicate L ∼ 3 in this system, and we observed less accuracy in l = 4. We demonstrated the origins of parameter dependencies appearing in the crossing behavior and the oscillations of the energy error curves. With raising the moment l we observed, smaller values of the damping parameter provided more accurate results and smoother behaviors with respect to cutoff length were obtained. These features can be explained, on the basis of the theoretical error analyses, such that the excess energy accuracy is improved with increasing l and that the total accuracy improvement within l ⩽ L is facilitated by a small damping parameter. Although the accuracy was fundamentally similar to the ion system, the bulk water system exhibited distinguishable quantitative behaviors. A smaller damping parameter was effective in all the practical cutoff distance, and this fact can be interpreted by the reduction of the excess subset. A lower moment was advantageous in the energy accuracy, where l = 1 was slightly superior to l = 2 in this system. However, the method with l = 2 (viz., the zero-quadrupole sum) gave accurate results for the radial distribution function. We confirmed the stability in the numerical integration for MD simulations employing the ZM scheme. This result is supported by the sufficient smoothness of the energy function. Along with the smoothness, the pairwise feature and the allowance of the atom-based cutoff mode on the energy formula lead to the exact zero total-force, ensuring the total-momentum conservations for typical MD equations of motion.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

Tunable multipole resonances in plasmonic crystals made by four-beam holographic lithography

Plasmonic nanostructures confine light to sub-wavelength scales, resulting in drastically enhanced light-matter interactions. Recent interest has focused on controlled symmetry breaking to create higher-order multipole plasmonic modes that store electromagnetic energy more efficiently than dipole modes. Here we demonstrate that four-beam holographic lithography enables fabrication of large-area plasmonic crystals with near-field coupled plasmons as well as deliberately broken symmetry to sustain multipole modes and Fano-resonances. Compared with the spectrally broad dipole modes we demonstrate an order of magnitude improved Q-factors (Q = 21) when the quadrupole mode is activated. We further demonstrate continuous tuning of the Fano-resonances using the polarization state of the incident light beam. The demonstrated technique opens possibilities to extend the rich physics of multipole plasmonic modes to wafer-scale applications that demand low-cost and high-throughput.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

A single-site multipole model for liquid water

Accurate and efficient empirical potential energy models that describe the atomistic interactions between water molecules in the liquid phase are essential for computer simulations of many problems in physics, chemistry, and biology, especially when long length or time scales are important. However, while models with non-polarizable partial charges at four or five sites in a water molecule give remarkably good values for certain properties, deficiencies have been noted in other properties and increasing the number of sites decreases computational efficiency. An alternate approach is to utilize a multipole expansion of the electrostatic potential due to the molecular charge distribution, which is exact outside the charge distribution in the limits of infinite distances or infinite orders of multipoles while partial charges are a qualitative representation of electron density as point charges. Here, a single-site multipole model of water is presented, which is as fast computationally as three-site models but is also more accurate than four- and five-site models. The dipole, quadrupole, and octupole moments are from quantum mechanical–molecular mechanical calculations so that they account for the average polarization in the liquid phase, and represent both the in-plane and out-of-plane electrostatic potentials of a water molecule in the liquid phase. This model gives accurate thermodynamic, dynamic, and dielectric properties at 298 K and 1 atm, as well as good temperature and pressure dependence of these properties.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2012

Can residuals of the solar system foreground explain low multipole anomalies of the CMB?

The low multipole anomalies of the Cosmic Microwave Background has received much attention during the last few years. It is still not ascertained whether these anomalies are indeed primordial or the result of systematics or foregrounds. An example of a foreground, which could generate some non-Gaussian and statistically anisotropic features at low multipole range, is the very symmetric Kuiper Belt in the outer solar system. In this paper, expanding upon the methods presented in [1], we investigate the contributions from the Kuiper Belt objects (KBO) to the WMAP ILC 7 map, whereby we can minimize the contrast in power between even and odd multipoles in the CMB, discussed in [2,3]. We submit our KBO de-correlated CMB signal to several tests, to analyze its validity, and find that incorporation of the KBO emission can decrease the quadrupole-octupole alignment and parity asymmetry problems, provided that the KBO signals has a non-cosmological dipole modulation, associated with the statistical anisotropy of the ILC 7 map. Additionally, we show that the amplitude of the dipole modulation, within a 2σ interval, is in agreement with the corresponding amplitudes, discussed in [4].

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Mixing bispectrum multipoles under geometric distortions

We derive general expressions for how the Alcock–Paczynski distortions affect the power spectrum and the bispectrum of cosmological fields. We compute explicit formulas for the mixing coefficients of bispectrum multipoles in the linear approximation. The leading-order effect for the bispectrum is the uniform dilation of all three wavevectors. The mixing coefficients depend on the shape of the bispectrum triplet. Our results for the bispectrum multipoles are framed in terms of the ‘natural’ basis of the lengths of three wavevectors but can be easily generalized for other bases and reduction schemes. Our validation tests confirm that the linear approximation is extremely accurate for all power spectrum multipoles. The linear approximation is accurate for the bispectrum monopole but results in sub-per cent level inaccuracies for the bispectrum quadrupole and fails for the bispectrum hexadecapole. Our results can be used to simplify the analysis of the bispectrum from galaxy surveys, especially the measurement of the baryon acoustic oscillation peak position. They can be used to replace numeric schemes with exact analytical formulae.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

CMB low multipole alignments across WMAP and Planck data releases

ABSTRACT The first observations of the cosmic microwave background (CMB) from NASA's Wilkinson Microwave Anisotropy Probe (WMAP) led to finding ‘alignment’ anomalies not expected from fluctuations in the isotropic cosmological model. We study the data of all 8 full-sky public releases since then to test for anomalous alignments and shapes of the first 60 multipoles, i.e. over the range $2\le l \le 61$. We use rotationally invariant and covariant statistics to test isotropy of all subsequent WMAP data releases, along with those from the ESA’s Planck mission. Anomalous alignments among the multipoles $l=1, 2, 3$ are very consistent and robust. More alignments are detected, some of them new, while significance is diluted by the large range of the search. Power entropy, a measure of the randomness of the multipoles, is consistently anomalous at about $2\sigma$ level or better across all data releases. It appears that the CMB is not as random as the cosmological principle predicts on large angular scales.

Patel, Sanjeet Kumar↗
DOE OSTI2014

Explicit expressions for totally symmetric spherical functions and symmetry-dependent properties of multipoles

Closed expressions for matrix elements < lm'|A(G)|lm >, where |lm > are spherical functions and A(G) is the average of all symmetry operators of point group G, are derived for all point groups (PGs) and then used to obtain linear combinations of spherical functions that are totally symmetric under all symmetry operations of G. In the derivation, we exploit the product structure of the groups. The obtained expressions are used to explore properties of multipoles of symmetric charge distributions. We produce complete lists of selection rules for multipoles Q(l) and their moments Q(lm), as well as of numbers of independent moments in a multipole, for any l and m and for all PGs. Periodicities and other trends in these properties are revealed.

Zapol, B.↗
DOE OSTI2020

Nuclear multipole responses from chiral effective field theory interactions

We probe nuclear multipole resonances in the framework of the random-phase approximation by using the interaction obtained from the chiral effective field theory. The three-nucleon force is included in a form of the in-medium two-nucleon interaction, which was derived from the chiral three-nucleon force. The isoscalar monopole, isoscalar dipole, isovector dipole, and isoscalar quadrupole resonances of the closed-shell 56,68,78 Ni have been investigated. The calculations reasonably reproduce the experimental multipole resonances of 56,68 Ni, and well describe the pygmy dipole resonance and dipole polarizability measured in 68 Ni. The multipole resonances of 78 Ni, including pygmy dipole resonance and dipole polarizability, are predicted. The detailed effects of the tensor force and three-body force are analyzed by dissecting the chiral interaction. In this work, we find that in general the tensor force effect on electric giant resonances is not as significant as the effect from the three-body force, although the tensor force provides more than half of the binding energy. The effect from three-body force is strong in light nuclei. Particularly, three-body force is crucial for the formation of the pygmy resonance in calculations.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2026

BICEP/Keck XXI: Constraints on early-Universe parity violation from multipole-dependent birefringence

We present the first constraints on multipole-dependent cosmic birefringence using CMB polarization data from the BK18 dataset, which combines observations from BICEP2, Array, and BICEP3 at frequencies of 95, 150, and 220 GHz. Photon coupling to an axionlike field leads to the rotation of CMB polarization, inducing nonzero cross-correlations. We show that a multipole-dependent rotation imprints a distinct signature in the polarization spectra that can be constrained. Specifically, we consider an early dark energy (EDE) scenario in which a pseudoscalar field couples to photons through a Chern-Simons interaction, generating a polarization rotation with multipole dependence. We introduce a phenomenological as a step function, obtaining constraints on the step function size consistent with zero, with uncertainties less than (68% CL). In addition, using multi-frequency , , and cross spectra, along with robust BICEP/ foreground treatment and likelihood framework, we derive constraints on the axion-photon coupling amplitude for several choices of EDE parameters. For the baseline best-fit value from the 2018 analysis, we obtain (68% CL), consistent with previous limits.

Ade, P A R↗
DOE OSTI2013

Multipole approximation of special nonlinear magnets for the IOTA ring

All existing focusing optics for accelerators are built to be "linear", i.e. the transverse focusing force is proportional to the particle displacement from the closed orbit. Such approach is used in order to make trajectories stable in the entire area of phase space, because the potential describing linear focusing is integrable and does not produce chaotic motion. Technically, the needed field is achieved with the help of dipole and quadrupole magnets. But the linear motion is unstable to perturbations in the focusing fields and excludes any betatron tune spread over particle's energy, which is beneficial for the beam stability. Moreover, the focusing strength depends on the particle's energy deviation from the designed one. These undesired effects are usually suppressed by the addition of non- linear elements, such as sextupoles and octupoles. Unfortunately, these additions make focusing potential nonintegrabe, leading to the limitation of stable phase space area. Recent research showed, that there exist non-linear potentials that, nevertheless, are fully integrable [1, 2]. Thus, there is an opportunity to create stable non-linear focusing optics that provides high betatron tune spread and mitigate magnetic field errors. This approach is to be implemented in electron storage ring called Integrable Optics Test Accelerator (IOTA) at Fermilab [3]. The main goal is to show the possibility to implement the nonlinear integrable system in a realistic accelerator design. Development of a magnet, which creates exact non-linear potential, is a technically complicated task, that takes much time and efforts. However, it is possible to expand the non- linear potential into superposition of multipoles of even orders (i.e quadrupole, octupole, etc.). Though this series is not convergent in the entire space, a few low order terms can be used to model the true potential in some limited area. In turn, each of the multipole potentials can be generated by speci ed electrical current in PCB coiled on the surface of storage ring's vacuum tube. Such approach is much cheaper than the creation of a magnet of solid iron with high current coil and faster than the development of \exact" non-linear magnet, but it can help identify some features of non-linear integrable optics application. The goal of this research work is to find analytically the angular distribution of electrical current on the surface of the vacuum tube, which creates inside it multipole potential with given parameters. Hereafter, there is the task to develop a model of flexible printed circuit board (PCB), which could be coiled around the vacuum tube in order to provide the desired current distribution.

Vorobev, Viktor↗
DOE OSTI2013

Individual NSLS-II Multipole Magnet Alignment by Laser Trackers

The magnet alignment of the 90 th and final multipole girder assembly was completed January 17 th , 2013. Our procedure of “rough” pre-alignment by laser trackers and final alignment by vibrating wire & “Differential Variable Reluctance Transducers” (DVRT) has been quite successful, achieving magnet to magnet alignment well under the 30 micron specification. The pre-alignment accuracy is verified by the initial scan from the vibrating wire. The alignment residuals can be calculated from the offsets of a best fit line. The results are impressive, especially considering the environment of the Building 902 Annex (which has a large bay door that would open during pre-alignment, without fail), the sloppiness of the magnets’ fiducials, and especially the fact that the girders were pre-aligned to their mechanical centers, but the vibrating wire reports the magnetic centers which may be considerably different. This result hints at the potential feasibility of alignment with laser trackers for replacement magnets of NSLS-II and maybe even all magnets of future accelerators. But, it is difficult to predict the achievable accuracy until the main three error sources (a.k.a. Survey group excuses) are eliminated: measuring environment, bad magnet fiducials, and mechanical/magnetic center differences. The test performed on a large aperture quadrupole on girder assembly C19G4 was designed to answer the following questions with the aforementioned error sources either eliminated or simulated: 1) How accurately can a NSLS-II multipole be realigned after repair or other disturbance given the existing fiducials? 2) How accurately can a replacement NSLS-II multipole be installed in situ if its magnetic center is known relative to good fiducials? 3) If the current NSLS-II spares are never magnetically fiducialized, how well can they be installed using only mechanical data? 4) Could the magnets of a future accelerator (with similar alignment specifications as NSLS-II) be aligned using commercially available metrology instruments? This paper will answer these questions.

36 MATERIALS SCIENCE↗