Multipole Interactions Research Articles

A new resource-rich carbonate luminescent host KNa5Ca5(CO3)8: A case study activated by Eu3+ for red-phosphor purpose

Exploring and designing new luminescent host with rich resource is beneficial to reducing the cost and the large-scale application of luminescent materials. Herein, a series of novel Eu3+-activated KNa5Ca5(CO3)8 red-emitting phosphors with different Eu3+ concentrations were successfully synthesized under mild hydrothermal condition. XRD and SEM characterizations show that all as-prepared KNa5Ca5(CO3)8:Eu3+ samples crystallize in a hexagonal structure, P63mc (No.186) space group, and show monodispersed hexagonal sheet morphologies, within a side length interval about of 2–3 μm and a thickness of about 0.6 μm. Eu3+ doping within 6 mol% concentration has little influence on the crystal structure of the host, as indicated by XRD, and FT-IR results, but thermal stability changes as shown by the TG-DTA. KNa5Ca5(CO3)8 phosphors with different Eu3+ concentrations can emit red-light under effectively excitation by near-UV light. CIE chromaticity coordinates of all KNa5Ca5(CO3)8:Eu3+ phosphors were calculated to locate in red region, and CCT of each sample is the range from 2815 to 3650 K, very close to the tungsten lamp light. Besides, concentration quenching phenomenon appearing at 5 mol% is found to be the result of electric multipolar interaction of Eu3+ activators, and the decay lifetime is about 1.63 ms, which almost unchanges with Eu3+ doping concentration. Importantly, thermal quenching analysis reveals good thermal stability with thermal activation energy of about 0.216 eV. The results reported here demonstrate that KNa5Ca5(CO3)8 is a promising fluorescent host for LEDs applications.

Nuclear transition matrix elements for neutrinoless double-β decay within Rp-violating squark-neutrino mechanism

Within the squark-neutrino mechanism of [Formula: see text]-violating SUSY, sets of 12 nuclear transition matrix elements (NTMEs) are calculated for the neutrinoless double-[Formula: see text] decay [Formula: see text] of [Formula: see text]Zr, [Formula: see text]Mo, [Formula: see text]Pd, [Formula: see text]Te and [Formula: see text]Nd isotopes. Specifically, four sets of HFB wave functions generated with four different parametrizations of the pairing plus multipolar two-body interactions, dipole form factor and three different parametrizations of the Jastrow short-range correlations are employed in the calculation of NTMEs with two possible prescriptions for the hadronization, namely the two-nucleon mode and the pionic mode. Without (with) Miller–Spencer parametrization of short-range correlation, uncertainties in average NTMEs [Formula: see text] (QBM), [Formula: see text] (NRQM), [Formula: see text] (FF3) and [Formula: see text] turn out be 11–18% (29–37%), 11–16% (27–31%), 5–12% (13–17%) and 3–13% (9–15%), respectively.

Active Plasmonics and Active Chiral Plasmonics through Orientation-Dependent Multipolar Interactions

While most active plasmonic efforts focus on responsive metamaterials to modulate optical response, we present a simple alternative based on applied orientation control that can likely be implemented for many passive plasmonic materials. Passive plasmonic motifs are simpler to prepare but cannot be altered post-fabrication. We show that such systems can be easily manipulated through substrate orientation control to generate both active plasmonic and active chiral plasmonic responses. Using gold nanocrescents as our model platform, we demonstrate tuning of optical extinction from -21% to +36% at oblique incidence relative to normal incidence. Variation of substrate orientation in relation to incident polarization is also demonstrated to controllably switch chiroptical handedness (e.g., Δg = ±0.55). These active plasmonic responses arise from the multipolar character of resonant modes. In particular, we correlate magnetoelectric and dipole-quadrupole polarizabilities with different light-matter orientation-dependence in both near- and far-field localized surface plasmon activity. Additionally, the attribution of far-field optical response to higher-order multipoles highlights the sensitivity offered by these orientation-dependent characterization techniques to probe the influence of localized electromagnetic field gradients on a plasmonic response. The sensitivity afforded by orientation-dependent optical characterization is further observed by the manifestation in both plasmon and chiral plasmon responses of unpredicted structural nanocrescent variance (i.e., left- and right-tip asymmetry) not physically resolved through topographical imaging.

Synthesis, neutron diffraction and photoluminescence properties of a whitlockite structured Ca9MgLi(PO4)7:Pr3+ phosphor

The whitlockite structured compound Ca9MgLi(PO4)7 has been investigated as a potential host material for down-conversion phosphors. However, no detailed crystal structure and crystallographic site occupancy have been reported for this compound previously. In this work, Pr3+ activated Ca9MgLi(PO4)7 phosphors were successfully synthesized, and the structure and photoluminescence properties were characterized. Upon neutron and X-ray powder diffraction data, the crystallographic site occupancies of M1-M5 in the whitlockite-type structure were carefully refined by the Rietveld method. The M1, M2 and M3 sites are all found to be 8-fold coordinated and fully occupied with Ca and Pr. The results also revealed that the M4 and M5 sites are co-occupied by Li–Mg and Ca–Mg, respectively. Photoluminescence (PL) study showed that the phosphor has a characteristic excitation band at the visible light region of 430–500 nm. Three peaks observed at 442, 468, and 484 nm in the excitation spectra can be ascribed to the 3H4→3P2, 3H4→3P1, and 3H4→3P0 transition of Pr3+, respectively. The phosphor doped with 0.02 Pr3+ gives the strongest emission peaked at 612 nm, with the CIE chromaticity coordinate of (0.613, 0.366). This reddish-orange emitting phosphor has good thermal stability that it retains 83% of its emission intensity at 200 °C. Energy transfer mechanism of the activators was analyzed to be dipole-dipole multipolar interaction. The present work demonstrates that Ca9MgLi(PO4)7:Pr3+ is a promising phosphor for blue-pumped white light-emitting diodes (WLEDs).

Fabrication of single-phase BaLaAlO4:Dy3+ nanophosphors by combustion synthesis

ABSTRACT To solve the problem of multiphased commercial light emitting diodes, a series of new single-phased BaLaAlO4:Dy3+ nanophosphors emitting cool white light were successfully fabricated via an energy-efficient approach, namely, solution combustion. X-ray diffractometry and Rietveld refinement revealed orthorhombic phase purity with P 212121(19) space group. Agglomerated orthorhombic nanoparticles (30–50 nm) were reported via scanning and transmission electron microscopy, whereas elemental distribution was confirmed by energy dispersive X-ray analysis. On excitation by 349 nm light, the nanophosphors showed peaks at 481 nm (blue color) on account of magnetic dipole transition and 573 nm (yellow color) by reason of electric dipole transition. Maximum intensity was observed for 3.0 mol% concentration of Dy3+ ions above which concentration quenching occurred due to electric multipolar interactions. Decay lifetime and energy band gap for optimized nanophosphor (BaLa1−x Dy x AlO4 (x = 3.0 mol%)) were obtained as 0.651 ms and 5.41 eV, respectively. CIE coordinates (0.297, 0.283), color purity (25.44 × 10−2), and correlated color temperature (CCT) value (8480 K) of optimal nanophosphor were found approximate to white coordinates of commercial LEDs and National Television System Committee, which proved their emission of cool white light and suggested their use in the outdoor lighting devices, i.e., white light emitting diodes.

Optical thermometry using Stark sublevels in charge compensated transition metal molybdate upconverting phosphors

A series of anorthic phased Er3+/Yb3+/Li+/K+: ZnMoO4 phosphors have been synthesized by solid state reaction method. Under 980 nm excitation, Er3+-Yb3+codoped phosphors exhibit intense green emission at ~531 nm/~553 nm corresponding to the 2H11/2/4S3/2 → 4I15/2 and weak blue and red emissions at ~408 nm, ~474 nm and ~656 nm corresponding to the 2H9/2 → 4I15/2, 4F7/2 → 4I15/2 and 4F9/2 → 4I15/2 transitions, respectively. The multipolar interaction is found responsible for the concentration quenching. Furthermore, with incorporating the charge compensating agents Li+/K+ the UC emission intensity as well as temperature sensing performance have been perceptibly improved compared to the Er3+-Yb3+ phosphors. The fluorescence intensity ratio (FIR) technique have been performed to the thermally coupled (2H11/2, 4S3/2) levels and Stark sublevels (2H11/2, 4S3/2(I) &2H11/2, 4S3/2(II)) to confirm the temperature sensing ability of the phosphors. The maximum sensitivity corresponding to the 2H11/2, 4S3/2(I) → 4I15/2 &2H11/2, 4S3/2(II) → 4I15/2 transitions from the Stark sublevels have been observed.

Luminescence properties of Dy3+ doped glass ceramics containing Na3Gd(PO4)2

Dy3+ doped precursor glass (PG) was prepared by melt-quenching, the glass ceramics (GCs) containing Na3Gd(PO4)2 phase were obtained via heating treatment. The heat treatment regime, luminescence properties of the samples were studied by differential scanning calorimetry (DSC), X-ray diffraction (XRD), scanning electron microscope (SEM), and photoluminescence spectrum (PL), respectively. Finally, heat treatment system of glass ceramics is 660°C, 2 h. Emission spectra show that there are two mainly peaks at 483 nm and 577 nm, emitting blue and yellow light. The optimal doping concentration of Dy2O3 was determined to be 0.7%, and the reason for the concentration quenching is electrical multipolar interaction. According to the chromaticity coordinates (CIE), it indicated that with the change of Dy2O3concentration, the luminescence color changed from blue light to warm white light. The result shows that the Dy3+ doped GCs containing Na3Gd(PO4)2 have potential application value in white light-emitting diode (W-LED).

New LHCb pentaquarks as hadrocharmonium states

New LHCb Collaboration results on pentaquarks with hidden charm1 are discussed. These results fit nicely in the hadrocharmonium pentaquark scenario.[Formula: see text] In the new data the old LHCb pentaquark [Formula: see text] splits into two states [Formula: see text] and [Formula: see text]. We interpret these two almost degenerated hadrocharmonium states with [Formula: see text] and [Formula: see text], as a result of hyperfine splitting between hadrocharmonium states predicted in Ref. 2. It arises due to QCD multipole interaction between color-singlet hadrocharmonium constituents. We improve the theoretical estimate of hyperfine splitting[Formula: see text] that is compatible with the experimental data. The new [Formula: see text] state finds a natural explanation as a bound state of [Formula: see text] and a nucleon, with [Formula: see text], [Formula: see text] and binding energy 42 MeV. As a bound state of a spin-[Formula: see text] meson and a nucleon, hadrocharmonium pentaquark [Formula: see text] does not experience hyperfine splitting. We find a series of hadrocharmonium states in the vicinity of the wide [Formula: see text] pentaquark that can explain its apparently large decay width. We compare the hadrocharmonium and molecular pentaquark scenarios and discuss their relative advantages and drawbacks.

Thermally stable orange-red emitting Ba2SiO4:Sm3+ phosphor: Synthesis and luminescence properties

A series of powder phosphors with the composition of Ba2-xSmxSiO4 are successfully synthesized by the solid-state reaction method in air. The X-ray powder diffraction patterns, energy disperse spectroscopy spectrum, elemental mapping images, decay curves, and luminescence properties are investigated. The excitation spectrum of Ba2SiO4:Sm3+ phosphor monitored at 598 nm is observed in the region from 200 to 525 nm with some distinct excitation peaks. The emission spectral bands of Ba2SiO4:Sm3+ phosphor with excitation at 402 nm are shown in the range of 425–750 nm due to the 4G5/2 → 6H5/2, 4G5/2 → 6H7/2, 4G5/2 → 6H9/2, and 4G5/2 → 6H11/2 transitions of Sm3+ ions, respectively. The optimal Sm3+ ion doping concentration is ~10 mol% in Ba2SiO4:xSm3+ phosphor. The concentration quenching mechanism is the electric multipole interaction. Ba2SiO4:Sm3+ phosphor has a good thermal stability and the thermal activation energy (Ea) value is ~0.2707 eV.

Field-Angle-Resolved Magnetic Excitations as a Probe of Hidden-Order Symmetry in CeB6

In contrast to magnetic order formed by electrons’ dipolar moments, ordering phenomena associated with higher-order multipoles (quadrupoles, octupoles, etc.) are more difficult to characterize because of the limited choice of experimental probes that can distinguish different multipolar moments. The heavy-fermion compound CeB6 and its La-diluted alloys are among the best-studied realizations of the long-range-ordered multipolar phases, often referred to as “hidden order.” Previously, the hidden order in phase II was identified as primary antiferroquadrupolar and field-induced octupolar order. Here, we present a combined experimental and theoretical investigation of collective excitations in phase II of CeB6. Inelastic neutron scattering (INS) in fields up to 16.5 T reveals a new high-energy mode above 14 T in addition to the low-energy magnetic excitations. The experimental dependence of their energy on the magnitude and angle of the applied magnetic field is compared to the results of a multipolar interaction model. The magnetic excitation spectrum in a rotating field is calculated within a localized approach using the pseudospin representation for the Γ8 states. We show that the rotating-field technique at fixed momentum can complement conventional INS measurements of the dispersion at a constant field and holds great promise for identifying the symmetry of multipolar order parameters and the details of intermultipolar interactions that stabilize hidden-order phases.6 MoreReceived 6 January 2020Revised 20 February 2020Accepted 5 March 2020DOI:https://doi.org/10.1103/PhysRevX.10.021010Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Physical SystemsHeavy-fermion systemsStrongly correlated systemsTechniquesInelastic neutron scatteringCondensed Matter, Materials & Applied Physics

Multipole and multimode engineering in Mie resonance-based metastructures

Abstract Benefited from the well-known Mie resonance, a plethora of physical phenomena and applications are attracting attention in current research on dielectric-based nanophotonics. High-index dielectric metastructures are favorable to enhance light-matter interaction in nanoscale with advantages such as low loss, optical magnetism, and multipolar responses, which are superior to their plasmonic counterpart. In this review, we highlight the important role played by Mie resonance-based multipolar and multimodal interaction in nanophotonics, introducing the concept of “multipole and multimode engineering” in artificially engineered dielectric-based metastructures and providing an overview of the recent progress of this fast-developing area. The scope of multipole and multimode engineering is restricted not only in multipolar interferences of meta-atom and meta-molecule but also in the nontrivial intermodal coupling (Fano resonance and bound states in the continuum), in the collective mode and the surface lattice mode appearing via periodic meta-lattices and aperiodic meta-assembly, in chiral enhancement via chiral and achiral dielectric metastructures, and in Mie resonance-mediated hybrid structures (Mie-plasmon and Mie-exciton). Detailed examples and the underlying physics of this area are discussed in-depth, in order to lead the multifunctional metastructures for novel applications in the future.

Eu3+-activated Ca3Mo0.2W0.8O6 red-emitting phosphors: A near-ultraviolet and blue light excitable platform for solid-state lighting and thermometer

Eu3+-activated Ca3Mo0.2W0.8O6 phosphors, which presented bright red emissions mainly from the 5D0 → 7F2 transition of Eu3+ ions upon the near-ultraviolet and blue light excitation, were successfully prepared by a traditional high-temperature technology. The crystal structure, phase constitution, morphology, electronic structure, photoluminescent behaviors, decay time, internal quantum efficiency and thermal stability of the resultant phosphors were investigated in detail. Under the excitation of 365, 393 and 465 nm, the optimal doping concentration was 9 mol% and the electrical multipolar interaction contributed to the non-radiative energy transfer between Eu3+ ions in Ca3Mo0.2W0.8O6 host lattices. Additionally, through using the prepared compounds, commercial blue/green-emitting phosphors and light-emitting diodes (LEDs), the fabricated white-LEDs emitted warm white light with excellent performance, such as good CIE coordinates, high color rending index and low correlated color temperature, when driven by 30 mA of injection current. Ultimately, according to the response of the decay time to the temperature, the thermometric performances of the resultant phosphors in the temperature range of 298–573 K were studied. The maximum sensor sensitivity of the studied phosphors reached up to 1.16% K−1 at 523 K. These achievements revealed that Eu3+-activated Ca3Mo0.2W0.8O6 phosphors were promising candidates for solid-state lighting and thermometer.

Octupolar versus Néel Order in Cubic 5d^{2} Double Perovskites.

We report time-of-flight neutron spectroscopy and neutron and x-ray diffraction studies of the 5d^{2} double perovskite magnets, Ba_{2}MOsO_{6} (M=Zn,Mg,Ca). These materials host antiferromagnetically coupled 5d^{2} Os^{6+} ions decorating a face-centered cubic (fcc) lattice and are found to remain cubic down to the lowest temperatures. They all exhibit thermodynamic anomalies consistent with a single phase transition at a temperature T^{*}, and a gapped magnetic excitation spectrum with spectral weight concentrated at wave vectors typical of type-I antiferromagnetic orders. However, while muon spin resonance experiments show clear evidence for time-reversal symmetry breaking below T^{*}, we observe no corresponding magnetic Bragg scattering signal. These results are shown to be consistent with ferro-octupolar symmetry breaking below T^{*}, and are discussed in the context of other 5d double perovskite magnets and theories of exotic orders driven by multipolar interactions.

Z-Shaped toroidal dipole planar terahertz metasurfaces

We proposed and fabricated a Z-shaped toroidal dipole terahertz metasurfaces, consisting of two incomplete U-shaped split rings, and the two incomplete U-shapes are rotationally symmetric with respect to the Z-axis. The effects of the structural parameters on the electromagnetic characteristics of toroidal resonances are investigated via simulation and experiment. The mechanism of toroidal dipole resonance is studied deeply by establishing an LC resonance model and calculating the multipoles scattering energy. The capacitance was highly depended on the value of lx, and the inductance was determined by the value of ly. Hence, the red shift was observed as the increasing of ly/lx. The resonant frequency of the toroidal dipole resonance is red shifted, as the ly/lx of the metallic pattern increases. The tunability of toroidal dipole resonant frequencies explores the interaction of toroidal multipoles with terahertz wave that could have extensive applications in the terahertz functional devices.

Effect of Eu3+ ion concentration on the structural and photoluminescence properties of Ba1.3Ca0.7SiO4 ceramic-based red phosphors for solid-state lighting applications

In this study, Ba1.3Ca0.7SiO4: Eu3+ red-orange phosphors were synthesized by the solution combustion method. The effect of varying Eu3+ molar percentages on the material properties including its intrinsic quantum efficiency (IQE) was investigated. XRD results showed synthesized phosphor is hexagonal T-phase Ba1.3Ca0.7SiO4 and the average crystallite size calculated using the Scherer’s formula was estimated to be ∼33 nm. SEM result showed the synthesized Ba1.3Ca0.7SiO4: Eu3+ phosphor had granular shaped and slightly agglomerated particles. The EDS shows that prepared samples contain Ba, Ca, Eu, Si, and O as expected. UV–vis measurement confirms the percent reflectance in the UV- region to be dependents on Eu3+ ions and the estimated bandgap vary between 3.80 and 4.23 eV. Photoluminescence emission measurements of all prepared samples at room temperature appeared to be entirely from inter configurational Eu3+ emission and depend both on the site symmetry as well as ion concentration. Hence, the peak centred at 592 nm is due to transition 5D0 →7F1, while the peak centred at 615 nm is due to transition 5D0 →7F2. The quadrupole-quadrupole multipolar interaction was found to be exclusively responsible for luminescence quenching. The Judd-Ofelt intensity parameters (Ω2, Ω4), asymmetry ratio (R0), and average decay lifetime of the nanocrystals showed dependence on concentration. High IQE values were obtained at low Eu3+ ion concentrations but the efficiency decreased with increasing ion concentration. The CIE color coordination confirmed that the emitted color fall in the strong orange-red region of the emission spectrum.

Synthesis, structure and luminescence of a high-purity and thermal-stable Sr9LiMg(PO4)7: Eu3+ red phosphor

Eu3+-activated Sr9LiMg(PO4)7 phosphors, which presented bright red emissions mainly from the 5D0→7F2 transition of Eu3+ ions upon the near-ultraviolet excitation, were successfully synthesized in ambient atmosphere. The crystal structure, phase constitution, photoluminescent behaviors, decay time, internal quantum efficiency and thermal stability of the resultant phosphors were investigated in detail. Eu3+ ions are found to tend to occupy multiple Sr2+ sites, which are 7, 8 and 10-coordinated. The optimal doping concentration is 7 mol% and the electrical multipolar interaction contributed to the non-radiative energy transfer between Eu3+ ions in Sr9LiMg(PO4)7 host lattices. Temperature-dependent PL spectra indicated Sr9LiMg(PO4)7: Eu3+ possess excellent emission and color stability at elevated temperature. Fabricated single-chromatic LED prototype emit bright red light under 20 mA bias current, which demonstrates that Sr9LiMg(PO4)7: Eu3+ phosphor is of great potential as converted phosphor in NUV LED application.

Interaction and Entanglement of a Pair of Quantum Emitters near a Nanoparticle: Analysis beyond Electric-Dipole Approximation.

In this paper, we study the collective effects which appear as a pair of quantum emitters is positioned in close vicinity to a plasmonic nanoparticle. These effects include multipole–multipole interaction and collective decay, the strengths and rates of which are modified by the presence of the nanoparticle. As a result, entanglement is generated between the quantum emitters, which survives in the stationary state. To evaluate these effects, we exploit the Green’s tensor-based quantization scheme in the Markovian limit, taking into account the corrections from light–matter coupling channels higher than the electric dipole. We find these higher-order channels to significantly influence the collective rates and degree of entanglement, and in particular, to qualitatively influence their spatial profiles. Our findings indicate that, apart from quantitatively modifying the results, the higher-order interaction channels may introduce asymmetry into the spatial distribution of the collective response.

Distinct antiblockade features of strongly interacting Rydberg atoms under a two-color weak excitation scheme

We present distinct antiblockade features of strongly interacting 64D5/2 Rydberg atoms employing a two-color excitation scheme. The first color (pulse A) is set to resonantly excite a few seed Rydberg atoms, each of which establishes a blockade region due to the long-range multipole interactions. The second color (pulse B) is blue detuned so that the multipole-interaction-induced shifts of certain atoms are well compensated to result in the antiblockade effect. We find in particular that a few seed atoms can lead to a remarkable difference of the Rydberg excitation in the presence of pulse B for a wide range of blue detuning. Relevant dynamics of this antiblockade excitation is also investigated by varying the pulse-B duration for a fixed blue detuning, further confirming the facilitation process of Rydberg excitation accompanied by a saturation effect. These experimental results can be well recovered by theoretical simulations based on a multilevel two-body model.

Synthesis and photoluminescence properties of Ba2Y3(SiO4)3F:Tm3+ blue-emitting fluorosilicate phosphors

Various novel Ba2Y3(SiO4)3F:Tm3+ blue-emitting fluorosilicate materials were achieved via solid-state synthesis. The structure and phase purity of prepared Ba2Y3(SiO4)3F:xTm3+ (x = 0.001-0.10 mol) were examined by X-ray powder diffraction. The surface morphology of Ba2Y3(SiO4)3F:0.01Tm3+ was studied by scanning electron microscopy. Photoluminescence properties were systematically explored under the monitoring emission (λem = 468 nm) and excitation (λex = 302 nm) spectra. The optimum mole ratio of as-synthesized phosphors was 0.01 mol. The concentration quenching mechanism in the Ba2Y3(SiO4)3F host was due to electric multipole interaction. Particularly, the chromaticity coordinates (0.1334, 0.0474) of Ba2Y3(SiO4)3F:0.01Tm3+ are near to those of the commercial BaMgAl10O17:Eu2+. These results validated the Ba2Y3(SiO4)3F:Tm3+ fluorosilicate phosphor can be used as a good blue-emitting candidate for W-LEDs.

A New Relatively Simple Approach to Multipole Interactions in Either Spherical Harmonics or Cartesians, Suitable for Implementation into Ewald Sums.

We present a novel derivation of the multipole interaction (energies, forces and fields) in spherical harmonics, which results in an expression that is able to exactly reproduce the results of earlier Cartesian formulations. Our method follows the derivations of Smith (W. Smith, CCP5 Newsletter 1998, 46, 18.) and Lin (D. Lin, J. Chem. Phys. 2015, 143, 114115), who evaluate the Ewald sum for multipoles in Cartesian form, and then shows how the resulting expressions can be converted into spherical harmonics, where the conversion is performed by establishing a relation between an inner product on the space of symmetric traceless Cartesian tensors, and an inner product on the space of harmonic polynomials on the unit sphere. We also introduce a diagrammatic method for keeping track of the terms in the multipole interaction expression, such that the total electrostatic energy can be viewed as a ‘sum over diagrams’, and where the conversion to spherical harmonics is represented by ‘braiding’ subsets of Cartesian components together. For multipoles of maximum rank n, our algorithm is found to have scaling of vs. for our most optimised Cartesian implementation.