Multipole Density Research Articles

Potential, field, and interactions of multipole spheres: Coated spherical magnets

We show that the energy, force, and torque between two spherically symmetric multipole density distributions are identical to those between two point multipoles, and apply this point-sphere equivalence to coated spherical dipole magnets. We also show that the potential and field of such a distribution are equivalent to those due to point multipoles located at the center of the distribution. We expand the inverse-distance potential in terms of harmonic (Hermite irreducible) tensors, whose properties enable us to express the potential energy, force, and torque for two arbitrary source distributions in a series of point-multipole interactions. This work generalizes recent work on interactions between uniformly magnetized dipole spheres [B. F. Edwards, D. M. Riffe, J.-Y. Ji, and W. A. Booth, Am. J. Phys. 85, 130 (2017)] to interactions between spherically-symmetric multipole spheres.

Sensitivity of 4D-STEM to Valence Electron Distribution Based on Multipole Density Formalism

Sensitivity of 4D-STEM to Valence Electron Distribution Based on Multipole Density Formalism

Alterations in heart rate variability are associated with abnormal myocardial perfusion

BackgroundAbnormalities in the autonomic nervous system may occur in ischemic heart disease, but the mechanisms by which they are linked are not fully defined. The risk of cardiac events is increased during morning hours. Studying the contributions of autonomic mechanisms may yield insights into risk stratification and treatment. We hypothesize that autonomic dysfunction, measured by decreased heart rate variability (HRV), associates with abnormal stress myocardial perfusion imaging (MPI). MethodsWe performed a cross-sectional study of the association between abnormal myocardial stress perfusion with HRV using 276 middle-aged veteran twins without known ischemic heart disease. The primary independent variable was cardiac autonomic regulation measured with 24-hour electrocardiogram (ECG) monitoring, using linear and non-linear (multipole density, or Dyx) HRV metrics. The primary outcome was abnormal perfusion (>5% affected myocardium) during adenosine stress on [13N]-ammonia myocardial perfusion imaging with positron emission tomography. ResultsThe mean (SD) age was 53 (3) years and 55 (20%) had abnormal perfusion. HRV (by Dyx) was reduced during morning hours in subjects with abnormal perfusion. At 7 AM, each standard deviation (SD) decrease in Dyx was associated a 4.8 (95% CI, 1.5 — 15.8) odds ratio (OR) for abnormal MPI. With Dyx < 2.0, the 7 AM OR for abnormal MPI was 11.8 (95% CI, 1.2 — 111.4). ConclusionsAutonomic dysfunction, measured by non-linear HRV in the morning hours, was associated with an increased OR of abnormal MPI. These results suggest a potentially important role of ECG-based biomarkers in risk stratification for individuals with suspected ischemic heart disease.

Collective multipole oscillations direct the plasmonic coupling at the nanojunction interfaces.

We present a systematic study of the effect of higher-multipolar order plasmon modes on the spectral response and plasmonic coupling of silver nanoparticle dimers at nanojunction separation and introduce a coupling mechanism. The most prominent plasmonic band within the extinction spectra of coupled resonators is the dipolar coupling band. A detailed calculation of the plasmonic coupling between equivalent particles suggests that the coupling is not limited to the overlap between the main bands of individual particles but can also be affected by the contribution of the higher-order modes in the multipolar region. This requires an appropriate description of the mechanism that goes beyond the general coupling phenomenon introduced as the plasmonic ruler equation in 2007. In the present work, we found that the plasmonic coupling of nearby Ag nanocubes does not only depend on the plasmonic properties of the main band. The results suggest the decay length of the higher-order plasmon mode is more sensitive to changes in the magnitude of the interparticle axis and is a function of the gap size. For cubic particles, the contribution of the higher-order modes becomes significant due to the high density of oscillating dipoles localized on the corners. This gives rise to changes in the decay length of the plasmonic ruler equation. For spherical particles, as the size of the particle increases (i.e., ≥80 nm), the number of dipoles increases, which results in higher dipole-multipole interactions. This exhibits a strong impact on the plasmonic coupling, even at long separation distances (20 nm).

Electron densities by the maximum entropy method (MEM) for various types of prior densities: a case study on three amino acids and a tripeptide

Dynamic model densities according to Mondal et al. [(2012), Acta Cryst. A68, 568-581] are presented for independent atom models (IAM), IAMs after high-order refinements (IAM-HO), invariom (INV) models and multipole (MP) models of α-glycine, DL-serine, L-alanine and Ala-Tyr-Ala at T ≃ 20 K. Each dynamic model density is used as prior in the calculation of electron density according to the maximum entropy method (MEM). We show that at the bond-critical points (BCPs) of covalent C-C and C-N bonds the IAM-HO and INV priors produce reliable MEM density maps, including reliable values for the density and its Laplacian. The agreement between these MEM density maps and dynamic MP density maps is less good for polar C-O bonds, which is explained by the large spread of values of topological descriptors of C-O bonds in static MP densities. The density and Laplacian at BCPs of hydrogen bonds have similar values in MEM density maps obtained with all four kinds of prior densities. This feature is related to the smaller spatial variation of the densities in these regions, as expressed by small magnitudes of the Laplacians and the densities. It is concluded that the use of the IAM-HO prior instead of the IAM prior leads to improved MEM density maps. This observation shows interesting parallels to MP refinements, where the use of the IAM-HO as an initial model is the accepted procedure for solving MP parameters. A deconvolution of thermal motion and static density that is better than the deconvolution of the IAM appears to be necessary in order to arrive at the best MP models as well as at the best MEM densities.

Electron densities by the maximum entropy method (MEM) for various types of prior densities: a case study on three amino acids and a tripeptide

Dynamic model densities according to Mondal et al. [(2012), Acta Cryst. A68, 568–581] are presented for independent atom models (IAM), IAMs after high-order refinements (IAM-HO), invariom (INV) models and multipole (MP) models of α-glycine, dl-serine, l-alanine and Ala–Tyr–Ala at T ≃ 20 K. Each dynamic model density is used as prior in the calculation of electron density according to the maximum entropy method (MEM). We show that at the bond-critical points (BCPs) of covalent C—C and C—N bonds the IAM-HO and INV priors produce reliable MEM density maps, including reliable values for the density and its Laplacian. The agreement between these MEM density maps and dynamic MP density maps is less good for polar C—O bonds, which is explained by the large spread of values of topological descriptors of C—O bonds in static MP densities. The density and Laplacian at BCPs of hydrogen bonds have similar values in MEM density maps obtained with all four kinds of prior densities. This feature is related to the smaller spatial variation of the densities in these regions, as expressed by small magnitudes of the Laplacians and the densities. It is concluded that the use of the IAM-HO prior instead of the IAM prior leads to improved MEM density maps. This observation shows interesting parallels to MP refinements, where the use of the IAM-HO as an initial model is the accepted procedure for solving MP parameters. A deconvolution of thermal motion and static density that is better than the deconvolution of the IAM appears to be necessary in order to arrive at the best MP models as well as at the best MEM densities.

ACCURACY OF THE MULTIPOLE EXPANSION OF DENSITY DISTRIBUTION IN THE PRESENCE OF OCTUPOLE DEFORMATION

The accuracy of multipole expansion of density distribution for deformed nuclei is tested. The interaction potential for a deformed-spherical pair of nuclei was calculated using the folding model derived from zero-range nucleon–nucleon (NN) interaction. We considered two spherical projectiles Ca 40 and Pb 208 scattered on U 238 deformed target nucleus. The error in the heavy ion (HI) potential resulting from using a truncated multipole density expansion is evaluated for each case in the presence of octupole deformation δ3 besides quadrupole δ2. We are interested in the value of error for R ≥ RT (touching distance). We found that for values of |δ3|≤0.1 the error at R = RT reaches reasonable values when six terms expansion is used. For |δ3| = 0.2, we calculated the Coulomb barrier parameters using realistic NN force and found that the large error present in six terms for zero range force decreases strongly to less than 1% when the zero range is added to finite range forces and Coulomb interaction to form the Coulomb barrier. It is noted that the negative value of octupole deformation parameters δ3 = -0.1 produce error at orientation angle θ equal in value to that produced at angle (180°-θ) for the positive values δ3 = 0.1. We also found that the error decreases as the mass number of the projectile nucleus increases.

On the accuracy of multipole expansion of deformed density distribution

The interaction potential for a deformed-spherical pair of nuclei is calculated using the folding model derived from different range nucleon–nucleon (NN) interactions. Five spherical projectiles of different mass numbers scattered on the 238U deformed target are considered. The error in the heavy ion (HI) potential by using the truncated multipole density expansion is evaluated for each case. We find systematic trends of the percentage error in the HI potential depending on the number of multipoles considered; this percentage decreases if the mass number of the projectile or the range of the NN force increases, and it becomes smaller for small values of the separation distance between two nuclei or when higher deformation parameters vanish. The maximum error in using the truncated density expansion is estimated in the case of two deformed interacting nuclei.

Mechanical and chemical spinodal instabilities in finite quantum systems.

Self-consistent quantum approaches are used to study the instabilities of finite nuclear systems. The frequencies of multipole density fluctuations are determined as a function of dilution and temperature for several isotopes. The spinodal region of the phase diagrams is determined, and it appears that instabilities are reduced by finite size effects. The role of surface and volume instabilities is discussed. It is indicated that the important chemical effects associated with mechanical disruption may lead to isospin fractionation.

Multipole modes and spin features in the Raman spectrum of nanoscopic quantum rings

We present a systematic study of ground state and spectroscopic properties of many-electron nanoscopic quantum rings. Addition energies at zero magnetic field (B) and electrochemical potentials as a function of B are given for a ring hosting up to 24 electrons. We find discontinuities in the excitation energies of multipole spin and charge density modes, and a coupling between the charge and spin density responses that allow to identify the formation of ferromagnetic ground states in narrow magnetic field regions. These effects can be observed in Raman experiments, and are related to the fractional Aharonov-Bohm oscillations of the energy and of the persistent current in the ring.

On the origin of topological differences between experimental and theoretical crystal charge densities

Topological analysis of experimental and theoretical (molecular and crystal) electron densities of p-nitroaniline and p-amino-p'-nitrobiphenyl reveals considerable discrepancies between experiment and theory for the bond critical points properties. Particularly large differences occur for the positive curvature along the bond path (lambda 3). The differences become somewhat smaller when more extended basis sets and correlation effects are introduced in the theoretical calculations. The effect of the crystal matrix on the properties of bond critical points is evaluated for the p-nitroaniline molecule using the 6-21G** and 6-31G** basis sets. The differences between the isolated molecule and the molecule in the crystal are too small to explain the quantitative disagreement between the theoretical and experimental topologies reported in the literature and found in the current study. For most bonds, the observed changes in the properties of the electron density agree well for both basis sets but some discrepancies are found for changes in lambda 3 for N-H and aromatic C-C bonds. When the theoretical densities are projected into the multipole density functions through refinement of the theoretical structure factors, the topological properties change and differences between theory and experiment are reduced. The main origin of the observed discrepancies is attributed to the nature of the radial functions in the experimental multipole model.

Wave-vector dependence of spin and density multipole excitations in quantum dots

We have employed time-dependent local-spin density-functional theory to analyze the multipole spin and charge density excitations in $\mathrm{GaAs}\ensuremath{-}{\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$ quantum dots. The on-plane transferred momentum degree of freedom has been taken into account, and the wave-vector dependence of the excitations is discussed. In agreement with previous experiments, we have found that the energies of these modes do not depend on the transferred wave vector, although their intensities do. Comparison with a recent resonant Raman scattering experiment [C. Sch\"uller et al., Phys. Rev. Lett. 80, 2673 (1998)] is made. This allows us to identify the angular momentum of several of the observed modes as well as to reproduce their energies.

The use of CCD area detectors in charge-density research. Application to a mineral compound: the alpha-spodumene LiAl(SiO3)2.

X-ray diffraction data sets collected on both Nonius and Siemens (Bruker) goniometers equipped with charge-coupled device (CCD) area detectors have been tested for the electron-density determination of the aluminosilicate mineral compound alpha-spodumene LiAl(SiO(3))(2), aluminium lithium silicon oxide. Data collection strategies, reflection intensity peak integration methods and experimental error estimates are different for the two instruments. Therefore, the consistency and quality of the two types of CCD measurements have been carefully compared to each other and to high-resolution data collected on a conventional CAD-4 point-detector diffractometer. Multipole density model refinements were carried out against the CCD data and the statistical factors analysed in terms of experimental weighting schemes based on the standard uncertainties of the diffraction intensities derived by the Nonius and Siemens software programs. Consistent experimental electron-density features in the Si-O-Si and Si-O-Al bridges were found from both CCD data sets. The net atomic charges obtained from the kappa refinements against each CCD data set are also in good agreement and quite comparable with the results of the conventional CAD-4 experiment.

Multipole analysis of the electron density in triphylite, LiFePO4, using X-ray diffraction data

The electron density distribution and 3d-orbital electron occupancies for the Fe atom in synthetic triphylite, LiFePO 4 , have been analysed using single-crystal X-ray diffraction data measured at T=298 K with Mo Kα (λ=0.71069 A) radiation to a resolution corresponding to (sin θ max /λ)=1.078 A -1 . Measurements of 3265 reflections gave 944 unique data [R int (I)=0.036] with I>2σ(I). For an atomic multipole density model fitted by least-squares methods R(F)=0.0174 for all unique reflections. The Fe atom 3d-orbital occupancies have been derived from the multipole population coefficients using point-group-specific relations

Electron Deformation Density at Temperatures around 20 K

Abstract A 20 K difTractometer for X-ray measurement is described. Electron density experiments at 15 K for oxalic acid and at 23 K for acetamide were carried through, leading to high-resolution experi-mental and multipole static density maps.

Magnetic fields, rotating atoms and the origin of diamagnetism

A qualitative difference between the electric and magnetic responses of matter to external fields is pointed out and is explained as being due to the different nature of the corresponding microscopic dipoles. The classical explanation of diamagnetism is reviewed and elaborated. Various approaches to Larmor's theorem are discussed. Some results of Marsh concerning the magnetic field of rotating charge distributions are extended and applied to the problem. The Langevin formula for the diamagnetic susceptibility is arrived at as well as a formula for the induced field at the nucleus. The general theorem that the field outside a spherical multipole density is a pure multipole field is presented.

Source equivalency based on the energy specification of order l

It is shown that the multipole expansion of electrostatic energy can be expressed in the form of energy specification S{We} =〈[P]l,∇(l)φ〉, where ∇(l) is a differential operator, whereas φ and [P]l represent an arbitrary test potential and equivalent reduced volume multipole density, respectively. Two electrostatic sources are l-equivalent if their energy specifications are identical. The formalism by means of which electrostatic multipole sources can be effectively handled is developed.

Refinement of metald-orbital occupancies from X-ray diffraction data

Expressions are derived relating the occupancies of transition-metal d orbitals in a trigonal distortion of an octahedral field to the populations of multipole density functions refined from X-ray diffraction data. The expressions are used to obtain the iron d-orbital configuration in cubic FeS2 by least-squares refinement of very high resolution single-crystal X-ray intensity measurements. The refined populations correspond to a large distortion of the iron electron density from spherical symmetry towards a low-spin configuration.

Experimental electron density distribution of molecular chlorine

The electron density distribution of solid chlorine is determined from high resolution X-ray diffraction intensity measurements at 90 K. Deformation densities are calculated from the observed X-ray data and parameters obtained from a high order refinement of data with sin ϑ/λ > 0·65 A-1. Refinements including the population of multipole density functions are used to calculate model deformation densities and the asymmetry of the electric field gradient. Inclusion of third and fourth cumulant thermal parameters results in a significant improvement in the fit to the experimental data. The deformation density reveals a torus of non-bonding density at 0·31–0·37 e/A3 encircling the Cl-Cl bond somewhat in the back of each of the Cl atoms. A double peak of 0·22 e/A3 is found in the bonding region of the experimental deformation density similar to the bonding density found in S8. This feature is not properly reproduced in density maps based on the atom-centred multipole functions. The calculated distribution of er...

Electron-density studies. V. The electron density in melamine (2,4,6-triamino-s-triazine) with and without exponent refinement

The electron density distribution in melamine has been studied with X-ray diffraction data and neutron structural parameters. The at-rest valence density is represented as a set of nuclear-centred multipole density functions with Slater-type radial functions. Two series of analyses were compared, the first with the radial exponents fixed at the standard molecular values and the second with these exponents as variable parameters. Exponent refinement allows a marked improvement in the fit of the model to the data. The population coefficients of the multipole terms are better defined when the exponents are optimized. On chemically similar atoms the populations of the monopole terms are inversely related to the exponents. The carbon atom parameters agree to high precision. Exponents for the ring and amine nitrogens differ and small differences within each set are related to the hydrogen-bond and packing environment. The exponents and electron density near the nucleus are relatively low for hydrogens involved in hydrogen bonding. The most significant deformation functions in the multipole expansion have a symmetry compatible with nearest-neighbour geometry. Differences between populations are related to distortions from the idealized geometry or to hydrogen-bonding interactions. The inclusion of a hydrogen dipole deformation term with a large exponent results in internally consistent populations which are correlated with N-H stretching amplitudes in the structure. This suggests that the convolution approximation is invalid at this level of structure refinement.