Fourier-Bessel Method Research Articles

The Fourier–Bessel method for the inverse scattering problem of cavities

In this paper, we study the inverse scattering problem and propose a new method to determine the shape of a cavity with Dirichlet boundary conditions from the knowledge of acoustic near-field measurements. The Fourier–Bessel method (FBM) is used to reconstruct the boundary of the cavity from scattered fields which are incited by point sources and measured on a closed curve inside the cavity. We not only analyze the stability but also present the numerical experiments to show the effectiveness of the FBM.

A Fourier–Bessel method with a regularization strategy for the boundary value problems of the Helmholtz equation

This paper is concerned with the Fourier–Bessel method for the boundary value problems of the Helmholtz equation in a smooth simply connected domain. Based on the denseness of Fourier–Bessel functions, the problem can be approximated by determining the unknown coefficients in the linear combination. By the boundary conditions, an operator equation can be obtained. We derive a lower bound for the smallest singular value of the operator, and obtain a stability and convergence result for the regularized solution with a suitable choice of the regularization parameter. Numerical experiments are also presented to show the effectiveness of the proposed method.

The Fourier–Bessel method for solving the Cauchy problem connected with the Helmholtz equation

This paper is concerned with the Cauchy problem connected with the Helmholtz equation. We investigate the denseness of Fourier–Bessel functions and propose a numerical method for approximating the solution to the problem. The convergence and stability are analyzed with truncated singular value decomposition method. Numerical experiments are also presented to show the effectiveness of our method.

2-D soft x-ray arrays in the EAST.

A high spatial and temporal resolution soft x-ray (SXR) imaging diagnostic has been installed in EAST for the study of magnetohydrodynamics activities and core high-Z impurity transport. Up to 122 lines of sight view the poloidal plasma from three directions (two up-down symmetrical horizontal arrays and one vertical array), which renders the diagnostic able to provide detailed tomographic reconstructions under various conditions. Fourier-Bessel method based on flux coordinates was employed for 2-D SXR tomographic reconstruction. Examples of several events measured by SXR diagnostic in EAST are shown, namely the crash patterns of sawtooth, periodical burst of edge localized modes, and the transport of high-Z intrinsic impurities.

Stability analysis of the Fourier–Bessel method for the Cauchy problem of the Helmholtz equation

This paper is concerned with the Cauchy problem connected with the Helmholtz equation in a smooth-bounded domain. The Fourier–Bessel method with Tikhonov regularization is applied to achieve a regularized solution to the problem with noisy data. The convergence and stability are obtained with a suitable choice of the regularization parameter. Numerical experiments are also presented to show the effectiveness of the proposed method.

Single-particle based helical reconstruction—how to make the most of real and Fourier space

The helical assembly is a fundamental organization principle of biomacromolecules. To determine the structures of helical filaments or tubes has been helped by the fact that many different views of the helical unit are present to reconstruct a three-dimensional image from a single helix. In this review, I present the current state of helical image reconstruction from electron cryo-micrographs by introducing Fourier-based processing alongside real-space approaches. Based on this foundation, I describe how they can be applied to determine the symmetry and high-resolution structure of helical assemblies. In the past, the main structure determination approach of helical assemblies from electron micrographs was the Fourier-Bessel method, which is based on a comprehensive theory and has generated many successful applications in the last 40 years. The emergence of the single-particle technique allowed segmented helical specimens to be treated as single particles, thus rendering new specimens amenable to 3D helical reconstruction and facilitating high-resolution structure analysis. However, helical symmetry determination remains the crucial step for a successful 3D reconstruction. Depending on the helical specimen, Fourier and real-space approaches or a combination of both provide important clues to establish the correct helical symmetry. I discuss recent developments in combining traditional Fourier-Bessel procedures with single-particle algorithms to provide a versatile and comprehensive approach to structure determination of helical specimens. Upon introduction of direct electron detectors, a series of near-atomic resolution structures from helical assemblies have become available. As helical organization is fundamental to many structural assemblies of the cell, these approaches to structure elucidation open up promising capabilities to study the underlying structures at atomistic resolution.

Image of electron densities from line and plane projections

We compare Fourier transforms with orthogonal polynomials techniques applied in reconstructing three-dimensional electron–positron momentum densities from two-dimensional angular correlation of annihilation radiation (2D-ACAR) spectra and electron momentum densities from one-dimensional Compton profiles (1D-CP). In the case of Fourier transforms, we show results for two different algorithms: filtered back projection and Fourier–Bessel method. These techniques are presented for 2D-ACAR spectra in Y, ErGa3 and model profiles.

Field Computation for Two-Dimensional Array Transducers with Limited Diffraction Array Beams

A method is developed for calculating fields produced with a two-dimensional (2D) array transducer. This method decomposes an arbitrary 2D aperture weighting function into a set of limited diffraction array beams. Using the analytical expressions of limited diffraction beams, arbitrary continuous wave (cw) or pulse wave (pw) fields of 2D arrays can be obtained with a simple superposition of these beams. In addition, this method can be simplified and applied to a 1D array transducer of a finite or infinite elevation height. For beams produced with axially symmetric aperture weighting functions, this method can be reduced to the Fourier-Bessel method studied previously where an annular array transducer can be used. The advantage of the method is that it is accurate and computationally efficient, especially in regions that are not far from the surface of the transducer (near field), where it is important for medical imaging. Both computer simulations and a synthetic array experiment are carried out to verify the method. Results (Bessel beam, focused Gaussian beam, X wave and asymmetric array beams) show that the method is accurate as compared to that using the Rayleigh-Sommerfeld diffraction formula and agrees well with the experiment.

Electron momentum-space densities and Fermi surface ofLi100−xMgx(0<~x<~40)alloys: Compton scattering experiment versus theory

Directional Compton profiles (CP's) of ${\mathrm{Li}}_{100\ensuremath{-}x}{\mathrm{Mg}}_{x}(0<~x<~40)$ were measured for (11)12 directions of the momentum transfer q with (0.14) 0.16 a.u. $(\mathrm{a}.\mathrm{u}.=\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{c}$ units: $\ensuremath{\Elzxh}=e=m=1)$ momentum space resolution using synchrotron x rays, monochromatized to (31) 59 keV (values in brackets refer to $x=0).$ The total experimental valence electron CP's, their second derivatives, and the associated directional differences were compared with the corresponding Korringa-Kohn-Rostoker--coherent-potential-approximation--local-density-approximation (KKR--CPA--LDA) computations. Discrepancies between measurements and calculations are partly traced back to an inadequate treatment of electron correlation effects within the LDA framework. The three-dimensional (3D) valence-electron momentum density, \ensuremath{\rho}(p), as well as the 3D occupation number density $N(\mathbf{k}),$ were reconstructed from experimental data using the Fourier-Bessel method. The reconstructed \ensuremath{\rho}(p)'s exhibit evidence of higher momentum components due to 110 umklapp processes, whose variations with x are in reasonable agreement with the KKR-CPA calculations. The experimental $N(\mathbf{k})$'s provide a direct measure of the Fermi-surface anisotropy and show the presence of necks on the 110 Brillouin-zone boundaries for $x>~13$ in good accord with 2D positron annihilation experiments.

On Density and Velocity Fields and β from the [ITAL]IRAS[/ITAL] PSC[CLC][ITAL]z[/ITAL][/CLC] Survey

We present a version of the Fourier-Bessel method first introduced by Fisher and coworkers and Zaroubi and coworkers with two extensions: (1) we amend the formalism to allow a generic galaxy weight that can be constant, rather than the more conventional overweighting of galaxies at high distances, and (2) we correct for the masked zones by extrapolation of Fourier-Bessel modes rather than by cloning from the galaxy distribution in neighboring regions. We test the procedure extensively on N-body simulations and find that it gives generally unbiased results but that the reconstructed velocities tend to be overpredicted in high-density regions. Applying the formalism to the PSCz redshift catalog, we find that β = 0.7 ± 0.5 from a comparison of the reconstructed Local Group velocity to the cosmic microwave background dipole. From an anisotropy test of the velocity field, we find that β = 1 cold dark matter models normalized to the current cluster abundance can be excluded with 90% confidence. The density and velocity fields reconstructed agree with the fields found by Branchini and coworkers on most points. We find a back infall into the Great Attractor region (Hydra-Centaurus region), but tests suggest that this may be an artifact. We identify all the major clusters in our density field and confirm the existence of some previously identified possible ones.

Electron momentum-space densities of Li metal: A high-resolution Compton-scattering study

Directional Compton profiles~CP’s ! of Li metal were measured for 11 directions of the momentum transfer q with 0.14 a.u. ~a.u.5atomic units: \5e5m51! momentum-space resolution using synchrotron radiation from the DORIS ~Doppel-Ring-Speicheranlage! storage ring monochromatized to 31 keV. Both the total valence-electron CP’s and the directional differences of the CP’s exhibit considerable deviations from the most recent density-functional calculations, performed by Sakurai et al. @Phys. Rev. Lett. 74, 2252 ~1995!# within the limits of the local density approximation. These discrepancies are attributed to self-energy effects connected with the excitation of so-called plasmaron modes. The three-dimensional ~3D! valence-electron momentum density, r~p!, as well as the 3D occupation number density N~k!, were reconstructed using the Fourier-Bessel method. The reconstructed r~p! exhibits clear evidence of higher momentum components due to 110 umklapp processes. The reconstructed N~k! enables a direct experimental access to the Fermi-surface anisotropy of Li, which was found to be 3.661.1%. The reconstructed N~k! for ki @001# was fitted to a model with the renormalization factor z as the only free parameter, which was found to be z50.160.1. @S0163-1829~96!07843-5#

The deuteron optical potential at low energies

The elastic scattering of vector-polarized deuterons has been measured near E d = 20 MeV on a variety of nuclei to investigate the effective deuteron-scattering potential at low energies. The analysis of the data has been performed within the conventional optical model as well as in the framework of the model-unrestricted Fourier-Bessel method. Whereas both methods result in scattering potentials which agree in their bulk properties, only the latter method leads to a statistically relevant description of the data with χ 2/ F≈1. The Fourier-Bessel potentials show features similar to those observed at 52 MeV: in contrast to what is expected from static folding, the real central potential has a small skin thickness comparable to that of nucleon potentials. In the absorptive potential the breakup contribution amounts to about 20%, in agreement with theoretical predictions. The empirical potentials are interpreted by single- and double-folding calculations.

Momentum-space calculation of vacuum polarisation effects for extended charge distributions

Vacuum polarisation potentials of extended charge distributions are calculated using an efficient Fourier-Bessel method for the evaluation of folding integrals. In order to obtain a well defined and a quantitatively correct picture of vacuum polarisation effects, these potentials are employed to yield vacuum polarisation charge densities via Poisson's equation of classical electrostatics.

Nuclear charge densities in the transition region: 192Os

Data from the first electron-scattering study of an Os-Pt transition region nucleus, 192Os, are analyzed in combination with moonic-atom data by using the model-independent Fourier-Bessel method. The results are discussed in terms of available IBA and BET model calculations. New Hartree-Fock (DME) calculations are presented as an example of the agreement achieved with microscopic-based theories.

Charge distribution of the neven sulphur isotopes from elastic electron scattering

Elastic electron scattering experiments on the isotopes 32,34,36S were performed covering a range in momentum transfer q = 0.5−2.6 fm −. The cross sections were analysed with the Fourier-Bessel method yielding model-independent charge distributions and their differences. The extracted rms radii follow approximately the systematics of even-even nuclei; this also holds for the gross features as expressed in dms radii and skin thicknesses.

Analyzing power and differential cross section at 9.9, 11.9 and 13.9 MeV for Ca(n, n)Ca

The analyzing power and differential cross section for elastic neutron scattering from calcium have been measured at 9.9, 11.9 and 13.9 MeV using the 2H(d, n) 3He source reaction and neutron time-of-flight (t.o.f.) techniques to detect the scattered neutrons. Polarized neutron beams were produced via the polarization transfer reaction 2H( d , n ) 3He at θ = 0°. The data have been corrected for finite geometry and multiple scattering effects. None of the global neutron-nucleus optical model parameter sets usually referred to in the literature reproduces the present cross-section and analyzing power data. Individual as well as energy-averaged fits of the data resulting from new optical model searches are presented. It is shown that the quoted uncertainties of a recent empirical determination of the real part Δ V c of the Coulomb correction term are probably underestimated. Our imaginary Coulomb correction term Δ W c agrees quite well with both a very recent empirical determination and theoretical studies. Although the quality of the fits to the data can be improved by adding l-dependent potentials to the general optical potential, no definite conclusions can be drawn from the present data as to whether or not l-dependent potentials are important in neutron-calcium scattering in the energy range investigated. The data have also been analyzed using a Fourier-Bessel series description of the real central optical potential. Comparing the X 2 values, the experimental data are better reproduced by the Fourier-Bessel method than by our Woods-Saxon optical model analyses. The Fourier-Bessel potentials obtained show strong deviations from the standard Woods-Saxon shape but are in good agreement with calculations using the nuclear structure approach.

Comparison between radial sensitivity of different strongly interacting probes

We investigate the radial sensitivity of different strongly interacting probes to neutron density distributions in nuclei. The experiments considered are elastic scattering of 104 MeV $\ensuremath{\alpha}$ particles, 1 GeV protons, 130 MeV pions, and also shifts and widths of pionic atom levels. The Fourier-Bessel method is used, thus avoiding any prior assumption on the neutron densities. To enable statistically meaningful comparisons between the different experiments, "pseudodata" are used, which are based on the real data. Although the region of most sensitivity is near the surface and is similar for each case, the $\ensuremath{\alpha}$ particles probe better the extreme surface and the protons probe better the interior. Pion scattering appears to be inferior to the other two scattering experiments because of the gradient terms in the potential. Surprisingly, there are some indications that ${\ensuremath{\pi}}^{+}$ could be better than ${\ensuremath{\pi}}^{\ensuremath{-}}$ in determining neutron densities. Pionic atom data are sensitive mostly to the surface region. A critical discussion of error analyses is presented.NUCLEAR REACTIONS Radial sensitivity to neutron density within optical model calculations of elastic $\ensuremath{\sigma}(\ensuremath{\theta})$ data analyzed for $\ensuremath{\alpha}$, $p$, ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$+$^{48}\mathrm{Ca}$ for 100-1000 MeV and of pionic atoms of $^{48}\mathrm{Ca}$.

Method for analysis of inelastic α-particle scattering

The analysis of inelastic scattering of strongly interacting particles from nuclei suffers from the model dependence introduced when specifying the form factors (coupling potentials), e.g. in the framework of a vibrational model. Moreover, the relation between strength and shape of the coupling potentials and corresponding quantities of the transition densities is not well established, and the interpretation in terms of transition rates and nuclear moments introduces additional uncertainties. In the present studies of inelastic scattering of 104 MeV α-particles from 50Ti and 52Cr we remove the constraints due to pre-chosen forms of the coupling potentials by applying the Fourier-Bessel method which is more flexible. It allows quite general shapes and provides realistic estimates of the uncertainties of the potentials and derived quantities (moments). Using the identities between the integral moments of a folded potential distribution and of the underlying nuclear matter distribution isoscalar transition rates have been derived.

Nuclear sizes ofCa40,42,44,48from elastic scattering of 104 MeV alpha particles. I. Experimental results and optical potentials

Differential cross sections for elastic scattering of 104 MeV $\ensuremath{\alpha}$ particles from $^{40,42,44,48}\mathrm{Ca}$ have been measured with high angular accuracy over a wide angular range. Optical model analysis based on a Fourier-Bessel description of the real potential reveals isotopic differences which, in particular for $^{48}\mathrm{Ca}$, indicate a small neutron skin.NUCLEAR REACTIONS $^{40,\phantom{\rule{0ex}{0ex}}42,\phantom{\rule{0ex}{0ex}}44,\phantom{\rule{0ex}{0ex}}48}\mathrm{Ca}$($\ensuremath{\alpha}$, $\ensuremath{\alpha}$), ${E}_{\ensuremath{\alpha}}=104$ MeV, measured $\ensuremath{\sigma}(\ensuremath{\theta})$; ${\ensuremath{\theta}}_{\mathrm{c}.\mathrm{m}.}=3\ensuremath{-}110\ifmmode^\circ\else\textdegree\fi{}$; enriched targets; optical model analysis using Fourier-Bessel method, deduced isotopic differences in the real potentials.

The charge-distribution differences of 209Bi, 208, 207, 206, 204Pb and 205, 203Tl investigated by elastic electron scattering and muonic X-ray data

Elastic electron scattering cross sections and cross-section ratios have been measured for the stable Bi, Pb and T1 isotopes. The data are analysed with the Fourier-Bessel method and the charge-distribution differences of the isotonic and isotopic pairs are presented as well as the Fourier-Bessel coefficients of the seven single nuclei. To improve the accuracy of the results, muonic X-ray data are incorporated into the analysis. The measured Δρ( r) are compared with the simple shell model, where the stretching due to the additional nucleons is taken into account, and with more sophisticated HF calculations.