Partial Wave Calculations Research Articles

Use of correlated double continuum wave functions for the determination of the triple differential cross section of the ionization of lithium by electron impact

Electron impact ionization of Lithium in its ground state is investigated theoretically. An ab-initio calculation is proposed which includes the correlated double continuum Brauner-Briggs-Klar BBK wave function [8]. The results are in good agreement with the existing experimental measurements and partial wave calculations for the symmetric case. Some interesting kinematical situations are explored for near future experiments and the similarity with the ionization of hydrogen (2s) studied.

Cross sections for elastic scattering of fast electrons and positrons by atoms

In order to provide reliable cross section data needed for Monte Carlo simulation of high-energy electron and positron transport, a simple procedure to evaluate cross sections for elastic scattering of these particles by neutral atoms is described. Our approach starts from the exact Mott cross section for scattering by a bare point nucleus. Screening effects are accounted for by means of the Born approximation. A correction, for which we give a simple analytical expression, is introduced to compensate for the inaccuracies of the Born approximation at low energies. This correction has been determined from an extensive partial wave calculation involving 20 elements within the periodic system and electrons and positrons with kinetic energies from 1 keV to 10 MeV. Finite nuclear size effects are introduced by means of the Born approximation and a simple nuclear charge distribution. The resulting differential cross sections are adequate for describing multiple elastic scattering of electrons and positrons with kinetic energies above ∼ZkeV, where Z is the atomic number of the target atom.

Quasielastic 40Ca(e,e'p) cross sections in a many-particle self-consistent Hartree model.

A fully distorted partial-wave calculation is presented for the reaction $^{40}\mathrm{Ca}$(e,e'p${)}^{39}$K which uses relativistic self-consistent Hartree wave functions for both the target nucleus and the residual nucleus. The final state interaction of the knocked-out nucleon is described by an optical potential and the Coulomb distortion of the electron wave functions is also included. We assume standard values for meson coupling constants and masses in the Hartree calculations and the optical potential used comes from global fits to elastic proton scattering; hence, this calculation is a description of (e,e'p) reactions with no free parameters. We examine the proton knocked-out from the ${\mathit{d}}_{3/2}$ state of $^{40}\mathrm{Ca}$ which leaves the residual nucleus $^{39}\mathrm{K}$ in its ground state. Agreement with the experimental data of NIKHEF is good and the calculated spectroscopic factor agrees very well with the one extracted from experiment. There seems to be no need to make the assumption that the ${\mathit{d}}_{3/2}$ state is appreciably depleted. We also calculate the (e,e'p) cross section from the 2${\mathit{s}}_{1/2}$ and 1${\mathit{f}}_{7/2}$ orbitals. For the 2${\mathit{s}}_{1/2}$ orbital the shape does not agree so well with the experimental data. The ${\mathit{f}}_{7/2}$ orbital leads to approximately the correct shape and we extract a spectroscopic factor of 3.7%.

On the theory and simulation of multiple elastic scattering of electrons

Multiple elastic scattering of electrons in matter is analyzed on the basis of accurate single scattering differential cross sections obtained from partial wave calculations. We give a brief derivation of Molière's multiple scattering theory that clarifies its physical content and points out its limitations. In particular, it is shown that transport mean free paths calculated from the Molière single scattering cross section differ significantly from the values obtained from partial wave calculations. We present a mixed simulation algorithm that overcomes most of the limitations of the currently available condensed Monte Carlo codes. This algorithm takes advantage of the fact that most of the collisions experienced by a high-energy electron along a given path length are soft, i.e. the scattering angle is less than a selected small value χs. The global effect of these soft collisions is described by using a multiple scattering approximation. Hard collisions, with scattering angle larger than χs, occur in a moderately small number and are described as in detailed simulations. This mixed algorithm can be applied to any single scattering differential cross section, it leads to the correct spatial distributions and it completely avoids problems related to boundary crossing. Moreover, when the single scattering law underlying Molière's theory is adopted, the algorithm can be formulated in a completely analytical way.

Numerical investigation of non-eikonal corrections to the Glauber model at intermediate energies.

The Glauber model with first- and second-order non-eikonal corrections has been applied to elastic nuclear scattering of 178- and 800-MeV protons, 180-MeV antiprotons, and 140-MeV \ensuremath{\alpha} particles. Reasonable agreement with the corresponding partial-wave calculations is obtained for the differential cross sections, polarizations, spin rotations, and reaction cross sections. Relativistic kinematics is taken into account in such a way that the relativistic Rutherford cross section is reproduced in both the Glauber model and the optical model.

Coherent scattering of 59.5-keV gamma rays by some medium and heavy elements.

Whole-atom differential coherent scattering cross sections were measured for 59.5-keV \ensuremath{\gamma} rays scattered at 90\ifmmode^\circ\else\textdegree\fi{} by Cu, Mo, Ag, Cd, Sn, Pr, Sm, Gd, Dy, Ho, Yb, Ta, W, Pb, Th, and U employing a reflection-geometry setup and an effective graded shielding arrangement. A 2-${\mathrm{cm}}^{3}$ hyperpure Ge detector was used for detecting coherently scattered \ensuremath{\gamma} rays. The measured cross sections are compared with the theoretical values based on the numerical partial-wave calculations for the available cases. Form factors corresponding to 3.395-\AA{} $^{\mathrm{\ensuremath{-}}1}$ photon-momentum transfer were extracted from the measured cross sections. The results were compared with the different formalisms of form-factor theory and their relative appropriateness is discussed.

Conductivity and reflectivity of a solid state density plasma from 1000 to 106 K

Abstract Theoretical results are given for the electrical conductivity of a hot dense plasma. The gradient formulation of density functional theory is applied to calculate the ionization states and electron-ion potentials. Elastic scattering cross sections are obtained by partial wave calculations. These are then combined with ionic static structure factors to compute electrical resistivity according to the extended Ziman formula. Calculation results are compared with qualitatively novel experimental data.

Another look at meson-exchange interactions

We calculate the nucleon structure in a two-phase model, assuming forlow Q2: meson dynamics, and forhigh Q2: quark-gluon dynamics, and find a low meson scale in accordance with electron-nucleon scattering. Using these form factors we show that with the increased importance of nucleon structure in the nucleon-nucleon interaction, higher-mass meson exchange (m>1 GeV) becomes important. We argue that these contributions can be taken into account by a sort of closure relation, leading to contact interactions of the physical nucleons. A one-boson-exchange potential is constructed including these direct nucleon structure interactions. Implications concerning the corresponding three-nucleon interaction are also discussed. A partial-wave calculation of neutron-proton scattering is performed showing satisfactory agreement with experiment. The model has several advantages over conventional bosonexchange models.

Measurement of the asymmetry in the emission of bremsstrahlung by transversely polarized electrons

Bremsstrahlung emitted by transversely polarized electrons shows a “right-left” asymmetry in spatial distribution. Measurements and remeasurements of this asymmetry are presented using electrons of 300 and 128 keV, respectively, impinging on a gold target. Within the accuracy of the experiment there are no significant discrepancies to the partial wave calculations of Tseng and Pratt.

Quantitative Energy-Filtered CRYO-EM of Hydrated Chromosome Fibers, Viruses, and Heavy Atoms

We have calculated and experimentally measured the extent to which energy filtration improves the S/N in phase- and scattering-contrast images of frozen-hydrated molecules. Multiple scattering calculations based on PW (partial wave) complex scattering amplitudes agree quantitatively with 80 KV image intensities of 50-500nm-thick ice. For an objective aperture of 2.5 nm-1 the PW elastic mean free path is 280 nm. From the ratio of filtered to unfiltered image intensities from 50-500 nm of ice at 80 KV, we calculate an empirical inelastic mean free path of 180 nm. The PW calculations show that Beer's Law should be obeyed for thicknesses up to five mean free paths, whether or not energy filtration is used. Thus the bright-field signal can be used to determine the absolute mass distribution of molecules embedded in thick ice layers.Energy filtration removes background intensity due to inelastic scattering (Fig.1a), and increases the scattering signal from molecules (becoming the sum of both elastic electrons excluded by the objective aperture and inelastic electrons excluded by the energy filter). As the result, the calculated contrast is 2-15 times greater in the Zeiss EM902 than in a TEM without energy filtration (Fig.1b). 80 KeV energy-filtered images of frozen-hydrated molecules should have a higher S/N than unfiltered images taken at intermediate or high voltage (Fig. 1c,d). TMV contrast and S/N is improved by filtration and agrees with calculations (Fig.1b,2). The advantages of energy-filtration are important to the microscopy of chromatin fibers, which are excluded from ice layers ≤200 nm and viruses, which are usually excluded from layers ≤100 nm. Radial density reconstructions from near-focus energy-filtered images of frozen-hydrated chromatin from Thyone sperm show that the centers are not hollow, but very dense, probably containing DNA (Fig.3). Energy-filtered images of icosahedral viruses in even the thinnest possible ice have a much improved S/N ratio, decreasing the dose required for analysis (Fig.2a,b).

η production from nuclei by π capture

In the framework of the distorted-wave impulse approximation, this paper discusses η production from charge exchange reaction of type A(π +, η)B. Calculations are presented for the reaction 13C(π +, η) 13N ∗ at pion energies 600–700 MeV. Distorted waves are calculated using the eikonal approximation and by exact partial wave calculations. It is shown that the eikonal approximation is quite good for the energy region considered. The importance of the behaviour of the bound state wave function is discussed. We find that use of exact bound state wave functions is essential for the calculation of cross sections at large momentum transfers.

Charge overlap effects for first-order molecule-molecule interactions, through high partial wave order, using the N2-N2interaction as a model

Results are presented for the individual partial wave components of the first-order non-expanded Coulomb energy for the N2-N2 interaction as a function of intermolecular separation R, for 3·5 ˇ- R ˇ- 9·5a 0 and for partial wave indices l a and l b corresponding to (l a + l b) ˇ- 8. These results permit the evaluation of the first-order Coulomb energy, its R -1 multipole expansion, and the charge overlap corrections to the multipole expansion, as a function of R, relative molecular configuration, and (l a + l b). Calculations of this type are used in a discussion of the relative importance of first-order charge overlap effects, and of the convergence of the partial wave calculations, as a function of R, configuration, and (l a + l b). The difficulties in obtaining reliable results for charge overlap, and related effects, are also discussed briefly.

A calculation of low-energy πN partial waves based on fixed- t analyticity

The real parts of πN partial waves from threshold to 500 MeV/ c and below threshold down to the nucleon-exchange cut are calculated from “partial-wave relations” which are obtained by a projection of fixed- t dispersion relations. The formalism proposed by earlier authors was modified such that the method can also be applied to the isospin-even amplitudes. Except for the partial waves P13, D13, D33 and D35, the results are compatible with the KH80 partial-wave solution. The agreement with our new fits to the partial-wave dispersion relation is reasonable, except for D33 and D35. Our results are useful as a starting solution and as constraints in partial-wave analyses. They show to what extent the “experimental” partial-wave amplitudes are compatible with various analyticity constraints. For further applications, we have calculated the coefficients of a threshold expansion which converges up to 320 MeV/ c, provided one considers the one-nucleon contribution separately.

Faddeev partial-wave calculations with a three-nucleon potential for the triton ground state

The triton binding energy, the percentages ofS, S′, andD states, and the asymptotic normalization constants are presented for 5-, 18-, 26-, and 34-channel Faddeev calculations with some realistic two-nucleon potentials and the Tucson-Melbourne three-nucleon potential. For the two-nucleon interaction, we take the Reid soft-core, the Argonne, the Paris, and the de-Tourreil-Rouben-Sprung potentials. The convergence properties of these calculations are discussed in detail. It turns out that five-channel calculations are decisively insufficient. We also discuss some asymptotic properties, such as the asymptotic normalization constants, the charge radius, and the momentum distribution in the long-wave-length region. It is shown that these quantities do not depend on the specific dynamics of the system, but only on the binding energy.

Elastic small-angle electron scattering by He, Ne, and Ar at 35 keV.

New experimental data are presented for the scattering of 35-keV electrons by He, Ne, and Ar in the limit of vanishing momentum transfer. Previous data of Geiger and Mor\'on-Le\'on show a strong forward peak with structures similar to a Fraunhofer diffraction pattern (shadow scattering) for elastic scattering of 15-25-keV electrons. No recent theory can explain this effect; therefore we repeated the experiment. The new data agree well with partial-wave calculations utilizing configuration interaction and Hartree-Fock potentials. The results are compared with the previous unexpected findings.

Connection between the bremsstrahlung tip and direct radiative recombination: Angular distributions and polarization correlations

We illustrate the possibility for calculation of bremsstrahlung tip angular destributions and polarization correlations from corresponding photoeffect results with an extension of the method for connecting bremsstrahlung tip and direct-radiative-recombination (DRR) cross sections developed by Lee and Pratt. The agreement between results obtained with this method and results from our numerical partial-wave calculations is, in general, satisfactory. We note features of the distributions and correlations, and once again point out the inadequacies of the nonrelativistic dipole approximation for the evaluation of bremsstrahlung or DRR angular distributions.

Calculation of free-free Gaunt factors in hot dense plasmas

We have obtained predictions for free-free Gaunt factors, using a full relativistic numerical partial-wave and multipole calculation of the bremsstrahlung process as a single electron transition in a potential. We study hot dense cesium plasmas, at various densities and temperatures, using the finite-temperature and -density Thomas-Fermi potential. We consider incident electron energies in the range 1-50 keV. Our results show that use of isolated neutral or ion Gaunt factors has only limited applicability in this environment, while use of the Elwert-Born approximation can lead to a factor-of-3 discrepancy for low incident electron energies.

Computation of 5–10 MeV bremsstrahlung at 0° from a 0.1 g cm −2 Au radiator

The computation of the bremsstrahlung spectrum at 0° from a 0.1 g cm −2 Au radiator struck by a 5–10 MeV beam from an electron linear accelerator is described. The computation is based on recent partial wave calculations of the integrated-over-angle bremsstrahlung cross-section, and includes energy losses, straggling and multiple scattering of the electrons in the radiator, the variation of the bremsstrahlung cross-section angular distribution with photon energy, and an incident electron energy spectrum appropriate for the analysed beam from an electron linear accelerator. The importance of the various components of the bremsstrahlung spectrum is quantified by determining the sensitivity of a typical threshold photofission cross-section unfolded from a yield curve to these components.

Computational studies of first-born scattering cross sections II. Moment-theory approach

Moment-theory methods are described for calculations of the high-energy electron impact-excitation and -ionization cross sections and closely related Van Hove correlation functions of atomic and molecular targets. The conventional Bethe-Born expressions are evaluated employing moment-theory techniques that avoid explicit construction of discrete and continuum target eigenspectra. Appropriately defined generalized-oscillator-strength spectral moments and corresponding polynomial recurrence coefficients are seen to furnish necessary and sufficient information for Born calculations. The convergence of partial-wave variational calculations of such moments and coefficients is investigated for hydrogenic targets using complete basis sets of square-integrable functions. Calculated spectral moments and recurrence coefficients, and previously devised Stieltjes-TschebyschefF techniques, are employed in constructing Gaussian and Radau quadratures that provide convergent approximations to the corresponding Bethe surface of discrete and continuum generalized oscillator strengths, and to the associated cross sections and correlation functions. Square-integrable principal pseudostates obtained from variational calculations of multipole spectral moments are shown to provide correctly normalized convergent approximations to the discrete and continuum transition densities appropriate for the target considered, clarifying the underlying basis of reliability and numerical stability of the moment-theory approach to Bethe-Born cross sections. The development should prove particularly useful in investigations of the Bethe surfaces, related scattering cross sections, and Van Hove correlation functions of molecular targets.

Partial-wave calculation of the residual electrical resistance of dilute binary alloys based on normal metals

Partial-wave calculation of the residual electrical resistance of dilute binary alloys based on normal metals