Method Of Phase Shift Analysis Research Articles

Properties of Pb208 predicted from the relativistic equation of state in the full Dirac space

Relativistic Brueckner-Hartree-Fock (RBHF) theory in the full Dirac space allows one to determine uniquely the momentum dependence of scalar and vector components of the single-particle potentials. In order to extend this new method from nuclear matter to finite nuclei, as a first step, properties of $^{208}\mathrm{Pb}$ are explored by using the microscopic equation of state for asymmetric nuclear matter and a liquid droplet model. The neutron and proton density distributions, the binding energies, the neutron and proton radii, and the neutron skin thickness in $^{208}\mathrm{Pb}$ are calculated. In order to further compare the charge densities predicted from the RBHF theory in the full Dirac space with the experimental charge densities, the differential cross sections and the electric charge form factors in the elastic electron-nucleus scattering are obtained by using the phase-shift analysis method. The results from the RBHF theory are in good agreement with the experimental data. In addition, the uncertainty arising from variations of the surface term parameter ${f}_{0}$ in the liquid droplet model is also discussed.

Study of electron impact elastic scattering from Kr@C60 and Xe@C60 using a fully relativistic approach

In the present work, a detailed study has been reported on electron impact elastic scattering from krypton (Kr) and xenon (Xe) atoms when confined in two different types of C60 potentials viz (a) hard annular square well (ASW) and (b) diffused Gaussian annular square well (GASW). The Dirac equations are solved using these potentials for encaged Kr and Xe in C60. First, bound state Dirac–Fock wave functions of these encaged Kr and Xe atoms are found by utilizing modified general relativistic atomic structure package and thereafter, the charge densities and static potentials of the endohedral Kr@C60 and Xe@C60 are obtained. Further, using these, the Dirac equations are solved by the relativistic partial wave phase shift analysis method and the scattering amplitudes in terms of phase shifts are obtained. Thereafter, the electron elastic differential and integrated cross sections of Kr@C60 and Xe@C60 along with the C60 are calculated in the range of 0.1–15 eV incident electron energies. Presently, no experimental and theoretical results are available to compare our electron scattering cross section results from Kr@C60 and Xe@C60; thus, we have shown the cross section results obtained from ASW and GASW potential and compared them.

Study of Positron Impact Scattering from Methane and Silane Using an Analytically Obtained Static Potential with Correlation Polarization

A detailed study of positron impact elastic scattering from methane and silane is carried out using a model potential consisting of static and polarization potentials. The static potential for the molecular target is obtained analytically by using accurate Gaussian molecular wavefunctions. The molecular orbitals are expressed as a linear combination of Gaussian atomic orbitals. Along with the analytically obtained static potential, a correlation polarization potential is also added to construct the model potential. Utilizing the model potential, the Schrödinger equation is solved using the partial wave phase shift analysis method, and the scattering amplitude is obtained in terms of the phase shifts. Thereafter, the differential, integrated and total cross sections are calculated. These cross-section results are compared with the previously reported measurements and theoretical calculations.

Fully relativistic study on electron impact elastic scattering from Nq+ (q = 1–3), Na+, Arq+ (q = 1–3, 7–8), and Xeq+ (q = 2–6, 8)

AbstractA study is presented on the elastic scattering of electrons from Nq+ (q = 1–3), Na+, Arq+ (q = 1–3, 7–8), Xeq+ (q = 2–6, 8) to understand the available experimental differential cross section results. A model potential approach has been utilized to describe the scattering process. The model potential includes the static, exchange, polarization and absorption potentials. The static potentialis obtained through the charge density calculated by obtaining ionic wave functions using multi‐configuration Dirac‐Fock (MCDF) approximation. Thereafter, the static potential is added to the suitable exchange, polarisation and absorption potentials to construct the spherically averaged complex optical potential. Using the obtained potential in the Dirac equations,these are solved with the partial wave phase shift analysis method and the differential cross sections are calculated. Results for different ions exhibit prominent interference structures in the energy versus cross section curves and show good agreement on comparison with the experimental results available in the selected energy ranges.

An approach to study electron and positron scattering from NH3 and PH3 using the analytic static potential

A detailed study of the elastic scattering of electrons and positrons from two C3v symmetrical molecules viz ammonia and phosphine molecules is presented. The partial wave phase shift analysis method with complex optical potential consisting of static, exchange, polarization and an imaginary absorption potentials has been used. In contrast to the previously used similar approaches, the present method is quite different in the sense that we have obtained the required static potential in an analytical form using correctly represented Gaussian molecular wave functions for NH3 and PH3 in our calculations. The differential, integrated, momentum transfer, absorption and total cross sections are calculated and reported in the incident electron and positron energy range of 10–500 eV. In order to test the applicability of our approach, the present results are compared with the available previous measurements and theoretical calculations and a good agreement is found.

Electroweak charge density distributions with parity-violating electron scattering

Parity-violating electron scattering (PVS) is an accurate and model-independent way to investigate the weak-charge density distributions of nuclei. In this paper, we study parity-violating electron scattering with the Helm model where the effects of spin-orbit currents on nuclear weak skins are taken into account. The conditions of two PVS measurements to constrain the surface thickness ${\ensuremath{\sigma}}_{W}$ of Helm weak-charge densities are investigated. According to the plane wave Born approximation, ${A}_{\mathrm{pv}}$ is expressed in terms of parameters of the corresponding Helm charge and weak-charge densities. After fitting the results of ${A}_{\mathrm{pv}}$ calculated from the phase-shift analysis method where the Coulomb distortion effects are incorporated, an empirical formula in terms of Helm model parameters for calculating ${A}_{\mathrm{pv}}$ is obtained. If two PVS measurements with different scattering angles are carried out, the modeled weak-charge density distributions with two parameters could be extracted from this empirical formula.

Theoretical Study of the Nuclear Charge Distributions of Tin Isotopes

Nuclear binding energies, charge radii and the charge distributions of even-even tin (Sn) isotopes are calculated using relativistic mean field theory, and the theoretical results are found to be in accordance with the experimental data. The nuclear charge form factors for Sn isotopes are calculated using the phase-shift analysis method. It is shown that the minima of the charge form factors shift upward and inward with an increase in the neutron number of the Sn isotopes.

Theoretical study of the central depression of nuclear charge density distribution by electron scattering

The charge form factors of elastic electron scattering for isotones with N = 20 and N = 28 are calculated using the phase-shift analysis method, with corresponding charge density distributions from relativistic mean-field theory. The results show that there are sharp variations at the inner parts of charge distributions with the proton number decreasing. The corresponding charge form factors are divided into two groups because of the unique properties of the s-states wave functions, though the proton numbers change uniformly in two isotonic chains. Meanwhile, the shift regularities of the minima are also discussed, and we give a clear relation between the minima of the charge form factors and the corresponding charge radii. This relation is caused by the diffraction effect of the electron. Under this conclusion, we calculate the charge density distributions and the charge form factors of the A = 44 nuclei chain. The results are also useful for studying the central depression in light exotic nuclei.

Central depression of nuclear charge density distribution

The center-depressed nuclear charge distributions are investigated with the parametrized distribution and the relativistic mean-field theory, and their corresponding charge form factors are worked out with the phase shift analysis method. The central depression of nuclear charge distribution of $^{46}\mathrm{Ar}$ and $^{44}\mathrm{S}$ is supported by the relativistic mean-field calculation. According to the calculation, the valence protons in $^{46}\mathrm{Ar}$ and $^{44}\mathrm{S}$ prefer to occupy the $1{d}_{3/2}$ state rather than the $2{s}_{1/2}$ state, which is different from that in the less neutron-rich argon and sulfur isotopes. As a result, the central proton densities of $^{46}\mathrm{Ar}$ and $^{44}\mathrm{S}$ are highly depressed, and so are their central charge densities. The charge form factors of some argon and sulfur isotopes are presented, and the minima of the charge form factors shift upward and inward when the central nuclear charge distributions are more depressed. Besides, the effect of the central depression on the charge form factors is studied with a parametrized distribution, when the root-mean-square charge radii remain constant.

Theoretical study of nuclear charge densities with elastic electron scattering

The calculated cross sections for elastic Coulomb electron scattering from the phase-shift analysis method and the eikonal approximation are compared. It is shown that the phase-shift analysis method can reproduce the experimental data very well for both light and heavy nuclei. Then the phase-shift analysis is used to investigate elastic electron scattering along the O and S isotopic chains, where the charge densities are obtained from relativistic mean-field theory. Results show that the minima of the differential cross section shift inward and upward with the increase of the neutron number.

Phase shift analysis of modulated photocurrent: A new approach to determining the energy scale

One of the major problems of the method of phase shift analysis of modulated photocurrent for studying the density of states in the energy gap of amorphous semiconductors has been the determination of the energy scale corresponding to this DOS profile. This study presents a new way of dealing with this problem. This new method is especially useful in the case where the DOS profile lacks a characteristic peak. A computer analysis is used to confirm the validity of this method and to demonstrate how it can be used.

Electronic structure and transport properties of GaAs-GaAlAs superlattices in high perpendicular electric fields

We present a study of the effect of high (∼1×105 V/cm) perpendicular electric fields on the electronic structure of a GaAs-Ga0.7Al0.3As superlattice represented by five quantum wells separated by 70-Å barriers. An exact solution is obtained by using the transfer-matrix technique to match the Airy functions at the interfaces. The method of phase-shift analysis is employed to evaluate the energy levels, the width of the Stark resonances, the real-space wave functions, and the time of flight associated with the field-induced phenomena. Wells of 50- and 70-Å width are used to elucidate the relationship between level depth and field. The quantum states above the semiclassical confining barrier, which have so far been represented by a continuum, are correctly accounted for. These states are mainly localized in the barrier material. It is shown that the real-space localization and bandwidth of these states are significantly altered by the external field. In particular, we evaluate the time delay for electrons above the barrier due to the field-induced change in the density of states, which is related to the presence of the above higher-lying confined states and which would be observed in current–voltage measurements.

Phase shift analysis of hyperon-nucleon elastic scattering using optimized polynomial expansion techniques

A relatively stable method of phase shift analysis of hyperon-nucleon scattering proposed by us is applied to Σ+p and Λp scattering. The analytic cutt-planes of analyticity of the helicity amplitudes are mapped into the interior of unifocal ellipses. The helicity amplitudes are then expressed as accelerated convergent expansions in the mapped variable. A definite economy is observed in the number of free parameters for fixed energy phase shift analysis of Σ+p and Λp scattering at 40 and 100 MeV and 100 MeV respectively. Twenty six more phase shifts and coupling parameters corresponding to higherJ values are also predicted.

Phase shift analysis of pion-4He scattering by optimal polynomial expansion

A novel method of phase shift analysis for the scattering of charged pion from4He nuclei is proposed. The nuclear amplitudefN(ϑ) has been parametrized in terms of a polynomial expansion in a conformally mapped variable, which is obtained by optimally exploiting the analytic property offN. The method exhibits a significant reduction in the number of free parameters required for the fixed energy phase shift analysis of the differential cross-section data. The nuclear amplitude thus constructed is then used to obtain phase shifts and inelastic parameters of all possible orders. Reliable values of the real and imaginary parts of the forward amplitude are also obtained.

Scattering of electrons by hydrogen atoms at intermediate energies: Elastic differential cross sections from 10 to 30 eV

The elastic scattering amplitude obtained from an open pseudostate calculation employing the algebraic variational method for partial waves of angular momentum $0<L<3$, is combined with phase shifts obtained in the extended polarization approximation for $4<L<99$ to obtain elastic differential cross sections for electron-hydrogen scattering for electron energies in the range 10-30 eV. Elastic angular distributions have been measured using a crossed-electron modulated-atom-beam method and absolute cross-section values determined using a phase-shift analysis method. The measurements include energies from 10 to 30 eV and angles from 10\ifmmode^\circ\else\textdegree\fi{} to 140\ifmmode^\circ\else\textdegree\fi{} with estimated experimental error limits of \ifmmode\pm\else\textpm\fi{} 17%. Within these error limits they are in agreement with the theoretical predictions except at 20 to 26 eV for angles greater than or equal to 120\ifmmode^\circ\else\textdegree\fi{}, where the theoretical values are significantly higher than allowed by these error limits.

Position-operator method for evaluating the shift of a totally reflected electron

A method based on expectation values of a position operator is developed for evaluating the spatial shift of a polarized Dirac electron undergoing total reflection. Longitudinal and transverse components of the shift, as well as the associated time delay, are obtained by considering the differences in the expectation values for the physically reflected wave packet and a purely geometrically reflected wave packet. For a number of position operators, this method is used to calculate the mean shift arising from a single reflection of an electron from a step-function potential. Eigenpolarization states are determined by means of a localization form-invariance argument. For these states, results obtained employing the Newton-Wigner position operator are shown to be identical to those obtained using the phase-shift analysis method. A comparison calculation giving different results is made using the current-flux method. The current-flux results are shown to be inconsistent with conservation of total angular momentum normal to the reflecting boundary.

Dynamical Properties of Quasiparticle Excitations in Metallic Copper

The results of experimental studies of the cyclotron masses associated with various orbits on the Fermi surface of copper are analyzed in terms of an interpolation scheme based on surfaces of constant energy that are constructed by the method of phase-shift analysis. Our results confirm that it is possible to construct surfaces of constant energy that are entirely consistent with experimental Fermi-surface and cyclotron-mass data. The variation of the quasiparticle velocity over the Fermi surface is determined from the experimental cyclotron masses. The anisotropy of the Fermi velocity is in excellent over-all agreement with the results of a similar calculation by Halse, and with recent experimental measurements by Doezema, Koch, and Strom. A comparison between the quasiparticle velocities and the velocities computed from a one-electron potential that has been constructed to fit optical and Fermi-surface data demonstrates that in copper the renormalization of the one-electron energy bands by the electron-phonon interaction varies significantly over the Fermi surface. We find that, for electronic states on the belly, the anisotropy of the renormalization factor is in good qualitative agreement with the predictions of a simple deformation-potential calculation, but that for states within a few degrees of the necks the deformation-potential approximation fails. Encouraged by the partial success of the deformation-potential calculation for virtual electron-phonon processes, we apply similar techniques to discuss the real processes that at low temperatures $T$ contribute a ($\frac{1}{{T}^{3}}$)-dependent term to the relaxation time for quasiparticle excitations. We propose a model to describe the variation of the electron-phonon relaxation time over the belly, but conclude that a first-principles calculation may be needed to investigate the relaxation time on the necks. At present there exist insufficient experimental data to allow a convincing test of the accuracy of the model, but it seems likely that our empirical deformation potential will be of value in the analysis of such data when they become available.

Phase-Shift Analysis of the Fermi Surface of Copper

In an earlier paper a method was described whereby the partial-wave phase shifts that characterize the interaction between the conduction electrons and the lattice in a metal may be derived from experimental Fermi-surface data. In the present paper we apply the method of phase-shift analysis to study the shape of the Fermi surface of copper, which is known to be strongly perturbed by the $d$-like energy bands that lie almost 2 eV below the Fermi level. By adjusting the values of the $s$, $p$, $d$, and $f$ phase shifts, and the Fermi-energy parameter, we construct a model Fermi surface on which the areas of the $〈100〉$ belly and the $〈111〉$ neck and belly orbits, the dog's bone, the four-cornered rosette, and the lemon orbit, are in good agreement with the results of precision measurements of the corresponding de Haas-van Alphen frequencies. The belly anisotropy of the model surface is also in good agreement with the experimental data, and the volume enclosed by the surface does not differ significantly from 1 electron/atom. The radii of the Fermi surface of copper in the (100) and (110) symmetry zones are determined to an accuracy of \ifmmode\pm\else\textpm\fi{}0.1%, and the results are in good agreement with the radii recently deduced by Halse by an independent technique. It is shown that the numerical values of the phase shifts are consistent with the position of copper in the Periodic Table. The local potential of Chodorow for ${\mathrm{Cu}}^{+}$ produces phase shifts that are in substantial agreement with the results of the present work. A simple nonlocal correction to the Chodorow potential is proposed, such that the Fermi surface derived from the modified potential is entirely consistent with the experimental data. The energies associated with certain optical transitions in metallic copper are computed from the modified potential, and are found to agree with the results of recent piezo-optical experiments to better than 0.2 eV. It is concluded that the method of phase-shift analysis is capable of representing accurately the form of the $d$-like electronic energy bands in metals, and that the modified Chodorow potential may well prove to be the best starting point for a full calculation of the band structure of copper in the vicinity of the Fermi level.

Electron-Ion Interaction and the Fermi Surfaces of the Alkali Metals

A method is proposed whereby the partial-wave phase shifts which characterize the scattering of plane waves by the ionic cores in a metallic lattice may be deduced from experimental Fermi-surface data. The method is applied to an analysis of currently available experimental data on the shapes of the Fermi surfaces of the alkali metals. Starting from the full many-body theory of conduction electrons in a metallic lattice, it is shown that the shape of the Fermi surface may be derived from a one-electron-like Schr\"odinger equation which involves a nonlocal effective potential. The augmented-plane-wave method is applied to solve the nonrelativistic Schr\"odinger equation for the shapes of the surfaces of constant energy in k space, in the approximation where the effective potential may be represented by an angular-momentum-dependent potential of muffin-tin form. The partial-wave phase shifts of the muffin-tin potential are adjusted to bring the area distortions of the computed surface into agreement with the experimental data. The phase shifts deduced in this way are shown to be largely independent of the radius of the muffin-tin sphere. It is found that the shapes of the Fermi surfaces of the alkali metals are systematically influenced by the position of each metal in the periodic table. Lithium shows a strong $p$ phase shift, which anticipates the onset of the $p$ resonance in the second row of the periodic table. The phase shifts in sodium are found to converge rapidly in angular momentum, presumably because the $p$-like component of the ionic potential is largely cancelled by the $p$-like core states, while the $d$-like component of the ionic potential is rather weak. Potassium, rubidium, and cesium show increasingly strong $d$ phase shifts, which are associated with the positions of these metals at the heads of the $3d$, $4d$, and $5d$ transition series. Only for potassium are the experimental data sufficiently accurate to show the influence of a small $f$ phase shift. The method of phase-shift analysis has several advantages as a technique by which the radial distortions of the Fermi surface of a metal may be deduced from experimental data on the anisotropy of its extremal cross-sectional area. These include rapid convergence of the series of phase shifts, the use of a model surface generated in a way closely related to a first-principles band-structure calculation, and the possibility of applying the technique both to nearly-free-electron metals and to metals of the transition series. The radial distortions of the Fermi surfaces of the alkali metals are computed, and the results are compared with those of earlier calculations. Finally, it is shown that our results impose criteria which must be satisfied by any effective potential of muffin-tin form that may be proposed to represent the effects of electron exchange and correlation in the alkali metals.

Monopole Excitation Corrections to Electron Elastic Scattering

The effect of the virtual excitation of nuclear levels on the elastic electron-scattering cross section is calculated by means of a phase-shift analysis method under the assumption that the inelastic excitation is represented by a single monopole transition and that the nuclear excitation energy can be neglected. The formulas are applied to the case of 250-MeV electrons and positrons incident on the nucleus of Ca. Effects of the order of 5% are found in the vicinity of the first diffraction minimum.