Three-body Wave Function Research Articles

A THEORETICAL STUDY ON ABSOLUTE TRIPLE DIFFERENTIAL CROSS SECTIONS FOR NEAR-THRESHOLD He+(e,2e) REACTIONS

A correlated three-body continuum wave function proposed by Brauner et al. has recently been modified further to allow for the three-body effects by introducing Sommerfeld parameters (Berakdar and Briggs),it is used to calculate the triple differential cross sections for electron impact ionization of He+ in coplanar equal-energy sharing and symmetric geometry. The initial channel wave function involves a Coulomb wave function due to the long range Coulomb attraction between the incident electron and the screened ionic nucleus. Since the present work deals with symmetric collisions, the electron exchang effect between the two continuum electrons has been taken into account properly. It is shown that the initial channel Coulomb field play a major role in determining the triple differential cross sections at near-threshold incident energy and small impact parameter values.

Pion electroproduction in the 3He(e,é π +) 3H reaction at intermediate energies

The differential coincident pion electroproduction cross section in the 3He(e,é π +) 3H reaction has been measured with high resolution with the three-spectrometer set-up at the Mainz Microtron (MAMI) electron accelerator. Measurements were performed at the four incident energies E 0 = 855, 675, 600, and 555 MeV at fixed four-momentum transfer Q 2 = 0.045 GeV 2, with the pions detected in parallel kinematics. This enables a separation of the measured cross section into the transverse and longitudinal structure functions by means of the Rosenbluth method. The experimental data are compared to model calculations, in which the elementary pion production amplitude includes the standard Born-amplitudes and also delta and higher resonance terms. Three-body Fadeev wave functions are used and the final state interaction of the outgoing pion is taken into account. The calculation describes the data only after medium modifications of the delta and of the pion are included.

Effective Sommerfeld parameters in the three-body Coulomb continuum problem

The three-body Coulomb continuum wave function as a product of three two-body Coulomb wave functions is modified by the introduction of an Ansatz for effective Sommerfeld parameters corresponding to the modification of a particular two-body Coulomb interaction by the presence of the third particle. The triple-differential cross sections for electron-impact ionization of atomic hydrogen at incident energies of 54.4 and 150 eV in asymmetric geometry are calculated. We generally find that this approach gives good agreement with experiment, though some small quantitative discrepancies remain.

Three-body halos. IV. Momentum distributions after fragmentation

Momentum distributions of particles from nuclear breakup of high-energy three-body halos are calculated incorporating effects of final state interactions. The same two-body interactions between the particles are used to calculate both the ground state structure and the final state of the reaction processes. The ground state wave function reproduces size and energy of the halo nucleus. First we give a general and detailed description of the method. Then we investigate the effect of final state interactions in neutron removal and core breakup reactions for one- and two-dimensional momentum distributions. We compute specifically core and neutron momentum distributions from ${}^{11}$Li fragmentation and compare those with available experimental data. We conclude that 20--30 % of the three-body wave function describing ${}^{11}$Li corresponds to a neutron-core relative $p$ state. The fragmentation data strongly suggest that ${}^{10}$Li has an $s$ state below 200 keV and a $p$ state around 500 keV.

Three-body models of electron-hydrogen ionization

Although the importance of electron-electron correlations in (e,2e) reactions has long been realized, untill recently only little theoretical progress had been made including these correlations. Theoretical studies using “three-body” models (models with explicit dependence on the electron-electron coordinate) are hampered by the difficult six-dimensional integral for the scattering amplitude. “Two-body” models, on the other hand, retain only electron-ion interactions and the standard techniques of partial-wave analysis greatly simplify the integral. In contrast, the wave functions used in three-body models give rise to two-center integrals, which are not efficiently evaluated using partial-wave methods. Therefore, a different method was sought for. We have found that direct six-dimensional numerical quadrature is efficient (about five minutes on a workstation to compute one scattering amplitude). As a result, we now have the capability of calculating (e,2e) triply-differential cross sections using three-body wave functions of arbitrary complexity. This allows for a severe test of three-body models. The utility of our method is demonstrated by using two recently reported analytical three-body wave functions in an (e,2e) calculation.

A comparative study on the spin asymmetry and integrated cross sections for the electron-impact ionization of atomic hydrogen

The spin asymmetry and integrated spin-unresolved cross sections for the electron-impact ionization of atomic hydrogen are studied. The influence of individual final-state interactions on these quantities is elucidated by investigating the collision process within the plane-wave impulse approximation, the first Born approximation and within an independent Coulomb particle model. The predictions of the latter approximation for the shape and magnitude of the spin asymmetry are in good agreement with experimental data over the entire measured energy range. The effects of final-state electronic correlations are investigated by the inclusion of the electronic Coulomb density-of-states factor as well as by employing a three-body Coulomb wave-function for the description of the final state. For ionization of hydrogenlike ions the calculated asymmetry parameter has a positive slope near threshold indicating that such an energy dependence is due to the electron-nucleus interactions. At threshold the results are analyzed in light of the Wannier theory of threshold ionization.

Asymptotic form of three-body (dtµ)+ and (ddµ)+ wave functions

In order to investigate a discrepancy between existing literature values for the normalization constant in the asymptotic form of three-body wave functions for (dtµ)+, we report the results of a new calculation of the normalization constants for this system as well as the related system (ddµ)+. These were obtained by fitting to accurate variational wave functions with special care being taken to describe the long-range behavior.

Parabolic-hyperspherical approach to the fragmentation of three-particle Coulomb systems.

In this work the three-body Coulomb continuum problem is reformulated in a set of parabolic-hyperspherical coordinates. It is shown how this procedure yields additional information on the angular and radial behavior of correlated three-body Coulomb continuum wave functions. For two electrons receding from a nucleus it is shown that three-body coupling is essential in describing the dependence of the fragmentation dynamics on the ratio of the electronic distances from the nucleus whereas the angular distribution of these electrons is less sensitive to this coupling. The spin asymmetry in the total cross section for electron-impact ionization of atomic hydrogen is analyzed and found to be sensitive to the radial correlation of the outgoing electrons. \textcopyright{} 1996 The American Physical Society.

Formation of metastable dt micro molecules in t micro(2s)-D2 collisions.

The formation process of metastable $\mathrm{dt}\ensuremath{\mu}$ molecules in ${D}_{2}\ensuremath{-}{T}_{2}$ mixtures in investigated. The $\mathrm{dt}\ensuremath{\mu}$ molecule exhibits a series of three-body resonances embedded in the $t\ensuremath{\mu}+d$ scattering continuum just below the $t\ensuremath{\mu}(2s)+d$ threshold. These states can be formed in collisions of excited $t\ensuremath{\mu}(2s)$ atoms with ${D}_{2}$ molecules, whereby the excess of binding energy is absorbed by the rovibrational degrees of freedom of ${D}_{2}$. We present a scattering-theoretic model for this process, and perform a numerical calculation of its cross section. The essential transition amplitudes are obtained from three-body wave functions for the $\mathrm{dt}\ensuremath{\mu}$ subcluster, and adiabatic wave functions for the entire hybrid system in the final channel. It is found that the effective formation rate is limited by the Auger-transition rates of the molecular complex formed. The calculated cross sections exhibit broad "peaks," with magnitudes large enough for the formation process to favorably compete with deexcitation of $t\ensuremath{\mu}(n=2)$ atoms via radiative and collisional processes. The formation of metastable $\mathrm{dt}\ensuremath{\mu}$ can therefore be one of the fastest processes depleting the $n=2$ levels of the $t\ensuremath{\mu}$ atom in hydrogen mixtures, of importance for low-energy muon science, electroweak physics, and muon catalyzed fusion.

Approximate analytical solution of the quantum-mechanical three-body Coulomb continuum problem.

In this work a symmetric representation of the three-body Coulomb continuum wave function is constructed that represents an exact asymptotic solution of the many-body Schr\odinger equation on a five-dimensional hypersphere of large hyperradius. Consequently, the wave function is shown to satisfy the Kato cusp conditions at all three two-body collision points. At finite distances the proposed solution is designed to account for properties of the total potential surface. In particular, dynamical stabilization due to the presence of ridge structure in the total potential (Wannier ridge) is encompassed in the present treatment. The behavior of the wave function at the total dissociation threshold is investigated. In order to allow for three-body interactions we linearly expand each two-body Coulomb potential in terms of all three two-body potentials. The expansion coefficients determine the amount of distortion of each two-body subsystem by the presence of the third particle and thus give direct information on the strength of three-body interactions. \textcopyright{} 1996 The American Physical Society.

Incorporation of Threshold Phenomena in Three-body Coulomb Continuum Wavefunctions

In this work a three-body Coulomb wavefunction for the description of two continuum electrons moving in the field of a nucleus is constructed such that the Wannier threshold law for double escape is reproduced and the asymptotic Coulomb boundary conditions as well as the Kato cusp conditions are satisfied. It is shown that the absolute value of the total cross section, as well as the spin asymmetry, are well described by the present approach. Further, the excess-energy sharing between the two escaping electrons is calculated and analysed in light of the Wannier theory predictions. This is the first time an analytical three-body wavefunction is presented which is asymptotically exact and capable of describing threshold phenomena.

An Approximate Three-Body Theory for Nuclear Reactions Induced by Weakly Bound Projectiles

At medium energies the conventional distorted waves Born approximation provides a much less reliable description of transfer reactions induced by composite projectiles. While use of the Johnson-Soper adiabatic method includes the dominant contributions from three-body break-up channels, recent experimental data have suggested the need for additional theoretical refinements. In this letter, an alternative approximation is developed for the treatment of break-up process, which presents a way of evaluating non- adiabatic corrections to the adiabatic model three-body wavefunction. However, the implementation of the model shows that such an approach cannot be used in the description of deuteron involving reactions.

H2O as a Three-Body system11The project supported by National Natural Science Foundation of China.

The few-body method for treating water molecules is presented. H2O can be regarded as a three-body system. The total three-body wavefunction is expanded in terms of the harmonic oscillator product states in the Jacobi coordinates. The binding energies are calculated by using a pairwise model potential in the variational approach. The geometry and modes of internal motion are analyzed by the shape density function. The results are E = -76.07 a.u, RHO = 0.970 Å and ∠HOH = 105.38°. These results are compared with some known calculations and experimental values.

Normalization of the asymptotic form of three-body (dt micro)+ and (dd micro)+ wave functions.

In order to investigate a discrepancy between existing literature values for the normalization constant in the asymptotic form of three-body wave functions for (dt\ensuremath{\mu}${)}^{+}$, we report the results of a calculation of the normalization constants for this system as well as the related system (dd\ensuremath{\mu}${)}^{+}$. These were obtained by fitting to accurate variational wave functions with special care being taken to describe the long-range (t\ensuremath{\mu})+d or (d\ensuremath{\mu})+d behavior. The implications of this reevaluation of the normalization constants are discussed in relation to the theoretical formation rates of molecules such as (dt\ensuremath{\mu})dee and (dd\ensuremath{\mu})dee, which are key intermediates in the muon-catalyzed fusion cycle.

Particle momentum distributions in an analytical three-body approach

Abstract An analytical approach with Yukawa-type three-body wave functions has been developed for calculations of the particle momentum distributions in light two-neutron halo nuclei. In the frame-work of the model the core and the neutron momentum distributions are well described by a lorentzian shape. The momentum distributions, the angular correlation between the valence neutrons and the ratio between the core and the neutron widths are analytically obtained and compared with experimental data for 11 Li and 14 Be.

Multinucleon effects in muon capture on 3He at high energy transfer.

Energy spectra of both protons and deuterons emitted following the capture of negative muons by [sup 3]He nuclei have been measured for energies above 15 MeV. A limited number of proton-neutron pairs emitted in coincidence were also observed. A simple plane wave impulse approximation (PWIA) model calculation yields fair agreement with the measured proton energy spectra, but underpredicts the measured rate of deuteron production above our energy threshold by a large factor. A more sophisticated PWIA calculation for the two-body breakup channel, based on a realistic three-body wave function for the initial state, is closer to the deuteron data at moderate energies, but still is significantly lower near the kinematic end point. The proton-neutron coincidence data also point to the presence of significant strength involving more than one nucleon in the capture process at high energy transfer. These results indicate that additional terms in the capture matrix element beyond the impulse approximation contribution may be required to explain the experimental data. Specifically, the inclusion of nucleon-nucleon correlations in the initial or final state and meson exchange current contributions could bring calculations into better agreement with our data. A fully microscopic calculation would thus open the possibility for a quantitative testmore » of multinucleon effects in the weak interaction.« less

Three-body Coulomb continuum problem.

A symmetric representation of the three-body Coulomb continuum wave function as a product of three two-body Coulomb wave functions is modified to allow for three-body effects whereby the Sommerfeld parameter describing the strength of interaction of any two particles is affected by the presence of the third particle. This approach gives excellent agreement with near-threshold absolute ([ital e],2[ital e]) ionization cross sections. In particular a recently observed deep minimum in noncoplanar geometry is reproduced for the first time.

Asymmetries in 100 MeV pi +/ pi --3H

We have measured cross sections and analyzing powers for [pi][sup +] and [pi][sup [minus]] elastic scattering from a polarized [sup 3]He target at [ital T][sub [pi]]=100 MeV. Measurements have been made over an angular range of 60[degree] to 110[degree] which extends over the region of the cross section minimum where spin effects in [pi]-nucleus scattering are predicted to be largest. This is evident from the large value of [ital A][sub [ital y]] observed in [pi][sup +] scattering from [sup 3]He near [theta][sub lab]=80[degree]. The [ital A][sub [ital y]] data for both [pi][sup +] and [pi][sup [minus]] scattering are qualitatively reproduced by a schematic PWIA model; however, agreement with the data is significantly improved when a full nonlocal DWIA reaction model employing realistic three-body wave functions is used. The asymmetry in [pi][sup [minus]]-[sup 3]He scattering resembles the asymmetry for [pi][sup [minus]] scattering from a free neutron near the cross section minimum and is about half that for [pi][sup +] scattering.

Triple differential ionisation cross-section of hydrogen atom bye �-impact including exchange

The Triple Differential Cross-Sections for ionisation of hydrogen atom bye± impact are calculated including exchange effect. A correlated three-body continuum wave function is used here for the final state. Attention is paid to equal energy sharing at the final channel where scattered and ejected energies are each 200 ev., and it is found that the exchange effect is remarkably high in this case. Present electron impact results are compared with the recent work of McCarthy et al. [1].

Quasielastic scattering of 11Li using realistic three-body wave functions.

The quasielastic scattering of $^{11}\mathrm{Li}$ from $^{12}\mathrm{C}$ at 60 MeV/nucleon is calculated in a four-body Glauber approximation. Different $^{11}\mathrm{Li}$ ground-state wave functions are used, including those calculated using Faddeev three-body models. The calculated quasielastic cross sections, including the ${2}^{+}$ and ${3}^{\mathrm{\ensuremath{-}}}$ states of $^{12}\mathrm{C}$ in a distorted wave Born approximation, reproduce the experimental data over most of the angular range, the differences between theoretical models being less than the quoted statistical errors on the available data.