Plane Wave Impulse Approximation Calculations Research Articles

Experimental and Theoretical Study on Electron Momentum Spectroscopy of SF6: Distorted-Wave and Vibrational Motion.

The vibrational and distorted-wave effects are usually invoked to explain the measured electron momentum profiles for molecular orbitals. The vibrational effect can be accounted for quantitatively by a harmonic analytical quantum mechanical approach within the plane-wave impulse approximation (PWIA). On the other hand, quantitative calculation considering the distorted-wave effect was available only recently by a multicenter-three-distorted-wave (MCTDW) method (Phys. Rev. A2022, 105, 042805). Here, we report a joint experimental and theoretical investigation on electron momentum spectroscopy of SF6. The experiments were performed using a high-sensitivity (e, 2e) spectrometer employing non-coplanar symmetric geometry with incident electron energy equal to 1200 eV + binding energy. The experimental electron momentum profiles are compared with theoretical calculations by the MCTDW method at equilibrium geometry and by the PWIA method both at equilibrium geometry and considering vibrational motions. For all the measured orbitals, large discrepancies were observed between the experiments and the PWIA calculations at equilibrium geometry. For the highest occupied molecular orbital 1t1g, the vibrational effect can partly explain the high intensity of the experimental momentum profile at low momenta. For the other orbitals, the influence of the vibrational effect is negligible. On the other hand, the MCTDW calculations improve the agreement with the experiments for all the observed orbitals, indicating that the distorted-wave effect plays an important role in reproducing the measured momentum profiles of SF6.

Ruling out Color Transparency in Quasielastic ^{12}C(e,e^{'}p) up to Q^{2} of 14.2 (GeV/c)^{2}.

Quasielastic ^{12}C(e,e^{'}p) scattering was measured at spacelike 4-momentum transfer squared Q^{2}=8, 9.4, 11.4, and 14.2 (GeV/c)^{2}, the highest ever achieved to date. Nuclear transparency for this reaction was extracted by comparing the measured yield to that expected from a plane-wave impulse approximation calculation without any final state interactions. The measured transparency was consistent with no Q^{2} dependence, up to proton momenta of 8.5 GeV/c, ruling out the quantum chromodynamics effect of color transparency at the measured Q^{2} scales in exclusive (e,e^{'}p) reactions. These results impose strict constraints on models of color transparency for protons.

Observation of strong relativistic and distorted-wave effects in (e,2e) electron-momentum spectroscopy of mercury

We report a measurement of the valence orbital-momentum profiles of mercury (Hg) using a high-sensitivity binary $(e,2e)$ electron-momentum spectrometer at incident energies of 600 and 1200 eV plus the binding energy. The $6{s}_{1/2}$ orbital and the spin-orbit components of $5{d}_{5/2}$ and $5{d}_{3/2}$ orbitals are well separated in the measured binding-energy spectra. The experimental momentum distributions for the individual orbitals and the branching ratio of $5{d}_{5/2}$ to $5{d}_{3/2}$ are obtained and compared with predictions from a plane-wave impulse approximation (PWIA) in which the orbital wave functions are calculated using the nonrelativistic (NR) and spin-orbital (SO) relativistic theories. The SO relativistic calculations are in better agreement with experiment than the NR, indicating clearly the importance of relativistic effects in the electronic structure of Hg. We also observe some discrepancies between experiment and PWIA calculations in the high-momentum region of $6{s}_{1/2}$ and the low-momentum region of $5d$ orbitals which display a dynamic dependence on the impact energy. These discrepancies become smaller at a higher energy of about 1200 eV and thus can be qualitatively assigned to the distorted-wave effect in the $(e,2e)$ reaction of Hg.

High-resolution electron-momentum spectroscopy of the valence orbitals of the benzene molecule

The binding-energy spectrum and electron-momentum spectra of valence orbitals of benzene were measured via a high-resolution electron-momentum spectrometer at incident energies of 600 and 1200 eV plus the binding energy. The experimental momentum profiles were compared with the results from a plane-wave impulse approximation (PWIA) calculation using Dyson orbitals obtained from symmetry-adapted-cluster configuration-interaction calculations. For the ${1e}_{1\mathrm{g}}, {3e}_{2\mathrm{g}}, {3e}_{1\mathrm{u}}$, and ${2b}_{1\mathrm{u}}$ orbitals, the discrepancy between the experimental distributions and the PWIA calculations was tentatively assigned to the distorted wave effects. The current high-resolution electron-momentum spectroscopy results provide the experimental benchmark for rigorous distorted-wave calculations about benzene in the future.

Cross sections and analyzing powers for(p,n)reactions onHe3andHe4at 346 MeV

The cross sections and analyzing powers for $(p,n)$ reactions on $^{3}\mathrm{He}$ and $^{4}\mathrm{He}$ have been measured at a bombarding energy of ${T}_{p}=346$ MeV and reaction angles of ${\ensuremath{\theta}}_{\mathrm{lab}}=9.{4}^{\ifmmode^\circ\else\textdegree\fi{}}\text{\ensuremath{-}}{27}^{\ifmmode^\circ\else\textdegree\fi{}}$. The energy transfer spectra for ${}^{3}\mathrm{He}(p,n)$ at large ${\ensuremath{\theta}}_{\mathrm{lab}}$ ($\ensuremath{\geqslant}{16}^{\ifmmode^\circ\else\textdegree\fi{}}$) are dominated by quasielastic contributions and can be reasonably reproduced by plane-wave impulse approximation (PWIA) calculations for quasielastic scattering. By contrast, the known $L=1$ resonances in $^{4}\mathrm{Li}$ are clearly observed near the threshold in the ${}^{4}\mathrm{He}(p,n)$ spectra. Because these contributions are remarkable at small angles, the energy spectra are significantly different from those expected for quasielastic scattering. The data are compared with the PWIA calculations, and it is found that the quasielastic contributions are dominant at large ${\ensuremath{\theta}}_{\mathrm{lab}}$ ($\ensuremath{\geqslant}{22}^{\ifmmode^\circ\else\textdegree\fi{}}$). The nuclear correlation effects on the quasielastic peak for ${}^{4}\mathrm{He}(p,n)$ are also discussed.

Measurement of the 2H(d,pn) reaction at 0 degrees at 270 MeV

The energy spectrum of the 2 H(d,pn) reaction at 0° at 270 MeV was obtained at RIKEN Accelerator Research Facility. The spectrum in the kinematical region where the resulting pn pair has the relative energy smaller than ∼2 MeV was analyzed by plane-wave impulse approximation calculations assuming the configurations of final state interaction with s-wave.

Extraction of the neutron magnetic form factor from quasielastic3He→(e→,e') atQ2=0.1−0.6 (GeV/c)2

We have measured the transverse asymmetry AT' in the quasielastic 3He→(e→,e') process with high precision at Q2 values from 0.1 to 0.6(GeV/c)2. The neutron magnetic form factor GMn was extracted at Q2 values of 0.1 and 0.2(GeV/c)2 using a nonrelativistic Faddeev calculation which includes both final-state interactions (FSI) and meson-exchange currents (MEC). Theoretical uncertainties due to the FSI and MEC effects were constrained with a precision measurement of the spin-dependent asymmetry in the threshold region of 3He→(e→,e'). We also extracted the neutron magnetic form factor GMn at Q2 values of 0.3 to 0.6(GeV/c)2 based on plane wave impulse approximation calculations.3 MoreReceived 9 May 2006DOI:https://doi.org/10.1103/PhysRevC.75.034003©2007 American Physical Society

An investigation of distorted wave effects in π * like molecular orbital by electron momentum spectroscopy

Sensitive images of orbital electron density for the ethane (C 2H 6) were investigated by using a newly developed electron momentum spectrometer with a wide range of experimental impact energies. Some ‘turn up’ effects in the momentum distributions of the 1E g and 3A 1g orbitals were observed at low and high momentum regions compared with plane wave impulse approximation calculations. Moreover, such discrepancies become smaller with the increase in the impact electron energy. The distorted wave effects could be a reasonable explanation for these observed discrepancies. While appropriate theoretical calculations using distorted wave in molecules could not be achieved at present.

Ionization excitation of helium by the(e,2e)reaction

The $(e,2e)$ cross sections for transitions to the $n=1$, 2, and 3 final state of ${\mathrm{He}}^{+}$ have been measured at impact energies of 1000 and $1600\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ by using a newly developed energy and momentum dispersive spectrometer. Binding energy spectra in the range of $8\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}86\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ and momentum profiles for the transitions to the ground and excited ion states of ${\mathrm{He}}^{+}$ are reported at different impact energies, and the experimental results are compared with plane wave impulse approximation calculations. The impact energy dependence of cross section ratios for the ${\mathrm{He}}^{+}\phantom{\rule{0.3em}{0ex}}n=1$, 2, and 3 final states are obtained. Some discrepancies are observed between experimental data and theoretical calculations.

(e,2e)and(e,3−1e)studies on double processes of He at large momentum transfer

The double processes of He in electron-impact ionization, single ionization with simultaneous excitation and double ionization, have been studied at large momentum transfer using an energy- and momentum-dispersive binary $(e,2e)$ spectrometer. The experiment has been performed at an impact energy of $2080\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ in the symmetric noncoplanar geometry. In this way we have achieved a large momentum transfer of $9\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$, a value that has never been realized so far for the study on double ionization. The measured $(e,2e)$ and $(e,3\ensuremath{-}1e)$ cross sections for transitions to the $n=2$ excited state of ${\mathrm{He}}^{+}$ and to doubly ionized ${\mathrm{He}}^{2+}$ are presented as normalized intensities relative to that to the $n=1$ ground state of ${\mathrm{He}}^{+}$. The results are compared with first-order plane-wave impulse approximation (PWIA) calculations using various He ground-state wave functions. It is shown that shapes of the momentum-dependent $(e,2e)$ and $(e,3\ensuremath{-}1e)$ cross sections are well reproduced by the PWIA calculations only when highly correlated wave functions are employed. However, noticeable discrepancies between experiment and theory remain in magnitude for both the double processes, suggesting the importance of higher-order effects under the experimental conditions examined as well as of acquiring more complete knowledge of electron correlation in the target.

Direct Observation of Distorted Wave Effects in Ethylene Using the (e,2e) Reaction

We report here the direct measurements of electron momentum distributions for ethylene using the (e,2e) reaction at different impact energies from 400 to 2400 eV. The "turn up" effects in the (e,2e) cross sections of the 1b(3g) orbital compared with the plane-wave impulse approximation calculations were observed at low and high momentum regions, and such discrepancies become smaller with the increase of the impact electron energies. It is suggested that the observed discrepancies are due to the distorted-wave effects in molecules, while appropriate theoretical calculations using distorted waves in molecules could not be achieved until now.

Scattering of GeV electrons in the framework of the relativistic hartree approximation

The high momentum transfer electron-nucleus scattering cross section is evaluated within the plane wave impulse approximation (PWIA) supplemented by the relativistic Hartree approximation (RHA). Binding effects on the struck nucleon are introduced through the scalar and vector meson-exchange fields within the framework of quantum hadrodynamics. This model naturally satisfies the current conservation, with the off-shell nucleons behaving as being on the mass shell but having an effective mass. The nucleon inelastic response is included via different parameterizations of the structure function measured at SLAC, while the smearing of the Fermi surface is introduced through a momentum distribution obtained from a perturbative nuclear matter calculation. Recent CEBAF data on inclusive scattering of 4.05 GeV electrons on 56Fe are well reproduced for all measured geometries by the first time. Scaling effects are analyzed as well, and the scaling properties of the nuclear response in terms of the y variable associated to the PWIA within the RHA framework are discussed. The theoretical scaling function obtained in our approach also significantly improves previous PWIA calculations within the multi-GeV electron scattering regime, and describes properly the CEBAF scaling data.

Polarization transfer and spin response functions of theH2(p⃗,n⃗)reaction at345MeV

Differential cross sections and a complete set of polarization observables have been measured in the quasielastic $^{2}\mathrm{H}(p,n)$ reaction at a bombarding energy of $345\phantom{\rule{0.3em}{0ex}}\text{MeV}$ and laboratory scattering angles of $16\ifmmode^\circ\else\textdegree\fi{}$, $22\ifmmode^\circ\else\textdegree\fi{}$, and $27\ifmmode^\circ\else\textdegree\fi{}$. The data are compared with plane-wave impulse approximation calculations employing an optimal factorization approximation. The agreement between the experimental and theoretical results validates these approximations in the present momentum- and energy-transfer regions. The experimental spin-longitudinal and spin-transverse response functions, ${R}_{L}$ and ${R}_{T}$, respectively, are deduced from the data. The obtained ${R}_{T}$ is consistent with that obtained from the quasielastic electron scattering. The theoretical calculations with the Reid soft core potential give good descriptions for ${R}_{L}$ and ${R}_{T}$, whereas the latter is slightly underestimated around the quasielastic peak.

Plane-wave impulse approximation extraction of the neutron magnetic form factor from quasielastic3He→(e→,e′)atQ2=0.3to0.6(GeV/c)2

A high precision measurement of the transverse spin-dependent asymmetry A_T' in ^3He(e,e') quasielastic scattering was performed in Hall A at Jefferson Lab at values of the squared four-momentum transfer, Q^2, between 0.1 and 0.6 (GeV/c)^2. A_(T') is sensitive to the neutron magnetic form factor, G_M^n . Values of G_M^n at Q^2 = 0.1 and 0.2 (GeV/c)^2, extracted using Faddeev calculations, were reported previously. Here, we report the extraction of G_M^n for the remaining Q^2 values in the range from 0.3 to 0.6 (GeV/c)^2 using a plane-wave impulse approximation calculation. The results are in good agreement with recent precision data from experiments using a deuterium target.

Valence shell orbital imaging in adamantane by electron momentum spectroscopy

Measurements of electron binding energy spectra and electron momentum profiles (electron momentum distributions) for the valence orbitals of the cage molecule adamantane (C 10H 16) using electron momentum spectroscopy are reported. Theoretical momentum profiles are calculated using the plane-wave impulse approximation (PWIA) within the target Hartree–Fock and also the target Kohn–Sham approximations (density functional theory, DFT) with the 4-31G and 6-311++G** basis sets and compared with the experimental results. The Hartree–Fock and the DFT calculations using the 6-311++G** basis set are in good quantitative agreement with the experimental momentum profiles. The experimental and the calculated orbital momentum distributions confirm the tentative orbital assignments made in the earlier published studies of adamantane by photoelectron spectroscopy. Small effects at low momenta, with the measured cross-section being above the PWIA calculations for molecular orbitals of e and t 1 symmetry, are attributed to distorted-wave effects.

Quasifree(e,e′p)Reactions and Proton Propagation in Nuclei

The ( e,e′p) reaction was studied on targets of C, Fe, and Au at momentum transfers squared Q2 of 0.6, 1.3, 1.8, and 3.3GeV2 in a region of kinematics dominated by quasifree electron-proton scattering. Missing energy and missing momentum distributions are reasonably well described by plane wave impulse approximation calculations with Q2 and A dependent corrections that measure the attenuation of the final state protons.Received 24 November 1997DOI:https://doi.org/10.1103/PhysRevLett.80.5072©1998 American Physical Society

Imaging of the HOMO electron density in Cr(CO) 6, Mo(CO) 6 and W(CO) 6 by electron momentum spectroscopy: a comparison with Hartree-Fock and DFT calculations

Studies of the electron density distributions in the t 2g HOMO orbitals (essentially metal valence d character) of the transition metal carbonyls Cr(CO) 6, Mo(CO) 6 and W(CO) 6 have been made using electron momentum spectroscopy (EMS). The EMS measurements provide a test of basis set effects and the quality of theoretical methods in the description of these large systems. The experimental momentum profiles are compared with theoretical spherically averaged momentum profiles calculated using the plane wave impulse approximation (PWIA) at the level of the target Hartree-Fock approximation (THFA) with a range of basis sets. Electron density maps for the HOMO orbitals in both momentum and position space have also been calculated for the oriented molecules at the Hartree-Fock level. Further comparisons of the measured momentum profiles to PWIA calculations are made in the target Kohn-Sham approximation (TKSA) using density functional theory (DFT) with the local density approximation and also with gradient corrected exchange correlation potentials. In addition to momentum profiles, other electronic properties have also been calculated by the various theoretical methods and compared to experimental values. The experimental momentum profiles for the HOMO orbitals of all three hexacarbonyls show significant cross-section at momenta below ∼0.5 au including a large contribution near p = 0 which is not expected from symmetry considerations or predicted by either the Target Hartree-Fock or the DFT target Kohn-Sham PWIA calculations. The discrepancies may be due to the significant vibrational excitation of the initial state metal carbonyl molecules expected at room temperature. However, theoretical studies of the Cr 3d and Mo 4d atomic systems show that significant low momentum components are introduced into the EMS cross-section when the PWIA treatment is replaced with the distorted wave impulse approximation (DWIA). While DWIA calculations are not possible for molecules the results for atomic Cr 3d and Mo 4d strongly suggest that the unusual effects observed experimentally in the M(CO) 6 HOMO (essentially Cr 3d, Mo 4d and W 5d in character) cross-sections are due to distortion of the electron waves in the collision process.

Analyzing powers for the 2H(p

We measured the analyzing power ${\mathit{A}}_{\mathit{y}}$ and the triple differential cross section ${\mathit{d}}^{3}$\ensuremath{\sigma}/d${\mathrm{\ensuremath{\Omega}}}_{\mathit{pd}}$${\mathrm{\ensuremath{\Omega}}}_{\mathit{ndE}\mathit{p}}$ for the $^{2}\mathrm{H}$(p\ensuremath{\rightarrow},pn${)}^{1}$H reaction at 200 MeV. Coplanar coincidence data were taken for all combinations of neutron angles ${\mathrm{\ensuremath{\theta}}}_{\mathit{n}}$=35\ifmmode^\circ\else\textdegree\fi{}, 45\ifmmode^\circ\else\textdegree\fi{}, or 55\ifmmode^\circ\else\textdegree\fi{} with proton angles ${\mathrm{\ensuremath{\theta}}}_{\mathit{p}}$=35\ifmmode^\circ\else\textdegree\fi{}, 45\ifmmode^\circ\else\textdegree\fi{}, or 52\ifmmode^\circ\else\textdegree\fi{}. Protons were detected with a \ensuremath{\Delta}E-E telescope with a 1000-\ensuremath{\mu}m silicon surface barrier \ensuremath{\Delta}E detector and a plastic scintillator E detector. Neutrons were detected with large-volume plastic scintillators at flight paths of 17.5 or 18 m. The overall neutron separation-energy resolution was about 3 MeV. Data are compared with plane-wave impulse-approximation calculations with a Hulth\'en deuteron wave function and p-n cross sections and analyzing powers obtained from N-N phase shifts. The agreement between these calculations and the data is generally good for the cross sections. The agreement for the analyzing powers is good near the point of zero recoil momentum. Our results suggest that the deuteron is a good ``neutron target'' for recoil momenta 100 MeV/c.

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

Proton-induced nucleon knockout from polarized 3He at 220 MeV.

A high pressure polarized $^{3}\mathrm{He}$ gas target has been used to measure spin observables and absolute cross sections for (p\ensuremath{\rightarrow},2p) and (p\ensuremath{\rightarrow},pn) nucleon knockout in quasielastic kinematics at 220 MeV. The neutron knockout spin observables exhibit large deviations from plane wave impulse approximation (PWIA) calculations and show the importance of rescattering corrections. At this energy, the proton knockout spin observables differ slightly from PWIA calculations, especially at low energy transfers (\ensuremath{\omega}), but this discrepancy may be resolved when final state interactions between nucleons of the two-body recoil scattering state are included. The large rescattering effects observed in the (p\ensuremath{\rightarrow},pn) results may have implications for experiments which consider polarized $^{3}\mathrm{He}$ as a polarized neutron.