Effective Momentum Approximation Research Articles

Coulomb corrections in the extraction of the proton radius

Multi-photon exchange contributions are important in extracting the proton charge radius from elastic electron–proton scattering. So far, only diagrams associated with the exchange of a second photon have been evaluated. At the low Q2 values relevant to the radius extraction, and especially the very low Q2 region to be probed by proposed measurements, higher order contributions may become important. We evaluate these corrections in the Effective Momentum Approximation, which includes the Coulomb interaction to all orders, and find small corrections with a strong Q2 dependence at low Q2 and large scattering angles. This suggests that the higher order terms may be important in the evaluation of the proton magnetic radius.

Coulomb distortion effects in quasi-elastic [formula omitted] scattering on heavy nuclei

The influence of the Coulomb distortion for quasi-elastic ( e , e ′ ) scattering on highly charged nuclei is investigated in distorted wave Born approximation for electrons. The Dirac equation is solved numerically in order to obtain exact electron continuum states in the electrostatic field generated by the charge distribution of an atomic nucleus. Different approximate models are used to describe the nucleon current in order to show that, at high electron energies and energy–momentum transfer, the influence of Coulomb distortions on ( e , e ′ ) cross sections can be reliably described by the effective momentum approximation, irrespective of details concerning the description of the nuclear current.

Focusing of high-energy particles in the electrostatic field of a homogeneously charged sphere and the effective momentum approximation

The impact of the strongly attractive electromagnetic field of heavy nuclei on electrons in quasi-elastic (e,e') scattering is often accounted for by the effective momentum approximation. This method is a plane wave Born approximation which takes the twofold effect of the attractive nucleus on initial and final state electrons into account, namely the modification of the electron momentum in the vicinity of the nucleus, and the focusing of electrons towards the nuclear region leading to an enhancement of the corresponding wave function amplitudes. The focusing effect due to the attractive Coulomb field of a homogeneously charged sphere on a classical ensemble of charged particles incident on the field is calculated in the highly relativistic limit and compared to results obtained from exact solutions of the Dirac equation. The result is relevant for the theoretical foundation of the effective momentum approximation and describes the high energy behavior of the amplitude of continuum Dirac waves in the potential of a homogeneously charged sphere. Our findings indicate that the effective momentum approximation is a useful approximation for the calculation of Coulomb corrections in (e,e') scattering off heavy nuclei for sufficiently high electron energies and momentum transfer.

Coulomb Effect of Positron for (e,e') Reactions in Quasielastic Region

The static Coulomb field of medium and heavy nuclei has been one of persistent problems in medium energy electron scattering. In particular, in the early 1990’s, Ohio group for the ðe; e0Þ reactions treated exactly the Coulomb distortion using partial wave expansion of the electron wave function, referred to the full distorted wave Born approximation (DWBA) in quasielastic region. This calculation is able to compare various nuclear models and furnish an invaluable check for several approximate methods of the Coulomb distortion. However, it is numerically challenging and computation time increases rapidly with higher incident electron energy. In particular, it is not possible to separate the cross section into the longitudinal and the transverse structure functions. In order to avoid the difficult problems related to the DWBA calculations, there are two approaches to treat approximately the electron Coulomb distortion. The first approach is known as the effective momentum approximation (EMA). This is that the electron momenta are modified by the value of the Coulomb potential at the center of target nucleus. This approximation describes the Coulomb distortion for light nuclei and for high incident electron energies but does not for heavy nuclei and for intermediate incident electron energies. Recently, there were experimental data measured from Saclay for C and Pb with positron beam. They improved the EMA calculation with the effective momentum p0 1⁄4 p ð2=3ÞVð0Þ, where Vð0Þ is the Coulomb potential at the origin. With this method, they claimed that if one uses this new EMA the total structure functions induced by positrons have roughly the same magnitude as those by electrons as a function of the energy transfer. As the other approach, from the middle of 1990’s, Kim and Wright have developed an approximate treatment of the electron Coulomb distortion. This approximation leads to an r-dependent momentum which includes the Coulomb distortion arising from the four potential A . The advantage of this approximation is that the separation of the cross section into a longitudinal term and a transverse term is possible. This approximation of the Coulomb distortion agreed with the full DWBA results (about 1–2%) near the peaks of the cross sections for heavy nucleus like Pb on ðe; e0Þ and ðe; e0pÞ reactions with the momentum transfer greater than approximately 300MeV/c. More recently, Kim and Wright improved the above method of the r-dependent momentum for the electron (or positron) current by averaging the phase shift of the leptons, referred to ad hoc approximation (see ref. 8 in detail). They compared the ad hoc approximation, the full DWBA, and the EMA calculations for the Coulomb distortion with a simple harmonic oscillation model. They showed that the total structure functions did not have the same magnitude because of the phase shift of the leptons even though they used the same kinematics as the Saclay experiments. In this report, we examine the Coulomb distortion effect of the positron for the cross section with the ad hoc approximation. In addition, we also investigate the positron Coulomb distortion effect at high incident positron energy of 2.02 GeV for several nuclei, C, Fe, and Au, which is the same kinematics as the SLAC data for the electron scattering. We use a relativistic single particle model for the bound state based on the –! model and apply the same potential for the outgoing nucleon as the bound state coupled with a free relativistic current operator. Since only electrons are observed in the ðe; e0Þ reaction, we do not take into account whether or not the ejected nucleons are ‘‘lost’’, which is no nucleon flux loss. This assumption guarantees current conservation and contains the final state interaction. In the plane wave Born approximation (PWBA) in which the electrons or the positrons are described as plane waves, the cross section for the inclusive ðe; e0Þ scattering can be written as d d d! 1⁄4 M q q4 RLðq; !Þ þ tan 2 q 2q2 ! RTðq; !Þ # ; ð1Þ

Extraction of structure functions from quasielastic electron scattering(e,e')off medium and heavy nuclei

Using a relativistic mean-field single particle knock-out model for (e,e') reactions on nuclei, we investigate approximate treatments of Coulomb distortion effects and the extraction of longitudinal and transverse structure functions. We show that an effective momentum approximation (EMA) when coupled with a focusing factor provides a good description of the transverse contributions to the (e,e') cross sections for electron energies above 300 MeV on 208Pb. This approximation is not as good for the longitudinal contributions even for incident electron eneriges above 1 GeV and if one requires very precise extraction of longitudinal and transverse structure functions in the quasielastic region it is necessary to utilize distortion factors based on a nuclear model and a more accurate inclusion of Coulomb distortion effects.

Coulomb distortion of relativistic electrons in the nuclear electrostatic field

Continuum states of the Dirac equation are calculated numerically for the electrostatic field generated by the charge distribution of an atomic nucleus. The behavior of the wave functions of an incoming electron with given asymptotic momentum in the nuclear region is discussed in detail and the results are compared to different approximations used in the data analysis for quasielastic electron scattering off medium and highly charged nuclei. It is found that most of the approximations provide an accurate description of the electron wave functions in the range of electron energies above 100 MeV typically used in experiments for quasielastic electron scattering off nuclei only near the center of the nucleus. It is therefore necessary that the properties of exact wave functions are investigated in detail in order to obtain reliable results in the data analysis of quasielastic (e, e'p) knockout reactions or inclusive quasielastic (e, e') scattering. Detailed arguments are given that the effective momentum approximation with a fitted potential parameter is a viable method for a simplified treatment of Coulomb corrections for certain kinematical regions used in experiments. Numerical calculations performed within the framework of the single-particle shell model for nucleons lead to the conclusion that our results are incompatible with calculations performed about a decade ago, where exact electron wave functions were used in order to calculate Coulomb corrections in distorted-wave Born approximation. A discussion of the exact solutions of the Dirac equation for free electrons in a Coulomb field generated by a point-like charge and some details relevant for the numerical calculations are given in the appendix.

A(e⃗,e′p⃗)Bresponses: From bare nucleons to complex nuclei

We study the occurrence of factorization in polarized and unpolarized observables in coincidence quasi-elastic electron scattering. Starting with the relativistic distorted wave impulse approximation, we reformulate the effective momentum approximation and show that the latter leads to observables which factorize under some specific conditions. Within this framework, the role played by final state interactions and, in particular, by the spin-orbit term is explored. Connection with the nonrelativistic formalism is studied in depth. Numerical results are presented to illustrate the analytical derivations and to quantify the differences between factorized and unfactorized approaches.

Analysis of polarizedO16(e⃗,e′p⃗)observables within the relativistic distorted wave impulse approximation

Recoil nucleon transferred polarization observables in coincidence quasielastic electron scattering are studied within the relativistic distorted wave impulse approximation. Results for response functions and polarization asymmetries are discussed for proton knockout from $p_{1/2}$, $p_{3/2}$ and $s_{1/2}$ shells in $^{16}$O. The impact of spinor distortion is examined by comparing the fully relativistic calculation with results obtained by projecting out the negative-energy components. In particular, a careful analysis of effects linked to the description of the bound and scattered relativistic nucleon wave functions is presented. The high sensitivity of some polarization observables to the dynamical enhancement of the lower components, already shown within the relativistic plane wave impulse approximation, is proven to be maintained in the relativistic distorted wave approach. Semi-relativistic approaches based on the effective momentum approximation are also studied. Finally, comparison with experimental data and a brief analysis of effects linked to medium modified form factors is presented.

Inclusive quasifree electrofission cross section for238U

We present results from a joint theoretical and experimental study of inclusive quasifree electrofission of ${}^{238}\mathrm{U}.$ The off-shell cross sections for the quasifree reaction stage have been calculated within the plane wave impulse approximation with distortion corrections included in the effective momentum approximation. Proton and neutron single-particle momentum distributions were calculated in the macroscopic-microscopic approach. The fissility for proton and neutron single hole excited states of the residual nuclei ${}^{237}\mathrm{Pa}$ and ${}^{237,238}\mathrm{U}$ was calculated within the compound nucleus model. Final state interaction corrections to residual nucleus excitation energy were calculated using the imaginary part of the optical potential. The total inclusive electrofission cross section was measured with high absolute precision, and all principal partial contributions are analyzed, in particular, the quasifree one.

Is the proton electromagnetic form factor modified in nuclei?

Guided by the recent experimental confirmation of the validity of the Effective Momentum Approximation (EMA) in quasi-elastic scattering off nuclei, we have re-examined the extraction of the Longitudinal and Transverse Response Functions in medium-weight and heavy nuclei. In the EMA we have performed a Rosenbluth separation of the available world data on 40Ca, 48Ca, 56Fe and 208Pb. We find that the Longitudinal Response Function for these nuclei is ”quenched” and that the Coulomb sum is not saturated, at odds with recent claims in the literature.

Quasi-free 238U [formula omitted] cross section in macroscopic–microscopic approach

We present the result of a theoretical study of inclusive quasi-free electrofission of 238U. The off-shell cross sections for the quasi-free reaction stage have been calculated within the Plane Wave Impulse Approximation (PWIA) using a macroscopic–microscopic description of the proton and neutron single particle momentum distributions. Electron wave function distortion corrections were included using the effective momentum approximation. The fissility for the proton single hole excited states of the residual nucleus 237Pa was calculated both with and without contributions of the pre-equilibrium emission of the particles. The fissility of the 237,238U residual nuclei was calculated within the compound nucleus model. Final state interaction correction to the fissility was calculated using a pure imaginary optical potential. The (e,e′f) cross sections, thus obtained, were compared with available experimental data.

Relativistic corrections in(γ,N)knockout reactions

We develop a fully relativistic distorted wave impulse approximation (DWIA) model for photonuclear reactions using the relativistic mean field theory for the bound state, and the Pauli reduction of the scattering state, which is calculated from a relativistic optical potential. Results for the ${}^{12}\mathrm{C}(\ensuremath{\gamma},p)$ and ${}^{16}\mathrm{O}(\ensuremath{\gamma},p)$ differential cross sections and photon asymmetries are displayed in a photon energy range between 60 and 257 MeV, and compared with nonrelativistic DWIA calculations. The effects of the spinor distortion and of the effective momentum approximation for the scattering state are discussed. The sensitivity of the model to different prescriptions for the one-body current operator is investigated. The off-shell ambiguities are large in $(\ensuremath{\gamma},p)$ calculations, and even larger in $(\ensuremath{\gamma},n)$ knockout.

Coulomb distortion in quasielastic (e,e′) scattering on nuclei: Effective momentum approximation and beyond

The role of the effective momentum approximation to disentangle Coulomb distortion effects in quasielastic (e,e′) reactions is investigated. The separation of the cross section in longitudinal and transverse components is discussed including higher-order DWBA corrections due to the focusing of the electron waves. The experimental studies performed, in the last few years, making use of different approximate treatments are shown to be sometime inconsistent. As a consequence some of the longitudinal and transverse responses, extracted from the inclusive cross sections cannot be considered reliable. A separation procedure based on the effective momentum approximation is discussed in connection with the recent experimental data on electron/positron quasielastic scattering on 12C and 208Pb.

Is the Coulomb sum rule violated in nuclei?

Guided by the experimental confirmation of the validity of the Effective Momentum Approximation (EMA) in quasi-elastic scattering off nuclei, we have re-examined the extraction of the longitudinal and transverse response functions in medium-weight and heavy nuclei. In the EMA we have performed a Rosenbluth separation of the available world data on $^{40}$Ca, $^{48}$Ca, $^{56}$Fe, $^{197}$Au, $^{208}$Pb and $^{238}$U. We find that the longitudinal response function for these nuclei is "quenched" and that the Coulomb sum is not saturated, at odds with claims in the literature.

Relativistic corrections in(e,e′p)knockout reactions

A consistent comparison between nonrelativistic and relativistic descriptions of the (e,e'p) reaction is presented. We use the nonrelativistic DWEEPY code and develop a fully relativistic model starting from the effective Pauli reduction for the scattering state and the relativistic mean field theory for the bound state. Results for the 16O(e,e'p) differential cross section and structure functions are compared in various kinematical conditions. A limit in energy of the validity of the nonrelativistic approach is established. The effects of spinor distortion and of the effective momentum approximation for the scattering state are discussed. A satisfactory agreement with data of differential cross sections, structure functions, and polarization observables is achieved.

Coulomb distortion measurements by comparing electron and positron quasielastic scattering off12Cand208Pb

Positron and electron quasielastic cross sections on ${}^{12}\mathrm{C}$ and ${}^{208}\mathrm{Pb}$ were measured at the Saclay linear accelerator (ALS) to test Coulomb corrections to the plane-wave Born approximation. The positron and electron response functions at the same effective kinematics were found to be equal within a 3% experimental uncertainty, which allows us to discriminate between different theoretical models of Coulomb distortions. Our result strongly supports the effective momentum approximation, even for nuclei as heavy as ${}^{208}\mathrm{Pb}$.

Effects of spinor distortion and density-dependent form factors upon quasifree16O(e→,e′p→)

We propose an effective current operator for nucleon electromagnetic knockout that incorporates spinor distortion and density-dependent nucleon form factors using an effective momentum approximation. This method can be used in a coordinate-space approach with either relativistic or nonrelativistic optical potentials and overlap functions. We studied these effects for the $^{16}O(\vec{e},e' \vec{p})$ reaction at Q^2 = 0.8 (GeV/c)^2. Spinor distortion substantially enhances the left-right asymmetry while reducing the ratio between sideways and longitudinal recoil polarization for p-shell knockout by about 5% for modest missing momenta. We also find that the density dependence of nucleon form factors suggested by a quark-meson coupling model reduces the polarization ratio further. Much larger effects are obtained for the s-shell than for p-shell. However, both effects are subject to much larger Gordon ambiguities than comparable nonrelativistic calculations.

E-p off-shell cross section in quasielastic (e,e'p) reactions.

By carrying out calculations on $^{208}\mathrm{Pb}$(e,e'p) reactions at different electron beam energies for fixed knocked-out proton kinematics, we examined whether, even in the presence of Coulomb distortions of the electron waves, the (e,e'p) cross section can still be separated into a reduced cross section ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{red}}$(${\mathit{p}}_{\mathit{m})}$ and the e-p off-shell cross section. We found that if one uses an effective momentum approximation in calculating the e-p off-shell cross section, one gets a reduced cross section which is independent of the electron kinematics within a few percent.

Longitudinal response functions for 40Ca from quasi-elastic electron scattering

Longitudinal response functions were extracted from quasi-elastic electron scattering data on 40Ca at angles of 45.5°, 90°, and 140° with bombarding energies ranging from 130 to 840 MeV and for constant three-momentum transfers of 300, 330, 370, 410 and 450 MeV/ c. Contrary to previously reported results, the present longitudinal response functions show no more than 20% missing strength when compared to the relativistic Fermi gas model. Calculations employing wavefunctions generated from a relativistic Hartree potential and using an effective momentum approximation yield results that are in much closer agreement with the experimental data and indicate essentially no longitudinal suppression.

Density of states and conductivity in highly impure semiconductors

AbstractThe evaluation of the density of states of a semiconductor disordered by a high density of impurities is considered by path integral methods. An effective momentum approximation and a cumulant approximation in momentum space are both used and graphical results for the density of states are presented. The results are applied to the evaluation of the static electrical mobility. It is found that this can be described by a Boltzmann like and non‐Boltzmann like term and graphical results for the variation with Fermi energy of both these terms are given.