Nonrelativistic Impulse Approximation Research Articles

Low-momentum-transfer nonrelativistic limit of the relativistic impulse approximation expression for Compton-scattering doubly differential cross sections and characterization of their relativistic contributions

The nonrelativistic (nr) impulse approximation (NRIA) expression for Compton-scattering doubly differential cross sections (DDCS) for inelastic photon scattering is recovered from the corresponding relativistic expression (RIA) of Ribberfors [Phys. Rev. B 12, 2067 (1975)] in the limit of low momentum transfer ($q\ensuremath{\rightarrow}0$), valid even at relativistic incident photon energies ${\ensuremath{\omega}}_{1}>m$ provided that the average initial momentum of the ejected electron $\ensuremath{\langle}{p}_{i}\ensuremath{\rangle}$ is not too high, that is, $\ensuremath{\langle}{p}_{i}\ensuremath{\rangle}$ $<m$. This corresponds to a binding energy ${E}_{b}<10$ keV. This $q\ensuremath{\rightarrow}0$ nr limit is simultaneous with the approach of the scattering angle $\ensuremath{\theta}$ to ${0}^{\ifmmode^\circ\else\textdegree\fi{}}$ ($\ensuremath{\theta}\ensuremath{\rightarrow}{0}^{\ifmmode^\circ\else\textdegree\fi{}}$) around the Compton peak maximum. This explains the observation that it is possible to obtain an accurate Compton peak (CP) even when ${\ensuremath{\omega}}_{1}>m$ using nr expressions when $\ensuremath{\theta}$ is small. For example, a $1%$ accuracy can be obtained when ${\ensuremath{\omega}}_{1}=1\phantom{\rule{0.3em}{0ex}}\text{MeV}$ if $\ensuremath{\theta}<{20}^{\ifmmode^\circ\else\textdegree\fi{}}$. However as ${\ensuremath{\omega}}_{1}$ increases into the MeV range, the maximum $\ensuremath{\theta}$ at which an accurate Compton peak can be obtained from nr expressions approaches closer to zero, because the $\ensuremath{\theta}$ at which the relativistic shift of CP to higher energy is greatest, which starts at ${180}^{\ifmmode^\circ\else\textdegree\fi{}}$ when ${\ensuremath{\omega}}_{1}<300$ keV, begins to decrease, approaching zero even though the $\ensuremath{\theta}$ at which the relativistic increase in the CP magnitude remains greatest around $\ensuremath{\theta}={180}^{\ifmmode^\circ\else\textdegree\fi{}}$. The relativistic contribution to the prediction of Compton doubly differential cross sections (DDCS) is characterized in simple terms using Ribberfors further approximation to his full RIA expression. This factorable form is given by $\mathrm{DDCS}=\mathit{KJ}$, where $K$ is the kinematic factor and $J$ the Compton profile. This form makes it possible to account for the relativistic shift of CP to higher energy and the increase in the CP magnitude as being due to the dependence of $J({p}_{\mathrm{min}},{\ensuremath{\rho}}_{\mathrm{rel}})$ (where ${p}_{\mathrm{min}}$ is the relativistic version of the $z$ component of the momentum of the initial electron and ${\ensuremath{\rho}}_{\mathrm{rel}}$ is the relativistic charge density) and $K({p}_{\mathrm{min}})$ on ${p}_{\mathrm{min}}$. This characterization approach was used as a guide for making the nr QED $S$-matrix expression for the Compton peak kinematically relativistic. Such modified nr expressions can be more readily applied to large systems than the fully relativistic version.

Phenomenological insight into the space-like proton elastic form factor using the deuteron impulse approximation

The analysis of the JLab proton polarization data on the ratio {mu}{sub p}G{sub Ep}(t)/G{sub Mp}(t) in the framework of the unitary and analytic model of nucleon electromagnetic structure implicates surprising nondipole behavior of the proton electric form factor G{sub Ep}(t) in the space-like region with the zero around the momentum transfer squared value t=-Q{sup 2}=-13 GeV{sup 2}. This result differs from the unpolarized cross-section measurements giving the dipole behavior of G{sub Ep}(t) by means of the Rosenbluth technique. Recent investigations demonstrate that inclusion of a two-photon mechanism could solve the problem, which plays an important role for the unpolarized cross section, but it is negligible for the polarization ratio. Here, investigating a size of a difference in a prediction of the deuteron elastic structure functions A(t),B(t) by both, the nondipole and the old dipole, behaviors of G{sub Ep}(t) in space-like region through the nonrelativistic impulse approximation, we have come to the conclusion that the two-photon mechanism plays much less of a role for the unpolarized electron-deuteron elastic scattering than for the unpolarized electron-proton elastic scattering.

Whole-atom Compton scattering cross-sections and individual shell cross-sections for the biological elements in the energy region from 1 to 4 keV

Whole-atom Compton scattering cross-sections are evaluated for selected biological elements in the energy region from 1 to 4keV and from 5keV to 10MeV, for the other elements. These cross-sections were calculated within the non-relativistic impulse approximation. The Compton profiles utilized here were taken from the tables of Biggs et al. Total cross-section for fixed incident photon energy were obtained by summing the sub-shell contributions.

Measurement of Compton scattering on bound electrons by the coincidence method

A review of results of the new method for measuring the Compton scattering on bound electrons in germanium, introduced by the presented authors, is given. It is based on the application of two detectors that operate in the coincidence mode. One detector is used as the scatterer and the other as the detector of scattered radiation. Two conditions, simultaneity of pulses from the two detectors and constant energy sum, result in very clean spectra in broad energy regions. Normalization of the Compton spectra to the Ge K X-ray escape peaks, which are measured simultaneously with the Compton spectrum, gives reliable double-differential Compton-scattering cross sections on an absolute scale. Several versions of the impulse approximation are compared to the cross sections obtained by the present method for incident photon energies in the range from 60 to 105 keV. The non-relativistic impulse approximation gives the best agreement to the experimental data. We point out the suitability of the new method for an investigation of the incoherent scattering function at small photon momentum transfer.

Analyzing powers for exclusive1s1/2proton knockout from light nuclei

Analyzing power data are reported for exclusive proton-induced proton knockout from the $1{s}_{1/2}$ states of 10 target nuclei ranging from $^{2}\mathrm{H}$ to $^{19}\mathrm{F}$ for an incident energy of 392 MeV. Compared to free proton-proton scattering, the data are significantly suppressed, the amount of suppression increasing monotonically as a function of increasing separation energies for the knocked-out protons. It is also possible to consider that this suppression increases monotonically as a function of the increasing effective mean density for most of the target nuclei, but data for $^{3}\mathrm{He}$ and $^{4}\mathrm{He}$ targets clearly deviate from such systematic change. The data are compared to model predictions based on the nonrelativistic and relativistic distorted-wave impulse approximation. Both relativistic plane wave model and a nonrelativistic distorted-wave model incorporating a relativistic correction associated with an effective nucleon mass predict a significant suppression of the analyzing power, but the magnitude of the suppression is not sufficient to explain the experimental data. However, a relativistic distorted wave model predicts values that are closer to the data, but the result is inconclusive in this work since recoil corrections are neglected.

Experimental determination of absolute-scale compton cross sections using the K X-ray escape and a comparison with three versions of the impulse approximation

Double-differential Compton cross sections at two incident photon energies of 68.9 and 70.8 keV (mercury Kα X-rays) at the scattering angle of about 172° were measured in germanium using the coincidence technique with a detector as the scatterer. The cross sections were determined by normalization of the Compton spectra to the peaks due to the escape of characteristic Kα and Kβ X-rays from the target detector. This new approach of determination of absolute-scale Compton cross sections can also be applied in widely used single-mode measurements (source–scatterer–detector assembly). Our analysis shows that the new method is especially convenient and accurate at lower incident photon energies above the K-edge in the target atoms. The experimental results are compared with the non-relativistic impulse approximation, the frequently used simplified version of the relativistic impulse approximation and the non-relativistic impulse approximation used with the relativistic expression for the atomic electron momentum in the direction of the photon momentum transfer. Contrary to our expectation, the non-relativistic impulse approximation clearly gives the best agreement with the experimental data in the region of the Compton peak.

Compton scattering cross-sections for individual subshells for a few elements of biological interest in the energy region 5 keV–10 MeV

Total Compton scattering cross-sections for the individual subshells of a few elements (H, C, N, O, Na, Mg, P, S, Cl, K, Ca, Fe) that are the predominant constituents of biological materials are evaluated for photon energies from 5 keV to 10 MeV. These cross sections were calculated within the nonrelativistic impulse approximation, utilizing tabulated values of the Compton profile. These data are expected to be of use in detailed simulations of photon transport in matter. Cross sections for the whole atom, obtained by summing over the subshell data, are shown to be in agreement with previous tabulations.

Relativistic versus nonrelativistic optical potentials inA(e,e’p)Breactions

We investigate the role of relativistic and nonrelativistic optical potentials used in the analysis of (e,e'p) data. We find that the relativistic calculations produce smaller (e,e'p) cross sections even in the case in which both relativistic and nonrelativistic optical potentials fit equally well the elastic proton-nucleus scattering data. Compared to the nonrelativistic impulse approximation, this effect is due to a depletion in the nuclear interior of the relativistic nucleon current, which should be taken into account in the nonrelativistic treatment by a proper redefinition of the effective current operator.

Relativistic impulse approximation treatment of the elastic scattering of 400 MeV pi +/- on 28Si.

The elastic scattering observables for 400 MeV [pi][sup [plus minus]] on [sup 28]Si are calculated using a relativistic impulse approximation Kemmer-Duffin-Petiau formalism to calculate the [pi][sup [plus minus]] nucleus optical potential, and the Kemmer-Duffin-Petiau wave equation of motion with the aforementioned potential. The results are compared with recent experiments and with the results of nonrelativistic impulse approximation treatments.

Ionization of two-electron systems by Compton scattering of a photon.

Photoionization of two-electron systems is studied. The nonrelativistic impulse approximation is applied to treat the Compton ionization contributions. For asymptotically high energies this treatment formally reduces to the generalized shake theory and predicts a limit for the ratio of double to single ionization different from that for photoeffect. At finite energies the ratio for He, which includes both Compton and photoeffect contributions, is marginally in agreement with experimental results but it disagrees with previous theoretical calculations.

Deuteron photodisintegration up to 200 MeV: (I). The theoretical model

We develop a theoretical model for deuteron photodisintegration by exploiting the identity connecting the matrix elements of the divergence of the internal-convection current density, the charge density for the nonrelativistic impulse approximation and the potential used to calculate the wavefunctions. This identity leads to a good approximation scheme for calculating the electric multipole amplitudes, which enables one-pion exchange to be treated consistently for both the electric and magnetic amplitudes. Full details are given of all the contributions to the multipole amplitudes which are taken into account in calculating the reaction observables.

Absolute measurement of the differential cross section for deuteron photodisintegration from 63 to 71 MeV.

The absolute differential cross section for the $^{2}\mathrm{H}$(\ensuremath{\gamma},p)n reaction has been measured using a large solid angle detector, with a tagged photon beam of mean energy 67 MeV. The data have been compared with nine different theoretical calculations of the cross section, which account for the data to varying degrees. Best agreement is obtained for models utilizing the nonrelativistic impulse approximation with a realistic nuclear potential, meson-exchange currents, and relativistic corrections.

Determining the radiofrequency dipole and quadrupole effects in quarter wave resonators

The asymmetry of a quarter wave resonator (QWR) around the beam axis z causes deflection and additional (de)focusing of the beam: these effects are immediately related to the first terms, the dipole and the quadrupole, in a Fourier-Bessel multipole expansion of the quasistatic potential φ; the non-relativistic impulse approximation for heavy ion acceleration is given. General features of multipole dependencies from beam phase and velocities are shown. Since the electric field square modules | E| 2 is measurable with the bead technique, an iterative determination of E x and E y from E z and the potential φ is proposed here; the noise contribution is not amplified at the shortest wavelength Fourier components, and it can be filtered. Field errors are reduced by imposing the fit to the correct analytic form inside the drift tubes, where the fields exponentially decrease. The preliminary set of data relative to a QWR with an ogiva central rod is treated as an example, showing that dipole effects have a zero near the beam velocities optimal for acceleration.

Proton-nucleus scattering and density dependent meson masses in the nucleus.

The influence of density dependent meson masses, proposed in many previous papers, on proton-nucleus scattering in the several hundred MeV region is investigated. Introduction of this density dependence is found to remove the nuclear radius discrepancy which persistently occurred in analyses with the nonrelativistic impulse approximation when empirical nuclear densities obtained from electron scattering were employed. Nonlocality, suggested by earlier derivations of the density dependent masses, had to be introduced. Taking the estimated range of nonlocality from the earlier calculations, agreement with experiment was significantly improved by this introduction. Effects on the spin-orbit interaction and imaginary part of the optical model are qualitatively discussed. The density dependent meson masses aid in increasing the strength of the spin-orbit interaction. Preliminary results calculated in a relativistic impulse approximation (IA2) formalism for the spin observables are shown and discussed.

Empirical density-dependent effective interaction for nucleon-nucleus scattering at 500 MeV.

We report new cross-section and analyzing-power data for the excitation by 498-MeV protons of all narrow normal-parity states of {sup 16}O below 12.1-MeV excitation. In addition, spin-rotation measurements for elastic scattering and depolarization measurements for the 1{sub 1}{sup {minus}}, 2{sub 1}{sup +}, and 3{sub 1}{sup {minus}} states of {sup 16}O have been performed. These data are used in conjunction with existing data for {sup 40}Ca to study medium corrections to the effective interaction for nucleon-nucleus scattering at 500 MeV. Systematic differences between the data and nonrelativistic impulse approximation calculations based upon either the free {ital t} matrix or a recent density-dependent effective interaction are interpreted within the framework of the local-density approximation. An empirical effective interaction has been constructed which parametrizes the density dependence of the medium modifications in a simple form amenable to phenomenological analysis of data.

Effective interaction for 16O(p,p') at Ep=318 MeV.

We report new cross-section and analyzing power data for the excitation by 318 MeV protons of 20 states of $^{16}\mathrm{O}$ below 14 MeV. The data for normal-parity isoscalar excitations are compared with calculations based upon the nonrelativistic impulse approximation using both density-independent and density-dependent interactions in the local-density approximation. These comparisons clearly indicate that density-dependent modifications of the effective interaction are required. Although the Paris-Hamburg, Nakayama-Love, and Ray interactions all predict significant medium modifications, none describes the data particularly well. Therefore, we fitted an empirical effective interaction to the data for five inelastic transitions simultaneously using a self-consistency cycle that maintains consistency between transition potentials and microscopic distorting potentials. This interaction describes the inelastic data as well as elastic data and data for $^{40}\mathrm{Ca}$ which were not included in the fitting procedure. Therefore, the local-density approximation is supported by the phenomenology. However, medium modifications of the effective interaction are substantially stronger than predicted and persist to low density. The optical potentials are similar to the Schr\"odinger-equivalent potentials from the relativistic IA2 model, showing that the equivalent density dependence due to virtual NN\ifmmode\bar\else\textasciimacron\fi{} pairs is comparable to the short-range repulsion fitted to inelastic-scattering data.

Polarized-proton elastic scattering from polarized 13C.

The p\ensuremath{\rightarrow}${+}^{13}$C\ensuremath{\rightarrow} (polarized target) elastic-scattering spin observables ${\mathit{A}}_{000\mathit{n}}$ (target analyzing power) and ${\mathit{A}}_{00\mathit{n}\mathit{n}}$ (spin-correlation parameter) were determined at 497.5 MeV over the laboratory angular range 12\ifmmode^\circ\else\textdegree\fi{}--30\ifmmode^\circ\else\textdegree\fi{} with statistical uncertainties typically \ifmmode\pm\else\textpm\fi{}(0.02--0.06). Results of distorted-wave Born approximation calculations, based on either the relativistic or the nonrelativistic impulse approximation, are in reasonable agreement with these new data.

Comparison of relativistic and nonrelativistic approaches to the collective model treatment of p+40Ca inelastic scattering.

Elastic and inelastic observables for the scattering of 500 MeV protons from $^{40}\mathrm{Ca}$ are calculated using either the first-order relativistic or nonrelativistic impulse approximation for the diagonal potentials and a collective model for the transition potentials. The calculations are done using either Dirac or Schr\"odinger coupled channel programs. The relativistic approach reproduces the inelastic observables better than the first-order nonrelativistic impulse approximation model.

The use of current conservation in the calculation of deuteron photodisintegration

We argue that it is not possible to use the current conservation law in its general form in the calculation of the electric multipole amplitudes for deuteron photodisintegration. It is possible, however, to use the well-known identity (for which we give the exact kinematics) relating the internal convection current density, the charge density for the nonrelativistic impulse approximation and the potential used to calculate the wavefunctions. We show that this identity leads to a reasonable approximation scheme for the calculation of the electric multipole amplitudes.

Analyzing powers for p

High quality analyzing power data ({Delta}{ital A}{sub {ital y}}{le}0.05) for 800 MeV {sup 208}Pb({ital {rvec p}},{ital p}{prime}) scattering to the 0{sup +} ground state and to the 3{sup {minus}} 2.61 MeV collective state were obtained for {theta}{sub c.m.}=22{degree} to 37{degree}, extending the range of momentum transfer out to 4.7 fm{sup {minus}1}. Theoretical predictions for these data were obtained using analyses based on both a nonrelativistic impulse approximation approach and a relativistic impulse approximation approach. For both the elastic and inelastic analyzing powers, the relativistic impulse approximation approach provided a better description at these large momentum transfers.