Coulomb Cross Section Research Articles

Coulomb and photo cross sections for nucleon emission from relativistic 16O projectiles

Coulomb cross sections for nucleon emission out of 16O projectiles scattered with laboratory energies of 2.1, 60 and 200 GeV/nucleon on various target nuclei are calculated in first-order perturbation theory by using the corresponding photo cross sections. The photo cross sections are obtained with an 1-particle-1-hole basis in the giant resonance region and with the quasi-deuteron model up to photon energies of 140 MeV. Whereas the 2.1 GeV/nucleon results agree well with the experimental data points, the 60 and 200 GeV/nucleon cross sections are roughly one quarter lower than the measured ones, because also photons with energies higher than 140 MeV contribute at these incident energies.

Coulomb and photo cross sections for nucleon emission by relativistic heavy ions and application to 40Ar on 89Y

The Coulomb-induced emission of nucleons by relativistic heavy ions is studied with respect to partial cross sections related to the various multipole orders of the electromagnetic field. Since the partial Coulomb cross sections can be factorized into the numbers of virtual photons and the photo cross sections, we first calculate the photo cross sections using the oscillator shell model and 1-particle-1-hole configurations. Then the Coulomb cross sections are obtained by folding the photo cross sections with the numbers of virtual photons. We apply the calculations to the Coulomb scattering of 40Ar ions on 89Y targets, where experiments at E lab = 1.8 GeV/ nucleon were carried out by Mercier et al. The contributions of the various multipole orders are discussed as functions of the projectile energy and the energy range of the emitted nucleons. Also methods for the separation of the most contributing multipole orders are developed.

Oxygen detection by non-Rutherford proton backscattering below 2.5 MeV

Oxygen detection by proton backscattering has been investigated. The oxygen detection sensitivity of 2.5 MeV proton backscattering is shown to exceed that of 4He backscattering by even a factor of about 15 depending on the matrix. The needed proton elastic scattering cross sections of oxygen for θ lab = 170° have been measured in the energy range E lab = 770–2480 keV relative to Ti and Sn elastic scattering cross sections using thin TiO 2 and SnO 2 samples. The angular dependence of the cross section was measured at energies E lab = 1790, 1990, 2191, and 2382 keV for backscattering angles. The experimental cross sections were found to be 1.1–5.7 times the pure Coulomb cross section. Theoretical calculations for the scattering cross sections were performed and their inapplicability to experimental purposes is demonstrated. Fits to experimental data are given.

Eikonal representation ofN-body Coulomb scattering amplitudes

A new technique for the construction of $N$-body Coulomb scattering amplitudes is proposed, suggested by the simplest case of $N=2$: Calculate the scattering amplitude in eikonal approximation, discard the infinite phase factors which appear upon taking the limit of a Coulomb potential, and treat the remainder as an amplitude whose absolute value squared produces the exact, Coulomb differential cross section. The method easily generalizes to the $N$-body Coulomb problem for elastic scattering, and for inelastic rearrangement scattering of Coulomb bound states. We give explicit results for $N=3 \mathrm{and} 4$; in the $N=3$ case we extract amplitudes for the processes (12)+3\ensuremath{\rightarrow}1+2+3 (breakup), (12)+3\ensuremath{\rightarrow}1+(23) (rearrangement), and (12)+3\ensuremath{\rightarrow}(12)\ensuremath{'}+3 (inelastic scattering) as residues at the appropriate poles in the free-free amplitude. The method produces scattering amplitudes ${f}_{N}$ given in terms of explicit quadratures over ${(N\ensuremath{-}2)}^{2}$ distinct integrands.

Implications of cyclotron features in the X-ray spectrum of Hercules X-1

The X-ray spectra from binary pulsars is investigated through the modification of physical processes due to an intense magnetic field. An effective scattering cross section for line photons and the Coulomb cross section for electron-ion collisions, treating the electrons relativistically, are calculated. The quantization of electron orbits in strong fields are discussed. The absorption cross sections and emission rates for cyclotron photons are calculated. An application of the results to the data from Hercules X-1 is examined.

Screening effects in pair production near threshold

Pair production cross sections near threshold have been calculated numerically in partial waves for a screened central potential, permitting a comparison of theory with recent experiments. Exchange effects are found to remain small. The level of agreement with experiment is variable and does not suggest any systematic trends. A more approximate theory of screening, energy-shift screening theory (EST), identifying screened cross sections with Coulomb cross sections of shifted spectrum energy ${V}_{0}$, is in agreement with the full calculation if the proper choice is made for the energy shift ${V}_{0}$; however, the procedure becomes more delicate as threshold is approached. A previous attempt to apply EST to threshold experiments made an incorrect choice for ${V}_{0}$.

A theory of local and global processes which affect solar wind electrons 2. Experimental support

We have extended the theoretical considerations of Scudder and Olbert (1979) (hereafter called paper 1) to show from the microscopic characteristics of the Coulomb cross section that there are three natural subpopulations for plasma electrons: the subthermals with local kinetic energy E <kTc, the transthermals with kTc <E <7kTc, and the extrathermals E >7kTc. We present experimental support from three experimental groups on three different spacecraft over a radial range in the interplanetary medium for the five interrelations projected in paper 1 between solar wind electron properties and changes in the interplanetary medium: (1) subthermals respond primarily to local changes (compressions and rarefactions) in stream dynamics; (2) the extrathermal fraction of the ambient electron density should be anticorrelated with the asymptotic bulk speed; (3) the extrathermal ‘temperature’ should be anticorrelated with the local wind speed at 1 AU; (4) the heat flux carried by electrons should be anticorrelated with the local bulk speed; and (5) the extrathermal differential ‘temperature’ should be nearly independent of radius within 1 AU. From first principles and the spatial inhomogeneity of the plasma we show that the velocity dependence of Coulomb collisions in the solar wind plasma produces a bifurcation in the solar wind electron distribution function at a transition energy E*. This energy is theoretically shown to scale with the local thermal (that is, approximately the core) temperature as E*(r) ≃7kTc (r). This scaling is observationally supported over the radial range from 0.45 to 0.9 AU by Mariner 10 data and Imp data acquired at 1 AU. The extrathermals, defined on the basis of Coulomb collisions, are synonymous with the subpopulation previously labeled in the literature as the ‘halo’ or ‘hot’ component. If the transition energy should be required to equal the polarization potential energy eΦ(r) for all radii beyond the observer, (as motivated by Mariner 10 data) the thermal electrons should obey a polytrope law with index γ=7/6. In this circumstance the polarization potential is equal to the specific enthalpy: eΦ(r)=[γ/(γ ‐ 1)] kTc (r). This relation probably does not obtain in the corona, inside r <10–20Rs (cf. paper 1). In the asymptotic spherically symmetric solar wind the inverse power law index for radial variation required for the thermal electrons should be α=1/3, which is consistent with the recent in situ determinations between 0.45 and ∼3 AU. We thus provide the first self‐consistent argument for associating E*(r) with eΦ(r). This theoretical asymptotic (r → ∞) thermal electron temperature variation is between the conduction dominated (2/7) and the inviscid solution (2/5). In the proximity of the ecliptic the ratio of the thermal electron Coulomb mean free path to scale length is shown to decrease with increasing radial distance. By contrast, over the solar poles the same asymptotic temperature and density variations imply that the Coulomb thermal mean free path will increase with increasing radial distance. A straightforward extension of our model to slow time dependent situations shows that the fundamental scaling of E* with temperature and, by inference, the polytrope law for thermal electrons is theoretically maintained within compression and rarefaction regions. An illustrative calculation is given of the extrathermal density's response in stream compressions near 1 AU. These calculations show that the extrathermal electron density can either increase or decrease depending on whether the core electrons compress more nearly as an isothermal gas or an adiabatic one. On the basis of these calculations and a reexamination of published data we suggest that the extrathermal density is probably enhanced within stream compression regions. This implies (as observed) that (1) the core electrons should compress more nearly as an isothermal (γ ≃1) rather than as an adiabatic (γ ≃5/3) gas and (2) a very large electron heat flux should be observed leaving the compression region along the magnetic field. We thus conclude that the global and local Coulomb effects discussed in paper 1 and elaborated here are essential aspects of the solar wind plasma as it is observed. The experimental support which the predictions of the theory in paper 1 enjoy strongly suggests that the further development of that theory is warranted with the ultimate objective of understanding the impact that global heat transport has on the physics of coronal expansions.

K-shell X-ray production for fluorine ions on target elements Ti to La

The production of target atomK-shell X-rays has been studied for 2 to 28 MeV fluorine ions incident on thin solid targets of 14 elements with atomic numbers Z2=22 to 57. Total X-ray production cross sections, energy shifts ofKα andKβ lines andKα/Kβ intensity ratios were measured with a Si(Li) detector. The results of cross section measurements are compared with theoretical predictions of inner shell ionization. In most cases, satisfactory agreement between measured cross sections and theoretical Coulomb ionization cross sections, corrected for the perturbation of the target atom by the projectile charge and for relativistic effects, was obtained.

The acceleration of minor ion species in the solar wind

This paper provides a comprehensive analysis of the dynamics of the flow of minor ion species in the solar wind under the combined influences of gravity, Coulomb friction (with protons), rotational forces (arising from the Sun's rotation and the interplanetary spiral magnetic field) and wave forces (induced in the minor ion flow by Alfven waves propagating in the solar wind). It is assumed that the solar wind can be considered as a proton-electron plasma which is, to a first approximation, unaffected by the presence of minor ions. In the dense hot region near the Sun Coulomb friction accelerates minor ions outwards against the gravitational force, part of which is cancelled by the charge-separation electric field. Once the initial acceleration has been achieved, wave and rotational forces assist Coulomb friction in further increasing the minor ion speed so that it becomes comparable with, or perhaps even exceeds, the solar wind speed. A characteristic feature of the non-resonant wave force is that it tends to bring the minor ion flow into an equilibrium where the radial speed matches the Alfven speed relative to the solar wind speed, whereas Coulomb friction and rotational forces tend to bring the flow into an equilibrium where the radial speed of the minor ions equals the solar wind speed. Therefore, provided that there is sufficient wave energy and Coulomb friction is weak, the minor ion speed can be ‘trapped’ between these two speeds. This inteststing result is in qualitative agreement with observational findings to the effect that the differential flow speed between helium ions and protons is controlled by the ratio of the solar wind expansion time to the ion-proton collision time. If the thermal speeds of the protons and minor ions are small compared to the Alfven speed, two stable equilibrium speeds can exist because the rapid decrease in the Coulomb cross-section with increasing differential flow speed allows the non-resonant wave force to balance Coulomb friction at more than one ion speed. However, it must be emphasized that resonant wave acceleration and/or strong ion partial pressure gradients are required to achieve radial speeds of minor ions in excess of the proton speed, since, as is shown in Section 4, the non-resonant wave acceleration on protons and minor ions are identical when their radial speeds are the same, with the result that, in the solar wind, non-resonant wave acceleration tends (asymptotically) to equalize minor ion and proton speeds.

Screened Coulomb scattering in the eikonal approximation

Scattering amplitudes and cross sections for screened Coulomb potentials are examined using the eikonal approximation. We use the method of $h$ transforms to show that if the screening function is sufficiently smooth, then the screened cross section ${\ensuremath{\sigma}}_{\ensuremath{\rho}}$ approaches the Coulomb cross section ${\ensuremath{\sigma}}_{C}$ as the screening radius $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\infty}$. On the other hand, for a sharply cut-off Coulomb potential ${\ensuremath{\sigma}}_{\ensuremath{\rho}}$ does not approach ${\ensuremath{\sigma}}_{C}$ as $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\infty}$. These results agree with results obtained earlier using the Born approximation.

Cross sections for screened potentials

We consider scattering by screened Coulomb and screened short-range potentials. We prove, in Born approximation, that the cross section for a sharply cutoff Coulomb potential does not converge to the Rutherford cross section as the screening radius ρ→∞. On the other hand, for certain ’’smooth’’ screening functions we show that the screened Coulomb cross section does approach the Rutherford cross section as ρ→∞. In the case of short-range potentials, we prove (exactly, not just in Born approximation) that the screened cross section approaches the unscreened cross section as ρ→∞, whether the screening function is smooth or not.

The transfer reactions induced by 38 MeV 19F ions on 27Al

The reactions induced by 38 MeV 19F on 27Al have been studied using the Tandem Van de Graaff accelerator at Harwell together with a mass identification system based on a combination of a Buechner magnetic spectrograph and position sensitive silicon detectors in the focal plane of the magnet. Comparison of the elastic scattering measurements with Coulomb cross sections provided an absolute normalisation to the measured reaction yields. Transfer reactions were measured as a function of emitted particle energy and angle for single nucleon pick-up and stripping and for 2, 3 and 4 particle stripping. Essential features of the experimental technique are outlined and the experimental results are presented. The significance of the energy spectra and angular distributions are discussed. The results of DWBA analyses of one-nucleon transfer reactions yield spectroscopic factors for levels in 28Si, 28Al and 26Al which are in fair agreement with those obtained in other studies.

Ion-ion collisions and element separation effects in the boundary region of a plasma surrounded by neutral gas

In the boundary region between a plasma and neutral gas, the temperature is low (some thousand degrees). Then the Coulomb cross-section is large compared with other collision cross-sections. The influence of ion-ion collisions in the boundary region is discussed. In particular a simple formula is derived for the separation between different kinds of constituents. This separation depends on different collision cross-sections for diffusion and seems to be especially large when resonant charge exchange collisions are involved. These diffusive processes of importance in impermeable plasmas are also compared with separation processes due to a difference in ionization potential, which is known to occur in permeable plasmas. Different isotopes have the same ionization potential and the same charge exchange cross-sections. The separation processes described above are, therefore, likely to give element, but not isotope, separation.

Plasma screening effect in ion-ion scattering

When a plasma is extremely dense or hot, the electrons lose their efficiency for stopping heavy charged particles. The main process for their slowing down then is the small-angle ion-ion scattering, where the bare Coulomb cross section becomes infinite as ${\ensuremath{\theta}}^{\ensuremath{-}4}$. The purpose of this paper is to evaluate the collective screening effects which make the ion Coulomb cross section finite. The calculations are performed in the random-phase approximation and show that the Coulomb cross section is essentially cut off for energy transfers smaller than $\ensuremath{\hbar}{\ensuremath{\omega}}_{p}$, where ${\ensuremath{\omega}}_{p}$ is the plasma frequency of the ions.

Intermediate energy e-Li+elastic scattering in eikonal approximation

The eikonal theory of electron-atom elastic scattering recently proposed by Joachain and Mittleman (1971) is applied to e-Li+ elastic scattering. Calculations were done in the energy range 200-500 eV. Considerable deviations from the Coulomb scattering were observed. The main features of the results are the occurrence of a minimum in the ratio of the calculated to the Coulomb cross section and the enhancement for large scattering angles.

A Useful Simple Recursion Formula to Account for Inelastic Collision Effects on the Electron Distribution in a Plasma

The competitive effects of inelastic electron-atom collisions and electron-electron interactions on a sufficiently isotropic electron distribution is considered for a low temperature monatomic plasma. It is presupposed that unbalanced collisional transitions between ground state and resonance level of the plasma atoms are the dominant disturbance effect on the electron distribution. The unbalance is assumed to be due to resonance radiation escape and excitation diffusion. An analytical solution, which permits the calculation of the electron distribution even well above the ionization energy of the atomic ground state, is presented. In particular, it is demonstrated that due to the rapid decrease of the Coulomb cross section with increasing electron energy, the solution for the high energy tail of the electron distribution can be related to its form at lower energies by a simple recursion formula. The latter enables an easy computation of the ionization coefficients even for the lowest atomic energy levels. The analytical solution and its high energy approximation are numerically evaluated for the example of a low temperature -low pressure cesium plasma.

Confirmation of the effect of field-dependent scattering centers on electron mobility in gallium arsenide

In an earlier paper, measurements of the electron mobility in GaAs as a function of electric field using a magnetoresistance technique were presented which revealed variations in the mobility due in part to field-dependent scattering centers. These centers were hypothesized to be multiply ionized acceptor impurities which by their large Coulomb cross section adversely affected the electron mobility in the material. Further studies using photoconductivity techniques at liquid-nitrogen temperatures to obtain the necessary sensitivity have shown that in the above case, acceptors, probably copper, exist in the material, and can be assumed to be the source of the observed field-dependent effect. The relatively high purity of these materials, as evidenced by their good performance as microwave sources, indicates that the concentration of the impurity is of the order of 1012/cm3.

Calculation of electron slowing down spectra--II. Classical cross sections.

Classical Coulomb cross sections, which make approximate allowance for binding of the atomic electrons, have been used for calculating electron slowing down spectra. The main results of binding is to increase the flux at low energy which improves agreement with experiment. Calculations were made for four specific cases of beta ray sources in aluminum, copper and gold for which experimental results are available. In a simplified, empirical version of the theory it was found that the calculations were in reasonable agreement with experiment down to about 300 eV for all three metals. However, attempts to improve the results by more detailed consideration of the electronic structure of the metals showed that the particular classical model on which these cross sections are based is not adequate.

Elastic Scattering ofO16byTi48,Ca40,Al27,C12,Li7, andLi6

A number of isotopically enriched light targets were bombarded with ${\mathrm{O}}^{16}$ ions of 36- to 48- MeV energy (lab), and ten elastic angular distributions were measured. All cross sections show strongly destructive interference of nuclear and Coulomb scattering at angles above 30 to 60\ifmmode^\circ\else\textdegree\fi{} c.m., and fall orders of magnitude below the Coulomb cross section. The data were fitted reasonably well with conventional optical-model calculations, and $〈{\ensuremath{\chi}}^{2}〉$ values between 0.5 and 5.2 could be obtained for all targets. Typically, four-parameter fits were adequate. For the heavier targets the parameter sets are not unique, but were found to have continuous well-depth---geometry ambiguities, which were investigated in some detail. A procedure to produce in situ clean targets of highly reactive elements is described.

Nucleon emission in electron scattering

Coincidence electron scattering experiments at large energy transfer in which an emitted nucleon is detected in addition to the electron are potentially very useful since they probe the detailed structure of the nucleus. The most important process which contributes in such an experiment is quasi-elastic scattering, and we review how the cross section for this process may be used to obtain single particle momentum distributions and binding energies. We also show that for certain regions the cross section comes only from processes involving nucleon-nucleon interactions and, thus, can be used to investigate such interactions in the nucleus and possibly the ground state correlations they produce. As an order of magnitude estimate, the Coulomb cross section in such a region for a hard-sphere Fermi gas is calculated to lowest order in the gas parameter. The results indicate that such experiments should be feasible. We also discuss the general form of the double coincidence cross section in Born approximation and show that it is a linear combination of four generalized form factors which can be separated experimentally. As is the case for non-coincidence experiments, this provides a method for extracting the Coulomb cross section.