Single Scattering Cross Section Research Articles

How important is the first inelastic collision in nuclear precompound reactions?

We derive an expression of the angle- and energy-integrated cross section σ 1 for single inelastic collisions in nucleon-nucleus reactions. σ 1 is related to the strength of the imaginary part of the optical potential. Using a phenomenological optical potential we calculate the ratio σ 1/σ R, where σ R is the total reaction cross section. The ratio σ 1/σ R is about 50% for a target as light as 12C and drops to about 30% for a heavy target. The calculated values agree with experimental values for Al(p, p')X and Al(p, n')X at 90 MeV. Most of the single scattering cross section σ 1 arises from partial waves which are located on the nuclear surface.

Multiple scattering — A different approach for any cross section and any scatterer thickness

A new and simple method is shown for obtaining the small-angle multiple (or plural) scattering distribution of an incident beam of charged particles. The same method works for any physically reasonable shape of single-scattering cross section and for any thickness of scatterer (i.e., average number of scatterings anywhere from ⪡1 to ⪢1). Results are given for two forms of single scattering. One is σ a ∼ (η 2 + η a 2) −3 2 , used by Snyder and Scott. The resulting scattering distribution agrees with theirs within 1% for 100 collisions. The other (hypothetical) is σ b∼exp(− η 2/ η b 2). For σ a the tails of the multiple scattering distribution asymptotically approach the single-scattering distribution, as is known. For σ b that does not happen. Graphs and tables show quantitatively how the multiple-scattering distributions depart from a Gaussian in the central region and from single scattering in the tails. The application to atom-atom scattering is considered, and the strong forward peaking in σ is shown not to be an important problem. The method starts with an infinitesimal thickness of scatterer and produces, by successive foldings, the projected scattering distributions for thickness of 2 m times the original thickness, m = 1,2,3,…. The inversion of the projected distribution is considered.

Monte Carlo calculation of the distributions of interatomic separation and relative velocity resulting from the dissociation of fast molecular ions by thin foils

A Monte Carlo program to calculate the distributions of separation and relative velocity resulting from the dissociation of fast molecular ions traversing thin foils, including effects of Coulomb explosion, multiple scattering, and differential energy loss, has been developed. Here it is used to explore the effect of the single-scattering cross section employed on the distributions which result. It is demonstrated that different single-scattering distributions---based upon different assumptions concerning atomic screening---give rise to differing separation distributions even though they result in essentially identical multiple-scattering distributions. Such effects must be taken into account in models of subsequent molecular or atomic recombination of the exiting particles.

Neutralization in surface scattering of low energy H + by polycrystalline graphite

Backscattering H + spectra were obtained as a function of incident H + energy (600 < E<2400 eV) and of total scattering and incidence angles. Using single scattering cross sections calculated for a screened Coulomb potential, the fraction f + of hydrogen backscattered as H + in the binary scattering edge was deduced. Models including only Auger neutralization at the surface failed to describe the results. A model including collisional neutralization and ionization which fits the data indicated that 95–98% of incident H + is neutralized by the collision while 2–9% of incident (neutralized) H + is re-ionized by the collision. The Auger neutralization at the surface is described by a characteristic velocity υ c −2 −10×10 6 cms −1 The empirical formula f +=5.1× 10 −10 υf⊥ +1.6×10 −10 υi⊥ using the initial (υ i⊥) and final(υ f⊥) perpendicular velocities gives are markably good fit to all data.

Measurements of differential single-scattering cross sections of heavy ions at MeV energies on solid targets

The differential screened Rutherford-scattering cross sections of 2-15 MeV Cl, 2 and 4 MeV Br, and 5 and 15 MeV Au ions on Au atoms have been measured, utilising a transmission-scattering geometry and targets consisting of very thin (2-10 AA) layers of gold evaporated onto 150-270 AA thick carbon foils. The measured cross sections are not influenced by multiple scattering, and a careful examination of the target composition during irradiation has been performed. In outline, the measurements confirm the scaling properties inherent in the theory of Lindhard et al. (1968). Some deviations from the t1/2 scaling as well as from the actual size of the Thomas-Fermi cross section are, however, observed at t1/2 values smaller than three.

Thermal-neutron critical multiple scattering via the Glauber approximation

Glauber's high-energy approximation for scattering is shown to provide a simple description of critical multiple scattering of thermal neutrons from liquids and ferromagnets. Sufficiently close to the critical point it provides a closed-form expression for the differential cross section which includes all orders of scattering and has the same form as the single-scattering cross section with a modified exponent for the wave-vector transfer. (AIP)

The ion-atom differential single-scattering cross section in the region of weak screening measured on solid targets

Abstract An accurate method for measuring the single-scattering cross section of ions on solid targets is described and used for 14–31 MeV 81Br ions scattered through 3–12° on gold targets. The results agree with the theory of Lindhard, Nielsen and Scharff except for the lowest t ½ values, where a small deviation is observed. Comparing the present data with earlier published measurements of 0.3–2 MeV H, He, and Li ions scattered on selenium and gold, we conclude that the theoretically predicted scaling laws are fulfilled in the region of weak screening of the Coulomb interaction.

Spatial distribution of backscattered electrons in the scanning electron microscope and electron microprobe

Monte Carlo calculations have been applied to the electron scattering theory in scanning electron microscopy. These calculations are based on a single-scattering model in which the screened Rutherford expression for single-scattering cross section and the continuous energy-loss equation of Bethe are used. The surface distribution of backscattered electrons is obtained for Al and Au targets at 20 keV, with 0° and 45° incidence. From these results the exit areas of the backscattered electrons are discussed and compared with experiment. The penetration depths of the backscattered electrons are given at 0° incidence. Everhart's model for backscattering was modified, taking into account the calculated results. The proportion of the singly backscattered electrons to the total backscattered electrons was estimated by the modified model. The result shows that about half of the total backscattered electrons are caused by large-angle single-scattering events. This is consistent with the previous results of Cosslett and Thomas.

Small-angle multiple scattering of ions in the screened Coulomb region: I. Angular distributions

Bothe's formula for the angular distribution of multiply scattered particles in the small-angle limit has been evaluated numerically for ion-atom potentials of Thomas-Fermi and Lenz-Jensen type, and analytically for power potentials. Special attention is paid to a correct transition from multiple- to single-scattering distributions at small target thicknesses. In general, we find good agreement with results of Meyer that are based on essentially the same physical model, but deviations in the distributions of up to ≈40% occur. Our tabulations comprise a larger range of reduced thicknesses than Meyer's, 0.001 ⩽ τ ⩽ 2000, with a view to both lower absolute thicknesses (e.g. a monolayer) and higher ion energies. Finally, an inversion formula for extracting single-scattering cross sections from measured multiple-scattering distributions is derived.

Multiple Scattering of Protons and Nitrogen Ions on Metal Foils

The angular distributions are obtained of multiply scattered particles for various impact energies (0.4∼1.5 MeV for protons and 1.3∼4.5 MeV for nitrogen ions), foils (aluminium, copper, silver and gold) and their thicknesses. The data agree fairly well with theories of Moliere and of Keil, Zeitler and Zinn with the use of Thomas-Fermi radius which takes account of both projectile and target atom. The slight deviations found when the mean number of scatterings in the foil is smaller than about 20 are attributed to an underestimation of Moliėre's single-scattering cross section in the small angle region. The theory of Nigam, Sundaresan and Wu shows a large discrepancy from the observations, and that of Meyer also gives a little deviation when the mean number of scatterings is smaller than about 10.

Differences in Single, Plural and Multiple Scattering of Electrons and Positrons through Intermediate Angles

Experiments have been done where the elastic scattering of a divergent beam of electrons and positrons in foils of Al, Cu, Ag and Pb have been studied. The difference between the scattered fractions of the two types of particles was measured as a function of the particle energy in the range 100-1 300 keV, foil thicknesses 0.2-10 mg/cm2, and incidence angles in the intervals 15°-30° and 30°-47°. The results have been compared to available theoretical treatments and a good agreement is found between the experiments with thin scatterers and the Mott formula for single scattering, while difficulties have occurred in attempts to get a theoretical representation of the scattering in thicker foils. The latter fact may be understood from the incompletenesses in the theoretical treatments of plural and multiple scattering, in the first place the neglect of spin-orbit effects and the absence of a calculation of the real crooked paths of the particles. However, a theoretical fit to the measured differences in scattering has been obtained by assuming a larger screening parameter for positrons than for electrons, but this assumption should only be considered as a mathematical way of obtaining a larger single scattering cross section for electrons than for positrons as required by the Mott formula.

The effects of multiple scattering in small-angle nuclear scattering experiments

The Molière theory of multiple scattering by a screened Coulomb field is applied to typical data from varioys laboratories on p-nucleus scattering at small angles, and it is shown that in many cases the systematic errors due to multiple scattering are much larger than the published errors at the smallest scattering angles. A brief account of the Molière theory is given, with an extension to take into account the effects of beam size, beam divergence and counter size, provided that, together, these can be represented by a Gaussian. The Moliére theory is extended to take into account deviations from the screened Coulomb field cross section caused by nuclear effects, provided that these deviations can be expressed as an additive correction to the screened Coulomb field cross section. Finally, a method is outlined for extracting single scattering cross sections from data containing contributions from multiple scattering, the method being based on the validity of the Molière theory as a first approximation.

Scattering of 2-Bev/cMuons in Carbon and Lead

The scattering cross section of high-energy $\ensuremath{\mu}$ mesons in carbon and lead has been measured, using a pure, monoenergetic beam of muons obtained with the Bevatron at the Lawrence Radiation Laboratory. Preparation, purification, and measured properties of the beam are described. The median momentum was 2.00\ifmmode\pm\else\textpm\fi{}0.03 Bev/c, the spread in momentum was not more than \ifmmode\pm\else\textpm\fi{}3.5%, and the effective contamination due to pions was 4.9\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$. During the experiment the total number of muons incident on the apparatus was 2.5\ifmmode\times\else\texttimes\fi{}${10}^{7}$. Counter hodoscopes recorded the muons scattered from 14.4 g/${\mathrm{cm}}^{2}$ of lead and from 27 g/${\mathrm{cm}}^{2}$ of carbon. Inelastic as well as elastic processes were accepted. Scattered particles were observed at angles up to 12\ifmmode^\circ\else\textdegree\fi{} (momentum transfer \ensuremath{\sim}400 Mev/c). The lead data cover the same range as those cosmic-ray experiments which have appeared to indicate an anomalously large scattering. No anomaly is found; the lead scattering agrees closely with the distribution calculated by Cooper and Rainwater for purely electromagnetic interactions. The carbon data permit a better comparison with theoretical expectations, since one is measuring the single-scattering cross section directly, and one can account for the effects of nuclear structure rather accurately, using electron-scattering data and a detailed theoretical analysis of Drell and Schwartz. The carbon scattering results, based on 300 events in the region 70 Mev/c-400 Mev/c momentum transfer, agree closely with the Drell-Schwartz theory. The upper limits which this result places on a nonelectromagnetic scattering cross section and on a muon form factor are discussed.

Theory of Multiple Scattering: Second Born Approximation and Corrections to Molière's Work

The formula given by Moli\`ere for the scattering cross section of a charged particle by an atom, on which has been based the formula for the "screening angle" ${\ensuremath{\chi}}_{\ensuremath{\alpha}}$ in his theory of multiple scattering, has been examined and found to contain an inconsistent approximation in all orders of the parameter ${\ensuremath{\alpha}}_{1}=\frac{\mathrm{zZ}}{137\ensuremath{\beta}}$ except the lowest (the first Born approximation). In the present work, the correct expression of Dalitz is used for the single-scattering cross section of a relativistic Dirac particle by a screened atomic field up to the second Born approximation. It is found that the effect of the deviation from the first Born approximation on the screening angle is much smaller than Moli\`ere's expression for this quantity would lead one to believe. This is so because the deviation from the first Born approximation is very small at the small angles that go into the definition of the screening angle. In Moli\`ere's work, all the effect of the deviation from the first Born approximation on the distribution function $f(\ensuremath{\theta})$ for multiple scattering is contained in the quantity $B$ which depends only on ${\ensuremath{\chi}}_{\ensuremath{\alpha}}$. In the present work, it is shown that in a consistent treatment of terms of various orders in ${\ensuremath{\alpha}}_{1}$, there exist additional terms of order $\frac{\mathrm{zZ}}{137}$ in the distribution function. These terms, which represent the second Born approximation, become important at large angles. Calculations have been carried out for the scattering of 15.6-Mev electrons by Au and Be. The $\frac{1}{e}$ widths of the distribution function obtained are in good agreement with the experimental result of Hanson et al., whereas Moli\`ere's theory gives too great a width compared with the experimental value in the case of Be.

Determination of Electron Polarization by Means of Mott Scattering

The results of a numerical calculation of the polarization asymmetry factor, S( THETA ) = ( sigma ( THETA )!/sup 1/2/, and the single scattering cross section for the scattering of electrons by the unscreened Coulomb fields of gold and aluminum at energies of 75 and 121 kev are presented. The results for gold at 121 kev are compared with the Mohr and Tassie calculations for the same element and energy which include the effects of screening. This comparison (indicating a Mott asymmetry about 50% greater for the screened field at angles near 165 deg ) suggests the desirability of more detailed investigation of screening effects in order to be confident of Mott scattering as a means of measuring the polarization of electrons is this energy region. (auth)

XXII. The large-angle scattering ofμ-mesons in iron

Abstract The large-angle scattering of μ-mesons in iron has been investigated by a method similar to that reported previously for lead. The measured large-angle distributions are compared with the theories of Molière, Olbert and a modification of Olbert's theory which is based on a single scattering cross section inferred from recent electron-scattering measurements. At energies from several hundred Mev to about 4 Bev the measured distribution, at large angles, is not consistent with present Coulomb scattering theories in which the effects of the finite nuclear dimensions are included. As in the lead experiment, the number of large angles observed is in rough agreement with Moliere's theory for a ‘point’ nucleus. Some properties of the so-called anomalous scattering, which follow from the lead and iron experiments, are discussed.

On the Theory of Multiple Scattering, Particularly of Charged Particles

The general theory of the elastic multiple scattering of particles with a strongly anisotropic scattering function is investigated without making the small-angle approximation. The rigorous transport equation is used and approximations are introduced at a later stage. The paper consists of four parts. In the first part the general formulation of the problem is given. The approximations involved in the existing theories of small-angle forward scattering are discussed in some detail. In the second part the spherical harmonic method is formulated in a manner so as to permit an explicit expression for the general nth approximation. There is an ambiguity both in (a) the way of defining successive approximations and in (b) the way of introducing approximate boundary conditions. We have chosen (b) to give the best approximation to the exact solution of the Schwarzschild-Milne problem. In the third part it is shown that our choice of (a) for the spherical harmonic method leads to the same final formulas as the gaussian quadrature method. The relation of these two methods is discussed in detail. In the fourth part the problem of anisotropic multiple scattering is reduced to a quasi-isotropic one by using a generalized Goudsmit-Saunderson type distribution function (defined also for back scattering) as a first approximation. Three different methods are given for forward scattering (including large angles). The first method is a perturbation treatment. The second method is based on the approximate delta-function character of the scattering function and employs a Fokker-Planck type development for the peaked part of the scattering function. The third method is a Liouville-Neumann type of iteration applied directly to the transport equation. For back scattering the second and third of these methods also apply. In addition a special method is developed, based on the smallness of the back single scattering cross section. The generalized Goudsmit-Saunderson distribution function is developed in powers of the thickness of the scatterer and it is shown that all three methods lead to the same single scattering tail. The three methods can be applied for any single scattering law. The screened Born-Rutherford law is introduced in some cases as an example. The relation of the present work to previous theories is discussed.