The statistical theory of multi-step compound and direct reactions

The theory of nuclear reactions is extended so as to include a statistical treatment of multi-step processes. Two types are distinguished, the multi-step compound and the multi-step direct. The wave functions for the system are grouped according to their complexity. The multi-step direct process involves explicitly those states which are open, while the multi-step compound involves those which are bound. In addition to the random phase assumption which is applied differently to the multi-step direct and to the multi-step compound cross-sections, it is assumed that the residual interaction will have non-vanishing matrix elements between states whose complexities differ by at most one unit. This is referred to as the chaining hypothesis. Explicit expressions for the double differential cross-section giving the angular distribution and energy spectrum are obtained for both reaction types. The statistical multi-step compound cross-sections are symmetric about 90°. The classical statistical theory of nuclear reactions is a special limiting case. The cross-section for the statistical multi-step direct reaction consists of a set of convolutions of single-step direct cross-sections. For the many step case it is possible to derive a diffusion equation in momentum space. Application is made to the reaction ¹⁸¹Ta(p, n)¹⁸¹W using the statistical multi-step compound formalism.     

Thermal neutron induced charged particle reactions in radioactive targets of37Ar,109Cd,125Xe,127Xe,and132Cs

Thermal neutron induced charged particle reactions in a radioactive target of³⁷Ar have been studied. Upper limits of the cross-sections for the (n, α) reaction in radioactive targets of¹⁰⁹Cd,^{125, 127}Xe, and¹³²Cs have been obtained. The isotopically pure targets were produced at the ISOLDE facility at CERN and irradiated with thermal neutrons at the high flux reactor of the Institute Laue-Langevin in Grenoble. Charged particles from (n, p) and (n, α) reactions in³⁷Ar were observed with cross-sections of 69±14b and 1970 ±330b, respectively. TheQ-values for these reactions were determined to be 1600 ±12 keV and 4630±7 keV, in agreement with existing mass data. The branching ratio Γ_α/Γ_p of the³⁷Ar capturing state was found to be 28.5±2.7. An upper limit of the cross-section for the (n, γα) reaction in³⁷Ar was obtained.     

Angular distributions and cross-sections of projectile-like fragments in the 19F + 159Tb reaction

The angular distribution of projectile-like fragments (PLFs) in the ¹⁹F + ¹⁵⁹Tb reaction have been measured at beam energy equal to 98MeV. Angular distributions of PLFs showed a systematic change with increasing mass transfer, starting from the peaking at grazing angle for heavier PLFs to very forward peaked angular distributions for lighter PLFs. Cross-sections of the different PLFs were obtained by integrating their centre-of-mass angular distributions. The PLF cross-sections have been compared with the incomplete fusion cross-sections obtained from the earlier measurement of the evaporation residue cross-section. Reduced cross-sections for lighter PLFs were observed to be higher compared to those observed in ¹⁹F + ⁶⁶Zn reaction at similar values of E _cm/V _b. Also, elastic scattering measurements were carried out to get information about the grazing angle and total reaction cross-section.     

Modified Glauber model for the total reaction cross-section of 12C + 12C collisions

Glauber's theory has been adopted to calculate the total heavy-ion reaction cross-sections at high energies. At relatively low energies, Glauber's total reaction cross-section has been modified in order to take into account the Coulomb field effect and is called modified Glauber model I. In addition to the Coulomb field effect, the nuclear effect has also been taken into account in the Glauber model and is called modified Glauber model II. An analytical expression for the transparency function for heavy-ion reactions, involving the nuclear densities of the colliding ions and the nucleon-nucleon cross- section, has been obtained within the framework of the modified Glauber models I and II. The transparency and the total reaction cross-sections of the ¹²C + ¹²C collisions are calculated at different bombarding energies. The obtained results are in good agreement with the experimental data and with previous theoretical calculations. Received: 26 January 2001 / Accepted: 15 October 2001     

Double folding model analysis of elastic scattering of halo nucleus 11Be from 64Zn

Calculations of elastic scattering cross-sections for ^9,10,11Be + ⁶⁴Zn at near-Coulomb barrier energy have been performed using a potential obtained from the double folding model and are compared with the experiment. In the framework of the double folding model, the nuclear matter densities of ^9,10,11Be projectiles and a ⁶⁴Zn target are folded with the complex energy-dependent effective M3Y interaction. The angular distributions of the differential cross-section for ^9,10Be scattering from ⁶⁴Zn at E _c.m≈ 24.5 MeV agree remarkably well with the data, while in case of ¹¹Be, calculations show a Coulomb–nuclear interference peak which is not observed in the data.     

Radiative π0 photoproduction on protons in the Δ+ (1232) region

The reaction γp → pπ ⁰γ′ has been measured with the Crystal Ball/TAPS detectors using the energy-tagged photon beam at the electron accelerator facility MAMI-B. Energy and angular differential cross-sections for the emitted photon γ′ and angular differential cross-sections for the π⁰ have been determined with high statistics in the energy range of the Δ⁺ (1232)-resonance. Cross-sections and the ratio of the cross-section to the nonradiative process γp → pπ ⁰ are compared to theoretical reaction models, having the anomalous magnetic moment κ _Δ as free parameter. As the shape of the experimental distributions is not reproduced in detail by the model calculations, currently no extraction of κ _Δ is feasible.     

On-line γ-ray investigation of the14N+27Al reaction at 30 MeV/u

The cross-sections of target-like residues following the¹⁴N+²⁷Al reaction at 30 MeV/u have been measured from in-beam and radioactiveγ-ray spectra. The recoil velocities of some fragments have been estimated from theγ-ray Doppler broadening.     

Multistep direct analysis of (p, α) reactions to the continuum

The double differential cross-sections for the (p,α) reaction have been measured for the isotope pairs^90,91Zr at 24.02 MeV and^142,143Nd at 23.1 MeV. The cross-sections are separated into compound nucleus and multistep direct components by the subtraction method and analyzed using the Hauser-Feshbach and Feshbach-Kerman-Koonin theories. The direct interaction is assumed to take place by the alpha-particle knockout process. The extent to which the unpaired neutrons in⁹¹Zr and¹⁴³Nd act as spectators in the reaction is studied and the role of pairing effects assessed.     

Measurement of 232Th(n, γ) and 232Th({n}, {2n}) cross-sections at neutron energies of 13.5, 15.5 and 17.28?MeV using neutron activation techniques

The ²³²Th(n, ??) reaction cross-section at average neutron energies of 13.5, 15.5 and 17.28?MeV from the ⁷Li(p, n) reaction has been determined for the first time using activation and off-line ??-ray spectrometric technique. The ²³²Th(n, 2n) cross-section at 17.28?MeV neutron energy has also been determined using the same technique. The experimentally determined ²³²Th(n, ??) and ²³²Th(n, 2n) reaction cross-sections from the present work were compared with the evaluated data of ENDF/BVII and JENDL-4.0 and were found to be in good agreement. The present data, along with literature data in a wide range of neutron energies, were interpreted in terms of competition between ²³²Th(n, ??), (n, f), (n, nf) and (n, xn) reaction channels. The ²³²Th(n, ??) and ²³²Th(n, 2n) reaction cross-sections were also calculated theoretically using the TALYS 1.2 computer code and were found to be in good agreement with the experimental data from the present work but were slightly higher than the literature data at lower neutron energies.     

Quasiclassical trajectory calculations of the isotopic effect on cross-sections of reactions O(1D) + HCl (DCl,TCl)

The paper reports on a quasiclassical trajectory (QCT) study of the O(¹D) + HCl (DCl, TCl) reactions, with emphasis on the isotopic effect on cross-sections and branching ratio. The calculations were performed on the ground 1 ¹ A' potential energy surface (PES). The cross-sections and branching ratios of the O(¹D) + HCl (DCl, TCl) reaction have been calculated at the collision energies (E _col) of 3–15 kcal/mol. Obvious differences have been found in the energy dependences of the reaction cross-sections and branching ratios among these isotopic reactions.     

UV Subshell photoionisation cross-section of atomic Zn,Cd, and Hg: Experiment and theory

Partial photoionisation cross-sections have been measured at photon wavelengths of 584, 304, 256 and 243 Å for the valence d and s shells of zinc, cadmium and mercury atoms. These cross-sections have also been calculated using the GIPM, a simple one-electron method which obtains the final-state potential by an inversion of the Schrödinger equation for the initial orbital of the photoelectron in the Hartree-Fock wave function. Comparison shows good qualitative agreement between theory and experiment. The quantitative agreement is typically within a factor of 2. Angular asymmetry parameters have also been calculated and the branching ratio for the splitting of the d-subshell cross-section into ²D_{$\text{5}\text{2}$} and ²D_{$\text{3}\text{2}$} contributions has been measured.     

Elastic scattering and fusion cross-sections in 7Li + 27Al reaction

With an aim to understand the effects of breakup and transfer channels on elastic scattering and fusion cross-sections in the ⁷Li + ²⁷Al reaction, simultaneous measurement of elastic scattering angular distributions and fusion cross-sections have been carried out at various energies (E _lab?=?8.0–16.0 MeV) around the Coulomb barrier. Optical model (OM) analysis of the elastic scattering data does not show any threshold anomaly or breakup threshold anomaly behaviour in the energy dependence of the real and imaginary parts of the OM potential. Fusion cross-section at each bombarding energy is extracted from the measured α-particle evaporation energy spectra at backward angles by comparing with the statistical model prediction. Results on fusion cross-sections from the present measurements along with data from the literature have been compared with the coupled-channels predictions. Detailed coupled-channels calculations have been carried out to study the effect of coupling of breakup, inelastic and transfer, channels on elastic scattering and fusion. The effect of 1n-stripping transfer coupling was found to be significant compared to that of the projectile breakup couplings in the present system.     

Interaction of loosely bound radioactive 7Be and stable 7Li with 9Be

Quasielastic scattering angular distributions have been measured for the ⁷Be + ⁹Be system at E _lab = 17 , 19 and 21MeV in the angular range $ \theta_{{cm}}^{}$ = 24^° - 57^° . An optical model (OM) analysis of these data has been carried out in order to extract optical potential parameters and reaction cross-sections. One-proton stripping cross-sections were also measured for this system at E _lab = 19 and 21MeV. These transfer angular-distribution data were compared with the finite-range distorted-wave Born approximation (FRDWBA) calculations. For the ⁷Li + ⁹Be system quasielastic scattering angular distributions were measured and emitted light charged particles were detected at E _lab = 15.75 , 24.00 and 30.00MeV in the angular range $ \theta_{{cm}}^{}$ = 7^° - 70^° . Fusion cross-sections were obtained by reproducing the measured $ \alpha$ -evaporation spectra from the compound nucleus at backward angles with the statistical model calculations. The ratios of the experimental fusion cross-sections to the total reaction cross-sections (obtained from OM analysis) were found to be small. This result suggests that the break-up process has a strong influence on the fusion process leading to a reduction in the fusion cross-section.     

Scattering of fast neutrons on238U,average energy and angular distributions of fission neutrons

Angular distributions of neutrons elastically and inelastically scattered from²³⁸U have been measured with a time-of-flight spectrometer at seven incident neutron energies between 1.5 and 5.5 MeV. Inelastic angular distributions for groups of unresolved levels are given for incoming neutron energies of 1.5, 1.9 and 2.3 MeV. The corresponding neutron cross-sections were obtained relative to then-p scattering cross-sections. The average energies and angular distributions of the fission neutrons were extracted from the measured fission neutron spectra at 1.5,1.9 and 2.3 MeV. Cross-section calculations based on a spherical optical model have shown to be inadequate to describe the neutron-nucleus interaction in case of strong nuclear deformation. The experimental reality may be better approached, instead, if the calculations are made using a potential which takes into account the deformation of the target nucleus. Some of the present measurements are interpreted in this theoretical perspective.     

Charged lepton production from iron induced by atmospheric neutrinos

The charged current lepton production induced by neutrinos in ⁵⁶Fe nuclei has been studied. The calculations have been done for the quasielastic as well as the inelastic reactions assuming Δ-dominance and take into account the effect of Pauli blocking, Fermi motion and the renormalization of weak transition strengths in the nuclear medium. The quasielastic production cross-sections for lepton production are found to be strongly reduced due to nuclear effects, while there is about 10% reduction in the inelastic cross-sections in the absence of the final-state interactions of the pions. The numerical results for the momentum and angular distributions of the leptons averaged over the various atmospheric-neutrino spectra at the Soudan and Gran Sasso sites have been presented. The effect of nuclear-model dependence and the atmospheric-flux dependence on the relative yield of μ to e has been studied and discussed.     

Angular distributions of elastic and quasi elastic heavy-ion collisions. pattern analysis

From classical in-beam spectroscopy on the¹⁰⁶Cd+¹²C reaction, unambiguousγ-ray assignments have been done for¹¹⁶Xe,¹¹⁶I and¹¹⁶Te nuclei. The ground band levels observed in¹¹⁶Xe have been compared to those of heavier even-even xenon isotopes and to the IBA theoretical predictions. In addition, using the newγ-ray assignments, evaporation cross-sections of the¹¹⁸Xe compound nucleus have been estimated and compared to the evaporation model calculations.     

Fluctuations in large-angle πp elastic scattering at 5 GeV/c

In a re-analysis of large-angle π^±p elastic scattering cross-sections at 5 GeV/c the data were divided in two momentum bins separated by approximately 80 MeV/c. For π^±p the differential cross-sections are found to fluctuate considerably between the two momentum bins, whereas for π^?p no fluctuations are observed. The result is interpreted as possible evidence for Ericson fluctuations.     

InclusiveK *0(890) and\bar K*^0 (890) production on beryllium byK − and π− at 175 GeV

The reactions \(K^ - Be \to {}^(\bar K^) *^0 (890)X,\pi ^ - Be \to {}^(\bar K^) *^0 (890)X\) , have been studied in a 175 GeV unseparated hadron beam in the kinematic range 0<x _F <1.0 andp _T ² <5 GeV². Integrated cross-sections and the dependence of the cross-sections on the longitudinal and transverse momentum are presented, together with quark counting rules predictions. The nuclear dependence ofK ^? fragmentation intoK ^*0(890) with respect to Feynmanx is investigated from hydrogen to beryllium.     

The cross-section investigation of (n, α) reaction on chromium isotopes

Cross-sections of the (n, α) reaction on chromium isotopes are of great importance for determining the radiation hardness of stainless steel, which is widely used in nuclear reactors. The EXFOR library contains a limited amount of the relevant experimental data, mainly for the reactions induced by neutrons with an energy around 14 MeV. There are no experimental data on the probability of the (n, α) reaction at the fission neutron energies. In this work we present results of investigating the ⁵⁰Cr(n, α) and ⁵²Cr(n, α) reactions. It is shown that the ENDF/B VII library evaluations for the ⁵⁰Cr(n, α) reaction are about 20 times as high as our experimental data. In addition, a structure that was predicted in none of the libraries is observed in the excitation function of the ⁵⁰Cr(n, α) reaction. The ⁵²Cr(n, α) reaction cross-sections at reactor neutron energies are obtained for the first time.     

Spectroscopy of198,199Po

Excited states in^198,199Po have been studied by in-beamγ-ray spectroscopy. These nuclei were produced in the 4n and 5n reaction channels for 115 MeV²⁰Ne on¹⁸³W-targets. Multipolarities and mixing ratios ofγ-transitions were determined from DCO-ratios, leading to spin/parity assignments. In¹⁹⁸Po,γ-cascades have been observed above several excited levels that are similar to the vibration-like structure built on the ground state. In¹⁹⁹Po, the coupling of the oddi13/2 neutron seems to favor the spherical shape of the even core; no levels resembling the intruder states of the even nuclei are found. A semi-quantitative evaluation of the cross-sections for nuclei produced in fission and multi-particle transfer is presented.