Approximate normalized wave functions for the finite-size Coulomb problem

Using perturbation methods developed previously for the finite-size energy shift, thes-state wave functions for a bound lepton in the Coulomb field of a nucleus with an arbitrary charge distribution are developed through order (Zα)². This technique allows a determination of the finite-size contribution to the normalization of the wave function up to order (Zα)², both for small separations of the lepton and nucleus and for the lepton outside the nucleus. General features of the wave function are discussea, including the transition of the Dirac problem to the (singular) point charge limit. A practical application is developed using these results.     

Division of nuclear charge in the thermal neutron fission of U235

The mean primary nuclear charges of fragments from thermal neutron fission of U²³⁵ as a function of initial mass in the range 88–105 have been determined from theK-ray energy spectra of the light fragments.K- rays were registered with an argon-methane filled proportional counter in coincidence with the pulses from a pair of semiconductor detectors for complementary fission fragments. The deviation of the mean primary charge of the fragments from the “unchanged charge density” value as compared to the density of the parent nucleus U²³⁶ was found to be 0.54±0.14 charge units independent of mass in the range 88–105. No closed shell effect on the mean primary charge was found. Within about 10^?9 sec after fission aK-X ray yield of 0.057±0.012 per fission in the light fragment group was measured. The yields are nearly independant of mass in the range 88–95 corresponding to a value of 0.04 per fragment and increase up to 0.09 in the mass range 95–104, the relative accuracy being 3 to 4%.     

Quasi-elastic nucleon transfer and single charge exchange in 48Ti + 42Ca collisions

Single nucleon transfer and single charge exchange have been studied for quasi-elastic collisions of ⁴⁸Ti and ⁴²Ca at E_lab = 240, 300 and 385 MeV. Specific features below 12 MeV excitation energy are displayed in all of the measured particle energy spectra for each channel. These have been accounted for in the case of nucleon transfer by a one-step direct transfer mechanism in which all of the available hole and bound particle states in the donor and acceptor nucleus, respectively, have been included. This core excitation model was extended to single charge exchange (SCX) two-step transfers in which a proton and a neutron are exchanged. The absolute magnitude and distribution of strength in the SCX channel was obtained. The contributions from direct charge exchange via the isovector parts of the NN interaction has also been investigated in a microscopic model and is found to contribute of the order of 1–5% to the charge exchange cross section. The distribution of Fermi and Gamow-Teller strength is discussed.     

Theory of mean primary charge distribution in low energy fission of even-even nuclei

A molecular model of fission is introduced in which the compound nucleus before scission is described. The ground state wave function is assumed to be a BCS wave function. The single particle wave functions are expanded in terms of eigenfunctions of the unperturbed spherical fragments. The BCS wave function is determined from the minimum condition for the total energy. The fragment masses, the centers of mass, and the total proton and neutron numbers are kept constant. The resulting BCS and Hartree-Bogoliubov equations are solved approximately within the framework of an extended Nilsson model. The numerical results for the charge distribution in low energy fission of²³⁶U are in agreement with experiments. The heavier (lighter) fragment has on the average 0.5 protons less (more) than expected on the basis of the socalled unchanged charge distribution (UCD). At magic configurations the charge distribution shows characteristic deviations from the average value due to the shell structure of the fragments. The charge distribution seems to result mainly from three competing effects:

1. a)
   from the level density of the unperturbed fragments in the vicinity of the Fermi energy.     
   
   

Primary mass and charge distributions in the reaction of 900 MeV 132Xe ions with 197Au

A stacked foil technique and radioehemical yield measurements were combined in order to study the reaction of ¹³²Xe ions with ¹⁹⁷Au at 1.3 times the interaction barrier energy. Evidence is presented for a preferential exchange of neutrons for energy losses of up to 100 MeV. The same amount of energy damping is required for the development of a drift in the position of the centroid of the element distribution and for the evolution of full correlations in the exchange of neutrons and protons. The widths of the charge distributions at fixed mass asymmetry rise slowly as the degree of energy damping increases. This is reminiscent of classical statistical fluctuations, although the widths at large excitation energies have also been reproduced quantum-mechanically in terms of the zero-point motion of a collective isovector mode analogous to the giant dipole resonance in spherical nuclei. All results concerning the initial evolution of charge and mass distributions seem to be dominated by a hindrance of proton exchanges for large internuclear distances.     

Langevin description of the charge distribution of fragments originating from the fission of excited nuclei

The charge distribution of fragments originatingfrom the fission of the ²³⁶U compound nucleus is calculated within a stochastic approach based on Langevin equations. The elongation coordinate, the neck-thickness coordinate, and the charge-asymmetry coordinate are chosen as collective variables. The friction parameter of the charge mode is calculated on the basis of two nuclear-viscosity mechanisms, that of one-body and that of two-body dissipation. It is shown that the Langevin approach is applicable to studying isobaric distributions. In addition, the charge distribution in question is studied as a function of the excitation energy of the compound nucleus and as a function of the coefficient of two-body viscosity.     

Excitation energy dependence of fragment characteristics for the photofission of 232Th

Independent and cumulative product yields were measured for the photofission of ²³²Th with bremsstrahlung with endpoint energies 6.5, 7.0, 8.0, 11.0, 12.0, and 14.0 MeV, applying γ spectrometric techniques on catcherfoils and pneumatically transported ²³²Th-samples. The independent heavy fragment yields for the fission of the ²³²Th compound nucleus at excitation energies in the vicinity of the fission barrier were deduced. Postneutron mass, isobaric charge, isotopic mass distributions, isotonic and elemental yield distributions and proton odd-even effects were obtained from these independent yields. In the mass distributions a maximum yield is observed for mass splits with heavy fragments in the region of A = 142, corresponding with a high production of Ba(Z = 56) - isotopes. A slightly increased yield is also observed for mass splits with heavy mass in the vicinity of A = 134. The latter effect increases with increasing compound nucleus excitation energy. The similarity between the mass distributions of the N = 142 fissioning systems ²³²Th, ²³⁴U and ²³⁶Pu is striking. For low excitation energy the proton odd-even effect in the element distributions amounts to 30%, while on the other hand no sizeable neutron odd-even effect could be deduced from the isotonic distributions. The proton odd-even effects remain constant up to compound nucleus excitation energies of about 7.85 MeV. For higher compound nucleus excitation energies the proton odd-even effect drops rapidly. A possible explanation of these observations in terms of pair breaking at the outer barrier is proposed.     

Nuclear charge distribution of mass-separated isobars from thermal-neutron-induced fission of 235U

The nuclear charge distribution of fission products with mass numbers A = 90, 91, 94, 99, 100, 101 and 104 provided by the mass separator “Lohengrin” was measured. Adjacent elements in the group of the light fission products could be separated by their different energy loss in a carbon absorber. The Z-yields were found to be strongly dependent on the kinetic energy of the fission products. The widths of the nuclear charge distributions are very small, in general, and strongly dependent on A as well as on the kinetic energy. The influence of the neutron evaporation and odd-even effects are clearly detected. An asymmetric nuclear charge distribution was found for A = 104 indicating the suppression of fission fragments with Z = 43. The average nuclear charges of the fission products at their average kinetic energy are in good agreement with the results from measurements of the number of β-decays and K X-ray measurements. The average nuclear charge of the isobar A = 132 was measured at its average kinetic energy with a calibrated secondary electron detector to be Z = 51.14 ± 0.15 which is in very good agreement with the radiochemical results. Thus previous physical measurements indicating a large independent yield for the doubly magic nucleus ¹³²Sn could not be confirmed.     

Nuclear charge distribution of heavy fission fragments from thermal-neutron-induced fission of235U

Fragments from thermal-neutron induced fission of²³⁵U have been separated by a mass spectrometer with respect to their masses and kinetic energies within 1 μsec. The separation principles are briefly described. For masses 130 to 139 amu the charge distributions have been determined by counting the number of beta tracks emitted from the individual mass selected fission fragments in a nuclear photographic emulsion. In another method, the average number of beta particles for each fragment mass is determined by use of a 4π-proportional counter. The mean nuclear charge as a function of mass is compared with other experimental results and theoretical curves. Contradictory to the radiochemical results, this experiment yields a dip in the mean nuclear charge versus mass curve at mass 132 amu corresponding to the doubly magic nucleus (N=82,Z=50)¹³²Sn. Recent theoretical calculations of Nörenberg are in agreement with this finding.     

Symmetric and asymmetric fission modes in proton-induced fission at 660 MeV of <Superscript>238</Superscript>U

Fission product cross sections of intermediate-energy fission of ²³⁸U were used in order to construct the charge and mass yield distributions. Enriched target of ²³⁸U was irradiated by proton beam with energy 660 MeV for several hours at the LNP Phasotron, Joint Institute for Nuclear Research (JINR), Dubna, Russia. The charge distribution of the fission fragments was analyzed for calculation of isobaric cross sections. The mass yield curves were expanded into symmetric and asymmetric components according multimodal fission approach. The fissility values of actinides were calculated at given proton energy. The obtained results have been compared to the same data for targets ²³⁷Np and ²⁴¹Am.     

Tungsten ions produced by infrared pulsed laser irradiation

Energetic ions have been obtained irradiating a tungsten target with a Q-switched Nd:Yag laser, 1064?nm wavelength, 9?ns pulse width, 900?mJ maximum pulse energy and power density of the order of 10¹⁰?W/cm². The laser-target interaction induces a strong metal etching with production of plasma in front of the target. The plasma contains neutrals and ions with high charge state. Time-of-flight measurements are presented for qualitative analysis of the ion production. A cylindrical electrostatic ion analyzer permits measuring of the yield of emitted ions, the charge state of detected ions and the ion energy distribution. Measurements indicate that, at a laser fluence of the order of 100?J/cm², the charge state may reach 9+ and the ion energy reaches about 5?keV. The ion energy distribution is given as a function of the charge state. Experimental results indicate that an electrical field is developed along the normal to the plane of the target surface, which accelerates the ions up to high velocity. The ion velocity distributions follow a “shifted Maxwellian distribution”, which the author has corrected for the Coulomb interactions occurring inside the plasma.     

Thermal multifragmentation of hot nuclei and liquid-fog phase transition

Multiple emission of intermediate-mass fragments (IMF) in the collisions of protons (up to 8.1 GeV), ⁴He (4 and 14.6 GeV), and ¹²C (22.4 GeV) on Au has been studied with the 4π setup FASA. In all the cases, thermal multifragmentation of the hot and diluted target spectator takes place. The fragment multiplicity and charge distributions are well described by the combined model including the modified intranuclear cascade followed by the statistical multibody decay of the hot system. IMF-IMF-correlation study supports this picture, giving a very short time scale of the process (≤70 fm/c). This decay process can be interpreted as the first-order nuclear “liquid-fog” phase transition inside the spinodal region. The evolution of the mechanism of thermal multifragmentation with increasing projectile mass was investigated. The onset of the radial collective flow was observed for heavier projectiles. The analysis reveals information on the fragment space distribution inside the breakup volume: heavier IMFs are formed predominantly in the interior of the fragmenting nucleus possibly due to the density gradient.     

Application of a two-step dynamical model to calculating properties of fusion-fission reactions

The two-step model of fusion-fission reactions that was proposed previously by the present authors is extended in such a way as to describe the multiplicity of light particles emitted from nuclearfission fragments. Calculations are performed for the reaction induced by ⁴⁸Ca + ²⁴⁴Pu collisions. The mass distribution of fragments, their mass-energy distribution, and the total multiplicity of neutrons and gamma rays are obtained for E _lab = 230, 238, 249, and 255 MeV. It is shown that the model reproduces qualitatively relevant experimental data. In order to attain quantitative agreement, it is necessary to take into account the angular momentum carried away by particles from the nucleus undergoing fission and various types of gamma rays emitted by the nucleus and its fission fragments.     

Electromagnetic energies of nuclei and the nuclear compton amplitude

The electromagnetic energy of a nucleus is derived in perturbation theory, which relates this quantity to the amplitude for the forward scattering of virtual photons on a nucleus (nuclear Compton amplitude). Using the gauge invariance of this amplitude, the energy is separated into Coulomb and transverse components. Our formalism, although basically nonrelativistic, admits corrections of order ($\text{v}\text{c}$)² to the nuclear charge operator. The energy is further separated into one-body terms, related to the n-p mass difference, and two-body terms which lead to the Breit interaction and the nuclear Lamb shift. These results are then related to electron scattering sum rules in the manner of Cottingham. Mesonic contributions to the electromagnetic energy are also discussed.     

Ternary fission within statistical approach

The ternary system with a light nucleus between two heavy fragments is assumed to appear from the binary configuration near scission. The formation of a third light nucleus in the binary system is considered. The calculated charge distributions in spontaneous ternary fission of ²⁵²Cf and in induced ternary fission of ⁵⁶Ni are compared with the available experimental data. The neutron multiplicity from the fission fragments is described. The fine structures of the TKE-mass distribution are predicted.     

The process γ+γ→ℓ+ℓ in coherent heavy ion collisions

Using a covariant formalism we calculate to leading order the 2-photon contribution to lepton pair production in coherent heavy ion collisions with-in the framework of EPA. The most significant result of our calculation is the presence of a prominent narrow threshold enhancement in thee ⁺ e ^? invariant mass distribution. The position of this peak is essentially energy independent, however, its height and width tend to increase with increasing energy, mass and charge of the colliding ions. Results are presented for configurations at the Bevalac and SPS machines which are then extrapolated down to the GSI energy region.     

Do electron scattering and muonic x-ray data really require a central depression in the charge distribution of208Pb?

The charge distribution of²⁰⁸pb calculated in the Hartree-Fock (HF) approach using the density dependent nucleon-nucleon interaction of Ehlers and Moszkowski is tested by simultaneous comparison with experimental data from 502 MeV elastic electron scattering and muonic atoms. In both cases the agreement is very good and nearly as good as the best fits with a phenomenological charge distribution of Fermi type, if the effect of the polarization of the nucleus due to the presence of the muon is properly taken into account. In contradiction to the Fermi fits the HF distribution shows a hump at the center of the nucleus.     

Mass and charge distributions in the very asymmetric thermal neutron induced fission of the odd-Z nucleus 242mAm

Yields of light fission products (A = 68, 70–84, 87, 88, 94, 96, 98, 102 and 106–108), their kinetic energies and nuclear charge distributions (A = 71–84, 87 and 88) in the thermal neutron induced fission of the odd-Z nucleus ^242mAm(Z = 95) were measured using the mass-separator Lohengrin at the Institute Laue-Langevin in Grenoble (France). The mass yield curve shows a fine structure at A = 70, probably due to shell and/or odd-even effects affecting also the nuclear charge distribution. The analysis of isotopic chain yields gives evidence for a very low excitation energy of the lightest fission fragments observed. A preferential formation of fragments with even Z is found for this odd-Z compound nucleus. Calculated values for the local odd-even effect are comparable with those for the neighbouring even-Z fissile nuclides and increase from 13% to 30% with increasing asymmetry of the mass split (A_L = 84 to 68 and Z_L = 35 to 28). The neutron odd-even effect shows a similar but less pronounced behaviour.     

Peripheral fragmentation of relativistic nuclei 11B in nuclear track emulsion

Data obtained from a nuclear track photoemulsion exposed to a beam of ¹¹B nuclei with a momentum of 2.75 GeV/c per nucleon are reported. Peripheral interactions where the total charge of particles emitted within the forward cone of relativistic fragmentation is equal to the charge of the projectile nucleus are analyzed to study the clustering of the ¹¹T nucleus. It is found that the three-body breakup of 2 + 2 + 1 charge configuration is a leading process. Tritons are revealed to play a crucial role in the most peripheral interactions of this type. Events interpreted as charge exchange of the 11B nucleus to excited states of the ¹¹C* nucleus above the nucleon-coupling threshold were observed for the first time. Prospects for studying the ¹¹C nucleus are discussed.     

Further studies of large collision residues in relativistic heavy-ion reactions with heavy nuclei

Target residue mass and charge distributions have been measured radiochemically for the reaction of 25.2 GeV ¹²C ions with Au and Pb. Enhanced product yields for A = 140?170 are found that are not present in reactions of Pb with GeV protons. The shape of the mass distribution in this region is used to define a pre-equilibrium product mass distribution.