Showing papers in "Physical Review C in 1972"

Hartree-Fock Calculations with Skyrme's Interaction. I. Spherical Nuclei

Abstract: Hartree-Fock calculations for spherical nuclei using Skyrme's density-dependent effective nucleon-nucleon interaction are discussed systematically. Skyrme's interaction is described and the general formula for the mean energy of a spherical nucleus derived. Hartree-Fock equations are obtained by varying the mean energy with respect to the single-particle wave functions of occupied states. Relations between the parameters of the Skyrme force and various general properties of nuclear matter and finite nuclei are analyzed. Calculations have been made for closed-shell nuclei using two rather different sets of parameters, both of which give good binding energies and radii for $^{16}\mathrm{O}$ and $^{208}\mathrm{Pb}$. Both interactions give good binding energies and charge radii for all closed-shell nuclei. Calculated electron scattering angular distributions agree qualitatively with experiment, and for one interaction there is good quantitative agreement. The single-particle energies calculated with the two interactions are somewhat different owing to a different nonlocality of the Hartree-Fock potentials, but both interactions give the correct order and density of single-particle levels near the Fermi level. They differ most strongly in their predictions for the energies of $1s$ single-particle states.

Density-matrix expansion for an effective nuclear Hamiltonian

Abstract: An expansion for the nuclear wave-function density matrix in relative and cm coordinates is developed such that the leading term is the corresponding nuclear-matter density matrix at the local neutron and proton density Truncation of all derivatives beyond second order yields an extremely simple form for the energy density which retains all the computational simplicity of the modified $\ensuremath{\delta}$ interaction and the Skyrme force, while maintaining contact with nuclear-matter theory based on a realistic interaction and reproducing the results of more-complicated density-dependent Hartree-Fock calculations

Three-Body Correlations in Nuclear Matter

Abstract: A momentum-space method is developed for the calculation of three-body terms in the Brueckner-Bethe method for nuclear matter. The method is similar to one used earlier for central $S$-wave potentials. Here we extend it to the full nuclear force, including tensor forces, spin-orbit forces, etc. Furthermore, we show how the method can be used to investigate the possibility of long-range correlations in nuclear matter by summing the generalized ring series. The numerical accuracy obtainable with various mesh parameters and cut-offs in momentum space, and with various truncations of partial-wave expansions, is thoroughly explored. Several angle-average approximations are used, and the estimated numerical accuracy in the three-body cluster energy is 10-15%. The method is applied to a central potential ${v}_{2}$, a semirealistic potential ${v}_{6}$ (Reid), which has a tensor force, and to the Reid potential, augmented by an interaction that is consistent with empirical scattering phase shifts in two-body partial waves with $j\ensuremath{\ge}3$. In all cases the three-body contribution to the energy is correctly given in order of magnitude by ${\ensuremath{\kappa}}_{2}{D}_{2}$, where ${D}_{2}$ is the two-body contribution and ${\ensuremath{\kappa}}_{2}$ is the usual convergence parameter of the Brueckner-Bethe method. The generalized ring series is found to converge rapidly, indicating that long-range correlations are not very important for the binding energy of nuclear matter. The Reid potential is found to saturate at the right energy but at too high a density.NUCLEAR STRUCTURE Method for solving Brueckner-Bethe three-body equations in nuclear matter developed and applied to the Reid potential.

Angular Momentum of Primary Products Formed in the Spontaneous Fission of Cf 252

Abstract: The measured intensities of intraband cascading transitions in the ground-state bands of 21 high-yield even-even fission products have been analyzed by two methods to determine the magnitude of the intrinsic angular momentum of the primary products formed in the spontaneous fission of $^{252}\mathrm{Cf}$. The first method was to quantitatively compare the intensities of the intraband transitions observed in fission with those reported in the literature for in-beam (particle, $\mathrm{xn}$) reactions for which the primary angular-momentum distribution was determined by optical-model calculations. The second method was based on a simple statistical-model analysis of the angular momentum distribution throughout the neutron evaporation and the pre-ground-state-band $\ensuremath{\gamma}$-ray transition phases of the deexcitation process. The two methods gave reasonably similar results, with the former method yielding a somewhat larger primary angular momentum for the fragments. The general conclusions from the statistical-model analysis are that: (1) The average angular momentum of the products is $\overline{l}\ensuremath{\approx}(7\ifmmode\pm\else\textpm\fi{}2)\ensuremath{\hbar}$; (2) the heavy fission products have $\ensuremath{\approx}20%$ greater angular momentum than the light products; (3) the more symmetric the mass division the lower the initial angular momentum; and (4) there are only small changes in angular momentum [$\ensuremath{\sim}(1\ensuremath{-}2)\ensuremath{\hbar}$] with changes in fragment kinetic energy. An important feature of these results is that the fragment angular momentum does not correlate with the number of neutrons evaporated by the fragment. Additional measurements have been made to study the angular distribution of individual prompt $\ensuremath{\gamma}$ rays. In all observed cases the ${2}^{+}\ensuremath{\rightarrow}{0}^{+}$ ground-state transitions were forward-peaked with respect to the fission axis, and this is consistent with the assumption that the angular momentum is aligned in a plane perpendicular to the direction of fission. The results are discussed in terms of a quasistatistical model in which the neck width at scission is approximately constant.

Prompt gamma rays emitted in the thermal-neutron-induced fission of 235 U

Abstract: The average number and average energy of $\ensuremath{\gamma}$ rays emitted within \ensuremath{\sim}5 nsec after fission have been determined as functions of fragment mass and as functions of fragment mass and total kinetic energy in two-dimensional representations. In a four-parameter experiment, energies of coincident pairs of fission fragments were measured with surface-barrier detectors and $\ensuremath{\gamma}$-ray energies were measured with a large NaI(Tl) detector, which was located 89 cm from a thin $^{235}\mathrm{U}$ target and positioned coaxially with the fragment detectors. The time difference between detection of a fission fragment and a $\ensuremath{\gamma}$ ray was measured to allow time-of-flight discrimination against fission neutrons. The $\ensuremath{\gamma}$-ray data were analyzed with a "weighting method" proposed by Maier-Leibnitz to deduce average numbers and energies of $\ensuremath{\gamma}$ rays from measured pulse heights. The Doppler shift in the laboratory angular distribution of $\ensuremath{\gamma}$ emission was utilized to obtain the number and energy of $\ensuremath{\gamma}$ rays as functions of single fragment mass. The results, for both average number and average energy as functions of single fragment mass, are characterized by a sawtooth behavior similar to that which is well known for neutron emission. The over-all average number and energy of $\ensuremath{\gamma}$ rays emitted per fission were found to be 6.51 \ifmmode\pm\else\textpm\fi{} 0.3 and 6.43 \ifmmode\pm\else\textpm\fi{} 0.3 MeV, respectively, giving an average photon energy of 0.99 \ifmmode\pm\else\textpm\fi{} 0.07 MeV.

Non-Hermitian Hamiltonians, Decaying States, and Perturbation Theory

Abstract: The conventional time-independent perturbation theory and projection-operator techniques are generalized to systems propagating under the influence of an effective non-Hermitian Hamiltonian, e.g., an optical-model potential. The mathematical complications which can arise due to lack of orthogonality and completeness for the eigenstates of such a Hamiltonian are discussed on an elementary level. Examples of the various techniques involved are taken from considerations of models for exotic atoms and cascade processes.

Relativistic Self-Consistent Meson Field Theory of Spherical Nuclei

Abstract: A relativistic self-consistent theory is used in conjunction with meson field potentials having the form of the generalized one-boson-exchange potentials (GOBEP) to construct a relativistic self-consistent meson field theory of nuclear structure. A simple GOBEP model with qualitative features of successful $N\ensuremath{-}N$ models, e.g., approximate cancellation of static terms arising from generalized (or regularized) scalar- and vector-meson fields, is used to calculate ground-state properties of the doubly-magic spherical nuclei $^{16}\mathrm{O}$, $^{40}\mathrm{Ca}$, $^{90}\mathrm{Zr}$, and $^{208}\mathrm{Pb}$, and one superheavy nucleus $^{298}114$. Good agreement is obtained between theoretical and experimental total binding energies and radial charge distributions. The isotopic shift in charge distributions between the isotopes $^{40}\mathrm{Ca}$ and $^{48}\mathrm{Ca}$ and the single-particle eigenvalues agree quite well with the experimental numbers. The absence of explicit correlation corrections, the relationship of this model to earlier meson-theoretic descriptions, and physical interpretation in terms of nucleon form factors and relativistic interactions are discussed.

Nonelastic interactions of nucleons and pi mesons with complex nuclei at energies below 3 GeV.

Abstract: Calculation of nonelastic interactions corresponding to continuum-state transitions using the intranuclear-cascade evaporation approach. Spallation yields, energy- and angle-dependent spectra, particle multiplicities, and nonelastic cross sections are calculated for incident nucleons and pi mesons with energies below 3 GeV on complex nuclei. Comparisons with experimental data are made, and, in general, the agreement is good. Discrepancies in these comparisons are discussed with respect to the deficiencies in the model.

Rb and Cs Isotopic Cross Sections from 40-60-MeV-Proton Fission of U 238 , Th 232 , and U 235

Abstract: The isotopic distributions of Rb and Cs from the fission of $^{238}\mathrm{U}$, $^{232}\mathrm{Th}$, and $^{235}\mathrm{U}$ induced by 40- to 60-MeV protons have been measured by means of an on-line mass spectrometer. The Rb isotopic distributions have a Gaussian shape, but those of Cs are somewhat asymmetrical. As the proton bombarding energy increases, the neutron-excess sides of the distributions remain approximately fixed while the neutron-deficient sides shift to lower mass numbers. The distributions also show significant variations with the neutron-to-proton ratio of the target. All the isotopic cross sections show a significant odd-even structure, with the formation of even neutron isotopes being favored. The effect is more pronounced for the neutron-rich isotopes. A similar structure is found to occur in Rb and Cs distributions from fission induced by thermal neutrons, 155-MeV protons, and 24-GeV protons, as well as in the Na and K cross sections from 24-GeV-proton reactions with a variety of targets. The odd-even effect in the Rb and Cs distributions can be accounted for by a 10 to 15% neutron-pairing effect in the prompt yields and a 2 to 3% pairing effect in the neutron emission. From the mean mass numbers of the Rb and Cs distributions, the average total number of emitted neutrons has been estimated for each reaction. This information, together with other results on neutron emission as a function of fragment mass, has allowed the mean mass numbers to be corrected for prompt neutron emission. These corrected values differ by about one mass unit from the values predicted by the unchanged-charge-density mechanism and are consistent with the same mechanism of charge division as that operating in thermal-neutron fission.

Distribution of Mass in the Spontaneous Fission of Fm 256

Abstract: The distribution of mass in the spontaneous fission of $^{256}\mathrm{Fm}$ has been investigated by the radiochemical determination of the fission yields for 28 mass chains. The mass distribution is asymmetric with a peak-to-valley ratio of \ensuremath{\sim}12. The average masses of the light and heavy groups are 111.9 and 141.0, respectively. The fission-yield data indicate a value of 3 \ifmmode\pm\else\textpm\fi{} 1 for $\overline{\ensuremath{ u}}$, the average number of neutrons emitted per fission. A comparison of the characteristics of the mass distribution for low-energy fission of nuclides ranging from $^{227}\mathrm{Th}$ to $^{256}\mathrm{Fm}$ is given.

Effects of Λ-Σ Coupling in 4 Λ H, 4 Λ He, and 5 Λ He

Abstract: We consider the effect on the binding energies of $_{\ensuremath{\Lambda}}^{4}\mathrm{He}\ensuremath{-}_{\ensuremath{\Lambda}}^{4}\mathrm{H}$ of inclusion of admixing the $\ensuremath{\Sigma}$ particle, based on a $\ensuremath{\Lambda}\ensuremath{-}N$ potential derived from $_{\ensuremath{\Lambda}}^{5}\mathrm{He}$. This mixing explains the discrepancy in Coulomb energy of $_{\ensuremath{\Lambda}}^{4}\mathrm{He}\ensuremath{-}_{\ensuremath{\Lambda}}^{4}\mathrm{H}$ and leads to a consistent set of potential parameters.

Simple formula for the general oscillator brackets.

Abstract: An explicit formula for Talmi-Moshinsky transformation brackets of unequal-mass particles is given which is the sum of simple expressions over five variables; it is especially suited for numerical calculations.

Independent Yields of Krypton and Xenon Isotopes in Thermal-Neutron Fission of U-235. Observation of an Odd-Even Effect in the Element Yield Distribution

Abstract: The fractional and independent fission yields of krypton isotopes (masses 87 to 94) and xenon isotopes (masses 137-143) produced in thermal-neutron fission of $^{235}\mathrm{U}$ were measured using a mass separator operating in an on-line mode with a fission source exposed to time-controlled neutron fluxes. The noble-gas isotopes were collected simultaneously at given times and each mass was measured separately. The distribution of the isotopic yields of each element was found to be Gaussian, in general agreement with Wahl's estimates. The total independent yields of krypton and xenon, as well as their complementary elements, barium and strontium, were found to indicate an odd-even effect in the element yield distribution. Even-$Z$ elements are formed in higher yields than their odd-$Z$ neighbors (20-50% higher); the even-$Z$ yields are also higher than Wahl's "normal" element yields (\ensuremath{\sim}12% higher for Kr and Ba and \ensuremath{\sim}40% for Xe and Sr). Half-life values of 20 nuclides produced in the decay of krypton and xenon isotopes were measured as well.

Analysis of the Total ( n , p ) Cross Sections Around 14 MeV with the Pre-Equilibrium Exciton Model

Abstract: The absolute value of ($n,p$) reaction cross sections, as given by the pre-equilibrium exciton model is estimated using Fermi gas approximation.A general agreement with experimental data, particularly in the gross $A$ dependence is obtained. The energy range considered is 10-20 MeV, and the nuclei are those with $Ag100$: In these ranges evaporation is negligible and the analysis is rather easy. The approximate life-time of one single-particle exciton in nuclear matter is also deduced.

( p , t ) Reaction on Even-Even N = Z Nuclei in the 2 s 1 d Shell

Abstract: The ($p,t$) reaction on the even-even $N=Z$ nuclei in the $2s1d$ shell has been used to study the energy levels of $^{18}\mathrm{Ne}$, $^{22}\mathrm{Mg}$, $^{26}\mathrm{Si}$, $^{30}\mathrm{S}$, $^{34}\mathrm{Ar}$, and $^{38}\mathrm{Ca}$. The energies of the excited states observed are reported along with spin and parity assignments when possible. Two-nucleon-transfer distorted-wave calculations were carried out. Comparisons are made with the shapes of the experimental angular distributions. It is found that the calculated shapes are primarily dependent upon the $L$ transfer and the optical-model parameters. The magnitudes of the calculated cross sections are found to depend strongly not only on the optical-model parameters, but also the bound-state parameters and the configuration mixing in the initial and final nuclear wave functions.

Population of 0 + States in Actinide and A~190 Nuclides by the (p,t) Reaction

Abstract: The ($p,t$) reaction has been studied for targets of $^{230,232}\mathrm{Th}$, $^{234,236,238}\mathrm{U}$, $^{242,244}\mathrm{Pu}$, $^{248}\mathrm{Cm}$, $^{182,184,186}\mathrm{W}$, and $^{196}\mathrm{Pt}$. Uniformly strong (\ensuremath{\sim}15% of ground-state cross section) transitions were found to populate excited ${0}^{+}$ states in all of the actinide nuclides. The observed ${0}^{+}$ states were previously known for five of the actinide nuclei, and in these cases the states were characterized by small $\ensuremath{\alpha}$-decay hindrance factors and large $E0$ matrix elements. They have been previously classified as $\ensuremath{\beta}$ vibrations. The strong excitation of the ${0}^{+}$ states in the ($p,t$) reactions, combined with all other available evidence, suggests that throughout the actinide mass range these ${0}^{+}$ states represent a stable collective excitation different in character from both the $\ensuremath{\beta}$ vibration and the most common formulation of the pair vibration. No such excitations were seen in the W-Pt region.

Beta Decay of the Al 26 Ground State

Abstract: The decay scheme of the ground state of $^{26}\mathrm{Al}$ has been reexamined by measurement of its $\ensuremath{\gamma}$-ray spectrum with Ge(Li) detectors. The $\ensuremath{\gamma}$-ray spectrum shows very strong positron annihilation radiation, a strong peak at 1808.65 \ifmmode\pm\else\textpm\fi{} 0.07 keV, a weak peak at 1129.67 \ifmmode\pm\else\textpm\fi{} 0.10 keV, and a very weak peak at 2938 keV. These $\ensuremath{\gamma}$-ray energies correspond to transitions from the first and second excited states of $^{26}\mathrm{Mg}$, which have been determined more accurately in this experiment. The $^{26}\mathrm{Al}$ branching ratios, calculated from the $\ensuremath{\gamma}$-ray relative intensities, are (82.1 \ifmmode\pm\else\textpm\fi{} 2.5)% positron emission to the 1809-keV level, (15.2 \ifmmode\pm\else\textpm\fi{} 2.5)% electron capture to that same level, and (2.7 \ifmmode\pm\else\textpm\fi{} 0.2)% electron capture to the 2938-keV level of $^{26}\mathrm{Mg}$. The 2938-keV state decays (10.2 \ifmmode\pm\else\textpm\fi{} 2.0)% directly to the ground state and (89.8 \ifmmode\pm\else\textpm\fi{} 2.0)% by cascade. These branching ratios combine with previous results to give a half-life of (7.16 \ifmmode\pm\else\textpm\fi{} 0.32) \ifmmode\times\else\texttimes\fi{} ${10}^{5}$ yr for the ground state of $^{26}\mathrm{Al}$.

Elastic pd scattering at 316, 364, 470, and 590 MeV in the backward hemisphere.

Abstract: The elastic pd differential cross section at center-of-mass angles between 91 and 164 deg was determined for 316, 364, 470, and 590 MeV proton scattering in a backward hemisphere. For the three largest energies, the cross sections were within 10% of each other at any given angle larger than 130 deg. The extrapolated 180 deg differential cross section remained nearly constant from 316 to 590 MeV.

Elastic scattering of 600-MeV protons from H, D, He-3, and He-4.

Abstract: Study of the elastic scattering of 600-MeV protons from light nuclei. Differential cross sections have been obtained for the scattering of protons from hydrogen, deuterium, helium-3, and helium-4. Polarization was measured for deuterium and He-4 nuclei. The p-p cross-section data are in excellent agreement with the predictions from the Livermore phase shifts. Small-angle p-D, p-He-3 elastic scattering data are compared with calculations based on the multiple-scattering theories of Watson and Glauber.