Comparison of neutron resonance spacings with microscopic theory for spherical nuclei

The nuclear level spacings determined from neutron resonance experiments for nuclei with 20 ≦ A ≦ 148 and 181 ≦ A ≦ 209 are compared with spacings calculated for spherical nuclei with a microscopic theory which includes the nuclear pairing interaction. Single particle levels of Seeger et al. and Nilsson et al. are used in the calculations. The gross features of the experimental data due to nuclear shells are reproduced with the microscopic theory. In addition, the absolute agreement between experiment and theory is reasonable (67 % of the 151 cases examined agree to within a factor of 2) in view of uncertainties in the experimental data, the theoretical single particle levels and the pairing strength. Values of the spin cutoff parameter σ²(E), calculated with a microscopic theory, are included also for several doubly even nuclei and discussed in terms of nuclear shells.     

Estimating the occupation probabilities of single-particle orbits in nuclei

Occupation probabilities of neutron and proton single-particle orbits are estimated for a number of spherical and close to spherical nuclei with 20 ≤ Z ≤ 50 and 20 ≤ N ≤ 82 near the Fermi energy. The estimates are made according to the formula of the BCS theory with single-particle energies calculated using the mean-field model with dispersive optical potential. The closeness of the occupation probabilities to 0 and 1 is demonstrated for nuclei with traditional and new magic numbers.     

Neutron single-particle characteristics of Ag isotopes in the dispersive optical model

The neutron dispersive optical potential for Ag isotopes is constructed for a wide range of variation in the N number. Good agreement with the experimental data on the scattering of neutrons by ¹⁰⁷Аg isotopes, and on single-particle energies and the probabilities of subshell occupation near the Fermi energy, is obtained for stable isotopes ^{107, 109}Ag using the dispersive optical model. Calculations that predict the evolution of the neutron single-particle energies up to the boundary of neutron stability are performed. It is shown that new magic number N = 56 (for Z = 40) disappears upon moving from Zr to Sn in an isotonic chain with N = 56, due to rapid deepening of level 1g _7/2.     

Second Coulomb energy differences of isospin triplets in the 2s-1d shell

Second Coulomb energy differences, which in the present case are proportional to the tensor Coulomb energy, are calculated for 0⁺, T = 1 ground states in the region 18 ≦ A ≦ 42 using a shell model that includes a pairing interaction. The calculation is done with a mathematical formalism that includes p-n pairs as well as p-p and n-n pairs. Besides an enhancement of proton-pair Coulomb energies, the pairing interaction is responsible for lowering the Coulomb energy of N = Z members of isospin triplets and also gives rise to an important term in the second energy difference. Using pairing strengths derived from fitting energy levels for mass-18 and mass-42 nuclei, results of the calculation reproduce experimental second energy differences extremely well.     

Nuclear level densities below 40 MeV excitation energy in the mass regionA ≃ 50

Consistent pre-equilibrium emission and statistical model calculations of fast neutron induced reaction cross sections are used to validate nuclear level densities for excitation energies up to 40 MeV in the mass regionA ?50. A “composed” level density approach has been employed by using the back-shifted Fermi gas model for excitation energies lower than 12 MeV and a realistic analytical formula for higher excitations. In the transition region from the BSFG model range to that of full applicability of the realistic formula, an interpolation between the predictions of the two models is adopted. The interpolation rule, suggested by microscopic level density calculations, has been validated through the comparison of the calculated and experimental cross sections.     

原子核的有限体积效应对高离化态离子能级和波函数的影响

在相对论理论框架下,分别在点电荷和两参数Fermi核模型近似下计算了高离化态类氢离子（Z=80—112）n=1—3壳层的波函数和能级. 分析了核有限体积效应对它们的影响. 在此基础上,给出了核有限体积效应对原子能级的修正公式. 同时,还进一步讨论了相对论效应和核有限体积效应之间的相互影响,发现对高Z元素相对论效应与核有限体积效应之间有很强的耦合. 关键词： 原子核有限体积效应 高离化态离子 相对论效应     

Analysis of fissionability data at high excitation energies

A level density formula that takes into account the smoothed volume, surface and curvature dependence of the single particle level density at the Fermi surface using the results of Balian and Bloch, is shown to be compatible with the level spacings found in neutron resonance data if complemented by a simple Ansatz for shell effects (due to Ignatyuk) and pairing effects. The three parameters involved, a scaling parameter, a shell damping energy and a pairing energy shift are compatible, respectively, with known nuclear radii, microscopic level density calculations and odd-even mass fluctuations. At excitation energies on the order of the neutron binding energy no evidence for an absolute level density problem or a different behaviour of level densities (collective contributions) for deformed nuclei as opposed to spherical nuclei is found. The proposed level density formula allows to calculate a priori macroscopic ratios of level densities, e.g. at the groundstate and at the saddle point, removing this important parameter from the analysis of fissionability data. As a first application, the fissionability of a number of actinide nuclei at excitation energies a few MeV above the fission barrier is analysed.     

Level density of a Fermi gas: average growth and fluctuations

We compute the level density of a two-component Fermi gas as a function of the number of particles, angular momentum, and excitation energy. The result includes smooth low-energy corrections to the leading Bethe term (connected to a generalization of the partition problem and Hardy-Ramanujan formula) plus oscillatory corrections that describe shell effects. When applied to nuclear level densities, the theory provides a unified formulation valid from low-lying states up to levels entering the continuum. The comparison with experimental data from neutron resonances gives excellent results.     

Is there neutron-proton pairing in medium heavy nuclei?

Present knowledge of the proton and neutron pairing energies Δ_p and Δ_n, deduced from nuclear masses, is reviewed and an attempt is made to find general trends in the data. The analysis shows that, besides the well-known smooth slow decrease with A, the pairing energies also contain a symmetry energy-like dependence on the neutron excess $\text{(N?Z)}\text{A}$. The trends is most pronounced in the shell 50<Z<82, 82<N<126 where both pairing energies decrease by a factor of almost two between the most neutron deficient and the most neutron rich nuclei. The same tendency persists in other mass regions, but is not a universal one. An empirical expression for Δ, that is more accurate than the value $\text{12A}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{MeV}$ usually assumed, is proposed.     

Direct experimental estimate of parameters that determine the cascade gamma decay of compound states of heavy nuclei

A method is proposed for simultaneously determining the interval of the most probable values of the density of levels excited in the radiative capture of slow neutrons and the sum of radiative E1 and M1 strength functions in the excitation-energy interval extending nearly up to the neutron binding energy. Experimental data on the intensities of two-step photon cascades between the compound state and a given low-lying level of the nucleus being studied are analyzed together with the total radiative widths of neutron resonances. Such an analysis can be performed for nuclei having an arbitrary level density, including deformed ones. The resulting data demonstrate that there are significant deviations from the predictions of commonly accepted level-density models—for example, the Fermi gas model—and specify the range of nuclei and the regions of their excitation energies where a further experimental investigation can furnish new important information about the properties of nuclear matter.     

Nuclear level densities and level spacing distributions: Part II

Extensive and complete nuclear level schemes and neutron resonance densities of 75 nuclides have been used to determine parameters of level density formulae, the spacing distribution of levels with equal spins and parities, and spacing correlations. The dependences of the level density parameters on A, on pairing energies and on shell effects are discussed. The spacing distribution is found to lie between a Poisson and Wigner distribution indicating a transitional character between regular and chaotic properties of the nuclear quantum system near the ground state.     

Nuclear level densities with collective rotations included

The level density is calculated from the single particle energies in a Woods-Saxon potential with pairing included in the BCS approximation. The collective rotations are included by addition of a rotational band on top of each of the intrinsic levels. The nuclei investigated have mass numbers in the region 100 ≦ A ≦ 253. At the ground state deformation and at the neutron separation energy for the nucleus in question we compare calculated and observed level densities. The dependence on the parameters in the model are investigated. Considering the uncertainties in these parameters the calculated results are believed accurate to within a factor of 3. The rotations contribute typically a factor of 40. They must be included for deformed and not for spherical nuclei. We underestimate systematically the level density by a factor of 4 with fluctuations around the average value by a factor of 3. The nuclei lighter than ¹³⁸Ba are an exception. We obtain around a factor 100 too few levels in the calculation.     

Level density parameters for the back-shifted fermi gas model in the mass range 40 < A < 250

The parameters a and Δ for the Fermi gas model with fictive ground state are determined for about 220 nuclei from experimental level densities at low excitation energy and at the neutron binding energy. In. agreement with previous results it is found that for most nuclei the fictive ground state is back-shifted relative to the conventionally shifted ground state as determined by the pairing energy. Shell effects are evident at the mass numbers 90, 140 and 208 for both the level density parameter a and the back-shift. A comparison is given with previous results and different experimental data on level densities.     

Level densities in the Pairing plus Quadrupole model for Erbium isotopes

The level densities, up to about 100 MeV of excitation energy, for even Erbium isotopes are computed using the Pairing plus Quadrupole model, in the framework of the Static Path Approximation (SPA). The resulting level densities are in reasonable agreement with the empiricalA/8 law below 40 MeV of excitation energy. At higher excitation energies (U?60 MeV) the level densities agree with the Fermi gas formula witha?A/10. The inclusion of small amplitude collective motion (RPA-SPA) does not improve the results over the SPA at high excitation energy, and gives small corrections to the ground state energy.     

Nuclear Level Density at High Spin and Excitation Energy

The intensive studies of equilibrium processes in heavy-ion reaction have produced a need for information on nuclear level densities at high energies and spins. The Fermi gas level density is often used in investigation of heavy-ion reaction studies. Some papers have claimed that nuclear level densities might deviate substantially from the Fermi gas predications at excitations related to heavy-ion reactions. The formulae of calculation of the nuclear level density based on the theory of superconductivity are presented, special attention is paid to the dependence of the level density on the angular momentum. The spin-dependent nuclear level density is evaluated using the pairing interaction. The resulting level density for an average spin of 52 ħ is evaluated for ¹⁵⁵Er and compared with experimental data. Excellent agreement between experiment and theory is obtained.     

Rearrangements in neutron shell closures far from stability

A survey of experimental results obtained at GANIL (Caen, Prance) on the study of the properties of very neutron-rich nuclei (Z = 6–20, A = 20–60) near the neutron drip line and resulting in an appearance of further evidence for the new magic number N = 16 is presented. Very recent data on mass measurements of neutron-rich nuclei at GANIL and some characteristics of binding energies in this region are discussed. Nuclear binding energies are very sensitive to the existence of nuclear shells and together with the measurements of instability of doubly magic nuclei ¹⁰He and ²⁸0 they provide information on changes in neutron shell closures of very neutron-rich isotopes. The behaviour of the two-neutron separation energies S_2n derived from mass measurements gives a very clear evidence for the existence of the new shell closure N = 16 for Z = 9 and 10 appearing between 2s_1/2 and ld_3/2 orbitals. This fact, strongly supported by the instability of C, N and O isotopes with N > 16, confirms the magic character of N = 16 for the region from carbon up to neon while the shell closure at N = 20 tends to disappear for Z ≤ 13. Decay studies of these hardly accessible short-lived neutron-rich nuclei from oxygen to silicon using the in-beam γ-ray spectroscopy are also reported.     

A systematic investigation of the (α, 2nγ) reaction on medium-heavy nuclei

Exclusive neutron spectra and angular distributions have been measured for 28–35 MeV (α, 2nγ) reactions on various nuclei in the 80 ≦ A ≦ 210 region. Pre-equilibrium processes dominate the 35 MeV (α, 2nγ) reaction mechanism in much of this region. Analysis of systematic variation in the neutron spectrum parameters shows that the reaction mechanism is strongly correlated with the target neutron excess parameter (N?Z/A. Analysis of the γ-decay of the entry states shows that well-defined incident angular momentum windows exist for the pre-etjuilibrium (α, 2nγ) reaction. These features are discussed in terms of various models for the reaction mechanism.     

Experimental study of pionic Ca and Ti and examination of the pion-nuclear potential for low-Z nuclei

We report the measurement of the strong interaction shifts and widths of the 3d → 2p transitions in pionic ^{40,42,43,44,48}Ca and ^46,48,50Ti. Using these new data in combination with earlier low-Z pionic data (6 ≦ Z ≦ 16), we have studied, in a purely phenomenological way, the pion-nuclear optical potential. Employing nuclear-structure information from Hartree-Fock calculations and measured charge densities, we have fitted the pion-nuclear potential parameters to the pionic atom data. The effect of adding an isovector dependence to the s- and p-wave two-nucleon terms has been investigated. We have also explored the sensitivity of these data to the value of the Lorentz-Lorenz parameter ξ. A small but definite sensitivity to the latter parameter has been observed with the best fit value of ξ being significantly greater than 1. We have used the phenomenological potentials to determine the neutron radii of the Ca and Ti isotopes and, in fact, find reasonable agreement with Hartree-Fock predictions and with the results of other hadronic probes.     

A new mass formula and new neutron and proton magic numbers in light nuclei

This mass formula explains gross features of the binding energy curves for all the elements from Li to Bi. It has no shell effects incorporated. Comparisons of separation energies computed from this formula and measured masses show extra-stability at N=6 (Z=3?8), Z=6 (N=6?9), N=14 (Z=7?10), Z=14 (N=14?19), N=16 (Z=7?8), Z=16 (N=24?26), loss of magicity at N=8 (Z=4), N=20 (Z=12?15) and quenching of N=50, 82, 126, Z=50 near driplines. Z=82 magicity rises at N=104 after strong quenching near N=107.