Nuclear Energy Surface Research Articles

From Heavy-Ion Elastic Scattering to Fission: A Unified Potential for the Description of Large-Scale Nuclear Collective Motion

In a generalization of the modified liquid-drop model obtained by requiring that two semi-infinite slabs of constant-density nuclear matter have minimum energy at zero separation, we calculate the generalized nuclear surface energy by means of a double volume integral of a Yakawa-plus-exponential folding function. This leads to a unified nuclear potential that is useful for describing heavy-ion elastic scattering and fusion, nuclear fission, and nuclear ground-state masses and deformations.

Properties of nuclear matter using the velocity-dependent effective potential ofs-wave interaction

Two forms of a velocity-dependent effective potential are used. The binding energy and the incompressibility of the nuclear matter are calculated. These are found to be in a good agreement with those obtained by others and with the experimental results. The single-particle potential, the effective massM* and the nuclear surface energy are also discussed and compared with those obtained by the others.

Determination of the spectroscopic quadrupole moments of 131Cs, 132Cs and 136Cs

Nuclear spectroscopic quadrupole moments of the radioactive isotopes 131Cs, 132Cs, and 136Cs have been determined from the hyperfine structure of the 6 2 P 3 2 state by the level crossing method. The results including a Sternheimer correction are: Q s ( 131 Cs) = −0.625(6) b , Q s ( 132 Cs) = +0.508(7) b , Q s ( 136 Cs) = +0.225(10) b . The quadrupole moments of all the Cs isotopes from A = 131 to A = 137 are recalculated. It is shown, that nuclear quadrupole moments of a specific isotope obtained from different atomic P-states only agree within the limits of error after application of the Sternheimer correction. The increase of Q s with decreasing neutron number conforms with other observations and theoretical calculations stating that for elements around Z = 55 nuclear deformation develops below N = 82. The staggering of the sign of Q s may be interpreted as consequence of an oblate-prolate degeneracy of the nuclear energy surface. Some magnetic moments have been slightly improved: μ I ( 132 Cs) = 2.219(7) μ N , μ I ( 136 Cs) = 3.705(15)μ N (corrected for diamagnetism).

Empirical analysis of the formula for the binding energy of nuclei

An accurate knowledge of the properties of the nuclear energy surface is important not only for practical reasons (to calculate nuclear masses, and to predict the properties of superheavy elements and nuclei far from the/3-stability line), but also to check and refine nuclear models. The well-known semiempirical formula for the binding energy due to Bethe and Weizsacker [1] has been subjected to a number of refinements and modifications in order to obtain better agreement with experiment [2-10]. The correction terms, introduced from theoretical considerations, fake into account the specific features of the distribution of nucleons in the nucleus (the diffuseness of the boundary, the nonuniformity of the distribution in the inner region, and the difference between the neutron and proton densities) [2-5], the effects of deformation and compressibility of the nuclei [6-10], Coulomb exchange interaction [3, 61, etc. However, because there is no rigorous theory of the nucleus the role of the various correction terms remain unclear. As far as their empirical basis is concerned, the introduction of correction terms into the equation for the binding energy, despite the addition of new parameters, does not lead to any appreciable improvement in the agreement with experiment. Nevertheless, the introduction of purely empirical corrections to the parity and the effects of the shells considerably improves the agreement with experiment [1, 3, 6, 11]. All this stimulated us to try to clarify the features of the structure of the formula for the binding energy, without previous resource to any model representations, and based exclusively on experimental data. It is well lmown that for any fixed mass number A the stability of the isobars falls, while the energy increases the further the nuclei are from the region of stability both on the neutron-excess side and on the proton-excess side. A detailed analysis of the experimental data given in [12-15], shows that within the limits of experimental error the isobar cross sections of the energy surface in the regions between the magic numbers of neutrons and protons are (for nuclei of each of the four types of parity) quadratic parabolas, so that the total energy of a nucleus with mass number A and charge Z can be represented in the form [12-14]

Effect of curvature on nuclear surface energy

Effect of curvature on nuclear surface energy

Adiabatic and non-adiabattc cranking models for the solution of the large amplitude time-dependent Hartree-Fock equations and the calculation of nuclear energy surfaces

An analysis is made of the device of forced vibrations to reduce the difficult large amplitude TDHF (time-dependent Hartree-Fock) problem to a simpler static problem. It is shown that cranking is a perfectly valid technique if the cranking field is defined by the criterion that the constrained static wave function be identical at a given instant of time to the freely oscillating TDHF wave function. On the basis of this criterion an exact general expression is derived for the cranking field and adiabatic and non-adiabatic cranking models are proposed as practical means of solving the large amplitude TDHF equations and calculating energy surfaces for fission and heavy ion reactions. The non-adiabatic model, which is favoured, corresponds to cranking the nucleus to move always in the direction of a local decoupled “normal mode” calculated from its unconstrained time-dependent equations of motion. The familiar Belyaev cranking model and the Baranger-Kumar model are discussed in the context of the new formalism.

Remarks concerning the nuclear surface and curvature energy : By W. Stocker; Sektion Physik, University of Munich, 8046 Garching, West Germany

Remarks concerning the nuclear surface and curvature energy : By W. Stocker; Sektion Physik, University of Munich, 8046 Garching, West Germany

Remarks concerning the nuclear surface and curvature energy

The A 1 3 term in the nuclear mass formula is studied. The curvature correction is investigated within the framework of the energy density formalism; the compression term is evaluated from a phenomenological point of view. The results are in fair agreement with the droplet model values obtained by Myers and Swiatecki.

Nuclear Surface Energy and Neutron-Star Matter

Direct calculations of the nuclear surface energy are made for a Hamiltonian containing the Skyrme nucleon-nucleon interaction. A plane surface separating nuclear matter and a neutron gas or a vacuum is considered in Hartree-Fock and Thomas-Fermi approximations. These surface energies are incorporated in the compressible liquid-drop model to obtain properties of neutron-star matter. The Hartree-Fock results lead to $Z$ values for the nuclei roughly constant at around $Z\ensuremath{\sim}36\ensuremath{-}38$.

Neutron star matter

The matter in neutron stars is essentially in its ground state and ranges in density up to and beyond 3 × 10 14 g/cm 3, the density of nuclear matter. Here we determine the constitution of the ground state of matter and its equation of state in the regime from 4.3 × 10 11 g/cm 3 where free neutrons begin to “drip” out of the nuclei, up to densities ≈ 5 × 10 14 g/ cm 3, where standard nuclear-matter theory is still reliable. We describe the energy of nuclei in the free neutron regime by a compressible liquid-drop model designed to take into account three important features: (i) as the density increases, the bulk nuclear matter inside the nuclei, and the pure neutron gas outside the nuclei become more and more alike; (ii) the presence of the neutron gas reduces the nuclear surface energy; and (iii) the Coulomb interaction between nuclei, which keeps the nuclei in a lattice, becomes significant as the spacing between nuclei becomes comparable to the nuclear radius. We find that nuclei survive in the matter up to a density ∼ 2.4 × 10 14 g/ cm 3; below this point we find no tendency for the protons to leave the nuclei. The transition between the phase with nuclei and the liquid phase at higher densities occurs as follows. The nuclei grow in size until they begin to touch; the remaining density inhomogeneity smooths out with increasing density until it disappears at about 3 × 10 14 g/cm 3 in a first-order transition. It is shown that the uniform liquid is unstable against density fluctuations below this density; the wave-length of the most unstable density fluctuation is close to the limiting lattice constant in the nuclear phase.

Systematics of beta-decay energies and the properties of the nuclear energy surface

Systematics of beta-decay energies and the properties of the nuclear energy surface

Deviations from theI(I+1)Law in Deformed Nuclei

Abstract : A natural extension of the methods used in computing the nuclear energy surfaces enables one to calculate deviations from the I(I+1) law corresponding to rigid rotors. Comparisons with the experimental data and with previous theoretical calculations are included. (Author)

Neutron and Proton Densities in Nuclei

A semiempirical investigation of neutron and proton densities in nuclei is made. Experimental values are assumed for the nuclear radius, binding energy, surface energy, surface thickness, and symmetry energy. It is found that the neutron and proton densities extend to approximately the same radius; these results do not depend sensitively on the input data. The nuclear potential extends \ensuremath{\sim}0.7\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ cm further than the material radius. An estimate of $\ensuremath{\lesssim}100A$ Mev is given for nuclear compressibility.

A Collective Description of the Surface Oscillation of Atomic Nuclei

Tomonaga's ingenious method of the description of collective surface motion is extended to the case of three dimensional nuclei. In this extension some difficulty appears which was not the case with Tomonaga's two dimensional example, and a method to keep out from such a difficulty is given in this paper. Applym;;. cnts method a Hamiltonian is derived whtch contains terms standing in one to one correspondence with those of Bohr's together with other correction terms. Brief calculation of the nuclear surface energy is also performed. The method of collective description mentioned here may be useful for many other problems where the collective motion takes place.

Nuclear Surface Energy and the Diffuseness of the Nuclear Surface

Nuclear Surface Energy and the Diffuseness of the Nuclear Surface

Berkelium and Californium Isotopes Produced in Neutron Irradiation of Plutonium

Californium isotopes of mass numbers 249, 250, 251, and 252 have been identified by mass spectrometric analyses of californium samples produced from neutron-irradiated plutonium. The half-lives of ${\mathrm{Bk}}^{249}$, ${\mathrm{Cf}}^{249}$, ${\mathrm{Cf}}^{250}$, ${\mathrm{Cf}}^{252}$, and ${\mathrm{Cf}}^{253}$ are 290\ifmmode\pm\else\textpm\fi{}20 days, 470\ifmmode\pm\else\textpm\fi{}100 years, 10.0\ifmmode\pm\else\textpm\fi{}2.4 years, 2.2\ifmmode\pm\else\textpm\fi{}0.2 years, and 18\ifmmode\pm\else\textpm\fi{}3 days, respectively. The alpha-particle energies of the most prominent alpha peaks for ${\mathrm{Cf}}^{249}$, ${\mathrm{Cf}}^{250}$, and ${\mathrm{Cf}}^{252}$ are 5.81\ifmmode\pm\else\textpm\fi{}0.03, 6.033\ifmmode\pm\else\textpm\fi{}0.010, and 6.117\ifmmode\pm\else\textpm\fi{}0.010 Mev, respectively. The plot of alpha-particle energies vs mass number for the isotopes of californium shows a break (i.e., the alpha energy of ${\mathrm{Cf}}^{252}$ is larger than the alpha energy of ${\mathrm{Cf}}^{250}$) indicating an irregularity in the nuclear energy surface. The alpha spectra of ${\mathrm{Cf}}^{250}$ and ${\mathrm{Cf}}^{252}$ exhibit characteristic even-even nuclide fine structure peaks. The beta-particle energy of ${\mathrm{Bk}}^{249}$ is 80\ifmmode\pm\else\textpm\fi{}20 kev. The spontaneous fission half-lives of ${\mathrm{Cf}}^{250}$ and ${\mathrm{Cf}}^{252}$ are 1.5\ifmmode\pm\else\textpm\fi{}0.5\ifmmode\times\else\texttimes\fi{}${10}^{4}$ and 66\ifmmode\pm\else\textpm\fi{}10 years, respectively. Lower limits of 2\ifmmode\times\else\texttimes\fi{}${10}^{8}$ and 5\ifmmode\times\else\texttimes\fi{}${10}^{6}$ years have been placed on the partial spontaneous fission half-lives of ${\mathrm{Bk}}^{249}$ and ${\mathrm{Cf}}^{249}$. The pile (Materials Testing Reactor at Arco, Idaho) neutron capture cross sections of ${\mathrm{Bk}}^{249}$, ${\mathrm{Cf}}^{250}$, ${\mathrm{Cf}}^{251}$, and ${\mathrm{Cf}}^{252}$ are 350, 1500, 3000, and 25 barns, respectively.

The Nuclear Surface Energy

The kinetic and potential parts of the nuclear surface energy are calculated on the basis of the statistical model of the nucleus taken to first order and using a gaussian interaction between particles. The resulting surface energy is found to be of the right order of magnitude, the calculated figure being in fact 55% of the empirical value.