W.J. Swiatecki

Equilibrium configurations of rotating charged or gravitating liquid masses with surface tension. II

Abstract The equilibrium configurations of rotating charged or gravitating liquid masses with surface tension are investigated. The objective is to discuss in a unified manner configurations of idealized atomic nuclei, rotating liquid drops and rotating idealized astronomical masses. The present paper formulates the problem generally but the applications discussed are mostly to nuclei. These include the estimates of the geometrical characteristics, of the rigid moments of inertia, and of the energies and fission barriers of idealized nuclei throughout the periodic table and with any angular momentum. The existence of super-deformed nuclei, stretched out into cylinderlike configurations by the centrifugal force, is discussed.

Average nuclear properties

Abstract A generalized treatment of average nuclear properties is presented. The theory is developed on two levels: First a refinement of the Liquid Drop Model, called the Droplet Model, is described. The degrees of freedom in this model, in addition to the usual shape variables, are variables specifying deviations from uniformity of the neutron and proton densities. The form of the Hamiltonian defining the Droplet Model, of which only the potential energy part is considered in this paper, is derived by expanding the volume, surface, and Coulomb energies in Taylor series around the standard Liquid Drop Model values. Such an expansion, designed to retain all terms in the total energy up to order A 1 3 , I 2 A 2 3 , and I 4 A , where I = (N − Z) A , turns out to contain eleven parameters, two of which may be eliminated. Four of the resulting nine parameters are the standard adjustable parameters of the Liquid Drop Model and five are new coefficients specifying various properties of nuclear systems. (Nuclear compressibility and the curvature correction to the surface tension are two examples.) Minimizing the Droplet Model potential energy with respect to density variations leads to equations, in closed form, specifying the separate neutron and proton radii and the density nonuniformities. The minimized energy expression leads to a refined Droplet Model Mass Formula with nine parameters. The second level at which average nuclear properties are treated is based on assuming a concrete model of a two-component saturating system, consisting of neutrons and protons interacting by velocity-dependent Yukawa forces (and Coulomb forces). When this model is treated in the Thomas-Fermi approximation a pair of coupled integral equations results, which can be used as the basis of a self-contained model of all average static nuclear properties. The solutions of these equations are discussed in the idealized situations of nuclear matter and semi-infinite nuclear matter, and for finite nuclei both with and without Coulomb energy. One result of these studies is the determination of the values of the five new Droplet Model parameters. Other results have to do with the nuclear density distributions, and the binding energies. The applications of the Droplet Model and Thomas-Fermi Model discussed in this paper include predictions concerning neutron and proton radii (in particular the presence of a neutron skin), the isotope effect in proton radii, the compression of the nucleus by the surface tension and the dilatation by the Coulomb energy, and the central depression in the densities caused by the Coulomb repulsion. Calculations are made for the surface curvature correction, for the surface symmetry energy, and for a modification to the volume symmetry energy at a large neutron excess. A revised estimate is made for the value of the symmetry energy of nuclear matter. Also treated is the question of whether or not neutron matter is bound, and some discussion is given of the spatial distribution, the energy dependence, and the composition dependence to be expected for nuclear optical model potentials on the basis of the statistical methods used in this paper.

One-body dissipation and the super-viscidity of nuclei☆

Abstract This is a study of a type of fluid dynamics dominated by a “one-body” dissipation mechanism expected to be relevant for an assembly of particles whose mean free paths are comparable to or larger than the size of the system. Two simple dissipation formulas are derived, one relevant for the process of nuclear fission and the other for nuclear collisions. The resulting predictions, free of adjustable parameters, are compared quantitatively with measured fission-fragment kinetic energies and qualitatively with nucleus-nucleus collision data. The one-body dissipation concept is also tested against classical and quantal computer studies of particles in a deforming potential well. This brings out special effects associated with the symmetries of the well and points to a macroscopic dynamics of nuclear deformations which, except for super-fluidity at very low temperatures, consists of a smooth background dominated by one-body dissipation (“super-viscidity” of nuclei), on which are superposed modifications due to symmetries and quantization.

Nuclear properties according to the Thomas-Fermi model

In order to formulate a statistical model of nuclear properties we combine the Thomas-Fermi assumption of two fermions per h3 of phase space with an effective interaction between nucleons that contains seven adjustable parameters. After allowing for shell effects, an even-odd correction and a congruence energy (“Wigner Term”), six of the seven parameters were fitted to 1654 ground state masses of nuclei with N, Z ⩾ 8, together with a constraint that ensures agreement with measured values of the nuclear surface diffuseness. The RMS deviation in the fit to masses was 0.655 MeV, and the calculated values exhibit no drastic discrepancies even for A = 3. Calculated sizes of nuclear charge distributions agree closely with measurements. Calculated fission barriers were compared with 40 measured values down to 75Br. For Z ⩾ 88 the agreement is almost perfect. For Z < 88 the trend of the measurements seems to confirm the expectation that the congruence energy should double its magnitude for strongly necked-in saddle-point shapes. A seventh (density-dependence) parameter in the effective interaction can be adjusted to ensure fair agreement with the measured energy-dependence of the optical model potential in the range from −70 MeV to 180 MeV. The model is used to predict properties of nuclear and neutron matter (including their compressibilities). A table of some 9000 calculated ground state masses of nuclei up to Z = 135 has been prepared.

The nuclear droplet model for arbitrary shapes

Abstract The Droplet Model of masses and density distributions, introduced in Ref. [1] for spherical configurations, is generalized to arbitrary shapes. Equations in closed form are given for the neutron and proton density nonuniformities induced by the electric forces, and also for the dependence of the neutron skin thickness on position on the nuclear surface. The formulas for the corrections to the nuclear energy associated with these effects are derived and this leads to a Droplet Model atomic mass formula which is presented with a preliminary set of coefficients adjusted to nuclear ground state masses and fission barriers.

STUDIES IN THE LIQUID-DROP THEORY OF NUCLEAR FISSION

Abstract In connection with nuclear fission we study the division of an idealized charged drop, using a simplified version of the liquid-drop model. The degrees of freedom essential to a discussion of the division of a charged drop and the separation of the fragments to infinity are taken into account: a fragment-separation coordinate, a mass-asymmetry coordinate, a deformation coordinate for each fragment and rotational coordinates for each fragment. To specify fragment deformation, the fragments are represented by spheroids; a nucleus prior to division is represented by two overlapping spheroids. The Hamiltonian for the idealized system consists of a sum of surface, Coulomb and kinetic energies. A study of the saddle-point energies and shapes calculated in this two-spheroid approximation indicates that the approximation is most useful for discussing the fission of elements lighter than about radium. On the basis of this model, we calculate probability distributions for certain observable characteristics of fission fragments at infinity — their total translational kinetic energy, mass, individual excitation energies and individual angular momenta. This is done by applying standard static, dynamical and statistical methods to the Hamiltonian for the system. The present treatment, for the most part, is classical; quantum mechanics is considered only in the statistical-mechanics discussion of the behaviour of the system near the saddle point. The predictions of the model are compared with existing experimental data for distributions in fragment mass and total translational kinetic energy for nuclei lighter than radium. The comparisons are made without the use of any adjustable parameters. The theory is capable of accounting for the magnitudes of the most probable values and widths of the experimental distributions, as well as some, but not all, finer details of the distributions. The dependence of the experimental distributions upon nuclear temperature, and the dependence of the experimental most probable kinetic energies upon the fissility parameter are also approximately reproduced by the calculations.

DYNAMICAL ASPECTS OF NUCLEUS-NUCLEUS COLLISIONS

Abstract This is an extension of the macroscopic theory of nucleus-nucleus reactions described by Swiatecki. The fusion or reseparation of two colliding nuclei is treated according to a schematic model based on the “chaotic regime dynamics” (liquid-drop potential energy plus one-body dissipation). Attention is focused on three hurdles or “milestone configurations” that a colliding system may be faced with: the touching configuration, the conditional saddle-point configuration at frozen mass asymmetry, and the unconditional saddle-point configuration. Semi-empirical formulae are derived for the “extra push” (the extra energy needed in some situations to carry the system from the first to the second hurdle) and for the “extra-extra push” (the energy needed to carry the system from the first to the third hurdle). The theoretical formulae are confronted with measurements of fusion and evaporation-residue cross sections. A discussion of the implications for super-heavy-element reactions is given, using the production of element 107 in the bombardment of 209 Bi with 54 Cr as a calibrating reaction.

THE DEFORMATION ENERGY OF A CHARGED DROP: PART V: RESULTS OF ELECTRONIC COMPUTER STUDIES

Abstract The results of electronic computer studies of equilibrium configurations of an idealized charged drop are presented. The symmetric family of saddle-point shapes has been traced as a function of the fissionability parameter x . The properties of the saddle-point shapes have been tabulated in the interval x = 0.30 to x = 1.00 in steps of 0.02. The appearance of these shapes changes from dumbbell-like for x ≲ 0.67 to cylinder-like for x ≳ 0.67. The transition is fairly rapid, but not discontinuous. The properties tabulated include the energies, moments of inertia, and quadrupole moments of the saddle-point shapes. In addition, the elastic constants (stiffnesses) of the symmetric saddle-point shapes for different types of symmetric and asymmetric distortions have been determined. The shapes were found to be stable against asymmetry down to x = 0.39 4 , at which point an asymmetric family of equilibrium shapes bifurcates. A simple formula is given which reproduces the calculated liquid-drop thresholds with fair accuracy.

A generalization of the Proximity Force Theorem

We generalize the Proximity Force Theorem of J. Blocki et al. (Ann. Phys. (N.Y.)105 (1977), 427) (valid for gently curved surfaces) to include surfaces that may have large curvatures (but are still characterized by small angles between relevant portions of the interacting surfaces). A general proof is given for the approximate continuity of the proximity force when a gap configuration goes over into a crevice after contact. Simple and some-what improved formulae are given for the universal proximity potential functions Φ and φ for gaps and crevices.

Dynamical hindrance to compound-nucleus formation in heavy-ion reactions☆

Abstract The model of a sharp-surfaced drop with one-body dissipation is used to map out the extra-extra push energy Exx(A1, A2) (i.e. the excess bombarding energy above the Coulomb barrier required to form a compound nucleus) in its dependence on the mass numbers A1, A2; of the colliding nuclei. The calculated two-dimensional function Exx(A1, A2) may be scaled approximately to a one-dimensional dependence on a mean fissility parameter x m = 2 3 x + 1 3 x e , where x is the total systems fissility and xe is the entrance channel “effective” fissility. Apart from indications of possible nuclear structure effects, the theoretical predictions seem consistent with experimental data on evaporation residue measurements for systems with A 1 ⋍ A 2 . For asymmetric systems the relation to experiment is unclear. A by-product is the calculation of nucleus-nucleus sticking times, which are generally found to be in the range around a few times 10−21 s, but could be longer in special cases.