W.D. Myers

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.

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.

Geometric properties of leptodermous distributions with applications to nuclei

Abstract Relationships between geometrical properties of leptodermous distributions are employed in the interpretation of experimentally determined nuclear density distributions and optical-model potential wells. It is found that nuclear sizes imply a density for neutral symmetric nuclear matter corresponding to r0 = 1.16 fm (krmf = 1.31 fm−1) and that the densities and potentials can be related to each other by means of a saturating two-body interac1tion.

Development of the semiempirical droplet model

Abstract The motivation behind the development of the droplet model is presented with the mass formula that results. Values for the 16 coefficients appearing in this new formula are given. Most of these values were taken from the literature or estimated on the basis of simple physical arguments. However, the four main (liquid-drop model) coefficients and some of the new coefficients introduced by the droplet model were determined by fitting to masses, deformations, fission barriers, and radii. Graphical comparison is made between measured values of a number of different nuclear properties and the corresponding droplet-model predictions. In addition, the predicted values of the mass defects of 8000 nuclides with N and Z ≥ 10 are given in the main table of this issue.

AN UPDATE ON DROPLET MODEL CHARGE DISTRIBUTIONS

Abstract The droplet-model expressions for calculating various moments of the nuclear charge distribution are given. There are contributions to the moments from the size and shape of the system, from the internal redistribution induced by the Coulomb repulsion, and from the diffuseness of the surface. A case is made for the use of diffuse charge distributions generated by convolution as an alternative to Fermi functions.

A model for high-energy heavy-ion collisions

Abstract A model is developed for high-energy heavy-ion collisions that treats the variation across the overlap region of the target and projectile in the amount of energy and momentum that is deposited. The expression for calculating any observable takes the form of a sum over a series of terms, each one of which consists of a geometric, a kinematic, and a statistical factor. The geometrical factors for a number of target projectile systems are tabulated.

Droplet model nuclear density distributions and single-particle potential wells

Abstract A description is given of the nuclear density distributions and single-particle potential wells that arise in the course of Thomas-Fermi calculations of average nuclear properties. Simple expressions are given for the calculation of the essential characteristics of these distributions, and it is shown how the results obtained here may be used to approximate the densities and potential wells in terms of Fermi functions.

Droplet-model theory of the neutron skin

The droplet-model theory of the neutron skin is reviewed and an elementary formula is derived for the associated difference between the RMS radii of the neutron and proton density distributions. The resulting predictions are compared with recent experimental estimates and with Hartree-Fock calculations. There appears to be no serious disagreement with most of the current, tentative data. Improved measurements should eventually make possible an independent estimate of the stiffness coefficient Q, governing the resistance of the nuclear surface against the formation of a neutron skin.

Droplet model isotope shifts and the neutron skin

Abstract The isotope and isotone shifts in the nuclear charge radius and the thickness of the neutron skin are calculated for nuclei along beta stability by means of the droplet model.