Macroscopic response of the nuclear surface

The macroscopic surface response of a semi-infinite Fermi liquid is considered. On the basis of Landau's kinetic equation, a dispersion relation is derived for the free surface modes. In the limit of weak quasiparticle interaction we find a surface mode with a purely imaginary frequency. This mode is shown to be critically damped if the stiffness is given by the excess surface energy and the friction by the one-body wall formula. By assuming a correspondence with the quantum relation between dissipation rate and response function, the macroscopic surface response function is obtained. The results are applied to inelastic scattering of protons from nuclei. While the qualitative behavior of the nuclear response is reproduced by the semi-infinite model, the decrease with energy is too rapid.     

Semi-infinite matter and the nuclear surface

The surface energy and thickness of semi-infinite nuclear matter are calculated by Swiatecki's variational method. Smooth, effective two-body interactions are used to study the relationship between the characteristics of each interaction and the predicted bulk surface properties. Both local and non-local Gaussian interactions are considered and are adjusted to either saturate infinite nuclear matter or to fit two-nucleon (singlet-even) phase shifts. Purely local potentials are found to give an excessively large surface energy ( ), whereas potentials having some non-locality (even if the non-locality is confined to short distances) reduce the surface energy to a reasonable value of (20–30) . The separate potential and kinetic energy contributions are examined in detail and are used to explain how the character of each interaction affects the surface properties. This model calculation shows that the surface energy is greatly influenced by non-locality. Also, we find that the predicted surface thickness tends to be too large (t ≈ 3.5 fm) for potentials that are adjusted to saturate and too small (t ≈ 1.8 fm) for potentials that are fitted to two-nucleon phase-shifts.     

Nonlinear evolution of the axisymmetric nuclear surface

We consider a uniformly charged incompressible nuclear liquid bounded by a closed surface. It is shown that the evolution of an axisymmetric surface Г(r, t) ≡ σ ? ∑(z, t) = 0, r = (σ, φ, z) can be approximately reduced to the motion of a curve in the (σ, z) plane. A nonlinear integro-diffrerential equation for the contour Σ (z, t) is derived. The contour Σ (z, t) and the local curvature are found to be a direct correspondence, which makes it possible to use methods of differential geometry to analyze the evolution of an axisymmetric nuclear surface.     

Role of the surface in nuclear pairing

Recent results obtained by solving the ab initio equation for the pairing gap in a slab of nuclear matter are reviewed. They conclusively prove a surface nature of nucleon pairing in nuclei. First, the effective pairing interaction in the vicinity of the slab surface makes a dominant contribution to the gap. Second, there is a sizable surface enhancement of the pairing gap Δ(X) itself. Finally, an analysis of spatial correlation features of the anomalous density matrix shows that the local correlation length at the slab surface is very small, about 1.5 fm, but, in the interior of the slab, it is rather large, about the slab thickness. These results are in qualitative agreement with those obtained recently by Pillet and his coauthors for spherical nuclei by using the phenomenological Gogny force. An increase in the concentration of Cooper pairs at the surface of nuclei was yet another result of their study, and our present analysis confirms this indirectly.     

Clustering in the region of nuclear surface

The aim of this report is to formulate the approach to describe evolution of a localized density fluctuation in the region of a nuclear surface.     

Description of the nuclear surface by moments

The parameters proposed by W. D. Myers to describe the radial shape dependence of the nuclear surface are shown to be as useful as they are natural. For spherical nuclei, the central radiusC, the charge radiusR, the quadratic radiusQ are redefined, and it is shown how they are interrelated by Myers' surface widthb, flair γ₃, crookedness γ₄, and the higher shape parameter γ₅. All these quantities are calculated for some special charge distribution functions. (For symmetric distributions, possessing a symmetry center atC, the odd surface momentsb ³γ₃ andb ⁵γ₅ vanish.) The connection of the surface moments Г_μ =b ^μ γ _μ with the volume momentsF _k that have been extracted by K. W. Ford and J. G. Wills from muonic atoms, is indicated.     

Electron wave functions at the nuclear surface

Finite size corrections of Bhalla and Rose are examined in the midway to the point-nucleus limit. A possible reason of poor convergence of their positron wave functions is suggested.     

The breathing mode and the nuclear surface

The role of the nuclear surface in the breathing mode of nuclei is analyzed. We discuss a simple model in which the density varies according to a scaling of the coordinates. We show that this model reproduces accurately the results of microscopic calculations in heavy nuclei, and we use it to estimate the contribution of the surface to the effective compression modulus of semi-infinite nuclear matter. The calculation is performed in the framework of an extended Thomas-Fermi approximation and using several effective interactions. It is shown that the surface energy is maximum with respect to variations of the density around saturation density. The reduction of the effective compression modulus due to the surface turns to be proportional to the bulk compression modulus. The magnitude of the effect is compared with results of RPA calculations. Other contributions to the effective compression modulus of finite nuclei are also discussed.     

Further studies of the macroscopic nuclear surface response

The macroscopic response of a semi-infinite Fermi fluid is reconsidered. By retaining the first two terms in the expansion of the normal surface stress in powers of the external frequency, we obtain an improved dispersion relation which yields explicit expressions for the friction coefficient and the inertial mass for the damped surface motion. The resulting response function has a finite integral and yields a good absolute reproduction of the response observed via inelastic proton scattering.     

Temperature dependence of nuclear surface properties

Thermal properties of nuclear surface are investigated in a semi-infinite medium. Explicit analytical expressions are given for the temperature dependence of surface thickness, surface energy and surface free energy. In this model the temperature effects depend critically on the nuclear incompressibility and on the shape of the effective mass at the surface. To illustrate the relevance of these effects we made an estimate of the temperature dependence of the fission barrier height.     

Remarks concerning the nuclear surface and curvature energy

The $\text{A}\text{1}\text{3}$ 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.     

Aspects of alpha-particle scattering and structure of the nuclear surface

The interaction of α particles above 5 MeV with a 2s-1d target is dominated by resonances. It cannot be described only in terms of a mean-field one-body potential. An analysis of the elastic α-particle scattering by ²⁸Si encourages the comprehension of the resonance states to be mainly fragments of a mixed-parity band. In the present article, the angular distributions of particles scattered by ³²S are analyzed in terms of such bands. The analysis of new data from an experiment made at Florida State University reveals the existence of states that do not belong to the above bands. This follows from a coupled-channel analysis of the elastic and inelastic (2⁺) cross sections. An α-particle structure at the nuclear surface is suggested.     

The deformation of the Fermi surface in polarized nuclear matter

Existence of the tensor components in the quasi-particle interaction implies the deformation of Fermi surfaces in spin and spin-isospin polarized nuclear matter. A set of linear equations is derived from which the values of the deformation parameters can be calculated from the Landau parameters of the spin dependent terms in the quasi-particle interaction in nuclear matter. The effect of the deformation on the spin and spin-isospin symmetry energy and on the single particle potential in nuclear matter is discussed. Numerical results are obtained for the Reid soft-core potential using suitable approximations.     

Single-particle density distributions of silver in the extreme nuclear surface

The single-particle wave functions of ¹⁰⁷Ag beyond the nuclear potential are calculated exactly. It is shown that the centrifugal barrier cannot be neglected in the region of the last classical turning point of the nuclear density. Thus K⁻ meson absorption by emulsions cannot be explained by the difference between the effect of the Coulomb barrier in heavy and light emulsion nuclei.     

The nuclear surface as a collective variable

In heavy nuclei where the thickness of the diffused edge is relatively small, a certain sharp effective surface can be defined which characterizes the shape of the nucleus, and it can be considered as a collective dynamic variable. It is shown that the problem of fluid dynamics can be simplified by reducing it to simple linearized equations for the dynamics in the nuclear interior and boundary conditions set at the effective dynamic sharp surface of the density distribution. These conditions are derived from the fluid dynamical equations. Transitional densities obtained from this simple model are compared with the numerical solution of fluid dynamical equations.