Nuclear Optical Potential Research Articles

Nuclear Effects in the Production ofπ+andπ−by Protons on Nuclei

We have modified an earlier semiclassical model of pion production by protons on nuclei by including various nuclear physics corrections. In addition to allowing for $pn$ charge exchange and realistic nuclear densities, we have corrected the two-body input cross sections for the effects of the Pauli principle, Fermi motion, and nuclear optical potentials. The agreement with experiment is similar to that obtained earlier. Ambiguities arising from the nuclear corrections are found to be larger than the effects of moderate variations of the nuclear surface parameters.

Criteria for the Elimination of Discrete Ambiguities in Nuclear Optical Potentials.

Criteria for the Elimination of Discrete Ambiguities in Nuclear Optical Potentials.

Pauli Principle and the Linked Cluster Expansion for the Nuclear Optical Potential

A linked cluster expansion is derived for the nuclear optical potential which includes already in lowest order, in addition to the Hartree-Fock potential, a term which has recently been shown to be large for nucleons scattering from light nuclei. It is shown that this additional term reproduces effects of the Pauli exclusion principle.

Spin-orbit component of optical model from nuclear matter — Spherical and deformed cases

A method of calculating the spin-orbit component of the nuclear optical potential from matter distributions is given. It is more direct than previous methods, and can be generalized to the case of deformed nuclei.

Criteria for the Elimination of Discrete Ambiguities in Nuclear Optical Potentials

The elastic scattering of composite projectiles by nuclei is considered in the semiclassical limit. Criteria based on this description are obtained for the incident energy and angular range of data necessary to eliminate discrete ambiguities in optical potentials for such projectiles. We demonstrate the effectiveness of these criteria for elastic $\ensuremath{\alpha}$-scattering data.

Evidence for energy dependence in the isospin dependent part of the nuclear optical model potential

Angular distributions and the excitation function of the total cross section for the 208Pb(p, n) charge exchange reaction are fitted with DWBA calculations. Isospin potentials which have the relative real volume and surface imaginary strengths found in the optical model studies of Becchetti and Greenlees provide adequate fits to the angular distributions. The excitation function indicates a strong energy dependence in the strength of the isospin potential.

On the collective contribution to the nuclear optical potential

The possible importance of collective effects on the nuclear optical potential, and upon the extraction of geometrical parameters of the matter distribution is discussed. The formal problem is delimited, and an empirical study is made for some cases.

Average nuclear properties

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 2A 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.

Charge exchange in nuclei following pion inelastic scattering

The effect of the inclusion of an isospin dependent term in the nuclear optical potential which can then cause charge exchange has been considered. An experiment determined the ratio, R, for positive to negative pions of 11C production from the interaction of 180 MeV pions with 12C, to be unity. There is a major discrepancy between this result and that predicted by a simple impluse approximation which gives a ratio of R = 3. The inclusion of charge exchange in a nuclear final state interaction has reduced the ratio R to less than 2.

Surface phenomena in Fermi liquids

We formulate a general theory of surface phenomena for a zero-temperature Fermi liquid. The similarities between Classical and Quantum surface motion are studied the comparison leading to an expression for the surface energy of the liquid. We show, within the time-dependent self-consistent field approximation, that the dynamical modes of the system include Landau's zero-sound waves as well as surface vibrations. Exploiting the close relation between “spurious modes” and surface waves, an approximate dispersion relation is derived for the latter. We show that the existence of a gap in the fluid's particle-hole spectrum is essential if surface modes are to propagate. Thus in a “normal” Fermi-fluid odd-parity vibrations propagate at sufficiently long wavelength while even-parity ones are damped. The effect of surface modes on the nuclear optical potential is studied. The propagation of such modes implies a peak in absorption at the surface of the nucleus. The observation of surface waves in 3He is also discussed.

Surface phenomena in Fermi liquids

The surface motion of a finite Fermi fluid is described quite generally. Waves are found to propagate only when there are gaps in the particle-hole spectrum. The effects of surface waves in the nuclear optical potential are studied.

Isospin dependence of the nuclear optical potential

The isospin-dependent term in the optical-model potential has been investigated by comparing results obtained from simultaneous fitting of neutron and proton elastic scattering data from 59Co and 209Bi at an energy approximately equal to 14.4 MeV. The coefficient of the nuclear symmetry parameter term is equal to 23.8±3 MeV if the fits were obtained by keeping fixed the imaginary part of the potential. If this part is allowed to vary and only the best fits are taken into consideration C = 22.8±4 MeV.

Generalized optical model for elastic and quasi-elastic scattering of protons

The isospin dependence of the nuclear optical potential is studied using projection formalism. Exchange effects are taken into account. Instead of Lane's phenomenological potential V 0+V 1 t s ̇ T , we find V 0+V 1(2t 3T 3− t s ̇ T) . The difference leads to a change of the relative sign between the wave functions p and n, describing the motion of the proton and a neutron, respectively. An explicit relation between the functions p, n and the full wave function is given. General expressions for the potentials V 0and V 1 are derived. They may serve as starting point for microscopic theory.

Nonlocal potentials and the energy dependence of the optical model for neutrons

The energy dependence of the nuclear optical model potential for neutrons is derived from a nonlocal formalism. The imaginary part of the potential changes continuously from purely surface absorption at zero energy to purely volume absorption at energies beyond 100 MeV. Most of the parameters are determined by demanding that the equivalent local potential at zero energy coincide with the Moldauer potential while the remaining parameters are varied to fit the total and nonelastic cross sections of various spherical nuclei between 30 and 150 MeV. The resulting potential also produces good fits to elastic angular distributions, and, when an appropriate spin-orbit potential is added, to polarization measurements in this energy region as well as the intermediate region between 7 and 24 MeV. This shows that the energy dependence could be described in terms of nonlocality of the interaction and provides a simple local potential which can be used over the whole energy range from 0 to 150 MeV and probably even beyond.

A spin-spin part of the low-energy nuclear optical potential

Recently some authors have introduced a phenomenological interaction between the spins of the incident nucleon and the target nucleus in the optical potential. However, due to the inavailability of suitable data the parameters of this potential remained undetermined. In the present work an upper limit is placed on the strength of this potential by considering a certain observed difference between the low-energy nuclear data of even and odd mass nuclei.

Scattering of low-energy K--mesons by nuclei

Nuclear scattering of K- mesons of an average energy of 21 MeV has been measured inG5 nuclear emulsion. An optical model calculation of the differential and total reaction cross-sections has been performed. Comparison of experiment and theory supports the conclusion reached at higher K- meson energies that the nuclear optical potential is attractive.

Radiative capture by excitation of collective vibrations: (I). Theory

Compound nucleus capture falls off rapidly with energy above a few MeV. Observed capture cross-sections remain quite large for energies up to 20 MeV or so. A recent paper attempted to explain this discrepancy in terms of a new model of capture, i.e. direct single particle transitions in the nuclear optical potential. However, cross-sections calculated with this model are too small by about one order of magnitude. We now propose an extension of the model in which the potential is not restricted to be spherically symmetric, implying that the incident particle is coupled to collective vibrations of the target, particularly the giant dipole resonance. Capture may occur when a target vibration is excited, and the particle enters a bound orbital. Rough estimates indicate that cross-sections of about the correct order of magnitude and energy dependence can be achieved. More accurate calculations are given in the following paper.

Scattering of 22.5-MeV Protons from Zirconium and the Symmetry Term in the Nuclear Optical Potential

Differential cross sections for the elastic scattering of 22.5-MeV protons were measured from isotopically enriched foils of ${\mathrm{Zr}}^{90}$, ${\mathrm{Zr}}^{91}$, ${\mathrm{Zr}}^{92}$, ${\mathrm{Zr}}^{94}$, and ${\mathrm{Zr}}^{96}$ over an angular range of 20 to 150\ifmmode^\circ\else\textdegree\fi{}. The ratio to the Rutherford cross section exhibits a smooth decrease in the angular position of the maxima and minima as the mass of the target nucleus is increased. The magnitude of these shifts of the maxima and minima is larger than expected on the basis of previously observed 22.2 -MeV proton elastic scattering on targets over a large range of $A$ values. This larger shift in the angular position of the features is in agreement with a dependence of the real nuclear potential well depth on the nuclear symmetry parameter, $\frac{(N\ensuremath{-}Z)}{A}$. A detailed optical model fit to the data yields a real nuclear potential of the form, $\ensuremath{-}V=[43.4+27.8\frac{(N\ensuremath{-}Z)}{A}]$ MeV.

(d, p) stripping reaction and (d, d) elastic scattering on Cr 50, Cr 52, Cr 53, Cr 54 and Fe 54, Fe 56 and Fe 57

A method is described for determining the absolute differential cross sections for reactions using a multi-angle magnetic analyser. The energy spectra and angular distributions of protons from the (d, p) reactions on the targets of enriched Cr 50, 52, 53, 54 and Fe 54, 55, 57 are studied. Altogether 153 levels of final nuclei are studied, including 48 new ones. The l and Θ 2 values are obtained for 93 levels, including 58 new ones. A well-pronounced dependence of the cross sections for the (d, p) reaction on the parameter of the target nucleus symmetry is established. The occupation numbers of the neutron 2p subshell are found from the treatment of the experimental data using the sum rule and the results are compared with other data concerning the the (d, t) reaction. The isobaric spin constant of the nuclear optical potential is estimated by the shift of single-particle neutron levels after the addition of two neutrons.

The energy and isotopic spin dependence of the nuclear optical model potential

The energy and isotopic spin dependence of the nuclear optical model potential