Narrow Results

By mapping the scalar Helmholtz equation (SHE) to the Schrödinger form we investigate the behavior of $\mathcal{𝒫}\mathcal{𝒯}$ optical structure when the refractive index distribution n admits variation in the longitudinal direction only. Interpreting the Schrödinger equation in terms of a superpotential we determine the supersymmetric partners for n. We also obtain new analytical solutions for the refractive index profiles and provide graphical illustrations for them.

The optical potential of an attractive nonrelativistic electron gas interacting with nuclear matter is determined on the basis of the concept of degenerate Fermi gas. In fact, the involved electrons are treated as three-dimensional quantum harmonic oscillators confined at the surface of a spherical (approximately ideal) potential well. Within this picture, the Fermi velocity is calculated as well as the spatial electron density at the surface of the potential well and the attractive force between the electron gas and the nuclear matter. In addition, considerations related to the Lippmann–Schwinger model are made.

Calculations of total reaction cross-sections and differential cross-sections using the modified Glauber model II for p–¹⁶O scattering in the energy range 100–497.5 MeV are compared with experimental data. The real parts of the nuclear central potential and spin-orbit potential were used to calculate the modified Glauber model II. The nuclear potential was constructed using two approaches: the Dirac-equation-based optical potential and the non-relativistic treatment. The phenomenological and analytical methods are used to calculate these two approaches. The strength of the real parts of central and spin-orbit potentials are normalized to the best fit to the data. The best values were obtained close to unity. Most of the calculations derived from the Dirac-equation-based optical potentials were more comparable with the experimental data than the non-relativistic calculations. This may be attributed to the delicate cancellation between the short-range repulsive and the long-range attractive contributions. The present investigation indicated that a good choice of the potential at given energy offers a reliable determination of the impact parameter estimated according to the modified Glauber model II.

The high energy single folding optical potential approximation is studied to calculate the differential cross-section for proton elastic scattering of ¹²C at 156 MeV and 1440 MeV and ¹²C for state 2⁺ (4.44 MeV) at 1440 MeV. A Gaussian nuclear density distribution was used for the proton and Gaussian and Brink nuclear density distributions for the ¹²C target. We used the following three effects to derive twelve different methods for the central optical potential: (i) Love and Franey and the Gaussian amplitudes, with the Brink and one-term Gaussian nuclear density distributions, (ii) Pauli correlation in the Gaussian amplitude with these densities, (iii) coupling channels on the differential cross-sections in proton elastic scattering of ¹²C at 1440 MeV with single channel calculations using these amplitudes, nuclear density distributions and Pauli correlation in the Gaussian amplitude. A new numerical technique was performed to solve the deformed optical potential equations using computational programs.

We have compared the binding energy of nuclear matter and the nucleon–nucleus optical potential, calculated in Brueckner theory starting from both the soft-core Urbana V-14 and the hard-core Hamada–Johnston internucleon potentials. Our results show that the real central part of the optical potential calculated from V-14 is about 10 MeV deeper than from the hard-core potential in the energy region 1–200 MeV. This greater depth mainly comes from the internucleon S- and D-states. In these states, the V-14 and Hamada–Johnston potentials give different phase shifts, the V-14 being in better agreement with experimental data. This difference is further enhanced by the Pauli principle in the calculation of the optical potential. Our analysis of proton–⁴⁰Ca differential cross-section and polarization data, in the energy region 30–200 MeV, shows that the optical potential calculated using V-14 is in better agreement with the data as compared with the Hamada–Johnston potential.

The nucleon scattering on even–even nuclei in the medium-energy region has been analyzed on the basis of microscopic optical potential (OP) obtained from nuclear-matter calculations with using effective density-dependent nucleon–nucleon interaction of Skyrme type with taking account of the rearrangement potential. Calculations have been performed for volume integrals and rms radii of nucleon–nucleus OP, for energy dependencies of total and total reaction cross sections of neutron– and proton–nucleus scattering and for differential cross sections of the elastic neutron scattering at several energies on various target nuclei. Comparison of the calculation results for the mentioned quantities with corresponding experimental data has been carried out, which has shown a principal possibility of their reasonable description in the framework of the model under consideration.

Proton elastic scattering at various incident energies is one method to study nuclear density distributions and nuclear radii. Single folding potential describing the p-scattering on ⁴⁰Ca over a broad energy range 9–48.4 MeV is constructed. The resulting potential does not need any renormalization to fit the measured elastic scattering angular distributions and total reaction cross-sections. Furthermore, correlation between volume integral and proton incident energy is discussed. Theoretical calculations are in a good agreement with existing experimental data.

The $(\mathrm{\text{p}},\mathrm{\text{2p}})$ reaction on ${}^{40}\mathrm{\text{Ca}}$ at incident proton energy of 300MeV is examined in the formalism of finite-range relativistic distorted-wave impulse approximation (FR-RDWIA). In comparison to conventional t-matrix model of Love–Franey, a new form of nucleon–nucleon t-matrix effective interaction is derived at 300MeV using Reid soft core potentials for isotopic spin one and taking into account the finite-range effects in the $\mathrm{\text{p}}-\mathrm{\text{p}}$ interaction at knockout vertex. In comparison to the conventional finite range nonrelativistic and relativistic formalism, the present formalism with a new version of $\mathrm{\text{p}}-\mathrm{\text{p}}$ t-matrix is effectively reproducing the shape of cross-section energy distributions for ${\mathrm{\text{1d}}}_{3/2}$, ${\mathrm{\text{1d}}}_{5/2}$ and ${\mathrm{\text{2s}}}_{1/2}$ states for asymmetric angle pair of $30^{\circ }$–$55^{\circ }$. Discrepancies between the experimental cross-section data and finite range theoretical calculations at ${\mathrm{\text{E}}}_p=300$MeV are reasonably resolved in the present approach. Without any adjustable parameter of bound state, the obtained spectroscopic factors are in reasonably good agreement with the relativistic and nonrelativistic theoretical predictions by $(\mathrm{\text{p}},\mathrm{\text{2p}})$, $(\mathrm{\text{e}},{\mathrm{\text{e}}}^{\prime }\mathrm{\text{p}})$ and $(\mathrm{\text{d}},{}^3\mathrm{\text{He}})$ analysis.

Alpha elastic scattering of $p$-nuclei was studied for calculating optical potentials. Choice of the $\alpha$-optical potentials is important to measure the reaction rates of $p$-process. ${}^{106}\mathrm{\text{Cd}}{(\alpha ,\alpha )}^{106}\mathrm{\text{Cd}}$ and ${}^{113}\mathrm{\text{In}}{(\alpha ,\alpha )}^{113}\mathrm{\text{In}}$ elastic scattering cross-section data were used to determine the potential parameter sets at $E_{\mathrm{\text{lab}}}=16.1\text{–}27$MeV for ${}^{106}\mathrm{\text{Cd}}$ and $E_{\mathrm{\text{CM}}}=15.59$, 18.8MeV for ${}^{113}\mathrm{\text{In}}$ system. A Wood–Saxon potential form factor is used for both real and imaginary parts. The potential parameters extracted in the present study exhibit a satisfactory result with respect to existing global potential parameters.

The method of impulse approximation is used to check the validity of the first-order optical potential for the elastic scattering problem of the neutron on the bound system, namely, ${\mathrm{\text{C}}}^{12}(n.n){\mathrm{\text{C}}}^{12},{\mathrm{\text{Ca}}}^{40}(n,n){\mathrm{\text{Ca}}}^{40}$ and ${\mathrm{\text{Pb}}}^{208}(n,n){\mathrm{\text{Pb}}}^{208}$at incident neutron energies of 155 and 225MeV. The optical potential is derived as the first-order term within the spectator expansion of a nonrelativistic multiple scattering terms using the Lippmann–Schwinger equation. The Modern realistic two-body potential ArgonneV18 in the momentum space was used as input in the Lippmann–Schwinger equation. The obtained results for the elastic differential cross-sections are in a good agreement with the experimental data taken from EXFOR Database for all studied targets at neutron energy above 200MeV. As the neutron energy decreases down to approximately 155MeV, the discrepancies with experimental data appear, which is in accordance with the impulse approximation formalism.

These lectures, held at a school at the Galileo Galilei Institute for Theoretical Physics, give a survey of the formalism, the methods and achievements of nuclear direct reaction theory. After recapitulating the principles of quantum scattering theory, projection techniques are used to reduce the complexities of nuclear reactions to the manageable level of a set of a few selected coupled channels interacting via effective operators. Direct reactions are introduced as fast reactions proceeding preferentially through the excitation of doorway components of the nuclear many-body configurations, corresponding to a separation of scales in energy and momentum transfer. The concept of the optical model is introduced and applied to elastic scattering on nuclei. The coupled channel T-matrix formalism is discussed for pion-induced reactions on the nucleon. Applications of direct reaction to theory are illustrated for transfer reactions, inelastic scattering, and single and double charge exchange reactions with light and heavy ions.

A microscopic approach was used to analyze the elastic scattering of protons on ${}^{10,11}$Be at incident energies of 9.1, 15.1, 39.1, and 59.3MeV/nucleon for ${}^{10}$Be and ${}^{11}$Be (38.4MeV/nucleon and 49.3MeV/nucleon) within the framework of single folding (SF) and São-Paulo (SP) potentials. Also, the energy dependence of the reaction cross-sections of the considered reactions is investigated. The study was performed using FRESCO code on the free parameters ($N_r,N_I,N_S,W_s,W_v,r_I$ and $a_I$). The DFPOT code was used to compute the SF and SP potentials microscopically for the real and imaginary parts of the optical potential (OP), respectively. The OP was derived using two different nuclear density distributions since ${}^{11}$Be as Gaussian–Gaussian (GG) $\rho \mathrm{\text{GG}}$ and Gaussian–Oscillator (GO) $\rho \mathrm{\text{GO}}$, and the effective N–N interactions were represented in the Gaussian form. The resulting differential cross-sections agreed well with the given experimental data.

The main questions which are still open concerning the optical model potential for α-particles at low energies are discussed with respect to the α-particle elastic scattering as well as α-induced reactions and α-particle emission. In order to understand the differences between the optical potential parameters used to describe the α-particle emission from excited compound residual nuclei, as compared to those determined by analysis of α-particle elastic scattering, the double-folding model is involved within semi-microscopic analysis of the α-particle elastic scattering on medium-mass nuclei, at energies below 30 MeV, and next involved within calculations of (n, α) reaction cross sections. The corresponding phenomenological optical potentials which match each other in the outer limit of the nuclear surface are shown to be described by a temperature-dependent shape of the nuclear density, which produces decreased central values and a larger diffuseness of the DF real potential.

Elastic scattering of ⁶He on both proton and carbon targets were measured at 82.3 MeV/nucleon. For both targets, the measured differential cross sections show a large enhancement at small angles relative to the Rutherford cross section, similar to those observed at lower energies for the scattering of halo nuclei. For the proton target, the experimental results were well reproduced by optical model calculations by using the global potential KD02 with a reduction of the real part, as well as by using the microscopic JLM potential with a reduction of imaginary part. For the carbon target, the overall structure of the cross section is reasonably reproduced by the optical model calculations.

The quasi-elastic scattering angular distribution of the proton drip line nucleus ¹⁷F on a ¹²C target was measured at 60 MeV. The experimental data have been compared with theoretical analysis based onto optical model and continuum discretized coupled channels (CDCC). The couplings between breakup and elastic scattering channels, and between inelastic and elastic scattering channels resulted weakly.