Elastic $^{16}$O+$^{12}$C scattering is known to exhibit the nuclear rainbow pattern at incident energies $E_\text{lab}\gtrsim 200$ MeV, with the Airy structure of the far-side scattering cross section clearly seen at medium and large angles. Such a rainbow pattern is well described by the deep real optical potential (OP) given by the double-folding model (DFM). At lower energies, the extensive elastic $^{16}$O+$^{12}$C scattering data show consistently that the nuclear rainbow pattern at backward angles is deteriorated by an oscillating enhancement of elastic cross section that is difficult to describe in the conventional optical model (OM). Given a significant $α$ spectroscopic factor predicted for the dissociation $^{16}$O$\toα+^{12}$C by the shell model and $α$-cluster models, the contribution of the elastic $α$ transfer (or the core-core exchange) to the elastic $^{16}$O+$^{12}$C scattering should not be negligible and is expected to account for the enhanced elastic cross section at backward angles. To reveal the impact of the elastic $α$ transfer, a systematic coupled reaction channels analysis of the elastic $^{16}$O+$^{12}$C scattering has been performed, with the coupling between the elastic scattering and elastic $α$ transfer channels treated explicitly, using the real OP given by the DFM. We found that the elastic $α$ transfer enhances the near-side scattering significantly at backward angles, giving rise to an oscillating distortion of the smooth Airy structure. The dynamic polarization of the OP by the coupling between the elastic scattering and elastic $α$ transfer channels can be effectively taken into account in the OM calculation by an angular-momentum (or parity) dependent potential added to the imaginary OP, as suggested by Frahn and Hussein 40 years ago.
It is shown that a dilute property of an $α$ particle condensate can be seen in the Airy structure of nuclear rainbow and prerainbow scattering. The dilute property of the Hoyle state of $^{12}$C with a developed $α$ cluster structure is discussed by studying refractive $^3$He+$^{12}$C and $α$+$^{12}$C scattering.
We present evidence for the nuclear ripples superimposed on the Airy structure of the nuclear rainbow, which is similar to the meteorological rainbow. The mechanism of the nuclear ripples is also similar to that of the meteorological rainbow, which is caused by the interference between the externally reflective waves and refractive waves. The nuclear ripple structure was confirmed by analyzing the elastic angular distribution in $^{16}$O+$^{12}$C rainbow scattering at $E_L$=115.9 MeV using the coupled channels method by taking account of coupling to the excited states of $^{12}$C and $^{16}$O …
Elastic scattering of alpha-particle and some tightly-bound light nuclei has shown the pattern of rainbow scattering at medium energies, which is due to the refraction of the incident wave by a strongly attractive nucleus-nucleus potential. This review gives an introduction to the physics of the nuclear rainbow based essentially on the optical model description of the elastic scattering. Since the realistic nucleus-nucleus optical potential (OP) is the key to explore this interesting process, an overview of the main methods used to determine the nucleus-nucleus OP is presented. The refractive …
The nuclear rainbow observed in the elastic $α$-nucleus and light heavy-ion scattering is proven to be due to the refraction of the scattering wave by a deep, attractive real optical potential. The nuclear rainbow pattern, established as a broad oscillation of the Airy minima in the elastic cross section, originates from an interference of the refracted far-side scattering amplitudes. It is natural to expect a similar rainbow pattern also in the inelastic scattering of a nucleus-nucleus system that exhibits a pronounced rainbow pattern in the elastic channel. Although some feature of the nucle…
Rainbows are generally considered to be caused by static refraction and reflection. A primary and a secondary rainbow appear due to refraction and internal reflection in a raindrop as explained by Newton. The quantum nuclear rainbow, which is generated by refraction in the nucleus droplet, only has a "primary" rainbow. Here we show for the first time evidence for the existence of a secondary nuclear rainbow generated dynamically by coupling to an excited state without internal reflection. This has been demonstrated for experimental $^{16}$O+$^{12}$C scattering using the coupled channel method …
The Dyson-Schwinger equations of QCD, truncated to ladder-rainbow level, are used to calculate meson form factors in impulse approximation. The infrared strength of the ladder-rainbow kernel is described by two parameters fitted to the chiral condensate and f_pi; the ultraviolet behavior is fixed by the QCD running coupling. This obtained elastic form factors F_pi(Q^2) and F_K(Q^2) agree well with the available data. We also calculate the rho to pi gamma and K* to K gamma transition form factors, which are useful for meson-exchange models.
A semiclassical analysis of an optical potential cross section is presented. The cross section considered is characterized by the appearance of an Airy-like pattern. This pattern is similar to that which is present in many cross sections, which fit the recent measurements of light heavy ion elastic scattering, and is considered as a manifestation of a rainbow phenomenon. The semiclassical analysis shows that, in the case considered, the oscillations arise from the interference between the contributions from two different terms of a multi-reflection expansion of the scattering function, and, th…
The existence of a supernumerary nuclear rainbow in inelastic scattering is reported. This is done by studying inelastic $^{16}$O scattering from $^{12}$C, exciting the $2^+$ (4.44 MeV) state of $^{12}$C and elastic scattering at the incident energies in the range 124 to 200 MeV, using the coupled channels method. An extended double folding potential is used. This is derived from realistic wave functions for $^{12}$C and $^{16}$O calculated with a microscopic $α$ cluster model and a finite-range density-dependent nucleon-nucleon force. Excitations to the $2^+$ (4.44 MeV), 3$^-$ (9.64 MeV) and …
The Pauli exclusion principle plays an important role in many-body fermion systems preventing them from collapsing by repulsion. For example, the Pauli principle causes a repulsive potential at short distances between two $α$ particles. On the other hand, the existence of nuclear rainbows demonstrates that the inter-nuclear potential is sufficiently attractive in the internal region to cause refraction. The two concepts of repulsion and attraction are seemingly irreconcilable. Contrary to traditional understanding, it is shown that the Pauli principle causes a {\it universal structural Pauli a…
The Airy structure in $^{16}$O+$^{14}$C rainbow scattering is studied with an extended double folding (EDF) model that describes all the diagonal and off-diagonal coupling potentials derived from the microscopic realistic wave functions for $^{16}$O using a density-dependent nucleon-nucleon force. The experimental angular distributions at $E_L$=132, 281 and 382.2 MeV are well reproduced by the calculations. By studying the energy evolution of the Airy structure, the Airy minimum at around $θ$=76$^\circ$ in the angular distribution at $E_L$=132 MeV is assigned as the second order Airy minimum $…
Inelastic $^{16}$O +$^{12}$C rainbow scattering to the $2^+$ (4.44 MeV) state of $^{12}$C was measured at the incident energies, $E_L$ = 170, 181, 200, 260 and 281 MeV. A systematic analysis of the experimental angular distributions was performed using the coupled channels method with an extended double folding potential derived from realistic wave functions for $^{12}$C and $^{16}$O calculated with a microscopic $α$ cluster model and a finite-range density-dependent nucleon-nucleon force.The coupled channels analysis of the measured inelastic scattering data shows consistently some Airy-like …
Quasi-elastic responses in nuclear matter and in $^{12}$C and $^{40}$Ca nuclei are calculated in ring approximation to investigate the finite size effects on the electromagnetic quasi-elastic responses. A method to simulate these effects in infinite systems calculations is proposed. The sensitivity of the results to the various terms of the residual interaction is studied. The results of nuclear matter RPA calculations are compared with those obtained in ring approximation to evidence the importance of the exchange terms.
The elastic properties of neutron star crusts are relevant for a variety of currently observable or near-future electromagnetic and gravitational wave phenomena. These phenomena may depend on the elastic properties of nuclear pasta found in the inner crust. We present large scale classical molecular dynamics simulations where we deform nuclear pasta. We simulate idealized samples of nuclear pasta and describe their breaking mechanism. We also deform nuclear pasta that is arranged into many domains, similar to what is known for the ions in neutron star crusts. Our results show that nuclear past…
A model to calculate nuclear responses considering short-range correlation effects is presented. The model is applied to the study of electromagnetic responses induced by one-body operators. We calculate one- and two-nucleon emission responses and cross sections of the 16O and 40Ca nuclei in the quasi-elastic region, and we compare them with experimental data.
The effects of nuclear re-interactions in the quasi-elastic neutrino-nucleus scattering are investigated with a phenomenological model. We found that the nuclear responses are lowered and their maxima are shifted towards higher excitation energies. This is reflected on the total neutrino-nucleus cross section in a general reduction of about 15% for neutrino energies above 300 MeV.
The problem of discontinuous functions and their representations in the form of Legendre polynomial series in quantum nuclear scattering theory is presented briefly. The problem is quite old yet not adequately explained in numerous Quantum Theory textbooks and sometimes not correctly understood by physicists. This problem might be quite important in interpretation and analyses of nuclear elastic scattering data and so requires to be stated in a more detailed and clear way.
The Airy structure of the nuclear rainbow and prerainbow in inelastic and elastic $α+^{16}$O scattering is studied with the coupled channel method using a folding potential derived from the microscopic wave functions of $^{16}$O. The theoretical calculations reproduce the characteristic energy evolution of the Airy minimum of the experimental angular distributions. The energy levels with $α$-cluster structure in $^{20}$Ne are reproduced well using the potentials determined from the analysis of scattering. It is shown that the emergence of the $K=0_3^+$ $α$-cluster band with core excitation at …
The existence of a secondary bow is confirmed for 13C+12C nuclear rainbow scattering in addition to the 16O+12C system. This is found by studying the experimental angular distribution of 13C+12C scattering at the incident 13C energy $E_L$=250 MeV with an extended double folding (EDF) model that describes all the diagonal and off-diagonal coupling potentials derived from the microscopic wave functions for 12C using a density-dependent nucleon-nucleon force. The Airy minimum at θ$ $\approx$70$^\circ$, which is not reproduced by a conventional folding potential, is revealed to be a secondary bow …
A nuclear-elastic $^9$Be+$^4$He reaction is investigated. A tangential literature overview of the proximity potential is presented, necessitated by an approximate method for calculating $l\text{-values}$. The model is surveyed intensively, from classical conception (liquid drop model) to physical implication (fusion barrier height), leading to a second-part calculation of the $^9$Be+$^4$He angular distribution. The results obtained are limited by the exclusion of phase variation, but provide enough accuracy to give order-of-magnitude estimates for nuclei interacting peripherally, in agreement …
The quasi-elastic neutrino-nucleus cross section has been calculated by using a Fermi gas model corrected to consider the re-scattering between the emitted nucleon and the rest nucleus. As an example of the relevance of this effect we show results for the muon production cross section on 16O target.
A systematic study of the in-medium correction factor ($F$) on nucleon-nucleon elastic cross section is performed within the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) model. The effects of the beam energy dependence of $F$ on the directed, elliptic flow and nuclear stopping in $^{197}$Au+$^{197}$Au collisions with energy ranging from $0.09$ to $0.8A$ GeV are explored. It is found that the directed, elliptic flow and nuclear stopping at relatively low energies are very sensitive to $F$, and the sensitivity gradually weakens with increasing beam energy. The beam energy dependent in-m…
The elastic scattering of the halo nucleus $^6$He from heavy targets at incident energies near the Coulomb barrier displays a marked deviation from the standard Fresnel-type diffraction behavior. This deviation is due to the strong Coulomb dipole breakup coupling produced by the Coulomb field of the heavy target, a specific feature of the nuclear structure of $^6$He. We have performed Continuum Discretized Coupled Channels calculations for the elastic scattering of $^{6}$He and $^6$Li from $^{58}$Ni, $^{120}$Sn, $^{144}$Sm, $^{181}$Ta and $^{208}$Pb targets in order to determine the range of $…
We explore the nuclear responses at intermediate energies, particularly in the spin longitudinal and spin transverse isovector channels, within the continuum random phase approximation framework. We also employ an extension of the standard random phase approximation to account for the spreading width of the single particle states through the inclusion of a complex and energy-dependent nucleon self-energy. The nuclear responses are then used as the basic ingredient to calculate hadronic reactions in the Glauber theory framework. Here both one and two-step contributions to the multiple scatterin…
Elastic electron scattering from oriented odd-Aaxially deformed nuclei is studied in the plane-wave Born approximation. The nuclear structure is described within a microscopic selfconsistent Skyrme deformed Hartree-Fock approximation with pairing correlations. The interference form factors between monopole and quadrupole Coulomb terms that characterize the nuclear response with aligned nuclear targets are shown to increase or decrease the unpolarized cross-section, depending on the sign of the quadrupole deformation. This feature provides valuable information on the nuclear deformation that ca…
Motivated by arguments of the nuclear core-layer model formulated in [S.I. Bastrukov, J.A. Maruhn, Z.Phys. A 335 (1990) 139], the macroscopic excitation mechanism of electric pygmy dipole resonance (PDR) is considered as owing its origin to perturbation-induced effective decomposition of nucleus into two spherical domains - undisturbed inner region treated as static core and dynamical layer undergoing elastic shear vibrations. The focus is placed on the imprinted in the core-layer model mechanism of emergence of the low-energy dipole electric resonant excitation as Goldstone soft mode of trans…
I investigate the elastic electron scattering off nuclei far from the stability line. The effects of the neutron and proton skins and halos on the differential cross sections are explored. Examples are given for the charge distribution in Sn isotopes and its relation to the neutron skin. The neutron halo in $^{11}$Li and the proton halo in $^{8}$B are also investigated. Particular interest is paid to the inverse scattering problem and its dependence on the experimental precision. These studies are of particular interest for the upcoming electron ion colliders at the GSI and RIKEN facilities.
We formulate microscopic optical potentials for nucleon-nucleus scattering from chiral two- and three-nucleon forces. The real and imaginary central terms of the optical potentials are obtained from the nucleon self energy in infinite nuclear matter at a given density and isospin asymmetry, calculated self-consistently to second order in many-body perturbation theory. The real spin-orbit term is extracted from the same chiral potential using an improved density matrix expansion. The density-dependent optical potential is then folded with the nuclear density distributions of 40Ca, 42Ca, 44Ca, a…
The model independent procedure of extracting charge density distributions from elastic electron scattering data is investigated. The charge density distributions are expanded on an orthonormal basis and the parameters of the expansions are fixed by the comparison with the experimental data. Two bases with different analytical properties (Fourier-Bessel and Hermite) are used. This allows us to disentangle the uncertainties coming from the choice of the expansion basis from those intrinsic to the extraction procedure. We design a set of tests to select the number of the expansion coefficients a…
The effects of the nuclear structure uncertainties on the description of processes induced by coherent scattering of neutrinos on nuclei are investigated. A reference calculation based on a specific nuclear model is defined and the cross sections and also the expected number of events produced by neutrinos generated by the explosion of a supernova in our galaxy, and by a spallation neutron source are evaluated. By changing the input parameters of the reference calculation their relevance on cross sections and on the number of the detected events is estimated. Seven spherical nuclei with differ…
We present a new procedure which enables to extract a scattering matrix $S(l)$ as a complex function of angular momentum directly from the scattering data, without any a priori model assumptions implied. The key ingredient of the procedure is the evolutionary algorithm with diffused mutation which evolves the population of the scattering matrices, via their smooth deformations, from the primary arbitrary analytical $S(l)$ shapes to the final ones giving high quality fits to the data. Due to the automatic monitoring of the scattering matrix derivatives, the final $S(l)$ shapes are monotonic and…
In order to clear up the sensitivity of the nucleus-nucleus scattering to the nuclear matter distributions of exotic halo nuclei, we have calculated differential cross sections for elastic scattering of the $^6$He and $^{11}$Li nuclei on several nuclear targets at the energy of 0.8 GeV/nucleon with different assumed nuclear density distributions in $^6$He and $^{11}$Li.
We formulate microscopic neutron-nucleus optical potentials from many-body perturbation theory based on chiral two- and three-body forces. The neutron self energy is first calculated in homogeneous matter to second order in perturbation theory, which gives the central real and imaginary terms of the optical potential. The real spin-orbit term is calculated separately from the density matrix expansion using the same chiral interaction as in the self energy. Finally, the full neutron-nucleus optical potential is derived within the improved local density approximation utilizing mean field models …
The driving terms in three-body theories of elastic and inelastic scattering of a charged particle off a bound state of two other charged particles contain the fully off-shell two-body Coulomb T-matrix describing the intermediate-state Coulomb scattering of the projectile with each of the charged target particles. Up to now the latter is usually replaced by the Coulomb potential, either when using the multiple-scattering approach or when solving three-body integral equations. General properties of the exact and the approximate on-shell driving terms are discussed, and the accuracy of this appr…
The last few years activity of the Italian community concerning nuclear physics with electroweak probes is reviewed.Inclusive quasi-elastic electron-scattering, photon end electron induced one- and two-nucleon emission are considered. The scattering of neutrinos off nuclei in the quasi-elastic region is also discussed.
Backward elastic electron scattering from odd-A nuclear targets is characterized by magnetic form factors containing precise information on the nuclear structure. We study the sensitivity of the magnetic form factors to structural effects related to the evolution and shape transitions in both isotopic and isotonic chains. Calculations of magnetic form factors are performed in the plane-wave Born approximation. The nuclear structure is obtained from a deformed self-consistent mean-field calculation based on a Skyrme HF+BCS formalism. Collective effects are included in the cranking approximation…
We investigate the screening effects on the nucleon-nucleon elastic cross section inside nuclear matter at zero and finite temperature. The N-N interaction is described phenomenologically via meson exchanges ($σ$, $π$ and $ω$) in the framework of a relativistic lagrangian model. The expressions for the in-medium meson propagators, which take into account the effects of matter polarization and renormalized vacuum polarization, are used in the calculation of the various meson contributions to the total cross section in the one-boson exchange approximation. The final expressions allow a satisfact…
The optical model potentials for nucleon-nucleus elastic scattering at $65$~MeV are calculated for $^{12}$C, $^{16}$O, $^{28}$Si, $^{40}$Ca, $^{56}$Fe, $^{90}$Zr and $^{208}$Pb in first order multiple scattering theory, following the prescription of the spectator expansion, where the only inputs are the free NN potentials, the nuclear densities and the nuclear mean field as derived from microscopic nuclear structure calculations. These potentials are used to predict differential cross sections, analyzing powers and spin rotation functions for neutron and proton scattering at 65 MeV projectile …
Elastic proton scattering on ${}^{3}{H}$ nucleus is studied inbetween p-% ${}^{3}{H}$ and n-${}^{3}{He}$ thresholds, in the energy region where $α$-particles first excited state is inbeded in the continuum. For this aim Faddeev-Yakubovski equations are solved in configuration space, fully considering effects due to the isospin breaking as well as rigorously treating Coulomb interaction. Different realistic nuclear Hamiltonians are tested, elucidating open problems in nuclear interaction description.
A new non-relativistic expansion in terms of the nucleon's momentum inside nuclear matter of the current for isobar electro-excitation from the nucleon is performed. Being exact with respect to the transferred energy and momentum, this yields new current operators which retain important aspects of relativity not taken into account in the traditional non-relativistic reductions. The transition current thus obtained differs from the leading order of the traditional expansion by simple multiplicative factors. These depend on the momentum and energy transfer and can be easily included together wit…
The Quasi-Elastic contribution of the nuclear inclusive electron scattering model developed in Nucl. Phys. A 627 (1997) 543 is extended to the study of electroweak Charged Current induced nuclear reactions, at intermediate energies of interest for future neutrino oscillation experiments. The model accounts for long range nuclear (RPA) correlations, Final State Interaction and Coulomb corrections. RPA correlations are shown to play a crucial role in the whole range of neutrino energies, up to 500 MeV, studied in this work. Predictions for inclusive muon capture for different nuclei, and for the…
We consider $K^-$ nucleon elastic and inelastic scattering in isospin symmetric nuclear matter. It is found that the proper description of the $Λ(1405)$ resonance structure in nuclear matter requires a self-consistent approach. Then the $Λ(1405) $ resonance mass remains basically unchanged, however, the resonance acquires an increased decay width as nuclear matter is compressed. The elastic and inelastic scattering amplitudes show important medium modifications close to the kaon nucleon threshold. We also construct the $K^-$-nuclear optical potential appropriate for $K^-$ atoms. Our microscopi…
Realistic nuclear structure calculations are presented for the event rates due to coherent elastic neutrino-nucleus scattering (CE$ν$NS), assuming neutrinos from pion-decay at-rest, from nuclear reactors and from Earth's interior. We focus on the currently interesting Germanium isotopes, $^{70,73,76}$Ge, which constitute detector materials of the recently planned CE$ν$NS experiments. We study in addition the potential use of $^{64,70}$Zn and $^{28}$Si isotopes as promising CE$ν$NS detectors. From nuclear physics perspectives, recently, calculations have been carried out within the framework of…
The structure of the exotic $^{8}$B nucleus is studied by means of elastic scattering, as well as its breakup on nuclear targets. We present microscopic calculations of the optical potentials (OPs) and cross sections of elastic scattering of $^{8}$B on $^{12}$C, $^{58}$Ni, and $^{208}$Pb targets at energies $20<E<170$ MeV. The density distributions of $^{8}$B obtained within the variational Monte Carlo (VMC) model and the three-cluster model (3CM) are used to construct the potentials. The real part of the hybrid OP is calculated using the folding model with the direct and exchange terms includ…
The influence of nuclear isospin mixing on parity-violating elastic electron scattering is studied for the even-even, N=Z nuclei 12C, 24Mg, 28Si, and 32S. Their ground-state wave functions have been obtained using a self-consistent axially-symmetric mean-field approximation with density-dependent effective two-body Skyrme interactions. Some differences from previous shell-model calculations appear for the isovector Coulomb form factors which play a role in determining the parity-violating asymmetry. To gain an understanding of how these differences arise, the results have been expanded in a sp…
The feasibility of a model-independent extraction of the forward strong amplitude from elastic nuclear cross section data in the Coulomb-nuclear interference region is assessed for $π$ and $K^+$ scattering at intermediate energies. Theoretically-generated "data" are analyzed to provide criteria for optimally designing experiments to measure these amplitudes, whose energy dependence (particularly that of the real parts) is needed for disentangling various sources of medium modifications of the projectile-nucleon interaction. The issues considered include determining the angular region over whic…
We investigate the use of proton-nucleus elastic scattering experiments using secondary beams of 6He and 8He to determine the physical structure of these nuclei. The sensitivity of these experiments to nuclear structure is examined by using four different nuclear structure models with different spatial features using a full-folding optical potential model. The results show that elastic scattering at intermediate energies (<100 MeV per nucleon) is not a good constraint to be used to determine features of structure. Therefore researchers should look elsewhere to put constraints on the ground sta…
The Coulomb-nuclear interference (CNI) has recently been used by the TOTEM Collaboration to analyse proton-proton elastic-scattering data from the LHC and to draw physics conclusions. This paper will present an eikonal calculation of the CNI effects performed to all orders of the fine structure constant, $α$. This calculation will be used as a reference to benchmark several widely-used CNI formulae and to verify several recent claims by other authors.
Based on the isospin-dependent Boltzmann-Uehling-Uhlenbeck(IBUU04) transport model, we explored effects of in-medium $NN$ elastic scattering cross section and nuclear symmetry energy on the sub-threshold pion production in $^{132}Sn+^{124}Sn$ reaction. We find that with decrease of incident beam energy, effects of in-medium $NN$ elastic scattering cross section on the $π^{-}/π^{+}$ ratio are larger than that of the symmetry energy although the latter may be also larger. While keeping the effect of symmetry energy, the double ratio of $π^{-}/π^{+}$ from neutron-rich and neutron-poor reaction sy…
