Airy Minima Research Articles

Primary rainbow and Airy minima in 12C + 12C elastic scattering with families of non-monotonic potentials*

The experimental differential cross section of the 12C + 12C elastic scattering over a wide laboratory energy scale 25 ⩽ E lab ⩽ 360 MeV are analyzed within the framework of the optical model (OM) using non-monotonic (NM) nucleus–nucleus potential. The real parts of the NM potentials are derived from the energy density functional (EDF) formalism with the sudden approximation embodying the Pauli exclusion principle and the imaginary parts are taken as phenomenology. The diffractive and the refractive scattering with Airy structures in the whole angular region of the 12C + 12C elastic scattering in the above mentioned energy range are found profoundly successful in OM with the NM nucleus–nucleus potentials in five of different potential families. All the families at lower energies converge at 240 MeV to the unique potential family starting with the EDF potential. This, in conjunction with our recent findings for the 16O + 16O system in Islam et al (2021 J. Phys. G 48 075109), not only verifies the Goldberg criterion for elimination of discrete potential ambiguities for NM potentials, but also establishes that this leads uniquely to the EDF family. The 90° excitation function is also successfully reproduced with extension using NM potentials. The near- and far-side (N/F) decomposition of total elastic scattering amplitude has also been studied using our NM potentials.

Optical Model Analysis of Elastic Scattering of <sup>16</sup>O + <sup>12</sup>C

This paper presents the optical model analysis of the elastic scattering of ¹⁶O + ¹²C at the incident energies of 608 and 1503 MeV using optical potentials derived from B3Y-Fetal effective interaction. Optical model (OM) analysis of the elastic data of this system at these incident energies has shown two of the four optical potentials, the DDB3Y1-Fetal (K=176 MeV) and BDB3Y1-Fetal (K=235 MeV), to give a better description of the elastic data than the BDB3Y2-and BDB3Y3-Fetal potentials, making them the best-fit folded potentials, in agreement with previous work done with the M3Y-Reid effective interaction in both identical and non-identical heavy ions. This is a conclusive and convincing confirmation of the concensus among Nuclear Physics scholars, based on OM analyses of heavy ions, that nuclear matter has an underlying soft equation of state. In addition, results of calculations herein have also shown the best-fit folded potentials, the DDB3Y1-and BDB3Y1-Fetal with-227.8 and-220.6 MeV at 608 MeV and-124.3 MeV and-120.5 MeV at 1503 MeV, respectively as the largest values at smaller inter-nuclear distances, to be in good agreement with their counterparts, the DDM3Y1-Reid and BDM3Y1-Reid, whose largest values at smaller inter-nuclear distances are-231.6 and-223.8 MeV at 608 MeV and-138.8 MeV and-134.2 MeV at 1503 MeV respec¬tively, in terms of magnitude, shape and trend. This is a further validation of the viability of the B3Y-Fetal, corroborating the findings of previous studies carried out with it. In the final analysis, the findings of this study have not shown the occurrence of distinctive features of refractive scattering such as Airy minima in the calculated cross sections at both 608 MeV and 1503 MeV in agreement with previous work.

Airy minima in 4He +28Si elastic and inelastic scattering

We present the results of the [Formula: see text]-matrix description of the [Formula: see text] elastic and inelastic scattering differential cross-sections between 40 and 240 MeV with allocation of Airy minima of various orders. The first-order Airy minima have been unambiguously identified in the differential cross-sections for elastic scattering and inelastic scattering to the first [Formula: see text] state of [Formula: see text] at energies [Formula: see text] MeV, and the second-order Airy minima — at [Formula: see text] MeV. The angular positions of these minima obey an inverse dependence on energy. The intensities of nuclear refraction and absorption also obey the same energy dependence.

Analysis of Elastic 16O + 40Ca Refractive Scattering at 214 MeV

The elastic scattering of 16O+40Ca system at the incident energy 214 MeV has been analyzed within the optical model (OM) potential using the phenomenological Woods-Saxon (WS) and the real optical potentials calculated in two different versions of the folded model based upon the α-α effective interaction and CDM3Y6 effective nucleon-nucleon (NN) interaction. The new cluster Fermi three-parameter (3PFC) density distribution has been extracted to calculate the DFC1 potential. The Airy oscillation observed recently in the elastic 16O+40Ca system at Elab = 214 MeV is well reproduced by using the DFC1 potential without renormalization factor. The observed minimum around θ ≈ 46° in the angular distribution with DFC1 approach was shown to be an Airy minimum, probably of the third order (A3), which is in agreement with the previous study. The predicted volume integrals and total reaction cross section are consistent with the previous analyses. This reflects the success of our DFC1 potential to reproduce a nice description of the Airy structure for HI elastic scattering.

The breakup coupling effects on the rainbow scattering of 6Li at 35 AMeV

To investigate the breakup coupling effects on the formation of the nuclear rainbow, the existing experimental data of the refractive elastic scattering of 6Li on 12C, 40Ca, 58Ni and 90Zr targets at 35 AMeV are analyzed. The continuum discretized coupled-channel (CDCC) method is adopted to calculate the elastic scattering angular distributions including the coupling effects from the breakup of 6Li into α and d. The one-channel (1-ch) calculations that switch off the breakup coupling are also performed for comparison. In the calculated angular distributions given by both approaches, the first Airy (A1) minimum shows a trend of moving to larger angles as the target mass increases. Comparing the calculated results through two approaches, one can find that the breakup coupling shifts the A1 minima to larger angles. This shows that the breakup coupling can boost the nuclear rainbow, although it also damps the elastic scattering cross sections. We also find that the stronger the breakup coupling is, the larger the A1 shift becomes.

Elastic and inelastic O16+C12 rainbow scattering within the coupled-channels mechanism

Angular distributions of the differential cross section of $^{16}\mathrm{O}+^{12}\mathrm{C}$ elastic and inelastic scattering in the energy range ${E}_{\mathrm{lab}}=181\ensuremath{-}1503$ MeV were fitted with the full cluster folded approach (DFC1) using the coupled-channels (CC) method by taking into account the effect of the low-lying states, ${0}^{+}$ and ${2}^{+}$ (${E}_{\mathrm{ex}}=4.44$ MeV) for the target nucleus. The cluster transition density is generated macroscopically by deforming the cluster ground state modified Gaussian density distribution of the target nucleus. The coupled-channels calculations with the fitted parameters for the elastic scattering provide a successful description of the inelastic experimental measurements. The obtained results are compared with the finding of the phenomenological Woods-Saxon potential. It was found that the characteristic features of the nuclear rainbow phenomena can be reproduced successfully by our CC calculations with both potentials but fail to reproduce the new Airy minimum of the nuclear rainbow structure. The deduced intrinsic quadrupole moment values, ${Q}_{20}$, derived from the analyses are compared with results of previous studies. The deformation lengths, ${\ensuremath{\delta}}_{20}$, obtained by CC calculations are in agreement with those obtained from the electromagnetic measurement values and those reported in previous studies.

Unified S-matrix analysis of Airy structures in α+24Mg elastic and inelastic scattering

Using the six-parameter [Formula: see text]-matrix model, we have obtained a simultaneous correct description of the [Formula: see text]Mg elastic and inelastic scattering differential cross-sections over the energy region [Formula: see text]–[Formula: see text] MeV, where typical nuclear rainbow and prerainbow patterns are observed. The Airy minima of the first-order have been unambiguously identified in the cross-sections for elastic scattering and inelastic scattering to the first 2[Formula: see text] state of [Formula: see text]Mg at [Formula: see text]–145[Formula: see text]MeV. Their angular positions obey an inverse dependence on energy, which is in line with the “rainbow” interpretation of the data. Within this interpretation, the scattering matrix and the deflection function for the system [Formula: see text]Mg at [Formula: see text]–240[Formula: see text]MeV show physically justified smooth variations with the projectile energy.

Further evidence for a dynamically generated secondary bow inC13+C12rainbow scattering

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 \theta$ $\approx$70$^\circ$, which is not reproduced by a conventional folding potential, is revealed to be a secondary bow generated dynamically by a coupling to the excited state 2+ (4.44 MeV) of 12C. The essential importance of the quadruple {\it Y2} term (reorientation term) of potential of the excited state 2+ of 12C for the emergence of a secondary bow is found. The mechanism of the secondary bow is intuitively explained by showing how the trajectories are refracted dynamically into the classically forbidden angular region beyond the rainbow angle of the primary rainbow.

Airy structure inO16+C14nuclear rainbow scattering

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 $\theta$=76$^\circ$ in the angular distribution at $E_L$=132 MeV is assigned as the second order Airy minimum $A2$ in contrast to the recent literature which assigns it as the third order $A3$. The Airy minima in the 90$^\circ$ excitation function is investigated in comparison with well-known $^{16}$O+$^{16}$O and $^{12}$C+$^{12}$C systems. Evolution of the Airy structure into the molecular resonances with the $^{16}$O+$^{14}$C cluster structure in the low energy region around $E_{c.m.}$=30 MeV is discussed. It is predicted theoretically for the first time for a non-$4N$ $^{16}$O+$^{14}$C system that Airy elephants in the 90$^\circ$ excitation function are present.

Refractive effects and Airy structure in inelasticO16+C12rainbow scattering

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 $\alpha$ 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 structure in the inelastic scattering cross sections for the first $2^+$ state of $^{12}$C, which is somewhat obscured and still not clearly visible in the measured data. The Airy minimum was identified from the analysis and the systematic energy evolution of the Airy structure was studied. The Airy minimum in inelastic scattering is found to be shifted backward compared with that in elastic scattering.

Study of the structure of the Hoyle state by refractive α-scattering

α + 12 C elastic and inelastic to the Hoyle state (0 + 2, 7.65 MeV) differential cross-sections were measured at the energies 60 and 65 MeV with the aim of testing the microscopic wave function (1) widely used in modern structure calculations of 12 C. Deep rainbow (Airy) minima were observed in all four curves. The minima in the inelastic angular distributions are shifted to the larger angles relatively those in the elastic ones, which testify the radius enhancement of the Hoyle state. In general, the DWBA calculations failed to reproduce the details of the cross sections in the region of the rainbow minima in the inelastic scattering data. However, by using the phenomenological density with rms radius equal 2.9 fm, we can reproduce the Airy minimum positions.

Unification of Airy structure in inelasticα+16O scattering andα-cluster structure with core excitation in20Ne

The Airy structure of the nuclear rainbow and prerainbow in inelastic and elastic $\alpha+^{16}$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 $\alpha$-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 $K=0_3^+$ $\alpha$-cluster band with core excitation at 7.19 MeV in $^{20}$Ne is intimately related to the emergence of the prerainbow and rainbow in inelastic scattering to the $^{16}$O($0^+_2$). It is found that the $\alpha$-cluster states with core excitation, the prerainbow and the rainbow in inelastic scattering are understood in a unified way as well as in the case of elastic scattering.

Observation of an Airy minimum in elastic and inelastic scattering of3He from12C at 50.5 and 60 MeV, andα-particle condensation in12C

Angular distributions for elastic and inelastic scattering of $^3$He from $^{12}$C were measured at energies 50.5 and 60 MeV. The Airy minimum of the prerainbow scattering was clearly observed in the angular distributions for the 0$_2^+$ (7.65 MeV) state of $^{12}$C (Hoyle state). The experimental results were analyzed with a coupled channels method with double folding potentials derived from the microscopic wave functions for the ground 0$_1^+$, 2$^+$ (4.44 MeV), 3$^-$ (9.64 MeV) and 0$_2^+$ states. The analysis supports the view that the Hoyle state is a three alpha particle condensate with a large radius of dilute matter distribution.

12C + 12C large angle elastic scattering at 240 MeV

The measurement of the differential cross sections of the 12C + 12C and the 13C + 12C elastic scattering are carried out at the 12C energy 240 MeV and the 13C energy 250 MeV to solve some interpretation problem of the 12C + 12C scattering data. Is it possible that the 1 st Airy minimum in 12C + 12C scattering data is missing? Our analysis shows that the positions of the 1 st Airy minima obtained in former measurements are confirmed.

Evidence for strong refraction ofHe3in an α-particle condensate

We have analyzed $^{3}$He scattering from $^{12}$C at 34.7 and 72 MeV in a coupled channel method with a double folding potential derived from the precise wave functions for the ground 0$^+$ state and $0_2^+$ (7.65 MeV) Hoyle state, which has been suggested to be an $\alpha$ particle condensate. It is found that strong refraction of $^3$He in the Hoyle state can be clearly seen in the experimental angular distribution at {\it low} incident energy region as an Airy minimum of the {\it pre-rainbow oscillations}.

Nuclear rainbow in the elastic scattering of 16O nuclei on carbon isotopes

An investigation of refractive effects in heavy-ion scattering is continued. For the elastic scattering of 16O nuclei on a 13C target at E(16O) = 132 MeV, the differential cross sections are measured for the first time in addition to previous measurements for targets from the carbon isotopes 12C and 14C. Airy structures that are similar for all isotopes and which have close cross sections are observed and are found to be consistent with the energy systematics of Airy minima that were obtained previously. The volume integrals of the real and imaginary parts of the optical potentials found in the present study are nearly identical for all of the isotopes considered here and are also in good agreement with the systematics obtained previously.

Airy minima in the scattering of weakly bound light heavy ions

We reanalyze the existing $^{6}\mathrm{Li}$ + $^{12}\mathrm{C}$ elastic scattering angular distributions for incident energies ranging from a few MeV to 318 MeV within the frame of the optical model. Despite the important breakup effects expected in the scattering of such a fragile projectile, the system is found to display a surprising transparency. Indeed the barrier-wave/internal-wave decomposition of the elastic scattering amplitude reveals that a substantial part of the incident flux that penetrates the nuclear interior reemerges in the elastic channel, and typical refractive effects, like Airy minima, are clearly identified in the angular distributions. Coupled channel calculations performed on $^{12}\mathrm{C}$($^{6}\mathrm{Li}$,$^{6}\mathrm{Li}'$)$^{12}\mathrm{C}$${}^{*}$(${J}^{\ensuremath{\pi}}={2}^{+},{E}_{x}=4.44$ MeV) angular distributions extending through the whole angular range confirm the existence of an important internal-wave contribution in the backward hemisphere. A similar transparency is observed in other systems of this mass region, such as $^{7}\mathrm{Li}$ + $^{12}\mathrm{C}$ or $^{6}\mathrm{Li}$ + $^{16}\mathrm{O}$. Finally, we examine recent $^{6}\mathrm{He}$ + $^{12}\mathrm{C}$ elastic scattering data obtained at 18 MeV by Milin et al. [Nucl. Phys. A730, 285 (2004)] and extending up to ${\ensuremath{\theta}}_{\mathrm{c.m.}}\ensuremath{\simeq}{85}^{\ifmmode^\circ\else\textdegree\fi{}}$, and we suggest additional measurements that could ascertain whether some transparency persists in the scattering of this radioactive projectile.

Evolution of Airy structure in 12C(12C, 12C)12C between 5 and 10 MeV/A

We investigate the properties of a global optical potential, which describes the 12C + 12C elastic-scattering data between 70 and 130 MeV, within the nearside/farside and barrier-wave/internal-wave decomposition techniques. Particular emphasis is laid on the discussion of the incomplete absorption features of this system, and especially on the properties of the Airy minima which are observed in the experimental $90^\circ$ excitation function. The complicated angular and energy evolution of the data is explained in terms of the interference of a small set of scattering subamplitudes with a much simpler behavior.

Consequences of Weak Absorption in Light Heavy-Ion Reactions: Molecular Resonances and Nuclear Rainbows

In the elastic angular distributions of the systems composed of tightly bound α-nuclei, 12C+12C, 16O+16O and 16O+12C, the higher-order Airy minima are observed in the energy region between 5 and 10 MeV per nucleon. The obtained optical potentials are deeply attractive and with incomplete absorption. The same potentials successfully describe the lower-energy (molecular resonances) and the higher-energy (first Airy minimum) data.

Tangential Velocity Corrections to a Second-Order Coulomb-Modified Eikonal Model for Heavy-Ion Elastic Scattering

We present the tangential velocity corrections to the second-order Coulomb-modified eikonal model at the distance of closest approach. It has been applied to elastic angular distributions of the 16 O +16 O system at E lab =350 and 480 MeV. The calculated results with tangential velocity show better agreements with the experimental data compared to those with asymptotic velocity. The Fraunhöfer oscillations observed in the elastic angular distributions can be explained by the strong interference between the near- and the far-side amplitudes. Airy structures can be shown by reducing the effective imaginary potential strength. It is found that the Airy minimum is more visible as the effective imaginary potential strength is reduced. Deep real potentials associated with rather weak imaginary ones are found to be essential to describe the refractive 16 O +16 O elastic scatterings at E lab =350 and 480 MeV.