Coupled-channel Calculations Research Articles

Study of sub-barrier fusion of 36S+50Ti,51V systems

A detailed comparative study of the sub-barrier fusion of the two near-by systems 36S+50Ti,51V was performed at the National Laboratories of Legnaro (INFN). Aim of the experiment was the investigation of possible effects of the non-zero spin of the ground state of the 51V nucleus on the sub-barrier excitation function, and in particular on the shape of the barrier distribution. The results sh w that the two measured excitation functions are very similar down to the level of 20 - 30 μb. The same is observed for the two barrier distributions. Coupled-channels calculations have been performed and are in good agreement with the experimental data. This result indicates that the low-lying levels in 51V can be interpreted in the weak-coupling scheme, that is, 51V(I) = 50Ti(2+)⊗ p(1 f7/2).

Fusion Hindrance and Pauli Blocking in 58Ni + 64Ni

58Ni +64Ni is the first case where the influence of positive Q-value transfer channels on sub-barrier fusion was evidenced, in a very well known experiment by Beckerman et al., by comparing with the two systems 58Ni + 58Ni and 64Ni+64Ni. Subsequent measurements on 64Ni + 64Ni showed that fusion hindrance is clearly present in this case. On the other hand, no indication of hindrance can be observed for 58Ni + 64Ni down to the measured level of 0.1 mb. In the present experiment the excitation function has been extended by two orders of magnitude downward. The cross sections for 58Ni + 64Ni continue decreasing very smoothly below the barrier, down to '1 µb. The logarithmic slope of the excitation function increases slowly, showing a tendency to saturate at the lowest energies. No maximum of the astrophysical S -factor is observed. Coupled-channels (CC) calculations using a Woods-Saxon potential and includinginelastic excitations only, underestimate the sub-barrier cross sections by a large amount. Good agreement is found by adding two-neutron transfer couplings to a schematical level. This behaviour is quite different from what already observed for 64Ni+ 64Ni (no positive Q-value transfer channels available), where a clear low-energy maximum of the S -factorappears, and whose excitation function is overestimated by a standard Woods-Saxon CC calculation. No hindrance effect is observed in 58Ni+ 64Ni in the measured energy range. This trend at deep sub-barrier energies reinforces the recent suggestion that the availability of several states following transfer with Q>0, effectively counterbalances the Pauli repulsion that, in general, is predicted to reduce tunneling probability inside the Coulomb barrier.

Enhanced calculations of fusion barrier distribution for heavy-ion fusion reactions using Wong formula

The effects of coupled-channels in heavy-ion fusion reaction for the system 40Ca + 192Os, 40Ca + 194Pt, and 48Ca + 197Au are discussed. The fusion cross-section σfus, the fusion barrier distribution Dfus and fusion probability Pfus are also discussed. The fusion barrier distribution is calculated using numerical three point, five point and Wong methods. Full quantum coupled-channels calculations are performed using CCFULL code with all order coupling to compare and with available experimental data. The χ² values for the case of no-coupling and coupling effects included shows clearly that the present calculations are in agreement with the experimental.

The Z_c structures in a coupled-channels model

The Z_{c}(3900)^pm /Z_c(3885)^pm and Z_{c}(4020)^pm are two charmonium-like structures discovered in the pi J/psi and D^*bar{D}^{(*)}+h.c. invariant mass spectra. Their nature is puzzling due to their charge, which forces its minimal quark content to be cbar{c} ubar{d} (cbar{c} dbar{u}). Thus, it is necessary to explore four-quark systems in order to understand their inner structure. Additionally, their strong coupling to channels such as pi J/psi and the closeness of their mass to D^*bar{D}^{(*)}-thresholds stimulates both a molecular interpretation or a coupled-channels threshold effect. In this work we perform a coupled-channels calculation of the I^G(J^{PC})=1^+(1^{+-}) sector including D^{(*)}bar{D}^{*}+h.c., pi J/psi and rho eta _c channels in the framework of a constituent quark model which satisfactorily describes a wide range of properties of (non-)conventional hadrons containing heavy quarks. The meson–meson interactions are dominated by the non-diagonal pi J/psi -D^*bar{D}^{(*)} and rho eta _c-D^*bar{D}^{(*)} couplings which indicates that the Z_{c}(3900)^pm /Z_c(3885)^pm and Z_{c}(4020)^pm are unusual structures. The study of the analytic structure of the S-matrix allows us to conclude that the point-wise behavior of the line shapes in the pi J/psi and Dbar{D}^* invariant mass distributions is due to the presence of two virtual states that produce the Z_c peaks.

Reactions with Exotic Nuclei at Near and Sub-barrier Energies

We will introduce some new results derived recently by the nuclear reaction group at China Institute of Atomic Energy, on the properties of the optical model potentials of neutron-halo 6He system and reaction mechanisms induced by proton-drip line nucleus 17F at energies around the Coulomb barrier. For the study of optical model potentials of exotic nuclear system, we proposed a novel method, i.e., the transfer reaction method. This method has been applied to extract the optical potentials of neutron-halo 6He+209Bi system by measuring the one-proton transfer reactions induced by 7Li on a 208Pb target. A complete picture of threshold anomaly behavior was obtained in the 6He+209Bi system for the first time, where a decreasing trend of the depth of the imaginary potential is observed in the deep sub-barrier region, and the reaction threshold energy is extracted. Moreover, results show thatthe dispersion relation is not applicable for this exotic nuclear system. Reaction mechanisms of proton-rich nuclear systems 17F+89Y and 58Ni were also studied at energies around the Coulomb barrier. Continuum discretized coupled-channels calculations indicate that the coupling effects of the continuum states of 17F+89Y is not significant. Thanks to the employment of a powerful ionization-chamber based detector array, the reaction products over a large Z in the systems of 17F+58Ni can be identified clearly. The data analysis is undergoing.

Fusion hindrance in light and heavy systems

The phenomenon of hindrance in sub-barrier heavy-ion fusion has been confirmed by several experimentalevidences and it is now recognised as a general phenomenon of heavy-ion fusion process. In many cases the signature of fusion hindrance lies in the trend of the logarithmic slope of the excitation function and of the S factor at low energies. The comparison with stadard Coupled-Channels calculations is a more quantitative evidence for its existence. In many medium-heavy systems the hindrance effect has been recognised with different features depending on the various couplings to the inelastic and transfer channels. Different theoretical appro ches have been proposed to explain the hindrance but the underlying physics is still a matter of debate. Hindrance is observed in light systems, independent of the sign of the fusion Q-value, with different features. In the case of the 12C + 30Si system the effect is smallbut it is clearly observed. Near-by cases show evidence for systematic behaviours. A very recent experiment has concerned the lighter case 12C + 24Mg where hindrance shows up clearly, because a maximum of the S factor appears already at a relatively high cross section σ=1.6mb. The consequences for the dynamics of stellar evolution have to be clarified by further experimental and theoretical work.

Feshbach resonances in potassium Bose-Bose mixtures

We present a detailed study of the scattering properties of ultracold mixtures of bosonic potassium atoms. We locate 20 previously unobserved Feshbach resonances in isotopic 39K-41K mixtures. These are assigned to s-wave molecular channels by comparison to an asymptotic bound state model and coupled channels calculations. Additional Feshbach resonances are studied in spin mixtures of a single potassium isotope, both in 39K and 41K. In particular, we characterize the parameters of a selected 39K Feshbach resonance by radio-frequency association of Feshbach molecules. Our results could be exploited to refine the model potentials for potassium scattering. Furthermore, these new Feshbach resonances enlarge the range of experiments possible with degenerate Bose-Bose mixtures.

Reproducing heavy-ion fusion cross sections at extreme sub-barrier energies with a simple formula

Heavy-ion fusion hindrance occurs at extreme sub-barrier energies. This behavior is well reproduced with a simple cross section formula, which was developed by Siwek-Wilczynska et al., based on a single-Gaussian distribution of fusion barrier heights, before the discovery of the hindrance phenomenon. This expression has not yet been widely used and referenced in the literature. An analysis by using this simple formula is presented for 29 systems, from 16O + 18O to 64Ni + 124Sn , all being measured down to less than 10μb. The agreement with the data is even better than the ones from sophisticated Coupled-channels calculations. This simple expression also applies to fusion reactions in lighter systems. The three parameters contained in this formula vary in a relatively smooth fashion over the whole mass range, and can be used to extrapolate cross sections or to obtain an estimate of the excitation function for systems which have not been measured. Extensions and restrictions of this method are also discussed.

Reaction mechanisms of the O18+Cu63 system at near-barrier energies

A precise quasielastic excitation function for the $^{18}\mathrm{O}+^{63}\mathrm{Cu}$ system has been measured at energies around the Coulomb barrier at ${\ensuremath{\theta}}_{\mathrm{LAB}}={161}^{\ensuremath{\circ}}$. The corresponding quasielastic barrier distribution has been derived. Two-neutron-, one-proton-, and \ensuremath{\alpha}-transfer-excitation functions have also been measured at the same energies and angle. Coupled reaction channels calculations were performed to describe the experimental data. Large-scale shell-model calculations were performed to derive most of the spectroscopic amplitudes. No surface imaginary potential was necessary for the interaction potential because almost all relevant reaction channels were explicitly included in the calculation. The theoretical results were compared to the experimental quasielastic barrier distribution and a very good agreement was achieved. The comparison of the coupled reaction channel calculations and data has put in evidence several important details of the reaction mechanism of the $^{18}\mathrm{O}+^{63}\mathrm{Cu}$ system. The collectivity of the $^{63}\mathrm{Cu}$ nucleus has important contribution to the reaction mechanism of this system, mainly due to its first $5/{2}^{+}$ and $7/{2}^{+}$ states. It was also observed a striking influence on the reaction dynamics of the $^{18}\mathrm{O}$(${2}^{+}$) state, the two-neutron transfer and the reorientation of the target ground-state spin. The best agreement to data was achieved when the nuclear matter diffuseness for the $^{18}\mathrm{O}$ was assumed equal to 0.60 fm, value that we have derived in a previous paper and that is 10% greater than the $^{16}\mathrm{O}$ diffuseness. Another significant result was that the two-neutron transfer process is much more relevant than the one-neutron-transfer process, which suggests that the pairing correlation could play an important role in the transfer process of this system.

Accurate calculations of weakly bound state energy and scattering length near magnetically tuned Feshbach resonance using the separable potential method.

We present a theoretical model for investigating the magnetically tuned Feshbach resonance (MTFR) of alkali metal atoms using the separable potential method (SPM). We discuss the relationship and difference between the SPM and the asymptotic bound state model. To demonstrate the validity of the SPM, we use it to calculate the weakly bound state energy and magnetically tuned scattering length for the 6Li-40K, 7Li2, and 6Li2 systems with narrow and broad Feshbach resonances. The results of the SPM calculations are in good agreement with those of coupled channel calculations and with experimental measurements for all three systems. The SPM, by simplifying the calculation of the two-body MTFR, is expected to simplify numerical computations for three-atom collisions in a magnetic field and the Feshbach-optimized photoassociation process.

Using a quadrupole deformed generalized Woods–Saxon plus spin-orbit potential to describe the polarized interaction 7Li→ +12C at 34MeV

An overall description of the cross-sections and tensor analyzing powers corresponding to the elastic interaction [Formula: see text] at [Formula: see text] has been obtained with a coupled channel analysis using the rotational model. Based on the high electric quadrupole moment of the [Formula: see text] projectile, an optical model potential equal to the sum of a quadrupole deformed generalized Woods–Saxon potential and a standard Spin-Orbit term has been used. The deformed part of the potential induces reorientations and transitions that involve the first four energy states of the [Formula: see text] nucleus. These states are supposed to belong to an inverted [Formula: see text][Formula: see text][Formula: see text] rotational band proposed in the literature. Partial reproductions of the inelastic interaction [Formula: see text] have also been achieved by using the same deformed potential with the same parameter values that describe the elastic interaction. Comparisons are made with previous results obtained from optical model potential and continuum-discretized coupled-channel calculations using the cluster-folding model. The use of a deformed scattering potential introduces a tensor term that allows the reproduction of the tensor analyzing powers.

Systematic study of proximity potentials for heavy-ion fusion cross sections

Within the framework of the empirical barrier distribution model (EBDM) introduced in our previous work (Wang et al. [20]), we have carried out a comprehensive theoretical study to find the best proximity potentials for reproducing the fusion excitation functions in 99 colliding systems with the condition 15≤Z1Z2≤1804 for the charge product of their participant nuclei. Coulomb barrier heights applied for EBDM approach are determined by 14 different versions of phenomenological proximity potentials. Comparing fusion cross sections by theoretical results and corresponding experimental data for our selected mass range suggests that the EBDM with proximity potential 1977 (Prox. 77) provides the best description for fusion data at above- and below-barrier energies. A comparison is also performed with the theoretical results based on the coupled-channel calculations.

Fusion cross sections for nucleonic-halo systems in solar systems

The behavior of total fusion cross sections for proton and neutron-halo systems at near and sub-barrier energies have been studied using coupled-channel calculations. We have done the fusion distribution by considering phonon excitations and rotational deformation. Comparing with available experimental data represents a good agreement for neutron halo systems. A comparison between the reduced fusion cross sections of these systems shows different behavior in sub-barrier energies.

Elastic scattering, inelastic excitation, and 1n pick-up transfer cross sections for B10 + Sn120 at energies near the Coulomb barrier

The $^{10}\mathrm{B}+^{120}\mathrm{Sn}$ reaction has been systematically studied at laboratory energies around the Coulomb barrier: ${E}_{\mathrm{LAB}}=31.5$, 33.5, 35.0, and 37.5 MeV. Cross sections for the elastic scattering and some reaction processes have been measured: excitation to the ${1}^{+}$ state of $^{10}\mathrm{B}$; excitation to the ${2}^{+}$ and ${3}^{\ensuremath{-}}$ states of $^{120}\mathrm{Sn}$; and the one-neutron pick-up transfer $^{120}\mathrm{Sn}(^{10}\mathrm{B},^{11}\mathrm{B})^{119}\mathrm{Sn}$. Coupled reaction channel (CRC) calculations have been performed in the context of the double-folding S\~ao Paulo potential. The theoretical calculations result in a good overall description of the experimental angular distributions. The effect on the theoretical elastic-scattering angular distributions of couplings to the inelastic and transfer states (through the CRC calculations) and to the continuum states (through continuum-discretized coupled-channels calculations) has been investigated.

Laser spectroscopic study of the B/B′2Σ+–X2Σ+ (9-0), (10-0), (11-0) and (18-0) bands of CaH

Calcium monohydride is an astronomically important molecule. Despite the long history of work on this molecule, discrepancies still exist between experiment and theory for the higher vibrational levels of the B/B′2Σ+ state. We report on rotational-vibrational-electronic transitions to the B/B′2Σ+ state which we have assigned to the B/B′2Σ+ (v = 9, 10, 11 and 18) – X2Σ+ (v = 0) transitions. The energy spacing of the higher lying vibrational levels of the B/B′2Σ+ state alternates strongly with vibrational quantum number as predicted by the coupled-channel calculations of Carlsund-Levin et al. (2002). We confirm that this alternation extends all the way from v = 9 to v = 20.

Energy evolution of channel coupling for the system Li6+Ni64 at near-barrier energies

Background: Different theoretical models have been used to understand the effects of coupling of direct reaction channels on reaction observables in recent years. However, very few attempts have been made to check the consistency of the model calculations in terms of comparing different experimental observables with the model predictions.Purpose: In the present work, coupled channel calculations are performed with transfer and breakup coupling for collision of the weakly bound stable projectile $^{6}\mathrm{Li}$ with the medium-mass target $^{64}\mathrm{Ni}$ at near-barrier energies. The main goal is to find the consistency of the model calculation in simultaneous reproduction of reaction observables like the elastic angular distributions, back-angle elastic excitation function, and fusion excitation function to find the effective energy regions of dominance of these couplings.Method: Both coupled reaction channel (CRC) and continuum discretized coupled channel (CDCC) calculations are performed using the code fresco. Transfer coupling is included through the CRC formalism. The CDCC scheme is used for breakup coupling and to distinguish the effect of resonant and nonresonant couplings.Results: In the case of the CRC calculation, the full coupling including inelastic scattering, $1n$ stripping, and $1p$ stripping gives the best description of the experimental data and particularly reproduces the data in the subbarrier energy region. It is observed that the $1p$-transfer channel has a larger impact on reaction observables compared to $1n$ transfer. On the other hand, for breakup nonresonant breakup has the dominant effect on the reaction observables, although resonant breakup has a larger cross section. The full breakup coupling reproduces the elastic scattering angular distributions and excitation function in the above-barrier region. But the scheme overpredicts these observables in the subbarrier region.Conclusion: CRC and CDCC calculations are complementary to each other. The energy dependencies of the relevant couplings for the $^{6}\mathrm{Li}+^{64}\mathrm{Ni}$ system around the barrier show that inelastic and transfer coupling are important at energies below the Coulomb barrier but continuum coupling is dominant in the above-barrier energy region.

Investigation of fusion hindrance in a soft asymmetric system deep below the barrier

A steeper fall of fusion excitation function, compared to the predictions of coupled-channel (CC) models, at energies below the lowest barrier between the reaction partners, is termed as deep sub-barrier fusion hindrance. This phenomenon has been observed in many symmetric and nearly symmetric systems. Different physical origins of the hindrance have been proposed though a complete understanding is yet to be achieved. This work reports the measurement of the fusion (evaporation residue) cross sections for the system 19F+181Ta, from above the barrier down to the energies where fusion hindrance is expected to come into play. CC calculation with standard Woods–Saxon potential gives a fair description of the fusion excitation function down to energies ≃14% below the barrier. This is in contrast with the observation of increasing fusion hindrance in asymmetric reactions induced by increasingly heavier projectiles, viz. 6,7Li, 11B, 12C and 16O. The most likely reason for non-observation of fusion hindrance in the present system is the ‘softness’ of the collision partners. The ‘threshold’ for the onset of fusion hindrance in reactions involving well-deformed nuclei is expected to be at an energy lower than that estimated from the systematics. It is also noted that the asymmetric reactions, which have not shown any signature of fusion hindrance within the measured energy range, are induced by projectiles with low α-particle break-up threshold and have positive Q-values for most light particle pick-up channels. Since CC calculation reproduces the fusion excitation functions for these systems, probable roles of projectile break-up and particle transfer in fusion deep below the barrier are not conclusively proven. Further measurements and inclusion of break-up and transfer channels within the framework of CC formalism would be of interest.

Effect of resonance states of the weakly bound 6Li on total fusion in reactions with targets 28Si, 96Zr and 209Bi

Continuum Discretized Coupled-Channel calculations of total fusion cross sections for reactions induced by the weakly bound nucleus 6Li with targets 28Si, 96Zr and 209Bi at energies around the Coulomb barrier are presented. In the cluster structure frame of 6Li → α+d, short-range absorption potentials are considered for the interactions between the α- and d-fragments with the targets. In the model, the effect of resonance (l = 2, Jπ = 3+, 2+, 1+) and non-resonance breakup states of 6Li on fusion is studied by using two approaches: (1) by omitting the resonance states from the full discretized CDCC breakup space and 2) by considering only the resonance discretized sub-space. Among other things, it is found that resonance breakup states produce strong repulsive polarization potentials that lead to fusion suppression. The particular effect on fusion of a single resonance is calculated. It is found that resonances 2+ and 1+ have a more significant role on incomplete fusion, while the 3+ on complete fusion.

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.

Fusion reaction studies for the Li6+Sn124 system at near-barrier energies

The complete and incomplete fusion along with one neutron stripping and pickup cross sections for the $^{6}\mathrm{Li}+^{124}\mathrm{Sn}$ system are measured using online and offline characteristic $\ensuremath{\gamma}$-ray detection techniques. The complete fusion (CF) cross sections at energies above the Coulomb barrier are found to be suppressed by $\ensuremath{\sim}34%$ compared to the coupled channel calculations. This suppression observed in complete fusion cross sections is found to be commensurate with the measured total incomplete fusion (ICF) cross sections. There is a distinct feature observed in the ICF cross sections, i.e., $d$ capture is found to be dominant than $\ensuremath{\alpha}$ capture at all the measured energies, contrary to the data available for $^{6}\mathrm{Li}+^{209}\mathrm{Bi},^{197}\mathrm{Au}$ systems. The total fusion cross section ratio between $^{6}\mathrm{Li}$ and $^{7}\mathrm{Li}$ induced reactions shows an increasing trend as the energy decreases below the barrier while it remains unity at above-barrier energies. A simultaneous explanation of complete, incomplete, and total fusion (TF) data is also obtained from the calculations based on the continuum discretized coupled channel method with short-range imaginary potentials. The cross section ratios of CF/TF and ICF/TF obtained from the data as well as the calculations shows the dominance of ICF at below-barrier energies and CF at above-barrier energies.