We present here a robust analytical model based on nuclear reaction theory for non-resonant fusion cross sections near Coulomb barrier. The astrophysical $S$-factors involving stable and neutron rich isotopes of C, O, Ne, Mg and Si for fusion reactions have been calculated in the centre of mass energy range of 2-30 MeV. The model is based on the tunneling through barrier arising out of nuclear, Coulomb and centrifugal potentials. Our formalism predicts precisely the suppression of $S$-factor at sub-barrier energies which are of astrophysical interest. The cross sections can be convoluted with Maxwell-Boltzmann distribution of energies to obtain thermo- or pycno- nuclear reaction rates relevant to nucleosynthesis at high density environments and stellar burning at high temperatures as well as for $^{34}$Ne $+$ $^{34}$Ne fusion occurring in the inner crust of accreting neutron stars.
We study the combined effects of vacuum polarization, relativity, Bremsstrahlung, and atomic polarization in nuclear reactions of astrophysical interest. It is shown that these effects do not solve the longstanding differences between the experimental data of astrophysical nuclear reactions at very low energies and the theoretical calculations which aim to include electron screening.
We derived analytic formulas of the effective S astrophysical S factor,S^eff for a non-resonant reaction of charged particles using a Taylor expension of the astrophysical S factor and a uniform approximation.The formulas will be able to generate generate more accurate approximation to S^eff than previous ones.
Semimicroscopic optical potentials for low energy proton reactions in mass 90-100 region have been obtained by folding the density dependent M3Y interaction with relativistic mean field densities. Certain parameters in the potential have been deduced by comparing calculated results with the data for elastic scattering. Low energy proton reactions in this mass region have been studied in the formalism with success. Rates of important astrophysical reaction in the mass region have been calculated.
We summarize recent ab initio studies of low-energy electroweak reactions of astrophysical interest, relevant for both big bang nucleosynthesis and solar neutrino production. The calculational methods include direct integration for np radiative and pp weak capture, correlated hyperspherical harmonics for reactions of A=3,4 nuclei, and variational Monte Carlo for A=6,7 nuclei. Realistic nucleon-nucleon and three-nucleon interactions and consistent current operators are used as input.
Microscopic optical potentials obtained by folding the DDM3Y interaction with the densities from Relativistic Mean Field approach have been utilized to evaluate S-factors of low-energy $(p,γ)$ reactions in mass 60-80 region and to compare with experiments. The Lagrangian density FSU Gold has been employed. Astrophysical rates for important proton capture reactions have been calculated to study the behaviour of rapid proton nucleosynthesis for waiting point nuclei with mass less than A=80.
We discuss the effect of electromagnetic environment on laboratory measurements of the nuclear fusion reactions of astrophysical interest. The radiation field is eliminated using the path integral formalism in order to obtain the influence functional, which we evaluate in the semi-classical approximation. We show that enhancement of the tunneling probability due to the radiation correction is extremely small and does not resolve the longstanding problem that the observed electron screening effect is significantly larger than theoretical predictions.
The cross sections for the deep sub-barrier fusion reaction of light nuclei are calculated within the theoretical framework of the selective resonant tunneling model. In this model, assumption of a complex square-well nuclear potential is invoked to describe the absorption inside the nuclear well. The theoretical estimates for these cross sections agree well with the experimentally measured values. The features of the astrophysical S-factor are derived in terms of this model. Present formalism appears to be particularly useful for the low energy resonant reactions between two charged nuclei.
The proton transfer ${\rm{^7Be}}$($d$,$n$)${\rm{^8B}}$ reaction at the energy of 4.5 MeV (c.m.) has been analysed within the modified DWBA. New estimates and their uncertainties are obtained for values of the asymptotic normalization coefficients for $p + {\rm{^7Be}} \to {\rm{^8B}}$, the astrophysical $S_{17}(0)$ factor and the s-wave $p + {\rm{^7Be}}$ scattering lengths.
The halo effect in the final states of astrophysically relevant direct neutron--capture reactions by neutron--rich nuclei is discussed. As an example, we calculate the cross sections for $^{18}$O(n,$γ$)$^{19}$O at thermonuclear and thermal energies.
We discuss the effects of non-inertial motion in reactions occurring in laboratory, stars, and elsewhere. It is demonstrated that non-inertial effects due to large accelerations during nuclear collisions might have appreciable effects nuclear and atomic transitions. We also explore the magnitude of the corrections induced by strong gravitational fields on nuclear reactions in massive, compact stars, and the neighborhood of black holes.
The astrophysical $S$-factor for the reaction $^7$Be(p,$γ$)$^8$B up to an energy of 2~MeV (c.m.) and the capture cross section of $^7$Li(n,$γ$)$^8$Li up to 1~MeV (c.m.) are calculated using the Direct Capture model (DC). Both calculations are in good agreement with experimental data.
In the frame of modified potential cluster model based on the classification of orbital states by Young diagrams and revised interaction potential parameters for the bound states of 7Be in 3He4He cluster model with forbidden states the astrophysical S-factor for the radiative capture reaction has been calculated from the 10 keV. Obtained results S(23 keV) = 0.561 keV b reproduce the latest experimental data at 23 keV. Calculated and parametrized reaction rate is compared to some results known in the range of temperatures from 0.05 to 5 T9.
The quantum diffusion approach is extended to low energy fusion (capture) reactions of light- and medium-mass nuclei. The dependence of the friction parameter on bombarding energy is taken into account. A simple analytic expression is obtained for the capture probability at extreme sub-barrier energies. The calculated cross-sections are in a good agreement with the experimental data. The fusion excitation functions calculated within the quantum diffusion and WKB approaches are compared and presented in the astrophysical $S$-factor representation.
We study the breakup of 9Li projectiles in high energy (28.5 MeV/u) collisions with heavy nuclear targets (208Pb). The wave functions are calculated using a single-particle model for 9Li, and a simple optical potential model for the scattering part. A good agreement with measured data is obtained with insignificant E2 contribution.
The total cross sections of the radiative proton capture on 8Li at astrophysical energies were considered in the framework of the modified potential cluster model with forbidden states, with the classification of the orbital cluster states according to Young diagrams. The recalculation of total cross sections for the 9Be(γ,p0)8Li photodisintegration was used as experimental data. In whole, it is possible to obtain the available data on cross sections at energies to 7.0 MeV. Astrophysical S-factors and reaction rates at the temperature range of 0.01 to 10 T9 were calculated. It was shown that t…
We study the effect of virtual Coulomb break-up, commonly known as the dipole polarizability, of the deuteron projectile on the astrophysical fusion reaction 3He(d,p)4He. We use the adiabatic approximation to estimate the potential shift due to the E1 transition to the continuum states in the deuteron, and compute the barrier penetrability in the WKB approximation. We find that the enhancement of the penetrability due to the deuteron break-up is too small to resolve the longstanding puzzle observed in laboratory measurements that the electron screening effect is surprisingly larger than theore…
The Feshbach theory of the optical potential naturally leads to a microscopic description of scattering in terms of the many-body self-energy. We consider a recent calculation of this quantity for 16O and study the possibility of applying it at astrophysical energies. The results obtained for the phase shifts and the 16O(p,γ) capture suggest that such studies are feasible but the calculations require some improvement geared to this specific task.
The stopping power of protons and deuterons in low energy collisions with helium gas targets is investigated with the numerical solution of the time-dependent Schrödinger coupled-channels equations using molecular orbital wavefunctions. It is shown that at low projectile energies the energy loss is mainly due to nuclear stopping, charge exchange, and the excitation of low energy levels in the target. The second and third mechanisms, called electronic stopping, dominate for E>200 eV. At lower energies it is also shown that a threshold effect is responsible for a quick drop of the energy loss. T…
We calculate the electron screening effect in low energy nuclear fusion reactions by taking non-adiabaticity of the tunnleing process into account. In order to investigate deviations from the adiabatic limit, we use the dynamical norm method, which has recently been developed by the present authors. Using $d$+D reaction as an example, we show that the screening energy never exceeds those estimated in the adiabatic approximation. Our calculations indicate that the non-adiabatic effect is important both in classically allowed and classically forbidden processes.
The total cross sections of the neutron radiative capture on 12B at astrophysical energies to the ground state of 13B have been calculated in the energy range of 10E-8 to 10 MeV within the framework of a modified potential cluster model with the classification of orbital states according to Young diagrams. Reaction rates in the temperature range of 0.01 to 10 T9 and their analytical parameterization were obtained. The calculated rates of 12B(n,gamma)13B excess the previous results by approximately to one order. Cross sections and reaction rates of 12C(n,gamma)13C are calculated and compared to…
We have calculated astrophysical reaction cross-sections for $(γ,α)$ reactions of some nuclei important for the calculation of $p$-process reaction-decay network. Reaction rates for $α$-induced reactions are calculated with the semi-microscopic optical potential constructed using double folding method, where nuclear density distributions for finite nuclei along with the effective nucleon-nucleon interaction are the important components of the folded potential. For this purpose density distributions of target nuclei are obtained from relativistic mean field approach. Astrophysical reaction cros…
We estimate the ratios of bare astrophysical S-factors at zero incident energy for proton and deuteron induced reactions in a model which assumes a compound nucleus formation probability plus a statistical decay. The obtained ratios agree well with available experimental values, as far as the reactions which have dominant s-wave entrance channel components are investigated. Due to its simplicity the model could be used as a guidance for predictions on reactions which have not been investigated yet.
The astrophysical S-factor for the process p7Be --> 8Bgamma of radiative capture by the 8B ground state is described up to 5 MeV, within the modified potential cluster model. The signatures of the S-factor resonances at 0.632, 2.18 and 3.36 MeV due to M1 and E2 transitions from the scattering of resonance 3P1, 3F3 and 3D2 waves to the ground 3P2 state are illuminated using total cross sections, astrophysical S-factors and reaction rates at temperatures from 0.01 to 10.0 T9. Experimental data on the S-factor are reproduced well in the range 100 keV to 3 MeV. The importance of the input of the 0…
Background: The ($α$,n) and ($α$,$γ$) reactions on $^{17,18}$O have significant impact on the neutron balance in the astrophysical $s$-process. In this scenario stellar reaction rates are required for relatively low temperatures below $T_9 \lesssim 1$. Purpose: The uncertainties of the $^{17,18}$O($α$,n)$^{20,21}$Ne reactions are investigated. Statistical model calculations are performed to study the applicability of this model for relatively light nuclei in extension to a recent review for the $20 \le A \le 50$ mass range. Method: The available experimental data for the $^{17,18}$O($α$,n)$^{2…
Within the asymptotic theory proposed by authors R. Yarmukhamedov and K.I. Tursunmakhatov [Phys. Rev. C (submitted 2019)] for the peripheral sub- and above-barrier transfer $A$($x$, $y$)$B$ reaction in the three-body ($A$, $a$ and $y$) model ($ x$= $y$ + $a$ and $B$= $A$ + $a$, and $ a$ is a transferred particle), the analysis of the experimental angular distributions of the differential cross sections is performed for the peripheral proton and triton transfer ${\rm{^9Be(^{10}B,\,^9Be)^{10}B}}$, ${\rm{^{16}O(^3He}},\,d{\rm{)^{17}F}}$ and ${\rm{^{19}F(}}p,\,α{\rm{)^{16}O}}$ reactions at above-a…
The structure of the Borromean nucleus $^9$Be ($α+α+n$) is addressed within a three-body approach using the analytical transformed harmonic oscillator method. The three-body formalism provides an accurate description of the radiative capture reaction rate for the entire temperature range relevant in Astrophysics. At high temperatures, results match the calculations based on two-step sequential processes. At low temperatures, where the particles have no access to intermediate two-body resonances, the three-body direct capture leads to reaction rates larger than the sequential processes. These r…
he velocity dependence of the stopping power of swift protons and deuterons in low energy collisions with hydrogen and helium gas targets is investigated with the numerical solution of the time-dependent Schroedinger coupled-channels equations using molecular orbital wavefunctions. At low projectile energies the stopping is mainly due to nuclear stopping, charge exchange of the electron, and excitation of the lowest levels in the target. The second and third mechanisms dominate at E < 200 eV. At lower energies it is also shown that a threshold effect is responsible for a quick drop of the ener…
We have studied the neutron-capture reactions 8Li(n,gamma)9Li and its role in the primordial nucleosynthesis. The n + 8Li --> 9Li + gamma reaction has a significant astrophysical interest because it includes one of the variants of chain of primordial nucleosynthesis processes of the Universe and thermonuclear reactions in type II supernova.
Precise nuclear reaction rates are needed for a detailed description of the production of elements in primordial nucleosynthesis and during the hydrostatic burning of stars to constrain the astrophysical models. The relevant reactions are extremely difficult to measure directly in the laboratory at the small astrophysical energies. In recent years direct reactions methods have been developed and applied to extract low-energy astrophysical S factors. The application of these methods requires a combination of experimental and theoretical efforts. This contribution focuses on the underlying react…
The $^{12}$C+$^{12}$C fusion reaction plays a significant role in the later phases of stellar evolution. For a better understanding of the evolution involved, one must understand the corresponding fusion-fission dynamics and reaction characteristics. In the present analysis, we have studied the fusion cross-section along with the S-factor for this reaction using the well-known M3Y and recently developed R3Y nucleon-nucleon (NN) potential along with the relativistic mean-field densities in double folding approach. The density distributions and the microscopic R3Y NN potential are calculated usi…
The value of the alpha spectroscopic factor (S_alpha) of the 6.356 MeV 1/2+ state of 17O is believed to have significant astrophysical implications due to the importance of the 13C(alpha,n)16O reaction as a possible source of neutron production for the s process. To further study this effect, an accurate measurement of the 13C(6Li,d)17O reaction at E_lab = 60 MeV has been performed recently by Kubono et al., who found a new value for the spectroscopic factor of the 6.356 MeV 1/2+ state of 17O based on a distorted wave Born approximation (DWBA) analysis of these data. This new value, S_alpha ap…
Accurate nuclear reaction rates are needed for primordial nucleosynthesis and hydrostatic burning in stars. The relevant reactions are extremely difficult to measure directly in the laboratory at the small astrophysical energies. In recent years direct reactions have been developed and applied to extract low-energy astrophysical S-factors. These methods require a combination of new experimental techniques and theoretical efforts, which are the subject of this presentation.
We formulate a many-body theory to calculate the cross section of direct radiative neutron capture reaction by means of the Hartree-Fock-Bogoliubov mean-field model and the continuum quasiparticle random phase approximation (QRPA). A focus is put on very neutron-rich nuclei and low-energy neutron kinetic energy in the range of O(1 keV) - O(1 MeV), relevant for the rapid neutron-capture process of nucleosynthesis. We begin with the photo-absorption cross section and the E1 strength function, then, in order to apply the reciprocity theorem, we decompose the cross section into partial cross secti…
We have reanalyzed reaction cross sections of 16N on 12C target. The nucleon density distribution of 16N, especially surface density distribution, was extracted using the modified Glauber model. On the basis of dilute surface densities, the discussion of 15N(n, γ)16N reaction was performed within the framework of the direct capture reaction mechanism. The calculations agreed quite well with the experimental data.
A new astrophysical reaction rate for $^{12}$C($α,γ$)$^{16}$O has been evaluated on the basis of a global R-matrix fitting to the available experimental data. The reaction rates of $^{12}$C($α,γ$)$^{16}$O for stellar temperatures between 0.04 $\leq$ $T_9$ $\leq$ 10 are provided in a tabular form and by an analytical fitting expression. At $T_9$ = 0.2, the reaction rate is (7.83 $\pm$ 0.35)$\times10^{15}$ $\rm{cm^3 mol^{-1} s^{-1}}$, where stellar helium burning occurs.
We have studied the neutron-capture reactions 10,11B(n, g) and the role of the 11B(n, g) reaction in seeding r-process nucleosynthesis. The possibility of the description of the available experimental data for cross sections of the neutron capture reaction on 10B at thermal and astrophysical energies, taking into account the resonance at 475 keV, was considered within the framework of the modified potential cluster model (MPCM) with forbidden states and accounting for the resonance behavior of the scattering phase shifts. In the framework of the same model the possibility of describing the ava…
We investigate the energy dependence of the astrophysical S factors for the reactions 7Be(p,gamma)8B, the primary source of high-energy solar neutrinos in the solar pp chain, and 16O(p,gamma)17F, an important reaction in the CNO cycle. Both of these reactions have predicted S factors which rise at low energies; we find the source of this behavior to be a pole in the S factor at a center-of-mass energy E = -E_B, the point where the energy of the emitted photon vanishes. The pole arises from a divergence of the radial integrals.
We study the effect of vacuum polarization in nuclear reactions of astrophysical interest. This effect has the opposite sign compared to the screening by the atomic electrons. It is shown that vacuum polarization further increases the longstanding differences between the experimental data of astrophysical nuclear reactions at very low energies and the theoretical calculations which aim to include electron screening, albeit by a small amount.
Nucleosynthesis processes involve reactions on several thousand nuclei, both close to and far off stability. The preparation of reaction rates to be used in astrophysical investigations requires experimental and theoretical input. In this context, two interesting aspects are discussed: (i) the relevant gamma transition energies in astrophysical capture reactions, and (ii) the newly discovered Coulomb suppression of the stellar enhancement factor. The latter makes a number of reactions with negative Q value more favorable for experimental investigation than their inverse reactions, contrary to …
The phase shift analysis, done on the basis of the known measurements of the differential cross-sections of the p13C elastic scattering at the energy range 250-750 keV, shows that it is enough to take into account only 3S1 wave in the considered energy region. The potential for the triplet 3S1 state in p13C system at the resonance energy 0.55 MeV corresponding to quantum numbers JpT = 1-1 as well as the potential for the 3P1 bound state of 14N were constructed on the basis of the obtained scattering phase shifts. The possibility to describe the experimental data of the astrophysical S-factor o…
Theoretical estimations for the astrophysical S-factor and the d(alpha,gamma)6Li reaction rates are obtained on the base of the two-body model with the alpha-d potential of a simple Gaussian form, which describes correctly the phase-shifts in the S-, P-, and D-waves, the binding energy and the asymptotic normalization constant in the final S-state. Wave functions of the bound and continuum states are calculated by using the Numerov algorithm of a high accuracy. A good convergence of the results for the E1- and E2- components of the transition is shown when increasing the upper limit of effecti…
The relevant energy ranges for stellar nuclear reactions are introduced. Low-energy compound and direct reactions are discussed. Stellar modifications of the cross sections are presented. Implications for experiments are outlined.
The Coulomb dissociation method to obtain the astrophysical S-factor, $S_{17}(0)$, for the $^7$Be($p,γ)^8$B reaction at solar energies is investigated by analysing the recently measured data on the breakup reaction $^{208}$Pb$(^8$B,$^7$Be$~p)^{208}$Pb at 46.5 MeV/A beam energy. Breakup cross sections corresponding to E1, $E2$ and $M1$ transitions are calculated with a theory of Coulomb excitation that includes the effects of the Coulomb recoil as well as relativistic retardation. The interplay of nuclear and Coulomb contributions to the breakup process is studied by performing a full quantum m…
In this review we have considered the possibility to describe the astrophysical S-factors of radiative capture reactions with light atomic nuclei on the basis of the potential two-cluster model by taking into account the splitting the orbital states according to Young's schemes. Within this model, interaction of the nucleon clusters is described by local two-particle potential determined by fit to the scattering data and properties of bound states of these clusters. Many-body character of the problem is taken into account under some approximation, in terms of the allowed or forbidden by the Pa…
A simple description for obtaining the parity distribution of nuclear levels in the pf + g9/2 shell as a function of excitation energy was recently derived. We implement this in a global nuclear level density model. In the framework of the statistical model, cross sections and astrophysical reaction rates are calculated in the Fe region and compared to rates obtained with the common assumption of an equal distribution of parities. We find considerable differences, especially for reactions involving particles in the exit channel.
The hindrance phenomenon of heavy-ion fusion cross sections at deep subbarrier energies often accompanies a maximum of an astrophysical $S$ factor at a threshold energy for fusion hindrance. We argue that this phenomenon can naturally be explained when the fusion excitation function is fitted with two potentials, with a larger (smaller) logarithmic slope at energies lower (higher) than the threshold energy. This analysis clearly suggests that the astrophysical $S$ factor provides a convenient tool to analyze the deep subbarrier hindrance phenomenon, even though the $S$ factor may have a strong…
The theoretical cross section calculations for the astrophysical $p$ process are needed because the most of the related reactions are technically very difficult to be measured in the laboratory. Even if the reaction was measured, most of the measured reactions have been carried out at the higher energy range from the astrophysical energies. Therefore, almost all cross sections needed for $p$ process simulation has to be theoretically calculated or extrapolated to the astrophysical energies. The $^{112}$Sn($α$,$γ$)$^{116}$Te is an important reaction for the $p$ process nucleosynthesis. The theo…
We have applied the Gamow shell model (GSM) in the coupled-channel representation to study the astrophysical $S$-factor for the proton radiative capture reaction of $^6$Li. Reaction channels are built by coupling the proton wave function expanded in different partial waves with the GSM wave functions of the ground state ($1^+$) and the excited states ($3^+_1$, $0^+_1$ and $2^+_1$) of $^6$Li. All relevant $E1$, $M1$, and $E2$ transitions from the initial continuum states in $^7$Be to the final bound states (${3/2}^-_1$ and ${1/2}^-_1$) states are included. It is found that the calculated total …
The astrophysical S-factor of the 4He-12C radiative capture is calculated in the potential model at the energy range 0.1-2.0 MeV. Radiative capture 12C(α,γ)16O is extremely relevant for the fate of massive stars and determines if the remnant of a supernova explosion becomes a black hole or a neutron star. Because this reaction occurs at low-energies the experimental measurements is very difficult and perhaps impossible. In this paper, radiative capture of the 12C(α,γ)16O reaction at very low-energies is taken as a case study. In comparison with other theoretical methods and available experimen…
A computer program is presented aiming at the calculation of bound and continuum states, reduced transition probabilities, phase-shifts, photo-disintegration cross sections, radiative capture cross sections, and astrophysical S-factors, for a two-body nuclear system. The code is based on a potential model of a Woods-Saxon, a Gaussian, or a M3Y, type. It can be used to calculate nuclear reaction rates in numerous astrophysical scenarios.
The relationship between the Coulomb displacement energy for the A=8, J=2+, T=1 state and the low-energy astrophysical S factor for the Be7(p,gamma)B8 reaction is discussed. The displacement energy is interpreted in a particle-hole model. The dependence of the particle displacement energy on the potential well geometry is investigated and is used to relate the particle displacement energy to the rms radius and the asymptotic normalization of the valence proton wave function in B8. The asymptotic normalization is used to calculate the astrophysical S factor for the Be7(p,gamma)B8 reaction. The …
In a previous publication [Atomic Data and Nuclear Data Tables 75, 1 (2000)], we had given seven parameter analytical fits to theoretical reaction rates derived from nuclear cross sections calculated in the statistical model (Hauser-Feshbach formalism) for targets with 9<Z<84 (Ne to Bi) and for a mass range reaching the neutron and proton driplines. Reactions considered were (n,gamma), (n,p), (n,alpha), (p,gamma), (p,alpha), (alpha,gamma), and their inverse reactions. On public demand, we present here the theoretical nuclear cross sections and astrophysical reaction rates from which those rate…
The majority of nuclear reactions in astrophysics involve unstable nuclei which are not fully accessible by experiments yet. Therefore, there is high demand for reliable predictions of cross sections and reaction rates by theoretical means. The majority of reactions can be treated in the framework of the statistical model (Hauser-Feshbach). The global parametrizations of the nuclear properties needed for predictions far off stability probe our understanding of the strong force and take it to its limit. The sensitivity of astrophysical scenarios to nuclear inputs is illustrated in the framework…
Microscopic calculations show a strong parity dependence of the nuclear level density at low excitation energy of a nucleus. Previously, this dependence has either been neglected or only implemented in the initial and final channels of Hauser-Feshbach calculations. We present an indirect way to account for a full parity dependence in all steps of a reaction, including the one of the compound nucleus formed in a reaction. To illustrate the impact on astrophysical reaction rates, we present rates for neutron captures in isotopic chains of Ni and Sn. Comparing with the standard assumption of equi…
Reacting astrophysical flows can be challenging to model because of the difficulty in accurately coupling hydrodynamics and reactions. This can be particularly acute during explosive burning or at high temperatures where nuclear statistical equilibrium is established. We develop a new approach based on the ideas of spectral deferred corrections (SDC) coupling of explicit hydrodynamics and stiff reaction sources as an alternative to operator splitting that is simpler than the more comprehensive SDC approach we demonstrated previously. We apply the new method to a double detonation problem with …
