Asymptotic normalization coefficients (nuclear vertex constants) for $p+^7Be\to ^8B$ and the direct $^7Be(p,γ)^8B$ astrophysical S-factors at solar energies
A new analysis of the precise experimental astrophysical S-factors for the direct capture $^7Be(p,γ)$ $^8B$ reaction [A.J.Junghans et al.Phys.Rev. C 68 (2003) 065803 and L.T. Baby et al. Phys.Rev. C 67 (2003) 065805] is carried out based on the modified two - body potential approach in which the direct astrophysical S-factor, $ {\rm S_{17}(E)}$, is expressed in terms of the asymptotic normalization constants for $p+^7Be\to ^8B$ and two additional conditions are involved to verify the peripheral character of the reaction under consideration. The Woods-Saxon potential form is used for the bound ($p+^7Be$)- state wave function and for the $p^7Be$- scattering wave function. New estimates are obtained for the $^{\glqq}$indirectly measured\grqq values of the asymptotic normalization constants (the nuclear vertex constants) for the $p+^7Be\to ^8B$ and $S_{17}(E)$ at E$\le$ 115 keV, including $E$=0. These values of $S_{17}(E)$ and asymptotic normalization constants have been used for getting information about the $^{\glqq}$indirectly measured\grqq values of the $s$ wave average scattering length and the $p$ wave effective range parameters for $p^7Be$- scattering.
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 normalization condition for the relativistic three nucleon Bethe-Salpeter and Gross bound state vertex functions is derived, for the first time, directly from the three body wave equations. It is also shown that the relativistic normalization condition for the two body Gross bound state vertex function is identical to the requirement that the bound state charge be conserved, proving that charge is automatically conserved by this equation.
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…
Explicit relations between the effective-range expansion and the nuclear vertex constant or asymptotic normalization coefficient (ANC) for the virtual decay $B\to A+a$ are derived for an arbitrary orbital momentum together with the corresponding location condition for the ($A+a$) bound-state energy. They are valid both for the charged case and for the neutral case. Combining these relations with the standard effective-range function up to order six makes it possible to reduce to two the number of free effective-range parameters if an ANC value is known from experiment. Values for the scatterin…
Owing to the presence of the Coulomb barrier at astrophysically relevant kinetic energies it is very difficult, or sometimes impossible, to measure astrophysical reaction rates in the laboratory. That is why different indirect techniques are being used along with direct measurements. Here we address two important indirect techniques, the asymptotic normalization coefficient (ANC) and the Trojan Horse (TH) methods. We discuss the application of the ANC technique for calculation of the astrophysical processes in the presence of subthreshold bound states, in particular, two different mechanisms a…
A new analysis of the modern precise measured astrophysical $S$ factors for the direct capture $^3He(α,γ)^7{\rm {Be}}$ reaction [B.S. Nara Singh {\it et al.}, Phys.Rev.Lett. {\bf 93}, 262503 (2004); D. Bemmerer {\it et al.}, Phys.Rev.Lett. {\bf 97}, 122502 (2006); F.Confortola {\it et al.}, Phys.Rev.C {\bf 75}, 065803 (2007), T.A.D.Brown {\it et al.}, Phys.Rev. C {\bf 76}, 055801 (2007) and A Di Leva, {\it et al.},Phys.Rev.Lett. {\bf 102}, 232502 (2009)] populating to the ground and first excited states of $^7{\rm Be}$ is carried out based on the modified two - body potential approach. New est…
A new analysis of the precise experimental astrophysical $S$-factors for the direct capture $^3He(α,γ)^7{\rm {Be}}$ reaction [B.S. Nara Singh et al., Phys.Rev.Lett. {\bf 93} (2004) 262503; D. Bemmerer et al., Phys.Rev.Lett. {\bf 97} (2006) 122502; F.Confortola et al., Phys.Rev. {\bf C 75} (2007) 065803 and T.A.D.Brown et al., Phys.Rev. {\bf C 76} (2007) 055801] populating to the ground and first excited states of $^7{\rm Be}$ is carried out based on the modified two - body potential approach in which the direct astrophysical $S$-factor, $S_{34}(E)$, is expressed in terms of the asymptotic norm…
The method of asymptotic normalization coefficients is a standard approach for studies of two-body non-resonant radiative capture processes in nuclear astrophysics. This method suggests a fully analytical description of the radiative capture cross section in the low-energy region of the astrophysical interest. We demonstrate how this method can be generalized to the case of three-body $2p$ radiative captures. It was found that an essential feature of this process is the highly correlated nature of the capture. This reflects the complexity of three-body Coulomb continuum problem. Radiative capt…
The $^{22}$Ne(p,$γ$)$^{23}$Na reaction in NeNa cycle plays an important role in the production of only stable sodium isotope $^{23}$Na. This nucleus is processed by the NeNa cycle during hot bottom burning (HBB) in asymptotic giant branch (AGB) stage of low metallicity intermediate mass stats (4 M$_O$ $\leq$ M $\leq$ 6 M$_O$). Recent measurements have addressed the uncertainty in the thermonuclear reaction rate of this reaction at relevant astrophysical energies through the identification of low lying resonances at E$_p$ = 71,105, 156.2, 189.5 and 259.7 keV. In addition, precise measurements o…
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 …
We show how the charge symmetry of strong interactions can be used to relate the proton and neutron asymptotic normalization coefficients (ANCs) of the one nucleon overlap integrals for light mirror nuclei. This relation extends to the case of real proton decay where the mirror analog is a virtual neutron decay of a loosely bound state. In this case, a link is obtained between the proton width and the squared ANC of the mirror neutron state. The relation between mirror overlaps can be used to study astrophysically relevant proton capture reactions based on information obtained from transfer re…
Recently to study the radiative capture $α+ d \to {}^{6}{\rm Li} + γ$ process a new measurement of the ${}^{6}{\rm Li}({\rm A\,150\,MeV})$ dissociation in the field of ${}^{208}{\rm Pb}$ has been reported in [F. Hammache {\it et al.} Phys. Rev ${\bf C 82}$, 065803 (2010)]. However, the dominance of the nuclear breakup over the Coulomb one prevented from obtaining the information about the $α+ d \to {}^{6}{\rm Li} + γ$ process from the breakup data. The astrophysical $S_{24}(E)$ factor has been calculated within the $α-d$ two-body potential model with potentials determined from the fits to the …
It has been suggested recently (Phys. Rev. Lett. 91, 232501 (2003)) that charge symmetry of nucleon-nucleon interactions relates the Asymptotic Normalization Coefficients (ANCs) of proton and neutron virtual decays of mirror nuclei. This relation is given by a simple analytical formula which involves proton and neutron separation energies, charges of residual nuclei and the range of their strong interaction with the last nucleon. Relation between mirror ANCs, if understood properly, can be used to predict astrophysically relevant direct proton capture cross sections using neutron ANCs measured…
Radiative capture of nucleons at energies of astrophysical interest is one of the most important processes for nucleosynthesis. The nucleon capture can occur either by a compound nucleus reaction or by a direct process. The compound reaction cross sections are usually very small, specially for light nuclei. The direct capture proceeds either via the formation of a single-particle resonance, or a non-resonant capture process. In this work we calculate radiative capture cross sections and astrophysical S-factors for nuclei in the mass region A<20 using single-particle states. We carefully discus…
Assuming that the ratio between asymptotic normalization coefficients of mirror states is model independent, charge symmetry can be used to indirectly extract astrophysically relevant proton capture reactions on proton-rich nuclei based on information on stable isotopes. The assumption has been tested for light nuclei within the microscopic cluster model. In this work we explore the Hamiltonian independence of the ratio between asymptotic normalization coefficients of mirror states when deformation and core excitation is introduced in the system. For this purpose we consider a phenomenological…
The $^{15}{\rm N}(p,γ)^{16}{\rm O}$ reaction provides a path from the CN cycle to the CNO bi-cycle and CNO tri-cycle. The measured astrophysical factor for this reaction is dominated by resonant capture through two strong $J^π=1^{-}$ resonances at $E_{R}= 312$ and 962 keV and direct capture to the ground state. Recently, a new measurement of the astrophysical factor for the $^{15}{\rm N}(p,γ)^{16}{\rm O}$ reaction has been published [P. J. LeBlanc {\it et al.}, Phys. Rev. {\bf C 82}, 055804 (2010)]. The analysis has been done using the $R$-matrix approach with unconstrained variation of all pa…
Bayesian inferences on covariant density functionals from multimessenger astrophysical data: The impacts of likelihood functions of low density matter constraints
The $N \to Δ$ weak vertex provides an important contribution to the one pion production in neutrino-nucleon and neutrino-nucleus scattering for $πN$ invariant masses below 1.4 GeV. Beyond its interest as a tool in neutrino detection and their background analyses, one pion production in neutrino-nucleon scattering is useful to test predictions based on the quark model and other internal symmetries of strong interactions. Here we try to establish a connection between two commonly used parametrizations of the weak $N \to Δ$ vertex and form factors (FF) and we study their effects on the determinat…
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…
Existing experimental data for the breakup of 8B at energies from 30 to 1000 MeV/nucleon on light through heavy targets are analyzed in detail in terms of an extended Glauber model. The predictions of the model are in excellent agreement with independent reaction data (reaction cross sections and parallel momentum distributions for core like fragments). Final state interactions have been included in the Coulomb dissociation component. We extract asymptotic normalization coefficients (ANC) from which the strophysical factor S17(0) for the key reaction for solar neutrino production, 7Be(p,gamma)…
Astrophysical S-factors at zero energy for the direct nuclear capture reactions $^{3}{\rm He}(α, γ)^{7}{\rm Be}$, $^{3}{\rm H}(α, γ)^{7}{\rm Li}$ and $^{7}{\rm Be}(p, γ)^{8}{\rm B}$ are estimated within the framework of two-body potential cluster model on the basis of extranuclear capture approximation of D. Baye and E. Brainis. The values of S(0)-factors have been calculated using two different potential models for each process, which were adjusted to the binding energies and empirical values of the asymptotical normalization coefficients from the literature. New values of S(0)-factors have b…
Low energy capture cross sections are calculated within a microscopic many-body approach using an effective Hamiltonian derived from the Argonne V18 potential. The dynamics is treated within Fermionic Molecular Dynamics (FMD) which uses a Gaussian wave-packet basis to represent the many-body states. A phase-shift equivalent effective interaction derived within the Unitary Correlation Operator Method (UCOM) that treats explicitly short-range central and tensor correlations is employed. As a first application the 3He(alpha,gamma)7Be reaction is presented. Within the FMD approach the microscopic …
Recently, a new $Δ$ method for the calculation of asymptotic normalization coefficients (ANC) from phase-shift data has been formulated, proved and used for bound states. This method differs from the conventional one by fitting only the nuclear part of the effective-range function which includes a partial phase shift. It should be applied to large-charge nuclei when the conventional effective-range expansion or the Padé-approximations using the effective-range function $K_l(k^2)$ fitting do not work. A typical example is the nucleus vertex $α+^{12}$C $\longleftrightarrow ^{16}$O. Here we apply…
The $α+d\rightarrow\, ^6{\rm Li}+γ$ radiative capture is studied in order to predict the $^6$Li primordial abundance. Within a two-body framework, the $α$ particle and the deuteron are considered the structureless constituents of $^6$Li. Five $α+d$ potentials are used to solve the two-body problem: four of them are taken from the literature, only one having also a tensor component. A fifth model is here constructed in order to reproduce, besides the $^6$Li static properties as binding energy, magnetic dipole and electric quadrupole moments, also the $S$-state asymptotic normalization coefficie…
The astrophysical factor $S_{17}$ for $^7$Be($p,γ$)$^8$B reaction is reliably extracted from the transfer reaction $^7$Be($d,n$)$^8$B at $E=7.5$ MeV with the asymptotic normalization coefficient method. The transfer reaction is accurately analyzed with CDCC based on the three-body model. This analysis is free from uncertainties of the optical potentials having been crucial in the previous DWBA analyses.
The $^{13}{\rm C}(α,n)^{16}{\rm O}$ reaction is considered to be the main neutron source responsible for the production of heavy nuclides (from ${\rm Sr}$ to ${\rm Bi}$) through slow $n$-capture nucleosynthesis ($s$-process) at low temperatures during the asymptotic giant branch (AGB) phase of low mass stars ($\lesssim 3-4\;{\rm M}_{\odot}$, or LMSs). In recent years, several direct and indirect measurements have been carried out to determine the cross section at the energies of astrophysical interest (around $190\pm40\;{\rm keV}$). However, they yield inconsistent results causing a highly unc…
Astrophysical models studying the origin of the p-nuclei require knowledge of the reaction rates of photodisintegrations and capture reactions. Since experimental data at astrophysically relevant energies are limited, reaction rate calculations rely on Hauser-Feshbach (HF) theory predictions. The HF theory requires nuclear physics input such as masses, level densities, $γ$-ray strength functions and proton-nucleus optical potentials (pOMP). The scope of this work is to improve a global semi-microscopic pOMP at energies relevant to the p-process. This is achieved by adjusting the normalization …
We present S factor data obtained from the Coulomb dissociation of 83 MeV/nucleon 8B, and analyze 7Be longitudinal momentum distributions measured at 44 and 81 MeV/nucleon using a potential model, first-order perturbation theory, and dynamical solution of the time-dependent Schroedinger equation. Comparing our results with independent continuum-discretized coupled channels calculations, we study the reaction model and beam energy dependence of the E2 contribution to the dissociation cross section. By fitting radiative capture and Coulomb breakup data taken below relative energies of 400 keV wi…
Angular distribution measurements of $^2$H($^7$Be,$^7$Be)$^2$H and $^2$H($^7$Be,$^8$B)$n$ reactions at $E_{c.m.}\sim$~4.5 MeV were performed to extract the astrophysical $S_{17}(0)$ factor using the asymptotic normalization coefficient (ANC) method. For this purpose a pure, low emittance $^7$Be beam was separated from the primary $^7$Li beam by a recoil mass spectrometer operated in a novel mode. A beam stopper at 0$^{\circ}$ allowed the use of a higher $^7$Be beam intensity. Measurement of the elastic scattering in the entrance channel using kinematic coincidence, facilitated the determinatio…
The astrophysical $^{3}{\rm He}(α, γ)^{7}{\rm Be}$ and $^{3}{\rm H}(α, γ)^{7}{\rm Li}$ direct capture processes are studied in the framework of the two-body model with the potentials of a simple Gaussian form, which describe correctly the phase-shifts in the s-, p-, d-, and f-waves, as well as the binding energy and the asymptotic normalization constant of the ground $p_{3/2}$ and the first excited $p_{1/2}$ bound states. It is shown that the E1-transition from the initial s-wave to the final p-waves is strongly dominant in both capture reactions. On this basis the s-wave potential parameters …
The astrophysical factor of the 8B(p,gamma)9C at zero energy, S18(0), is determined from three-body model analysis of 9C breakup processes. The elastic breakup 208Pb(9C,p8B)208Pb at 65 MeV/nucleon and the one-proton removal reaction of 9C at 285 MeV/nucleon on C and Al targets are calculated with the continuum-discretized coupled-channels method (CDCC) and the eikonal reaction theory (ERT), respectively. The asymptotic normalization coefficient (ANC) of 9C in the p-8B configuration extracted from the two reactions show good consistency, in contrast to in the previous studies. As a result of th…
Low-energy doublet neutron-deuteron scattering is described on the basis of the triton bound and virtual state parameters - the energies and the nuclear vertex constants of these states. The van Oers-Seagrave formula is derived from the Bargmann representation of the S matrix for a system having two states. The presence of a pole in this formula is shown to be a direct corollary of the existence of a low-energy triton virtual state. Simple explicit expressions for the nd scattering length and for the pole of the function $k\cot δ$ are obtained in terms of the triton bound and virtual state par…
It is known from the scattering theory that the phase-shift of elastic collision does not provide a unique potential to describe the bound state of the two-particle system. The bound state wave function is the most crucial input for various nuclear processes bearing astrophysical significance. In this paper, we emphasize on the important role of the asymptotic normalization coefficients for description of the $^{13}\rm{N}\longrightarrow {^{13}\rm{C}}+β^++ν_e$ reaction. Using experimental data of the elastic scattering phase-shift of proton and neutron on $^{12}\rm{C}$, we find the asymptotic n…
A new dispersion (asymptotic) theory is proposed for the peripheral sub- and above-barrier charged particle transfer $A(x,y)B$ reaction in the three-body ($A$, $a$ and $y$) model where $ x= y +a$ and $B=A+a$, and $ a$ is a transferred particle. It is based on the combination of the dispersion theory and the conventional DWBA method. The explicit forms have been derived for the exact three-body pole amplitude and differential cross section in which the contribution of the three-body ($A$, $a$ and $y$) Coulomb dynamics of the transfer mechanism in the peripheral partial amplitudes, corresponding…
We discuss the use of one-nucleon breakup reactions of loosely bound nuclei at intermediate energies as an indirect method in nuclear astrophysics. These are peripheral processes, therefore we can extract asymptotic normalization coefficients (ANC) from which reaction rates of astrophysical interest can be inferred. To show the usefulness of the method, three different cases are discussed. In the first, existing experimental data for the breakup of 8B at energies from 30 to 1000 MeV/u and of 9C at 285 MeV/u on light through heavy targets are analyzed. Glauber model calculations in the eikonal …
