Asymptotic Normalization Coefficients Research Articles

Bayesian estimation of theSfactor and thermonuclear reaction rate forO16(p,γ)F17

The $^{16}$O(p,$\gamma$)$^{17}$F reaction is the slowest hydrogen-burning process in the CNO mass region. Its thermonuclear rate sensitively impacts predictions of oxygen isotopic ratios in a number of astrophysical sites, including AGB stars. The reaction has been measured several times at low bombarding energies using a variety of techniques. The most recent evaluated experimental rates have a reported uncertainty of about 7.5\% below $1$~GK. However, the previous rate estimate represents a best guess only and was not based on rigorous statistical methods. We apply a Bayesian model to fit all reliable $^{16}$O(p,$\gamma$)$^{17}$F cross section data, and take into account independent contributions of statistical and systematic uncertainties. The nuclear reaction model employed is a single-particle potential model involving a Woods-Saxon potential for generating the radial bound state wave function. The model has three physical parameters, the radius and diffuseness of the Woods-Saxon potential, and the asymptotic normalization coefficients (ANCs) of the final bound state in $^{17}$F. We find that performing the Bayesian $S$ factor fit using ANCs as scaling parameters has a distinct advantage over adopting spectroscopic factors instead. Based on these results, we present the first statistically rigorous estimation of experimental $^{16}$O(p,$\gamma$)$^{17}$F reaction rates, with uncertainties ($\pm 4.2$\%) of about half the previously reported values.

Compatibility of the asymptotic normalization coefficient for the C14→B13+p overlap extracted from the C14(B11,C12)B13 reaction with C14(d,He3)B13 data

When the asymptotic normalization coefficient (ANC) for the ${}^{\mathrm{A}+1}(Z+1)\ensuremath{\rightarrow}{\phantom{\rule{0.16em}{0ex}}}^{\mathrm{A}}Z+p$ overlap is extracted from data for proton pickup from the ${}^{\mathrm{A}+1}(Z+1)$ nucleus, the value obtained may strongly depend on the degree of completeness of the modeling of the reaction mechanism, particularly when heavy-ion probes are employed. Taking the specific case of the ANC for the $^{14}\mathrm{C}\ensuremath{\rightarrow}^{13}\mathrm{B}+p$ overlap, an analysis of data for the heavy-ion $^{14}\mathrm{C}(^{11}\mathrm{B},^{12}\mathrm{C})^{13}\mathrm{B}$ reaction showed that the distorted wave Born approximation (DWBA) yields a significantly larger ANC than the coupled reaction channel (CRC) technique, incorporating multistep reaction paths. In this paper we present an analysis of data from the literature for the light-ion $^{14}\mathrm{C}(d,^{3}\mathrm{He})^{13}\mathrm{B}$ pickup reaction and show that the ANC extracted from the heavy-ion data using the CRC technique is entirely compatible with the light-ion data when used in either DWBA or more complete coupled discretized continuum channel plus CRC calculations. The present results also show that the ANC extracted from the light-ion reaction is essentially unaffected by multistep reaction paths, in contrast with the heavy-ion case.

Elastic α−C12 scattering at low energies with the resonant 22+ and 23+ states of O16

The elastic $\alpha$-$^{12}$C scattering for $l=2$ at low energies is studied in an effective Lagrangian approach. We explicitly include two resonant $2_2^+$ and $2_3^+$ states of $^{16}$O in the scattering amplitudes and construct an $S$ matrix assuming that three amplitudes, the non-resonant part of the amplitude which has the sub-threshold $2_1^+$ state of $^{16}$O and those of the two resonant states, are represented by the summation of corresponding parts of the phase shift. Then, we fit the parameters in the $S$ matrix to the phase shift data by imposing three conditions at very low energies where the phase shift data are not available. By using the fitted parameters, we calculate the asymptotic normalization coefficients (ANC) of the $2_1^+$ state of $^{16}$O and find that the previously reported small and large values of the ANC can be reproduced depending on the imposed conditions, but we obtain large error bars for the large ANC values which are reported from the $\alpha$ transfer reactions.

Performing Bayesian Analyses With AZURE2 Using BRICK: An Application to the 7Be System

Phenomenological R-matrix has been a standard framework for the evaluation of resolved resonance cross section data in nuclear physics for many years. It is a powerful method for comparing different types of experimental nuclear data and combining the results of many different experimental measurements in order to gain a better estimation of the true underlying cross sections. Yet a practical challenge has always been the estimation of the uncertainty on both the cross sections at the energies of interest and the fit parameters, which can take the form of standard level parameters. Frequentist (χ2-based) estimation has been the norm. In this work, a Markov Chain Monte Carlo sampler, emcee, has been implemented for the R-matrix code AZURE2, creating the Bayesian R-matrix Inference Code Kit (BRICK). Bayesian uncertainty estimation has then been carried out for a simultaneous R-matrix fit of the 3He (α,γ)7Be and 3He (α,α)3He reactions in order to gain further insight into the fitting of capture and scattering data. Both data sets constrain the values of the bound state α-particle asymptotic normalization coefficients in 7Be. The analysis highlights the need for low-energy scattering data with well-documented uncertainty information and shows how misleading results can be obtained in its absence.

Erratum to: Determination of Asymptotic Normalization Coefficients by Analytic Continuation of Differential Cross Sections

An Erratum to this paper has been published: https://doi.org/10.1134/S106377882233002X

Determination of Asymptotic Normalization Coefficients by Analytic Continuation of Differential Cross Sections

A method is discussed for determining the asymptotic normalization coefficients (ANC) using the analytic continuation of experimental differential cross sections for nuclear transfer reactions to the pole point of the reaction amplitude with respect to the variable costheta, where theta is the scattering angle. Within the framework of this method, using experimental data on the differential cross section for the reaction {}^{12}textrm{C}(0^{+};0textrm{ MeV})(d,p){}^{13}textrm{C}(1/2^{+};3.09textrm{ MeV}) the ANC for the channel {}^{13}textrm{C}(1/2^{+};3.09textrm{ MeV})to{}^{12}textrm{C}(0^{+};0textrm{ MeV})+n was determined. When determining the ANC, the corrections caused by the Coulomb interaction in the initial, final and intermediate states of the reaction were taken into account. It is shown that these corrections significantly affect the extracted ANC value.

Extracting the asymptotic normalization coefficient for the C14→B13+p overlap from the C14(B11,C12)B13 reaction

The availability of a comparatively complete data set for the elastic and inelastic scattering of a 45-MeV beam of $^{11}\mathrm{B}$ ions from a $^{14}\mathrm{C}$ target, together with an angular distribution for the $^{14}\mathrm{C}(^{11}\mathrm{B},^{12}\mathrm{C})^{13}\mathrm{B}$ proton pickup reaction enables a determination of the $\ensuremath{\langle}^{14}\mathrm{C}\ensuremath{\mid}^{13}\mathrm{B}+p\ensuremath{\rangle}$ asymptotic normalization coefficient (ANC) using the coupled reaction channel (CRC) technique. The complete nature of the data set allows two-step contributions to the reaction mechanism to be controlled while a set of $\ensuremath{\langle}^{12}\mathrm{C}\ensuremath{\mid}^{11}\mathrm{B}+p\ensuremath{\rangle}$ overlaps obtained from a consistent analysis of ($d,^{3}\mathrm{He}$) and ($e,{e}^{\ensuremath{'}}p$) data fixes the projectile overlaps. We find that the level of completeness of the modeling of the reaction mechanism has a significant effect on the value obtained for the ANC, with a distorted wave Born approximation analysis yielding a significantly larger ANC than the full CRC calculation. The final result obtained, when compared with the theoretical value calculated within the source term approach, is in accord with trends for proton removal from similar $p$-shell nuclei.

Application of ANCs for calculation of β+ decay of 17F nucleus

Calculations of half-life in the single-particle model of the nucleus generally give a lower value than the measured data. This difference in the case of beta decay of the 17F nucleus is ∼13%. Calculating many-particle wave functions, taking into account all possible contributions from the interaction between the particles of the nucleus is a very difficult task. We have applied the method of two-particle clustering taking into account spectroscopic factors for the parent and daughter nuclei and have obtained a good agreement with the experimental values of logft and the half-life of the beta decay of 17F. For this purpose, we have used the fitting procedure to find the proton and neutron asymptotic normalization coefficients (ANCs) for the 17F and 17O nuclei, respectively.

Trojan Horse Investigation for AGB Stellar Nucleosynthesis

Asymptotic Giant Branch (AGB) stars are among the most important astrophysical sites influencing the nucleosynthesis and the chemical abundances in the Universe. From a pure nuclear point of view, several processes take part during this peculiar stage of stellar evolution thus requiring detailed experimental cross section measurements. Here, we report on the most recent results achieved via the application of the Trojan Horse Method (THM) and Asymptotic Normalization Coefficient (ANC) indirect techniques, discussing the details of the experimental procedure and the deduced reaction rates. In addition, we report also on the on going studies of interest for AGB nucleosynthesis.

Asymptotic normalization coefficients in nuclear reactions and nuclear astrophysics

Asymptotic normalization coefficients in nuclear reactions and nuclear astrophysics

Halo effective field theory analysis of one-neutron knockout reactions of Be11 and C15

Background: One-nucleon knockout reactions provide insightful information on the single-particle structure of nuclei. When applied to one-neutron halo nuclei, they are purely peripheral, suggesting that they could be properly modeled by describing the projectile within a Halo Effective Field Theory (Halo-EFT). Purpose: We reanalyze the one-neutron knockout measurements of $^{11}$Be and $^{15}$C-both one-neutron halo nuclei-on beryllium at about 60MeV/nucleon. We consider Halo-EFT descriptions of these nuclei which already provide excellent agreement with breakup and transfer data. Method: We include a Halo-EFT description of the projectile within an eikonal-based model of the reaction and compare its outcome to existing data. Results: Excellent agreement with experiment is found for both nuclei. The asymptotic normalization coefficients inferred from this comparison confirm predictions from \emph{ab initio} nuclear-structure calculations and values deduced from transfer data. Conclusions: Halo-EFT can be reliably used to analyze one-neutron knockout reactions measured for halo nuclei and test predictions from state-of-the-art nuclear structure models on these experimental data.

Indirect determination of the astrophysical S factor for the Li6 ( p,γ)Be7 reaction using the asymptotic normalization coefficient method

Background: The Li 6 ( p , γ ) Be 7 cross section influences a variety of astrophysical scenarios, including big-bang and stellar nucleosynthesis. In recent years, conflicting results of direct measurements have been published, reporting contradictory low-energy trends. Purpose: To shed light on the contradiction between the existing data sets, the reaction was studied using the asymptotic normalization coefficient (ANC) technique which was up-to-now never used for this reaction. Methods: To derive the ANC, the Li 6 ( He 3 , d ) Be 7 transfer reaction, studied at the Department of Physics and Astronomy of the University of Catania and at the John. D. Fox Superconducting Accelerator Laboratory at Florida State University, was re-analyzed, focusing on the proton transfer mechanism [the α transfer process is discussed by Kiss et al. [Phys. Lett. B 807, 135606 (2020)]. The energy of the He 3 beam impinging on a Li 6 target was E lab = 3 MeV and E lab = 5 MeV. The yield of the emitted deuterons was measured with high precision by using silicon Δ E - E telescopes. Results: From the DWBA analysis of the angular distributions of the emitted deuterons populating the ground ( E * = 0.0 MeV; 3 2 − ) and the first excited ( E * = 0.429 MeV ; 1 2 − ) states of Be 7 , the ANCs for the Li 6 + p → Be 7 system were deduced. Furthermore, the recently measured Li 6 ( p , γ ) Be 7 reaction cross sections [Piatti , Phys. Rev. C 102, 052802 (2020)] were also analyzed within this theoretical framework. Excellent agreement was found between ANC values derived indirectly and those determined from the direct data, which strengthens the conclusion of the present work. The astrophysical S factor—at energies characterizing the Sun—for the Li 6 ( p , γ ) Be 7 reaction was calculated using the weighted mean of the experimentally derived ANC values. Conclusions: The result of the present comprehensive study supports the extrapolation of Piatti [Phys. Rev. C 102, 052802 (2020)], Dong [J. Phys. G Nucl. Partic. 44, 045201 (2017)], and Gnech and Marcucci [Nucl. Phys. A 987, 1 (2019)], and thus disfavors the conclusions drawn by He [Phys. Lett. B 725, 287 (2013)] and Xu [Nucl. Phys. A 918, 61 (2013)].

Study of the Influence of Different Methods of Taking into Account the Coulomb Interaction on Determining Asymptotic Normalization Coefficients within the Framework of Exactly Solvable Model

It is shown that the Schrödinger equation for the sum of the potential of a square well and the Coulomb potential of a uniformly charged sphere admits an analytical solution for arbitrary values of the orbital angular momentum. An explicit form of this solution has been found. Using the obtained solution, the influence of the Coulomb interaction for both point and distributed nuclear charges on the values of asymptotic normalization coefficients for various nuclear systems is investigated. It is shown that taking into account the non-point distribution of the nuclear charge has little effect on the calculated values of the asymptotic normalization coefficients, provided that the binding energy of the system is assumed to be fixed.

A new Δh method and its application to the asymptotic normalization constants of the 16O bound states

A new Δh method for finding the asymptotic normalization coefficient (ANC) was recently proposed by the author (2021). It was shown that the denominator of the re-normalized scattering amplitude f˜l includes the factor dl(E)=Δl(E)+hr(E)−h(η), where Δl(E) is the nuclear interaction part of the effective-range function (ERF) and Δh(E)=hr(E)−h(η) is the difference of the related Coulomb terms. Here hr(E)=Reh(η) for E>0, η=1/aBk is the Sommerfeld parameter, aB is the Bohr radius. The equation dl(E)=0 determines the f˜l poles at E=−ε (ε is a binding energy). In the present paper to calculate Δh(−ε) the function hr(E) is analytically continued to E<0. For this the series of hr(E) in powers of (aBk)2 which converges if (aBk)2<1 should be used. It is found that the first dominant term (aBk)2/12 of the asymptotic series has(E) is the same for hr(E) and h(η) (at E<0 for h(η)) when E→0. The subtraction of this dominant term from both hr(E) and h(η) simplifies the Δh(−ε) calculation. Here the Δh method applies to the ground and first excited S-wave bound states of 16O (16O↔4He+12C) which meet the condition |aBk|2<1. For 16O the standard ERF method does not work due to the large product Z1Z2 of the 4He and 12C charges. For the P− and D- wave 16O states the binding energies ε are so small that the approximate Δ method, when dl(E)≈Δl(E), is valid. ANCs for the ground and first excited P-wave bound states of 7Be (7Be↔3He+4He) are also calculated by polynomial fitting the sum Δl(E)+hr(E) up to E2 in an analogy with the ERF method. The results for Δh and EFR methods are close to each other. For this system aBκ>1 and the Δh method has no advantages over the standard ERF method. The Δh method opens up a new direction for systems with large Z1Z2 values when the ERF method no longer works.

A new Δ method for bound state asymptotic normalization coefficients with a finite limit of the nuclear part of the effective-range function at zero energy

A new bound state Δ method for finding an asymptotic normalization coefficient (ANC) is proposed. This is valid, while the standard effective-range function (ERF) Kl(k2) method does not work when the product of colliding particles charges increases. The denominator of the strict expression of the re-normalized scattering amplitude f˜l includes the factor dl(E)=Δl(E)+hr(E)−h(η), where η=1/aBk, aB is the Bohr radius. In the physical region, the Coulomb term hr=Reh(η). So an analytical continuation of hr(E) from the physical region to E≤0 can be found by fitting hr(E) for E≥0. The related analytical continuation of h(η) consists in a simple sign change, E→−E, using an explicit dependence of h(η) on E. This is important that for E≤0, abs(Δh)=abs(h−hr)=0 at E=0, and it increases when abs(E) does. Thus, a new real equation, dl(−ε)=0, is obtained for a binding energy ε. It is applied to find a residue W of f˜l at the bound state pole E=−ε, the nuclear vertex constant (NVC) and (ANC). The Coulomb-nuclear phase shift δlcs, cotδlcs and a finite limit of the nuclear part Δl(k2) of Kl(k2) are also derived for an arbitrary orbital momentum l when E→0. It is shown that cotδlcs has an essential singularity at zero energy, but Δl(k2) does not. The explicit finite limit of Δl(k2) when E→0 is found using the expression for Kl(k2). These results are in agreement with those for the S-wave scattering, which are widely accepted. The ANCs for the ground and first excited bound states for the vertex Be7↔3He4He are calculated using the proposed new method, and are compared with those for the approximate method when dl(E)=Δl(E) proposed by Ramírez Suárez and Sparenberg (2017). The fit of Δl(E) is found from the experimental phase-shift input data and the additional equation dl(−ε)=0.

On the Convergence of Oscillator Basis Calculations

The convergence of bound-state calculations performed via the oscillator basis expansions by means of locating $$S$$ -matrix poles for bound states within the HORSE and SS-HORSE approaches is examined. The convergence in question is studied both in the case of a sharp truncation of the potential matrix in the harmonic-oscillator space and in the case of smoothed matrix elements of the potential. As a result, a new method of extrapolation to the case of the infinite-dimensional model space is proposed. This method makes it possible to predict, on the basis of variational calculations, binding energies and asymptotic normalization coefficients for bound states to a high accuracy and to estimate the uncertainties of these predictions.

Asymptotic characteristics of decay channels of states of light nuclei in the ab initio approach

A new convenient method for precise theoretical calculations of quantities of traditional theory of nuclear reactions such as widths of resonances (including sub-threshold), and asymptotic normalization coefficients is proposed. This method may be considered as a step on the road to full theoretical ab initio description of light nuclei spectroscopic data. As an illustration of this method the computational results for all relevant two-body channels for all known and some theoretically predicted states of 7Li nucleus are shown. Well-proven on a large amount of data Daejeon16 potential was used in the calculations. The most part of the results turn out to be in a good agreement with the experimental data contained in spectroscopic tables.

Si26(p,γ)P27 direct proton capture by means of the asymptotic normalization coefficients method for mirror nuclei

The $^{26}\mathrm{Si}(p,\ensuremath{\gamma})^{27}\mathrm{P}$ reaction might be relevant in understanding the $^{26}\mathrm{Si}$ depletion and $^{26}\mathrm{Al}$ production in stars. The asymptotic normalization coefficients (ANC) for the $^{26}\mathrm{Si}(p,\ensuremath{\gamma})^{27}\mathrm{P}$ reaction were extracted earlier from the reanalysis of $^{26}\mathrm{Mg}(d,p)^{27}\mathrm{Mg}$ and $^{26}\mathrm{Mg}(t,d)^{27}\mathrm{Mg}$ reactions, in which the deduced ANC's show significant discrepancies. In this work, a dedicated ($d,p$) measurement is presented: the ANC for the $^{27}\mathrm{Mg}\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}^{26}\mathrm{Mg}+n$ virtual decay is deduced from the $^{26}\mathrm{Mg}(d,p)^{27}\mathrm{Mg}$ reaction populating the ground and the first excited state of $^{27}\mathrm{Mg}$ using the distorted wave Born approximation (DWBA). The charge symmetry properties for mirror nuclei have been used to calculate the ANC for the direct capture $^{26}\mathrm{Si}+p\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}^{27}\mathrm{P}$ populating the ground state of $^{27}\mathrm{P}$. By means of the same formalism, the ${\mathrm{\ensuremath{\Gamma}}}_{p}$ width for the first excited state has also been deduced. The reaction rate is also updated using recent values for the ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\gamma}}/{\mathrm{\ensuremath{\Gamma}}}_{p}$ ratio measured by [Sun et al., Phys. Rev. C 99, 064312 (2019); Sun et al., Phys. Lett. B 802, 135213 (2020)].

Isospin triplet A=12: search for states with enhanced radii

Two independent methods: Asymptotic Normalization Coefficients (ANC) and Modified Diffraction Model (MDM) were applied to new and existing experimental data to search for states with enhanced radii in isobaric analogue states (IAS) of 12B and 12N. The ANC analysis of the 11B(d,p)12B reaction at Ed = 21.5 MeV has confirmed neutron halo existence for two states of 12B: 2−, 1.67 MeV and 1−, 2.62 MeV [1]. Some new results were obtained for higher excited states: halo-like states were observed for 0+, 2.72 MeV and 3−, 3.39 MeV states. It should be mentioned that the last one is unbound state, which is 19 keV above the neutron emission threshold and in both states last neutron has a non-zero orbital momentum (ln=1 and ln=2 correspondingly). We propose to use MDM to study isobaric analogue states of 12B in the mirror 12N nucleus and apply it to analyze the (3He,t) reaction data. The existing in the literature data are not completed enough to make definite conclusion about halo in the 2− and 1− states of 12N. This fact simulated us to carry out a new experiment on the 12C(3He,t)12N reaction at the end of 2018. The measurements were conducted at the University of Jyväskylä (Finland) using the K130 cyclotron to produce a 3He beam at E(3He)=40 MeV. The differential cross sections of the 12C(3He,t)12N reaction were measured in the c.m. angular range 8°–70°. Preliminary results for angular distributions are obtained. MDM analysis of preliminary (3He,t) experimental data is done.

Isospin triplet A=14: search for states with enhanced radii

This article is devoted to study of isobar-analogue states 1− in triplet A=14: 14C-14N-14O. Previously signs of neutron halo in the 1−, 6.09 MeV state of 14C were obtained by two independent groups. In this article we propose to study neighbouring nuclei 14N and 14O using the Modified diffraction model (MDM) method and the method of Asymptotic normalization coefficients (ANC). Methods were applied to experimental differential cross sections of 14C(α,α)14C scattering and reactions 13C(3He,d)14N and 14N(3He,t)14O. MDM and ANC gave practically similar within errors radii for the studied 1− states: the 6.09 MeV state in 14C – 2.7±0.1 fm, the 8.06 MeV state in 14N – 2.7 ± 0.1 fm, the 5.17 MeV state in 14O – 2.6 ± 0.2 fm. Moreover, the signs of proton halo in the 1− state of 14N were obtained for the first time.