Asymptotic Normalization Coefficients Research Articles

Asymptotic normalization coefficients for Be7+p→B8 from the peripheral Be7(d,n)B8 reaction and their astrophysical application

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.

Radii of the bound states in 16N from the asymptotic normalization coefficients**Supported by National Natural Science Foundation of China (11505117, 11490560, 11475264, 11321064, 11375269), Natural Science Foundation of Guangdong Province (2015A030310012), 973 program of China (2013CB834406) National key Research and Development Province (2016YFA0400502)

The asymptotic normalization coefficients (ANCs) of the virtual decay 16N → 15N + n are extracted from the 15N(7Li, 6Li)16N reaction populating the ground and first three excited states in 16N. The root-mean-square (rms) radii of the valence neutron in these four low-lying 16N states are then derived by using the ANCs. The probabilities of the valence neutron staying out of the core potentials are found to be 31% ± 8%, 58% ± 12%, 32% ± 8%, and 60% ± 12%. The present results support the conclusion that a one-neutron halo may be formed in the 16N first and third excited states, while the ground and second excited states do not have a one-neutron halo structure. However, the core excitation effect has a strong influence on the one-neutron halo structure of the ground and first excited states in 16N.

Erratum: Asymptotic normalization coefficients of resonant and bound states from the phase shifts forααandα12Cscattering [Phys. Rev. C93, 014612 (2016)

Erratum: Asymptotic normalization coefficients of resonant and bound states from the phase shifts for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>α</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>α</mml:mi><mml:mn>12</mml:mn></mml:msup><mml:mi mathvariant="normal">C</mml:mi></mml:mrow></mml:math>scattering [Phys. Rev. C<b>93</b>, 014612 (2016)

Analytic continuation of scattering data to the region of negative energies for systems that have one and two bound states

An exactly solvable potential model is used to study the possibility of deducing information about the features of bound states for the system under consideration (binding energies and asymptotic normalization coefficients) on the basis of data on continuum states. The present analysis is based on an analytic approximation and on the subsequent continuation of a partial-wave scattering function from the region of positive energies to the region of negative energies. Cases where the system has one or two bound states are studied. The α+d and α+12C systems are taken as physical examples. In the case of one bound state, the scattering function is a smooth function of energy, and the procedure of its analytic continuation for different polynomial approximations leads to close results, which are nearly coincident with exact values. In the case of two bound states, the scattering function has two poles—one in the region of positive energies and the other in the region of negative energies between the energies corresponding to the two bound states in question. Padeapproximants are used to reproduce these poles. The inclusion of these poles proves to be necessary for correctly describing the properties of the bound states.

Primordialα+d→Li6+γreaction and second lithium puzzle

During the Big Bang, ${}^{6}{\rm Li}$ was synthesized via the ${}^{2}{\rm H}(\alpha,\gamma){}^{6}{\rm Li}$ reaction. After almost 25 years of the failed attempts to measure the ${}^{2}{\rm H}(\alpha,\gamma){}^{6}{\rm Li}$ reaction in the lab at the Big Bang energies, just recently the LUNA collaboration presented the first successful measurements at two different Big Bang energies [M. Anders {\it et al.}, Phys. Rev. Lett. {\bf 113}, 042501 (2014)]. In this paper we will discuss how to improve the accuracy of the direct experiment. To this end the photon's angular distribution is calculated in the potential model. It contains contributions from electric dipole and quadrupole transitions and their interference, which dramatically changes the photon's angular distribution. The calculated distributions at different Big Bang energies have a single peak at $\sim 50^{\circ}$. These calculations provide the best kinematic conditions to measure the ${}^{2}{\rm H}(\alpha,\gamma){}^{6}{\rm Li}$ reaction. The expressions for the total cross section and astrophysical factor are also derived by integrating the differential cross section over the photon's solid angle. The LUNA data are in excellent agreement with our calculations using a potential approach combined with a well established asymptotic normalization coefficient for ${}^{6}{\rm Li} \to \alpha +d$. Comparisons of the available experimental data for the $S_{24}$ astrophysical factor and different calculations are presented. The Big Bang lithium isotopic ratio ${}^{6}{\rm Li}/^{7}{\rm Li} = (1.5 \pm 0.3)\times 10^{-5}$ following from the LUNA data and the present analysis are discussed in the context of the disagreement between the observational data and the standard Big Bang model, which constitutes the second Lithium problem.

Indirect methods in nuclear astrophysics

We discuss recent developments in indirect methods used in nuclear astrophysics to determine the capture cross sections and subsequent rates of various stellar burning processes, when it is difficult to perform the corresponding direct measurements. We discuss in brief, the basic concepts of Asymptotic Normalization Coefficients, the Trojan Horse Method, the Coulomb Dissociation Method, (d,p), and charge-exchange reactions.

Asymptotic normalization coefficients of resonant and bound states from the phase shifts forααandαC12scattering

Recently we have published a paper [Irgaziev, Phys. Rev. C 91, 024002 (2015)] where the $S$-matrix pole method (SMP) which is only valid for resonances has been developed to derive a new explicit expression for the asymptotic normalization coefficient (ANC), and is applied to the low-energy resonant states of nucleon$+\alpha$ and $\alpha+^{12}\rm{C}$ systems. The SMP results are compared with the effective-range expansion method (EFE) results. In the present paper the SMP and EFE plus the Pad\'e-approximation are applied to study the excited 2$^+$ resonant states of $^{8}\rm{Be}$. A contradiction is found between descriptions of the experimental phase shift data for $\alpha\alpha$ scattering and of the $^{8}\rm{Be}$ resonant energy for 2$^+$ state. Using the EFE method, we also calculate the ANC for the $^{8}\rm{Be}$ ground 0$^+$ state with a very small width. This ANC agrees well with the value calculated using the known analytical expression for narrow resonances. In addition, for the $\alpha+^{12}\rm{C}$ states1$^-$ and 3$^-$ the SMP results are compared with the Pad\'e-approximation results. We find that the Pad\'e-approximation improves a resonance width description compared with the EFE results. The EFE method is also used to calculate the ANCs for the bound $^{16}\rm{O}$ ground 0$^+$ state and for the excited 1$^-$ and 2$^+$ levels which are situated near the threshold of $\alpha+^{12}\rm{C}$ channel.

Reaction rate of the 13C(α,n)16O neutron source using the ANC of the -3 keV resonance measured with the THM

The s-process is responsible of the synthesis of most of the nuclei in the mass range 90 ≤ A ≤ 208. It consists in a series of neutron capture reactions on seed nuclei followed by β-decays, since the neutron accretion rate is slower than the β-decay rate. Such small neutron flux is supplied by the 13C(α,n)16O reaction. It is active inside the helium-burning shell of asymptotic giant branch stars, at temperatures < 108 K, corresponding to an energy interval of 140–230 keV. In this region, the astrophysical S (E)-factor is dominated by the −3 keV sub-threshold resonance due to the 6.356 MeV level in 17O. In this work, we have applied the Trojan Horse Method (THM) to the 13C(6Li,n16O)d quasi-free reaction to extract the 6.356 MeV level resonance parameters, in particular the asymptotic normalization coefficient . A preliminary analysis of a partial data set has lead to , slightly larger than the values in the literature. However, the deduced 13C(α, n)16O reaction rate is in agreement with most results in the literature at ∼ 108 K, with enhanced accuracy thanks to our innovative approach merging together ANC and THM.

How well do we understand7Be +p→8B +γ? An Effective Field Theory perspective

We have studied the 7 Be( p , γ ) 8 B reaction in the Halo effective field theory (EFT) framework. The leading order (LO) results were published in Ref. [1] after the isospin mirror process, 7 Li( n , γ ) 8 Li, was addressed in Ref. [2]. In both calculations, one key step was using the final shallow bound state asymptotic normalization coefficients (ANCs) computed by ab initio methods to fix the EFT couplings. Recently we have developed the next-to-LO (NLO) formalism [3], which could reproduce other model results by no worse than 1% when the 7 Be-p energy was between 0 and 0:5 MeV. In our recent report [4], a different approach from that in Ref. [1] was used. We applied Bayesian analysis to constrain all the NLO-EFT parameters based on measured S -factors, and found tight constraints on the S -factor at solar energies. Our S ( E = 0 MeV) = 21.3 ± 0.7 eV b. The uncertainty is half of that previously recommended. In this proceeding, we provide extra details of the Bayesian analysis, including the computed EFT parameters’ probability distribution functions (PDFs) and how the choice of input data impacts final results.

Halo effective field theory constrains the solar 7Be + p → 8B + γ rate

We report an improved low-energy extrapolation of the cross section for the process Be7(p,γ)B8, which determines the 8B neutrino flux from the Sun. Our extrapolant is derived from Halo Effective Field Theory (EFT) at next-to-leading order. We apply Bayesian methods to determine the EFT parameters and the low-energy S-factor, using measured cross sections and scattering lengths as inputs. Asymptotic normalization coefficients of 8B are tightly constrained by existing radiative capture data, and contributions to the cross section beyond external direct capture are detected in the data at E<0.5 MeV. Most importantly, the S-factor at zero energy is constrained to be S(0)=21.3±0.7 eVb, which is an uncertainty smaller by a factor of two than previously recommended. That recommendation was based on the full range for S(0) obtained among a discrete set of models judged to be reasonable. In contrast, Halo EFT subsumes all models into a controlled low-energy approximant, where they are characterized by nine parameters at next-to-leading order. These are fit to data, and marginalized over via Monte Carlo integration to produce the improved prediction for S(E).

Asymptotic normalization coefficients and radiative widths

The asymptotic normalization coefficient (ANC) is an important quantity in the calculation of radiative width amplitudes, providing limits on the radiative width. Here we present some examples showing the connection between the ANC and radiative width. In particular, the radiative width of the $E1$ transition $^{17}\text{F}(1/{2}^{\ensuremath{-}},\phantom{\rule{0.16em}{0ex}}{E}_{x}=3.104\phantom{\rule{4pt}{0ex}}\mathrm{MeV})$ to $^{17}\mathrm{F}(1/{2}^{+},{E}_{x}=0.495\phantom{\rule{4pt}{0ex}}\mathrm{MeV})$ reported by Rolfs [Nucl. Phys. A 217, 29 (1973)] is $(1.2\ifmmode\pm\else\textpm\fi{}0.2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$ eV. Meanwhile the ANC for the first excited state in $^{17}\text{F}$ puts a lower limit on the radiative width, which is $(3.4\ifmmode\pm\else\textpm\fi{}0.50)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$ eV. Such a strong disagreement between the measured radiative width and the lower limit imposed by the ANC calls for a new measurement of this radiative width. Other examples are also considered.

Determination of astrophysical 7Be(p, γ)8B reaction rates from the 7Li(d, p)8Li reaction

The 7Be(p, γ)8B reaction plays a central role not only in the evaluation of solar neutrino fluxes but also in the evolution of the first stars. Study of this reaction requires the asymptotic normalization coefficient (ANC) for the virtual decay 8B g.s. → 7Be + p. By using the charge symmetry relation, we obtain this proton ANC with the single neutron ANC of 8Li g.s. →7Li + n, which is determined with the distorted wave Born approximation (DWBA) and adiabatic distorted wave approximation (ADWA) analysis of the 7Li(d, p)8Li angular distribution. The astrophysical S-factors and reaction rates of the direct capture process in the 7Be(p, γ)8B reaction are further deduced at energies of astrophysical relevance. The astrophysical S-factor at zero energy for direct capture, S 17(0), is derived to be (19.9 ± 3.5) eV b in good agreement with the most recent recommended value. The contributions of the 1+ and 3+ resonances to the S-factor and reaction rate are also evaluated. The present result demonstrates that the direct capture dominates the 7Be(p, γ)8B reaction in the whole temperature range. This work provides an independent examination to the current results of the 7Be(p, γ)8B reaction.

Spectrum of three-body bound States in a finite volume.

The spectrum of a bound state of three identical particles with a mass m in a finite cubic box is studied. It is shown that in the unitary limit, the energy shift of a shallow bound state is given by ΔE=c(κ^{2}/m)(κL)^{-3/2}|A|^{2}exp(-2κL/sqrt[3]), where κ is the bound-state momentum, L is the box size, |A|^{2} denotes the three-body analog of the asymptotic normalization coefficient of the bound state wave function, and c is a numerical constant. The formula is valid for κL≫1.

Theoretical analysis of the astrophysical S-factor for the capture reaction α + d → 6Li + γ in the two-body model

Theoretical estimates for the astrophysical S-factor and the rate of the reaction d(α, γ)6Li were obtained on the basis of the two-body model involving an α−d potential that has a simple Gaussian form and which describes correctly S-, P-, and D-wave phase shifts, the binding energy, and the asymptotic normalization coefficient for the S-wave bound state. The wave functions for the bound and continuum channels were calculated with the aid of the highly precise Numerov algorithm. The results for the contributions of the E1 and E2 transition components reveal a good convergence as the upper limit in the effective integrals increases up to 40 fm. The results obtained for the astrophysical S-factor and the rate of the reaction d(α, γ)6Li in the temperature range of 106 K ≤ T ≤ 1010 K agree well with the results of the calculations performed by A.M. Mukhamedzhanov and his coauthors [Phys. Rev. C 83, 055805 (2011)] by using the known asymptotic form of the wave function at low energies and a complicated two-body potential at higher energies.

Constraining the 6.05 MeV 0+ and 6.13 MeV 3- cascade transitions in the 12C(α,γ)16O reaction using the asymptotic normalization coefficients.

The 12C(α,γ)^16O reaction plays a fundamental role in astrophysics and needs to be known with accuracy better than 10%. Cascade γ transitions through the excited states of 16 O are contributing to the uncertainty. We constrained the contribution of the 0+ (6.05 MeV) and 3- (6.13 MeV) cascade transitions by measuring the asymptotic normalization coefficients for these states using the α-transfer reaction 6 Li(12C,d)^16O at sub-Coulomb energy. The contribution of the 0+ and 3- cascade transitions at 300 keV is found to be 1.96 ± 0.3 and 0.12 ± 0.04 keV b for destructive interference of the direct and resonance capture and 4.36 ± 0.45 and 1.44 ± 0.12 keV b for constructive interference, respectively. The combined contribution of the 0+ and 3- cascade transitions to the 12C(α,γ)16O reaction cross section at 300 keV does not exceed 4%. Significant uncertainties have been dramatically reduced.

Resonance-state properties from a phase shift analysis with theS-matrix pole method and the effective-range method

Asymptotic normalization coefficients (ANCs) are fundamental nuclear constants playing an important role in nuclear physics and astrophysics. We derive a new useful relationship between ANC of the Gamow radial wave function and the renormalized (due to the Coulomb interaction) Coulomb-nuclear partial scattering amplitude. We use an analytical approximation in the form of a series for the nonresonant part of the phase shift which can be analytically continued to the point of an isolated resonance pole in the complex plane of the momentum. Earlier, this method which we call the $S$-matrix pole method was used by us to find the resonance pole energy. We find the corresponding fitting parameters for the $^5\rm{He},\,^5\rm{Li}$, and $^{16}\rm{O}$ concrete resonance states. Additionally, based on the theory of the effective range, we calculate the parameters of the $p_{3/2}$ and $p_{1/2}$ resonance states of the nuclei $^5\rm{He}$ and $^5\rm{Li}$ and compare them with the results obtained by the $S$-matrix pole method. ANC values are found which can be used to calculate the reaction rate through the $^{16}\rm{O}$ resonances which lie slightly above the threshold for the $\alpha^{12}\rm{C}$ channel.

Spectroscopic study of the exotic nucleusP25

Motivated by the importance of $^{25}\mathrm{P}$ for the two-proton decay of $^{26}\mathrm{S}$ and for searches of the mirror analog of the island of inversion near $N=16$, we present the first predictions for the spectroscopy of the exotic isotope $^{25}\mathrm{P}$ obtained in the shell model, a potential model, and a microscopic-cluster model. All models predict $^{25}\mathrm{P}$ to be unbound, with an energy in the range $0.78--1.03$ MeV, which favors previous mass systematics over more recent revisions. We show that $^{25}\mathrm{P}$ possesses a rich low-lying spectrum that should be accessible by experimental studies. All of the predicted states below 7 MeV, except one, are narrow. Many of them are built on the excited-core states of ${}^{24}\mathrm{Si}$ for which the Coulomb barrier is raised. For decays into the ${}^{24}\mathrm{Si}(\text{g.s.})+p$ channel we determined the proton widths based on their link to the asymptotic normalization coefficients (ANCs) of their mirror analogs in ${}^{25}\mathrm{Ne}.$ We determine these ANCs from the analysis of the transfer reaction ${}^{24}\mathrm{Ne}{(d,p)}^{25}\mathrm{Ne}$. The proton widths for decay into excited-state channels are obtained in model calculations. The only broad state is the intruder $3/2{}^{\ensuremath{-}}$, the mirror analog of which has been recently observed in ${}^{25}\mathrm{Ne}.$ The ${}^{25}\mathrm{P}(3/{2}^{\ensuremath{-}})$ energy is lower than that in ${}^{25}\mathrm{Ne}$, suggesting that the island of inversion may persist on the proton-rich side. All excited states of $^{25}\mathrm{P}$ have at least two decay modes and are expected to populate variously the ${2}_{1,2}^{+}$ and ${4}^{+}$ states in ${}^{24}\mathrm{Si}$, which then decay electromagnetically.

Analytic continuation as a bridge between continuum and bound states

The problem of obtaining characteristics of bound nuclear states from con- tinuum states data is discussed. It is shown that the ambiguities due to the existence of phase-equivalent potentials can be resolved by using the analytic properties of scattering amplitudes. The methods of determination of asymptotic normalization coefficients and vertex constants are considered. The asymptotic normalization coefficients for 6 Li in the α+d channel are found by analytic continuation of the two-channel effective range expan- sion. The account of inelastic channels within the effective range approach is discussed.

Determination of neutron halo radii in the first excited states ofC13andBe11with the asymptotic normalization coefficients method

An analysis of the differential cross sections of the $^{12}\mathrm{C}(d,p)^{13}\mathrm{C}$ (ground state, or g.s., $1/{2}_{1}^{+}$) and $^{10}\mathrm{Be}(d,p)^{11}\mathrm{Be}$ (g.s., $1/{2}_{1}^{\ensuremath{-}}$) reactions at the incident deuteron energies of 11.8, 25.9, and 30 MeV for the first system, and 12.0 and 25.0 MeV for the second system has been performed by applying the coupled-reaction-channels method. The spectroscopic factors ${S}_{\text{expt}}$, asymptotic normalization coefficients (ANCs), and rms radii of the last neutron in the ground states and the first excited states of the stable $^{13}\mathrm{C}$ and the radioactive $^{11}\mathrm{Be}$ nuclei were extracted. The existence of neutron halos in the first excited state of $^{13}\mathrm{C}$ and the ground state of $^{11}\mathrm{Be}$ is confirmed. The rms radii of halos in these states were found to be $5.72\ifmmode\pm\else\textpm\fi{}0.16$ and $8.0\ifmmode\pm\else\textpm\fi{}0.2$ fm, respectively, in agreement with experimental results. We showed that a halo is also formed in the 0.32 MeV $1/{2}_{1}^{\ensuremath{-}}$ excited state of $^{11}\mathrm{Be}$. The halo radius in this state was determined for the first time and was found to be ${R}_{h}=4.65\ifmmode\pm\else\textpm\fi{}0.20$ fm.

Asymptotic normalization coefficients and spectroscopic factors from deuteron stripping reactions

We present the analysis of three deuteron stripping reactions, $^{14}\mathrm{C}(d,p)^{15}\mathrm{C},^{58}\mathrm{Ni}(d,p)^{59}\mathrm{Ni}$, and $^{116}\mathrm{Sn}(d,p)^{117}\mathrm{Sn}$ using the combined method [A. M. Mukhamedzhanov and F. M. Nunes, Phys. Rev. C 72, 017602 (2005)], in which each reaction is analyzed at low and significantly higher energies. At low energies all these reactions are peripheral and the experimental asymptotic normalization coefficients (ANCs) are determined with accuracy about $10%$. At higher energies we determine the spectroscopic factors (SFs) by fixing the normalization of the peripheral parts of the reaction amplitudes governed by the ANCs found from the low-energy data. The combined method imposes a strict limitation on the variation of the geometrical parameters of the single-particle potential, which can be arbitrarily taken in the standard approach. By checking the compatibility of the ANCs and SFs using the combined method we reveal the flaw in the contemporary nuclear reaction theory in treating the nuclear interior, which is the most crucial part in the determination of the SFs.