Connection between effective-range expansion and nuclear vertex constant or asymptotic normalization coefficient

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 scattering length, effective range, and form parameter are determined in this way for the $^{16}$O+$p$, $\alpha+t$ and $\alpha+^3$He collisions in partial waves where a bound state exists by using available ANCs deduced from experiments. The resulting effective-range expansions for these collisions are valid up to energies larger 5 MeV.Comment: 17 pages, 6 figure

Coulomb renormalization and ratio of proton and neutron asymptotic normalization coefficients for mirror nuclei

Asymptotic normalization coefficients (ANCs) are fundamental nuclear constants playing important role in nuclear reactions, nuclear structure and nuclear astrophysics. In this paper the physical reasons of the Coulomb renormalization of the ANC are addressed. Using Pinkston-Satchler equation the ratio for the proton and neutron ANCs of mirror nuclei is obtained in terms of the Wronskians from the radial overlap functions and regular solutions of the two-body Schr\"odinger equation with the short-range interaction excluded. This ratio allows one to use microscopic overlap functions for mirror nuclei in the internal region, where they are the most accurate, to correctly predict the ratio of the ANCs for mirror nuclei, which determine the amplitudes of the tails of the overlap functions. Calculations presented for different nuclei demonstrate the Coulomb renormalization effects and independence of the ratio of the nucleon ANCs for mirror nuclei on the channel radius. This ratio is valid both for bound states and resonances. One of the goals of this paper is to draw attention on the possibility to use the Coulomb renormalized ANCs rather than the standard ones especially when the standard ANCs are too large.Comment: 20 pages, 14 figure

Combined method to extract spectroscopic information

Spectroscopic factors (SF) play an important role in nuclear physics and astrophysics. The traditional method of extracting SF from direct transfer reactions suffers from serious ambiguities. We discuss a modified method which is based on including the asymptotic normalization coefficient (ANC) of the overlap functions into the transfer analysis. In the modified method the contribution of the external part of the reaction amplitude, typically dominant, is fixed and the SF is determined from fitting the internal part. We illustrate the modified method with $(d,p)$ reactions on ${}^{208}{\rm Pb}, {}^{12}{\rm C}$, and ${}^{84}{\rm Se}$ targets at different energies. The modified method allows one to extract the SF, which do not depend on the shape of the single-particle nucleon-target interaction, and has the potential of improving the reliability and accuracy of the structure information. This is specially important for nuclei on dripline, where not much is known.Comment: accepted in Phys. Rev. C, 4 pages and 2 figure

Quantum Monte Carlo calculations of spectroscopic overlaps in A≤7A \leq 7 nuclei

We present Green's function Monte Carlo calculations of spectroscopic overlaps for $A \leq 7$ nuclei. The realistic Argonne v18 two-nucleon and Illinois-7 three-nucleon interactions are used to generate the nuclear states. The overlap matrix elements are extrapolated from mixed estimates between variational Monte Carlo and Green's function Monte Carlo wave functions. The overlap functions are used to obtain spectroscopic factors and asymptotic normalization coefficients, and they can serve as an input for low-energy reaction calculations

Astrophysical SS factor for the 15N(p,γ)16O{}^{15}{\rm N}(p,\gamma){}^{16}{\rm O} reaction from RR-matrix analysis and asymptotic normalization coefficient for 16O→15N+p{}^{16}{\rm O} \to {}^{15}{\rm N} + p. Is any fit acceptable?

The $^{15}{\rm N}(p,\gamma)^{16}{\rm O}$ 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 $J^{\pi}=1^{-}$ 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 $^{15}{\rm N}(p,\gamma)^{16}{\rm O}$ 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 parameters including the asymptotic normalization coefficient (ANC). The best fit has been obtained for the square of the ANC $C^{2}= 539.2$ fm${}^{-1}$, which exceeds the previously measured value by a factor of $\approx 3$. Here we present a new $R$-matrix analysis of the Notre Dame-LUNA data with the fixed within the experimental uncertainties square of the ANC $C^{2}=200.34$ fm${}^{-1}$. Rather than varying the ANC we add the contribution from a background resonance that effectively takes into account contributions from higher levels. Altogether we present 8 fits, five unconstrained and three constrained. In all the fits the ANC is fixed at the previously determined experimental value $C^{2}=200.34$ fm${}^{-1}$. For the unconstrained fit with the boundary condition $B_{c}=S_{c}(E_{2})$, where $E_{2}$ is the energy of the second level, we get $S(0)=39.0 \pm 1.1$ keVb and normalized ${\tilde \chi}^{2}=1.84$, i.e. the result which is similar to [P. J. LeBlanc {\it et al.}, Phys. Rev. {\bf C 82}, 055804 (2010)]. From all our fits we get the range $33.1 \leq S(0) \leq 40.1$ keVb which overlaps with the result of [P. J. LeBlanc {\it et al.}, Phys. Rev. {\bf C 82}, 055804 (2010)]. We address also physical interpretation of the fitting parameters.Comment: Submitted to PR

Generalized Faddeev equations in the AGS form for deuteron stripping with explicit inclusion of target excitations and Coulomb interaction

Theoretical description of reactions in general, and the theory for $(d,p)$ reactions, in particular, needs to advance into the new century. Here deuteron stripping processes off a target nucleus consisting of ${A}$ nucleons are treated within the framework of the few-body integral equations theory. The generalized Faddeev equations in the AGS form, which take into account the target excitations, with realistic optical potentials provide the most advanced and complete description of the deuteron stripping. The main problem in practical application of such equations is the screening of the Coulomb potential, which works only for light nuclei. In this paper we present a new formulation of the Faddeev equations in the AGS form taking into account the target excitations with explicit inclusion of the Coulomb interaction. By projecting the $(A+2)$-body operators onto target states, matrix three-body integral equations are derived which allow for the incorporation of the excited states of the target nucleons. Using the explicit equations for the partial Coulomb scattering wave functions in the momentum space we present the AGS equations in the Coulomb distorted wave representation without screening procedure. We also use the explicit expression for the off-shell two-body Coulomb scattering $T$-matrix which is needed to calculate the effective potentials in the AGS equations. The integrals containing the off-shell Coulomb T-matrix are regularized to make the obtained equations suitable for calculations. For $NN$ and nucleon-target nuclear interactions we assume the separable potentials what significantly simplifies solution of the AGS equations.Comment: 34 pages, 13 figure

{Once more about astrophysical SS factor for the α+d→6Li+γ\alpha + d \to {}^{6}{\rm Li} + \gamma reaction

Recently to study the radiative capture $\alpha + d \to {}^{6}{\rm Li} + \gamma$ 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 $\alpha + d \to {}^{6}{\rm Li} + \gamma$ process from the breakup data. The astrophysical $S_{24}(E)$ factor has been calculated within the $\alpha-d$ two-body potential model with potentials determined from the fits to the $\alpha-d$ elastic scattering phase shifts. However, the scattering phase shift itself doesn't provide a unique $\alpha-d$ bound state potential, which is the most crucial input when calculating the $S_{24}(E)$ astrophysical factor at astrophysical energies. In this work we emphasize an important role of the asymptotic normalization coefficient (ANC) for ${}^{6}{\rm Li} \to \alpha + d$, which controls the overall normalization of the peripheral $\alpha + d \to {}^{6}{\rm Li} + \gamma$ process and is determined by the adopted $\alpha-d$ bound state potential. We demonstrate that the ANC previously determined from the $\alpha-d$ elastic scattering $s$-wave phase shift in [Blokhintsev {\it et. al} Phys. Rev. {\bf C 48}, 2390 (1993)] gives $S_{24}(E)$, which is at low energies about 38% lower than the one reported in [F. Hammache {\it et al.} Phys. Rev ${\bf C 82}$, 065803 (2010)]. We recalculate also the reaction rates, which are also lower than those obtained in [F. Hammache {\it et al.} Phys. Rev ${\bf C 82}$, 065803 (2010)].Comment: 6 pages and 2 figure

Bound, virtual and resonance SS-matrix poles from the Schr\"odinger equation

A general method, which we call the potential $S$-matrix pole method, is developed for obtaining the $S$-matrix pole parameters for bound, virtual and resonant states based on numerical solutions of the Schr\"odinger equation. This method is well-known for bound states. In this work we generalize it for resonant and virtual states, although the corresponding solutions increase exponentially when $r\to\infty$. Concrete calculations are performed for the $1^+$ ground and the $0^+$ first excited states of $^{14}\rm{N}$, the resonance $^{15}\rm{F}$ states ($1/2^+$, $5/2^+$), low-lying states of $^{11}\rm{Be}$ and $^{11}\rm{N}$, and the subthreshold resonances in the proton-proton system. We also demonstrate that in the case the broad resonances their energy and width can be found from the fitting of the experimental phase shifts using the analytical expression for the elastic scattering $S$-matrix. We compare the $S$-matrix pole and the $R$-matrix for broad $s_{1/2}$ resonance in ${}^{15}{\rm F}$Comment: 14 pages, 5 figures (figures 3 and 4 consist of two figures each) and 4 table

A new insight into the observation of spectroscopic strength reduction in atomic nuclei: implication for the physical meaning of spectroscopic factors

Experimental studies of one nucleon knockout from magic nuclei suggest that their nucleon orbits are not fully occupied. This conflicts a commonly accepted view of the shell closure associated with such nuclei. The conflict can be reconciled if the overlap between initial and final nuclear states in a knockout reaction are calculated by a non-standard method. The method employs an inhomogeneous equation based on correlation-dependent effective nucleon-nucleon (NN) interactions and allows the simplest wave functions, in which all nucleons occupy only the lowest nuclear orbits, to be used. The method also reproduces the recently established relation between reduction of spectroscopic strength, observed in knockout reactions on other nuclei, and nucleon binding energies. The implication of the inhomogeneous equation method for the physical meaning of spectroscopic factors is discussed.Comment: 4 pages, accepted by Phys. Rev. Let

Asymptotic normalization coefficients of alpha-particle removal from 16^{16}O(3−,2+,1−3^-,2^+,1^-)

Asymptotic normalization coefficients (ANC) determine the overall normalization of cross sections of peripheral radiative capture reactions. In a recent paper [Blokhintsev et al., Eur. Phys. J. A 58, 257 (2022)], we considered the ANC $C_0$ for the virtual decay $^{16}$O$(0^+; 6.05$ MeV)$\to \alpha+^{12}$C(g.s.). In the present paper, which can be regarded as a continuation of the previous, we treat the ANCs $C_l$ for the vertices $^{16}$O$(J^\pi)\to \alpha+^{12}$C(g.s.) corresponding to the other three bound excited states of $^{16}$O ($J^\pi=3^-$, $2^+$, $1^-$, $l=J$). ANCs $C_l$ ($l=3,\,2,\,1$) are found by analytic continuation in energy of the $\alpha^{12}$C $l$-wave partial scattering amplitudes, known from the phase-shift analysis of experimental data, to the pole corresponding to the $^{16}$O bound state and lying in the unphysical region of negative energies. To determine $C_l$, the scattering data are approximated by the sum of polynomials in energy in the physical region and then extrapolated to the pole. For a more reliable determination of the ANCs, various forms of functions expressed in terms of phase shifts were used in analytical approximation and subsequent extrapolation.Comment: arXiv admin note: substantial text overlap with arXiv:2208.0958