1 INTRODUCTION

Asymptotic normalization coefficients (ANC) determine the asymptotics of nuclear wave functions in binary channels at distances between fragments exceeding the radius of nuclear interaction [1]. In terms of ANCs, the cross sections of peripheral nuclear transfer reactions are parameterized. ANCs are of particular importance for nuclear astrophysics. They determine the overall normalization of cross sections of radiative capture reactions at astrophysical energies [2–4]. Unlike binding energies, ANCs cannot be directly measured. To determine them on the basis of experimental data, a certain theoretical analysis is required. In particular, the values of ANCs can be extracted by comparing the absolute values of experimental cross sections of nuclear transfer reactions with theoretical ones, calculated within the framework of the distorted wave Born approximation. Some other methods for determining ANCs from experimental data are described in [5].

A common way to extract ANC \(C_{abc}\) from experimental data on the elastic scattering of particles \(b\) and \(c\) is the extrapolation in the center-of-mass energy \(E\) of the partial-wave scattering amplitude (obtained by the phase-shift analysis) to the pole corresponding to the bound state \(a\) and lying in the unphysical region (\(E<0\)). Usually, for such a procedure, the effective range function (ERF method) or the \(\Delta\) function defined below in Eq. (12) (\(\Delta\)-method) are approximated at positive energies by a polynomial or a rational function and then analytically continued to the pole.

However, both methods have their drawbacks (see, e.g. [6] and references therein). The \(\Delta\)-method suggested in [7] is not quite strict and correct from the point of view of mathematics. Moreover, this method is applicable only to systems with sufficiently large electric charges and masses. On the other hand, the ERF method, although formally rigorous, is practically suitable only for the lightest nuclear systems due to the presence of a large background of purely Coulomb terms.

In this regard, in the present work, we point out the possibility of obtaining information about ANCs using the \(R\)-matrix formalism which is often used to analyze experimental data on binary nuclear reactions.

The paper is organized as follows. Section 2 presents the general formalism of the method used. Section 3 is devoted to the derivation of formulas expressing the scattering amplitude in terms of the \(R\)-matrix and allowing analytical continuation of the scattering amplitude to the region of negative collision energies. In Section 4, the proposed approach is applied to find the ANC for the channel \({}^{16}\textrm{O}\to\alpha+{}^{12}\)C. The results obtained are briefly discussed in Section 5.

Throughout the paper we use a system of units in which \(\hbar=c=1\).

2 BASIC FORMALISM

In this section we recapitulate basic formulas which are necessary for the subsequent discussion.

2.1 Coulomb–Nuclear Scattering Amplitude

The full amplitude of the elastic scattering of particles \(b\) and \(c\) in the presence of the Coulomb and short-range (nuclear) interactions is written as the sum of the pure Coulomb and Coulomb–nuclear amplitudes (the Coulomb interaction is taken to be repulsive):

$$f(\mathbf{k})=f_{\textrm{C}}(\mathbf{k})+f_{\textrm{NC}}(\mathbf{k}),$$
(1)
$$f_{\mathrm{C}}(\mathbf{k})=\sum\limits_{l=0}^{\infty}(2l+1)\frac{\exp(2i\sigma_{l})-1}{2ik}P_{l}(\cos\theta),$$
(2)
$$f_{\textrm{NC}}(\mathbf{k})=$$
$${}\sum\limits_{l=0}^{\infty}(2l+1)\exp(2i\sigma_{l})\frac{\exp(2i\delta_{l})-1}{2ik}P_{l}(\cos\theta).$$
(3)

Here \(\mathbf{k}\) is the relative momentum of \(b\) and \(c\), \(\sigma_{l}\) and \(\delta_{l}\) are the pure Coulomb and Coulomb–nuclear phase shifts, respectively.

The behavior of the Coulomb–nuclear partial-wave amplitude \(f_{l}=(\exp(2i\delta_{l})-1)/2ik\) is irregular near \(E=0\). Therefore, one has to introduce the renormalized Coulomb–nuclear partial-wave amplitude \(\tilde{f}_{l}\):

$$\tilde{f}_{l}=\exp(2i\sigma_{l})\frac{\exp(2i\delta_{l})-1}{2ik}$$
$${}\times\left[\frac{l!}{\Gamma(l+1+i\eta)}\right]^{2}e^{\pi\eta},\quad\textrm{or}$$
(4)
$$\tilde{f}_{l}=\frac{\exp(2i\delta_{l})-1}{2ik}C_{l}^{-2}(\eta),$$
(5)

where

$$\eta=Z_{b}Z_{c}e^{2}\mu/k$$
(6)

is the Coulomb (Sommerfeld) parameter for the \(b+c\) scattering state with the relative momentum related to the energy by \(k=\sqrt{2\mu E}\), \(\mu\), \(Z_{b}e\), and \(Z_{c}e\) are the reduced mass and the electric charges of \(b\) and \(c\). \(C_{l}(\eta)\) is the Coulomb penetrability factor (or Gamow factor)

$$C_{l}(\eta)=\left[\frac{2\pi\eta}{\exp(2\pi\eta)-1}v_{l}(\eta)\right]^{1/2},$$
(7)
$$v_{l}(\eta)=\prod_{n=1}^{l}(1+\eta^{2}/n^{2})\quad(l>0),\quad v_{0}(\eta)=1.$$
(8)

It was shown in [8] that the analytic properties of \({\tilde{f}}_{l}\) on the physical sheet of \(E\) are analogous to the ones of the partial-wave scattering amplitude for the short-range potential and \({\tilde{f}}_{l}\) can be analytically continued into the negative-energy region.

The amplitude \(\tilde{f}_{l}\) can be represented in the following equivalent forms:

$$\tilde{f}_{l}=\frac{1}{kC_{l}^{2}(\eta)(\cot\delta_{l}-i)},$$
(9)
$$\tilde{f}_{l}=\frac{1}{v_{l}\Delta_{l}(E)-ikC_{l}^{2}(\eta)},$$
(10)
$$\tilde{f}_{l}=\frac{k^{2l}}{\tilde{\Delta}_{l}(E)-ik^{2l+1}C_{l}^{2}(\eta)},$$
(11)
$$\Delta_{l}(E)=kC_{0}^{2}(\eta)\cot\delta_{l},$$
(12)
$$\tilde{\Delta}_{l}(E)=k^{2l+1}v_{l}C_{0}^{2}(\eta)\cot\delta_{l}$$
$${}=k^{2l+1}C_{l}^{2}(\eta)\cot\delta_{l}.$$
(13)

If the \(b+c\) system involves the bound state \(a\) with the binding energy \(\varepsilon=\varkappa^{2}/2\mu>0\), then the amplitude \(\tilde{f}_{l}(E)\) has a pole at \(E=-\varepsilon\). The residue of \(\tilde{f}_{l}\) at this point is expressed in terms of the ANC \(C_{l}\):

$${\textrm{res}}\tilde{f}_{l}(E)|_{E=-\varepsilon}\equiv\lim_{\begin{subarray}{c}E\to-\varepsilon\end{subarray}}[(E+\varepsilon)\tilde{f}_{l}(E)]$$
$${}=-\frac{1}{2\mu}\left[\frac{l!}{\Gamma(l+1+\eta_{b})}\right]^{2}C_{l}^{2},$$
(14)

where \(\eta_{b}=Z_{b}Z_{c}e^{2}\mu/\varkappa\) is the Coulomb parameter for the \(b+c\) bound state \(a\).

2.2 Basic \(R\)-Matrix Formalism

The \(R\)-matrix theory was introduced in [9]. A detailed exposition of this theory can be found in review articles [10, 11]. The main idea of the \(R\)-matrix approach to binary nuclear reactions is the existence of such a distance \(r=a\) between the centers of mass of two fragments that at \(r>a\) the only type of interaction between the fragments is the central Coulomb interaction \(V_{c}(r)\). In what follows, we will be interested in the single-channel case of purely elastic scattering. In this case, the \(R\)-matrix for a partial-wave state with orbital momentum \(l\) is defined as follows (spin variables are omitted for brevity)

$$R_{l}(E)=\left.\varphi_{lE}(a)\right/\left(r\frac{d\varphi_{lE}(r)}{dr}\right)_{r=a},$$
(15)

where \(\varphi_{lE}(r)=r\psi_{lE}(r)\), \(\psi_{lE}(r)\) is a radial wave function describing the relative motion of two fragments with energy \(E\). Using this definition, one can prove that the \(R\)-matrix for a partial-wave state with orbital momentum \(l\) has the form [10, 11]:

$$R_{l}(E)=\sum\limits_{\lambda}\dfrac{\gamma_{l\lambda}^{2}}{E_{l\lambda}-E}.$$
(16)

Here \(E_{l\lambda}\) are the energy eigenvalues corresponding to some complete set of radial wave functions \(u_{l\lambda}(r)\) defined on the interval \(0\leq r\leq a\). \(\gamma_{l\lambda}\) is called the reduced width amplitude and is determined by the formula

$$\gamma_{l\lambda}=\left(\dfrac{1}{2\mu a}\right)^{1/2}u_{l\lambda}(a),$$
(17)

where \(\mu\) is a reduced mass of colliding nuclei. Knowing the \(R\)-matrix determines the \(S\)-matrix.

The functions \(u_{l\lambda}(r)\) satisfy the boundary condition

$$\left.\dfrac{r}{u_{l\lambda}}\dfrac{du_{l\lambda}}{dr}\right|_{r=a}=B_{l},$$
(18)

where \(B_{l}\) is a constant which does not depend on \(\lambda\). An important property of the \(R\)-matrix formalism is the fact that, although \(R_{l}\) depends on \(B_{l}\), the partial \(S\)-matrix and, in particular, phase shifts, do not depend on \(B_{l}\). However, the choice of \(B_{l}\neq 0\) can improve the quality of the description of experimental data using the \(R\)-matrix. The formulas given below, which relate the \(R\)-matrix to the observed quantities, are written for the case of \(B_{l}=0\) for brevity, but they remain valid at \(B_{l}\neq 0\) if the following substitution is made in them [10, 11]:

$$R_{l}^{-1}\longrightarrow R_{l}^{-1}+B_{l}.$$
(19)

For an exact \(R\)-matrix, the number of terms on the right-hand side of Eq. (16) is infinite, and the values of the parameters included in (16) can, in principle, be found by solving the Schrödinger equation if the interaction between the nuclei is known. However, in practical applications one is limited to a finite sum in (16), and the quantities \(a\), \(B_{l}\), \(\gamma_{l\lambda}\) and \(E_{l\lambda}\) are treated as parameters that are fitted to the scattering data.

2.3 Notation Used in the Following Sections

Let us give the notation used for brevity in the following sections.

$$F_{c}\equiv F_{l}(\eta,kr),\quad F^{\prime}_{c}=\dfrac{dF_{c}}{dr};$$
$$G_{c}\equiv G_{l}(\eta,kr),\quad G^{\prime}_{c}=\dfrac{dG_{c}}{dr},$$
(20)

\(F_{l}(\eta,kr)\) and \(G_{l}(\eta,kr)\) are regular and irregular Coulomb functions, respectively [12].

$$W_{c}\equiv W_{-\eta_{b},l+1/2}(2\varkappa r),\quad W^{\prime}_{c}=\dfrac{dW_{c}}{dr},$$
(21)

\(W_{\alpha,\beta}(x)\) is a Whittaker function.

$$S^{+}=[a(F_{c}F^{\prime}_{c}+G_{c}G^{\prime}_{c})/(F^{2}_{c}+G^{2}_{c})]_{r=a},$$
(22)
$$S^{-}=(aW^{\prime}_{c}/W_{c})_{r=a},$$
(23)
$$P^{+}=[ka/(F^{2}_{c}+G^{2}_{c})]_{r=a}.$$
(24)

Plus and minus superscripts in Eqs. (22)–(24) mean regions \(E>0\) and \(E<0\), respectively.

$$\phi_{c}=\arctan\left(\frac{F_{c}}{G_{c}}\right)_{r=a}.$$
(25)

3 CONTINUATION OF \(\tilde{F}_{L}(E)\)TO THE REGION OF NEGATIVE ENERGIESIN THE FRAMEWORK OF THE \(R\)-MATRIX FORMALISM

In the framework of the \(R\)-matrix approach, the phase shift in the physical domain (\(E\geq 0\)) is related to the \(R\)-matrix as follows [10]

$$\delta_{l}(E)=\arctan\frac{P^{+}}{R_{l}^{-1}-S^{+}}-\phi_{c}.$$
(26)

Note that in Eq. (26), as well as in subsequent equations (28)–(30), (32)–(37), it is understood that in the arguments of the functions included in these equations \(r=a\).

Based on Eq. (26), one can calculate the amplitude \(\tilde{f}_{l}\). However, Eq. (26) is not suitable for continuation of \(\delta_{l}(E)\) and \(\tilde{f}_{l}(E)\) to the region \(E<0\) due to the irregular behavior of the functions included in it at \(E\to-0\). Therefore, in this paper, the authors propose to use other expressions for \(\delta_{l}\) and \(\tilde{f}_{l}\) in the framework of the \(R\)-matrix formalism.

Since there is no nuclear interaction in the \(R\)-matrix approach at \(r>a\), the radial wave function \(\varphi_{lE}(r)\) at \(r>a\) can be written as

$$\varphi_{lE}(r)=A(F_{c}\cos\delta_{l}+G_{c}\sin\delta_{l}),$$
(27)

where \(A\) is some constant. Hence, according to the definition (15) the \(R\)-matrix assumes the form

$$R_{l}(E)=\varphi_{E}(a)/(a\varphi^{\prime}_{E}(a))$$
$${}=\frac{F_{c}\cos\delta_{l}+G_{c}\sin\delta_{l}}{a(F_{c}^{\prime}\cos\delta_{l}+G_{c}^{\prime}\sin\delta_{l})}.$$
(28)

It follows from (28) that

$$\cot\delta_{l}=-\frac{G_{c}-aR_{l}G_{c}^{\prime}}{F_{c}-aR_{l}F_{c}^{\prime}}.$$
(29)

Thus the quantity \(\tilde{\Delta}_{l}(E)\) (13) within the \(R\)-matrix formalism is of the form

$$\tilde{\Delta}_{l}(E)=k^{2l+1}C_{l}^{2}(\eta)\cot\delta_{l}$$
$${}=k^{2l+1}C_{l}^{2}(\eta)\left(-\frac{G_{c}-aR_{l}G_{c}^{\prime}}{F_{c}-aR_{l}F_{c}^{\prime}}\right).$$
(30)

The function \(\tilde{\Delta}_{l}(E)\) expressed using Eq. (30) has an essential singularity at zero energy, which creates problems for its continuation to the region \(E<0\). To overcome these problems, we introduce the modified Coulomb functions

$$F_{c1}=(F_{c}/r)/(C_{l}(\eta)k^{l+1}),$$
$$G_{c1}=-(G_{c}/r)(C_{l}(\eta)k^{l}).$$
(31)

Substituting (31) into (30) gives

$$\tilde{\Delta}_{l}(E)=\frac{aR_{l}G_{c1}^{\prime}+(R_{l}-1)G_{c1}}{aR_{l}F_{c1}^{\prime}+(R_{l}-1)F_{c1}}.$$
(32)

However, the function \(G_{c1}\) still has a singularity at zero energy, so we introduce one more function

$$G_{c2}=G_{c1}-ik^{2l+1}C_{l}^{2}(\eta)F_{c1}.$$
(33)

In terms of \(G_{c2}\), Eq. (32) is rewritten as

$$\tilde{\Delta}_{l}(E)$$
$${}=\frac{aR_{l}G^{\prime}_{c2}+(R_{l}-1)G_{c2}}{aR_{l}F_{c1}^{\prime}+(R_{l}-1)F_{c1}}+ik^{2l+1}C_{l}^{2}(\eta).$$
(34)

In (34), the part of \(\tilde{\Delta}_{l}(E)\) containing an essential singularity is represented by the second term on the right-hand side which has the explicit form; the first term does not have this singularity. As a result, denoting the denominator of expression (11) for \(\tilde{f}_{l}(E)\) by \(D_{l}(E)\), we obtain

$$D_{l}(E)=\tilde{\Delta}_{l}(E)-ik^{2l+1}C_{l}^{2}(\eta)$$
$${}=\frac{aR_{l}G_{c2}^{\prime}+(R_{l}-1)G_{c2}}{aR_{l}F_{c1}^{\prime}+(R_{l}-1)F_{c1}}.$$
(35)

The quantity (35) does not have an essential singularity at \(E=0\) and, therefore, can be continued to the region \(E<0\) up to the pole of \(\tilde{f}_{l}(E)\) at \(E=-\varepsilon=-\varkappa^{2}/2\mu\) at which \(D_{l}(-\varepsilon)=0\).

For the case when \(B_{l}\neq 0\) in the boundary condition (18), Eq. (35) can be rewritten in a more convenient form:

$$D_{l}(E)=\frac{aG_{c2}^{\prime}+(1-R_{l}^{-1}-B_{l})G_{c2}}{aF_{c1}^{\prime}+(1-R_{l}^{-1}-B_{l})F_{c1}}.$$
(36)

In accordance with Eqs. (14) and (15) and rules for finding the residue at the pole of a function, ANC \(C_{l}\) is determined by the formula

$$C_{l}^{2}$$
$${}=-2\mu\left[\frac{\Gamma(l+1+\eta_{b})}{l!}\right]^{2}\left.\dfrac{(-1)^{l}\varkappa^{2l}}{dD_{l}(E)/dE}\right|_{E=-\varepsilon}.$$
(37)

An \(R\)-matrix, written in the form of Eq. (16) with a finite number of terms, in the general case can lead to several bound states. The binding energies and ANCs for these states are found by continuing the \(R\)-matrix \(R_{l}(E)\) from the \(E>0\) region to the \(E<0\) region using Eqs. (36) and (37). The binding energies and ANCs (as well as the positions and widths of resonances at positive energy) generally depend on all terms of the sum in (16), although some terms may contribute much more to their values than others. However, there is one exception to this general statement that is important for finding ANCs and needs to be mentioned.

The position of the pole of the amplitude \(\tilde{f}_{l}(E)\) corresponding to the bound state energy \(E_{b}\) is found from condition \(D_{l}(E_{b})=0\), which, according to (36), is equivalent to the condition

$$aG_{c2}^{\prime}(E_{b})+(1-R_{l}^{-1}(E_{b})-B_{l})G_{c2}(E_{b})=0.$$
(38)

If we put \(R_{l}^{-1}(E_{b})=0\) in (38), then (38) becomes the condition for \(B_{l}\)

$$B_{l}=\left(1+a\frac{G_{c2}^{\prime}(E_{b})}{G_{c2}(E_{b})}\right).$$
(39)

Using the connection between the Coulomb functions \(F_{l}\) and \(G_{l}\) and the Whittaker function \(W_{-\eta_{b},l+1/2}\) [12], it is easy to show that the condition (39) is equivalent to the condition \(B_{l}=-S^{-}(E_{b})\), where \(S^{-}\) is defined in (23).

Condition \(R_{l}^{-1}(E_{b})=0\) at \(E_{b}<0\) means that in one of the terms in the sum (16) \(E_{l\lambda}=E_{b}\). Using the above considerations in reverse order, we conclude that if, when choosing the value (39) for \(B_{l}\), in one of the terms in (16) \(E_{l\lambda}=E_{b}\), then the \(R\)-matrix leads to the existence of a bound state with a binding energy \(\varepsilon=-E_{b}>0\). Moreover, the ANC \(C_{l}\) for this bound state depends only on the value of \(\gamma_{l\lambda}\) corresponding to this \(E_{l\lambda}\) and is expressed as [4, 11]

$$C_{l}=(2\mu aN)^{1/2}\gamma_{l\lambda}/W_{-\eta_{b},l+1/2}(2\varkappa_{B}a),$$
$$N^{-1}=1+\gamma_{l\lambda}^{2}\left[\dfrac{dS^{-}(E)}{dE}\right]_{E=-\varepsilon}.$$
(40)

In this case, the ANC can be found directly, without continuation from the region of positive energies, but the continuation method has a wider scope of applicability and can be applied to an arbitrary \(R\)-matrix with an arbitrary number of bound states with the same quantum numbers. Moreover, even for this particular case, employing a continuation from the region \(E>0\) can serve to analyze the quality of the \(R\)-matrix and its applicability for finding the ANC.

4 CONTINUATION OF THE R-MATRIX AND FINDING THE ANC FOR \({}^{4}\textrm{He}-^{12}\)C SYSTEM

One of the most important astrophysical reactions is the radiative capture of \(\alpha\) particles by \({}^{12}\)C. The \({}^{12}\)C\((\alpha,\gamma)^{16}\textrm{O}\) reaction is activated during the helium burning stages of stellar evolution. It determines the relative abundance of \({}^{12}\)C and \({}^{16}\textrm{O}\) in stellar cores. As noted in the Introduction, the overall normalization of cross sections of radiative capture reactions at astrophysical energies is determined by the values of corresponding ANCs. This implies the importance of knowing ANCs for the channel \({}^{16}\textrm{O}\to\alpha+{}^{12}\)C.

The \({}^{4}\)He–\({}^{12}\)C system has several bound states corresponding to the nuclear-stable levels of the \({}^{16}\)O nucleus. There are a number of phase-shift analyzes for this system. The results of the latest of them, presented in the papers [13] and [14], were obtained on the basis of the \(R\)-matrix formalism.

Consider, as an example, the \(D\)-channel of the \({}^{4}\)He–\({}^{12}\)C system in which there is a bound state with a binding energy \(\varepsilon=0.2450\) MeV corresponding to the level \({}^{16}\)O(\(2^{+};6.9171\) MeV). For this channel, the \(R\)-matrix given in the paper [14] contains 5 terms:

$$R_{l}(E)=\frac{0.473^{2}}{-0.2450-E}+\dfrac{(2.43\times 10^{-2})^{2}}{2.684-E}$$
$${}+\dfrac{(8.95\times 10^{-2})^{2}}{4.387-E}+\dfrac{2.60^{2}}{44.5-E}+\dfrac{0.128^{2}}{5.978-E}.$$
(41)

In this equation, as in the following Eq. (42), the values of \(E_{l\lambda}\) and \(\gamma_{l\lambda}\) are given in units of MeV and MeV\({}^{1/2}\), respectively. As can be seen from (41), the experimental binding energy \(\varepsilon=0.2450\) is explicitly included in the first term of the sum.

Figure 1 shows the graph of the calculated function \(\textrm{Re}\,D_{2}(E)\). At \(E<0\) \(D_{l}(E)\) is real and \(\textrm{Re}\,D_{2}(E)\) coincides with \(D_{2}(E)\). The solid curve in Fig. 1 is calculated by Eq. (36) using the \(R\)-matrix (41) with \(a=5.5\) fm and \(B_{2}\) determined by Eq. (39). For this \(R\)-matrix, the ANC found by (40) is \(C_{2}=1.42\times 10^{5}\) fm\({}^{-1/2}\). The dashed curve in Fig. 1 was obtained by means of the polynomial approximation of some function of \(\tilde{\Delta}_{2}(E)\) made in [6] on the basis of the phase-shift analysis performed in [13]. This curve leads to \(C_{2}=1.55\times 10^{5}\) fm\({}^{-1/2}\) which is close to the result obtained within the \(R\)-matrix approach.

Fig. 1
figure 1

Re\(D_{2}(E)\). The meaning of the curves is explained in the text.

Full size image

However, the advantage of the method proposed in the present paper is that it can be used to find the ANC for an arbitrary \(R\)-matrix. As an example, we use the \(R\)-matrix for the same \(D\)-state of the \({}^{4}\)He–\({}^{12}\)C system from [15], which has the form

$$R_{l}=\dfrac{0.77}{-1.511-E}+\dfrac{0.0255}{4.338-E}+\dfrac{9.02}{30.0-E}.$$
(42)

Eq. (42) corresponds to \(a=5.43\) fm and \(B_{2}=-0.942\).

This \(R\)-matrix, in contrast to \(R\)-matrix (41), does not have a term corresponding to the energy of the bound state, so the method for finding the ANC based on Eq. (40) is not applicable. However, for the \(R\)-matrix (42), it is possible to calculate the function \(D_{2}(E)\) using Eq. (36), find its zero in the negative energy region, corresponding to the binding energy, and then determine the ANC based on Eq. (37). The values of the binding energy and ANC found in this way are 0.242 MeV and \(1.47\times 10^{5}\) fm\({}^{-1/2}\), respectively, which practically coincides with the corresponding values obtained on the basis of the \(R\)-matrix (41).

5 CONCLUSION

In this paper, we propose to use the \(R\)-matrix formalism to determine the asymptotic normalization coefficients. The method is based on the analytic continuation of the \(R\)-matrix constructed to describe the elastic nuclear collision at positive energies (\(E>0\)) to the pole point of the scattering amplitude corresponding to the energy of the bound state and lying in the region \(E<0\). The formula is derived, expressing the partial-wave scattering amplitude through the \(R\)-matrix and modified Coulomb functions. This formula allows, when continuing the amplitude to the region \(E<0\), to overcome difficulties associated with irregular behavior of the scattering amplitude near \(E=0\) if the Coulomb interaction is present. The proposed method was tested on the example of determining the ANC for the channel \({}^{16}\textrm{O}\to\alpha+{}^{12}\)C.

In this work, we considered the question of determining the ANC using \(R\)-matrix, the parameters of which were adjusted according to experimental data on elastic scattering. However, the opposite approach is also possible. Let us assume that the ANC value for the system under consideration is known. It might be obtained by some other method, for example, from the analysis of experimental cross sections for peripheral nuclear transfer reactions in the framework of the distorted wave Born approximation. In that case, this ANC can be used in fitting the \(R\)-matrix parameters by choosing the appropriate energy \(E_{l\lambda}\) and reduced width \(\gamma_{l\lambda}^{2}\) values in one of the terms in the \(R\)-matrix expression (16).