Microscopic optical potentials from a Green’s function approach

We follow rather closely the formulation given by Feshbach in Ref. [29]. The Schrödinger equation for the system consisting of an incident nucleon impinging on a target of mass A is given by,

$\displaystyle\left(E-H_{A}(\xi)-V(\mathbf{r}_{n},\xi)-T\right)\Psi(\mathbf{r}_{n},\xi)=0.$

(1)

Here ${\mathbf{r}_{n}}$ is the projectile coordinate, $T$ is the incident nucleon kinetic energy, $H_{A}(\xi)$ is the target nucleus many-body Hamiltonian that satisfies $H_{A}(\xi)\phi_{i}(\xi)=\epsilon_{i}\phi_{i}(\xi)$, with $\epsilon_{i}$ and $\phi_{i}(\xi)$ being the eigenenergies and eigenstates of the target nucleus, respectively, and $\xi$ denotes all relevant coordinates of the $A$ nucleons in the target. We expand the wavefunction $\Psi$ in terms of a complete set of eigenstates $\phi_{i}(\xi)$,

$\displaystyle\Psi(\mathbf{r}_{n},\xi)=\phi_{0}(\xi)\chi_{0}(\mathbf{r}_{n})+\sum_{i\neq 0}\phi_{i}(\xi)\chi_{i}(\mathbf{r}_{n}),$

(2)

where $\chi_{i}$ are the (unknown) wavefunctions describing the projectile states. The states with $i\neq 0$ correspond to the inelastic scattering. Let us now project (1) on the nuclear intrinsic states $\phi_{i}(\xi)$, obtaining the set of coupled equations,

	$\displaystyle\left(E_{i}-U_{ii}(\mathbf{r}_{n})-T\right)\chi_{i}(\mathbf{r}_{n})=\sum_{j\neq i}U_{ij}(\mathbf{r}_{n})\chi_{j}(\mathbf{r}_{n}),$
	$\displaystyle\mathrm{with}\quad i,j=0,1\dots$		(3)

where channel energies verify $E_{i}=E-\epsilon_{i}$. The coupling potentials are defined as

$\displaystyle U_{ij}(\mathbf{r}_{n})=\int\phi^{*}_{j}(\xi)V(\mathbf{r}_{n},\xi)\phi_{i}(\xi)\,d\xi.$

(4)

In order to obtain the optical potential, we single out the ground state of the target by rewriting the system of equations (II.1) as,

		$\displaystyle\left(E-U_{00}(\mathbf{r}_{n})-T\right)\chi_{0}(\mathbf{r}_{n})=\sum_{i\neq 0}U_{0i}(\mathbf{r}_{n})\chi_{i}(\mathbf{r}_{n}),$		(5)
		$\displaystyle\sum_{j\neq 0}\left(E_{i}\delta_{ij}-U_{ij}(\mathbf{r}_{n})-T\delta_{ij}\right)\chi_{j}(\mathbf{r}_{n})=U_{i0}(\mathbf{r}_{n})\chi_{0}(\mathbf{r}_{n}),$		(6)
		$\displaystyle\mathrm{with}\quad i\neq 0,$

where we have set $\epsilon_{0}=0$ and, to arrive at Eq. (6), we have taken all the terms besides $j=0$ in the sum in Eq. (II.1) to the left hand side. We now define the Green’s function $G^{Q}$ restricted to the excited states of the target ($Q$-space),

$\displaystyle G^{Q}_{ij}(E)=\lim_{\eta\to 0^{+}}\big{(}(E_{i}-T+i\eta)\delta_{ij}-U_{ij}(\mathbf{r}_{n})\big{)}^{-1},\quad i,j\neq 0.$

(7)

Since the vanishing parameter $\eta$ is positive and the coupling potentials $U_{ij}(\mathbf{r})$ are real, the poles of the analytic continuation of the Green’s function on the complex $E$ plane does not have poles on the upper half-plane of $E$. Eq. (6) can now be formally solved,

$\displaystyle\chi_{i}(\mathbf{r}_{n})=\sum_{j\neq 0}G^{Q}_{ij}(E)\,U_{j0}(\mathbf{r}_{n})\,\chi_{0}(\mathbf{r}_{n}).$

(8)

Substituting Eq. (8) in (5), we obtain a decoupled equation for the elastic channel wavefunction $\chi_{0}$,

		$\displaystyle\left(T+U_{00}(\mathbf{r}_{n})+\sum_{i,j\neq 0}U_{0i}(\mathbf{r}_{n})\,G^{Q}_{ij}(E)\,U_{j0}(\mathbf{r}^{\prime}_{n})-E\right)$
		$\displaystyle\times\chi_{0}(\mathbf{r}_{n})=0.$		(9)

The non-local, complex, and energy-dependent OP is identified with the effective potential in the above equation,

	$\displaystyle\mathcal{V}(\mathbf{r,r^{\prime}},E)$	$\displaystyle=$	$\displaystyle U_{00}(\mathbf{r})+\sum_{i,j\neq 0}U_{0i}(\mathbf{r})G^{Q}_{ij}(\mathbf{r,r^{\prime}},E)U_{j0}(\mathbf{r^{\prime}})$		(10)
		$\displaystyle=$	$\displaystyle U_{00}(\mathbf{r})+V_{\text{dpp}}(\mathbf{r,r^{\prime}},E),$

where we have dropped the subindex ${n}$, and have defined the dynamic polarization potential,

$\displaystyle V_{\mathrm{dpp}}(\mathbf{r,r^{\prime}},E)=\sum_{i,j\neq 0}U_{0i}(\mathbf{r})G^{Q}_{ij}(\mathbf{r,r^{\prime}},E)U_{j0}(\mathbf{r^{\prime}}).$

(11)

This dynamic polarization potential has, in general, an angular-momentum dependence determined by the multipolar decomposition of the coupling potentials and the Green’s function.

Since the real coupling potentials $U_{ij}(\mathbf{r})$ are finite and energy-independent the above optical potential will have the same poles as the Green’s function (7) as a function of the complex energy, and will thus be analytical in the upper half-complex plane. As a consequence, the optical potential is dispersive and the energy-dependence of the real and imaginary parts are connected through the Kramers-Kronig dispersion relations. (see [33], see also Eq. (2.26) of [29]).

The calculations presented in this work are based in an explicit implementation of Eq. (10) in a truncated space of excited states of the composite system. The energies, spins, and parities of the target or of the composite nucleus, as well as the static (energy-independent) couplings (4), are provided by a structure model of choice. However, we still need to compute the Green’s function defined in Eq. (7), i.e., the propagator in the space of excited states of the target.

A possible way to obtain this propagator is based on its spectral decomposition (Lehmann representation), which involves a sum over the discrete spectrum of the restricted $Q$-space Hamiltonian and an integral over the continuum [16]. This approach requires the diagonalization of the propagator $G^{Q}$. In addition, one has to deal with the branch cut present in the continuum, which introduces a divergence in the integral for positive energies. This problem has been addressed in different ways, such as by introducing a small but finite value in the $\eta$ regularization parameter in (7) [23], or by making the spectral decoposition in the Berggren basis [27].

Other approaches are based on providing an approximation for the potential defining the propagator. The simplest option is to assume that the nucleon is propagating freely in $Q$-space, in which case $G^{Q}$ is diagonal and equal to the free Green’s function [34]. The authors of [32] use instead the zero-order static potential $U_{00}$ to model the propagation of the nucleon in all the excited states, also assuming in this case that the Green’s function is diagonal. One of the advantages of this calculation with respect to the free propagation is to account for the effect of single-particle resonances (“shape” resonances) characteristic of the potential $U_{00}$ (see [32]). In this study, we will take an additional step by assuming that the intermediate system propagates according to the “full” optical potential $\mathcal{V}$ defined in (10).

We will thus assume that the $Q$-space propagator is diagonal and independent of the excited states,

$\displaystyle G^{Q}_{ij}(E)=\widetilde{G}(E-\epsilon_{i})\delta_{ij},\;(\text{for all }i,j),$

(12)

and that the Green’s function and the optical potential are connected by the two following equations,

	$\displaystyle\mathcal{V}(\mathbf{r,r^{\prime}},E)$	$\displaystyle=U_{00}(\mathbf{r})+\sum_{i\neq 0}U_{0i}(\mathbf{r})\widetilde{G}(\mathbf{r,r^{\prime}},E-\epsilon_{i})U_{i0}(\mathbf{r^{\prime}})$
	$\displaystyle\widetilde{G}(\mathbf{r,r^{\prime}},E)$	$\displaystyle=\lim_{\eta\to 0}\left(E-T-\mathcal{V}(\mathbf{r,r^{\prime}},E)+i\eta\right)^{-1},$		(13)

to be solved self-consistently. Since the optical potential is complex, the present approach accounts for absorption effects during the propagation in the virtually populated intermediate states.

The self-consistency of Eqs (II.1) is enforced by the implementation of an iterative scheme, in which an iteration $\mathcal{V}^{(n+1)}$ of the optical potential is obtained from the previous $n$th iteration of the Green’s function. Schematically,

		$\displaystyle\mathcal{V}^{(0)}(E)=U_{00},$
		$\displaystyle\mathcal{V}^{(1)}(E)=U_{00}+\sum_{i}U_{0i}\,\widetilde{G}^{(0)}(E-\epsilon_{i})\,U_{i0}$
	$\displaystyle\vdots$
		$\displaystyle\mathcal{V}^{(n+1)}(E)=U_{00}+\sum_{i}U_{0i}\,\widetilde{G}^{(n)}(E-\epsilon_{i})\,U_{i0},$		(14)

where the Green’s function is updated at every stage using

$\displaystyle\widetilde{G}^{(n)}$

$\displaystyle(E-\epsilon_{i})=\lim_{\eta\to 0}\left(E-\epsilon_{i}-T-\mathcal{V}^{(n)}(E)+i\eta\right)^{-1}.$

(15)

This approach requires the development of a numerical procedure able to implement the above operator inversion for non-local potentials, which will be introduced in Sect. II.2. It is to note that, although each intermediate state is propagated at the correct energy $E-\epsilon_{i}$, we make the approximation of always evaluating the optical potential at the same bombarding energy $E$. Although we have checked that the impact of this approximation is negligible after the first 2 iterations for some specific cases, a full understanding and validation of this procedure is still pending. The convergence of this iterative procedure is monitored by computing the volume integral of the optical potential

$\displaystyle J^{(n)}=\int\mathcal{V}^{(n)}(\mathbf{r,r^{\prime}})d\mathbf{r}d\mathbf{r^{\prime}}$

(16)

at each iteration for both the real and the imaginary part (see Fig. 8).

The partial wave decomposition of the Green’s function can be calculated in a standard way as the product of the regular ($F_{l}$) and irregular ($G_{l}$) solutions of the radial wave equation constructed with the optical potential (see, e.g., [32, 35]),

		$\displaystyle\widetilde{G}_{l}(\mathbf{r},\mathbf{r}^{\prime},E)=\frac{\mu}{\hbar^{2}rr^{\prime}}\Big{(}\frac{F_{l}(\eta,kr_{<})H_{l}^{(+)}(\eta,kr_{>})}{\mathcal{W}(r_{<})}$
		$\displaystyle+\frac{F_{l}(\eta,kr_{<})H_{l}^{(+)}(\eta,kr_{>})}{\mathcal{W}(r_{>})}\Big{)}\sum_{m}Y_{l}^{m}(\hat{r})Y_{l}^{m*}(\hat{r}^{\prime})$		(17)

where $\hbar^{2}k^{2}/{2\mu}=E$ and $H^{(+)}_{l}=G_{l}+iF_{l}$ has the asymptotic behavior of an outgoing Coulomb spherical wave. The radial coordinate $r_{<}$ ($r_{>}$) is the smallest (largest) among $r$ and $r^{\prime}$. The Sommerfeld parameter $\eta=\mu Z_{t}Z_{p}/\hbar^{2}k$ accounts for the effect of the Coulomb potential associated with the charges $Z_{t}$ and $Z_{p}$ of the target and projectile, and $\mu$ is the reduced mass of the system. The above expression is a generalization with respect to the local potential case. More specifically, the Wronskian

$\displaystyle\mathcal{W}=\frac{\partial F_{l}(\eta,kr)}{\partial r}G_{l}(\eta,kr)-\frac{\partial G_{l}(\eta,kr)}{\partial r}F_{l}(\eta,kr)$

(18)

corresponding to a non local potential is not constant, in contrast with the Wronskian for a local potential, which has the constant value $\mathcal{W}=k$.

We thus need to obtain the numerical solution to the radial integro-differential wave equations

		$\displaystyle\left(-\frac{\hbar^{2}}{2\mu}\frac{d^{2}}{dr^{2}}+V_{C}(\eta,r)+\frac{\hbar^{2}l(l+1)}{2\mu r^{2}}-E\right)F_{l}(kr)$
		$\displaystyle+\int\mathcal{V}_{l}(r,r^{\prime},E)F_{l}(kr^{\prime})\,dr^{\prime}=0,$		(19)

and

		$\displaystyle\left(-\frac{\hbar^{2}}{2\mu}\frac{d^{2}}{dr^{2}}+V_{C}(\eta,r)+\frac{\hbar^{2}l(l+1)}{2\mu r^{2}}-E\right)G_{l}(kr)$
		$\displaystyle+\int\mathcal{V}_{l}(r,r^{\prime},E)G_{l}(kr^{\prime})\,dr^{\prime}=0,$		(20)

where $V_{C}$ is the Coulomb potential of a uniformly charged sphere (throughout this work, the Coulomb radius is the same as $U_{00}$ radius), and $\mathcal{V}_{l}$ are the terms of the multipolar expansion of the optical potential. For positive energies, the two linearly independent solutions verify the usual Coulomb asymptotic conditions in terms of the nuclear ($\delta_{l}$) and Coulomb ($\sigma_{l}$) phase shifts,

	$\displaystyle F_{l}(\eta,kr)\xrightarrow[r\to\infty]{}\sin(kr-l\pi/2-\eta\ln{(2kr)}+\sigma_{l}+\delta_{l}),$
	$\displaystyle G_{l}(\eta,kr)\xrightarrow[r\to\infty]{}\cos(kr-l\pi/2-\eta\ln{(2kr)}+\sigma_{l}+\delta_{l}).$		(21)

For negative energies, the asymptotic behavior correspond to Whittaker functions,

		$\displaystyle F_{l}(\eta,\kappa r)\xrightarrow[r\to\infty]{}(2\kappa r)^{\eta}e^{\kappa r},$
		$\displaystyle G_{l}(\eta,\kappa r)\xrightarrow[r\to\infty]{}(2\kappa r)^{-\eta}e^{-\kappa r},$		(22)

with $\kappa=\sqrt{2\mu|E|}/\hbar$ and Sommerfeld parameter now defined as $\eta=\mu Z_{t}Z_{p}/\hbar^{2}\kappa$.

In order to solve Eq. (II.2) for the regular solution, we follow the procedure described in [36]. The method is based on the inversion of the Hamiltonian matrix in the basis defined by the Lagrange functions

$\displaystyle\varphi_{i}(r)=(-1)^{N+i}\left(\frac{r}{ax_{i}}\right)\sqrt{ax_{i}(1-x_{i})}\frac{P_{N}(2r/a-1)}{r-ax_{i}}$

(23)

defined in the interval $(0,a)$, where the $P_{N}(r)$ are Legendre polynomials, and $N$ is the number of elements of the basis, which are also the number of Lagrange points that constitute the radial grid. These points $x_{i}$ are the $N$ roots of the equation

$\displaystyle P_{N}(2x_{i}-1)=0.$

(24)

In order to make it Hermitian in the finite interval and to enforce the boundary conditions, the Hamiltonian is complemented with the Bloch operator [37],

$\displaystyle\mathcal{L}_{a}=\frac{\hbar^{2}}{2\mu}\delta(r-a)\frac{d}{dr}.$

(25)

This procedure for obtaining the regular solution for a non-local potential also provides the scattering phase shifts, obtained by matching with the appropriate asymptotic behavior. All the elastic scattering cross sections presented in this work have been obtained in this way. For a more detailed description of the numerical procedure, we refer to Ref. [36].

Since the irregular solution does not vanish at the origin, Eq. (II.2) cannot be solved within the basis (23), as all the basis functions verify $\varphi_{i}(0)=0$. We use instead the alternative basis

$\displaystyle\tilde{\varphi}_{i}(r)=(-1)^{N+i}\sqrt{ax_{i}(1-x_{i})}\frac{P_{N}(2r/a-1)}{r-ax_{i}}$

(26)

defined in an interval $(\epsilon,a)$, where $\epsilon>0$ is a small value of the radius, set to avoid the divergence of the irregular solution at $r=0$.

The numerical parameters used in this work are checked for the convergence of the results. In addition, the upper limit $a$ of the radial interval has to be larger than the range of the optical potential, so that the asymptotic expressions (II.2) hold. In the context of the applications presented here, we found $a=30$ fm as the size of the Lagrange radial box, and $N=100$ Lagrange points to be sufficient.

To validate our method, we benchmark against the results presented in [32]. Following this reference, we adopt the 10 vibrational states and couplings in ⁴⁰Ca presented in Table 1 as the structure model for the derivation of the $p+^{40}$Ca OP presented in this earlier work, to which we will add the $n+^{40}$Ca one. The coupling strengths are given in terms of $\beta_{J}$ deformation parameters that can be linked to the electromagnetic transition strengths from the excited states to the ground states, and are extracted from $(\alpha,\alpha^{\prime})$ measurements [39, 40]. The spin-orbit coupling effects are neglected.

More specifically, the $U_{00}$ term of the optical potential is taken to be a local, Woods-Saxon potential with a depth $U_{0}=50$ MeV, a diffuseness $a=0.75$, and a radius $R_{0}=1.12A^{1/3}$ fm. A small absorptive potential is added in terms of an imaginary potential with the same geometry and depth $W_{0}$=2 MeV. The static, central potential can then be written as:

$\displaystyle U_{00}(r)=\frac{-(U_{0}+iW_{0})}{1+\mathrm{exp}[(r-R_{0})/a]}=U(r)+iW(r),$

(27)

plus the Coulomb potential of a uniformly charged sphere of radius 1.12 $A^{1/3}$ fm. The couplings of the ground state with each of the 10 states (labeled by $i$) are modeled in terms of surface-peaked transition densities,

$\displaystyle U_{0i}(r)=(2\lambda+1)^{-1/2}\,r\,\frac{dU_{00}(r)}{dr}Y^{\mu*}_{\lambda}(\hat{r}),$

(28)

where $\lambda,\mu$ are the angular momentum and its projection of the state $i$. Following Ref. [32], we use the weak coupling approximation, where the Green’s function in the definition of the dynamic polarization potential (11) is diagonal.

Our benchmark is in good agreement with Ref. [32] (Fig. 1), small deviations being likely due to numerical differences in the implementation that cannot be tracked down with the available information in the original reference. The importance of the dynamic polarization potential is illustrated by the comparison with the calculation made with $U_{00}$ alone (dotted blue line in Fig. 1), which yields significant deviations from the experimental data. In particular, the lack of absorption results in a large overestimation of the elastic cross section. The inclusion of all 10 collective states of ⁴⁰Ca brings the calculation much closer to the measured points.

The inclusion of a small imaginary part in the static potential (27) introduces a non-dispersive element to the OP. To examine its contribution to the elastic scattering cross sections, we show in Fig. 2 a calculation without the imaginary part, $W(r)$, of the static potential. We also show in this figure the effect of implementing the iterative scheme (14) by comparing the result obtained by computing the Green’s function with real part of the static potential (1 iteration), to the calculation obtained after convergence has been reached. We observe that the the small imaginary part in the static potential included by the authors of Ref. [32] has a small effect in the elastic cross section, indicating that most of the absorption comes from the non-local imaginary part of $V_{\mathrm{dpp}}$.

After iterating the potential and achieving convergence of the volume integral for each of the partial waves, the cross section at forward angles sees a small change, getting slightly closer to the experimental values. The effect of the iterations is more significant at backward angles, where the converged potential smooths out the last dip at $160^{\circ}$ (Fig. 2).

In addition to reproducing the proton elastic scattering presented in Ref. [32], we calculate the neutron elastic scattering cross sections for 30.3 MeV projectile energy using the same 10 vibrational states in ⁴⁰Ca, by simply removing the Coulomb term from the static potential (Fig. 3). Similar to the proton scattering, the values at forward angles do not change drastically when the potential is iterated. However, there is a larger difference at $\theta>90^{\circ}$ and the dips are smoother with the converged potential, which agrees better with the experimental data.

The idea behind the inclusion of a small imaginary part in the static potential by the authors of Ref. [32] is to account for absorption processes not included in the dynamic polarization potential. They were thus addressing the fact that the reaction cross section for 30.3 MeV protons on ⁴⁰Ca might not have been completely exhausted by the inelastic excitation of the 10 collective states present in their model. In principle, the implementation of the framework with an exhaustive description of the spectrum should mitigate the need for any added absorption, and provide an overall better description of elastic scattering. To explore this possibility, we derive in this section a $n+^{24}$Mg OP making use of the input provided by a ²⁵Mg valence shell model calculations.

The spectrum of ²⁵Mg (around 600 states up to about 15 MeV of excitation energy) has been calculated using the PSDPF interaction [42], built upon an inert ⁴He core. The valence nucleons occupy the p, sd and pf shells, therefore allowing for the description of both positive and negative parity states. This choice of an extended model space allows for excitations from the $p$ to $sd$ shell, as well as the $sd$ to $pf$ shell, ensuring an exact separation of the center of mass coordinate. In order to model the static potential $U_{00}(r)$, we use a simple real Woods-Saxon shape (27) with parameters $U_{0}=48.5$ MeV, $R_{0}=3.69$ fm, $a=0.65$ and $W_{0}=0.$ These values reproduce the experimental neutron separation energy of ²⁵Mg by binding a $1d_{5/2}$ single particle state by 7.33 MeV.

We need to define now the couplings of the incoming neutron with each one of the excited states of ²⁵Mg. We will first assume that the coupling matrix (4) is diagonal. The coupling potentials $U_{0i}(r)$ connecting the ground state to each excited state $i$ are taken to be a central real volume Woods-Saxon with the same radius and diffuseness as $U_{00}(r)$, but with a depth adjusted to reproduce the binding energy of each state. These couplings are then multiplied by a coupling strength taken to be the spectroscopic amplitude (overlap) $S_{i}$ of the corresponding state with the ground state of ²⁴Mg. We then obtain an $l$-dependent OP, with the following radial part

	$\displaystyle\mathcal{V}_{l}({r,r^{\prime}}$	$\displaystyle,E)=U_{00}(r)$
		$\displaystyle+\sum_{\begin{subarray}{c}i\neq 0\\ \ell_{i}=l\end{subarray}}|S_{i}|^{2}U_{0i}(r)g_{\ell_{i}}({r,r^{\prime}},E-\epsilon_{i})U_{i0}(r^{\prime}),$		(29)

where $\ell_{i}$ is the orbital angular momentum of the state $i$, and

	$\displaystyle g_{\ell_{i}}(r,r^{\prime},E)=\frac{\mu}{\hbar^{2}}\Big{(}\frac{F_{\ell_{i}}(\eta,kr_{<})H_{\ell_{i}}^{(+)}(\eta,kr_{>})}{\mathcal{W}(r_{<})}$
	$\displaystyle+\frac{F_{\ell_{i}}(\eta,kr_{<})H_{\ell_{i}}^{(+)}(\eta,kr_{>})}{\mathcal{W}(r_{>})}\Big{)}$		(30)

is the radial part of the Green’s function (II.2). The couplings being central, scattering partial waves only couple to ²⁵Mg states with the same angular momentum. Since the shell model calculation used here includes the p, sd and pf shells, only the $\ell=0,1,2,$ and $3$ partial waves are involved in the calculation of the dynamic polarization potential.

With this prescription, the contribution $\sigma_{i}$ of each state to the reaction cross section can be associated with the one-particle transfer cross section connecting the incoming scattering state with the corresponding final ²⁵Mg state, expressed in the $prior$ representation (see, e.g., [43]). This choice of the couplings might seem somehow heuristic, but, in this work, we explore the results obtained with this simple prescription, before taking a more microscopic approach for the couplings in future works. In particular, a more microscopic systematic prescription would be to use the calculated transition densities. This is what has been done, for example, in [23], albeit in a weak coupling approximation for a spectrum limited to the harmonic nuclear response (RPA).

We observe larger spectroscopic factors at the lower-lying positive and negative parity states (Fig. 4), making their contribution the strongest in the construction of the optical potential. We note that, even though at higher energies the spectroscopic factors become orders of magnitude smaller (Fig. 4, inset) their contribution is not necessarily negligible as the density of states increases exponentially.

The resulting energy-dependent dynamic polarization potential, $V_{\mathrm{dpp}}$, is non-local in nature and has both real and imaginary terms (Fig. 5). We calculate it at each neutron bombarding energy and add the static local $U_{00}$ term to obtain the full optical potential. The periodicity of $V_{\mathrm{dpp}}$ with respect to $r$ and $r^{\prime}$ is related to the periodicity of the regular and irregular solutions that enter in the expression of the Green’s function in Eq. (II.2).

Our main results are presented in Fig. 6. We show angular differential elastic scattering $n+^{24}$Mg cross sections for 5 different incident energies. In our calculations, all processes contributing to the formation of an excited ²⁵Mg compound nucleus state contribute to the imaginary part of the optical potential, and thus to the reaction cross section and the depletion of the elastic channel. A fraction of these compound nucleus states can decay emitting a neutron with an energy equal to the incident one, contributing to the compound elastic cross section. Since these events cannot be experimentally distinguished from elastic scattering, we have added incoherently to our calculations the compound elastic contribution calculated with the Hauser-Feshbach code YAHFC [44]. The resulting comparison with experimental data, as well as with the curves obtained with the Koning-Delaroche global optical potential [8] are remarkable given that our calculations contain no phenomenological imaginary terms. In order to be completely consistent, the Hauser-Feshbach calculation should have been performed using our optical potential instead of the default Koning-Delaroche used in YAHFC. We leave for future works a more consistent treatment of the neutron decay following the compound nucleus formation.

We illustrate the role of the dynamic polarization potential by comparing the cross section obtained with the full optical potential (III.2) to the one making use of the static potential $U_{00}(r)$ only (Fig. 7). The effects of the absorption introduced by the coupling to the many-body states of ²⁵Mg, as well as the renormalization of the real part of the optical potential associated with the virtual population of these states, is very apparent, and brings the calculated elastic scattering cross section much closer to the experimental values.

An important aspect of a unified description of structure and reactions is the treatment of the continuum. In this work, the structure of ²⁵Mg has been calculated in a harmonic oscillator basis, so the computed spectrum is fully discrete and bound, even above the experimental neutron separation energy of 7.33 MeV. However, in our approach we interpret the shell model states above the neutron separation energy as resonances, and the Green’s function $\widetilde{G}(E)$ exhibits the correct asymptotic behavior as a function of the energy, determined by the corresponding irregular solution $G_{l}$ in (II.2). Therefore, an arbitrary scattering state obtained acting on a free wave $\phi_{0}$ with the Möller operator $\Omega(E)$

$\displaystyle\Psi(E)=\Omega(E)\phi_{0}(E)=\left(1+\widetilde{G}(E)\mathcal{V}(E)\right)\phi_{0}(E),$

(31)

will exhibit the desired resonance behavior at the poles of the Green’s function, with widths determined by the penetrability factors multiplied by the corresponding spectroscopic factors. This approach, in which the discrete shell model spectrum above the particle emission threshold is interpreted as many-body resonances that are coupled to the continuum, is reminiscent of the continuum shell model formalism [45].

We now turn our attention to the convergence of the iterative scheme described in (14) by computing the volume integral (16) for each relavant partial wave as a function of the iteration number. For of 7.6 MeV incident neutrons, we observe that the calculations are fully converged after 15-20 iterations for all the partial waves, and both for the real and the imaginary part of the optical potential (Fig. 8).

An important aspect of the numerical convergence of the calculation is associated with the truncation of the Hilbert space. Aside from the consideration of the total number of states provided by the shell model calculation, the problem at hand offers a natural distinction between states situated above and below the considered bombarding energy. Although most calculations to date neglect the contribution of states above the bombarding energy, their effect in the renormalization of the real part of the optical potential should be addressed. These states in the general expression (III.2) correspond to negative values of the argument $E-\epsilon_{i}$ of the Green’s function, which then acquires an exponentially decaying asymptotic behavior (see Eq. (II.2)). We show in Fig. 9 the elastic scattering cross section for 3.4 MeV incident neutrons, calculated with and without the inclusion of states above the bombarding energy. The results are almost identical, which supports the notion that not including them in the calculation is a reasonable approximation.