Optical potentials for the rare-isotope beam era

Most energy and security applications require high-quality evaluated cross sections for neutron-induced reactions up to 20–30 MeV, such as those present in evaluated nuclear data libraries like ENDF/B-VIII.0 [128]. The highest-priority targets are actinides and structural/engineering materials. High-quality optical models provide the starting point for reliable descriptions of reactions needed for modern comprehensive evaluations of neutron induced reactions (e.g. see a recent evaluation of U-238 and U-235 neutron induced reactions [129] adopted into the ENDF/B-VIII.0 library [128]). Because of the high societal impact of these applications, many (but not all) relevant cross sections have been thoroughly examined, and sophisticated uncertainty quantification techniques developed and applied [130–132]. The extensive high-quality experimental data collected to inform the relevant cross sections mean that phenomenological models are in their range of validity, provided sufficient physics—such as selection of the appropriate reaction model and detailed structural information—are considered.

There are still needs for OMP development for materials used in next-generation reactor architectures. In some designs such as those involving molten salts, potentials are needed for isotopes beyond typical structural materials and actinides. Experimental data suitable for improving a phenomenological potential on these isotopes are not always available, especially because the bulk of single-nucleon scattering measurements were conducted more than thirty years ago. Also relevant are reactions on light elements which are important both in their own right and because they can provide important constraints on reverse reactions that are difficult to access experimentally but important for neutron economy (e.g. ¹⁶O${\left(n,\alpha \right)}^{13}{\rm{C}}$ to improve knowledge of ¹³C${\left(\alpha ,n\right)}^{16}{\rm{O}}$ or vice-versa).

An important application is the production of isotopes used in medical diagnosis and treatment [133]. The two major methods for isotope production are (1) reactor-based fission, capture, and (n, p) reactions and (2) charged-particle-induced reactions using fast charged-particle and/or neutron beams [134–136]. For reactor-based isotope production, neutron-induced reaction cross sections similar to those in the energy and security applications are important, so there can be significant overlap in the theoretical and experimental tools needed to constrain these cross sections [135]. For charged-particle-induced reactions, much higher energies are required: up to tens of MeV in commercial cyclotrons such as those found in hospital settings, and up to hundreds of MeV for dedicated accelerator facilities such as Los Alamos Neutron Science Center (LANSCE).

Isotope production requires not only nuclear physics information which can be partially obtained via an optical model, but also efficient chemical separation techniques that allow the produced isotope to be extracted. As such, the most important reactions are those that change the number of protons and thus the chemistry of the target, facilitating chemical separation and increasing the specific activity of the product. For charged-particle beams, the charge-exchange reactions (p, xn), (d, xn), (α, xn) are the most important [137–141]. In a similar vein, the most important neutron-induced reactions in the reactor-based setting are (n, p), fission, or multi-step reactions with subsequent β- or α-decay, yielding a change in element of the reaction products [135]. In some cases a direct production of the parent radionuclide is feasible (e.g. ¹⁰⁰Mo (n, 2n) reaction is used to produce ⁹⁹Mo parent for production of the very important ⁹⁹Tc generators) [139]. In each of these cases, secondary particle reactions often contribute significantly to reaction yields, compounding the importance of proper reaction modelling to consider all relevant channels. Because of the inherently complicated nature of these reactions, it should be noted that the OMP provides only the first ingredient required for reliable predictions of the needed cross sections, and that additional information—such as from a statistical reaction and pre-equilibrium model—is essential. As currently-available OMPs have been developed to describe p, n, d, or α scattering, several OMPs across a wide range of energies may be needed simultaneously to describe secondary particle production and follow-on reactions [15].

A further medical application is radiotherapy, where cross section information at energies up to 250 MeV/nucleons are important. For example, in a proton radiotherapy procedure, the dose delivered in the beam entrance region (before the Bragg peak) depends heavily on the elastic scattering of protons on tissue, which is one of the easiest quantities to cleanly predict using a suitable OMP and reaction code. However, this information must be combined with atomic and radiological data that are often absent or highly uncertain. As is the case in compound nuclear reactions that combine nuclear structure and OMP information, knowing the relative uncertainty of the OMP versus other reservoirs of uncertainty (e.g. atomic data) can help focus efforts on reducing the most impactful uncertainties first. There are also important quantities that the OMP cannot fully inform, for example, activation data or the production cross section for positron-emitting radionuclides (e.g. ¹¹C). As uncertainty quantification is still developing for many other types of physics data entering medical and energy/security applications, better OMP uncertainty assessments can potentially provide a methodological guidepost for other fields.

The interaction of radiation with space-based systems is important to understand for both national security and industry. Radiation shielding in space is a balance between maximizing protection and minimizing the amount of shielding material, as there is a strong cost motivation to minimize the overall weight of the system [142]. In addition, radiation effects on electronic systems are often difficult to directly study in space-like environments [143]. Relevant models for both these cases often lack experimental data [142]. Much like in other applications outlined above, secondary particle production contributes significantly to the relevant doses. Knowledge of neutron production from proton-induced reactions is very important and double-differential cross sections of produced neutrons may be needed. In addition, gamma rays produced by inelastic scattering can potentially have a large impact on overall dose delivered to a space-based system (for example, an electronics circuit) by incoming neutrons [143]. Given that these reaction cross sections are often poorly known, predictive and uncertainty-quantified OMPs, in concert with other theoretical inputs, can have a significant impact. In particular, a recent survey of nuclear data needs for space radiation protection has identified significant shortcomings in experimental data [144]. Previous work has been done using optical models to predict nuclear fragmentation processes relevant for space applications [145, 146].

Early developments of optical potentials sought to describe single-nucleon cross sections phenomenologically, using a local, complex, one-body potential (1)–(2) with parameters varying smoothly in E and A, analogous to the refractive index for an absorptive medium. By the 1960s, enough cross section data had been collected to contemplate training a 'global' optical potential suitable for predicting elastic scattering cross sections across a broad range of nuclei and energies, with Becchetti and Greenlees the first to do so [12]. Increases in computational power and the size of experimental databases led to potentials with a growing number of free parameters and better empirical performance, including the spherical CH89 [13] (40 ≤ A, 10 ≤ E ≤ 65 MeV) and the Koning and Delaroche [14] (24 ≤ A ≤ 209, 1 keV ≤E ≤ 200 MeV) potentials, which remain widely used. Most recent phenomenological efforts include additional physics, such as enforcing dispersivity, including non-locality, and/or describing deformation via a coupled-channels approach, each of which can improve the accuracy needed for applications [15]. To simplify interpretation in terms of nuclear asymmetry, many single-nucleon potentials adopt a Lane-consistent form where the OMP is partitioned into isoscalar and isovector components so that the neutron and proton OMPs differ only by a change in sign of the asymmetry-dependent components [98, 147]. Besides the high-visilibity global efforts, over the decades many hundreds of experimental papers have included isotope- or region-specific optical-potential analyses to assess the impact of their newly collected data (for example, [148]). However, the basic formula for developing a phenomenological OMP remains essentially unchanged since the 1950s: select a suitable collection of functional forms dependent on A, E, and/or δ, compile experimental scattering data, predict reaction observables given an OMP, and optimize OMP parameters according to χ² minimization.

As an example, consider the Koning–Delaroche OMP [14], one of the most widely used OMPs since its introduction in 2003. The nominal range of validity in energy is 1 keV to 200 MeV. Both global and several local versions are available, spanning near-spherical systems with 24 ≤ A ≤ 209. To train the OMP, the authors used hundreds of proton and neutron differential elastic scattering and analyzing power data sets from 27 ≤ A ≤ 209 collected from the 1950s to the 1990s, as well as proton reaction cross section data and neutron total cross section data on natural and isotopic targets. The potential itself includes six subterms with Woods–Saxon-like radial dependence separated from their energy dependence, with a total of forty-six free parameters. Optimization was done using a combination of 'computational steering' and χ² minimization, whereby a user manually guided potential parameters until predicted cross sections were visually close to experimental values, then invoked a simulated annealing algorithm to find a parameter-vector optimum. The success of the KD global OMP in reproducing its training data, particularly the elastic-scattering angular distributions and neutron total cross sections above the resolved-resonance region, as well as its ease of use, have led to its widespread adoption as a default OMP in reaction codes such as talys [149, 150] and Finite Range with Exact Strong COuplings (fresco) [151].

There are limitations of this approach that are discussed in more detail in section 3.3.

The Dispersive Optical Model (DOM), first introduced by Mahaux and Sartor [8–10], is an optical model which makes use of a dispersion relation (4) that relates the imaginary part of the potential to its real part over all energies. Equation (4) is a very powerful constraint that provides a variety of advantages over non-dispersive optical models, helping to reduce the number of model parameters (e.g. see [152]) and to achieve a better description of neutron-induced cross sections for energies below ∼5 MeV [153] compared to traditional OMPs like Koning–Delaroche [14]. Dispersion integrals given by equation (4) can be solved numerically [154] or analytically for selected imaginary potentials [10, 155, 156]. Several different dispersive OMPs have been derived (starting from equation (4)) for a variety of use-cases. One class of dispersive OMPs was derived to describe deformed nuclei assuming a rigid-rotor structure like Rh, Au, W, Ta, Hf and actinides [157–159]. Another was derived for spherical nuclei that are soft relative to vibrations assuming a soft-rotator structure like Fe, Ni, Cr [160] and Zr [15]. Yet another class of dispersive coupled-channel potentials has been used to describe both elastic and inelastic scattering data in a broad energy range up to 200 MeV. These derived dispersive OMPs [101, 161–163] have been shown to be approximately Lane consistent, i.e., the same OMP holds for incident neutrons and protons [164, 165] and the parametrization becomes isospin dependent. The dispersion relation has also been used to provide a consistent description of both bound and scattering data in spherical nuclei [11, 166] in some cases allowing 'data-driven' extrapolations to the drip lines [167, 168].

We now briefly discuss efforts that augment the data set constraining the OMP to negative energies by taking in structure information (charge density, energy levels, particle number, etc), in addition to the elastic-scattering data corresponding to positive energies. As mentioned in the Introduction, the optical potential can be interpreted as the irreducible self-energy ${{\rm{\Sigma }}}^{\star }(r,r^{\prime} ;E)$, in the Green's function formalism. Moreover, ${{\rm{\Sigma }}}^{\star }(r,r^{\prime} ;E)$ generalizes the exact nucleon–nucleus optical potential of Feshbach to include both bound and scattering states (a more detailed discussion can be found in section 3.2.2). Connecting the optical potential to the Green's function, along with utilizing the dispersion relation in equation (4), allows for a complete description of the nucleus over both the positive- and negative-energy domains [11, 166]. Adjusting OMP parameters to describe data using equation (4) guarantees that the irreducible self-energy stays well-defined [169, 170]. Currently, there are DOM fits using this Green's function formalism for spherical targets ^16,18O, ^40,48Ca, ^58,65Ni, ^112,124Sn, and ²⁰⁸Pb for −200 MeV < E < 200 MeV [169–172]. In principle, a dispersive optical potential can be applied in the same mass-number and energy range as a typical non-dispersive potential (such as KD) can. Work is currently underway to implement a global parametrization of a fully-dispersive optical potential for spherical targets [173].

Noting that Hartree–Fock potentials are already inherently nonlocal, it was demonstrated that spatial nonlocality of the self-energy including its imaginary part must be treated explicitly in order to describe properties below the Fermi energy [166]. To satisfy the dispersion relation in equation (4), it is at present assumed that the energy dependence of the imaginary part is the same for all spatial coordinates, which simplifies the numerical effort. Typically, optical potentials approximate the spatial nonlocality with an energy dependence [5]. However, this added energy dependence does not satisfy the dispersion relation, which would lead to an incorrect description of the negative-energy observables. Thus, the spatial nonlocality is treated explicitly with the so-called Perey–Buck form [5]

$$\begin{eqnarray}&&{ \mathcal U }(\vec{r},\vec{r}^{\prime} ;E)=U\left(\frac{r+r^{\prime} }{2};E\right){{\rm{e}}}^{-\tfrac{{\left(\vec{r}-\vec{r}^{\prime} \right)}^{2}}{{\beta }^{2}}}{\pi }^{-3/2}{\beta }^{-3},\end{eqnarray} \tag{ 17 }$$

where β is a nonlocality parameter which controls how much strength is distributed off the diagonal. It is worth noting that the functional form in equation (17) is chosen out of convenience, but is capable to represent essential features of microscopic potentials [174, 175]. With this treatment of the nonlocality, along with the dispersion relation in its subtracted form, quantities such as particle numbers, charge densities, and ground-state binding energies are included in the DOM fit. This allows for data-informed predictions of quantities such as the neutron skin of ⁴⁸Ca and ²⁰⁸Pb (see [169, 171, 172, 176] for more details).

The ability to describe both bound and scattering states of a nucleus is particularly useful in the description of stripping, transfer and knockout reactions needed to fully utilize FRIB [177, 178]. As a specific example, we briefly present the DOM calculation of ⁴⁰Ca ${\left(e,e^{\prime} p\right)}^{39}{\rm{K}}$ cross sections [170]. This reaction, measured at NIKHEF [179], can be described using a distorted-wave impulse approximation (DWIA), which assumes that the virtual photon exchanged by the electron couples to the same proton that is detected and that the final-state interaction can be described using an optical potential [180, 181]. The ingredients of the DWIA therefore require a distorted wave describing the outgoing proton at the appropriate energy and an overlap function for the removed proton and its associated spectroscopic factor. The DOM allows for a consistent DWIA analysis in that the bound state wave function, spectroscopic factor, and outgoing proton distorted wave can all be provided from the same self-energy. The resulting momentum distributions, shown in figure 6, came straight from the DOM self-energy—the ⁴⁰Ca${\left(e,e^{\prime} p\right)}^{39}{\rm{K}}$ data was not used in the DOM fit. The spectroscopic factors coming directly from the DOM self-energy show a good description of the data—thus updating the previously obtained spectroscopic factors (found by scaling the previous DWIA analysis to match the data points). Furthermore, this analysis was also done for ⁴⁸Ca${\left(e,e^{\prime} p\right)}^{47}{\rm{K}}$ providing a new perspective on the quenching of spectroscopic factors [82, 182]. These results demonstrate how, provided that sufficient elastic-scattering and structure data is available to constrain the fit, the DOM potential applied in the appropriate reaction theory is a powerful way to consistently describe knockout reactions and aspects of transfer reactions relevant to FRIB science.

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Uncertainty quantification (UQ) for OMPs is an emerging topic, with the majority of publications on the topic dating from the last five years. The earliest systematic attempt at OMP UQ was by the authors of the Chapel Hill globalOMP [13], who used a bootstrap method to assess that the variances of parameters in their OMP were very small (on the order of a percent). An attempt to estimate uncertainties of potential parameters was discussed at the IAEA RIPL project, where rough estimates of geometry and potential depth uncertainties were given [15]. Recent analyses using Bayesian techniques [86, 131, 176, 183, 184] reveal much larger uncertainties (tens of percent for elastic-scattering observables and up to a hundred percent for single-nucleon transfer cross sections) indicating that the statistical assumptions used for training phenomenological potentials can impact predictions as strongly as the data used for training. Due to the absence of global OMPs with well-calibrated parametric uncertainties, OMP users often resort either to tuning OMP parameters by hand or to performing ad hoc UQ by comparing predictions from multiple OMPs—neither of which is easily extrapolated to the high-asymmetry regime that will be probed at FRIB. Recently, Pruitt, Escher and Rahman have developed an extension of the global spherical proton and neutron OMPs of KD [14] and CH89 [13] with uncertainty quantification of the potential parameters, called KDUQ and CHUQ [185]. The mass and energy range of validity are the same as the original KD and CH89, i.e 24 < A < 209 and 0.001 MeV < E < 200 MeV for KDUQ and 40 < A < 209 and 10 MeV < E < 65 MeV for CHUQ. This new development will support the quantification of uncertainties in reaction observables. A natural next step to improve the constraint of these parametrizations is to enforce dispersion relations, include data on highly asymmetric systems and bound observables.

Feshbach formulation. One of the approaches used to integrate microscopic nuclear structure information into the construction of optical potentials is by using the Feshbach formulation [19], where the optical model potential for a nucleon scattering energy E is given by equation (10). Equation (10) can be obtained from the set of coupled differential equations written in terms of the reaction channels [19]. By solving this set of equations, the Green's function matrix G_jk can be diagonalized in the space of the excited states j, k ≠ 0, but one can also use the weak coupling approximation [19, 186–189], which neglects the couplings between excited states, i.e. ${{ \mathcal V }}_{{jk}}={{ \mathcal V }}_{0k}{\delta }_{j0}$. In this case the coupling potential is expressed in terms of an 'arrow' matrix (figure 7), and equation (10) becomes

$$\begin{eqnarray}&&{V}_{\mathrm{opt}}={{ \mathcal V }}_{00}+\displaystyle \sum _{j\ne 0}{V}_{0j}{G}_{{jj}}{{ \mathcal V }}_{j0},\end{eqnarray} \tag{ 18 }$$

where only the diagonal elements of the Green's function (8) enter. The coupling potentials and the Green's functions in the second term of equation (18) can be provided by nuclear structure calculations.

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The method works best for the low-energy region where nuclear structure models can provide reliable and converged calculations (approximately <50 MeV). The versatility of this approach lies in the variety of structure calculations it can accommodate. However, the description of compound nucleus reactions, in which the relevant target-nucleon states are statistical in nature, is an open challenge.

Nuclear structure method. Another method is to construct the potential from a phenomenological effective NN interaction using the Green's function formalism. It is called the Nuclear Structure Method (NSM). The Green's function formalism allows the hierarchization of correlations and avoids double countings. Antisymmetrization due to the fermionic nature of nucleons is taken into account. The NSM was first proposed for realistic NN interactions by N Vinh Mau in the early 1970s [190]. Then it has been recast in order to be used with density functionals such as Skyrme or Gogny [191].

When dealing with a spherical target nucleus (without pairing), the NSM potential is made of two contributions: the Hartree–Fock potential and the Random Phase Approximation potential. The mean-field term is energy-independent. Its exchange term (Fock term) turns out to be nonlocal when using a finite-range interaction. The RPA contribution is a polarization contribution. The absorption results from taking into account the coupling to inelastic channels when the target nucleus is excited. Such excitations are described in the RPA formalism. This term is non-local, energy dependent and complex. The NSM can be interpreted as a Feshbach potential with consistent ingredients as the same functional is used all along the calculation. Hence in equation (18), V₀₀ would be the equivalent of the Hartree–Fock potential in the NSM whereas V_0i 's would be provided by RPA.

A pioneering application of the method at lowest order with the Skyrme functional has demonstrated the ability of the Hartree–Fock potential to grasp the main features of the real part of the optical potential below 50 MeV incident energy [192]. Follow-up studies have included second-order terms with particle–particle (pp) and particle–hole (ph) correlations. The pp correlations are implicitly contained in the phenomenological NN interaction. particle–hole correlations are then taken into account explicitly through the Random Phase Approximation (RPA). In the beginning of the 1980s, several groups have worked on the NSM and related methods mostly using Skyrme interactions [191, 193, 194]. More refined versions of the NSM have been proposed including both inelastic excitations and (n, p) charge exchange [195–197]. The NSM has been used as well to determine α-nucleus potentials [198–200]. In the last decade, there has been a renewed interest in the NSM with Skyrme [201–203] and Gogny interactions [204–206]. The NSM describes with relative success, for both neutron and proton projectiles, the scattering off target nuclei such as: ¹⁶O [203, 207, 208], ⁴⁰Ca [193, 195, 203, 204, 206], ⁴⁸Ca [203, 206] and ²⁰⁸Pb [203, 208], for incident energy below 50 MeV. A calculation fully handling continuum and self-consistency has been proposed with Skyrme interactions [207]. The method has been applied to neutron scattering off ¹⁶O below 30 MeV. This approach is particularly interesting because it allows one to circumvent the pitfall of RPA calculations in a harmonic oscillator (HO) basis that requires the introduction of ad hoc escape and damping widths [204]. Methods close to the NSM have also been used to describe proton inelastic scattering [209].

The NSM works well for incident energies below 50 MeV. Thus the method is complementary to g-matrix approaches in terms of energy range. In its current versions, it is limited to target nuclei well-described within RPA, typically double-closed shell nuclei. However, the extended reach of energy density functional based on structure calculations (pairing, deformation and odd number of nucleons) [210] suggests that further versions of the NSM will be suitable for a wide range of target nuclei. Some recent attempts have extended the approach to scattering off target nuclei with pairing using Hartree–Fock–Bogolyubov (HFB) formalism [211, 212]. The NSM will then be extended to include pairing correlations within the HFB formalism with Quasiparticle-Random-Phase Approximation (QRPA) on top of it. These new developments will eventually allow for the description of nucleon scattering off deformed target nuclei with pairing.

We mention here also approaches that employ Skyrme [213–217] or Gogny [218] functionals for infinite nuclear matter (see more extensive discussion of such methods in section 3.2.4). The optical potential is then obtained using the local density approximation (LDA) with a consistent density. This approach allows a satisfactory description of the elastic scattering observables for energies up to 100 MeV. In this context, there have been several attempts to fit new Skyrme functionals adding scattering constraints to the more usual structure ones [215–217, 219].

Optical potentials from effective Hamiltonians. While it is alluring to use a single interaction (e.g. Chiral) or pseudo–interaction (e.g. Gogny or Skyrme functionals) to construct the nuclear structure properties for the ground state, excitations, and subsequently the optical potential, it is also possible to combine effective Hamiltonians and interactions without lack of generality. This strategy has the advantage of potentially reducing computational costs or increasing the many-body expansion with respect to a calculation that treats every component on equal footing.

Over the years, several approaches have used different effective interactions and microscopic methods to construct the optical potential. The approach based on Hartree–Fock [191, 220] consists in defining the real part of a local potential based on an appropriate Skyrme functional. It is extended with additional couplings and absorption [208, 221]. Eventually, one can consider the optical potential as arising from particle–vibration coupling, even extended in the continuum [209]. Another prominent example is the nuclear field theory, where single-particle and collective degrees of freedom are combined, eventually employing phenomenological coupling instead of a single consistent Hamiltonian or functional [222]. The nuclear field theory have been lately expanded, both in functional form [223], and using an effective coupling between multipolar vibrations and mean field [224]. The coupling between degrees of freedom has been recast as the solution to the Dyson equation [225], closely relating several structure and reaction observables [226–229]. However, it is difficult for an explicit Dyson procedure to treat cases where symmetry breaking is prominent, e.g. deformed nuclei, both in terms of guaranteeing symmetry restored final states and adequate computational costs.

The generator coordinate method (GCM) can tackle symmetry restoration in both even and odd nuclei. It is also appealing for its analogy with the resonating group method [230]. Developing effective Hamiltonians has the advantage of both formal consistency within the projection procedure that is difficult to maintain using functionals and simplifying the numerical calculations, all with excellent agreement with experimental structure observables [231]. The GCM has been used to calculate scattering properties of stable and exotic nuclei in several cases [230, 232, 233]. There is further work ongoing in connecting the microscopic structure description in the GCM and the reaction observables in the form of the construction of microscopic optical potential for deformed nuclei.

As discussed in previous sections, the irreducible self-energy Σ^⋆(E) generalizes the exact nucleon–nucleus optical potential of Feshbach to include both bound and scattering states [8, 23, 234]. Diagonalizing the self-energy leads to the one-body Green's function, also known as the propagator (see equation (12)). Hence, many-body Green's function theory provides a well-grounded connection between structure and reactions [21] and it enables the direct computations of the self-energy based on the best accurate ab initio methods. We present the state-of-the-art frontiers and challenges of the Green's function approach in the following.

Ab initio computations of the self-energy for finite nuclei can be approached in two ways: (a) either by direct application of propagator theory to calculate its Feynman diagram expansion, as done in the Self- consistent Green's Function (SCGF) [235–237]; or (b) inverting the propagators computed using an ab initio wave function approach, as done in the coupled-cluster method (CCM) [238], the No-Core Shell Model (NCSM), and the Symmetry-Adapted No-Core Shell Model (SA-NCSM) [239]. Ab initio methods can construct the optical potential from chiral interactions, or other effective field theory (EFT) forces so that they provide a direct link to the underlying symmetry and symmetry-breaking patterns of quantum chromodynamics. They also provide a systematic approach to quantify theoretical uncertainties arising from the nuclear force and the controlled many-body approximations. Typical calculations for finite nuclei involve large but truncated model spaces that lead to a discretization of the scattering continuum. In most cases, the greatest challenges relate to dealing with such discretization and to handling a large number of degrees of freedom needed to resolve the dynamics at several scattering energies [240, 241]; a very demanding task if compared to typical successful ab initio computations of low-energy nuclear structure.

Direct computations of the self-energy. The SCGF approach involves computing a converged series of Feynman diagrams based on prescriptions that are aimed at preserving conservation laws. Modern nuclear physics applications exploit the Nambu–Gorkov formulation to include pairing in spherical open-shell nuclei [242] and follow the algebraic diagrammatic construction (ADC) technique to devise a systematically improvable hierarchy of many-body truncations [237]. Up to third order, or ADC(3), accurate ground state observables and low-energy spectroscopy are achieved for several chains of isotopes near the oxygen, calcium, nickel, and tin regions [243–246]. The Nambu–Gorkov approach has been applied only at second order for open shells nuclei and it is now being implemented to the previously unavailable ADC(3) level [247, 248].

In SCGF theory the self-energy is naturally split into a mean-field part, denoted as Σ^(∞), and a dynamic contribution, $\widetilde{{\rm{\Sigma }}}(E)$, which is energy dependent and accounts for coupling to the virtual inelastic channels that give rise to the dispersion relation (4). Exploratory SCGF computations of optical potentials are reported in [174, 236]. The Σ^(∞) from SCGF agrees qualitatively well with direct scattering computations with the no core-shell model with resonating group method (NCSM/RGM, see [1] for a recent review and references therein) when virtual excitations of the target are suppressed, as seen in [249] and illustrated in figure 8 which compares the n-¹⁶O phase shifts obtained with both methods. To reach a more predictive description of single-particle bound states and resonances, we include virtual states of the target nucleus. By including low-lying excitations of ¹⁷O, the no-core-shell model with continuum (NCSMC) accounts effectively for virtual target excitations and leads to two bound states, namely 1/2⁺ and 5/2⁺. For SCGF, virtual excitations, contained in $\widetilde{{\rm{\Sigma }}}(\omega )$, correct single-particle states and generate a large number of narrow resonances across all scattering energies. Results for low energy states are shown in [249] and there is qualitative accord with the observation.

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In the SCGF approach, the biggest challenge for an ab initio theory is calculating $\widetilde{{\rm{\Sigma }}}(\omega )$. The SCGF-ADC(3) construction contains correlations from all two-particles one-hole (2p1h) and one-particle two-holes (1p2h) configurations and has a direct impact on the absorption of the optical model. This is demonstrated for elastic neutron scattering off ¹⁶O by the right-hand side of figure 8, where all 2p1h doorway states contribute to the solid blue line and are then gradually frozen until only the mean-field Σ^(∞) remains (dashed green line) [249]. The 2p1h states become insufficient already at intermediate energies, where more complex configurations must enter into play. Truncations well beyond third order, ADC(n) with n ≫ 3, will most likely resolve this problem but will require groundbreaking advances in nuclear many-body theory to automatically generate and efficiently sample the exponentially growing number of diagrams.

Inversion of propagators using ab initio wave functions. The indirect approach first evaluates the one-body Green's function in configuration space,

$$\begin{eqnarray}&&{G}_{\alpha \beta }(E)=\lt{{\rm{\Phi }}}_{0}\left|{a}_{\alpha }\frac{1}{E-({ \mathcal H }-{\epsilon }_{0})+{\rm{i}}\eta }{a}_{\beta }^{\dagger }\right|{{\rm{\Phi }}}_{0}\gt+\,\lt{{\rm{\Phi }}}_{0}\left|{a}_{\beta }^{\dagger }\frac{1}{E+({ \mathcal H }-{\epsilon }_{0})-{\rm{i}}\eta }{a}_{\alpha }\right|{{\rm{\Phi }}}_{0}\gt.\end{eqnarray} \tag{ 19 }$$

where α and β label the single-particle states and with η → 0⁺. One then inverts equation (12) for each scattering energy E to calculate the optical potential. To compute G_{α β} (E), the completeness of eigenstates $\left|{{\rm{\Psi }}}^{A\pm 1}\right\rangle$ can be used, which requires computationally intensive calculations of many eigenstates, but in practice, the inverse Hamiltonian operator in equation (19) is evaluated using one of a few available Lanczos algorithm methods [240, 250] (see also [237, 251, 252] and references therein). Note that if one evaluates equation (19) in the Lehmann representation using the completeness over $\left|{{\rm{\Psi }}}^{A\pm 1}\right\rangle$ , one is evaluating the overlap functions $\langle {{\rm{\Psi }}}_{n}^{A+1}| {a}_{\alpha }^{\dagger }| {{\rm{\Phi }}}_{0}\rangle$ that are in fact solutions. In this case, the approach is equivalent to computing a discretized set of scattering waves and then solving an inverse scattering problem.

The viability of the propagator inversion scheme was demonstrated for oxygen and calcium isotopes using the particle attached and removed CCM, as discussed in [238] and is being applied within NCSM frameworks [239, 253]. The SA-NCSM provides useful features for nucleon–nucleus scattering such as its suitability describing deformation. Ab initio descriptions of spherical and deformed nuclei up through the calcium region are now possible in the SA-NCSM [254–256] without the use of interaction renormalization procedures and effective charges. It has also been shown that the SA-NCSM can use significantly reduced model spaces as compared to the corresponding ultra-large conventional NCSM model spaces without compromising the accuracy of results for various observables. This allows the SA-NCSM to accommodate larger model spaces needed for clustering, collective, and continuum degrees of freedom, and to reach heavier nuclei such as ²⁰Ne [255, 257], ²¹Mg [258], ²²Mg [259], ²⁸Mg [260], as well as ³²Ne and ⁴⁸Ti [261]. Moreover, the construction of self-energies for light nuclei starting from the NCSM/RGM and its extension to the NCSMC are also currently being developed and are of interest as scattering states are included explicitly in the many-body basis. As the NCSMC reproduces low-energy scattering and bound-state observables [262–269] for light nuclei, the optical potentials derived within this theory, along with the reach of the symmetry-adapted RGM (SA-RGM) to intermediate-mass nuclei [256, 270], are expected to be accurate in a similar range of energies and masses.

Challenges and opportunities. Computationally, the most demanding task is the accurate evaluation of the self-energy for all relevant scattering energies. Methods that scale polynomially with the mass number, such as SCGF and CCM, are presently limited to simple excitations (e.g. 2p1h and 2h1p) throughout the energy range and converge with respect to the model space up to ≈160 MeV [174, 271]. However, more complex configurations that are important at intermediate energies are missing (see figure 8). The NCSM family of methods is complementary and it performs a truncation based on the number of HO excitations, which has two advantages. First, correlated multiple particle–hole configurations are well included at low energies (e.g. SA-NCSM can capture giant resonances, which is important to describe scattering from 1 to 15 MeV per nucleon) but high scattering energies may pose a challenge. Second, it ensures the exact separation of the motion of the center of mass.

Indeed, calculations of the one-body Green's function that use laboratory coordinates may pose issues due to spurious center-of-mass motions in both the target and the A ± 1 systems. Model spaces in typical ab initio computations are sufficiently large to decouple the intrinsic and the center of mass wave functions [272]. Moreover, this separation is even exact for NCSM calculations with HO-excitation truncations [273]. Nevertheless, the zero point motion of the center of mass is still present and it can be a source of spuriosity. Johnson discussed the technical difficulties with expressing a self-energy in a pure laboratory system [274, 275] and suggested that a proper optical potential theory should be expressed in Jacobi coordinates. Center of mass corrections are seen to be sizeable for light nuclei, such as ¹⁶O but become quickly negligible at larger masses where the self-energy approaches the one in the laboratory frame. We note that the NCSM/RGM and NCSMC methods routinely compute scattering among observables by handling these center-of-mass corrections [1]. A similar development could also be valuable for reformulating the SCGF self-energy in proper relative projectile-target coordinates.

A related technical question is the discretization of the scattering spectra due to the finite model spaces. Both the Green's function and the optical potential (i.e. the self-energy) develop a real and an imaginary part in the continuum. For practical applications, the finite size of the model space implies a set of discrete poles both in equation (19) and in the spectral representation for Σ^⋆(E). The correct continuum spectrum is recovered only taking the limit η → 0 while at the same time letting the density of intermediate states diverge (as per the complete set of configurations in an infinite model space). Ideally, one would like to use a finite η to impose a width as big as the distance between two neighboring levels, and check that predictions for observables are unaffected by variations of η around such central value. Note that the technical issue of handling the η → 0 limit is more compelling for the inversion propagator approaches, since the diverging poles in equation (19) can lead to instabilities in the inversion process. This has been studied with the CCM method using Berggren bases with the continuum [276, 277]. For all cases, however, it should be clear that the choice for η sets the energy resolution of the optical potential being computed. A higher resolution requires a higher density of intermediate states, posing stronger demand on the ab initio method being employed.

To conclude, constructing the self-energy starting from microscopic computations with a single realistic Hamiltonian allows for a consistent description of the target structure and reaction dynamics, to derive a nucleon–nucleus optical potential and calculate elastic scattering observables. Contrary to phenomenological approaches whose applicability is limited by the reliability of extrapolations, microscopic optical potentials rely only on the knowledge of nuclear interactions and can be built in principle for any nucleus accessible by the theory. Such nucleon–nucleus optical potentials could be also used for microscopic descriptions of (d, p) and (d, n) [42].

Although elegant and with controlled approximations, ab initio methods are computationally intensive, they require suitable approximations and still have to face important challenges. In all cases, the quality of constructed microscopic potentials will reflect the current status of high performance computing resources and the accuracy of the many-body approach used. In particular, the coupling to possible intermediate states strongly influences the absorption from the elastic channel, i.e. the magnitude of the elastic scattering cross section. Moreover, the diffraction pattern, i.e. the position of the minima in the elastic-scattering angular distribution, is determined by the root mean square radius of the target posing important requirements on the quality of the realistic nuclear force used. The most compelling issue is to advance ab initio methods to reach complete and stable description of intermediate configurations. Novel approaches as those discussed in [278, 279] will be key to reach full predictive power at medium energies.

The theoretical approach to the elastic scattering of a nucleon from a nuclear target pioneered by Watson [17, 280], made familiar by Kerman, McManus, and Thaler (KMT) [18], and further developed as spectator expansion of multiple scattering theory [281–284] is receiving renewed interest as an approach to the optical potential that can combine advances in nuclear structure with, e.g. chiral nucleon–nucleon (NN) interactions. A theoretical motivation for the spectator expansion derives from our present inability to calculate the full many-body problem when the projectile energy exceeds about 40–50 MeV (see sections 3.2.1 and 3.2.2). In this case, an expansion is constructed within a multiple scattering theory assuming that two-body interactions between the projectile and one of the nucleons in the target nucleus play the dominant role. In the spectator expansion the leading (first) order term involves two-body interactions between the projectile and one of the target nucleons, the next-to-leading (second) order term involves the projectile interacting with two of the target nucleons, and so forth. Hence, this expansion derives its ordering from the number of target nucleons interacting directly with the projectile, while the residual target nucleus remains 'passive'. Due to the many-body nature of the free propagator for the nucleon-target system, there is an additional aspect to consider in the ordering of the spectator series. The expansion of chiral NN forces not only leads to two-body forces but naturally introduces three-body forces at next-to-next-to-leading order. The latter will not contribute to the leading order in the spectator expansion. The calculation of an optical potential relies on basic input quantities. For the leading order those are fully-off-shell NN amplitudes (or t-matrices), representing the current understanding of the NN force, and fully-off-shell one-body density matrices representing the current understanding of the ground state of the target nucleus. For any higher order, additional input like 3N amplitudes and two-body density matrices for the target will be needed.

The standard approach to elastic scattering of a strongly interacting projectile from a target of A-particles is the separation of the Lippmann–Schwinger equation for the transition amplitude T

$$\begin{eqnarray}&&T=V+{{VG}}_{0}(E)T\end{eqnarray} \tag{ 20 }$$

into two parts, namely an integral equation for T

$$\begin{eqnarray}&&T=U+{{UG}}_{0}(E){PT},\end{eqnarray} \tag{ 21 }$$

where U is the optical potential operator defined by a second integral equation

$$\begin{eqnarray}&&U=V+{{VG}}_{0}(E){QU}.\end{eqnarray} \tag{ 22 }$$

In the above equations the operator $V={\sum }_{i=1}^{A}{v}_{0i}$ consists of the two-body NN potential v_0i acting between the projectile and the ith target nucleon. The free propagator G₀(E) for the projectile-target system is given by

$$\begin{eqnarray}&&{G}_{0}(E)=\frac{1}{E-{H}_{0}+{\rm{i}}\eta }\end{eqnarray} \tag{ 23 }$$

with η → 0⁺. Though most applications use targets with 0⁺ ground states, there is no need for this to be the case [285]. In fact, to develop optical potentials valid for exotic nuclei, a variety of targets with different ground state spin configurations will need to be considered.

The operators P and Q in equations (20) and (21) are projection operators with P + Q = 1, and P being defined such that equation (21) becomes a one-body equation. In this case, P is conventionally taken to project on the elastic channel, such that [G₀, P] = 0, and is defined as P = ∣Φ₀〉〈Φ₀∣/〈Φ₀∣Φ₀〉. With these definitions, the transition operator for elastic scattering can be defined as T_el = PTP, in which case equation (21) can be written as

$$\begin{eqnarray}&&{T}_{{el}}={PUP}+{{PUPG}}_{0}(E){T}_{{el}}.\end{eqnarray} \tag{ 24 }$$

The choice of the projector P fixes the scattering problem to be considered. A projection onto the target ground state is the appropriate choice to derive an optical potential describing the elastic scattering of a nucleon from a target nucleus, but when considering e.g. inelastic scattering in a coupled-channel approach, this P-space should contain the excited states under consideration. To our knowledge, this has not been attempted in a multiple scattering approach.

The expression for the optical potential in equation (22) contains the projection operator Q and thus, even in the leading order term where U is defined as $U={\sum }_{i=1}^{A}{\tau }_{0i}$, the quantity τ_0i cannot readily be identified with a NN amplitude derived in free space. Working in momentum space, it is straightforward to formulate an integral equation for the Watson optical potential [286]

$$\begin{eqnarray}&&{\tau }_{0i}={v}_{0i}+{v}_{0i}{G}_{0}(E)Q{\tau }_{0i}={\hat{\tau }}_{0i}-{\hat{\tau }}_{0i}{G}_{0}(E)P{\tau }_{0i},\end{eqnarray} \tag{ 25 }$$

where ${\hat{\tau }}_{0i}$ is the NN t-matrix given as a solution of a regular two-body Lippmann–Schwinger equation, in which only the many-body Green's function G₀(E) needs to be considered. The standard impulse approximation turns this Green's function into a two-body propagator. It should be noted that the above equations follow in a straightforward derivation and correspond to the first-order Watson scattering expansion [17, 280]. The integration of equation (25) taking into account contributions from the Q space corresponds to an averaging over inelastic channels and thus should only be applied for energies higher than ∼30–40 MeV. Unfortunately, a similar formulation as in equation (25) cannot be made in coordinate space. Here the closest to treating the operator Q is the averaging suggestion made by Kerman, McManus, and Thaler [18] leading to the KMT factor (A-1)/A in the optical potential. [286] showed that the explicit treatment of the operator Q is especially important for scattering from very light nuclei, where the KMT factor is not close to one. The importance of an explicit treatment of Q for nuclei far off the valley of stability needs to be explored. Studies of reaction cross sections of the helium isotopes at energies below 100 MeV revealed that treating Q exactly or via KMT did not lead to major differences: however, not treating Q at all caused discrepancies of more than 10% in the reaction cross section.

A further equally important consideration for obtaining the optical potential is to find a solution to equation (22), which still has a many-body character due to the propagator G₀(E). The standard impulse approximation assumes closure, i.e. ignores target excitations. For projectile energies above ∼80 MeV this is generally assumed to be a good approximation and errors have not been studied yet. In the impulse approximation and at leading order, the nucleon–nucleus optical potential for a certain kinetic energy E is given by equation (6). Let us now discuss a possible extension of this formalism to include effects of the antisymmetrization and to go beyond the impulse approximation.

The treatment of Pauli antisymmetry effects follows the philosophy growing out of the early work of Watson [287, 288] and developed via the spectator expansion in [289]. In the lowest order the two-body antisymmetry is achieved through the use of two-body t-matrices which are themselves antisymmetric in the two 'active' variables (corresponding to the weak binding limit in [288]).

Going beyond the impulse approximation in the spirit of the spectator expansion means consider that more nucleons of the target are active. At the next order in the expansion, one needs to consider that two nucleons of the target i and j are active. The optical potential would depend on propagators ${{ \mathcal G }}_{0j}$ that have the structure of three-body channel Green's functions. One can describe these propagators within a single particle description as

$$\begin{eqnarray}&&{{ \mathcal G }}_{0j}(E)=\displaystyle \frac{1}{E-{h}_{0}-{h}_{j}-\displaystyle \sum _{i\ne j}{v}_{{ij}}-{H}^{j}+i\eta },\end{eqnarray} \tag{ 26 }$$

where η → 0⁺, H^j is the residual target Hamiltonian involving (A − 1) particles (excluding particles 0 and j), h_j is the kinetic energy operator for nucleon j, and v_ij is the interaction between target nucleons i and j. One can then project this Green's function onto a fixed number of eigenstates of the residual target Hamiltonian H^j . Due to the presence of v_ij an exact solution will require fully-off-shell two-body density matrices for the target nucleus as well as three-body dynamics. Two-body density matrices from ab initio structure models are calculable in principle, so there is an opportunity to consistently estimate the contribution of the next order in the spectator expansion, and thus have a better understanding of its convergence as function of projectile energy as well as mass number. This will be a very challenging enterprise. A first attempt with nuclear densities derived from HFB mean field calculations for heavier nuclei was made in [290, 291] where the interaction v_ij was taken as the corresponding nuclear mean field. The result of this study showed that at projectile energies above 100 MeV the second-order correction is almost negligible, while starting to be evident in the spin observables at energies below 100 MeV. At about 50 MeV, the second-order correction is quite visible in the differential cross section.

Another approach to take into account the beyond-leading-order effects of three-body forces was implemented in [292] by constructing a density-dependent NN interaction that treats the 3N force in an approximate way [293, 294]. For energies above 100 MeV, few effects were noticeable in the differential cross sections, though some effects were observed in the analyzing power and spin rotation function.

Explicit calculation of the leading order term in the Watson approach. For explicit calculations of reaction observables from the leading order term in the spectator expansion, equation (6), one needs both, structure information (fully-off-shell one-body density matrices) and reaction information (NN amplitudes). Current ab initio calculations of multiple scattering theory are limited in their applicable mass range due to the available ab initio structure inputs. To reach target nuclei beyond the A ∼ 40 range, one-body density matrices will need to be calculated from other structure models, e.g. SCGF, CCM and In-Medium Similarity Renormalization Group (IM-SRG).

Recent work [295] showed that including the spin of the struck target nucleon has an effect on the elastic scattering spin observables for neutron-rich systems, which implies consistent calculations incorporating this term may be necessary to study nuclei off the valley of stability. This additional term requires both, a scalar and spin-dependent one-body density matrix, and guarantees that the scalar (Wolfenstein A), vector (Wolfenstein C), and tensor (Wolfenstein M, G, H, and D) parts of the NN interaction are included ²¹ . For J = 0 to J = 0 transitions, it has been shown that only A, C, and M contribute due to parity invariance, though this likely holds for other transitions between the same spin states. Future work to develop ab initio treatments of inelastic scattering in this framework are becoming possible and will allow for further study of the tensor (Wolfenstein M, G, H, and D) parts of the NN interaction.

In addition to the theoretical uncertainties arising from the spectator expansion of the multiple scattering theory (e.g. next-to-leading-order effects for three-body forces, as well as energy-dependencies of the NN t-matrix [298])—which are expected to be small in the energy regime above 60 MeV—there are additional theoretical uncertainties propagating from the nuclear-structure calculations. This includes both model-related uncertainties (e.g. any residual dependence on the size of the model space), as well as uncertainties from the underlying NN interaction (e.g. uncertainties associated with truncating a chiral effective field theory at a certain order, dependence on the fits of the low-energy constants (LECs), and others). Different approaches can be taken to address each of these uncertainties, but an accounting of them is necessary to extract reliable information about reaction observables.

Explicit calculation of the leading order term in the g-matrix approach. This approach starts directly from the general expression for the leading-order term of equation (6) and realizes that from quite general considerations [299] the two-body (NN) amplitude ${\hat{\tau }}_{\alpha }$ can be recast as

$$\begin{eqnarray}&&\langle \vec{k}^{\prime} \vec{p}^{\prime} | {\hat{\tau }}_{\alpha }(E)| \vec{k}\ \vec{p}\rangle =\int \ \frac{{\rm{d}}\vec{z}}{{\left(2\pi \right)}^{3}}\ {{\rm{e}}}^{{\rm{i}}\vec{z}\cdot (\vec{W}^{\prime} -\vec{W})}\ {g}_{\vec{z}}\left[\frac{1}{2}(\vec{W}^{\prime} +\vec{W});\vec{b}^{\prime} ,\vec{b}\right],\end{eqnarray} \tag{ 27 }$$

where ${g}_{\vec{z}}$ represents a reduced interaction at the local coordinate $\vec{z}$. In this expression $\vec{W}=\vec{k}+\vec{p}$, and $\vec{b}=(\vec{k}-\vec{p})/2$, the prior total and relative two-body momenta, respectively. The same applies to the post momenta, denoted by primed marks. Additionally, the $\vec{z}$ coordinate is given by the average $\vec{z}=(\vec{r}^{\prime} +\vec{s}^{\prime} +\vec{r}+\vec{s})/4$, the center of gravity of the four coordinates of the two-particles, $\vec{r}$, $\vec{r}^{\prime}$, $\vec{s}$, $\vec{s}^{\prime}$.

Assuming a density-dependent NN effective interaction, in [299] it is demonstrated quite generally that the folding potential in momentum space can be expressed as the sum of two terms,

$$\begin{eqnarray}&&U(\vec{k}^{\prime} ,\vec{k};E)=\displaystyle \sum _{\alpha }\int {\rm{d}}\vec{P}\ {\hat{\rho }}_{\alpha }(\vec{q};\vec{P})\ {\hat{\tau }}_{0}(E+{\epsilon }_{\alpha })+{U}_{1}(\vec{k}^{\prime} ,\vec{k};E),\end{eqnarray} \tag{ 28 }$$

where

$$\begin{eqnarray}&&\begin{array}{l}{U}_{1}(\vec{k}^{\prime} ,\vec{k};E)=\\ \quad -\displaystyle \sum _{\alpha }{\int }_{0}^{\infty }\,{\rm{d}}z\,\frac{4\pi {z}^{3}}{3}\int \frac{{\rm{d}}\vec{P}}{{\left(2\pi \right)}^{3}}\int {\rm{d}}\vec{q}^{\prime} \,{\hat{\unicode{x00237}}}_{1}(z| \vec{q}^{\prime} -\vec{q}| )\ {\rho }_{\alpha }(\vec{q}^{\prime} ;\vec{P}){\partial }_{z}g({\rho }_{z},E+{\epsilon }_{\alpha }).\end{array}\end{eqnarray} \tag{ 29 }$$

Here ${\hat{\unicode{x00237}}}_{1}(x)=3{j}_{1}(x)/x$, with j₁(x) being the spherical Bessel function of order 1 and ${\rho }_{\alpha }(\vec{q};\vec{P})={\varphi }_{\alpha }^{\dagger }(\vec{P}+\tfrac{1}{2}\vec{q}){\varphi }_{\alpha }(\vec{P}-\tfrac{1}{2}\vec{q})$, with φ_α the target single-particle wave function with energy _α . Note that in the sum only occupied states should be considered. While ${\hat{\tau }}_{0}$ represents the momentum-space free t-matrix (as present in the KMT term of the optical potential [300–302]), the fully off-shell g-matrix can be modeled with the infinite nuclear matter Brueckner–Hartree–Fock g-matrix. Assuming weak isospin asymmetry, the gradient term ∂_z g = [∂g/∂ρ][∂ρ/∂z], is evaluated at a local isoscalar density ρ_z . The resulting nonlocal potential $U(\vec{k}^{\prime} ,\vec{k};E)$ has been applied to nucleon elastic scattering, as reported in [303, 304] An interesting interpretation of equation (29) for U₁ is that intrinsic medium effects take place mostly at the surface of the target, as modulated by ∂_z ρ, the gradient of the density [303].

Finally, let us emphasize that there are other similar approaches which construct in coordinate space nucleon- and nucleus-optical potentials, folding microscopic neutron and proton densities with nucleon–nucleon effective interactions [305–312].

Compared to finite nuclei, infinite homogeneous nuclear matter represents a conceptually simpler physical system to study. In particular, calculations of the nucleon optical potential in nuclear matter avoid many of the technical difficulties and practical limitations faced when computing the nucleon optical potential directly in a finite system. When combined with a local density approximation, the nuclear matter approach can also be used to construct nucleon–nucleus optical potentials, provided that the isoscalar and isovector densities of the target nuclei are known. A significant advantage is that the nucleon optical potential in nuclear matter only needs to be computed once over a wide range of densities and proton fractions and then may be applied across large regions of the nuclear chart. Hence, the nuclear matter approach is naturally suited for the construction of global optical potentials, which will be vital for the future of reaction theory for rare isotopes. However, the assumptions of the nuclear matter approach that allow for the ease of constructing global nucleon–nucleus optical potentials also omit phenomena such as surface effects, resonances, and spin–orbit interactions. The nuclear matter approach also tends to produce an overly absorptive imaginary term at high energies. Some of these shortcomings may be straightforwardly remedied while for others the solution remains unclear, for more details see [313]. Ultimately, the quality of theoretical predictions for reaction cross sections from optical potentials derived within the nuclear matter approach must be assessed by comparisons to experimental data.

The framework for utilizing nuclear matter calculations of the optical potential for finite nuclei was built by Jeukenne, Lejeune and Mahaux in the late 1970s [113, 114]. They implemented the Local Density Approximation (LDA)

$$\begin{eqnarray}\begin{array}{ccl}U{\left(E;r\right)}_{\mathrm{LDA}} & = & V{\left(E;r\right)}_{\mathrm{LDA}}+{iW}{\left(E;r\right)}_{\mathrm{LDA}}\\ & = & V{\left(E;{k}_{f}^{p}(r),{k}_{f}^{n}(r)\right)}_{\mathrm{NM}}+{iW}{\left(E;{k}_{f}^{p}(r),{k}_{f}^{n}(r)\right)}_{\mathrm{NM}}^{\prime} \end{array}\end{eqnarray} \tag{ 30 }$$

which relates the optical potential at a given position in the nucleus with the optical potential of nuclear matter (denoted by NM) with the same local density and isospin asymmetry through the neutron ${k}_{f}^{n}(r)$ and proton ${k}_{f}^{p}(r)$ Fermi momenta. A key finding of [113] is that the LDA is insufficient for reproducing elastic-scattering data, which requires a modification called the Improved Local Density Approximation (ILDA) that takes into account the nonzero-range of the nuclear force

$$\begin{eqnarray}&&U{\left(E;r\right)}_{\mathrm{ILDA}}=\frac{1}{{\left(t\sqrt{\pi }\right)}^{3}}\int U{\left(E;r^{\prime} \right)}_{\mathrm{LDA}}{{\rm{e}}}^{\tfrac{-| \vec{r}-\vec{r}^{\prime} {| }^{2}}{{t}^{2}}}{{\rm{d}}}^{3}r^{\prime} .\end{eqnarray} \tag{ 31 }$$

The ILDA introduces a Gaussian smearing of the optical potential over the range of densities probed across the length scale t, typically chosen to be around t ∼ 1.2 fm, the effective range of the nuclear force. The most important consequence is that the optical potential in the interior of the nucleus changes little, while the surface diffuseness of the optical potential increases due to finite-range effects.

Whitehead–Lim–Holt global optical potential. Recent advances [314, 315] in the nuclear matter approach to constructing microscopic nucleon–nucleus optical potentials incorporate consistent two-body and three-body forces [316] at various orders in the chiral expansion. The nucleon self-energy in nuclear matter is calculated in the framework of many-body perturbation theory (MBPT) [317, 318], which has already been used to produce accurate models of the nuclear equation of state [319–323]. In addition to MBPT, there are other many-body frameworks for microscopically calculating the self-energy. One notable example is the work of Rios in SCGF theory [324].

In MBPT, the first-order (or Hartree–Fock) contribution to the nucleon self energy in isospin-symmetric nuclear matter is given by

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{2N}^{(1)}(k)=\displaystyle \sum _{1}\langle \vec{k}\ {\vec{h}}_{1}{{ss}}_{1}{{tt}}_{1}| {\bar{V}}_{2N}| \vec{k}\ {\vec{h}}_{1}{{ss}}_{1}{{tt}}_{1}\rangle {n}_{1},\end{eqnarray} \tag{ 32 }$$

where $\vec{k},s,t$ are the momentum, spin, and isospin of the projectile, n₁ is the occupation probability θ(k_f − h₁) for a filled state with momentum ${\vec{h}}_{1}$ below the Fermi surface, and the summation is over intermediate-state momenta ${\vec{h}}_{1}$, spins s₁, and isospins t₁.

The second-order perturbative contributions to the nucleon self energy in symmetric nuclear matter are expressed as

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{2N}^{(2a)}(k;E)=\displaystyle \frac{1}{2}\displaystyle \sum _{123}\displaystyle \frac{| \langle {\vec{p}}_{1}{\vec{p}}_{3}{s}_{1}{s}_{3}{t}_{1}{t}_{3}| {\bar{V}}_{2N}| \vec{k}{\vec{h}}_{2}{{ss}}_{2}{{tt}}_{2}\rangle {| }^{2}}{E+{\epsilon }_{2}-{\epsilon }_{1}-{\epsilon }_{3}+i\eta }{\bar{n}}_{1}{n}_{2}{\bar{n}}_{3},\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{2N}^{(2b)}(k;E)=\displaystyle \frac{1}{2}\displaystyle \sum _{123}\displaystyle \frac{| \langle {\vec{h}}_{1}{\vec{h}}_{3}{s}_{1}{s}_{3}{t}_{1}{t}_{3}| {\bar{V}}_{2N}| \vec{k}{\vec{p}}_{2}{{ss}}_{2}{{tt}}_{2}\rangle {| }^{2}}{E+{\epsilon }_{2}-{\epsilon }_{1}-{\epsilon }_{3}-i\eta }{n}_{1}{\bar{n}}_{2}{n}_{3},\end{eqnarray} \tag{ 34 }$$

where the occupation probability for particle states above the Fermi momentum is ${\bar{n}}_{i}=\theta ({k}_{i}-{k}_{f})$. The second-order contributions Σ^(2a) and Σ^(2b) are energy-dependent and complex. In equations (33) and (34), the single-particle energies E and should be computed self-consistently according to

$$\begin{eqnarray}&&E(k)=\displaystyle \frac{{k}^{2}}{2M}+\mathrm{Re}\,{\rm{\Sigma }}(k;E(k)).\end{eqnarray} \tag{ 35 }$$

The use of chiral nuclear forces provides several advantages to the phenomenological nuclear forces of the past. By virtue of being an effective field theory, chiral makes a concrete connection to the underlying theory of quantum chromodynamics through its symmetries. Furthermore, chiral nuclear forces are calculated in a perturbative expansion that allows for a direct method of uncertainty quantification by assessing order-by-order convergence [325].

The wide applicability of the nuclear matter approach and uncertainty quantification capabilities of chiral EFT were employed in [326] to construct both the first microscopic global optical potential and the first global optical potential with uncertainty quantification. Both of these advances are central to the development of reaction theory for the rare-isotope beam era, where theoretical predictions for thousands of exotic isotopes will be needed to drive scientific discovery and answer fundamental science questions in nuclear astrophysics. Present microscopic optical potentials have sizable uncertainties, which may be reduced within a Bayesian framework that incorporates experimental nucleon–nucleus scattering and reaction data in Bayesian likelihood functions. In [326], five separate global optical potentials were generated from a set of chiral potentials of different order in the chiral expansion and with varied momentum-space cutoffs. These global optical potentials are expressed in terms of Woods–Saxon functions that are parametrized in terms of energy, target mass, and target isospin asymmetry (E, A, δ). Assuming the five optical potentials are drawn from a multivariate Gaussian distribution in the space of optical potential parameters, one can then propagate statistical uncertainties to scattering observables. This multivariate Gaussian distribution of nucleon–nucleus optical potentials is referred to as the Whitehead–Lim–Holt (WLH) global optical potential.

In figure 9, the real (left panel) and imaginary (right panel) terms of the WLH optical potential for n+⁴⁰Ca at E = 5 MeV and E = 100 MeV are shown in coordinate space. As the energy increases, the real term decreases in depth while the imaginary term increases in depth. Both terms have larger uncertainties as the energy increases reflecting the enlarging uncertainties of chiral EFT at high energies. These potentials can be easily applied in open source reaction codes to propagate uncertainties to reaction observables. The WLH global optical potential yields good reproductions of experimental elastic scattering cross sections for projectile energies of E ≲ 150 MeV. Total reaction cross sections calculated from WLH are in good agreement with data up to moderate energies, and then overestimate data at larger energies due to an overly absorptive imaginary term. Beyond energies of E ≲ 150 MeV, predictions of the WLH optical potential are expected to have greater discrepancies with data along with larger uncertainties. Predictions of WLH for a wide range of reactions are shown in section 5.

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G-matrix solutions of the Brueckner–Bethe–Goldstone. The nuclear matter approach also gives access to direct inelastic scattering observables. In this case, the effective interaction used to build the microscopic optical potential also serves to build the transition potentials that enter the definition of the relevant Distorted Wave Born Approximation (DWBA) or coupled-channels equations. For instance, nucleon elastic and inelastic scattering were modeled from g-matrix solutions of the Brueckner–Bethe–Goldstone equations in nuclear matter—two well-known examples are the Melbourne [327] and the Santiago g-matrices [328]—and one-body density matrices through the calculation of non-local optical and transition potentials (an example of the application to inelastic scattering with the Melbourne g-matrix and RPA beyond mean-field approach is given in [329, 330]).

JLM folding potential. As these approaches have proven less suited at incident energy below 30–50 MeV, one pragmatic solution to cover the missing low-energy range, quite important for energy applications and experimental programs at RIBs, is to still rely on effective interactions derived from nuclear matter calculations but which are slightly renormalized to account for selected scattering observables. One such approach is the JLM folding model mentioned above, which has been extensively used to describe elastic and inelastic scattering of protons, neutrons, and composite particles within the double folding method, for both spherical and deformed targets. A global Lane-consistent parametrization of the JLM interaction was given by Bauge et al in 2001 [98] by adjusting the interaction to reproduce many elastic scattering and charge-exchange observables between 1 keV and 200 MeV. Many reactions were studied with this parametrization starting with HFB ground state densities and transition densities from the QRPA nuclear structure calculations. Recent examples are the determination of inelastic scattering to discrete states and to the continuum for neutron scattering below 30 MeV off spherical [331], and axially-deformed targets such as actinides within the coupled-channel framework [332], as well as the modeling of proton inelastic scattering off unstable targets [333]. The method's ability to provide accurate reaction observables is mostly related to the quality of the nuclear structure input, so it was intensively used to challenge structure theory with hadron scattering observables. Despite its phenomenological content, the method has displayed good predictive capabilities especially for direct inelastic scattering, as no inelastic observables are used to constrain the interaction. However, its phenomenological aspect makes the method's precision hard to improve beyond the use of better nuclear structure input, and it relies on simplified nuclear matter calculations with old-fashioned bare interactions. Moreover, resulting potentials from the JLM folding model are local and non-dispersive, while the optical potential is known to be non-local and to obey dispersion relations. The spin–orbit component is ad hoc—it does not stem from an underlying nuclear matter calculation—and uses a simple form factor given as a derivative of the microscopic density. This approach could thus be revisited starting from modern nuclear matter calculations such as those described above.

One aspect of inelastic scattering that deserves attention is the rearrangement correction in [334], which has a large renormalization effect on inelastic cross sections [335]. This correction, which stems from the density dependence of the effective interaction used for inelastic scattering, is still now applied in an ad hoc manner when folding models are used. We stress that this correction, which has been known for a long time to induce modifications as large as the difference between the t- and the g-matrix [334], should be described from more fundamental principles in order to reach a better description of inelastic scattering within the microscopic framework of folding and full-folding approaches.