Dispersive Optical Model (DOM) and Exotic Nuclei

A long-standing collaboration between the experimental faculty in radiochemistry (Bob Charity and Lee Sobotka) and our group aims to investigate the role of correlations in nuclei beyond the mean field as a function of nucleon asymmetry (δ=N-Z/A). These efforts were initiated in Phys. Rev. Lett. 97, 162503 (2006) (see Publ.105), Phys. Rev. C76, 044314 (2007) (see Publ.108), and were continued in Phys. Rev. C83, 064605 (2011) (see Publ.119). The dispersive optical model (DOM) was originally developed by Mahaux and Sartor and provides an excellent framework to connect and analyze elastic nuclear reactions and nuclear structure data that can be represented by the single-particle Green's function. The DOM utilizes a subtracted dispersion relation linking negative and positive energy domains with emphasis on the physics near the Fermi energy (the subtraction point) but extending to 200 MeV scattering energies above and the complete domain of negative energies.

In the original version with local potentials, it requires functional forms for local absorptive potentials at positive and negative energies, as well as a real potential at the Fermi energy referred to by Mahaux and Sartor as the Hartree-Fock (HF) potential because its functional form can be linked to results of empirical HF calculations. Since the HF potential is intrinsically nonlocal, it has in the past been transformed into a local but energy-dependent potential to facilitate the numerical effort and as such has been implemented in most DOM applications. Unfortunately, such a procedure compromises the normalization of the solutions of the Dyson equation which acts as the Schrödinger equation of a particle or hole in the medium under the influence of the DOM potential. We resolved this problem by replacing this normalization-distorting energy dependence of the local HF potential by a nonlocal and energy-indepent HF potential. The corresponding analysis was published earlier in Phys. Rev. C82, 054306 (2010) (see Publ.118). With this restoration it is possible to describe correlations beyond the mean-field in the DOM framework without resorting to the approximate expressions developed by Mahaux and Sartor. The corresponding quantities are the nucleon spectral function, one-body density matrix, natural orbits, momentum distribution, etc. The DOM can therefore properly describe ground-state properties of nuclei as a function of nucleon asymmetry in addition to standard ingredients like elastic nucleon scattering data and level structure. Predictions of nucleon correlations at larger nucleon asymmetries can then be made after data at smaller asymmetries constrain the potentials that represent the nucleon self-energy. An example is provided by the constraints provided by elastic scattering data on stable Sn nuclei. These exhibit an increasing surface absorption of protons when neutrons are added to the system (see Publ.119). A simple extrapolation for Sn isotopes beyond stability then generates predictions for increasing correlations of minority protons with increasing neutron number. Such predictions can be investigated by performing experiments with exotic beams. The predicted neutron properties for the double closed-shell 132Sn nucleus exhibit similar correlations as those in 208Pb as expressed in terms of spectroscopic factors. Further exploration of the neutron drip line in Sn nuclei clarifies that the proton spectral strength of the last occupied g9/2 orbit exhibits a sharp decline of the spectroscopic factor when the neutron drip line and the corresponding continuum is near in energy. We demonstrate however that the loss of strength in the main peak is accompanied by a shift of this strength into the nearby continuum which may be accessible experimentally. The resulting paper summarizing these results has been published in Eur. Phys. J. A50:23 (2014) (see Publ.124).

DOM potentials can also be utilized for the description of transfer reactions. The adiabatic wave approximation (ADWA) developed by Johnson and Soper in the seventies can be employed for the description of the (d,p) reaction and has the advantage that it includes the deuteron breakup channel. Furthermore, the deuteron optical potential is described by the sum of the neutron and proton optical potentials at half the incident deuteron energy. The DOM also supplies overlap functions to discrete final states with one neutron added. A collaboration between the reaction group of Filomena Nunes at NSCL/MSU and our DOM effort in St. Louis was therefore a natural development and resulted in a recent publication Phys. Rev. C84, 044611 (2011) (see Publ.121). These results show great promise for the unambiguous extraction of spectroscopic information for transfer reactions in inverse kinematics but require the consideration of nonlocal potentials as discussed below.

The properties of a nucleon that is strongly influenced by the presence of other nucleons have traditionally been studied in separate energy domains. Positive energy nucleons are described by fitted optical potentials mostly in local form. Bound nucleons have been analyzed in static potentials that lead to an independent-particle model modified by the interaction between valence nucleons as in traditional shell-model calculations. The link between nuclear reactions and nuclear structure is provided by considering these potentials as representing  different energy domains of one underlying nucleon self-energy as implemented in the DOM. So far the main stumbling block to describe ground-state properties pertaining to nuclear structure has been the need to utilize the approximate expressions for the properties of nucleons below the Fermi developed by Mahaux and Sartor. These expressions correct for the normalization-distorting energy dependence of the HF potential but do not adequately deal with the spectral properties when the absorptive potentials are substantial. As discussed above, it is possible to restore the proper treatment of nonlocality in the HF contribution, to overcome the problem of the normalization distortion. When the traditional local form of the absorptive potentials are maintained, it is however impossible to generate a good description of the nuclear charge density or even particle number. Recent work in our group established the microscopic content of the self-energy due to long-range (see Phys. Rev. C84, 034616 (2011) Publ.120) and short-range correlations (see Phys. Rev. C84, 044319 (2011) Publ.122) demonstrating that  nonlocal absorptive potentials are theoretically well-founded if not unavoidable.

We have therefore for the first time treated the nonlocality of the absorptive potentials for the nucleus 40Ca with the aim to include all available data below the Fermi energy that can be linked to the nucleon single-particle propagator while maintaining a correct description of the elastic-scattering data. The result is a DOM potential that can be interpreted as the nucleon self-energy constrained by all available experimental data up to 200 MeV. Such a self-energy allows for a consistent treatment of nuclear reactions that depend on distorted waves generated by optical potentials as well as overlap functions and their normalization for the addition and removal of nucleons to discrete final states. The re-analysis of such reactions may further improve the consistency of the extracted structure information. Extending this version of the DOM to N different from Z will allow for predictions of properties that require the simultaneous knowledge of both reaction and structure information since at present few weakly-interacting probes are available for exotic nuclei. It is in this sense that we aim at continuing to establish detailed links between the physics of the continuum and structure information below the Fermi energy because these domains are even more strongly coupled for exotic nuclei than for stable systems.

The chosen form of the nonlocal absorption adheres to the traditional treatment, i.e. a Gaussian form is chosen as suggested long ago by Perey and Buck. The description of elastic scattering data for both protons and neutrons is of the same quality as our earlier work with local potentialsdiscussed above. We also find the same quality description of total and reaction cross section for neutrons and protons, respectively. Nevertheless, the introduction of nonlocality has important consequences since it introduces an explicit orbital-angular-momentum dependence that generates very different distorted waves and therefore may generate different results when nuclear reactions are analyzed that require their knowledge. This feature already applies to the (e,e'p) reaction which has been employed to extract spectroscopic factors for the removal of valence protons. In the analysis of such reactions  the NIKHEF group has always utilized nonrelativistic local optical potentials. The Madrid group has shown that a relativistic optical potential generates spectroscopic factors that are 10-15% larger than those obtained by the NIKHEF group. Furthermore their work suggests that this is mainly due to the different treatment of nonlocality. We view this discrepancy as an essential future test of the DOM in which we plan to reanalyze these data with our nonlocal  DOM potentials. Meanwhile our results for the spectroscopic factors are consistent with those obtained with relativistic optical potentials since we obtain 0.78 for the 1s1/2 and 0.76 for the 0d3/2 protons, respectively.

The introduction of nonlocality in the absorptive potential has essential benefits for the convergence of the particle number as a function of the orbital angular momentum and brings it in line with ab initio results obtained in Phys. Rev. C84, 044319 (2011) allowing about 1-2% of the particles with orbital angular momentum larger than 5. More importantly, it is now possible to generate an accurate fit to the nuclear charge density. We have found it easier to obtain this result when we replaced the surface contribution of the HF potential by a wine bottle generating Gaussian centered at the origin in accord with similar results obtained with Green's function Monte Carlo studies of overlap functions by the Argonne group. A new constraint was introduced in the fit of properties below the Fermi energy by considering the spectral function for the removal of high-momentum protons as obtained at Jefferson Lab for Al and Fe nuclei. As these data per proton are essentially identical, we have employed them to constrain the distribution of high-momentum protons. While generating a reasonable description of these data, we obtain a modest 10.6% of the protons occupying momenta above 1.4 fm-1 in the 40Ca ground state. The presence of about 10% of high-momentum nucleons in the ground state is quite consistent with earlier ab initio work of our group and others employing different methods provided the underlying nucleon-nucleon interaction was not too soft.

Employing the energy or Koltun sum rule in the form given by Dieperink and DeForest, then yields a binding energy of 7.91 MeV/A much closer to the experimental 8.55 MeV/A than found in our earlier work  published in Phys. Rev. C82, 054306 (2010). The constrained presence of the high-momentum nucleons is responsible for this change. The 7.91 MeV/A binding obtained here represents the contribution to the ground-state energy from two-body interactions including a kinetic energy of 22.64 MeV/A and was not part of the fit. This empirical approach therefore leaves about 0.64 MeV/A attraction for higher-body interactions about 1 MeV/A  less than the Green's function Monte Carlo results of the Argonne group for light nuclei. We have published these results in Phys. Rev. Lett. 112, 162503 (2014) and posted the paper on the arXiv (1312:5209) (see
Publ.125). In addition we have deposited supplementary material there to provide the detailed parameters that are contained in the present fit (see arXiv:1312.4886).