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 ^{132}Sn nucleus exhibit similar correlations as those in ^{208}Pb
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 g_{9/2} 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 ^{40}Ca 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 1s_{1/2} and 0.76 for the 0d_{3/2} 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 ^{40}Ca 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).