The 2015 CEA/DAM–Santiago prospective study of the Gogny-D1S microscopic optical potential demonstrated parameter-free elastic nucleon scattering from the closed-shell nucleus 40^{40}Ca and identified, as its midterm objective, the extension to axially deformed targets. That extension requires four linked developments: an axially deformed Hartree–Fock–Bogoliubov (HFB) mean-field and pairing potential in coordinate space; target excitations from the quasiparticle random-phase approximation (QRPA) in the intrinsic frame; angular-momentum projection within the rotational approximation to construct the monopole and multipole coupling potentials; and the solution of coupled-channel equations with the resulting nonlocal, energy-dependent, complex microscopic potentials. A decade of follow-up work has produced a nonlocal coupled-channel solver only for spherical spin-0 targets, practical deformed coupled-channel calculations only with local folding potentials, and projected-generator-coordinate-method nonlocal optical potentials for deformed nuclei that still omit explicit inelastic channel coupling. A self-consistent nonlocal microscopic treatment of elastic and direct-inelastic scattering from axially deformed nuclei, validated against data, remains open.

Background

The optical-model potential (OMP) is the standard tool for describing elastic nucleon–nucleus scattering: an effective, generally complex one-body potential in which the real part refracts and the imaginary part absorbs the projectile, the latter accounting schematically for all non-elastic reaction channels. Global phenomenological OMPs such as that of Koning and Delaroche are fitted to existing data, which limits their predictive power for nuclei or energies far from the fitted region. Microscopic OMPs instead derive the potential from an effective nucleon–nucleon (NN) interaction together with a nuclear-structure calculation, so that structure and reactions are treated consistently and cross sections become a probe of the underlying interaction.

Several levels of microscopic treatment exist. Folding models based on nuclear-matter effective interactions (the Jeukenne–Lejeune–Mahaux or JLM potential, and g-matrix folding — the g-matrix being the effective in-medium NN scattering matrix, for which the Melbourne parametrization is widely used —) approximate the potential as local and density-dependent. The nuclear structure method (NSM) of Vinh Mau and collaborators goes further: the projectile self-energy (the one-body mass operator of Green's-function theory) is built from intermediate particle–hole excitations of the target — a particle–vibration-coupling mechanism — with the excited states computed in the random phase approximation (RPA, the small-amplitude collective oscillations of the closed-shell mean field) generated by the same interaction. Blanchon, Dupuis and Arellano implemented the NSM with the Gogny D1S interaction — a finite-range effective NN force widely used in nuclear structure — and showed (Phys. Rev. C 91, 014612 (2015)) that neutron and proton elastic scattering from 40^{40}Ca is reproduced up to about 30 MeV incident energy with no adjustable parameter, while neutron total cross sections are underestimated below 10 MeV, indicating missing absorption mechanisms.

Most nuclei, however, are deformed. For an axially deformed target the optical potential is not a single radial function: the projectile couples to the rotational band (the ladder of states of increasing angular momentum generated by rotations of the intrinsic deformed state), so a coupled-channel treatment is required in which elastic and inelastic channels are linked by multipole transition potentials. The self-consistent microscopic program for this case, laid out in the source paper (Eur. Phys. J. A 51, 165 (2015), §6.3), needs: (i) an axially deformed Hartree–Fock–Bogoliubov (HFB) description — mean field plus pairing field — carried out in coordinate space; (ii) intrinsic-frame QRPA (the RPA generalized to superconducting, open-shell systems) for the deformed target's excited states; (iii) angular-momentum projection — restoring the rotational symmetry that the deformed intrinsic state breaks, here within the rotational approximation — to convert intrinsic quantities into laboratory-frame monopole (diagonal) and multipole (coupling) potentials; and (iv) solution of the coupled-channel scattering equations when the potentials are nonlocal. A nonlocal potential is an integral operator acting on the scattering wave function through a kernel U(r,r′)U(r,r') in two radial coordinates, rather than a local function U(r)U(r); every NSM or g-matrix microscopic potential is intrinsically nonlocal, while standard coupled-channel codes (such as Raynal's ECIS) were built for local potentials, which is why the source paper singles out 'the problem of solving coupled equations with nonlocal potentials'. In practice, deformed nuclei are currently handled with local folding potentials of the JLMB family (the semi-microscopic, local-density-approximation (LDA) descendant of the JLM potential) folded with deformed HFB densities and QRPA transition densities, an approximation whose normalizations are phenomenological. A distinct route has recently been opened by the Lund group using the projected generator coordinate method (PGCM — a configuration mixing of symmetry-broken mean-field states with particle-number and angular-momentum restoration), which has produced genuinely nonlocal microscopic optical potentials for deformed nuclei, so far without explicit channel coupling.

Problem Statement

Extend the self-consistent microscopic optical-potential framework — of the type demonstrated for spherical closed-shell nuclei with the Gogny D1S interaction in the nuclear structure method (NSM, the particle–vibration-coupling self-energy construction of Vinh Mau and collaborators) — to axially deformed target nuclei, and use it to predict nucleon elastic and direct-inelastic scattering from deformed nuclei. Concretely, for an axially deformed target one must: (1) construct the deformed Hartree–Fock–Bogoliubov mean-field and pairing potentials in coordinate space; (2) generate the target's excited states in the intrinsic frame with the quasiparticle random-phase approximation (QRPA, the superfluid extension of RPA to paired systems) built on that deformed HFB solution; (3) perform angular-momentum projection — within the rotational approximation, which restores good angular momentum from the intrinsic deformed state using rotational-band kinematics — to obtain the laboratory-frame monopole optical potential and the multipole coupling potentials between the channels of the ground-state rotational band; and (4) solve the resulting coupled-channel scattering equations, whose potentials are nonlocal, energy-dependent and complex integral operators in the two radial coordinates ⟨r∣U∣r′⟩\langle r|U|r'\rangle, since existing coupled-channel machinery assumes local potentials. The objective is met when the predicted elastic and direct-inelastic differential cross sections and total cross sections for one or more axially deformed target nuclei are obtained without fitted renormalization or other phenomenological adjustment of the microscopic potentials — in the sense of the spherical prototype, whose only conventional input was a fixed, non-fitted width assigned to the intermediate excited states — and agree with the measured values, within their quoted experimental uncertainties, over the energy range where data exist (including the low-energy absorption content, where the spherical prototype underestimated neutron total cross sections below about 10 MeV). A demonstration on a single axially deformed target suffices to resolve the task. These four components are jointly required: the structure input, its projection, and the nonlocal coupled-channel solver are links of one pipeline whose common deliverable is a parameter-free microscopic description of scattering from deformed nuclei.

References

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