An exactly solvable framework was proposed (Wang, Pan, Launey, Luo, Draayer, Nucl. Phys. A 950 (2016) 1) in which an axially deformed Nilsson mean field combined with an exactly solvable pairing Hamiltonian — modified standard pairing (MSP) or nearest-level pairing (NLP) — is projected onto states of good total angular momentum, both exactly and through an approximate scheme restricted to intrinsic states with K=0. It describes low-lying spectra and electromagnetic properties of 18^{18}O, 18^{18}Ne, 20^{20}Ne, and 24^{24}Mg, but exhibits systematic failures (underpredicted yrast B(E2), tens-of-MeV overprediction of some higher levels). These limitations define an open extension program, which this problem records: (i) incorporate isospin-dependent (isovector, proton-neutron) pairing into the deformed, projected framework and determine to what extent exact solvability survives; (ii) quantify, against exact projection where feasible and against data, where the K=0 truncation breaks down as a function of excitation energy, K-mixing, and mass region; (iii) settle the effective-charge/core-polarization calibration so that quadrupole observables are reproduced systematically; and (iv) develop controlled truncation schemes making nuclei with many valence particles tractable. None of these components has been addressed in this framework; the isovector-pairing component has progressed only in spherical mean fields and seniority-zero subspaces.

Background

In nuclear structure, deformed nuclei are often described by breaking rotational symmetry at the mean-field level and restoring it afterwards. The Nilsson model is the classic single-particle mean field for an axially deformed nucleus: it yields deformed orbitals that are doubly degenerate time-reversed pairs (Kramers pairs) labeled by the projection KK of the single-particle angular momentum on the symmetry axis. Such an intrinsic state is not an eigenstate of total angular momentum JJ, whereas measured level energies, quadrupole moments, and transition rates refer to states of good JJ. Angular momentum projection restores this symmetry: projection operators built from rotation operators and Wigner DD-functions map intrinsic (body-fixed) states onto laboratory-frame states with definite JJ and projection MM. The quantum number KK is the projection of JJ on the intrinsic symmetry axis; a rotationally invariant state is a superposition of all KK components, and restricting to K=0K=0 components is a common, drastic simplification valid only for low-lying states of well-deformed axially symmetric systems.

Pairing correlations between like nucleons are the other key ingredient of low-energy nuclear structure. The standard pairing Hamiltonian (constant pairing strength GG among all levels) is exactly solvable: its eigenstates take the Richardson-Gaudin Bethe-ansatz form of products of generalized pair creation operators whose spectral parameters satisfy coupled nonlinear algebraic equations, and the Heine-Stieltjes correspondence provides the underlying polynomial formulation (Pan, Li, Zhang, Draayer, Phys. Rev. C 88, 034305 (2013); Guan et al., Phys. Rev. C 86, 024313 (2012)). Exact treatments of pairing are valuable because Bardeen-Cooper-Schrieffer (BCS) approximations fail for finite systems and small level spacing, and because exact intrinsic spectra and eigenstates are precisely what a projection formalism needs as input.

The source paper (Wang, Pan, Launey, Luo, Draayer, Nucl. Phys. A 950 (2016) 1) combined these threads. It projected an intrinsically deformed Hamiltonian — the axially deformed Nilsson mean field plus one of two exactly solvable pairing models — onto good JJ: the modified standard pairing (MSP), which adds level-dependent on-site pairing strengths GiiG_{ii} to constant-strength pair scattering and remains Richardson-Gaudin solvable, and the nearest-level pairing (NLP), whose pair-scattering matrix is tridiagonal (dominant between adjacent levels) and whose eigenstates follow from diagonalizing a finite tridiagonal matrix. Both the exact projection (all intrinsic states and all KK) and an approximate projection restricted to K=0K=0 intrinsic states were formulated. For the two-valence-particle nuclei 18^{18}O and 18^{18}Ne, exact and approximate projections give comparable rms deviations for low-lying levels and B(E2) values; for 20^{20}Ne and 24^{24}Mg only the K=0K=0 scheme is numerically feasible, and it reduces, e.g., the 24^{24}Mg configuration space from 245025 to 28503 intrinsic states. Here B(E2) denotes the reduced probability for an electric-quadrupole transition, conventionally quoted in Weisskopf units (W.u.), a natural single-particle scale; a uniform scale factor ρ2\rho_2 on the quadrupole operator (ρ2=5\rho_2=5 exact, ρ2=3\rho_2=3 approximate) plus effective proton and neutron charges were introduced to compensate the truncated model space. The calculations reproduce the pattern of low-lying spectra and some quadrupole moments, but with characteristic systematic discrepancies: the energy ratio R4/2=E(41+)/E(21+)R_{4/2}=E(4_1^+)/E(2_1^+) is underestimated in exact-projection fits, some higher excited levels come out tens of MeV too high, and many yrast-band B(E2) strengths are underpredicted (for 20^{20}Ne, B(E2;21+→0g+)B(E2;2_1^+\to 0_g^+) calculated 13.77 W.u. versus 20.3(10) W.u. experiment; B(E2;41+→21+)B(E2;4_1^+\to 2_1^+) calculated 4.11 W.u. versus 22(2) W.u.), while some non-yrast transitions are overpredicted by orders of magnitude. To preserve exact solvability the authors treated proton-proton and neutron-neutron pairing only: isospin-dependent (isovector, proton-neutron) pairing and residual proton-neutron interactions were omitted, although they are expected to matter, e.g. for level splittings in 20^{20}Ne. The ab initio symmetry-adapted no-core shell model (SA-NCSM), a large-basis diagonalization approach that uses SU(3) symmetry to organize the many-body basis, serves as one modern benchmark for the same nuclei.

Problem Statement

Within the angular-momentum-projected Nilsson mean-field plus exactly solvable pairing framework of Wang et al. (2016), establish a quantitatively reliable laboratory-frame spectroscopy of deformed nuclei by resolving the following open extensions of the framework: (1) incorporate isospin-dependent pairing — isovector T=1T=1 interactions including neutron-proton J=0J=0 pairs, and residual proton-neutron interactions — into the deformed Nilsson mean field, carry both through the angular momentum projection, and determine to what extent exact solvability survives; (2) quantify the validity boundary of the K=0K=0 approximate projection — its error relative to exact projection and to experiment as a function of excitation energy, importance of K≠0K\neq 0 components, deformation, and mass region — beyond the low-lying states of 18^{18}O, 18^{18}Ne, 20^{20}Ne, and 24^{24}Mg where it was validated; (3) resolve the systematic underprediction of yrast B(E2) strengths and related quadrupole observables; and (4) develop controlled truncation or solver schemes that keep the projected configuration space tractable for nuclei with many valence particles, where the intrinsic-space dimension grows combinatorially (already 245025 intrinsic states before projection for 24^{24}Mg). These four tasks serve one common objective — a good-JJ spectroscopy of deformed nuclei whose pairing is treated exactly where possible and whose truncations carry quantified error bars. All four extensions are jointly required components of that objective, yet each is independently resolvable and carries its own completion standard: (1) a controlled isovector treatment with quantified error and an explicit account of how much exact solvability is lost; (2) a K=0K=0 error map spanning the stated axes — excitation energy, KK-mixing, deformation, and mass region — beyond the four nuclei already validated; (3) a systematic quadrupole description obtained under a single effective-charge/core-polarization calibration fixed once and validated out-of-sample, rather than refitted per observable; and (4) a controlled truncation or solver scheme demonstrated on a heavier many-valence-particle nucleus with quantified error.

References

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