Nonobservable nature of the nuclear shell structure: Meaning, illustrations, and consequences

Background: The concept of single-nucleon shells constitutes a basic pillar of our understanding of nuclear structure. Effective single-particle energies (ESPEs) introduced by French [Proceedings of the International School of Physics “Enrico Fermi,” Course XXXVI, Varenna 1965, edited by C. Bloch (Academic Press, New York, 1966)] and Baranger [Nucl. Phys. A 149, 225 (1970)] represent the most appropriate tool to relate many-body observables to a single-nucleon shell structure. As briefly discussed in Duguet and Hagen [Phys. Rev. C 85, 034330 (2012)], the dependence of ESPEs on one-nucleon transfer probability matrices makes them purely theoretical quantities that “run” with the nonobservable resolution scale $\lambda$ employed in the calculation.

Purpose: Given that ESPEs provide a way to interpret the many-body problem in terms of simpler theoretical ingredients, the goal is to specify the terms, i.e., the exact sense and conditions, in which this interpretation can be conducted meaningfully.

Methods: While the nuclear shell structure is both scale and scheme dependent, the present study focuses on the former. A detailed discussion is provided to illustrate the scale (in)dependence of observables and nonobservables and the reasons why ESPEs, i.e., the shell structure, belong to the latter category. State-of-the-art multireference in-medium similarity renormalization group and self-consistent Gorkov Green's function many-body calculations are employed to corroborate the formal analysis. This is done by comparing the behavior of several observables and of nonobservable ESPEs (and spectroscopic factors) under (quasi) unitary similarity renormalization group transformations of the Hamiltonian parametrized by the resolution scale $\lambda$.

Results: The formal proofs are confirmed by the results of ab initio many-body calculations in their current stage of implementation. In practice, the unitarity of the similarity transformations is broken owing to the omission of induced many-body interactions beyond three-body operators and to the nonexact treatment of the many-body Schrödinger equation. The impact of this breaking is first characterized by quantifying the artificial running of observables over a (necessarily) finite interval of $\lambda$ values. Then the genuine running of ESPEs is characterized and shown to be convincingly larger than the one of observables (which would be zero in an exact calculation).

Conclusions: The nonobservable nature of the nuclear shell structure, i.e., the fact that it constitutes an intrinsically theoretical object with no counterpart in the empirical world, must be recognized and assimilated. Indeed, the shell structure cannot be determined uniquely from experimental data and cannot be talked about in an absolute sense as it depends on the nonobservable resolution scale employed in the theoretical calculation. It is only at the price of fixing arbitrarily (but conveniently) such a scale that one can establish correlations between observables and the shell structure. To some extent, fixing the resolution scale provides ESPEs (and spectroscopic factors) with a quasi-observable character. Eventually, practitioners can refer to nuclear shells and spectroscopic factors in their analyses of nuclear phenomena if, and only if, they use consistent structure and reaction theoretical schemes based on a fixed resolution scale they have agreed on prior to performing their analysis and comparisons.

Figure 1

Shell-model calculation of selected Ca isotopes performed in the $pf$ valence space with GXPF1 [11] and KB3G [12] empirical interactions. (a) ESPEs. (b) First $2^{+}$ excitation energy.

Figure 2

Comparison between the Fermi gap in the ESPE spectrum and the first $2^{+}$ excitation energy in ${}^{22,24}\mathrm{O}$. Results are obtained from a microscopic shell model [18, 19, 20] based on realistic $2N$ and $3N$ interactions. Calculations are displayed for three different values of the resolution scale characterizing the $2N$ interaction; see Ref. [20, 21] and the text for details.

Figure 3

Self-consistent Gorkov Green's function calculation of ${}^{74}\mathrm{Ni}$ with a realistic $2N$ chiral interaction [32]. (Left) Spectral strength distribution for one-neutron addition (above the dashed line) and removal (below the dashed line) processes. (Right) Baranger ESPEs.

Figure 4

Ground-state binding energies of ${}^{14\text{–}24}\mathrm{O}$ computed for $\lambda =1.88$ (open symbols), $2.00$ (crosses), and $2.24{\mathrm{fm}}^{-1}$ (solid symbols). (a) Lowest order (i.e., HFB) calculation for two cases, keeping the sole $2N$ interaction and incorporating both starting and induced $3N$ interactions. (b) G-SCGF second-order results without and with $3N$ forces (as above) and MR-IM-SRG(2) calculation with the full $2N+3N$ Hamiltonian.

Figure 5

One-neutron separation energies with dominant spectroscopic factors versus neutron ESPEs in ${}^{16,20,22,24}\mathrm{O}$. Each level is displayed for $\lambda =1.88$ (open symbols), $2.00$ (crosses), and $2.24 {\mathrm{fm}}^{-1}$ (solid symbols). Results are displayed for both HFB and second-order G-SCGF calculations. (a) One- and two-body operators are retained in the initial and transformed Hamiltonians. (b) One-, two-, and three-body operators are retained in the initial and transformed Hamiltonians.

Figure 6

Residual spreads of separation energies and ESPEs in HFB and second-order G-SCGF calculations with a $2N+3N$ Hamiltonian. Differences between $\lambda =1.88$ and $2.24 {\mathrm{fm}}^{-1}$ calculations are displayed for each spin-parity state. For one-neutron separation energies, states with spectroscopic factor larger than $30\%$ are retained. All states displayed in Fig. 5 are plotted indistinctly. Going from panel (a) to panel (c), one notices (i) a large reduction of the scale dependence and (ii) the expected compression of the strength owing to the inclusion of the coupling to fluctuations. Comparing panels (b) and (d) makes clear that none of these two features is reflected in the ESPEs, i.e., in the underlying shell structure.

Figure 7

Two-nucleon shell gap versus the ESPE Fermi gap in ${}^{14,16,22,24}\mathrm{O}$. Each quantity is displayed for $\lambda =1.88$ (dashed lines, open symbols), 2.00 (dotted lines, no symbols), and $2.24 {\mathrm{fm}}^{-1}$ (solid lines, solid symbols). Results are displayed for both HFB $\left({\delta }_{2n}^{\text{HFB}},\mathrm{\Delta }{e}^{\text{cent,HFB}}\right)$ and the MR-IM-SRG(2) $\left({\delta }_{2n},\mathrm{\Delta }{e}^{\text{cent}}\right)$ truncation scheme. One-, two-, and three-body operators are retained in the initial and transformed Hamiltonians.

Figure 8

Spectroscopic factors associated with one-neutron addition and removal process on the ground states of ${}^{14,16,18,20,22,24}\mathrm{O}$ computed as a function of the associated separation energy. For each final state, results obtained for $\lambda =1.88,2.00,2.24{\mathrm{fm}}^{-1}$ are joined by solid lines. One-, two-, and three-body operators are retained in the (initial and) transformed Hamiltonians. (a) Results obtained at the HFB level. (b) Results obtained from second-order G-SCGF calculations.