Controlled microscopic prediction of the nuclear pairing gap: the effective pairing interaction, finite-nucleus magnitude and its mass-number dependence, and the infinite-matter limit (the unfinished quantitative program of the Bohr–Mottelson–Pines superconductivity analogy)

In 1958 Bohr, Mottelson, and Pines proposed that the energy gap in the intrinsic excitation spectra of even-even nuclei, the quenching of rotational moments of inertia, and the odd-even staggering of nuclear masses all arise from Bardeen–Cooper–Schrieffer-type pairing of nucleons in time-reversed orbitals, and they identified what this proposal left open: the gap magnitude is exponentially sensitive to the poorly known product $N(0)V$ of the single-particle level density and the effective pairing matrix element; the approach of the gap to the infinite-nucleus (nuclear-matter) limit was undetermined; and the effect of interaction components beyond the idealized constant equal-and-opposite-momentum channel was unquantified. Six decades of work confirmed the analogy and built a successful phenomenology (BCS and Hartree–Fock–Bogoliubov theory with pairing strengths fitted to odd-even mass data, gaps of order 1 MeV), clarified the finite-size criterion for nuclear superfluidity, and computed infinite-matter gaps — but did not deliver the controlled microscopic prediction the source called for: ab initio chiral-EFT calculations still underpredict empirical gaps, microscopic effective pairing interactions still require phenomenological addenda, and infinite-matter gaps remain method-dependent at roughly the factor-of-two level. The surviving open problem is to derive the effective pairing interaction microscopically — including medium polarization and coupling to shape and collective degrees of freedom — and thereby predict the nuclear pairing gap with controlled uncertainties, both in finite nuclei across the nuclear chart and in the infinite-matter limit.

Background

A nucleus is a self-bound finite Fermi system of $A$ nucleons (protons and neutrons) held together by the strong interaction. In the independent-particle ('shell') picture the nucleons occupy discrete single-particle orbitals, and near the Fermi surface — the boundary between occupied and empty orbitals — the mean spacing $\delta$ between consecutive orbitals of a given species is small, of order $50\,A^{-1}$ MeV in the estimate of the source paper, i.e., a fraction of an MeV for a heavy nucleus. Even-even nuclei (even numbers of both protons and neutrons) posed a classical puzzle for this picture: their lowest 'intrinsic' excitations (excitations of the nucleons themselves, as opposed to collective rotational and vibrational motion of the whole deformed nucleus) typically lie near 1 MeV in the deformed rare-earth and actinide regions, several times larger than the ~ $\delta/2$ expected when two nucleons are simply promoted across the Fermi surface, and the low-lying two-particle configurations predicted by the independent-particle picture are systematically absent. In addition, the moments of inertia inferred from the rotational band energies of deformed nuclei are strongly reduced relative to rigid-body values, and nuclear binding energies display a characteristic odd-even mass staggering (nuclei with an odd nucleon number are systematically less bound than the average of their even neighbors), signaling an extra pairing energy.

The microscopic theory of superconductivity of Bardeen, Cooper, and Schrieffer (BCS, 1957) explains how an arbitrarily weak attraction between fermions near a Fermi surface is amplified: particles in time-reversed orbitals (equal and opposite momenta and spins, the $(k,-k)$ channel) bind into Cooper pairs, and the many-body ground state becomes a pair condensate separated from its excitations by an energy gap. Bohr, Mottelson, and Pines (1958) proposed that the same mechanism operates in nuclei: the residual interaction between nucleons is attractive in the time-reversed channel, and the resulting pairing correlations open a gap in the intrinsic spectrum, quench the moment of inertia, and generate the odd-even mass stagger. From the Cooper instability they estimated the correlated energy scale

$\epsilon \;\approx\; \Delta_c\,\exp\!\big[-1/(N(0)V)\big],$

where $N(0)$ is the density of single-particle levels per unit energy at the Fermi surface, $V$ is the effective pairing matrix element assumed constant within an energy window $\Delta_c$ of order the Fermi energy, and $\epsilon$ is identified with the gap. With the contemporary estimate $N(0)V\sim 1/5$ , the correlated interval around the Fermi surface is far narrower than the interaction window, so pairing strongly reorganizes the low-energy spectrum while barely affecting the total binding — consistent with observation. The same exponential form, however, makes quantitative prediction fragile: modest uncertainty in $N(0)V$ translates into large uncertainty in the gap. The paper explicitly left open the controlled estimate of $N(0)$ and $V$ in the finite, shape-dependent nuclear environment, the limiting behavior of the gap as $A\to\infty$ (a putative 'infinite nucleus', i.e., infinite uniform nuclear matter) and the system size at which the correlated rearrangement would wash out shell structure, the role of interaction components beyond the idealized constant $(k,-k)$ channel (finite range, spin–isospin dependence, coupling to shape degrees of freedom), and a quantitative microscopic criterion for when many-state coupling yields a gap exceeding the single-particle level spacing.

The subsequent literature developed along four lines. First, S. T. Belyaev (1959) applied the BCS formalism to nuclei with realistic two-body matrix elements, computing gaps, spectra, and moments of inertia, and showed that in a system with a discrete spectrum pairing requires a minimum interaction strength — the first finite-size criterion. Second, BCS and its self-consistent extension, Hartree–Fock–Bogoliubov theory (HFB, in which the mean field and the pairing field are determined variationally together and the elementary excitations are Bogoliubov quasiparticles, coherent particle–hole superpositions), became the standard description; with pairing strengths fitted to odd-even mass differences, this phenomenology reproduces the gap systematics (gaps of order 1 MeV, weakening near closed shells, modified by deformation), but it is a fit, not a microscopic prediction. Third, the infinite-matter limit was computed by solving the BCS gap equation in uniform nuclear or neutron matter (an idealized infinite system of nucleons; neutron matter is the pure-neutron case) with realistic interactions: the like-particle $^1S_0$ channel (spin-singlet, isospin-triplet S-wave pairing) supports a gap of a few MeV at sub-saturation density (below the equilibrium density of uniform nuclear matter, about 0.16 fm $^{-3}$ ), but polarization corrections — screening of the pairing interaction by particle–hole excitations and collective modes, precisely the 'additional interactions' prong of the source question — were found in Landau-theory-based calculations to suppress the gap by factors of two to three, while quantum Monte Carlo calculations find only a modest suppression; the quantitative value remains method-dependent. Fourth, the conceptual status of the criterion was sharpened: nuclei are mesoscopic systems in which the gap is only of the order of the mean single-particle level spacing or somewhat larger, a crossover regime where fluctuations of the pairing field matter and BCS mean field is not asymptotically controlled; exact solutions of schematic pairing Hamiltonians show that pairing correlations persist below the mean-field critical strength, and pairing competes with quadrupole deformation, which splits the single-particle spectrum (Nilsson orbitals) and thereby reshapes the effective level spacing that the gap must exceed.

Modern ab initio nuclear theory starts from chiral effective field theory (χEFT) parametrizations of the two- and three-nucleon forces, with their low-energy constants fit to few-body and scattering data, and solves the many-body problem with controlled, systematically improvable truncations (coupled cluster, in-medium similarity renormalization group, Gorkov–Green's-function methods for open-shell (partially filled shell) superfluid nuclei). It reproduces ground-state and separation energies of medium-mass nuclei rather well — but systematically underpredicts the empirical pairing gaps, a deficit now debated under the title 'where has the nuclear pairing gone?'. The quantitative core of the 1958 program thus remains open, at a standard of 'controlled' that has risen from phenomenological fits to first-principles prediction.

Problem Statement

Starting from the microscopic two- and three-nucleon interaction, determine the effective pairing interaction acting between nucleons in time-reversed orbitals in the finite, shape-dependent nuclear environment — including, beyond the idealized constant $(k,-k)$ channel of the 1958 Bohr–Mottelson–Pines analysis, the medium-polarization (screening) corrections, the exchange of collective surface modes, finite-range and spin–isospin components, and the coupling to deformation, every one of which the effective interaction must either incorporate or, if neglected, be shown to change the gap by less than the target uncertainty — and use it to predict, with controlled and quantified uncertainties, the nuclear pairing gap and its system-size dependence. Concretely: (i) predict the magnitude of the pairing gap across the nuclear chart, as confronted with odd-even mass staggering (e.g., three-point mass differences, the standard finite-difference estimators of the pairing gap from binding energies) and the systematics of first intrinsic excitations, including its dependence on shell filling and deformation — which supplies the quantitative microscopic criterion for when many-state pairing correlations produce a gap exceeding the mean single-particle level spacing, the criterion the source paper asked for; and (ii) determine the behavior of the gap in the approach to the infinite-nucleus limit, i.e., a converged, method-independent value of the pairing gap in infinite nuclear matter (with neutron-matter $^1S_0$ calculations as the best-controlled benchmark) and its density dependence. The single overarching objective is a microscopic, controlled quantitative theory of the nuclear pairing gap; the requested results in (i) and (ii) are jointly supporting deliverables of that one objective, exactly as posed by the source paper's call for 'improved, controlled estimates' of the pairing parameters and the gap they produce. The target is a theory in which empirical pairing gaps are reproduced without any pairing-strength parameter fitted to pairing or mass-staggering data, with an uncertainty budget derived a priori from the truncation orders, interaction parametrization, and missing correlations (this a priori budget is the target uncertainty), so that the gaps agree with the odd-even-mass-staggering systematics to within the accuracy with which the empirical gap is itself extracted from those data (about 0.1–0.2 MeV), and in which the infinite-matter gap is consistent across independent microscopic methods (e.g., quantum Monte Carlo versus polarization-corrected gap-equation approaches) to within that same a priori budget.

References

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