Halo phenomenon in finite many-fermion systems. Atom-positron complexes and large-scale study of atomic nuclei

Halo phenomenon in finite many-fermion systems.
Atom-positron complexes and large-scale study of atomic nuclei

V. Rotival vincent.rotival@polytechnique.org DPTA/Service de Physique Nucléaire - CEA/DAM Île-de-France - BP12 - 91680 Bruyères-le-Châtel, France National Superconducting Cyclotron Laboratory, 1 Cyclotron Laboratory, East Lansing, MI 48824, USA    K. Bennaceur bennaceur@ipnl.in2p3.fr Université de Lyon, F-69003 Lyon, France; Institut de Physique Nucléaire de Lyon, CNRS/IN2P3, Université Lyon 1, F-69622 Villeurbanne, France CEA, Centre de Saclay, IRFU/Service de Physique Nucléaire, F-91191 Gif-sur-Yvette, France    T. Duguet thomas.duguet@cea.fr CEA, Centre de Saclay, IRFU/Service de Physique Nucléaire, F-91191 Gif-sur-Yvette, France National Superconducting Cyclotron Laboratory, 1 Cyclotron Laboratory, East Lansing, MI 48824, USA Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
July 14, 2019
Abstract

The analysis method proposed in Ref. Rotival and Duguet (2009) is applied to characterize halo properties in finite many-fermion systems. First, the versatility of the method is highlighted by applying it to light and medium-mass nuclei as well as to atom-positron and ion-positronium complexes. Second, the dependence of nuclear halo properties on the characteristics of the energy density functional used in self-consistent Hartree-Fock-Bogoliubov calculations is studied. We focus in particular on the influence of (i) the scheme used to regularize/renormalize the ultra-violet divergence of the local pairing functional, (ii) the angular-momentum cutoff in the single-particle basis, as well as (iii) the isoscalar effective mass, (iv) saturation density and (v) tensor terms characterizing the particle-hole part of the energy functional. It is found that (a) the low-density behavior of the pairing functional and the regularization/renormalization scheme must be chosen coherently and with care to provide meaningful predictions, (b) the impact of pairing correlations on halo properties is significant and is the result of two competing effects, (c) the detailed characteristics of the pairing functional has however only little importance, (d) halo properties depend significantly on any ingredient of the energy density functional that influences the location of single-particle levels; i.e. the effective mass, the tensor terms and the saturation density of nuclear matter. The latter dependencies give insights to how experimental data on medium-mass drip-line nuclei can be used in the distant future to constrain some characteristics of the nuclear energy density functional. Last but not least, large scale predictions of halos among all spherical even-even nuclei are performed using specific sets of particle-hole and particle-particle energy functionals. It is shown that halos in the ground state of medium-mass nuclei will only be found at the very limit of neutron stability and for a limited number of elements.

halo; EDF; HFB; pairing
pacs:
2.10.Gv, 21.10.Pc, 21.30.Fe, 21.60.Jz

I Introduction

The formation of halos is a quantum phenomenon caused by the possibility for non-classical systems to expand in the classically forbidden region Hansen and Jonson (1987); Hansen et al. (1995); Tanihata (1996); Tanihata and Kanungo (2003); Jensen et al. (2004). Indeed, weakly-bound systems can extend well beyond the classically-allowed region, as has been theoretically predicted or experimentally observed for molecules (He-He-Li et al. (2006), He Schöllkopf and Toensies (1996); Nielsen et al. (1998); Grisenti et al. (2000), HeHe Bressanini et al. (2002)…), atom-positron complexes (Be, PsLi, PsHe…) Mitroy (2005a) and hypernuclei (H) Cobis et al. (1997). In nuclear physics, where the study of halos was initiated, efforts are still devoted to reach a better understanding of the structure and reaction properties of such exotic systems. For instance, the existence of halos in borromean systems or excited states of mirror nuclei still raises questions Chen et al. (2005). In light nuclei, it was found that a cluster picture is at play, and known halo systems are accurately described by two- Fedorov et al. (1994); Nunes et al. (1996a) or three-body Zhukov et al. (1993); Fedorov et al. (1994); Nunes et al. (1996b); Bang (1996) models, where one or two nucleons evolve around a tightly-bound core. This leads to a classification in terms of one-nucleon halos (Be Tanihata et al. (1985a); Fukuda et al. (1991); Zahar et al. (1993), Bazin et al. (1995); Kanungo et al. (2002), Ne Kanungo et al. (2003); Jeppesen et al. (2004)…) and two-nucleon halos (He Zhukov et al. (1993), Li Tanihata et al. (1985b, b)…).

Since the discovery of the anomalous cross-section of Li Tanihata et al. (1985b, b), one of the compelling questions relates to the existence of a mass limit beyond which the formation of halos is inhibited. On the proton-rich side, it is believed that the Coulomb interaction prevents the formation of halos beyond  Jensen and Riisager (2000). However, this could be put into question as non-trivial effects may come into play Liang et al. (). From a theoretical standpoint, halos in medium to heavy mass nuclei can be studied through relativistic or non-relativistic Hartree-Fock-Bogoliubov calculations Ring and Schuck (1980); Bender et al. (2003) performed in the context of energy density functional (EDF) methods. On the experimental side, the next generation of radioactive ion beam facilities (FAIR at GSI, RIBF at RIKEN, REX-ISOLDE at CERN, SPIRAL2 at GANIL…) might be able to assess the position of the neutron drip-line up to about  whi (2006). Although this would be an astonishing accomplishment, it will not allow the study of most of potential medium-mass halos. Still, the (distant) future confrontation of theoretical results with experimental data will provide crucial information that can be used to constrain theoretical models.

One difficulty resides in the absence of tools to characterize halo properties of finite many-fermion systems in a quantitative way. Light nuclei constitute an exception considering that the quantification of halo properties in terms of the dominance of a cluster configuration and of the probability of the weakly-bound clusters to extend beyond the classical turning point is well acknowledged Fedorov et al. (1993); Jensen and Riisager (2000); Riisager et al. (2000); Jensen and Zhukov (2001). Existing definitions and tools applicable to systems constituted of tens of fermions are too qualitative, the associated observables are incomplete and have led to misinterpretations in nuclei Rotival and Duguet (2009).

To improve on such a situation, a new quantitative and model-independent analysis method was proposed in Ref. Rotival and Duguet (2009). The method uses universal properties of the internal one-body density to extract, in a model-independent fashion, the part of the density that can be identified as a halo. Two criteria have been introduced to characterize halo systems in terms of (i) the average number of fermions participating in the halo and (ii) the influence of the latter on the system extension. The results deduced from EDF calculations of medium-mass nuclei have underlined the likely formation of a collective halo at the neutron drip-line of chromium isotopes. The neutron density of those nuclei displays a spatially decorrelated region built out of an admixture of and overlap functions (orbitals). The significant contribution of orbitals with orbital angular-momentum is at variance with the standard picture in light nuclei Riisager et al. (1992); Mizu et al. (1997).

Given such an analysis method, important questions can now be addressed. The versatility of the method must be tested which we do in the first part of our study by applying it to many-fermion systems of different scales computed with various many-body techniques : atom-positron/ion-positronium complexes on the one hand and light/heavy nuclei on the other. In the second part of the article, we focus on the influence of (i) the scheme used to regularize/renormalize the ultra-violet divergence of the local pairing functional, (ii) the angular-momentum cutoff in the single-particle basis, as well as (iii) the isoscalar effective mass, (iv) saturation density of nuclear matter and (v) tensor terms Lesinski et al. (2007) characterizing the particle-hole part of the energy functional. More generally, the different ways pairing correlations impact halo nuclei are studied, e.g. the anti-halo effect Bennaceur et al. (1999, 2000) or the potential decorrelation of orbitals from the pairing field Hamamoto and Mottelson (2003, 2004); Hamamoto (2005).

The present paper is organized as follows. The analysis method proposed in Ref. Rotival and Duguet (2009) is briefly recalled in Sec. II whereas its versatility is highlighted in Sec. II.2. In Sec. III, technical aspects of Skyrme-EDF calculations are provided and the dependence of halo predictions on some their ingredients is pointed out. Section IV is devoted to discussing the effect of pairing correlations on the formation of halos. Then, the sensibility of halo properties to the characteristics of the particle-hole part of the functional is studied. A large scale study of potential halos among all spherical medium-mass nuclei is proposed in Sec. V. In conclusions, we discuss how halo systems could help constraining the nuclear EDF and to which extent the present results can be related to data generated in the distant future by radioactive ion beam facilities.

Ii Characterization of halo systems

ii.1 Analysis method

Anew quantitative analysis-method of halos in finite many-fermion systems was proposed in Ref. Rotival and Duguet (2009). The starting point is a model-independent decomposition of the internal one-body density of spherical many-fermion systems in terms of spectroscopic amplitudes and their radial components  Van Neck et al. (1993); Van Neck and Waroquier (1998); Shebeko et al. (2006)

(1)

The appearance of a halo in the -body system, i.e. the part of the density that is spatially decorrelated from an a priori unknown core, was shown in Ref. Rotival and Duguet (2009) to be related to the existence of three typical energy scales in the excitation spectrum of the -body system. From a practical viewpoint, and using the internal one-body density as the only input, the method allows the extraction of the radius beyond which the halo, if it exists, is located.

With the radius at hand, two quantitative halo factors are introduced. First, the average number of fermions participating in the halo can be extracted through

(2)

Second, the contribution of the halo region to the root mean square radius of the system is also extracted

(3)

ii.2 Versatility of the method

To illustrate the analysis method and its versatility, we now apply it to the results of many-body calculations performed for three different systems: light nuclei studied through coupled-channels calculations Nunes et al. (1996a, b), medium-mass nuclei described through single-reference energy density functional calculations and atom-positron/ion-positronium complexes computed with the fixed-core stochastic variational method Varga and Suzuki (1995, 1997); Suzuki and Varga (1998); Ryzhikh et al. (1998a); Mitroy and Ryzhikh (2001–2007).

ii.2.1 Light nuclei

To check the consistency of the method in a situation where core and halo densities are explicitly computed, the values of and have been extracted from coupled-channels calculations of light nuclei. This will serve a basis for cases where the total density only is accessible.

Calculations are performed for a core+neutron system for which the internal dynamics of the core is taken into account Nunes et al. (1996a, b) and the total Hamiltonian reads as

(4)

To provide adequate nuclear quadrupole couplings, a deformed Woods-Saxon potential in the core rest frame is considered

(5)
(6)

where is the core quadrupole deformation. The total wave-function is expanded in a basis of eigenstates of the total angular momentum using a separation of the core internal motion, with eigenstates associated to the energies , from the neutron relative motion. The resulting coupled-channels equations for the loosely bound nucleon wave function read

(7)

and are solved in a Sturmian basis.

Calculations have been performed for two nuclei: the well-established one-neutron halo Be, whose total density and core+halo decomposition are plotted in Fig. 1, and the stable nucleus C as a control case. The results are summarized in Tab. 1, where the two criteria are evaluated for core and total densities.

Core Total
 [fm]  [fm]  [fm]
C
Be
Table 1: Values of and for C (stable) and Be (one-neutron halo). The criteria are applied to results obtained through coupled-channels calculations Nunes et al. (1996a, b).
Figure 1: (Color Online) Core, halo and total neutron density of Be obtained from coupled-channels calculations Nunes et al. (1996a, b). The value of and the tolerance margin are indicted by vertical lines.

For Be, is found to be compatible with the condition to have one order of magnitude difference between core and halo densities at that radius Rotival and Duguet (2009). The tail-to-core ratio is slightly different from ten, partly because the core is represented by a gaussian profile with the wrong asymptotic. In any case, the ideal value of still lies within the allowed theoretical error. This validates the method on a realistic system. The halo parameter shows that around neutron reside in the decorrelated region in average. The reason why one only finds a fraction of a neutron within the halo region is because the wave function of the ”halo nucleon” lies partly inside the volume of the core. The denomination of one-neutron halo is somewhat misleading from that point of view. The influence of the halo on the nuclear extension is large, of about  fm out of a total root-mean-square (r.m.s.) radius of  fm. Note that such a value is very close to the value extracted experimentally when going from Be to Be Tanihata (1988). Last but not least, and are found to be negligible for the two control cases considered, i.e. C and the core density of Be. This illustrates the ability of the method to discriminate between halo and non-halo systems.

ii.2.2 Medium-mass nuclei

In Ref. Rotival and Duguet (2009), the formation of neutron halos in Cr and Sn isotopes has been investigated through self-consistent EDF calculations performed using the SLy4 Skyrme functional complemented with a density-dependent mixed-type pairing functional (see Sec. III.2.1 and Sec. III.2.2). The value of is determined using as an input the one-body density obtained from a symmetry-breaking HFB state, which is assumed to map out the nuclear internal density (see discussion in Ref. Rotival and Duguet (2009)).

The (perturbative) excitation spectra of the last three bound odd Cr isotopes () shown in Ref. Rotival and Duguet (2009) display optimal energy scales as far as the formation of a halo is concerned. The values of and found for the last four bound even Cr isotopes, Cr being the predicted neutron drip-line nucleus for the parametrization used, are listed in Tab. 2. Beyond the neutron shell closure, the steep increase of both and is the signature of a halo formation. A decorrelated region containing up to neutron appears at the limit of stability. Such a number of neutrons participating in the halo is small relatively to the size of the system but comparable in absolute value to the number found in light halo nuclei, as recalled in Sec. II.2.

The contribution of the halo region to the nuclear extension reaches about  fm in Cr. On the one hand, such a value is significant in comparison with the total neutron r.m.s. radius of those systems. On the other hand, it corresponds to about one-third of the value found for Be. It is likely that the formation of halos is hindered as the mass increases because of the increased collectivity. We will come back to that in section V. Of course, one should not disregard the fact that single-reference EDF calculations miss certain long-range correlations which, once they are included, might slightly change the picture.

One interesting feature discussed in Ref. Rotival and Duguet (2009) is that the kink of the r.m.s. radius at , which had been interpreted as a halo signature in previous works, is partly due to a plain shell effect associated with the sudden drop of the two-neutron separation energy. As a matter of fact, the quantity allows the disentanglement of shell and halo effects in the increase of the neutron r.m.s. radius across . For isotopes further away from their drip-line, e.g. across in tin isotopes, the use of demonstrates that the kink of the neutron r.m.s. radius is entirely produced by the shell effect, whereas its further increase beyond is related to the growth of the neutron skin, not to the appearance of a halo Rotival and Duguet (2009). Generally speaking the quantity does not incorporate the contribution from the neutron skin and only characterizes the halo part of the density profile, e.g. spatially decorrelated neutrons.

          [fm]
Cr
Cr
Cr
Cr
Table 2: Values of and for the last bound even-even Cr isotopes, as predicted by the {SLy4+REG-M} functional. Results slightly differ from the ones shown in Paper I because of the different codes used to perform the EDF calculations. The difference remains very small, of the order of the third significative digit.

A decomposition of the halo region in terms of canonical states shows that it contains equal contributions from neutron and orbitals. The possibility for states to participate in the halo was not rejected from the outset. Eventually, the loosely bound shell strongly contributes here because of its rather large occupation and intrinsic degeneracy. The latter observations point towards the formation of a collective halo in drip-line Cr isotopes, formed by an admixture of several overlap functions, and call for a softening of the restrictions spelled out in light nuclei Jensen et al. (2004).

The energy scales necessary to the halo formation are not seen in the spectra of drip-line tin isotopes and the weak kink observed in the neutron r.m.s. radius at is nothing but a shell effect, as mentioned above. In contrast with previous studies based on the Helm model  Helm (1956); Rosen et al. (1957); Raphael and Rosen (1970); Friedrich and Voegler (1982); Mizutori et al. (2000), no collective halo is identified in tin isotopes, as the decorrelated region is found to have almost no influence on the matter extension Rotival and Duguet (2009).

ii.2.3 Atom-positron/ion-positronium complexes

In atomic physics, valence electrons of neutral atoms can be located at large distances from the core. Because of the very long range of the Coulomb interaction, the penetration of the wave-function into the classically forbidden part of the potential as the separation energy of the system becomes small cannot be interpreted as a halo formation Jensen et al. (2004). However, a positron can be attached to a neutral atom by the polarization potential, which can be parameterized as

(8)
(9)

where is the core polarization constant and a cutoff distance. In this case, the decay of the potential at large distances does not ensure that particles are able to tunnel through the potential barrier. It was found that several atom-positron complexes can exist Ryzhikh and Mitroy (1997); Ryzhikh et al. (1998b, a); Bromley and Mitroy (2001, 2002a, 2002b, 2002c); Mitroy et al. (2002), and have been identified as having halo characteristics Mitroy (2005a). To quantify such an observation, values of and are evaluated for such systems.

The Hamiltonian of the atom-positron system with valence electrons reads with normalized units (Ryzhikh et al. (1998b, a)

(10)

where is the positron position vector, is the relative position of two valence electrons, whereas the direct and exchange potentials between valence electrons and the core are computed exactly in the Hartree-Fock approximation. The two-body polarization potential is defined as

(11)
  Atom   Asympt.          []    []    [%]    [%]  [at. units]    [at. units] Ref.
Be +Be Mitroy (2005a)
Mg +Mg Mitroy and Ryzhikh (2001–2007)
Cu +Cu Bromley and Mitroy (2002c)
He Ps+He Mitroy (2005b)
Li Ps+Li Mitroy (2004)
Table 3: Results of the halo analysis for various atom-positron systems evaluated with the fixed-core stochastic variational method. The columns provide: neutral atom symbol, asymptotic form, total number of electrons , halo factors and , total matter r.m.s. radius , relative proportions and of electrons and positrons in the halo region, ground state energy of the +A complex, and binding energy with respect to the corresponding dissociation threshold. All length units are normalized to the Bohr radius , and energy values are in atomic units ( at. unit  eV).
Figure 2: (Color Online) Valence electron, positron, core and total density for the Be complex evaluated with the fixed-core stochastic variational method. All length units are normalized to the Bohr radius . The value of and the tolerance margin are indicated by vertical lines.
Figure 3: (Color Online) Same as in Fig. 2 for the +Li system.

When the system +A is bound, its asymptotic behavior can correspond to (i) a neutral core plus a positron , or (ii) a charged core and a neutral positronium complex Ps, depending on the relative binding energies of those configurations. Calculations are performed with the fixed-core stochastic variational Method (FCSVM) Varga and Suzuki (1995, 1997); Suzuki and Varga (1998); Ryzhikh et al. (1998a); Mitroy and Ryzhikh (2001–2007), in a basis of explicitly correlated gaussians for the individual wave-functions. The basis is taken large enough to correctly reproduce the asymptotic behavior of the +A or Ps+A systems. The results of such calculations for +Be, corresponding to the Be complex, are presented in Fig. 2, where the separation at large distances between a weakly bound positron and a core composed of the electrons is visible. As a result, a positron extended tail appears. In Fig. 3 are also displayed the results for the +Li system, which corresponds to a PsLi complex. Indeed, one observes that the density tail is composed of almost identical and components.

The results of the analysis, performed for several atom-positron complexes, are presented in Tab. 3. The separation energy in the appropriate channel (+A or Ps+A) is small compared to the ground state energy of the complex. The situation regarding the energy scales at play is very favorable as far as the formation of halos is concerned. It is also possible to evaluate the composition of the halo region in terms of the proportion of electrons and of positrons .

The values of and demonstrate the existence of halos in +Be, +Mg and +Cu which strongly affect the system extensions. For example, the spatially decorrelated part of the density accounts for about half of the total r.m.s. radius in +Be, although it contains only particle in average. In those cases, a positron halo is predicted as the halo region is almost exclusively built from the positron wave-function (). For +Li and +He, extremely large values for and are extracted, demonstrating that one is dealing in these cases with gigantic ion-positronium halos ().

Considering the values of and , one realizes that atom-positron and ion-positronium complexes display more extreme halo structures than nuclei. This is of course due to the nature of the interactions at play in such systems.

ii.3 Universality of the phenomenon

As demonstrated, the method developed in Ref. Rotival and Duguet (2009) can be applied successfully to finite many-fermion systems of very different scales. This relates to the fact that the method relies on a model-independent analysis of the internal one-body density . In all cases, a fraction of the constituents may extend far out from the core and influence strongly the size of the system.

It happens that halo systems display scaling properties which do not depend on their dimension and constituency. This can be characterized by the extension of the halo wave-function as a function of the separation energy , and those quantities can be made dimensionless using as a scale the classical turning point for the interaction potential of interest and the reduced mass of the systems  Fedorov et al. (1993, 1994); Riisager et al. (2000); Jensen et al. (2004). The generic asymptotic scaling laws of the two-body system are extracted using a finite spherical well, and depend on the angular momentum of the weakly-bound overlap function Fedorov et al. (1993), as seen in Fig. 4. Results for light nuclei obey rather well such universal scaling laws. For few-body systems, it is commonly admitted that halos appear when , which corresponds to a probability greater than for the weakly-bound nucleon to be in the forbidden region Jensen et al. (2004).

The results obtained for medium-mass nuclei can also be displayed in Fig. 4 and compared with the generic scaling laws. However, the dimensionless quantities have to be redefined. For medium-mass systems the halo r.m.s. radius is evaluated, by analogy with Ref. Riisager et al. (2000), through

(12)

where is the total neutron r.m.s. radius, while the core r.m.s. radius is approximated by (see Eq. (3)). The reduced mass is taken as the effective isoscalar nucleon mass , while the classical turning point of the central part of the one-body potential is evaluated, by analogy with the finite-well potential, as Fedorov et al. (1993)

(13)

where as we are interested in neutron halos.

In fig. 4, the last bound Cr isotopes are located in-between the and scaling curves, with Cr being closer to the curve than Cr. This is consistent with the admixture of orbitals that builds the corresponding halos, as discussed in Sec. II.2.2 Rotival and Duguet (2009). The neutron density of most medium-mass halo nuclei does not extend as much as those of few-body systems such as Be. Still, Cr and Cr display few-body-like halo properties and the ratio does exceed for Cr. On the contrary, the extension of neutron-rich tin isotopes is not significant enough in regard with their separation energy to be characterized as halo systems. This is consistent with the findings of Paper I.

It would be of interest to place atom-positron complexes in Fig. 4. However, the many-body method used to compute their properties does not allow an easy extraction of the corresponding classical turning point. As a result, the corresponding results do not appear in Fig. 4.

Figure 4: (Color Online) Universality of halos features. The dimensionless extension of the halo is plotted against the dimensionless separation energy for EDF calculations of medium-mass nuclei (filled symbols) and experimental results for few-body systems (open symbols). Generic scaling relationships obtained from a finite spherical well are given in solid lines, while the gray-shaded area corresponds to the halo are, as defined by the commonly-used criterion for few-body halo nuclei .

Iii Technical aspects

The objectives of the remaining part of the present study are (i) to predict which spherical medium-mass nuclei might display halo features, (ii) to check the dependence of the results on several ingredients of the numerical implementation, (iii) to probe the sensitivity of the predictions to the characteristics of the many-body treatment and (iv) to study the specific impact of pairing correlations on halo nuclei.

In the present section, and because halos are extreme systems whose asymptotic must be properly accounted for, we study the dependence of our predictions on certain technical features of the many-body calculation. To do so, ingredients of single-reference EDF calculations are briefly recalled at first.

iii.1 HFB equations

Spherical symmetry is assumed throughout the rest of the paper and spin/isospin indices are sometimes omitted for simplicity. Calculations are performed using a code that takes advantage of the so called “two-basis method” to solve the HFB equations Gall et al. (1994). Thanks to the spherical symmetry, the HFB equation are solved for each block separately in the basis that diagonalizes the single-particle field

(14)

where and are the expansion coefficients of the upper and lower parts of the HFB spinor on the basis , whereas denotes the pairing field. The two-basis method authorizes to perform calculations in very large boxes. The nucleus is put in a spherical box such that wave functions are computed up to a radial distance , with vanishing boundary conditions (Dirichlet) imposed. The value of has to be chosen large enough as to ensure convergence of the calculations (see Sec. III.4). The differential equation to find the is solved on a discrete mesh of step size  fm using the Numerov algorithm Dahlquist and Björck (1974); Bennaceur and Dobaczewski (2005).

In Eq. (14) the upper bound of the sum over the index is not specified. In an actual calculation, the sum is truncated by keeping states up to a certain maximum energy . Its actual value ranges from several MeV up to hundreds of MeVs depending on the method used to tackle the ultra-violet divergence of the local pairing functional (see section III.2.1). In addition to the energy cut, a truncation on the number of partial waves kept in the basis is implemented. In principle, all wave functions below should be kept, but this makes the computation time rather long whereas the wave functions with very high angular momenta will not contribute to the nuclear density. Nonetheless, such a truncation must not be too drastic, in particular for loosely bound nuclei. Checks of convergence with respect to , and of observables of interest in halo nuclei are discussed in Sec. III.4.

iii.2 Energy density functional

iii.2.1 Particle-hole channel

The Skyrme part of the EDF takes, in the case of spherical nuclei, the standard form Bender et al. (2003)

(15)

where , and denote the normal, kinetic and spin-orbit densities, respectively. The parameter is equal to zero or one, depending on whether tensor terms are included or not in the functional, while the index labels isoscalar () and isovector () densities. For protons, the particle-hole part of the EDF is complemented with a Coulomb term whose exchange part is treated within the Slater approximation Bender et al. (2003).

To study the effect of specific features of the particle-hole functional on the formation of halos, a set of Skyrme functionals characterized by different properties is used in the present study: (i) SLy4 stands as a reference point, (ii) SIII displays a different density dependence which leads to a too high infinite matter incompressibility , (iii) T6 has an isoscalar effective nucleon mass , providing a denser single-particle spectrum, (iv) SKa has a low isoscalar effective mass and a different density dependence (density-dependent term with an exponent of instead of ), (v) the functional “” has been specifically adjusted for the present work with the same procedure as for SLy4 but with the constraint , (vi) the parameterizations “” have also been adjusted specifically, with different nuclear matter saturation densities , (vii) T21 to T26 incorporate tensor terms that differ by their neutron-neutron couplings Lesinski et al. (2007). The parameterizations “” and “” have been adjusted using the procedure of Ref. Lesinski et al. (2007) which amounts to reproducing (i) the binding energies and charge radii of Ca, Ni, Zr, Sn and Pb, (ii) the binding energy of Sn, and (iii) the equation of state of pure neutron matter Wiringa et al. (1995) and other standard properties of symmetric nuclear matter as well as the Thomas-Reiche-Kuhn enhancement factor of the isovector giant dipole resonance.

Infinite matter properties of all used parameterizations are summarized in Tab. 4. The isovector effective mass (related to the Thomas-Reiche-Kuhn enhancement factor ) is significantly different for these parameterizations but its effect on static properties of nuclei is rather small Lesinski et al. (2006).

Ref.
SLy4 0.25 Chabanat et al. (1997, 1998)
SIII 0.53 Beiner et al. (1975)
0.25 Lesinski ()
0.25 Lesinski ()
0.25 Lesinski ()
0.25 Lesinski ()
T6 0.00 Tondeur et al. (1984)
SKa 0.94 Köhler (1976)
T21-T26 0.25 Lesinski et al. (2007)
Table 4: Infinite matter properties of Skyrme functionals used in the present study: saturation density  [fm], bulk compressibility  [MeV], isoscalar effective mass , Thomas-Reiche-Kuhn enhancement factor and energy per particle at saturation  [MeV].

iii.2.2 Particle-particle channel

The local pairing functional used

(16)

derives from a Density-Dependent Delta Interaction (DDDI) with the same strength V0 for neutron-neutron and proton-proton pairing. In addition, two parameters and control the spatial dependence of the effective coupling constant. With designating the saturation density of infinite nuclear matter, a zero value of corresponds to a pairing strength that is uniform over the nuclear volume (“volume pairing”) while corresponds to pairing strength which is stronger in the vicinity of the nuclear surface (“surface pairing”). A value corresponds to an intermediate situation (“mixed-type pairing”). The parameter is usually set to one. Values correspond to stronger pairing correlations at low density. In the present work, we are interested in varying such empirical parameters over a large interval of values to quantify how much the characteristics of the pairing functional impact halo systems. Note finally that, all along this work, the strength is chosen so that the neutron spectral gap  Dobaczewski et al. (1996) equals  MeV for Sn. Such a value of provides reasonable gaps in Ca, Sn and Pb regions.

To compensate for the ultra-violet divergence of the pairing density generated by the local pairing functional, a common procedure consists of regularizing all integrals at play through the use of a cutoff Dobaczewski et al. (1984), e.g. on quasiparticle energies . Pairing functionals using such a regularization scheme are referred to as “REG” in the following. In particular REG-S, REG-M and REG-V denote regularized surface-, mixed- and volume-type pairing functionals, respectively. If the parameter differs from 0, 1/2 and 1 or if differs from 1 the functional is generically noted as REG-X.

Using such a regularization method, the pairing strength is adjusted for each cutoff , the latter being eventually taken large enough for observables to be insensitive to its precise value. A widely used value is  MeV Dobaczewski et al. (1984, 1996); Borycki et al. (2006). As the density dependence of the pairing functional is made more surface-peaked, the pairing strength increases, for a fixed value of , as is exemplified in Fig. 5. This is a consequence of the fitting procedure that uses a single nucleus to adjust the overall strength. Indeed, if the pairing strength is peaked at the nuclear surface, individual gaps decrease, especially for well-bound orbitals residing in the nuclear interior. To compensate for this effect and maintain the same value of in one given nucleus, the overall pairing strength has to be increased.

Figure 5: (Color Online) Pairing strength as a function of and for regularized REG-X and in-medium renormalized REN-X pairing functionals.

As an alternative to the sharp cut-off regularization, one can identify and throw away the diverging part of the pairing density through the use of an auxiliary quantity that diverges in the same way but from which the divergence can be subtracted analytically. We follow here the procedure introduced in Refs. Bulgac (); Bulgac and Yu (); Bulgac and Yu (2002); Bulgac (2002); Yu and Bulgac (2003). Using such a renormalization procedure, results become independent of reaches about a few tens of MeVs Bulgac (2002). In the following, calculations of finite nuclei are performed using a conservative value of  MeV (see Sec. III.4.3). Pairing functionals combined with the renormalization scheme are referred to as “REN” in the following. In particular REN-S, REN-M and REN-V denote renormalized surface-, mixed- and volume-type functionals, respectively. If the parameter differs from 0, 1/2 and 1 or if differs from 1 the functional is generically noted as REN-X.

iii.3 Regularization versus renormalization scheme

It was observed in Ref. Dobaczewski et al. (2001) that combining the sharp cut-off regularization method with an extreme surface pairing functional might lead to unrealistic matter and pairing densities. As our goal is to study the influence of the pairing functional attributes on halo properties, it is of importance to characterize such a feature in more detail. As a matter of fact, it can be shown that extreme surface REG-X functionals lead to a spurious Bardeen-Cooper-Schrieffer (BCS) Bose-Einstein-Condensation (BEC) phase transition of infinite matter. On the contrary such a nonphysical transition, associated with the use of a pairing functional that (wrongly) predicts a bound di-neutron system in the zero-density limit Baldo et al. (1995), does not occur with extreme surface REN-X functionals Rotival (2008).

In the present section, we briefly explain that such a spurious feature is in one-to-one correspondence with producing unphysical matter and pairing densities through HFB calculations of finite nuclei. To do so, properties of the last bound Cr isotopes are calculated using different pairing functionals (i) (), which covers from volume to standard surface pairing functionals, and (ii) (), which correspond to extreme surface pairing functionals.

When the regularization scheme is used, one first observes that the position of the drip-line seems to change drastically for extreme surface pairing functionals. For instance, Cr is predicted to be bound against neutron emission  Dobaczewski et al. (1984) for . However, the situation is more subtle than it looks at first. As exemplified by Fig. 6 for Cr, nuclei are in fact no longer bound as a gas of low-density superfluid neutron matter develops as decreases. The normal and pairing (neutron) densities grow at long distances and become uniform beyond a radius  fm for which is precisely the critical value for which infinite nuclear matter undergoes an unphysical BCS-BEC transition Rotival (2008). It becomes in fact energetically favorable for the nucleus to drip bound di-neutrons and create a superfluid gas.

Two remarks are at play at this point. First, it is important to use enough partial waves in the single-particle basis to describe the gas properly. Otherwise, one may not resolve it and conclude that the nucleus displays unusually extended normal and pairing densities Dobaczewski et al. (2001). Here, calculations are performed including all partial waves up to . This is of course an extreme situation and less partial waves, but not too few, have to be used to describe genuine halos as is explained in the next section. Second, the appearance of a uniform gas of bound di-neutrons is driven by the neutron-neutron pairing interaction and not by the proximity of the continuum. As a matter of fact, the fictitious nucleus whose densities are displayed on Fig. 6, and which is held by the box, is still bound against single-nucleon emission (). If we were to increase , the system would turn into a gas of di-neutrons with binding energy  Baldo et al. (1995).

Figure 6: (Color Online) Neutron normal and pairing densities plotted in logarithmic scale for Cr and several regularized REG-X pairing functionals. The calculations are done in a box of  fm and using all partial waves up to .
Figure 7: (Color Online) and parameters for Cr isotopes and different regularized REG-X pairing functionals.

Finally, one concludes that using a functional which wrongly predicts the existence of a bound di-nucleon state in the channel in the vacuum translates into the creation of a spurious low-density di-neutron gas surrounding finite nuclei. Such an observation is of importance regarding the analysis of halo systems as it signals that the use of strongly surface-peaked pairing functionals, combined with the regularized scheme, might lead to un-physical predictions. The danger resides in particular in the use of pairing functionals which are not obviously un-physical, i.e. for which the di-neutron gas is not yet fully developed111Most probably because of inappropriate numerical parameters.. In such cases, the calculation will lead to wrongly predict the existence of gigantic halos, e.g. see in Fig. 7 the unreasonable values of and predicted for the last bound Cr isotopes by REG-X functionals with .

From that point of view, the renormalization scheme is safer as it prevents the problem from happening. Indeed, normal and pair neutron densities remain localized and evolve very little as decreases Rotival (2008). On the other hand, it is crucial to point out that no problem occurs with the regularization scheme either as long as the low-density part of the pairing functional is physically constrained, e.g. are adjusted as to reproduce pairing gaps calculated in infinite nuclear matter through ab-initio calculations Garrido et al. (1999); Matsuo (2006). Eventually, the standard fitting strategy used here is the real source of the potential problem rather than the regularization method itself.

iii.4 Convergence checks

Several basis truncations are introduced under the form of (i) a box of finite radius , (ii) an angular-momentum cutoff for each isospin and (iii) a continuum energy cutoff , to accelerate the convergence of the calculations. Such truncations are physically motivated as (i) the nucleus is localized in space (ii) high-lying unoccupied states are not expected to contribute to ground state properties. However, the values of the truncation parameters have to be carefully chosen not to exclude meaningful physics. As a result, the convergence of observables of interest has to be checked, which we do now. All tests are performed for the chromium isotopes using the {SLy4+REN-M} functional.

iii.4.1 Box radius

The evolution of the halo factors, which are the most critical observables related to halo properties are represented in Figs. 8 for Cr as a function of , for angular-momentum truncations and , and an energy cutoff  MeV. The pairing strength is not refitted for each bin, since the overall effect is found to be negligible.

Figure 8: (Color Online) Evolution of and as a function of the box radius for Cr.

For small box sizes, and are not fully converged and increase with , being the most sensitive quantity. A box radius  fm, for which convergence is achieved for all observables of interest, is used in the following.

iii.4.2 Angular-momentum truncation

The choice of and is critical because the partial-wave truncation impacts the way the continuum is represented in the calculations. Indeed, high- orbitals can contribute to nuclear properties such as the pairing gaps. To quantify such an effect, we calculate, in the canonical basis, the probability distribution of particle and pair occupations as a function of the single-particle angular-momentum

(18)

The neutron distribution is shown in Fig. 9 for the halo nucleus Cr and different angular momentum truncations . It is found that the strength is mostly distributed over states with . As a result, such an occupation distribution is converged, at least to first approximation, for . The corresponding distribution (Fig. 10) extends much further towards high values. This could be expected as is maximum for deeply bound canonical states whereas is peaked around the Fermi level and decay slower as the canonical energy increases above . Correspondingly, the local pair density extends further out in space than the normal density  Dobaczewski et al. (1984). As one goes to drip-line (halo) nuclei in particular, extends very far out and requires many partial waves to be well represented. This was already highlighted in Ref. Matsuo et al. (2005). It is clear from Fig. 10 that a minimal cutoff of is needed to achieve a reliable enough description of the pair distribution. One sees in particular that for (i) the missing strength at high is wrongly redistributed over and , making those states more paired and thus more localized, whereas (ii) some of the strength of the states is transferred to states. Considering that and states are precisely those building up the halo in Cr, the latter two effects associated with using a two small angular momentum cutoff inhibits artificially the formation of the halo. The results are qualitatively identical for the regularization scheme: in both cases, rather high values of are needed to properly describe the continuum.

Figure 9: (Color Online) Distribution of neutron canonical states occupations as fractions of the total strength for Cr computed with {SLy4+REN-M} functionals and different angular momentum truncations .
Figure 10: (Color Online) Same as Fig. 9 for the distribution of .

Indeed, this is what is observed in Fig. 11 where and are given for Cr as a function of . Both quantities reach a plateau around .

Figure 11: (Color Online) Evolution of and in Cr and {SLy4+REN-M} functionals for different angular momentum truncations .

The previous analysis shows that HFB calculations with too small values of and cannot be trusted at the limits of stability if one is interested in detailed information about potential halos. Of course, considering the ultimate experimental accuracy achievable for matter r.m.s., one should not be too extreme as far as the required convergence is concerned. From a theoretical perspective, and considering the theoretical error bars on the determination of and , it is necessary to include partial waves up to .

iii.4.3 Energy cutoff

The value of the energy cutoff in the quasiparticle continuum is an important parameter of the calculation.

For regularized pairing functionals, the values of and must be taken large enough that, including a renormalization of the coupling strength through the re-fitting of data, the observables of interest do not depend on their precise values. It was found that smaller basis truncations could be used for the REG case than for the REN case, as convergence is reached faster, as exemplified in Figs. 12 and 13. For the REG case, convergence for the ground state energy as well as for the neutron pairing gap is almost achieved for and MeV.

For renormalized functionals, the situation is more subtle. First, must be large enough for the result to be independent of its value Bulgac and Yu (). However, it must be remembered that the field theory renormalization scheme subtracts a diverging part on the basis that all partial waves below a certain energy cutoff have been included. Thus, for a given (high enough) , the angular momentum truncation must be large enough to prevent the counter term from removing contributions of states that were not considered in the first place. This is illustrated in Figs. 14 and 15 which display the binding energy and neutron spectral gap of Cr as a function of the angular momentum cutoff , for fixed values of . Note that all values considered are large enough to obtain converged observables. However, one sees that increasing the energy cutoff necessitates a larger number of partial waves to reach the converged values for both the energy and the gap. Consequently, it can be counter-productive to use a safe but unnecessarily large energy cut as it results in the necessity to also increase . On the other hand, the proper description of certain physical phenomena such as halos intrinsically requires a large number of partial waves as discussed in the previous section. In such a case, one first fixes and makes sure to use a coherent energy cutoff. In the present work, we use  MeV and . This corresponds to an conservative choice as  MeV and would be sufficient.

Figure 12: (Color Online) Binding energy as a function of the angular momentum cutoff for fixed values of as obtained from {SLy4+REG-M} functional for Cr.
Figure 13: (Color Online) Same as in Fig. 12 for the neutron spectral gap.
Figure 14: (Color Online) Same as in Fig. 12 for the {SLy4+REN-M} functional.
Figure 15: (Color Online) Same as in Fig. 13 for the {SLy4+REN-M} functional.

Iv Impact of pairing correlations

iv.1 Pairing anti-halo effect

In the presence of pairing correlations, the asymptotic of the one-body neutron density takes a different form from the one it has in the EDF treatment based on an auxiliary Slater determinant Rotival and Duguet (2009). Indeed, the decay constant reads as , with , being the lowest quasiparticle excitation energy. Considering a canonical state lying at the Fermi level near the drip-line (), one finds that . Therefore, in first approximation, paired densities decrease faster than unpaired ones and pairing correlations induce an anti-halo effect by localizing the one-body density Bennaceur et al. (1999, 2000); Yamagami (2005).

To evaluate the quantitative impact of this effect, drip-line Cr isotopes have been calculated with and without explicit treatment of pairing correlations. In the latter case, a filling approximation Perez-Martin and Robledo (2008) is used for incomplete spherical shells. In both cases, the SLy4 Skyrme functional is used. When including pairing, a mixed-type pairing is added. A comparison between neutrons single-particle levels is represented in Fig. 16 for the last bound nuclei, Cr being predicted to be bound when pairing correlations are excluded from the treatment. This is interesting in itself as it shows that pairing correlations can change the position of the drip-line and modify in this way the number of halo candidates over the nuclear chart. There is only little difference between the canonical energies in the two cases222The canonical basis is identical to the eigenbasis of is the zero-pairing limit.. However, the values of the halo criteria and are significantly different, i.e. the neutron halo is significantly quenched in Cr when pairing is omitted whereas the situation is reversed in the lighter isotopes, as seen in Fig. 18.

Figure 16: (Color Online) Neutron single-particle energies along the Cr isotopic chain. Left: eigenvalues of obtained when omitting pairing correlations in the calculation. The non-resonant continuum spectrum has been omitted while resonant states with positive are drawn as discrete states for convenience. Right: canonical energies obtained including pairing correlations through a mixed-type REG-M functional. The Skyrme SLy4 functional is used in both cases. The conventions used for single-particle states labeling are identical to Ref. Rotival and Duguet (2009) and are recalled in Fig. 17. The Fermi level () is set to the last filled orbital for the EDF treatment based on an auxiliary Slater determinant.
Figure 17: (Color Online) Conventions used in all the figures for the labeling of individual states and of the chemical potential.

Such results underline that pairing correlations affect halos in two opposite ways. Pairing (i) inhibits the formation of halos through the anti-halo effect (ii) enhance the formation of halos by scattering nucleons to less bound states with smaller decay constants. For example, the anti-halo effect dominates in Cr and Cr whereas the promotion of neutrons into the weakly bound makes the halo to be more pronounced in Cr when pairing is included.

When pairing is omitted, the number of nucleons in the orbital increases linearly as one goes from Cr to Cr. At the same time, the shell becomes more bound as a result of self-consistency effects. This explains the negative curvature of as one goes from Cr to Cr, while is indeed almost linear. In Cr, two effects contribute to the very pronounced halo that is predicted in the absence of pairing correlations (i) the gets fully occupied whereas (ii) the pairing anti-halo effect is inoperative. The results discussed above are further illustrated in Fig. 19 where the contributions of different single-particle states to the halo are shown.

Figure 18: (Color Online) Halo criteria and for Cr isotopes, obtained through spherical HF (solid lines) and HFB (dashed lines) calculations with the Skyrme SLy4 functional. For HFB calculations, mixed-type REG-M pairing is used.
Figure 19: (Color Online) Contributions of individual orbital to the halo: EDF calculation of the least bound Cr isotopes with (bottom) and without (top) pairing. The orbital corresponds to vertically-dashed areas, and orbital to filled ones. The other contributions are found to be negligible.
Cr (pairing)
 [MeV]
Cr (no pairing)
 [MeV]
Cr (pairing)
 [MeV]
??????
Table 5: , individual contributions to the halo (see Paper I for the definition of ), single-particle canonical energies and occupations for (i) Cr with pairing (ii) Cr without pairing, and (iii) Cr with pairing but using the single particle spectrum calculated in the absence of pairing.

To specifically extract the pairing anti-halo effect, we now perform a toy model calculation of a fictitious Cr nucleus. Filling the single-particle wave-functions obtained from the calculation of Cr without pairing correlations with the canonical occupations obtained from the calculation of Cr with pairing, we extract from each orbital333Single-particle states extracted from the calculation of Cr without pairing cannot be used because the essential orbital belongs to the continuum and has plane wave asymptotic in this case.. Such a procedure allows one to isolate, in semi-quantitative manner, the net change of due to the difference in the asymptotic of paired orbitals keeping their occupation fixed. Doing so, it is found that the contribution of the state to the halo is smaller in Cr than in Cr by about . For the orbital the suppression of is about (the value of is also different for Cr and Cr, but this does not affect the results significantly). As a result, one sees that if the anti-halo effect were ineffective, the scattering of particles into higher-lying orbitals would bring from to . Instead, is only increased to in the full fledged calculation of Cr with pairing correlations, i.e. the anti-halo effect reduces the potential increase by .

To conclude, there is no simple answer as to whether pairing correlations enhance or hinder the formation of halos. The net result depends on structure details of the particular nucleus of interest Grasso et al. (2001, 2006).

iv.2 Decorrelation from the pairing field

An additional effect might come into play as far as the role of pairing in the formation of halos is concerned. Very weakly bound orbitals are expected to decouple from the pairing field as the neutron separation energy goes to zero Hamamoto and Mottelson (2003, 2004); Hamamoto (2005). As a result, such an orbital would not be subject to the anti-halo effect and may develop a very long tail.

The signature of this effect can be seen in the single-particle occupation profile. The canonical occupations of all neutron single-particle states in all drip-line Cr isotopes are gathered in Fig. 20 and plotted as a function of , where is the neutron canonical energy and the neutron Fermi level. Those occupations are compared to the BCS formula

(19)

calculated using the maximum/minimum neutron spectral gap found among all drip-line Cr isotopes. The calculations have been performed using the {SLY4+REG-M} functional.

Figure 20: (Color Online) Neutron canonical occupations as a function of for all drip-line Cr isotopes. The calculations are done with the {SLY4+REG-M} functional. The profiles computed from the BCS-type formulæ  using the minimum (dashed line) and maximum (dashed-dotted line) spectral neutron gaps among all those isotopes are shown for reference. Conventions for labeling individual states are given in Fig. 17.

The -wave occupation probability follows closely the BCS-type profile calculated using the minimal spectral gap. The high- orbitals follow well the BCS-type profile computed with the large spectral gap. This corroborates the trend discussed in Refs. Hamamoto and Mottelson (2003, 2004); Hamamoto (2005) and underlines that orbitals with are less paired than high- ones. This is also confirmed by looking at individual gaps given in Fig. 21. The state displays a smaller gap than other orbitals as it approaches the Fermi level (from above), the latter reaching the continuum threshold.

On the other hand, the canonical gap of the state remains significant as it crosses the Fermi level and the anti-halo effect is still in effect for that orbital. In fact, the critical decoupling of orbitals from the pairing field, discussed in Refs. Hamamoto and Mottelson (2003, 2004); Hamamoto (2005) through schematic HFB calculations, becomes fully effective only when the Fermi level and the state energy are both of the order of a few keVs. Such extreme situations (i) are not reached in realistic systems Hamamoto () (ii) would require an accuracy on the predicted value of the separation energy which is far beyond present capacities of EDF calculations.

Figure 21: (Color Online) Individual neutron canonical gaps computed for drip-line Cr isotopes with the {SLY4+REG-M} functional and plotted as a function of . Conventions for individual states are given in Fig. 17.

iv.3 Importance of low densities

The values of and entering the pairing functional strongly affect the spatial localization of the pairing field, and thus the gaps of weakly-bound orbitals lying at the nuclear surface.

In previous studies Dobaczewski et al. (2001), it was found that the size of a neutron halo could change by one order of magnitude when the pairing functional evolves from a volume to a extreme-surface type. However (i) the evaluation of the halo size was performed through the Helm model, the limitations of which have been pointed out in Ref. Rotival and Duguet (2009), and (ii) the standard regularization scheme was used with extreme-surface pairing functionals which, as discussed in Sec. III.3, leads to unsafe predictions of halo properties. The renormalization scheme is used in the present section as it prevents any un-physical feature from appearing with extreme surface pairing functionals. The properties of the last bound Cr isotopes are evaluated for different pairing functionals: (i) and , and (ii) and .

Figure 22: (Color Online) parameter for Cr isotopes calculated using renormalized REN-X pairing functionals with different form factors (see text). Results for regularized REG-X functionals diverge for very strong surface pairing correlations and are represented in dashed lines.
Figure 23: (Color Online) Same as Fig. 22 for .

Overall, neutron canonical energies evolve very little with . The evolution of canonical pairing gaps with is presented in Fig. 24. For surface-enhanced pairing functionals, well-bound orbitals residing in the center of the nucleus become less paired. On the other hand, pairing gaps of states close to the Fermi level increase as the effective pairing strength becomes more important at the nuclear surface444Keeping the neutron spectral gap unchanged in Sn, and as some orbitals become less paired, others display larger individual gaps.. Considering the theoretical error bars, the values of and