Modeling $^{19}$B as a $^{17}$B-n-n three-body system in the unitary limit

TL;DR

This paper models the $^{19}$B nucleus as a three-body system near the unitary limit, predicting bound states and resonances, and discusses connections to Efimov physics, with implications for understanding neighboring isotopes.

Contribution

It introduces a three-body model for $^{19}$B near the unitary limit, predicting resonances and exploring Efimov physics relevance.

Findings

Successfully describes $^{17}$B ground state with two-body potentials.

Predicts two low-lying resonances in $^{19}$B.

Discusses potential links to Efimov physics.

Abstract

We present a model description of the bound $^{17}$B isotope in terms of a $^{17}$B-n-n three-body system where the two-body subsystems $^{17}$B-n and n-n are unbound (virtual) states close to the unitary limit. The $^{17}$B ground state is well described in terms of two-body potentials only, and two low-lying resonances are predicted. Their eventual link with the Efimov physics is discussed. This model can be naturally used to describe the recently discovered resonant states in $^{20,21}$B.

Equations

$$V_{n^{17}\text{B}}(r)=V_r\left(e^{-\mu r}-e^{-\mu R}\right)\frac{e^{-\mu r}}{r}$$

$$V_{n^{17}\text{B}}(r)=V_c(r)+({\vec{S}}_n\cdot {\vec{S}}_{{}^{17}\text{B}})V_{ss}(r)$$

$$2({\vec{S}}_n\cdot {\vec{S}}_{{}^{17}\text{B}})=S(S+1)-\frac{9}{2}$$

$$4V_{n^{17}\text{B}}^{(S=1)}$$

$$4V_{n^{17}\text{B}}^{(S=2)}$$

$$V_i(r)=V_r^{(i)}\left(e^{-\mu r}-e^{-\mu R}\right)\frac{e^{-\mu r}}{r}i=c,ss$$

$$V_r^{(c)}=\frac{1}{8}\left(3V_r^{(1)}+5V_r^{(2)}\right)V_r^{(ss)}=\frac{1}{2}\left(V_r^{(2)}-V_r^{(1)}\right)$$

$$V_{nn}=V_R\frac{e^{-{\mu }_{R}r}}{r}-V_A\frac{e^{-{\mu }_{A}r}}{r}$$

Full text

I Introduction

The interaction of a low-energy neutron ($n$) with a nucleus of mass number $A$ (with spin-parity $J^{\pi}_{A}$) is a balance between the attractive folded neutron-nucleus potential $V_{nA}$ and the effective Pauli repulsion between the incident and target neutrons. For $A=1$, all the $n$-$A$ spin ($S=J_{A}\pm 1/2$) and isospin ($T$) channels are attractive: this attraction manifests through the large and negative values of the singlet scattering lengths ($a^{nn}_{0}=-18.5$ fm and $a^{np}_{0}=-23.5$ fm) and by the existence of the 2H bound state in the only triplet $n$-$p$ (${}^{3}S_{1}$) channel KRS_ADNDT_1991 .

The first consequence of Pauli repulsion already manifests for $A=2$, in the neutron scattering on 2H in the $S=3/2^{+}$ channel. The scattering length in this channel is positive, $a_{3/2}=6.35$ fm, and reflects a strong repulsion, while the $S=1/2^{+}$ channel, despite having a positive scattering length $a_{1/2}=0.65$ fm, remains attractive accommodating the 3H bound state. For $A=3$ and $T=1$ both spin channels are repulsive with $a_{0}=5.2$ fm and $a_{1}=4.8$ fm. The same happens in the case of $n$-3He (isospin mixture) with $a_{0}=5.9$ fm (its real part) and $a_{1}=3.1$ fm. The only existing reaction with $A=4$, $n$-4He, is also repulsive with $a_{1/2}=2.61$ fm.

When increasing the neutron number in the target, the valence neutrons start filling $p$ (and higher angular momentum) orbitals, the Pauli principle becomes less constraining and the net balance is again attractive with the corresponding negative values of $a_{S}$, starting from the 7Li ground state $J^{\pi}=3/2^{-}$ with $a_{2}=-3.6$ fm. This is also true for 8He with $a_{S}\sim-3$ fm Johansson ; AlFalou and 9Li with $a_{S}\sim-14$ fm AlFalou , although in these cases the $n$-$A$ total spin $S$ was not determined.

For even higher numbers of neutrons, the virtual state of the $n$-$A$ system starts approaching the threshold. A spectacular consequence of this trend manifests with the 17B isotope, where the $n$-17B virtual state is located at the extreme vicinity of 18B ground-state threshold. This effect is reflected by the largest value of the neutron-nucleus scattering length observed so far, $a_{S}\sim-100$ fm MSU_2012 . However, the limited resolution and acceptance of the experiment did not allow to fix a lower bound for the scattering length MSU_2012 , and only an upper bound $a_{S}<-50$ fm was determined.

Despite this experimental uncertainty, the potentially huge $n$-17B scattering length confers a unique and intriguing character to 19B, a two-neutron Borromean halo nucleus B17-19_RMS exhibiting a weakly bound core-$n$-$n$ structure in which all the two-body subsystems are unbound Borromean . Moreover, due to its extremely weak binding ($2n$ separation energy of $S_{2n}=0.14\pm 0.39$ MeV B19_mass ), 19B has no bound excited states. As such, the ground state can be seen as an extended three-body system resulting from the very large scattering lengths of its two two-body components (of the order of $-20$ and $-100$ fm respectively).

The aim of this short note is to present a dynamical model describing simultaneously both structures, $n$-17B scattering and the ground state of 19B, which could be easily applicable to the description of the heavier boron isotopes 20,21B recently observed in Ref. 20B_PRL121_2018 . The model is based on a two-body $n$-17B local potential, built to this aim, supplemented with a realistic $n$-$n$ interaction.

The same dynamical approach was used in Ref. Mazumdar2000 to describe 19B. However, it overbound the ground state and predicted bound excited states due to the purely attractive $n$-17B non-local interaction. Similar models, based on EFT renormalized zero-range $n$-core interaction plus a three-body force, were applied to another Borromean nucleus, 22C 22C_Yamashita_2011 ; 22C_Phillips_2013 . Although the large scattering length values involved in the 19B subsystems make natural its description in terms of EFT zero-range plus three-body forces, we have preferred in view of further applications to rely only on two-body finite-range interactions. A more consistent attempt to link the $n$-$A$ low-energy parameters (LEP) with the reaction cross sections with the target nucleus was proposed in Refs. Blanchon_ NPA739_2004 ; Blanchon_NPA784_2007 .

Due to the experimental uncertainties in the relevant observables, respectively the $n$-17B scattering length and 19B binding energy, we will explore the possible range of those parameters up to the unitary limit. Finally, our model predicts the first excited states of 19B to be of resonant character, in agreement with experimental observations Gibelin_FB22 .

II Neutron-nucleus potential

It is assumed that a low-energy neutron approaching 17B feels a short-range repulsion due to the Pauli principle with respect to the 12 neutrons in the 17B core, plus a loose attraction due to the folded $n$-$A$ interaction. A simple form accounting for these facts can be:

[TABLE]

where $R$ is a hard-core radius, hindering the penetration of the incoming neutron in the nucleus, and $\mu$ is a range parameter for the folded $n$-17B potential. An educated value can be $R=3$ fm, which corresponds to the r.m.s. matter radius of 17B B17-19_RMS , and we take $\mu=0.7$ fm*-1* corresponding to the pion mass.

Once the range $\mu$ and the size $R$ are fixed, the potential depends on a single strength parameter $V_{r}$, which will be adjusted to reproduce the scattering length $a_{S}$. The numerical values along this work correspond to $m_{n}=939.5654$ MeV, $m_{{}^{17}{\text{B}}}=15879.1$ MeV, i.e. a $n$-17B reduced mass $m_{R}=887.0771$ MeV and ${\hbar^{2}/2m_{R}}=21.9473$ MeV$\cdot$fm2.

The dependence of the $n$-17B scattering length $a_{S}$ on the strength parameter $V_{r}$ within this model is displayed in Fig. 1. The $n$-17B potentials of Eq. (1) corresponding to $a_{S}=-50,-100,-150$ fm (dashed lines in Fig. 1) are displayed in Fig. 2. As one can see, despite the large variation of $a_{S}$ the potentials quickly saturate when approaching the unitary limit. For example, when $a_{S}$ varies from $-50$ to $-100$ fm the potential changes by only a few tens of keV, and at this scale it looks almost independent beyond those values.

Until now we have ignored any spin-spin effect in the $n$-17B interaction (spin symmetric approximation). In fact 17B is a $J^{\pi}=3/2^{-}$ state and can couple to a neutron in two different spin states, $S=1^{-},2^{-}$. If the resonant scattering length is due to $S=2^{-}$ state, as it is assumed in MSU_2012 , there is no reason for the $S=1^{-}$ to be resonant as well. To account for this eventual asymmetry in further calculations, we have introduced a spin-dependent interaction by assuming for each spin $S$-channel a potential $V_{n^{17}{\text{B}}}^{(S)}$ having the same form as Eq. (1) and driven by the corresponding strength parameter $V_{r}^{(S)}$. This can be cast in a single expression using the standard operator form:

[TABLE]

in terms of the spin-spin operator:

[TABLE]

and the central $V_{c}$ and spin-spin $V_{ss}$ components which are expressed in terms of $V_{n^{17}{\text{B}}}^{(S)}$ by inverting the linear system obtained from Eq. (2) for $S=1$ and $S=2$:

[TABLE]

It follows from that, that the central ($V_{c}$) and spin-spin ($V_{ss}$) components have the same form as Eq. (1):

[TABLE]

with the strength coefficients given by:

[TABLE]

Using this $n$-17B potential, the interest of our model lies in the description of the more complex systems composed by 17B and several neutrons in terms of two-body interactions. The first step in this direction is 19B, which will be the subject of the next section.

III 19B results

The 19B nucleus will be described as a 17B-$n$-$n$ three-body system. The $n$-17B potential of Eq. (1), supplemented by a $n$-$n$ interaction, will constitute the three-body Hamiltonian. This description is physically justified by the large values of the scattering length in each two-body subsystem.

For the $n$-$n$ potential, we have chosen two different models: the Bonn A potential Bonn_A and a charge dependent (CD) version of the semi-realistic MT13 interaction MT_NPA127_1969 . The Bonn A potential, being charge independent, provides the $n$-$n$ low-energy parameters $a_{nn}=-23.75$ fm and $r_{nn}=2.77$ fm and acts in all partial waves.

We have built a CD version of MT13 starting from the original parameters of the NN singlet state (that is $V_{R}=1438.720$ MeV$\cdot$fm, $\mu_{R}=3.11$ fm, $\mu_{A}=1.55$ fm):

[TABLE]

and adjusting the strength of the attractive term to $V_{A}=509.40$ MeV$\cdot$fm in order to reproduce the $n$-$n$ LEP $a_{nn}=-18.59$ fm and $r_{nn}=2.93$ fm, in close agreement with the experimental values. In order to cross-check the results, the three-body problem was solved independently by using two different formalisms: Faddeev equations in configuration space LC_3n_PRC72_2005 ; FE_3B_20054 and the Gaussian Expansion Method Hiya03 .

We have first computed the 19B ground-state energy $E(^{19}$B) as a function of the scattering length $a_{S}$ in the spin-symmetric case, that is with $V_{r}^{(1)}=V_{r}^{(2)}=V_{r}$. Results, measured with respect to the 17B-$n$-$n$ threshold (i.e. $-S_{2n}$), are displayed in Fig. 3. They concern two different versions of the model: (i) a purely $S$-wave interaction both in $V_{n^{17}{\text{B}}}$ and in $V_{nn}$ (solid blue line) and (ii) letting the interaction of Eq. (1) act in all partial waves (solid black line) supplied with the Bonn A model for $V_{nn}$.

In view of these results, the following remarks are in order:

• •

The quantum numbers of 19B ground state are $L=0$ and $S=3/2$, which are separately conserved, and so $J^{\pi}(^{19}\text{B})=3/2^{-}$. Notice that since the total angular momentum is $L=0$ the total parity is given by the intrinsic parity of 17B, which is a $3/2^{-}$ state.

• •

In the range of $a_{S}$ values compatible with the experiment ($a_{S}<-50$ fm MSU_2012 ) and in both versions of our model, 19B is bound. Its binding energy decreases when $a_{S}$ increases and the binding disappears for a critical value of the scattering length $a_{S}^{c}$ which slightly depends on the model version: $a_{S}^{c}\approx-30$ fm for (i) and $a_{S}^{c}\approx-15$ fm for (ii).

• •

The 19B binding energy is compatible with the experimental value $E=-0.14\pm 0.39$ MeV B19_mass for both versions of the model and in all the range of $a_{S}$, starting from $a_{S}=-50$ fm until the unitary limit in the $n$-17B channel, i.e. $a_{S}\to-\infty$. This limit (dotted lines) corresponds to $E_{u}=-0.081$ MeV in version (i) (blue) and $E_{u}=-0.185$ MeV in version (ii) (black). Note that in both cases the ground-state energy for $a_{S}=-150$ fm is only $\sim 20$–30 keV distant from the unitary limit.

• •

It is interesting to consider also the unitary limit in the $n$-$n$ channel, i.e. $a_{nn}\to-\infty$. This is realized, in the purely $S$-wave model version (i), by setting $V_{A}\approx 531.0$ MeV in the $n$-$n$ potential of Eq. (6). The full unitary result, where both $a_{S}=a_{nn}\to-\infty$, is indicated by the dashed blue line in this figure. It corresponds to a 19B energy $E_{uu}=-0.160$ MeV, compatible with the experimental result. The ab-initio nuclear physics in the $S$-wave unitary limit of the NN interaction was recently considered in Ref. Unitary_NP_PRL118_2017 . It was claimed that the gross properties of the nuclear chart were already determined by very simple NN interactions, provided they ensure $a_{0}=a_{1}\to-\infty$. The 19B ground state constitutes a nice illustration of this remarkable property, though at the level of cluster description and based on other dynamical contents.

• •

The main difference between version (i) and (ii), solid blue and black lines in Fig. 3 respectively, is essentially due to the contribution of the higher angular-momentum terms in $V_{n^{17}{\text{B}}}$. In the present state of experimental knowledge, this contribution is totally uncontrolled but we have made an attempt to quantify it by assuming $V$ to be the same in all partial waves. For $a_{S}=-150$ fm this accounts for about 60 keV of extra binding. The difference due to the $n$-$n$ interaction is smaller, and mainly given by the fact that Bonn A model is charge independent and has a larger $a_{nn}$ absolute value. Once this is corrected, the difference in the 19B binding energy at $a_{S}=-150$ fm reduces to 8 keV, which is mainly accounted for the higher partial waves in the $n$-$n$ interaction. As expected from the Effective Field Theory arguments, the details of the interaction turn out to be negligible for a system close to the unitary limit.

The spatial probability amplitude, i.e. the squared modulus of the total wave function $\Psi(r,R)$ in terms of the Jacobi coordinates, is represented in Fig. 4 (upper panel), for $a_{S}=-100$ fm and $E=-0.130$ MeV in model version (ii). For the sake of comparison, we have computed the same amplitude and in the same scale of another two-neutron halo nucleus, 6He ($E=-0.97$ MeV), using the $\alpha$-$n$-$n$ three-body model from Ref.  Hiyama_he6 . It is displayed in the lower panel of the same Fig. 4. Due to the very weak binding energy, the wave function of 19B is much more extended. Unlike 6He wave function it displays very large asymmetry, being strongly elongated in the 17B-$(nn)$ direction. An exceptional feature of 19B is the presence of the void in the few fm space around the center of mass of the three clusters 17B-$n$-$n$. This feature establishes 19B as a veritable two-neutron halo nucleus having a molecule-like structure with three centers.

Besides providing a satisfactory description of the 19B ground state, our model accommodates two broad resonances. Letting the interaction of Eq. (1) act in all partial waves with $a_{S}=-150$ fm ($V_{r}$=4089.9 MeV$\cdot$fm) and adopting the $S$-wave model for the $n$-$n$ interaction, two resonant states are found for total angular momentum $L=1$ and $L=2$, with energies $E_{1^{-}}\approx 0.24(2)-0.31(4)i$ MeV and $E_{2^{+}}\approx 1.02(5)-1.22(6)i$ MeV. The parameters of these resonances however depend on the value of $a_{S}$, which remains unknown. Their width display also a strong dependence on the size parameter ($R$) of the $n$-17B interaction, that we fixed once at all to $R=3$ fm: if $R$ is increased, keeping $a_{S}$ fixed, the resonance widths decrease. Further experimental data are needed for a fine tuning of these parameters.

In fact, several resonances in the continuum of 19B have been recently observed Gibelin_FB22 , although their precise energy and quantum numbers have not been determined yet. It is worth noting that despite its simplicity, our model of the $n$-17B interaction is able to account for the 18B virtual state, the 19B ground state and two resonances without any need of three-body forces. This follows from the strong resonant character of both $n$-17B and $n$-$n$ channels as well as the large spatial extension of the 19B ground state.

If we introduce the spin dependence of the interaction, Eq. (2), and assume the non-resonant scattering length $a_{1}$ to be smaller in absolute value than $a_{2}$, the weaker potential leads to 19B binding energies smaller than those in Fig. 3. For example, if we fix $a_{2}=-150$ fm the 19B binding energy decreases when $a_{1}$ varies from the spin-symmetric case $a_{1}=a_{2}=a_{S}$ up to some critical value $a_{1}^{c}$, still negative, beyond which 19B is no longer bound. The results are displayed in Fig. 5.

The very existence of this critical value $a_{1}^{c}$, as well as its negative sign, is independent of the model version (blue and black solid lines) and the size parameter $R$, that was also varied here to check the robustness of this result (dashed blue lines). In all the cases that we have explored, its value lies within $-7<a_{1}^{c}<-3$ fm (Fig. 5). On the other hand, in order to reach the critical value $a_{1}^{c}$ an extremely strong spin-spin interaction is required with $V^{(1)}_{r}/V^{(2)}_{r}\sim 0.6$, for the case $a_{2}=-150$ fm. We may hardly find any physical arguments to support the existence of such a strong spin-dependence in the $n$-17B interaction. Therefore, we conclude that the value of $a_{1}$ should be also negative and not small, and that 19B remains bound even when the spin symmetry is broken.

The proximity of the $n$-17B interaction to the unitary limit strongly suggests that 17B could be a genuine nuclear candidate to exhibit the Efimov physics NaidonEndo , that is the existence of a family of bound states whose consecutive energies are scaled by a universal factor $f^{2}$. This is what would happen by setting $a_{1}=a_{2}=a_{nn}=-\infty$ as in the blue dashed line of Fig. 3. However the 17B-$n$-$n$ system, representing a light-light-heavy structure, turns out to be quite an unfavorable case to exhibit the sequence of excited Efimov states due to the requirement of a very large factor $f$. However, in the real world, as well as in our model, 19B has only one bound state and it is governed by three different scattering lengths, from which only the $n$-$n$ scattering length is relatively well known and can be fixed to its experimental value. For the case when only the $n$-17B interaction is tuned the universal factor turns out to be $f\approx 2000$ NaidonEndo ; Hove_Efimov . It follows that the appearance of the first $L=0^{+}$ excited state of Efimov nature in 19B would manifest only when the $n$-17B scattering length reaches several thousands of fm. We conclude that, independently of the particular value of $a_{S}$, it is highly unlikely to observe any Efimov excited state in 19B. On the other hand, the universal features related to Efimov physics Mario are however genuinely preserved in this system.

It is worth mentioning that an hypothetical molecular-like 17B-17B-$n$ system would constitute a favorable heavy-heavy-light structure in order to generate Efimov states NaidonEndo . However, the long-range part of the effective hyper-radial interaction in this system would be dominated by the repulsive Coulomb interaction between the 17B nuclei, largely exceeding the $1/r^{2}$ tail of Efimovian attraction.

IV Summary

Motivated by the recent experimental results on heavy boron isotopes MSU_2012 ; Gibelin_FB22 ; 20B_PRL121_2018 , we have developed a model to describe the main phenomenological facts of these neutron-rich nuclei. It is based on a $n$-17B effective interaction which, supplemented with a realistic $n$-$n$ potential, provides a satisfactory description of 19B ground and resonant states in terms of a 17B-$n$-$n$ three-body system.

The key ingredient of the model is the proper description of the extremely shallow virtual state of 18B, which manifests through a very large and negative value of the $n$-17B scattering length, $a_{S}<-50$ fm, in the $S=2$ channel MSU_2012 .

It is worth noticing that despite its simplicity, the model is able to account for the $n$-17B virtual state, the 19B ground state and two broad resonances without introducing any three-body force. Such a possibility is certainly due to the strong resonant character of the $n$-17B and $n$-$n$ channels as well as the small binding energy and large spatial extension of 19B. In the spin-independent approximation, 19B is bound for $a_{S}<-30$ fm. Its binding energy is compatible with the experimental value for all the range of $a_{S}$ until the unitary limit $a_{S}\to-\infty$.

We have considered the stability of our 19B results when a spin-spin interaction term is introduced in $V_{n^{17}{\text{B}}}$. By fixing a resonant value in the $S=2$ channel ($a_{2}=-150$ fm), we found that the system remains bound for a large domain of the $S=1$ scattering length $a_{1}<a_{1}^{c}<0$. The existence of this negative critical value $a_{1}^{c}$, beyond which the binding disappears, is independent of the model parameters, and its particular value depends only slightly on them. One has typically $a_{1}^{c}\sim-5$ fm. Such a large asymmetry of $a_{S}$, however, requires unphysically large differences among the potential strengths of the two spin channels.

Work is in progress to extend the application of this model to describe the recently observed resonances in 20B and 21B 20B_PRL121_2018 in terms of 17B+$3n$ and +$4n$, respectively, by solving the corresponding four- and five-body equations using the techniques developed in Ref. LHC_PLB791_2019 . On the other hand, the precise measurement of the main experimental observables related to this problem, such as the 18B scattering length, the 19B binding energy, and the energy and quantum numbers of the first 19B resonant states, would provide a more stringent constraint to the parameters of the present model, and could eventually suggest the introduction of $n$-17B $P$-waves and/or 17B-$n$-$n$ three-body forces.

Acknowledgments

The authors thank Valérie Lapoux and Anna Corsi for helpful discussions on the experimental $n$-$A$ scattering lengths. We were granted access to the HPC resources of TGCC/IDRIS under the allocation 2018-A0030506006 made by GENCI (Grand Equipement National de Calcul Intensif). This work was supported by French IN2P3 for a theory project “Neutron-rich light unstable nuclei” and by the Japanese Grant-in-Aid for Scientific Research on Innovative Areas (No.18H05407).