aa r X i v : . [ nu c l - t h ] S e p Hyperon halo structure of C and B isotopes
Ying Zhang,
1, 2, ∗ Hiroyuki Sagawa,
2, 3, † and Emiko Hiyama
4, 2, ‡ Department of Physics, School of Science,Tianjin University, Tianjin, 300354, China RIKEN Nishina Center, Wako, Saitama 351-0198, Japan Center for Mathematics and Physics, the University of Aizu,Aizu-Wakamatsu, Fukushima 965-8580, Japan Department of Physics, Kyushu University, Fukuoka, 819-0395, Japan (Dated: September 29, 2020)
Abstract
We study the hypernuclei of C and B isotopes by Hartree-Fock model with Skyrme-type nucleon-nucleon and nucleon-hyperon interactions. The calculated Λ binding energies agree well with theavailable experiment data. We found halo structure in the hyperon 1 p -state with extended wavefunction beyond nuclear surface in the light C and B isotopes. We also found the enhanced electricdipole transition between 1 p - and 1 s -hyperon states, which could be the evidence for this hyperonhalo structure. PACS numbers: 21.80.+a, 21.60.Jz, 21.10.Gv. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION Since the halo structure of Li was observed in 1985 [1], the halo phenomena have beenstudied intensively experimentally and theoretically [2–5] in nuclei near and beyond theneutron and also proton drip lines. The halo nuclei are characterized by its extended densityprofile far beyond the nuclear surface region. Very much enhanced electric dipole transitionshave been also observed in several halo nuclei as an unique phenomenon associated with theextended halo wave function [6]. As a theoretical model, for lighter nuclei such as He and Li, the framework of core+ n + n three-body model has been adopted often to describe socalled ”Borromean system”, in which one nucleon+core system has never been bound, butonly two nucleon+core system makes a bound nucleus [7, 8]. For sd -shell neutron-rich nucleisuch as Ne isotope, some halo states have been found [9]. In addition, deformed structurewith larger β has been observed in these systems [10]. In the nuclei so far discussed, one ortwo nucleons will contribute to create the halo structure. When one goes to heavier nuclei,for instance, in neutron-rich Ca and Zr isotopes, theoretically in Refs. [11–16], giant halonucleus is predicted, in which several neutrons contribute to make halo nuclei.Let us consider hypernuclei consisting nuclei and a hyperon, especially a Λ particle. Someauthors pointed out that there were possibility to have halo states in lighter systems [17, 18]:In H, the observed binding energy is 0 .
13 MeV with respect to deuteron+Λ threshold,which is a very weakly bound state and then this system has a Λ halo structure with respectto deuteron [17]. One of the present authors (E. H.) pointed out that neutron or protondensities in the ground state of He, excited states of He and Li with isospin T = 1 havebeen enhanced with the framework of He + N + N three-body model [18]. Thus the studyof halo structure in Λ hypernuclei has been focused on lighter hypernuclei with A ≤
7. Inthis paper, we focus on the possibility to have a halo structure in more heavier Λ hypernucleisuch as Boron or Carbon isotopes with A ≥
8. Especially, in
C, we have observed dataof 2 positive-parity and 2 negative-parity states: the ground state, 1 / +1 , and either of 3 / +1 or 5 / +1 positive-parity excited state, 3 / − and 1 / − negative-parity excited states. Thedominate component of two negative-parity states is C ⊗ Λ(1 p ) configuration. Amongthese observed states, 3 / − and 1 / − states are important to extract the information onΛ N spin-orbit force: they measured the spin-orbit splitting energy of 1 / − -3 / − to be 0.15MeV [19, 20]. Furthermore these states are weakly bound by about 1 MeV with respect2o C + Λ threshold. This means that we have a chance to find Λ halo structure in Cisotopes. Therefore, in this paper, we focus on Λ halo states for this hypernuclear state. Inaddition, in C isotope, experimentally, a long isotope chain from C to C was observed.Considering this station, we study the ground states and the excited states (C ⊗ Λ(1 p )) ofC hypernuclei systematically with Hartree-Fock model using nucleon-nucleon and nucleon-hyperon interactions and discuss on the halo structure of hypernuclei and the possibility toobserve these halo structure by the calculation of B ( E
1) from the Λ(1 p ) states to the groundstate with Λ(1 s ).For this calculation, we use the Skyrme-Hartree-Fock model [21], which is commonlyadopted for the description of the gross properties of the nuclei in a broad region of masstable. The original Skyrme model has no strangeness degree of freedom. In 1981, Rayetintroduced the Skyrme-type Λ N interaction to describe the hypernuclei within the Skyrmemodel [22]. Since then, many Skyrme-type Λ N interactions were proposed based on realistichyperon-nucleon interactions, stimulated by many hypernuclear data [23–30]. With theseinteractions, the hypernuclear structures have been investigated extensively [31–34]. Butmost of these investigations did not include the Λ N spin-orbit interaction, since it wasexpected to be rather small. In this paper, we will adopt the Skyrme-type Λ N interaction [26]obtained by the G − matrix calculation from the one-boson-exchange potential with a reducedΛ N spin-orbit coupling strength which can reproduce the spin-orbit splitting of the 1 p statesin C [19]. The method is also applied to neighboring Boron isotopes to discuss p -wave halostructure of a Λ hyperon. These studies are performed for the first time with this framework.Organization of the present paper is as follows: In Section II, the Method is explained.The results are discussed in Sec. III and finally we summarize in Sec. IV. II. THEORETICAL FRAMEWORK
Hypernuclei of C and B isotopes are studied by using HF model with Skyrme-type
N N and N Λ interactions. The model is extended to describe systematically from light to heavyhypernuclei including the hyperon degree of freedom. In the Skyrme model, the two-body
N N interaction [35] reads, v NN ( r − r ) = t (1 + x P σ ) δ ( r − r ) + 12 t (1 + x P σ ) (cid:2) k ′ δ ( r − r ) + δ ( r − r ) k (cid:3) + t (1 + x P σ ) k ′ · δ ( r − r ) k + iW ( σ + σ ) · k ′ δ ( r − r ) × k , (1)3here k = ( −→ ∇ − −→ ∇ ) / i is the relative momentum operator acting on the wave functionson the right and k ′ = − ( ←− ∇ − ←− ∇ ) / i acting on the left, P σ = (1 + σ · σ ) / N N interaction is also introducedas v den − NN ( r , r , r ) = 16 t (1 + x P σ ) δ ( r − r ) ρ α (cid:18) r + r (cid:19) , (2)where α is the power of density dependence. The Skyrme-type three-body force is equivalentto the interaction (2) with choices of x = 1 and α = 1 for HF calculations.The Skyrme-like two-body Λ N interaction is taken as [26] v Λ N ( r Λ − r N ) = t Λ0 (1 + x Λ0 P σ ) δ ( r Λ − r N ) + 12 t Λ1 (cid:2) k ′ δ ( r Λ − r N ) + δ ( r Λ − r N ) k (cid:3) + t Λ2 k ′ δ ( r Λ − r N ) · k + iW Λ0 k ′ δ ( r Λ − r N ) · ( σ N + σ Λ ) × k (3)with an effective density-dependent Λ N force v den − Λ N ( r Λ , r N , ρ ) = 38 t Λ3 (1 + x Λ3 P σ ) δ ( r Λ − r N ) ρ γ (cid:18) r Λ + r N (cid:19) , (4)where γ is the power of density dependence.The total energy functional can be separated into two parts, E = Z d r ( H N + H Λ ) , (5)where H N is the hamiltonian density only related with the nucleons, and H Λ is the one withΛ hyperon degree of freedom. The nucleon hamiltonian density H N can be written as H N = ~ m N τ N + 12 t (cid:18) x (cid:19) ρ N − t (cid:18) x + 12 (cid:19) ( ρ n + ρ p )+ 14 (cid:20) t (cid:18) x (cid:19) + t (cid:18) x (cid:19)(cid:21) ρ N τ N + 14 (cid:20) − t (cid:18)
12 + x (cid:19) + t (cid:18)
12 + x (cid:19)(cid:21) ( ρ n τ n + ρ p τ p )+ 116 (cid:20) t (cid:18) x (cid:19) − t (cid:18) x (cid:19)(cid:21) ( ∇ ρ N ) − (cid:20) t (cid:18)
12 + x (cid:19) + t (cid:18)
12 + x (cid:19)(cid:21) (cid:2) ( ∇ ρ n ) + ( ∇ ρ p ) (cid:3) + 116 (cid:2) ( t − t ) (cid:0) J n + J p (cid:1) − ( t x + t x ) J N (cid:3) + 112 t (cid:18) x (cid:19) ρ α +2 N − t (cid:18)
12 + x (cid:19) ρ αN (cid:0) ρ n + ρ p (cid:1) + 12 W ( ∇ ρ N · J N + ∇ ρ n · J n + ∇ ρ p · J p ) + H coul . . (6)4n Eq. (6) and the following, we define the baryon density ( B = n, p, Λ) ρ B ( r ) = X i,σ n i | φ i,B ( r , σ ) | , (7)the kinetic energy density τ B ( r ) = X i,σ n i | ∇ φ i,B ( r , σ ) | , (8)and the spin density J B ( r ) = − i X i,σ,σ ′ n i φ ∗ i,B ( r , σ ) [ ∇ × σ φ i,B ( r , σ ′ )] , (9)where φ i,B ( r , σ ) is the wave function of the single-particle state, and n i is the correspondingoccupation number, which is defined by n i = v i (2 j +1). The occupation probability v i of thesingle-particle state i will be determined by either HFB, BCS or the filling approximationdepending on the model. In Eq. (6), the nucleon total densities are defined as ρ N = ρ n + ρ p , τ N = τ n + τ p , and J N = J n + J p .The hamiltonian density with Λ can be written as [29] H Λ = ~ m Λ τ Λ + t Λ0 (cid:18) x Λ0 (cid:19) ρ Λ ρ N + 14 (cid:0) t Λ1 + t Λ2 (cid:1) ( τ Λ ρ N + τ N ρ Λ )+ 18 (cid:0) t Λ1 − t Λ2 (cid:1) ∇ ρ Λ · ∇ ρ N + 12 W Λ0 ( ∇ ρ N · J Λ + ∇ ρ Λ · J N )+ 38 t Λ3 (cid:18) x (cid:19) ρ γ +1 N ρ Λ (10)As a first step, we assume the spherical symmetry for the hypernucleus, and the pairingcorrelation is not considered explicitly, but the filling approximation is adopted for theoccupation probability v i from the bottom of potential to the Fermi energy in order. Thesingle-particle wave function for nucleons and Λ can be written as φ i,B ( r σ ) = R i,B ( r ) r Y ljm (ˆ r σ ) , i = ( nljm ) and B = ( n, p, Λ) , (11)where R i,B ( r ) is the radial wave function, and Y ljm (ˆ r σ ) is the vector spherical harmonics.A parameter set SkM* [36] is chosen as the Skyrme N N interaction. With this parameterset, the J terms in the nucleon hamiltonian density H N (6) are neglected for parameterfitting procedure. We examine also other Skyrme parameter sets SLy4, SIII and SkO’, butthe present results are essentially not changed by the other Skyrme parameter sets. The Λ N interaction is chosen as the ’set V’ in Ref. [26] which is fitted to the Λ potential energy in5uclear matter obtained by the G -matrix calculation from the one-boson-exchange potential.In particular, this parameter set included the Λ N spin-orbit interaction W Λ0 = 62 MeV fm .However, we found the obtained spin-orbit splitting of the 1 p states in C is too largecompared to the experiment data 0 .
152 MeV [19]. Therefore, we use a reduced value W Λ0 =4 . instead, and obtain a realistic spin-orbit splitting 0 .
155 MeV of 1 p states in C.The center of mass correction is considered simply by multiplying the factor1 − m N / ( Am N + m Λ ) and 1 − m Λ / ( Am N + m Λ ) in front of the mass terms ~ / m N and ~ / m Λ respectively. The binding energy of Λ particle can be calculated by B Λ = E A − E Λ A +1 , (12)where E A is the total energy of the nucleus with A nucleons, and E Λ A +1 is the total energyof the hypernucleus with one additional Λ. 6 II. RESULTS AND DISCUSSIONSA. Hypernuclei of C isotopes
We first discuss C-isotopes since the spin-orbit splitting of hyperon states was observedonly in
C. The HF results of hypernuclei of C isotopes from C to
C are tabulated in TableI. As it is expected, the binding energy of Λ(1 s )-state increases for heavier C isotopes sincethe N Λ potential is deeper for the heavier isotopes. The agreements between the calculatedand experimental B Λ are surprisingly good as shown in the last column of Table I. The Cwas also observed by C( π + , K + ) C reaction [37]. The binding energy was determined as B Λ = 11 . ± . ± B Λ = 11 . ± .
12 MeV. The Λ(1 p )-state was also observed at E x = 9 . ± .
14 MeV,which is somewhat lower than HF values in Table I. In ref. [20], the excitation energies ofthe Λ(1 p / ) and Λ(1 p / ) states were obtained as E x = 10 . ± . ± E x = 10 . ± . ± E x =11.345 and 11.190 MeV for Λ(1 p / ) and Λ(1 p / ) states and show reasonableagreement. The spin-orbit interaction of N Λ is taken as much smaller value than the the G -matrix results. The HF gives the spin-orbit splitting of p − hyperon orbits in C,∆ ε (Λ(1 p / ) − Λ(1 p / )) = 0 . , (13)while the experimental value is ∆ ε (Λ(1 p / ) − Λ(1 p / )) = 0 . N Λ channel W Λ0 = 4 . ismore than a factor 20 smaller than the N N spin-orbit coupling strength W = 120 MeVfm .The spin-orbit splitting is almost constant in C isotopes, i.e., from ∆ ε (Λ( p / ) − Λ( p / )) =0 . C to ∆ ε (Λ( p / ) − Λ( p / )) = 0 . C.The rms radii of Λ(1 s )- and Λ(1 p )-orbits in C isotopes are listed in Table I. The Λ(1 p )-orbits are either quasi-bound (resonance) states or loosely-bound states. Especially, the rmsradii of p -orbits show a peculiar halo nature in C and
C similar to the halo state in nucleisuch as Li and Be. The wave functions of Λ(1 s / )- and Λ(1 p / )-states in C are drawnin Fig. 1. The enhancement of r.m.s. radii of Λ(1 p )-state is about 60% compared with theΛ(1 s )-orbit. Thus we can conclude to find the Λ(1 p ) halo structure in C. For
C and
Chypernuclei, the Λ(1 p ) states have very small binding energies and show the similar halostructure to that of C. 7he total binding energies and matter r.m.s. radii r Arms of C isotopes are tabulated in TableII. The HF result gives reasonable binding energies in nuclei near C, where deformationeffect might play a minor role. On the other hand, in the lighter and heavier isotopes, thedeformation effect will contribute to increase the total binding energies, which remains to bestudied. The mass radii of C-isotopes are observed by heavy-ion reactions [38, 39] and listedin Table II. The calculated results reproduce reasonably well the experiment values exceptthe neutron halo nuclei C and C. The r.m.s. radii of cores of corresponding hypernucleiare also listed as r coreArms . In comparison between r Arms and r coreArms , we can point out shrinkageor expansion effect of core nucleus in hypernucleus. For Λ(1 s ) hyperon case, we can seesmall shrinkage effect of the core, 0 . − .
02 fm, from light to heavy C isotopes. For Λ(1 p )hyperon case, it is interesting to see an expansion effect of the core for nuclei A ≤
13, butquantitatively it is even smaller than the shrinkage effect of Λ(1 s ) hyperon in the samenucleus. r [fm] -6 -5 -4 -3 -2 -1 R i , Λ [ f m − ] C Λ1/2 Λ1/2
FIG. 1: The square of single Λ wave function R i, Λ of Λ(1 s /
2) and Λ(1 p /
2) states in the hyper-nucleus C. B. Hypernuclei in B isotopes
The calculated results of hypernuclei of B isotopes are listed in Table III. Compared withthe experimental value of B Λ (exp) ∼ B [40], the calculated result is quite8easonable to be B Λ (HF)=10.8 MeV. The potential depth is becoming deeper for heavierisotopes and the binding energy of Λ(1 s / )-orbit increases from 8.97 MeV in B to 14.50MeV in
B. The halo structure of 1 p -orbits can be also seen in light B isotopes, especially in B and
B. The wave functions of Λ(1 s / )- and Λ(1 p / )-states in B are drawn in Fig. 2.The wave functions in
B are essentially identical to those of
C. Th spin-orbit splittings inB isotopes show a similar feature to that in C isotopes; ∆ ε (Λ(1 p / ) − Λ(1 p / )) = 0 . B and ∆ ε (Λ(1 p / ) − Λ(1 p / )) = 0 .
129 MeV for a heavier isotope
B. Two Λ(1 p )-shellstates were also observed in [40], as J π = (1 +1 or 2 +1 ) and (2 +2 or 3 +1 ) states, which areconsidered as coupling states of 3/2 − ground state of B and Λ(1 p / ) or Λ(1 p /
2) states.Since the spin-spin interaction of N Λ is not included in the present HF calculations, we cannot predict precisely the energy splitting of 1 + , + and 3 + states. However, the HF excitationenergies of Λ(1 p ) states E x ∼ . E x (exp)=10 . ± .
05 and 10 . ± .
03 MeV for J π = (1 +1 or 2 +1 ) and(2 +2 or 3 +1 ) states, respectively. r [fm] -6 -5 -4 -3 -2 -1 R i , Λ [ f m − ] B Λ1/2 Λ1/2
FIG. 2: The square of single Λ wave function R i, Λ of Λ(1 s /
2) and Λ(1 p /
2) states in the hyper-nucleus B. C. Electic dipole transition in Hypernuclei
We will study the electric dipole transition between hyperon 1 p - and 1 s -state. Electro-magnetic transitions may provide precise information of hyperon wave functions in quanti-9ative manner. Suppose the hypernucleus is initially in the excited state, e.g., Λ is in the1 p orbit, it will decay to the ground state 1 s orbit. This E B ( E J i → J f ) = 3 e π h f | r | i i (2 j f + 1) j f j i − (14)where e Λ is the effective charge for Λ hyperon and the integration h f | r | i i can be calculatedby the radial wave functions of the initial and final single-Λ state as h f | r | i i = Z ∞ R f, Λ ( r ) rR i, Λ ( r ) dr. (15)Since hyperons Λ have no electric charges, the effective charge in Eq. (14) is given as e ( E = − ZM Λ e/ ( AM N + M Λ ) (16)due to the recoil of the core nucleus [42].The calculated B ( E
1) values are listed in Tables I for C isotopes and III for B isotopes,respectively. The values are larger in light isotopes than those in heavier nuclei becauseof the effective charge in Eq. (16). The B ( E p / → s / )=0.1036 e fm of hyperonconfigurations in C corresponds to 0.29 B W ( E B W ( E
1) is the Weisskopf unit(single-particle unit) of electric dipole transition in A = 13 nucleus. The decay half life t / is estimated as t / = ln 2 T ( E
1) = 2 . × − sec , (17)where T is the decay rate, T ( E
1) = 1 . × ( E x ) B ( E
1) = 2 . × sec − . (18)The T ( E
1) is evaluated to be 1.505 × sec − for the transition (Λ(1 p / ) → Λ(1 s / )) in B and the half life is estimated as t / = 4.60 × − sec.In halo nuclei without Λ degree of freedom, the largest B ( E
1) transition between dis-crete states is observed in 2 s / → p / transition in Be [43]; B ( E
1; 2 s / → p / ) =0 . ± . e fm = 0 . ± . B W ( E B ( E p / ) → Λ(1 s / )) of hyperon configurations in C. Notice these B ( E
1) in halo nuclei(hypernuclei) are 2-3 order of magnitude larger than normal B ( E − e fm . The B ( E
1) strength of halo nuclei was studied also by the Coulomb breakup10eactions, which measure the excitation from the halo state to the continuum. In thesereactions, the B ( E
1) value was found B ( E . ± . e fm in Be [44] and B ( E . ± . e fm in C [45]. Systematic measurements of electromagnetictransitions in Λ(1 p ) states may give us a peculiar nuclear structure information includingthe characteristic features of hyperon halo wave functions.We do not discuss details of the total binding energies of B isotopes since B-isotopesare odd-even or odd-odd nuclei so that spin-spin interaction might play an important role,which is missing in the present Skyrme EDF. IV. SUMMARY AND FUTURE PERSPECTIVES
In this work, we calculated the Λ single-particle states systematically in the C and Bisotopes using the HF approach with the Skyrme-type Λ N interaction derived from the G − matrix calculation of the one-boson-exchange potential. We tuned the strength of Λ N spin-orbit interaction by fitting to the observed spin-orbit splitting data of 1 / − − / − statesin C. The Λ binding energies thus obtained agree with the available experiment data quitewell for the C and B hypernuclei. In the light hypernuclei − C and − B, we found veryweakly bound excited 1 p orbits for Λ hyperon, which could have much extended density andlarge r.m.s radii compared with the ground 1 s state. Furthermore, we calculated B ( E E p states to the ground 1 s state, which is a challenging open problem for the future experiment.On the other hand, with more neutrons, the Λ levels become more deeply bound, so thatthe hyperon halo structure disappears. Acknowledgments
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1) value of the transition from the excited Λ( p )-state to theground Λ( s )-state. Experimental data are taken from ref. [46, 47].Nucleus Λ( nlj ) e s . p . (MeV) B Λ (MeV) r Λrms (fm) B ( E
1) ( e fm ) B Λ (exp) (MeV) C 1 s / − .
478 7 .
600 2 . C 1 s / − .
662 8 .
821 2 . C 1 s / − .
615 9 .
864 2 . C 1 s / − .
433 10 .
787 2 .
139 10 . ± .
10 [46]1 p / − . − .
379 3 .
679 1 . × − p / − . − .
239 3 .
604 1 . × − C 1 s / − .
156 11 .
618 2 .
144 11.69 ± p / − .
782 0 .
273 3 .
464 1 . × − p / − .
936 0 .
428 3 .
410 1 . × − C 1 s / − .
563 12 .
181 2 .
172 12.19 ± p / − .
357 1 .
010 3 .
355 9 . × − p / − .
506 1 .
160 3 .
317 9 . × − C 1 s / − .
941 12 .
688 2 . p / − .
911 1 .
697 3 .
287 8 . × − p / − .
055 1 .
842 3 .
259 8 . × − C 1 s / − .
292 13 .
083 2 . p / − .
357 2 .
187 3 .
252 7 . × − p / − .
500 2 .
331 3 .
228 7 . × − C 1 s / − .
633 13 .
467 2 . p / − .
792 2 .
666 3 .
226 6 . × − p / − .
935 2 .
809 3 .
206 6 . × − C 1 s / − .
962 13 .
839 2 . p / − .
216 3 .
133 3 .
207 6 . × − p / − .
357 3 .
274 3 .
189 6 . × − C 1 s / − .
281 14 .
200 2 . p / − .
629 3 .
587 3 .
192 5 . × − p / − .
769 3 .
727 3 .
177 5 . × − C 1 s / − .
590 14 .
549 2 . p / − .
031 4 .
028 3 .
182 5 . × − p / − .
170 4 .
167 3 .
168 5 . × − C 1 s / − .
890 14 .
887 2 . p / − .
422 4 .
457 3 .
174 4 . × − p / − .
559 4 .
595 3 .
162 4 . × − C 1 s / − .
038 15 .
097 2 . p / − .
648 4 .
742 3 .
191 4 . × − p / − .
787 4 .
881 3 .
178 4 . × − C 1 s / − .
176 15 .
291 2 . p / − .
853 5 .
000 3 .
208 4 . × − p / − .
992 5 .
140 3 .
195 4 . × − ABLE II: Properties of isotopes A C : the calculated total binding energy B cal (MeV), the massrms radius r Arms (fm), and the mass rms radius of the core r coreArms (fm) in hypernucleus A +1Λ C.The experimental data of B exp are taken from Ref. [48]. The experimental uncertainty in theparentheses is given in unit of keV. The experimental data of r Arms are taken from Refs. [38, 39].Nucleus B cal (MeV) B exp (MeV) r Arms (fm) r Arms (exp) (fm) Λ( nlj ) r coreArms (fm) C 31 .
078 24 . . s / . C 46 .
633 39 . . s / . C 62 .
278 60 . . s / . C 77 .
855 73 . . s / . p / . p / . C 93 .
330 92 . . ± s / . p / . p / . C 100 .
716 97 . . ± s / . p / . p / . C 108 .
421 105 . . ± s / . p / . p / . C 110 .
946 106 . . ± s / . p / . p / . C 113 .
752 110 . . ± s / . p / . p / . C 116 .
835 111 . . ± s / . p / . p / . C 120 .
193 115 . . ± s / . p / . p / . C 123 .
819 116 . . ± s / . p / . p / . C 127 .
707 119 . . ± s / . p / . p / . C 130 .
470 119 . . s / . p / . p / . C 133 .
175 119 . . ± s / . p / . p / . ABLE III: The same as Fig. I, but for B isotopes. The p -states are unbound in − B. Experi-mental data of B Λ are taken from refs. [40, 49].Nucleus Λ(( nlj ) e s . p . (MeV) B Λ (MeV) r Λrms (fm) B ( E
1) ( e fm ) B Λ (exp) (MeV) B 1 s / − .
750 6 .
670 2 . B 1 s / − .
917 7 .
892 2 . B 1 s / − .
877 8 .
968 2 .
128 8.1 ± B 1 s / − .
712 9 .
932 2 . B 1 s / − .
457 10 .
805 2 .
137 11.38 [40]1 p / − . − .
386 3 .
674 8 . × − p / − . − .
245 3 .
599 8 . × − B 1 s / − .
843 11 .
375 2 . p / − .
787 0 .
364 3 .
503 7 . × − p / − .
925 0 .
502 3 .
454 7 . × − B 1 s / − .
205 11 .
885 2 . p / − .
331 1 .
061 3 .
402 6 . × − p / − .
465 1 .
195 3 .
367 6 . × − B 1 s / − .
544 12 .
277 2 . p / − .
765 1 .
549 3 .
352 5 . × − p / − .
898 1 .
682 3 .
323 5 . × − B 1 s / − .
876 12 .
660 2 . p / − .
193 2 .
028 3 .
315 5 . × − p / − .
325 2 .
160 3 .
291 5 . × − B 1 s / − .
203 13 .
034 2 . p / − .
613 2 .
497 3 .
287 4 . × − p / − .
744 2 .
628 3 .
266 4 . × − B 1 s / − .
522 13 .
399 2 . p / − .
026 2 .
956 3 .
265 4 . × − p / − .
156 3 .
086 3 .
247 4 . × − B 1 s / − .
834 13 .
754 2 . p / − .
430 3 .
403 3 .
249 3 . × − p / − .
559 3 .
532 3 .
233 4 . × − B 1 s / − .
138 14 .
099 2 . p / − .
825 3 .
839 3 .
236 3 . × − p / − .
952 3 .
966 3 .
222 3 . × − B 1 s / − .
276 14 .
306 2 . p / − .
034 4 .
112 3 .
254 3 . × − p / − .
163 4 .
241 3 .
239 3 . × − B 1 s / − .
406 14 .
497 2 . p / − .
224 4 .
360 3 .
271 3 . × − p / − .
353 4 .
489 3 .
257 3 . × −2