arXiv:nucl-th/0501052v3 17 Feb 2006 n tbence.Tpclcssare neighbor- cases to Typical compared nuclei. large quite stable are ing radii unsta- halo-type are their they and The as of interest ble one special of unbound. any are nuclei is that Borromean subsystems These property two-body 4]. the 3, their have 2, [1, systems intensive been three-body has study their and mon h cores, The oysbytm r not; are subsystems body tla ulosnhss sacutro three in of importance cluster a paramount is of nucleo-synthesis, the resonance, stellar is This example MeV. known well 6.8 A in resonance exist. Hoyle also nuclei stable of e ( tem ihteunbound the with sta h n eto eaainenergy, separation configuration this one subsystems of the two feature that other Another is The bound. weakly subsystems, unbound. nuclei are two-body halo is its of of di-neutron, type one the new only This that feature well. the as has halo-type several a in neu- which is two configuration cases to system, new a Further, the acquires nucleus down, fashion. trons Borromean in gradual halo develops a the halo in reach the nuclei that Borromean discovered the have We . usinw aemd uvyo h stpso several of this isotopes answer the the to of in order survey nuclei a In made nucleus. are have we stable question given more a as to formed added are nuclei Borromean halo hnta ftetwo-neutron, the of that than situation e ad Sxheiin 00.Nt htn Borromean no that Wal- Note Nuclear 2000). the edition, in (Sixth contained Cards let here about studied information In available isotopes the isotopes. the use normal shall other we follows the what by shared is configuration where nuclei halo Borromean orma uli ete aoo o,aeqiecom- quite are not, or halo them be nuclei, Borromean nipratiseta a ob drse shwthe how is addressed be to has that issue important An n ersnsanurn.Broenectdstates excited Borromean neutron). a represents E c n , eto-oefr on ytm sadowyt orma s Borromean to doorway a as system) bound a form neutron-core 23 ASnmes 57.j 57.n 41.q 21.60.-n 24.10.Eq, 25.70.Mn, 25.70.Jj, numbers: PACS e w far to so contrast nucleus the in halo reveal transfers, configuration. two-neutron could over Samba-halo targets one-neutron heavy of of with example nuclei excellent halo an Samba is fm 6.07 E < nttt eFıia nvriaed ˜oPuo ..6631 C.P. S˜ao Paulo, de F´ısica, Universidade de Instituto 4 and N eepottepsiiiyo e ofiuain nthree-bo in configurations new of possibility the exploit We eand He ora oBroenhl uli h ab configuration Samba the nuclei: halo Borromean to doorway A 2 n 8 12 ea w-oysubsystems. two-body as Be p htpeal ntenw“doorway” new the in prevails that 27 ta xiaineeg fabout of energy excitation an at C selrgo swl stefluorine the as well as region -shell 9 r fsc aue nparticular in nature, such of are F iaebudbttetretwo- three the but bound are Li 6 etoTenc eosail 22-0 ˜oJsedsCam S˜ao Jos´e dos 12228-900 T´ecnico Aeroespacial, Centro e+ He nvriaeEtda alsa E 80-1 tpv,SP Itapeva, 18409-010 CEP Paulista, Estadual Universidade 5 eand He eatmnod ´sc,IsiuoTco´gc eAeron´ Tecnol´ogico de F´ısica, Instituto de Departamento E 238 n E 2 U. E > n 10 ncnrs othe to contrast in , 11 iadthe and Li i[]and [5] Li 2 n fcus the course Of . E n 4 ssmaller is , Dtd coe 2 2018) 22, October (Dated: enuclei, He 6 nn .T Yamashita T. M. e[6]. He .S Hussein S. M. .Frederico T. sys- 23 ihahl-ieo 77sadahl aisof radius halo a and s 37.7 of half-life a with N ,9 0 1 2 3 4 5 7 9 Both 19. 17, 15, 14, 13, 12, 11, 10, 9, 8, stp xssi xgn(e,hwvr e.[1]). Ref. however, (see, oxygen in exists ol eBroenhl uli( nuclei halo Borromean be would isotope isi .5f hl ula aisi .1f,t be to fm, 2.91 is radius normal of nuclear fm while 2.50 to fm compared 5.15 is dius h ora ye hyare of They 8] [7, nuclei have type. We halo doorway Samba below. the for given candidates are five calculation identified radius our of tails oihax[]woepotdytaohrtp fhalo of hypertriton type the as another such yet systems, exploited three-body who [9] Robicheaux 27 e a earte oe“gttd n h three-body the subsys- and two-neutron “agitated” bound. the more remains rather of system a motion be the can the that tem of fact distinguished (two is the nucleus bound) in by Samba tightly fashion the are “harmonic” subsystems core, rather subsys- two-body the a of two-body in presence move bound the to a has Whereas, has that nucleus bound, tem systems. halo tightly halo Tango Tango is the systems subsystems such two-body called the and of one only ab aonce.The nuclei. halo Samba strengths h lcrmgei iscaino hs rgl systems through fragile measured these usually of the is dissociation in that electromagnetic quantity nuclei the important exotic an of is study distribution dipole The ntecluaino h ioemti lmn.The distribution, strength for- element. dipole analytical matrix simple the a for dipole of state mula derivation the continuum the for final of allows and the model calculation state for ground the employed type the is in Yukawa wave describe A a plane to as a used core. neutrons is the two function against the wave rough vibrates treats a that get model cluster to This employed [10]. usually estimate model cluster simple a .Tenm ab atyi nprdb h okof work the by inspired is partly Samba name The F. sa xml ecnie h oo isotopes: boron the consider we example an As ehv acltdterdcddpl transition dipole reduced the calculated have We a a on eetyfrteBorromean the for recently found was hat 15 ,CP03590Sa al,Brasil S˜ao Paulo, 05315-970 CEP 8, uieehneetadadominance a and enhancement lusive B stedowycngrto.Tehl ra- halo The configuration. doorway the is B h uinblwtebriro the of barrier the below fusion The yhl uli-Smatp (the - type Samba - nuclei halo dy ( E )adterdio hs v addtsfor candidates five these of radii the and 1) ses h nuclei The ystems. o,Brasil pos, autica, Brasil , B ( E )swr acltdusing calculated were 1)’s 12 A Be, 12 5nces h de- The nucleus. 15 = Be, 19 15 oes) The so). more B B, 15 B, 20 dB 17 C, Λ 3 and B ( H E 23 1 1) where , and N /dE A 19 = ∗ B . 2 in the field of a heavy target such as 208Pb. Integrating 1.6 ∗ ∗ dB(E1)/dE over E gives the B(E1) value. The cluster circles - Be model gives a simple formula for this squares - B triangles - C 3e2 2Z 2 1 B(E1) = , (1) diamond - N 16πµ  A  E2n 1.4 stars - F where µ is the reduced mass of the core and the 2n clus- ter. The factor 2Z/A corresponds to the number of neu- 0 trons in the halo, 2; the charge of the core, Z and the R/R of the whole , A. Note that 1.2 the two neutron is inversely propor- tional to the average distance between the core center 2 and the three-body center-of-mass (CM), hrc−CM i, which is used as the healing distance of the clusterq model 12 wave function. For, e.g., Be we find for B(E1) the 1.0 value 0.043 e2fm2 (another calculation gives 0.05-0.06 2 3 4 5 6 2 2 2 2 e fm [11]) to be compared with 0.051(13) e fm for the T experimental value [12] and to 0.61 e2fm2 for the well Z developed halo in the Borromean nucleus 6He. Thus one should be able to get reasonably reliable and apprecia- FIG. 1: Halo radii of our candidates for two-neutron halo ble dissociation cross section for this Samba nucleus (the nuclei in isotopes indicated in the legend. The full symbols are the Samba type nuclei (results from Table I) and the open yield or production cross section of these nuclei should symbols are the Borromean nuclei. All the points are divided be larger than those of the Borromean halo nuclei [14], by the the isotope radii, R0, calculated assuming a normal rendering the experiment quite feasible). It is worth men- nature of the isotope. Tz is the isospin projection. tioning here that for Tango halo nuclei, where the halo is a pair of proton-neutron, the B(E1) is similar to Eq. (1) except for the energy E2n, which is replaced by Epn and the factor (2Z/A)2 which is replaced by [(A − 2Z)/A]2. Conspicuous in the figure is lack of indication of one- This will render the B(E1) for Tango halo nuclei a fac- neutron halo nuclei whose radii are quite large and de- 2 11 19 tor [(A − 2Z)/2Z] smaller than that of a corresponding viant from R0. These nuclei, such as Be, C and others Samba or Borromean nucleus if E2n is maintained equal have been shown to have a well developed one-neutron to Epn. halo. We did not show these halo nuclei as the thrust In figure (1) we show the halo radii of our candidates of our work here is on three-body, two-neutron halo nu- for two-neutron halo nuclei as a function of the isospin clei. Also absent in the figure is the, what would be, projection Tz = (N − Z)/2. These radii can be esti- Borromean halo nucleus following the Samba one, 23N. mated from the neutron-CM root mean square radius We are tempted to predict that such a nucleus, 25N, may h¯2 2 exist, though we have no information about En, E2n and hrn−CM i estimated as ρ mnS3 (see Refs. [7] and [8]). q q its life time. From the above considerations and we can The quantity rho is adimensional. For our purposes ρ is 20 obtained calculating the average (R(14Be)+ R(17B))/2 clearly rule out C as a Samba halo nucleus and accord- ingly 22C as a Borromean halo nucleus. 2 and associating this to squareroot hrn−CM i above. 14 17 q The “doorway” aspect of the Samba halo nuclei is quite Here R( Be)=3.74 fm and R( B)=3.8 fm, from evident especially in the circles, squares and stars. They Ref. [2], S3 is the three-body energy with respect to the always precedes the final Borromean halo nucleus in the binding energy of the neutron in the two-body subsys- chain of isotopes. It would be indeed very interesting to tem, S3 = E2n(A) − En(A − 1) for Samba nuclei, where perform Coulomb dissociation experiment on, e.g. 23N, En(A − 1) is the neutron separation energy in the bound and 27F to investigate the dipole strength distribution A − 1 system, and S3 = E2n for Borromean nuclei. With and also the longitudinal momentum distribution to as- this value of ρ we calculate all the radii shown in the table sess the halo nature of these Samba systems. The nucleus and figure. The full symbols (see Table I) are the radii 12Be has a rather subtle shell model structure with the of Samba type halo nuclei calculated exactly within the ground state containing (1s )2, (1p )2 and possibly three-body model described below and the open symbols 1/2 1/2 (1d )2 configurations, making it less likely to be a clear are the radii of the Borromean halo nuclei calculated as 5/2 cut Samba type halo nucleus. above with ρ set equal 1.35. Both radii appear divided by the radii calculated assuming a normal nature of the In Table I we have collected several physical quantities 1/3 isotopes, namely, R0 =1.013A fm (the factor 1.013 fm of the following Samba type nuclei (treated as a n−n−c was taken to reproduce the experimental radius of 12C of three-body system): 12Be, 15B, 20C, 23N and 27F. The R= 2.32 fm [13]). lower frame shows the results for the core radius Rcore = 3

1.013(A − 2)1/3, the total radius taken to be of a zero range. Therefore, in this model the only physical scales used as input are directly related to observables: the nn scattering length, the energies E 2 2 A − 2 2 n R ∼ hr − i + R , (2) rA n CM A core and E2n. It is worth mentioning that the subtracted three-body equations are obtained through an elimina- B(E1,0 → 1) (eq. (1)), and the half-life, T1/2, of the tion procedure that involves the renormalization of the nuclei. delta function interaction strength so that the large mo- mentum divergence alluded to above is removed. In a nut 2 nucleus E E2 r R0 n n h n−CM i shell, one takes the two-body matrix t(E) for the delta ′ 2 (MeV) (MeV) p (fm) (fm) interaction, λδ(~r−~r ), at an energy, −µ(2), and identify it 12 2 Be 0.504 3.669 4.81 2.75 with the renormalized coupling strength λR(−µ(2)). The 15B 0.973 3.734 5.15 2.96 delta potential t-matrix at any energy can be obtained 20C 0.191 3.462 4.00 3.26 through usual Lippmann-Schwinger manipulations as 23 N 1.200 3.672 6.07 3.41 −1 2 −1 2 tR(E) = λR(−µ ) +2π (µ(2) + ik). (3) 27F 1.041 2.412 8.94 3.60 (2)

nucleus Rcore R B(E1,0 → 1) T1/2 where the index R will refer to a renormalized t- or T - (fm) (fm) (e2fm2) matrix. 12 The three-body T -matrix can be handled in a similar Be 2.18 2.80 0.043 21.3 ms 15 fashion with the above subtracted two-body matrix. The B 2.38 2.91 0.037 9.87 ms subtracted three-body T -matrix can be written as 20C 2.66 2.82 0.013 14 ms 23 2 2 N 2.80 3.22 0.024 37.7 s TR(E)= TR(−µ(3))+ TR(−µ(3))[G0(E) 27 > F 2.96 3.75 0.051 200 ns 2 −G0(−µ(3)) TR(E). (4) TABLE I: Physical quantities of the Samba halo nuclei given i in the first column. The second and third columns are, re- It is a simple matter to show that TR above is spectively, the n−c and the n−n−c energies used to calculate 2 independent on the value of the subtraction energy the n − CM root mean-square radii, hr − i. R0 is the 2 2 n CM −µ(3), namely dT (E)/dµ(3) = 0. This is clear by nucleus radius and Rcore is the core radius,p R (eq. (2)) is the 2 construction. Formally, this independence of T (E) average between R0 and hr i. B(E1,0 → 1) is given n−CM on µ can be shown by resorting to the identity, T (3) by eq. (1). 1/2 is the nucleusp half-life. 2 dT (E)/d(E) = −T (E)G0(E)T (E), which is clearly also 2 2 valid for dT (−µ(3))/dµ(3). Thus we have, using Eq. (4), The calculated radii of the Samba nuclei in the three- body model are a bit larger than the measured one (e.g. dTR(E) dTR(E) for 12Be, R = 2.59 ± 0.06 fm see Ref. [15]). We trace 2 =2µ(3) =0. (5) exp dµ dµ(3) this small discrepancy to the neglect of Pauli blocking (3) effect which tends to make the nuclear potential between The second relation in the above formula is a Callan- the core and the neutrons less attractive at short dis- Symanzik type, commonly used in renormalization pro- tances. In order to not change the three-body binding cedure of field theories. energy the system needs to shrink a little to better feel Armed with the above invariance of TR(E) with re- the nuclear attraction. gards to the subtraction energy and the removal of the The radii in Table I were calculated using the three- ultraviolet divergence of the delta potential two-body t- body formalism of Refs. [7, 16]. In these references, matrix, the three-body T -matrix can be used to construct subtracted Faddeev equations for the three-body system the corresponding wave function needed to calculate ma- n − n − c are used. The subtraction energy which is re- trix elements among which is the root mean-square ra- 2 2 quired in the model is taken to be µ(3) (E = −µ(3) is dius of a given nucleus [7]. The mean-square distances 2 2 an arbitrary subtraction point where the T -matrix (from (hrn−CM i and hrc−CM i) and the coupled subtracted in- this point a small t will be used when we refer to the two- tegral equations for the Faddeev spectator components body t-matrix and a capital one when we refer to the are obtained from expressions given in Refs. [7, 8]. 2 three-body T -matrix), T (−µ(3)), should be known. A Before ending, we should mention what is expected of more detailed description about our subtraction method the Samba halo nuclei when they are used to induce reac- can be found in Refs. [17]). The motivation behind the tions on heavy targets. In particular, at very low energies subtraction is the use in the model of a delta function po- (in the vicinity of the Coulomb barrier) we expect that tential for the nn interaction which yields divergent two- the Samba halo nucleus to fuse with, say, 208Pb, with a body t-matrix at large momenta (short distances). Thus probability which is not so much affected by the breakup the Faddeev equations for the three-body system must be coupling, and may exhibit a halo enhancement, contrary appropriately subtracted. The n − c interaction is also to what was found in the 6He+238U studied in Ref. [18]. 4

In further contrast to 6He, the Samba nucleus would ex- nuclei [18, 19], may come into light with the Samba nu- hibit a one-neutron transfer process competing with the clei. two-neutron one. The one-neutron transfer cross section could be larger than the two-neutron one, depending on Acknowledgements: We thank Professor P.G. Hansen the extension of the corresponding configuration in the for comments and suggestions. The work of MTY is Samba halo nuclei. A specific Samba nucleus that we suppported by FAPESP, that of TF partly by FAPESP suggest to investigate experimentally is 23N which has a and the CNPq and that of MSH partly by FAPESP, life time of 37.7 s and a halo radius of 6.07 fm. The en- the CNPq and the Instituto do Milˆenio de Informa¸c˜ao hancement of the fusion, sought for in vain in Borromean Quˆantica-MCT, all Brazilian agencies.

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