By comparing respectively the interaction between 3He atoms and the
neutron-neutron one, we examine the possibility for both fermionic systems to
form small clusters.Comment: Contributed talk to 17th International IUPAP Conference on Few-Body
  Problems in Physics, Durham, North Carolina, 5-10 June 2003. Published in the
  proceedings (Elsevier, 2004) pp. S106-S10

Carbonell, J.

Lazauskas, R.

English

arXiv

By comparing respectively the interaction between 3He atoms and the neutron-neutron one, we examine the possibility for both fermionic systems to form small clusters

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Small clusters of fermionsJ. Carbonell, R. LazauskasTo cite this version:J. Carbonell, R. Lazauskas. Small clusters of fermions. Gloeckle W. 17th International IUPAPConference on Few-Body Problems in Physics Few-Body 17, Jun 2003, Durham, United States.Elsevier, pp.S106-S108, 2004. <in2p3-00022254v2>HAL Id: in2p3-00022254http://hal.in2p3.fr/in2p3-00022254v2Submitted on 2 Dec 2004HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.democrite-00022254, version 2 - 3 Dec 20041Small clusters of fermionsJ. Carbonell and R. lazauskasaaLab. Physique Subatomique et Cosmologie, 53 Av. Martyrs, 38026 Grenoble, FranceSeveral theoretical studies [1] indicate that there is no any reasonable chance for 3n and4n to be bound. GANIL experiment [2] suggesting tetraneutron has not been confirmedbut these no-result had however the merit of pushing theoreticians at work to clarifythe possible existence of small n-clusters. If the situation seems clear for A=3 and 4,it is less well established for a larger number of neutrons. The search of multi-neutronresonances raises also some interest and a recent experiment scanning the 4n continuumin the d(8He,6 Li)4n reaction reports a 2-3 MeV width structure [3].In this issue it is enlightening to make a parallel with a similar, better-known, fermionicsystem: the 3He atomic clusters. Since small 3He droplets can exist [4], should we expectsome stability islands (see figure below) in the continuum neutron states going from N=2to N=∞. If yes, where? If not, why?        2 3 42 3 435 He clusters3unboud neutron matter He matter3?The two-body interactions of these systems look at first glance quite similar. We havecompared in Fig. 1 the n-n AV18 [5] and the 3He-3He Aziz [6] S-wave potentials. Theyhave been rescaled by the corresponding masses Mh¯2and the resulting length units, as wellas the inter-particle distance d are fm and A˚ respectively. Corresponding low energyparameters are given in Table 1. In the n-n case we have also considered Reid93 [7] andMT13 [8] potentials, the latter being adjusted to reproduce the experimental scatteringlength. None of these system supports a bound dimer (a < 0) but 3He seems less favorableTable 1Low energy n-n and 3He-3He parameters.n-n(fm) He-He(A˚)Av18 Reid MT13 Exp. Aziz 91a -18.49 -17.54 -18.59 −18.59± 0.4 -7.24r0 2.85 2.94 2.75± 0.1 13.5ηc 1.08 1.09 1.10 1.30to make clusters. This can be seen by calculating the critical values of the scaling factorηc introduced in the potential V(η)(r) = ηVnn(r), which bounds a dimer. For n-n, thisvalue is around ηc = 1.08 whereas for He-He is sensibly greater ηc = 1.30 (see Table 1).20 1 2 3 4 5d−3−2−10123M V HeHe Aziz91       (A)nn    AV18  1S0 (fm)Figure 1. Comparison between n-n and3He-3He potentials.0 10 20N11.522.53ηc??Figure 2. Critical value of the scaling fac-tor for n (circles) and 3He (squares)When examining larger systems we first consider the bosonic case. Despite the absenceof dimers, ”bosonic” neutron trimers and tetramers do exist with binding energies Bn3 ≈1 MeV and Bn4 ≈ 10 MeV. This is however not the case for atomic 3He, suggestingonce more that neutron clusters should be favored with respect to atomic 3He. Theyall disappear when the Pauli principle is imposed, but can reappear – as in 3He – if thenumber of interacting particles is increased. The existence of small clusters results thusfrom a compromise between an attractive pairwise interaction and the effective Paulirepulsion.In order to study such a balance, we have investigated in several directions [9]: (i)scaling factor in Vnn, (ii) three-neutron interactions (TnI), (iii) influence of n-n P- waves(iv) confining the system in an harmonic oscillator (HO) trap and (iv) ”dimer”-”dimer”scattering. It is always possible to bind 3- and 4-n states by modifying the usual n-nand/or TnI models but the violation has to be very strong, producing serious anomalies.Thus, in case (i) one needs a large scaling factor ηc ∼ 3. Using ad-hoc TnI, one gets avery compact object in which NN force becomes even repulsive. Keeping the usual S-waven-n potential, the enhancement of n-n P-waves required to bound a 3- or 4-n system issuch that they become themselves resonant!We have confined N=2,3,4 n in an HO trap with fixed frequency ω and size parameterb =√h¯mω. HO is the only external field in which ”internal” and ”center of mass” energiescan be properly separated. In absence of n-n forces the ”internal” energies are knownanalytically but can be obtained as well by solving the N-body ”internal” problem, withpairwise HO potential of frequency(ω√N). The effect of n-n interaction has been evaluatedby solving the internal problem withVij =12m(ω√N)2r2ij + Vnn(rij)and calculating the difference between the pure HO energy and the HO+Vnn one: BN =E(N)HO −E(N)HO+nn.Results concerning the ground state are given in Table 2 for several values of b. Somecomments are in order: (i) there is a clear indication of paring effect when going from3Table 2Binding energies BN of N neutrons in an HO trap with size parameter b.b=2 b=3 b=4N Jpi E(N)HO BNBNNBNEHOE(N)HO BNBNNBNEHOE(N)HO BNBNNBNEHO2 0+ 15.55 6.34 3.17 0.41 6.91 3.13 1.56 0.45 3.89 1.81 0.93 0.473 32−41.47 9.74 3.25 0.23 18.43 4.41 1.47 0.24 10.36 2.55 0.85 0.254 0+ 67.39 15.30 3.58 0.23 29.95 7.40 1.69 0.25 16.82 4.31 1.08 0.26N=2 → 3 → 4 (ii) one has always B4 > 2B2, suggesting an effective attraction betweendineutrons (iii) the binding energy per particle increases when going from N=2 to N=4(iv) the ratio BNEHOtends to a constant value independent of b. The preceding results tendto indicate that there is a benefit when going from 2 to 4. The main difference in respectto 3He has been found in the role of P-waves. Their influence in n case is attractivebut very small whereas they significantly contribute to the 3He binding energy (∼ 40%).The reason for such a different behaviour is the hard core radius of the correspondingpotentials, which differ by a factor 3 (see Fig. 1). The centrifugal barrier is one order ofmagnitude smaller in 3He and the effective potential is, contrary to n case, still attractivein regions where it can play a role. This difference can be dramatic in binding largerfermion systems.Despite the negative results quoted in [1], the question of larger neutron clusters meritssome attention. At present, the strongest argument against their existence are the mean-field results, all concluding to an unbounded infinite nuclear matter but it should bepossible to reach the same conclusion ”from below”, i.e. from a systematic study of few-neutron systems. This can be performed by studying the N-dependence of the criticalscaling factor ηc, calculated simultaneously for n and3He. Our technology allow us toreach only N=4 with the results displayed on Fig. 2. The value η(n)c makes a large jumpwhen passing from N=2 to N=3 but starts to decreases when going from N=3 to N=4. Isthat a pure numerical accident or, as in 3He, an indication of a descent towards the η = 1axis?. More powerful methods could go far beyond N=4 and draw a definite conclusionon this problem.REFERENCES1. S.C. Pieper, Phys. Rev. Lett. 90 (2003) 252501; contrib. to FB17 and refs. therein2. F.M. Marques et al, Phys. Rev. C65, 044006 (2002).3. D. Baumel, private communication4. R. Guardiola, J. Navarro, Phys. Rev. Lett. 84, 1144 (2001).5. R.B. Wiringa, V.G.J. Stoks, R. Schiavilla, Phys. Rev. C51 (1995) 386. R.A. Aziz and M.J. Slaman, J. Chem. Phys. 94 (1991) 80477. V.G. Stoks, R.A. Klomp, C.P. Terheggen, J.J. de Swart, Phys. Rev. C49 (1994) 29508. R.A. Malfliet, J. Tjon, Nucl. Phys. A127 (1969) 1619. R. Lazauska, PhD. Universite´ de Grenoble (2003)

Small clusters of fermions

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