A round up of the present status of the conjecture that n alpha nuclei form
an alpha-particle condensate in excited states close to the n alpha threshold
is given. Experiments which could demonstrate the condensate character are
proposed. Possible lines of further theoretical developments are discussed.Comment: 6 page

A. Tohsaki

G. Röpke

H. Horiuchi

P. Schuck

T. Yamada

Y. Funaki

English

arXiv

Schuck, P.

Horiuchi, H.

Roepke, G.

Tohsaki, A.

HAL-IN2P3

Alpha-particle condensation in nuclei

Peter Schuck

Hisashi Horiuchi

Gerd Röpke

Comptes Rendus Physique

theof furthertioncules alpha.s nécessaireseirthe recentlyrised by allms canng;uidity.elopednucleigine that,lyefollowingC. R. Physique 4 (2003) 537–540Exotic nuclei/Les noyaux exotiquesAlpha-particle condensation in nucleiPeter Schucka, Hisashi Horiuchib, Gerd Röpkec, A. Tohsakida Institut de physique nucléaire, 91406 Orsay cedex, Franceb Department of Physics, Kyoto University, Kyoto 606-8502, Japanc FB Physik, Universität Rostock, 18051 Rostock, Germanyd Department of Fine Materials Engineering, Shinshu University Ueda 386-8567, JapanPresented by Guy LavalAbstractA round up of the present status of the conjecture thatnα nuclei form anα-particle condensate in excited states close tonα threshold is given. Experiments which could demonstrate the condensate character are proposed. Possible linestheoretical developments are discussed.To cite this article: P. Schuck et al., C. R. Physique 4 (2003). 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved.RésuméCondensation de particules alpha dans les noyaux. Cet article fait le point sur la conjecture concernant la condensaen particules alpha des noyaux excités à des énergies proches du seuil de désintégration du noyau en plusieurs partiDes expériences permettant de mettre en évidence un tel phénomène sont proposées. Les développements théoriquepour étayer une telle hypothèse sont passés en revue.Po r citer cet article : P. Schuck et al., C. R. Physique 4 (2003). 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved.In this article we want to report on the present status of our recent conjecture [1] that at least lightnα-nuclei may showaround the threshold fornα disintegration, bound or resonance states which are of theα-particle gas type, i.e., they can bcharacterised by a self-bound quite dilute gas of almost unperturbedα-particles, all in relatives-states with respect to therespective center of mass coordinates and thus forming a Bose condensed state. Such a state is quite analogous todiscovered Bose condensates of bosonic atoms formed in magnetic traps. Also there the Bose condensate is characteBosons occupying the lowest-wave state of the mean field potential. Of course, in the atomic case the number of atobe enormous whereas in the nuclear case we, so far, only consider a handful ofα-particles. However, as soon asn 1, one canexpect condensation characteristics and therefore relatively fewα’s may be sufficient. The situation is similar to nuclear pairithere also a very limited number of Cooper pairs are involved and yet most nuclei manifest clear signatures of superflLet us briefly repeat what led us to our conjecture. It is well known that the only nucleus which shows a well devα-particle structure in its ground state is8Be. This singular feature was probably not given the attention it deserves. Otherwith three, four,. . . , nα-particles collapse in their ground states to a much denser system where theα-particles strongly overlapand probably loose almost totally their identity (see, however, discussion below). It seems a very natural idea to imawhen thesenα nuclei are expanded, at some low density individual almost unperturbedα-particles reappear, forming a weakself-bound gas-like state with, at most, pairwise twoα-particle correlations of the8Be-type. Sinceα-particles are Bosons, wconjectured that these low density states form a Bose condesate [1]. At the moment this conjecture is backed by theE-mail address: Schuck@ipno.in2p3.fr (P. Schuck).1631-0705/03/$ – see front matter 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rightsreserved.doi:10.1016/S1631-0705(03)00056-2538 P. Schuck et al. / C. R. Physique 4 (2003) 537–540esest.one oferefore,our theorynd staterver [5].end 0t recent].lumelations fore lowestat theemds value,d states ofes that theight.uredributioneavy ionrespect toof thendeedexplainer-pairCooperondensatelatterertzen and.findings. Based on the idea of anα-particle condensate, we made the following very simple variational ansatz for thnαwavefunction:〈r1σ1τ1, . . . , r4nσ4nτ4n|Φ(nα)〉 =A[e− 2B2 (X21+···+X2n)φ(α1) · · ·φ(αn)], (1)whereXi = (1/4)∑n rin is the center of mass coordinate of thei-th α-clusterαi . The internal wavefunction of theα-clusterαi isΦ(αi)= exp[−(18b2) 4∑m>n(rim − rin)2].The wavefunction of Eq. (1) is totally antisymmetrised by the operatorA. It is to be noted that the above wavefunction expresthe state wherenα clusters occupy the same 0s harmonic oscillator orbit exp[−(2/B2)X2] with B an independent widthparameter. For example, ifB is of the size of the whole nucleus whereasb remains more or less at the freeα-particle value, thenthe wavefunction (1) describes annα condensed state forn 1. On the other hand, forB = b (1) is a pure Slater determinanWe took the two parametersB andb of (1) as Hill–Wheeler coordinates and calculated for spherical geometry the 0+ spectrumof 8Be, 12C, 16O. Of course, the total center of mass was eliminated. For the effective nucleon–nucleon force theTohsaki [2] was adopted who adjusted parameters of a sum of two body and three body Gaussian forces toα−α catteringdata andα-particle properties long time ago. The Coulomb force was and is, of course, also included. The force, thressembles an effective force of the Gogny type [3], since the 3-body part gives rise to a density dependence. Thusdoes not contain any adjustable parameter. The solution of the Hill–Wheeler equation yields one 0+ state for8Be, two for12C,and three for16O. Very close agreement with experiment has been obtained (within some hundred keV) for the grouenergy of8Be, for the second 0+ in 12C (with respect to the 3α threshold energy), and for the fifth 0+ in 16O (with respectto the 4α threshold energy). For further details, please consult our papers, either published [1,4] or on the preprint seThe fact that we get correctly, without any adjustable parameters, three states close to thenα desintegration threshold in thesnuclei, is very nontrivial. For example, it is well known that it is very difficult to obtain the correct position of the seco+state in the12C [6]. Only sophisticated resonating group calculations have achieved this [6]. On the other hand, mosAMD calculations, though clearly showing the 3α structure of the second 0+ state in12C, miss its energy by several MeV [7Concerning the radii, our calculations show that12C and16O have, in the threshold states, approximately three times the voof the one of their ground states. This confirms the dilute gas structure of these states. Furthermore, deformed calcu8Be were also performed and preliminary results show that the wavefunction (1) allows one to reproduce the thremembers of the rotational spectrum of the8Be very satisfactority [5]. These successes of our theory make us believe thα-condensed states are real and we conjecture that they may exist even for quite high numbers ofα-particles. For example, onmay speculate of 10α-particles in40Ca. One may also think of a40Ca+ 40Ca head on heavy ion collision. Initially the systegets compressed. If the energy is just right, the decompression may stall approximately around theα-cond nsate density anthe whole system may decay intoα-particles via the coherent state. Of course, there is temperature, and depending on itmore or lessα’s may be broken. Such scenarios are nice dreams. But how to prove that these experimentalnα threshold statesreally have a Bose condensate character? The difficulty comes from the fact that the condensation occurs in excitethese nuclei. To prove Cooper pair condensation one makes a pair transfer from ground state to ground state and secross section is enhanced. But how to make anα-particle transfer into anα threshold state? Nevertheless, other methods mexist. Imagine that one excites the condensate, for example to a 2+ state. In12C such a state exists at∼3 MeV above thresholdThe nucleus may then decay inton= 3 α’s which can be detected with a multiparticle detector like INDRA. Their measvelocity distribution should be proportional to the Fourier transform of the excited condensate density. The velocity distwill therefore be rather narrow, if it corresponds to a wide density distribution, as assumed for the condensate. In hreactions rotational states are excited. A rotating condensate should have a strongly reduced moment of inertia withthe rigid body value [8]. Other experiments may be imagined to put the condensate character of thenα threshold states intoevidence. An indirect indication that the 02+-state in12C has extended character may come from the recent measurementspin-orbit splitting of of the extra neutron in13C when added to this state (von Oertzen and Bohlen, this special issue). Ithe spin-orbit splitting in13C corresponding to the 0+2 -state in12C is only half of the one of that corresponding to the 0+1 -state.A wider and flatter single particle potential in this excited state with respect to the one for ground state could possiblythis fact.Adding neutrons to theα-condensate is another very interesting subject. Adding neutrons to ordinary Coopcondensates quickly destroys superfluidity due to the blocking effect [8]. This also holds true for neutron–protonpairing [9]. However, adding neutrons to a low density Bose condensed gas of deuterons does not affect the deuteron cbecause the Pauli principle is not effective [9]. With theα-particle condensate we may well have an analogue to thesituation of the deuterons. Even on the contrary, some neutron excess may help to stabilise the condensate (von OBohlen, this special issue) making the question of the upper limit of the number of condensedα’s an even more intriguing oneRemember that8Be is unbound whereas9Be is bound!P. Schuck et al. / C. R. Physique 4 (2003) 537–540 539henergy. Leta nucleussolutionbecausear shellction iss.etheless,nsissituationsfnsityations recentlyapairingns mustogyer theral MeVwouldn then adjustedt higher.acanentthecannctionrorkedting theLet us now come to the question to what extend even the ground state ofnα-nuclei may show superfluid characteristics. Tcontour plotsE(B,b)= 〈Φnα |H |Φnα〉/〈Φnα |Φnα〉 show [1] that the absolute minima of the energy surfaces have an egain of several Mev with respect to the HF-limitB = b. However, one must be careful with the conclusion about this factus discuss the issue with the more familiar situation of standard Cooper pairing. In nuclear physics we are used to callsuperfluid, when the BCS equations have a non-trivial solution with a finite gap (contrary to infinite matter where a BCS-always exists for arbitrary weak attractive interaction, in finite systems the BCS equations not always have a solutionof finite level spacings and shell fluctuations of the single particle level density). As a rule, therefore, nuclei at or neclosure show no pair coherence whereas open shell nuclei are superfluid. The BCS wavefunction|BCS〉 = exp(P+)|vac〉, withP+ = ∑k zka+k a+k̄ the Cooper pair creation operator, does not conserve particle number. A better variational wavefuntherefore the so-called number projected BCS wavefunction|PBCS〉 = (P+)N/2|vac〉 whereN is, e.g., the number of neutronPerforming a minimisation of the energy with respect to the parameterszk with |PBCS〉, we will find that a non-trivial solutionalways exists, i.e., even doubly magic nuclei will experience a gain in energy with respect to the HF-approach. Nonwe will not consider magic nuclei as superfluid. This illustrates that|PBCS〉 theory allows one to treat two body correlatioin nonsuperfluid nuclei as well as it contains the superfluid case. We recognise that ourα-c ndensed wavefunction (1)analogous to number projected BCS and therefore may describe situations with four particle correlations as well aswith α-particle condensation. We believe that (1) largely describesα-like correlation in the ground states ofnα nuclei, whereasthe threshold states can be considered as condensates. This statement is corroborated by our recent investigations oα-particleconsensation in infinite nuclear matter [10]. There it turns out thatα-particle consensation ceases to exist once the dereaches values where theα’s start to overlap substantially. However, in finite nuclei this needs more carefull investigbecauseα’s on the surface of, say,40Ca in its ground state, may still experience low density whereas in the core theα’s arestrongly overlapping. In this respect, an intriguing fact of a pronounced even–odd effect in the number of deuterons habeen revealed by Schmidt et al. [11]. Indeed light nuclei with an even number of ‘deuterons’ (nα nuclei) have systematicallylower binding energy than those with an odd number of ‘deuterons’. This rather pronounced ‘even–odd’ effect hints toof ‘deuterons’. As a matter of fact sometime ago we have already attempted to describeα-particle condensation within a Bosopicture of nucleon pairs in applying a bosonic BCS approach to the IBMII model [12]. However, still further investigationbe performed before drawing any definite conclusions. For example, areα-particle transfer reactions enhanced innα nuclei?Does the band of ground states energies of neighbouringnα nuclei show rotational or vibrational character? This in analto pair vibrations and pair rotations, well known from the physics of ordinary pairing [8]. Another item could be whethrotational ground state band of, e.g.,24Mg shows a rigid or a superfluid moment of inertia.Let us also make some further remarks about the fact that our wavefunction (1) yields an energy minimum sevelower than the HF limit [1]. One may ask the question, whether present Skyrme- or Gogny-type of HF-calculationsalso get a lowering of several MeV whenα-type of correlations were included. We think that the answer is ‘yes’ but theparameters of the forces would have to be readjusted. On the other hand, the force of Tohsaki [2] used here has beewith correlations included. It therefore is not astonishing that our energy minima lie close to experiment and the HF-limiFurther light on these questions could be shed in formulating a theory ofα-particle condensation with a coherent,α-particlenon conserving, state|Φα〉, rather than with our number projected state|Φnα〉 of Eq. (1). For this, we have to introduceBogoliubov transformation among Fermionpairs, rather than of single Fermions like in the case of Cooper pairing. Thisbe done by introducing quasi-pair operatorsQ+ν =∑k,k′ [Xνk,k′a+k a+k′ −Yνk,k′ak′ak] and demanding the existence of a coherstate byQν |Φα〉 = 0. Minimising the average energyS1 =〈Φα∣∣[Qν [H ′,Q+ν ]]∣∣Φα 〉/〈Φα∣∣[Qν,Q+ν ]∣∣Φα 〉 (2)with respect to theX, Y amplitudes leads to the eigenvalue problem〈[δQ, [H ′,Q+ν ]]〉 =E〈[δQ,Q+ν ]〉 (3)with H ′ = H − µN . For a two body Hamiltonian the double commutator contains one and two body densities. SinceX,Y amplitudes form a orthonormal set, the relationQ+ = Xa+a+ − Yaa can be inverted and then the two body densitiesbe expressed in terms ofX, Y and one body densities. The latter can be calculated from a single particle Green’s fuconsistent with the whole scheme. This leads to anα-particle (four-body) gap equation and anα-particle order parameteκ1234= 〈Φα |a1a2a3a4|Φα〉 in analogy to pairing of single Fermions. To good approximation the vacuum is given by|Φα〉 ∼ exp[∑z1234a+1 a+2 a+3 a+4]|vac〉 (4)with z1234= (YX−1)1234. Partly, this formalism, called Self Consistent Quasi-particle RPA (SCQRPA) has been wout in [13]. In general, the numerical solution of the corresponding equations is very involved. However, approxima540 P. Schuck et al. / C. R. Physique 4 (2003) 537–540evalueross–bers oficularly ofcludesto theKluwer: Phys.g.wavefunctions by parametrised Gaussians as in (1), such calculations may seem feasable. Indeedz1234corresponds to a singlα-particle wavefunction in (1):z∼ κ ∼ exp[− 2B2X2]Φ(α) (5)The problem will be to obtain from this ansatz forz the corresponding expressions forX andY . This would allow one totreat practically any number ofα-particles, whereas the present number projected ansatz (1) is limited to a rather lowof n because the explicit antisymmetrisation quickly increases in complexity. Another way to treat large numbers ofα’s i todevelope an effective theory, considering theα’s as elementary particles, i.e., structureless bosons. Equations of the GPitaevskii-type or corresponding Hill–Wheeler equations may then be developed. The possibility to treat large numα-particles is important in order to decide where are the limits of a self-bound gas ofα-particles.A further speculation related to our studies above is that there may also exist condensates of heavior clusters, part16O nuclei. For example,48Cr might possess an excited 0+ state close to the three16O threshold analogous to the 0+2 in 12C,or 64Gd may show a four16O gas like state. Probably also partial Bose condensates exist. For example exciting32S a state witha16O− core+4 loosely boundα-particle may appear. Such states are being analyzed by M. Brenner et al. [14]. This conthe round up of the present status ofα-particle condensation in nuclei. We indicated present theoretical results hintingexistence of such states innα nuclei. We discussed consequences of such states, open problems, and perspectives.AcknowledgementsWe thank Y. Funaki and T. Yamada for fruitful discussions and contributions to this work.References[1] A. Tohsaki, H. Horiuchi, P. Schuck, G. Röpke, Phys. Rev. Lett. 87 (2001), 192501-1.[2] A. Tohsaki, Phys. Rev. C 49 (1994) 1814.[3] J. Decharge, D. Gogny, Phys. Rev. C 21 (1980) 1568.[4] P. Schuck, A. Tohsaki, H. Horiuchi, G. Röpke, in: W. Nazarewicz, D. Vretenar (Eds.), The Nuclear Many-Body Problem 2001,Academic, 2002, pp. 271–278.[5] Y. Funaki, H. Horiuchi, A. Tohsaki, P. Schuck, G. Röpke, nucl-th/0207022.[6] Y. Fujiwara, H. Horiuchi, K. Ikeda, M. Kamimura, K. Kato, Y. Suzuki, E. Uegaki, Prog. Theor. Phys. Suppl. 68 (1980) 29.[7] H. Horiuchi, private communication.[8] P. Ring, P. Schuck, The Nuclear Many Body Problem, Springer, 1980.[9] U. Lombardo, P. Nozières, P. Schuck, H.J. Schulze, A. Sedrakian, Phys. Rev. C 64 (2001) 064314.[10] G. Röpke, A. Schnell, P. Schuck, P. Nozières, Phys. Rev. Lett. 80 (1998) 3177.[11] J.J. Gaimard, K.H. Schmidt, Nucl. Phys. A 531 (1991) 709.[12] Y.K. Gambhir, P. Ring, P. Schuck, Phys. Rev. Lett. 51 (1983) 1235.[13] J. Dukelsky, P. Schuck, Phys. Lett. B 387 (1996) 233.[14] V.V. Lazarev, S.E. Belov, M. Brenner, K.A. Gridnev, T. Kurtukian Nieto, D.D. Caussyn, N.R. Fletcher, J.A. Liendo, S.H. Myers, inof Atomic Nuclei, 2000, Proceedings of the International Conference “Clustering Phenomena in Nuclear Physics”, St Petersbur

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