Generic strong coupling behavior of Cooper pairs in the surface of superfluid nuclei N. Pillet(a), N. Sandulescu(a,b,c), P. Schuck(c,d) (a)DPTA/Service de Physique nucl´eaire, CEA/DAM Ile de France, BP12, F-91680 Bruy`eres-le-Chˆatel, France (b) Institute of Physics and Nuclear Engineering, 76900 Bucharest, Romania (c) Institut de Physique Nucl´eaire, CNRS, UMR8608, Orsay, F-91406, France (d) Universit´e Paris-Sud, Orsay, F-91505, France (Dated: February 4, 2008) With realistic HFB calculations, using the D1S Gogny force, we reveal a generic behavior of concentration of small sized Cooper pairs (2-3 fm) in the surface of superfluid nuclei. This study confirmsand extendspreviousresults given in theliterature that use more schematic approaches. Theopportunitiesofferedbythenewradioactivebeam these relative scattered pieces of information withstand 7 facilities to study the properties of weakly bound nuclei a general study of superfluid nuclei using one of the 0 with large neutron skins or halos triggered new interest mostperformantHFBapproaches,thatisemployingthe 0 for the issue of space correlations induced by the forma- finite range Gogny D1S-interaction [12]. As a matter 2 tion of Cooper pairs. The spatial correlations of Cooper of fact we will see that many of the earlier findings n pairsinsuperfluidnucleihavenotbeenextensivelystud- are qualitatively or even quantitatively confirmed. a ied in the past, but nevertheless a certain number of in- Indeed, we will show that the strong concentration of J vestigations,someratherearly,doexist. Mostlythiswas pair probability of small Cooper pairs in the nuclear 9 doneforthesingleCooperpairproblem. Forexamplethe surface is a quite general and generic feature and that 2 rmsdiameteroftheextraneutronpairin18Oisshownas nuclear pairing is much closer to the strong coupling a function of the nuclear radius by Ibarra et al. [1]. One regime [8, 13] than previously assumed. 1 v sees a strong minimum in the nuclear surface, indicating We will start by explaining shortly how the spatial 6 anrmsseparationbetweenthetwoactiveneutronsofthe properties of nuclear pairing are investigated within the 8 order of 2-3 fm. A similar behaviour was found later by HFB approach and then we shall present our results 0 Catara et al. [2] and Ferreira et al. [3] for a neutron pair and conclusions. For further understanding of the 1 in 206Pb and 210Pb. More recently, there are also many phenomena, a simple semiclassical analytic model for 0 investigations of the single Cooper pair problem in the nuclear pairing will be also considered. 7 halo nucleus 11Li [4, 5]. 0 / One of the rare papers where spatial correlations of It is well known [11] that pairing correlations can be h Cooper pairs are investigated in superfluid nuclei is the adequatelystudiedwiththeCooper-pairprobability κ2, t | | - one of Tischler et al. [6] where the probability distribu- where (in standard notation): l c tion of the pairs is shown as a function of the center of nu massR= 21|~r1+~r2|andtherelativedistanceofthenucle- κσ,σ′(~r1,~r2)=<HFB|ψσ′(~r2)ψσ(~r1)|HFB > , (1) : onsinthepairsr =|~r1−~r2|with(~r1,~r2)thecoordinates is the anomalous density matrix or pairing tensor in r- v of two nucleons. They showedthat in the open shell iso- space,calculatedwiththeHFB-wavefunction. Following Xi tope 114Sn one also finds Cooper pairs with short range Refs. [2, 6], we shall also consider the quantity: space correlations, like in one pair systems. They con- ar firmedalsothefindingofCataraetal,i.e.,theimportant P(R,r)=R2r2|κ(R,r)|2, (2) roleplayedbytheparitymixingforinducing shortrange which is the pair probability averagedover the angle be- space correlations. Mostof those older workswere,how- tweenR~ and~r andmultiplied by the phase spacefactors ever,done using rather schematic models and/orpairing R2r2. This quantity is importantsince it determines the forces. There exists, however, one study with a realistic two-particle spectroscopic factor [14] and other expecta- pairingforce(i.e.,theGognyinteraction)byBarrancoet tion values of two-body operators (e.g., pairing energy). al. [7], dedicated to nuclei embedded in a neutron gas, a Let us mention that Eq.(2) is formally the same for the system found in the inner crust of neutron stars. One of single Cooper pair problem [1, 2, 3, 5] and for the case the first systematic analyses of strong di-neutron spatial of Cooper pairs in a condensate [6]. correlations induced by the pairing interaction was done In whatfollows,we shall considerthe HFB expressionof recently by Matsuo et al.[8], using a zero range pairing the pairing tensor κ in center of mass and relative coor- force. The study of nuclear surface pairing properties dinates given by: was alsothe aim of severalhalfinfinite matter investiga- tions[9], [10]. Itwasfoundthatthepairdensityreaches 1 κ(R~,~r)= κl1j1 out further than the ordinarydensity but neither the lo- 4π n2,n1 X cal coherence length nor the probability distribution of n1,n2,l1j1 2l+1 (3) the pairs were calculated. ( )l( )1/2u (r/√2)u (√2R) nl Nl × − 2l X 1 nNl The aim of the present work is to verify how much all P (cosrˆR)<nlNl;0n l n l ;0> , l 1 1 2 1 × | 2 0.178 0.213 0.167 0.200 R(fm)10246800 522O 1000000000000000..............000000001111111234567801234512356789023456 R(fm)10246800 560Ca1000000000000000..............000000011111111245689023467837037036036036 R(fm)024680 2 1044Sn60000000000000000................000000000000111101233456788901236431976420975320 0 2 1204Sn60000000000000111................00123445678890121974208642086420 0 2 1284Sn60000000000000000................000000000000000101123345566788906295174062951740 r(fm) 0.292 r(fm) 0.382 10 00..225764 10 00..333548 r(fm) r(fm) r(fm) 0.237 0.310 8 60Ni 00..220119 8 104Sn 00..226826 0.183 0.239 FIG.2: |κ(R,r)|2calculatedwithHFB-D1Sfor104Sn,120Sn, m) 6 00..114664 m) 6 00..129115 128Sn. Scale has been multiplied bya factor of 106. R(f 4 00..111208 R(f 4 00..114637 0.091 0.119 2 0.073 2 0.095 0.055 0.072 0 00..001387 0 00..002448 throughout the nuclear radius. However, the distribu- 0 5 10 0 5 10 tion in R is rather different in the three isotopes. The r(fm) r(fm) difference comes from the localisation properties of the 0.754 0.278 0.707 0.261 single-particleshellswhichareclosesttothechemicalpo- 10 0.659 10 0.243 8 120Sn 000...556161852 8 212Pb 000...122902196 taernotuinadl. RT=hu5s,fmtheinpr1o2n8Sounnicsedduceontcoentthreatsiuornfaocfe|κlo(Rca,lris)a|2- 0.471 0.174 m) 6 00..347274 m) 6 00..113597 tion of the single-particle wave function 1h11/2, which R(f 4 00..238330 R(f 4 00..110242 becomes much closer to the chemical potential in this 0.236 0.087 2 0.188 2 0.070 isotope compared to lighter ones. One can also notice 0.141 0.052 0.094 0.035 thatin120,128Snthepairprobabilityhasasizeablevalue 0 0.047 0 0.017 0 5 10 0 5 10 for small values of R, which comes mainly from the con- tribution of the state 3s to pairing correlations. If r(fm) r(fm) 1/2 wehavehadchosenthe neutrondeficientPb-isotopes,in FIG.1: P(R,r)calculatedwithHFB-D1Sfor22O,60Ca,60Ni, which there is no s-state in the major shell, one would 104Sn, 120Sn, 212Pb. Scales have been multiplied by a factor rather see a depression of pair probability at the origin. of 102. Therefore, to say where the Cooper pairs are preferen- tially locatedin nucleiis a somewhatsubtle questionbe- cause the answer depends rather strongly on the shell where < nlNl;0n l n l ;0 > is the Brody-Moshinski structure (see also [16]). The shell structure dependence 1 1 2 1 bracket, u (r) ar|e the radial wave functions of the har- of κ(R,r)2 is largely washed out by the phase factor nl | | monicoscillatorandκlj isthematrixofthepairingten- r2R2 when P(R,r) is calculated. We shall make further n′n sor for a given angular momentum lj. As defined here, investigationsonthisissueinafuturework. However,we the latter has anintrinsic parity( )l. The HFB calcula- want to point out again that in all expectations values, tions are performed with the D1S−Gogny force [15]. In like e.g. the pairing energy and spectroscopic factors, it thecalculationsabasiswith15harmonicoscillatorshells is P(R,r) which counts and not κ(R,r)2. | | have been considered. The contour-lines of P(R,r) for ThefactthattheCooperpairswithsmallsizearecon- various superfluid nuclei are shown in FIG.1. The strik- centratedinthesurfacecanbealsoseenfromthedepen- ing feature is that for all these nuclei the same scenario, dence of the coherence length on the center of mass of with only slight modulations, emerges: the probability the pairs. The coherence length is defined as: P(R,r) is strongly concentrated in the surface with a small diameter of the pairs of the order of 2 3fm. ( r4 κ(R,r)2dr)1/2 − ξ(R)= | | (4) In FIG.1, we show nuclei close to the neutron drip-line (R r2 κ(R,r)2dr)1/2 (60Ca) as well as nuclei closer to stability. Seemingly, R | | there is no essentialdifference in the behavior of P(R,r) It is shown for various nuclei in FIG.3. One sees well between very neutron-rich nuclei and stable ones. This defined and pronounced minima at ξ 2 3fm for R ∼ − fact explains why one finds in all the superfluid nuclei a of the order of the surface radius. As we have already high probability for two-neutron transfer reactions. mentioned, a small coherence length in the case of a To conclude from the strong concentration of P(R,r) single Cooper pair has already been found previously inthe surfacethatCooperpairsaremostlysittinginthe for 18O in Ref.[1]. It is also the case for the Cooper surface of the nucleus may be, however, a bit mislead- pair in 11Li [4, 5]. Our calculations did not allow to ing. In FIG.2, we show the pair probability κ(R,r)2) go much beyond the minima because of the employed without the factor R2r2 which enters in the|probab|il- harmonic oscillator basis what becomes inaccurate far ity P(R,r). One can notice that for all tin isotopes outside the nuclear radius. However, the position of theCooperpairshaveverysmallextensioninr-direction the minima is always clearly identified and seen to be 3 14 10 K2 K2 12 22O 8 e o *10-2 60Ca 10 61004NSin R(fm) 46 00000.....667784826033210 (fm) 8 122102SPnb 1020 000000......444556048260987654 6 8 r(fm) 2 Ke K0 120Sn K2 000...233937100 0.252 4 m) 6 000...112371543 2 R(f 4 000...000159776 -0.022 -0.061 0 1 2 3 4 5 6 7 8 2 -0.100 R(fm) 0 0 5 10 15 200 5 10 15 20 FIG. 3: Coherence length ξ(R) for 22O,60Ca, 60Ni, 104Sn, r(fm) r(fm) 120Sn, 212Pb. FIG.4: ContributionsofdefiniteparitytoP(R,r)calculated with HFB-D1S for 120Sn. similar in all cases. What is surprisingis that the size of the Cooper pairs starts to decrease already well inside, see also the study in [6]). So to grasp the full physics around R = 2fm. Moreover, the decrease towards the of nuclear pairing it is very important to work in a large surface is approximately linear. configuration space, comprising several shells below and above the active one (see also [8]). In order to demonstrate that the strong concentration Inordertounderstandinmoredetailwherethisextraor- of small Cooper pairs in the surface of the nuclei is not dinary concentration effect from even-odd parity mix- a trivial effect, we decompose κ(R,r) in a part κ (R,r) ing comes from, let us consider a very simple model. e whichcontainsonlyevenparitywavefunctionsandapart We got inspired by the Thomas-Fermi model presented κ (R,r) which contains only the odd parity ones, i.e., in Ref.[3] where the anomalous density matrix is given o κ(R,r)=κe(R,r)+κo(R,r). InFIG.4,weshowwhatare by κTF(R,r) ∼ kF(R)j0(kF(R)r) with j0(x) a spheri- the probability distributions for Pe(R,r), Po(R,r) and cal Bessel fonction, k (R)= 2m(µ V(R)) Θ(µ V) F ~2 P (R,r) in the case of 120Sn. The quantity P (R,r) q − − eo eo the local Fermi momentum, µ the chemicalpotential (or corresponds to the interference term 2κ κ . From FIG.4 e o Fermienergy),andV(R)isphenomenologicalmeanfield one can see that selecting only either even or odd parity potential. Itcanbe shown[18]thataslightlymoreelab- states in κ(R,r) has a strong delocalisationeffect on the orate semiclassical version can be written as Cooper pairs: they are democratically distributed with respecttoaninterchangeofRandrvariables(oneshould m κsc(R,r)= dE κ(E) k (R) j (k (R)r) , (5) notice that in Eq.(3) the symmetry between R and r in- ~2π2 Z E 0 E volvesafactor2,whichcomesthroughthestandarddefi- nition of Brody-Moshinskytransformation). So no small where k (R) is the local momentum at energy E, ob- E Cooper pairs in the nuclear surface are prefered at all tainable from k in replacing µ by E, and κ(E) is the F in those cases. The concentration only shows up, when continuum version of the κ’s for the individual quantum even and odd parity states are mixed. This is clearly re- levels: κ(E) = ∆(E)/(2 (E µ)2+∆(E)2). We see vealed in looking at the interference term P (R,r). We that for very small ∆′s, opne ge−ts back the TF model [3]. eo see that it is negative for regions close to the r-axis and However, for realistic gap values the distribution of κ’s positive close to the R-axis. We checked that this sce- isveryimportant,otherwisethe concentrationeffectwill nario stays the same for all other superfluid nuclei con- not show up. For ∆(E) we adjust a Fermi function to sidered. This scenario was also nicely described in the representonaveragethegap-valuesoftheindividualsin- papers by Catara et al. [2] and Tischler et al. [6]. Mix- gle particle levels. An example can be seen on FIG.4 of ingofparitiesnaturallyoccursinheavynucleibecauseof Ref.[20]. Inthepresentwork,wehavefittedthe function the presence of intruder states of unatural parity in the ∆(E)onHFB-D1Sresultsfor120Sn. Agoodfitfunction main shells of given parity. However, as seen in FIG.1 is given by ∆(E) = 4/[1+exp(E µ)/20] (all numbers − firstpanel,the concentrationofCooperpairsalsooccurs are in MeV). For the mean field potential V(R) we take in such light nuclei as the oxygen isotopes where no in- the Woods-SaxonformofRef. [17]. The chemicalpoten- truders are present. This means that pairing in nuclei tial µ is determined, as usual, via the particle number is sufficiently strong so that κ grabs contributions from condition. several main shells, allowing for parity mixing even in InFIG.5 weshowthe correspondingsemiclassicalprob- light nuclei. If one artificially restricted the pairing con- ability Psc(R,r). We see qualitatively good agreement figurations,e.g. in22O,tothes-dshell,thencertainlyno with the quantal HFB-results, for instance in what con- concentrationeffect atall wouldbe seen(in this respect, cerns the concentrationofsmall Cooper pairsin the sur- 4 10 K2 K2 In conclusion, we showed that Cooper pairs in su- 8 e o *10-2 perfluid nuclei preferentially are located with small size R(fm) 46 00000.....667784826033210 (p2ro−fit3ffrmom) inthteheCsouorpfaecreprhegenioonm.eTnhoenr,e,thtahteyism, awxiitmharlley- 0.604 spect to the neutron-neutron virtual S-state in the vac- 0.565 2 0.526 0.487 uum (rms 12 fm, [5]), strong extra binding occurs, as 0.448 100 2*K*K K2 00..347009 longasthe densityis nottoo high. Further tothe center e o 0.330 8 0.291 ofthenucleusthestrongereffectoftheorthogonalisation 0.252 m) 6 000...112371543 of the pair with respect to the denser core-neutrons per- R(f 4 -0000....000021592776 teuxrpbasndthseagpaaiinr w[5a]v.eTfuhnactttiohnis.siImt pstlea,rtpshtyosicoasclliyllaatpepaenald- 2 120Sn --00..100601 ing and generic picture, is so pronounced, has come as a 0 surprise. It is certainly important for the interpretation 0 5 10 15 200 5 10 15 20 of pair transfer reactions. Most of these facts had al- r(fm) r(fm) readybeenrevealedinthepastforspecificexamplesand FIG. 5: Parities contributions Psc(R,r) calculated with the schematicmodelsandforces. We think, itisthe meritof semiclassical model for 120Sn. this work that it demonstrates with realistic HFB calcu- lationsusingthefiniterangeD1Sforcethegenericaspect of strong coupling features of singlet isovector pairing in face. We also show in FIG.5 the parity projected prob- nuclei. These features are in agreement with the ones abilities. As in FIG.4, one sees the strong delocalisa- recently put forward by Matsuo et al. [8]. Let us men- tion effect. In our model this can be understood analyt- tion that the strong coupling features revealed here are ically. Parity projection can be written as κ (~r ,~r )= somewhatcontrarytotheoldbelieve[22]thatthecoher- e/o 1 2 1[κ(~r ,~r ) κ(~r , ~r )] = 1[κ(R~,~r) κ(~r,2R~)] (see encelengthofnuclearpairsisofthe sameorderorlarger R2ef. [119]2). ±We t1he−ref2ore see2that good±par2ity implies, thanthenucleardiameter. Onthecontrary,amuchmore up to a scale factor, a symmetrisation in coordinates R diverse local picture has emerged. This may also be the and r. This is general and can be investigated analyti- reasonfortherathergoodsuccesofLDAfornuclearpair- cally in the TF model. We also calculate the coherence ingfoundinthe past[21]. Inspite ofthestrongcoupling lengthsemiclassicaly. We findqualitatively the samebe- aspects revealed in this work, we hesitate to say that havior as in the quantal calculation of FIG.3. there is Bose-Einstein condensation (BEC) of isovector The analytic model also allows to quickly grasp the sig- Cooper pairs, since this, strictly speaking, occurs only nificance of the coordinates used by Matsuo et al. [8]. for(ininfinitematter)negativechemicalpotential,what There, one takes a reference particle at position ~r on meanstrue binding. However,µneverturnsnegativefor 1 the z-axis, i.e. ~r =z ~e . Moving the second particle on isovector pairing in infinite nuclear or neutron matters. 1 1 z the z-axis we see that for P two symmetric peaks at Nuclear isovector pairing is just in the transition region e/o ~r = z ~e and ~r = z ~e appear whereas for the total from BEC to BCS. 2 1 z 2 1 z probability only one−peak on the side of the test parti- Acknowledgements : We would like to thank Marc cle appears. 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