aa r X i v : . [ nu c l - t h ] J u l Probing neutron correlations through nuclear break-up
Marl`ene Assi´e
1, 2 and Denis Lacroix Institut de Physique Nucl´eaire, Universit´e Paris-Sud-11-CNRS/IN2P3, 91406 Orsay, France GANIL, Bd Henri Becquerel, BP 55027, 14076 Caen Cedex 5, France
The effect of initial correlations between nucleons on the nuclear break-up mechanism is studied. Aquantum transport theory which extends standard mean-field approach is developed to incorporateshort range pairing correlation as well as direct nucleon-nucleon collisions. A time evolution of thenuclear break-up from a correlated system leading to the emission of two particles to the continuumis performed. We show that initial correlations have strong influence on relative angles betweenparticles emitted in coincidence. The present qualitative study indicates that nuclear break-upmight be a tool to infer the residual interaction between nucleons in the nuclear medium.
PACS numbers: 21.60.Jz, 25.60.Je, 24.10.CnKeywords: mean-field, correlations, nuclear reactions, nuclear break-up
Nuclei are self-bound systems formed of fermions in-teracting through the strong nuclear interaction. Whilemany facets of nuclei could be understood in term of in-dependent particle motion, some aspects reveal internalcorrelations [1]. We consider here the so-called break-upprocess leading to the emission of nucleons to the contin-uum. Numerous dedicated models have been developedto account for this mechanism [2]. Among them, timedependent models based on the independent particle hy-pothesis have been shown to provide a good description ofthe nuclear as well as Coulomb break-up [3, 4]. These ap-proaches, by neglecting two-body correlations could how-ever not provide appropriate theories when two nucleonsare emitted from the same nucleus [5, 6, 7]. Interferom-etry measurements are being now analyzed using ratherschematic models [7] and more elaborated theories. arehighly desirable.The aim of the present work is twofold: (i) developa microscopic quantum transport theory which incorpo-rates effects beyond mean-field like pairing correlationsand/or direct nucleon-nucleon scattering in the medium.(ii) present a qualitative study of nuclear break-up andshow that this mechanism can be a tool of choice for thestudy of correlations in nuclei. Similar challenges to (i)are being now addressed in strongly correlated electronicsystems using Time-Dependent Density Functional The-ory (TDDFT) [8, 9]. The Energy Density Functional(EDF) [10] shares many aspects with DFT and is ex-pected to provide a universal treatment of static and dy-namical properties of nuclei [10, 11]. Current EDFs startfrom an effective interaction (of Skyrme or Gogny type)to provide an energy functional, denoted E ( ρ ), where ρ isthe one-body density matrix. Then, guided by the Hamil-tonian case, equations of motion are written in terms ofthe one-body density evolution given by i ~ ∂ t ρ = [ h [ ρ ] , ρ ],where h [ ρ ] ≡ ∂ E ( ρ ) /∂ρ denotes the mean-field Hamil-tonian. To account for the richness of phenomena innuclear dynamics [12], different extensions of mean-fieldhave been proposed starting from the evolution: i ~ ∂ρ∂t = [ h [ ρ ] , ρ ] + Tr [ v c , C ] (1) where v c denotes the effective vertex in the correlationchannel, Tr ( . ) is the partial trace on the second parti-cle. C denotes the two-body correlation defined fromthe two-body density ρ as C = ρ − ρ ρ (1 − A ).The indices refer to the particle on which the operator isapplied (see for instance [11, 12]) while A is the per-mutation operator. Eq. (1) is generally complementedby the correlation evolution: i ~ ∂C ∂t = [ h [ ρ ] + h [ ρ ] , C ] + B + P + H , (2)where again the indices in h refer to the particle to whichthe Hamiltonian is applied. Expression of B and P and H , which can be found in [12, 13] are guided by theBBGKY hierarchy [14]. These terms describe in-mediumcollisions, pairing and higher order effects respectively.When three-body correlations are neglected, the abovetheory reduces to the so-called Time Dependent Den-sity Matrix (TDDM) theory [13]. Coupled equations (1)and (2) have been directly applied to giant resonances inref. [15] and more recently to fusion in ref. [16]. How-ever, applications are strongly constrained by the sizeof the two-body correlation matrix involved. In addi-tion, similarly to ref. [17], we encountered difficultiesto obtain numerical convergence towards a stable corre-lated system. Therefore, an appropriate approximationshould be made to render the TDDM theory more ver-satile. Keeping only B and projecting out the effectof correlation onto the one-body evolution leads to theso-called Extended TDHF theory with a non-Markoviancollision term [12, 18]. Keeping only P and assumingseparable correlations leads to the Bogoliubov extensionof the TDHF, i.e. TDHFB [19]. Both theories have beenrecently applied but require an [18, 20].A different approximation is used here. We are inter-ested in nuclei at low excitation where pairing plays animportant role. Similarly to TDHFB, we group single-particles into pairs, denoted by { α, ¯ α } , where | ¯ α i is ini-tially the time-reversed state of | α i . We then assume thatonly components of v c and C between such pairs aredifferent from zero. This approximation, called hereafterTDDM P , leads to important simplifications: (i) the num-ber of correlation matrix components to be calculated issignificantly reduced; (ii) the term H cancels out. Inthe basis where ρ is diagonal with occupation numbers n α , i.e. ρ = P α | α i n α h α | , the evolution reduces to i ~ ∂ t | α i = h [ ρ ] | α i ; ˙ n α = 2 ~ X γ ℑ ( V αγ C γα ) (3) i ~ ˙ C αβ = V αβ ((1 − n α ) n β − (1 − n β ) n α )+ X γ V αγ (1 − n α ) C γβ − X γ V γβ (1 − n β ) C αγ (4)where V αβ ≡ h α ¯ α | v c (1 − A ) | β ¯ β i , C αβ ≡ h α ¯ α | C | β ¯ β i and where the degeneracy of time-reversed states, i.e. n α = n ¯ α has been used.The TDDM P incorporates correlations in the dynamicsbut also could be used to initialize a correlated system.A correlated nucleus is obtained in two steps. First theev8 code [21] is used to obtain single particle states whichminimizes the Skyrme EDF using SIII interaction. Sec-ond, the TDHF3D code [22], has been updated to incor-porate both the evolution of correlation (Eq. (2)) as wellas its coupling to the one-body density (Eq. (1)). Equa-tions are integrated using a second order Runge-Kuttamethod where quantities on the right side of Eqs. (3-4)are estimated at time t + ∆ t/ t to t +∆ t . Note that this method insures the properreorganization of the self-consistent mean-field when cor-relations built up. In particular, the mean-field polariza-tion due to correlations is accounted for. In this secondstep, following refs [15, 17], we make use of the Gell-Mann-Low adiabatic theorem [23] and switch on adiabat-ically the residual interaction by v c ( t ) ≡ v c (cid:0) − e − t/τ (cid:1) .The residual interaction is set to [10, 24]: v c ( ~r , ~r ) = v (cid:18) − α h ρ ( ~R ) /ρ i β (cid:19) δ ( ~r − ~r ) (5)where ~R ≡ ( ~r + ~r ) /
2. The different parameters setsused here are given in table I with ρ = 0 .
16 fm − . force v α β Attractive -300 1/2 1Repulsive +300 1/2 1Volume -159.6 0 -Surface -483.2 1 1Mixed -248.5 1/2 1TABLE I: Parameters of the different residual interactionsused in this work.
Applications are performed in a three dimensionalCartesian mesh of size (80 fm) with a step of 0.8 fmand a time step of 0.45 fm/c. For τ = 300 fm/c, a verygood convergence of the adiabatic method, much betterthan for the full TDDM case [17], has been achieved.For this first application of TDDM P , we consider theisotopic oxygen chain with an α core while correlations build up between neutrons belonging to the spd shells.After convergence, correlated systems have occupationnumbers different from 0 and 1 and non-zero correlationenergy given by E corr = Tr( v c C ). Occupation num-bers are displayed in Fig. 1. TDDM P goes beyond purepairing theory like HFB due to the inclusion of two par-ticles - two holes (2p-2h) terms. This is clearly illus-trated by the doubly magic O nucleus where correla-tions do not cancel out and are in relatively good agree-ment with experimental observations of Refs. [26]. Intable II, an estimate of the pairing gap defined as [27]∆ ≡ E corr / P α p n α (1 − n α ) is compared to the fullTDDM and to the HFB cases. For the sake of compar-ison, the same core ( O) and the same force as in ref.[27] has been used. We see that our results are glob-ally in agreement with the HFB and full TDDM resultsvalidating the approximation made in TDDM P . FIG. 1: Single-particle occupation numbers in oxygen iso-topes as a function of single particle energies. Lines representFermi functions fits. Open squares correspond to experimen-tal occupation probabilities in O [26].TDDM [27] TDDM P HFB [25] O -3.1 MeV -3.5 MeV -3.3 MeV O -2.7 MeV -3.1 MeV -3.4 MeVTABLE II: Effective pairing gaps for − O deduced respec-tively from TDDM [27] and TDDM P are compared to HFBresults [25]. Correlated systems initialized with TDDM P are thenused to study the nuclear break-up leading to the emis-sion of two neutrons in coincidence. Intuitively, the twofollowing scenarii have been proposed [28]: (i) if the twoneutrons are initially close in position, both will feel thestrong short range nuclear attraction of the reaction part-ner and will be emitted simultaneously at small relativeangles (ii) if the two nucleons are far away in r-space,only one will undergo nuclear break-up. Then, the othernucleon might eventually be emitted isotropically fromthe daughter nucleus. Accordingly, large relative anglesare expected between the two nucleons transmitted tothe continuum in this sequential emission. To confirmthis intuitive picture, the nuclear break-up dynamics isstudied for an oxygen impinging on a Pb target.
FIG. 2: (Color online) One body density for three differentsteps of the dynamical evolution for an O +
Pb calcula-tion at 40 A.MeV. The circle represents the
Pb projectile.
The correlated nucleus is first initialized at the centerof a 3D mesh of size (80 fm) . Concerning the dynamicalstep, the collision is simulated treating the collision part-ner as a one-body time-dependent external perturbation,by replacing first equation in (3) by: i ~ ∂ t | α ( t ) i = { h [ ρ ( t )] + V P ( ~r, t ) } | α ( t ) i (6)where V P ( ~r, t ) stands for the projectile perturbation.Since we are considering here neutron emission, Coulombeffect is neglected and we assume that V P is a mov-ing Woods-Saxon potential given by V P ( ~r, t ) = V / (1 +exp { ( | ~r − ~r ( t ) | − R Pb ) /a } ) where ~r ( t ) corresponds tothe lead center of mass position. A simple straight linetrajectory is used for ~r ( t ) corresponding to an impactparameter of 11 fm (grazing condition). The parameter R Pb is set to 7.11 fm which corresponds to the Pbequivalent sharp radius while V =-50 MeV and a = 0 . O, part of the nucleons (inside thecircles) have been transferred while the rest is emitted tocontinuum. The last component corresponds to nuclearbreak-up emission and has already been understood in [4]and observed experimentally [29]. At the end of the evo-lution, nucleons from the inert core remaining in O ornucleons transferred to the collision partner are removedfrom the following analysis. The relative angles be-tween two nucleons emitted in coincidence are then recon-structed from the correlation written in momentum spaceas: C ( ~p , ~p ) = P αβ ˜ φ α ( ~p ) ˜ φ ¯ α ( ~p ) C α ¯ αβ ¯ β ˜ φ β ( ~p ) ˜ φ ¯ β ( ~p ), ) ( a . u . ) q ( C Initial (degrees) q ) ( a . u . ) q ( C Final ) ( a . u . ) q ( C (degrees) q ) ( a . u . ) q ( C FIG. 3: Left: Relative angle correlation between neutronsat initial (top) and final (bottom) time of the evolution foran O initialized with an attractive (full line) or repulsive(dashed line) residual interaction. Right: Initial (top) andfinal (bottom) relative angle correlation using the three dif-ferent residual interactions : a “Volume” (dashed line), “Sur-face” (full line) or “Mixed” (dashed-dotted line) residual in-teraction. All calculations are performed for an impact pa-rameter of b = 11 fm. where the ˜ φ α denotes the fraction of the final time single-particle states emitted to the continuum. The distribu-tion of relative angles, ¯ C ( θ ), is then deduced by sum-ming up contributions of all possible couples ( ~p , ~p ).To test the two scenarii, i.e. strong initial spatial cor-relation or anti-correlation, the “Attractive” and “Repul-sive” interaction have been used. The “Repulsive” inter-action is used to mimic the “cigar” like configurations,where the two correlated nucleons are well separated andon opposite side with respect to the core. Such a con-figuration is being now investigated experimentally usingcoulomb and/or nuclear break-up in lighter nuclei [6, 28].The initial values of ¯ C ( θ ) are given in top-left panelof Fig. 3. At initial time, the attractive force (close neu-trons in r-space) leads to large relative momentum (andtherefore large relative angles). On the contrary, the re-pulsive force corresponds to small initial relative angles.In bottom-left panel, ¯ C ( θ ) is displayed at the final timeof the evolution. Several important remarks could bedrawn. First ¯ C ( θ ) is largely modified compared to theinitial ones underlying the importance of dynamical ef-fects. Therefore, experiments where two neutrons aredetected in coincidence [6] could only be used if propertransport model are developed. Second, we see that ourcalculation indeed confirms the intuitive picture. Stronginitial correlation in space leads to small relative angleemission (solid line) while for initially well separated nu-cleons, relative angles are much larger. Comparison ofour model results with experiments requires to sum upthe contribution of different impact parameters. It isworth mentioning that the shape of the angular correla-tion presented in Fig. 3 does not change much as theimpact parameter or Wood-Saxon parameters are mod-ified. Therefore calculated cross-sections behave quali-tatively as in Fig. 3. In addition, experiments gener-ally present the ratio C exp = P ( θ ) /P ( θ ) P ( θ ) where P ( θ ) is the probability to emit two particles with a rel-ative angle θ . ¯ C ( θ ) calculated in this article is thedifference between the correlated emission and the inde-pendent emission. It can be compared directly to theexperiment by normalizing it to the independent emis-sion and adding 1. In conclusion, experiments dedicatedto the study of two nucleon emission to the continuumdue to the nuclear break-up used in parallel with ded-icated transport models can provide a valuable methodfor the study of internal correlations in nuclei.We further investigated the sensitivity of relative an-gles with the initial correlations by comparing three real-istic residual interactions called “Volume”, “Surface” and“Mixed” in table I leading to the same scattering length[24]. Corresponding initial and final angular correla- tions are respectively shown in top-right and bottom-right panel of Fig. 3. Although all forces correspond tothe scenario (i), sizeable differences are observed in theamplitude of ¯ C ( θ ). Therefore, coincidence measure-ment of nucleon emission should be seriously consideredin the near future as a tool to further constraint residualinteraction used nowadays in EDF theories. It shouldhowever be kept in mind that a quantitative compari-son with experiments requires a careful analysis of thepresent model parameters which is underway. Acknowledgments
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