Extracting dynamics in the fusion of neutron-rich light nuclei
R.T. deSouza, Varinderjit Singh, S. Hudan, Z. Lin, C.J. Horowitz
aa r X i v : . [ nu c l - e x ] J u l Extracting dynamics in the fusion of neutron-rich light nuclei
R. T. deSouza, ∗ Varinderjit Singh, and S. Hudan
Department of Chemistry and Center for Exploration of Energy and Matter, Indiana University2401 Milo B. Sampson Lane, Bloomington, Indiana 47408, USA
Z. Lin
Department of Physics and Center for Exploration of Energy and Matter, Indiana University2401 Milo B. Sampson Lane, Bloomington, Indiana 47408 USA andDepartment of Physics, Arizona State University,450 E. Tyler Mall, Tempe, AZ 85287-1504 USA
C. J. Horowitz
Department of Physics and Center for Exploration of Energy and Matter, Indiana University2401 Milo B. Sampson Lane, Bloomington, Indiana 47408, USA (Dated: July 23, 2020)The dependence of fusion dynamics on neutron excess for light nuclei is extracted. This is accom-plished by comparing the average fusion cross-section at energies just above the fusion barrier for − C + C with measurements of the interaction cross-section from high energy collisions. Theexperimental results indicate that the fusion cross-section associated with dynamics increases withincreasing neutron excess. Calculations with a time-dependent Hartree-Fock model fail to decribethe observed trend.
PACS numbers: 21.60.Jz, 26.60.Gj, 25.60.Pj, 25.70.Jj
Nuclei are extremely interesting quantal systems. De-spite a limited number of constituent particles, they man-ifest collective dynamics. This collective dynamics is ob-served in many forms including the giant multipole res-onances [1], shape coexistence [2], and the quintessentialcase of fission [3]. Although typically associated withthe structure and reactions of mid-mass and heavy nu-clei, collectivity for very light nuclei has recently beenreported [4]. Nuclear fission and nuclear fusion provideexamples in which collective degrees of freedom undergosubstantial change as the reaction proceeds. Of partic-ular interest is the role of collectivity for neutron-richnuclei as for these nuclei the dependence of the dynam-ics on the asymmetry between the neutron and protondensities can be probed. Fusion reactions provide a pow-erful means to assess the response of neutron-rich nucleito perturbation. As fusion involves the interplay of therepulsive Coulomb and attractive nuclear potentials, byexamining fusion for an isotopic chain one probes theneutron density distribution and how that density dis-tribution evolves as the two nuclei approach and overlap[5–7]. In the following manuscript we propose a novel per-spective for investigating the role of collective dynamicsin fusion. Moreover, we present the dependence of thefusion dynamics on n/p asymmetry for the first time in-cluding the indication that for light nuclei fusion dynam-ics increases with increasing neutron number.It is well established that measurement of the interac-tion cross-section, σ I , in high energy collisions is an effec-tive means to invesitgate the spatial extent of the mat-ter distribution. At the high incident energy that theseexperiments were conducted at, one expects the sudden approximation to be valid. Hence, the measured inter-action cross-section, σ I provides a direct measure of theextent of the matter distribution. Systematic comparisonof these cross-sections for lithium isotopes revealed thehalo nature of Li [9, 10]. Presented in Fig. 1a are the in-teraction cross-sections of carbon isotopes with a carbontarget. Measurements for A ≥
12 were made at E/A ∼ σ I with neutron excess, (N-Z).Closer examination of Fig. 1a provides an indicationof the impact of shell structure on σ I . The dependenceof σ I on neutron excess for 12 ≤ A ≤
14 is weak as is thedependence for 16 ≤ A ≤
18. Between C and C oneobserves a jump in σ I from a value of ∼
850 mb to ∼ with N=8 and the population of the sd-shell indicatingthat the shell structure of the neutron-rich isotopes isobservable through measurement of σ I for an isotopicchain.Through comparison with a Glauber model, the rmsmatter radii of these nuclides has been extracted [13] andis presented in Fig. 1b. The extracted matter radii arecompared to the results of relativistic mean field (RMF)[14, 15] calculations using the NL3 and FSUGOLD inter-actions. In contrast to the widely used NL3 interaction,the FSUGOLD corresponds to a softer interaction andconsequently results in a slightly larger matter radius. Itis noteworthy that the NL3 interaction provides a some-what more accurate description of the extracted radii as C C + A Ι σ (LBL) Ι σ (a) Ι RMF (NL3)RMF (FSUGOLD)TDHF (SLy4) (b)
N-Z − − ( m b ) Ι σ ( f m ) R M S m R FIG. 1. Panel a): Dependence of the interaction cross-section σ I on neutron excess for various carbon isotopes. Taken from[8]. Panel b): The dependence of the matter radius on neutronexcess is compared with the predictions of RMF and TDHFmodels. (a) C C C C C (b) Radius (fm) ) ( nu c l eon /f m N ρ ) ( nu c l eon /f m P ρ FIG. 2. (Color online) Density distributions of neutrons andprotons (panels a and b respectively) predicted by the RMFmodel for carbon isotopes using the NL3 interaction. compared to the FSUGOLD interaction. Although theFSUGOLD interaction provides a slightly larger radiusfor all isotopes as compared to the NL3 interaction, thedependence on neutron excess is essentially the same.Presented in Fig. 2 are the neutron and proton densitydistributions for various carbon isotopes predicted by theRMF model using the NL3 interaction. Examination ofthe neutron density distributions reveals that with in-creasing neutron number, the tail of the neutron density C C + (a) C C + (b) C C + Fusion σ 〉 Fusion σ〈 (c) C C + RMF-SP(FSUGOLD)RMF-SP(NL3)TDHF (SLy4)RMF-SP(FSUGOLD)RMF-SP(NL3)TDHF (SLy4) (d) (MeV) c.m. E ( m b ) σ FIG. 3. (Color online) Fusion excitation functions for − C+ C. Experimental data are compared with the results of aRMF-SP model using the FSUGOLD and NL3 interactions.Predicted fusion cross-sections with a TDHF model with aSLy4 interaction are indicated. The fusion excitation functionfor C (not shown) is comparable to that of C. Data takenfrom [16, 17]. distribution extends further as expected. For the pro-ton density distributions, for N > Z, the largest observablechange is the decrease in the central density with increas-ing neutron number. This decrease in the central densityis correlated with a slight increase in the tail of the protondistribution due to the attractive nuclear force of the va-lence neutrons. This influence of the additional neutronson the charge radii has been experimentally establishedfor the carbon isotopic chain through measurement of thecharge changing cross-section [13]. Distributions calcu-lated with the FSUGOLD interaction exhibit the sametrends as those presented.To examine the response of the neutron and protondensity distributions to the perturbation involved in acollision we investigate the evolution of the fusion cross-section with increasing neutron number. We specificallyexamine the fusion cross-section for A C+ C at near bar-rier energies where long interaction times allow an adia-batic response of the density distributions. Using a novelactive target approach the fusion excitation functions forthese reactions was measured by the ANL group [18, 19]. C C + A (a) F σ Ι σ (LBL) Ι σ RMF-SP(NL3)TDHF (SLy4)RMF-SP(NL3)TDHF (SLy4) (b)
Expt.TDHF (SLy4) ( m b ) 〉 Ι σ 〈 ( m b )
17 14 〉 F σ 〈 ( m b ) 〉 D Y N A M I C S σ 〈 − N-Z
FIG. 4. Panel a) Comparison of the dependence of the averageabove-barrier fusion cross-section, <σ F > , and the interac-tion cross-section, σ I , on neutron excess. Experimental re-sults are compared with the predictions of the RMF-SP andTDHF models. Panel b) Dependence of the average cross-section due to dynamics on neutron excess. This active target approach is particularly well suitedto studying reactions with low-intensity beams and al-lowed measurement of the fusion excitation function withbeam intensities as low as 500 ions/s. Depicted in Fig. 3are the cross-section data for , , , C that have beentaken from [17] along with C cross-sections measuredusing the same approach but at higher beam intensitybeams [16]. While an improved analysis, primarily a re-duction of background arising from scattered beam, re-sulted in publication of revised cross-sections for , C[17, 20], systematic comparison of the neutron numberdependence of the fusion cross-section for the isotopicchain has not been published. The measured fusion exci-tation functions for , C + C are in good agreementwith those published in the literature [21] providing con-fidence in this approach to measure the fusion excitationfunction.Having established that the RMF model provides areasonably accurate description of the matter radii of thecarbon isotopes, we utilize the predicted RMF densitydistributions together with the Sao Paulo (SP) model[22] to predict the fusion cross-section. The fusion cal- culations were performed at energies of E/A = 2-3 MeVand the results are depicted in Fig. 3 as the solid andlong-dashed lines. While in the case of C the modeloverpredicts the experimental results, in the remain-der of the cases the agreement is reasonable. Calcula-tions with FSUGOLD consistently predicts larger cross-sections than those with NL3, consistent with the largerradii for FSUGOLD observed in Fig. 1b. It is notewor-thy that this increase in the cross-section for FSUGOLDas compared to NL3 is typical of the entire above-barrierregime. Although for C and C the RMF-SP(NL3)calculations provide a better description of the excitationfunctions, in the case of C, the RMF-SP(NL3) calcula-tions underpredict the measured cross-sections. For Ca better description is achieved using the FSUGOLD in-teraction. This trend indicates that the RMF-SP for agiven interaction, either NL3 or FSUGOLD, does notexhibit the same dependence on neutron number as theexperimental data. We therefore explore the dependenceof the average fusion cross-section on neutron excess. De-picted in Fig. 3 as the blue bar is the value of the averageexperimental cross-section. The bar also indicates the en-ergy interval of 14 MeV ≤ E c.m. ≤
17 MeV over which theaverage was calculated and the resulting quantity is des-ignated h σ F i . For − C the average cross-section isclearly representative of the above-barrier cross-section.In the case of C the choice of energy interval could re-sult in an average cross-section that is uncharacteristicalylow and therefore unrepresentative of the above-barrierbehavior. The upper energy limit of 17 MeV is dictatedby limited data for C motivating future measurement.The average above-barrier fusion cross-sections, h σ F i , are juxtaposed in Fig. 4a with the measured in-teraction cross-sections, σ I . One observes that for Cthe fusion cross-section and the σ I are essentially thesame. For N > Z however, the fusion cross-section dependsmore strongly on neutron excess than σ I does. Since thedependence of σ I on increasing neutron number indicatedthe inherent growth in the size of the matter radius withincreasing neutron number, the increased cross-section inthe case of fusion reflects the impact of dynamics in thefusion process. Moreover, the larger slope for fusion ascompared to σ I indicates that this dynamics increaseswith increasing neutron excess.Evidence for the dependence of the fusion dynamicson neutron number is also evident in Fig. 4a by compar-ing the experimental data with the RMF-SP(NL3) cal-culations. For N ≤ Z the value of <σ F > predicted bythe RMF-SP(NL3) model is essentially constant whilefor N > Z it increases approximately linearly. It is clearthat the experimental data exhibit a stronger dependenceon neutron excess than the RMF-SP model. This com-parison reinforces the increasing importance of dynam-ics with increasing neutron excess. Calculations withthe FSUGOLD interaction (not shown) though yieldinga slightly larger cross-section, exhibit the same increasewith neutron excess for N > Z.A key observation from Fig. 4a is the similarity of theslopes of the RMF-SP(NL3) calculations for N > Z andthat for σ I with N ≥ Z. This similarity of the two slopesarises from the fact that the σ I measures the size of thenucleus and the RMF-SP with the frozen density distri-butions is intrinsically related to the same quantity. Assuch, the quantity σ I provides a key experimental refer-ence from which to examine fusion dynamics. We there-fore present in Fig. 4b the difference between the aver-age above-barrier fusion cross-section and the interactioncross-section namely, h σ DY NAMICS i = h σ F i - σ I . Thisquantity represents the average cross-section due to thefusion dynamics. Although the span of neutron excessfor the fusion data presented is presently limited, the ex-perimental data manifests a linear behavior of the fusiondynamics on neutron excess.To investigate the dependence of dynamics on neu-tron number, we performed calculations with a time-dependent Hartree-Fock (TDHF) model. On generalgrounds the TDHF approach is well-suited to describ-ing the large-amplitude collective motion associated withfusion. Advances in theoretical and computational tech-niques allow TDHF calculations to be performed on a3D Cartesian grid thus eliminating artificial symmetryrestrictions [23]. Although in the sub-barrier regime inorder to accurately describe the fusion cross-sections itis necessary to perform density constrained TDHF (DC-TDHF) calculations [24–26] to obtain the heavy-ion po-tentials [27], at the above-barrier energies considered inthis work direct TDHF calculations can be performed byinitiating collisions for increasing impact parameters un-til the maximum impact parameter for fusion is reached.In practice this was done with an impact parameter pre-cision of 0.01 fm. Calculations were performed usingthe TDHF model, Sky3D [28] with a SLy4 interaction[29, 30]. Fusion calculations were performed for collisionof , , C projectiles with a C target. Due to the sen-sitivity of the fusion cross-section to pairing [26], exacer-bated for odd-A nuclei, and the deformed ground statesof some even-A isotopes, calculations were restricted tothese cases.Presented in Fig. 1b as the open squares are the RMSmatter radii predicted by the TDHF model for C and C. The radii predicted by the TDHF model are inreasonable agreement with the RMF calculations andonly slightly larger than those extracted experimentally.The fusion excitation functions predicted by TDHF for C+ C and C+ C are shown in Fig. 3 as the shortdashed line. The TDHF model systematically overpre-dicts the measured fusion excitation functions. Thisoverprediction has been previously reported for similarsystems [18, 26]. More interesting is the dependenceof the average fusion cross-section predicted by TDHFon neutron excess evident in Fig. 4. A linear behav-ior of the TDHF predicted cross-section on neutron ex- cess is observed indicating that neither any additionalenhancement or supression of dynamics is predicted de-spite the extreme neutron-richness of C. This linearbehavior manifests the same slope as the dependence of σ I on neutron excess defined by , , , , C indicat-ing that for the TDHF model dynamics provides essen-tially a constant increase to the cross-section above thegeometric size. The magnitude of this increase due tothe dynamics in the TDHF model is ≈
280 mb. Thelarger value of σ I for the weakly-bound nucleus C hasbeen associated with the halo nature of its unpaired neu-tron [13]. Due to the absence of data for σ I in thecase of C, h σ DY NAMICS i can only be calculated in theTDHF model for C and C. In Fig. 1b the depen-dence of h σ DY NAMICS i on neutron excess for the TDHFmodel arises purely from the near constancy of σ I , aconsequence of shell structure. The strong dependenceof h σ DY NAMICS i on neutron excess clearly exhibited inFig. 4b by the experimental data is not described by theTDHF model. This result suggests that an importantaspect of the dynamics is not incorporated in the model.Examination of the fusion cross-section at above-barrier energies for an isotopic chain is a powerful toolfor investigating nuclear dynamics. Comparison of av-erage fusion cross-sections just above the barrier withthe interaction cross-section, σ I , at high energies wherethe sudden approximation is valid allows extraction ofnot just the fusion dynamics but the dependence of thedynamics on neutron excess. For the light nuclei con-sidered in this work, a widely-accepted dynamical modelof fusion, namely a time-dependent Hartee-Fock model,fails to describe the dependence of the dynamics on neu-tron excess. By investigating this dynamics for the mostneutron-rich nuclei, valuable insight into the dynamics ofextremely asymmetric nuclear matter can be gained.We gratefully acknowledge helpful discussions on boththe general topic of fusion as well as the TDHF modelwith S. Umar (Vanderbilt University). This work wassupported by the U.S. Department of Energy underGrant No. DE-FG02-88ER-40404 (Indiana University).CJH is supported in part by U.S. DOE grants DE-FG02-87ER40365 and de-sc0018083. ZL gratefully ac-knowledges support from National Science Foundationunder PHY-1613708 and DOE grant de-sc0019470 (Ari-zona State University). ∗ [email protected][1] F. E. Bertrand, Ann. Rev. Nucl. Sci. , 457 (1976).[2] A. Gade and S. Liddick,J. Phys. G: Nucl. Part. Phys , 024001 (2016).[3] L. Meitner and O. Frisch, Nature , 239 (1939).[4] C. Morse et al. , Phys. Lett. B , 227 (2018).[5] Varinderjit Singh, J. Vadas, T. K. Steinbach, B. B.Wiggins, S. Hudan, R. T. deSouza, Zidu Lin, C. J. Horowitz, L. T. Baby, S. A. Kuvin, Van-dana Tripathi, I. Wiedenh¨over, and A. S. Umar,Phys. Lett. B , 99 (2017).[6] J. Vadas, V. Singh, B. B. Wiggins, J. Huston, S. Hu-dan, R. T. deSouza, Z. Lin, C. J. Horowitz, A. Chbihi,D. Ackermann, M. Famiano, and K. W. Brown,Phys. Rev. C , 031601(R) (2018).[7] S. Hudan, R. T. deSouza, A. S. Umar, Zidu Lin, andC. J. Horowitz, Phys. Rev. C , 061601 (2020).[8] A. Ozawa, T. Suzuki, and I. Tanihata, Nucl. Phys. A , 32 (2001).[9] I. Tanihata et al. , Phys. Lett. B , 380 (1985).[10] I. Tanihata et al. , Phys. Rev. Lett. , 2676 (1985).[11] A. Ozawa et al. , Nucl. Phys. A , 599 (2001).[12] A. Ozawa et al. , Nucl. Phys. A , 63 (1996).[13] R. Kanungo et al. , Phys. Rev. Lett. , 102501 (2016).[14] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. , 1(1986).[15] P. Ring, Prog. Part. Nucl. Phys. , 193 (1996).[16] S. Almaraz-Calderon et al. ,EPJ Web of Conferences , 01001 (2015).[17] P. F. F. Carnelli et al. ,Nucl. Instr. Meth. A , 197 (2015).[18] P. F. F. Carnelli et al. ,Phys. Rev. Lett. , 192701 (2014).[19] P. F. F. Carnelli, Measurement of fusion cross sectionsof carbon isotopes using an active target , Ph.D. thesis,Universidad Nacional De General San Martin, Comision Nacional de Energia Atomica, Buenos Aires, Argentina(2014).[20] K. Rehm, Private Communication.[21] D. G. Kovar et al. , Phys. Rev. C , 1305 (1979).[22] L. R. Gasques, L. C. Chamon, D. Pereira, M. A. G.Alvarez, E. S. Rossi, C. P. Silva, and B. V. Carlson,Phys. Rev. C , 034603 (2004).[23] A. S. Umar and V. E. Oberacker,Phys. Rev. C , 054607 (2006).[24] A. S. Umar, V. E. Oberacker, and C. J. Horowitz,Phys. Rev. C , 055801 (2012).[25] R. T. deSouza, S. Hudan, V. E. Oberacker, and A. S.Umar, Phys. Rev. C , 014602 (2013).[26] T. K. Steinbach, J. Vadas, J. Schmidt, C. Haycraft,S. Hudan, R. T. deSouza, L. T. Baby, S. A. Kuvin,I. Wiedenh¨over, A. S. Umar, and V. E. Oberacker,Phys. Rev. C , 041603(R) (2014).[27] A. S. Umar and V. E. Oberacker,Phys. Rev. C , 021601(R) (2006).[28] B. Schuetrumpf, P.-G. Reinhard, P. D. Steven-son, A. S. Umar, and J. A. Maruhn,Computer Physics Communications , 211 (2018).[29] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, andR. Schaeffer, Nucl. Phys. A , 710 (1997).[30] F. Douchin and P. Haensel,Astronomy and Astrophysics380