Microscopic calculations of 6He and 6Li with real-time evolution method
aa r X i v : . [ nu c l - t h ] F e b Noname manuscript No. (will be inserted by the editor)
Microscopic calculations of He and Li with real-timeevolution method
Q. Zhao a,1 , B. Zhou , M. Kimura , H. Motoki , Seung-heon Shin Nuclear Reaction Data Centre (JCPRG), Hokkaido University, Sapporo 060-0810, Japan Department of Physics, Hokkaido University, Sapporo 060-0810, Japan Institute of Modern Physics, Fudan University, Shanghai 200433, China Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japanthe date of receipt and acceptance should be inserted later
Abstract
The low-lying cluster states of He ( α +n+n)and Li ( α +n+p) are calculated by the real-time evo-lution method (REM) which generates basis wave func-tions for the generator coordinate method (GCM) fromthe equation of motion of Gaussian wave packets. The0 + state of He as well as the 1 + , 0 + and 3 + states of Li are calculated as a benchmark. We also calculatethe root-mean-square (r.m.s.) radii of the point matter,the point proton, and the point neutron of these states,particularly for the study of the halo characters of thesetwo nuclei. It is shown that REM can be one construc-tive way for generating effective basis wave functions inGCM calculations.
Keywords α cluster, halo nuclei, r.m.s. radius The light nuclei have been studied within the view ofthe cluster feature for more than five decades [1,2,3],and various nuclear theories have been developed forthe study of nuclear clustering [4,5,6]. By assuming thecluster structure, various cluster states of light nucleihave been investigated explicitly [7,8,9]. However, asthe number of the constituent clusters and nucleons in-creases or nuclear system becomes dilute, the numberof required basis wave functions increases very quickly.Therefore, a method which can efficiently sift out thebasis is highly desired. For this purpose, many effortshave been made, such as the stochastic sampling [10,11,12] and the imaginary-time development method [13].Recently, a newly time-dependent many-body the-ory has been developed in Refs. [14,15,16] for the cal-culations of Be and C isotopes. This real-time evolution a e-mail: [email protected] method (REM) generates the basis wave function usingthe equation of motion (EOM) which has been appliedin the study of heavy-ion collisions [17,18,19], but nowis found to be very effective in searching the basis wavefunctions for the microscopic calculations because of itsergodic nature.We intend to apply the REM on the 0 + ground stateof He nucleus ( α +n+n). It is a Borromean nucleus con-sisting of loosely bound and spatially extended three-body systems, typically composed of a compact coreplus two weakly bound neutrons (n+n+core) [20,21,22]. These properties can lead to the huge computa-tional difficulties despite of its simple physical struc-ture. Meanwhile, the low lying states of Li ( α +n+p)also can be a good comparison, where the more com-pact states (1 + and 3 + states with T = 0) and thedilute state (0 + state with T = 1) present simultane-ously. In this study, aiming to explore the applicabilityof REM, we will calculate the 0 + ground state of Heas well as the low lying states of Li to reproduce thehalo and un-halo properties of these states.This paper is organized as follows: Section 2 explainsthe framework of the wave function and the real-timeevolution method (REM). The numerical results includ-ing the energy and the root-mean-square (r.m.s.) radiusare presented and discussed in Sec. 3. The conclusionis summarized in Sec. 4. H = A X i =1 ˆ t i − ˆ T c.m. + A X i 65 MeV, V = 61 . c = 1 . 80 fm and c = 1 . 01 fm.We take the G3RS potential [24,25] as the spin-orbitinteraction, V ls = V ( e d r − e d r ) ˆ P ˆ L · ˆ S . (3)The strength parameter V is set to be 2000 MeV. TheGaussian parameters d and d are set to be 5 . − and 2 . 778 fm − , respectively.2.2 Generator coordinate methodIn the current work, the single-particle wave function φ ( r , Z ) are expressed in a Gaussian form multiplied bythe spin-isospin part χ τ,σ as φ ( r , Z ) = ( 2 νπ ) / exp[ − ν ( r − z √ ν ) + 12 z ] χ τ,σ . (4)Here the coordinate Z represents the generator coordi-nates, which includes the three-dimensional coordinate z for the spatial part of the wave function as well asthe spinor a and b for the spin part χ σ = a |↑i + b |↓i .In this work, the spinor a and b are also regarded astime-dependent variables which will be generated simi-larly to the spatial coordinates as introduced later. Theharmonic oscillator parameter b = p / (2 ν ) = 1 . 46 fm,which is same with that used in Refs. [11,26].We describe the He and Li as the α -cluster plustwo valence nucleon systems in the wave function. Thusthe corresponding wave function can be written as Φ ( Z , Z , z α ) = A{ φ ( r , Z ) φ ( r , Z ) Φ α ( r − , z α ) } . (5)Here Φ α is the wave function of the α -cluster with (0 s ) configuration. φ are the single-particle wave functionsas introduced above, which are used to describe the va-lence nucleons in He and Li. Thus, the coordinates r and r represent the real spatial position of valence nu-cleons while r − are for the nucleons in the α -cluster.Within the framework of generator coordinate method(GCM), the final wave function is the superposition ofthe basis wave functions with different sets of generatorcoordinates ( Z , Z , z α ): Ψ = X i f i ˆ P J π MK Φ i ( Z ,i , Z ,i , z α,i ) (6)where ˆ P J π MK is the parity and the angular momentumprojector. The generator coordinates Z can be obtainedby solving the equation of motion in REM as explainedin the next subsection. The corresponding coefficients f i will be determined by the diagonalization of the Hamil-tonian.2.3 Real-time evolution methodIn the quantum system, the wave function should sat-isfy the Schrodinger equation at all times. Thus, thetime-dependent variational principle holds for the in-trinsic wave function mathematically: δ Z dt h Φ ( Z , Z , Z α ) | i ~ d/dt − ˆ H | Φ ( Z , Z , Z α ) ih Φ ( Z , Z , Z α ) | Φ ( Z , Z , Z α ) i = 0(7)Regarding the coordinate Z as the function of the time t , we obtain the equation of the motion (EOM) as i ~ X j =1 , ,α X σ = x,y,z,a C iρjσ dZ jσ dt = ∂ H int ∂Z ∗ iρ (8) H int ≡ h Φ ( Z , Z , Z α ) | ˆ H | Φ ( Z , Z , Z α ) ih Φ ( Z , Z , Z α ) | Φ ( Z , Z , Z α ) i (9) C iρjσ ≡ ∂ ln h Φ ( Z , Z , Z α ) | Φ ( Z , Z , Z α ) i ∂Z ∗ iρ ∂Z jσ (10)By following the EOM, from an initial wave function at t = 0, the sets of the generator coordinates ( Z , Z , z α )for GCM can be yielded as a function of time t . Theensemble of the basis wave functions Φ i ( Z ,i , Z ,i , z α,i )denoted by these sets of the generated coordinates willhold the information of the quantum system. Thus, ef-fective basis can be generated. In practical calculations, we choose the proper ini-tial excitation energy (the definition can be found inRef. [14,15]) for obtaining various cluster configurationsin the evolution. To avoid the clusters or valence nucle-ons move to unphysical regions, the rebound conditionis imposed in our REM calculations. By following thework in Ref. [18], we add a potential barrier to theHamiltonian during the REM procedure with the form: V reb = k X i f ( | R i − R c.m. | ) f ( x ) = ( x − d ) θ ( x − d ) R i = Re( z i ) √ ν , R c.m. = 46 R α + 16 X j =1 R j . (11)Here R i and R c.m. represent the spatial position of the i th valence nucleon and the center of mass, respectively,so that | R i − R c.m. | is the distance between them. Be-cause of the step function θ ( x − d ), the evolving valencenucleon will face potential barrier when it is d fm farfrom the center of mass, and be smoothly pushed backin later evolution. We set the strength of the potentialbarrier k = 6 MeV/fm , which determines how rapidlythe height of the barrier increases. This value is notphysically important as long as it is not too large ortoo small. The rebound radius parameter d is set to be8 fm in our calculations, which is large enough for thecurrent work.We perform the above REM process for the intrinsicwave function of He and obtain an ensemble of basis.This ensemble of basis are used for both the calculationsof He and Li. We firstly show the energy spectra for the low-lyingstates of He and Li nuclei in Fig. 1. The experi-mental data and the corresponding results in the refer-enced works [11,26] are also included for comparison.It should be noted that we are using the same Hamil-tonian and the same form of the basis wave functions.In Fig. 1, it clearly shows that our REM method pro-vides the almost consistent results for the 0 + states of He and Li nuclei as the references. Besides, for the1 + ground state and the 3 + excited state of Li, thewave functions from our REM procedure provide bet-ter results than the reference work, which means thatwe have found more sufficient wave function throughthe evolution with the EOM. These results support thevalidity of the REM. Furthermore, it should be notedthat we are using one ensemble of the basis for both ofthe He and Li calculations, it is interesting that one -4-3-2-10 E n e r gy fr o m t h e t h r e s ho l d ( M e V ) -4-3-2-10He Li He Li He Li Ref. REM Exp. +++++ +++ + ++ + Fig. 1 The energy spectra of He and Li. Ref. denotes theresults from the reference works [11,26]. REM denotes theresults from the current work. Exp. denotes the correspondingexperimental data [27]. The energy is measured relative to the α + n + n threshold, which is set as 0 level with the dottedline. For the calculated results of both in this work and thereference works, the energy of He is − . 57 MeV, while it is − . 30 MeV in the experimental data. EOM can reproduce both the T = 0 states and T = 1states, and it indicates that the REM may have the po-tential for the investigation of the isospin mixing statesin the future study.Next we shall check the accuracy of our calculations.We show the energy convergences with the increasingnumber of basis in Fig. 2. It shows that the huge numberof the basis have been included and the binding ener-gies of all these states are well converged. These resultsprove that the number of basis in our calculations is suf-ficient to converge the energy results. Furthermore, wecan see that the converged results of 1 + and 3 + statesin our calculation are much lower than the results fromthe reference works. It denotes that the REM proce-dure have found more effective basis, which should beincluded to the total wave function.It is also an essential topic to investigate the haloproperty of the He nucleus as well as the Li nucleus.The 0 + ground state of He is the well known two-neutron halo. Likewise, the 0 + excited states of Li alsohas the controversial halo property [28]. To investigatethe halo property in these two nuclei, we calculate theroot-mean-square (r.m.s.) radii of He and Li with the number of basis -32-30-28-26-24 E n e r gy ( M e V ) He (0 ) REM Li (0 ) REM Li (3 ) REM Li (1 ) REM He (0 ) Ref. Li (0 ) Ref. Li (3 ) Ref. Li (1 ) Ref. ++++++++ Fig. 2 The energy convergence of He and Li from the REM calculations concerning the successive addition of bases. Thedash lines are the corresponding results from the reference works [11,26]. number of basis r . m . s . r a d i u s (f m ) He (0 ) + 0 200 400 600 800 1000point matter (0 )point matter (1 )point matter (3 )point matter in Ref. Li +++ Fig. 3 The r.m.s. radii of He and Li from the REM calculations concerning the successive addition of bases. The dottedlines denote the results in the reference works [11,26]. Table 1 The numerical results of the 0 + ground state of He, as well as the 1 + , 3 + and 0 + states of Li from the calculationsof REM. Energy (MeV) Point matter (fm) Point proton (fm) Point neutron (fm) He (0 + ) -28.37 2 . 71 2 . 03 2 . Li (1 + ) -30.92 2 . 65 2 . 66 2 . Li (3 + ) -29.87 2 . 42 2 . 43 2 . Li (0 + ) -27.58 2 . 79 2 . 81 2 . wave function from REM. The corresponding resultsare shown in Fig. 3.In the left panel of this figure, the calculated r.m.s.radii of point matter, point proton and point neutronof the 0 + state of He are 2 . 71 fm, 2 . 03 fm, and 2 . He. From the rightpanel of Fig. 3), one can also find that the r.m.s. radiusof point matter of the 0 + state of Li (2 . 79 fm) is larger than the radii of its 1 + (2 . 65 fm) and 3 + (2 . 42 fm). Itimplies that the 0 + state of Li can be treated as a halostate, which is consistent with the experimental conclu-sion [28]. These results show that the halo property ofthese states can be naturally included in the ensembleof the basis from the REM. Comparing with the ref-erence works, we notice that our results on the r.m.s.radii are larger than the results in the reference works,which are denoted by the dotted lines in Fig. 3. It in- dicates that our ensemble of basis from REM includesthe basis, where valence nucleons spread far from thecore, so that we provide more dilute structure for thehalo states of He and Li nuclei than theirs.In the end, the detailed numerical results are sum-marized in Table 1. The current results should be themost accuracy calculation on He and Li nuclei withinthe GCM framework. We perform the calculations for He and Li nuclei witha recently developed model named REM, which cangenerate the ergodic ensemble of the basis wave func-tions. During this work, we generate the basis wavefunctions from the procedure of REM and superposethem to construct the total wave functions. The con-verged results for the energy and the r.m.s. radius ofthe 0 + state of He as well as the 1 + , 0 + and 3 + statesof Li nuclei have been obtained in this work. The haloproperties of He and Li are well described in the cur-rent work, which indicates that the REM can searchthe basis more efficiently. The current works on Heand Li nuclei could be the most accurate calculationswithin the GCM framework to date. The benchmarkcalculations performed in this work can be instructivefor further calculation with REM. 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