Residue cross sections of 50 Ti-induced fusion reactions based on the two-step model
Ling Liu, Caiwan Shen, Qingfeng Li, Ya Tu, Xiaobao Wang, Yongjia Wang
aa r X i v : . [ nu c l - t h ] D ec EPJ manuscript No. (will be inserted by the editor)
Residue cross sections of Ti-induced fusion reactions based onthe two-step model
Ling Liu , Caiwan Shen , Qingfeng Li , Ya Tu , Xiaobao Wang , and Yongjia Wang College of Physics Science and Technology, Shenyang Normal University, Shenyang 110034, Liaoning, China School of Science, Huzhou University, Huzhou 313000, Zhejiang, ChinaReceived: date / Revised version: date
Abstract. Ti-induced fusion reactions to synthesize superheavy elements are studied systematically withthe two-step model developed recently, where fusion process is divided into approaching phase and forma-tion phase. Furthermore, the residue cross sections for different neutron evaporation channels are evaluatedwith the statistical evaporation model. In general, the calculated cross sections are much smaller than thatof Ca-induced fusion reactions, but the results are within the detection capability of experimental facil-ities nowadays. The maximum calculated residue cross section for producing superheavy element Z = 119is in the reaction Ti+
Bk in 3 n channels with σ res (3 n ) = 0 .
043 pb at E ∗ = 37.0 MeV. PACS.
The synthesis of superheavy nuclei is a hot study field innuclear physics, and it has obtained much progress exper-imentally and theoretically in recent years. Up to now,with the detecting of element Z = 117 in Dubna in 2010[1], the superheavy elements Z = 110 −
118 have been allsynthesized [2,3,4]. Theoretical supports for these verytime-consuming and very high-expensive experiments areextremely vital for choosing the suitable target-projectilecombinations and the optimum incident energy, and forthe estimation of residue cross sections.In synthesis of superheavy elements with proton num-ber Z = 114 − Ca as a projectile and actinide as a target is adopted.However, it comes increasingly difficult to synthesize heav-ier elements with projectile Ca. Maybe the last super-heavy element which can be produced in the reaction with Ca is the element with Z = 118 since the target heavierthan Cf is too difficult to be obtained. Thus to produceheavier elements, heavier projectiles such as Ti, Cr, Fe, Ni should be used.Nevertheless, it is well known that in heavy ion inducedreactions the deep-inelastic and quasi-fission processes arethe dominant reaction channels because of the strong fu-sion hindrance, thus the fusion probability is much smallerthan the light ion induced reactions. In the reactions with Ca and actinide targets the probability of fusion relativeto quasi-fission is less than 10%, and the ratio decreasesfor more symmetrical target-projectile combinations [5]. a Corresponding author (email: [email protected] )
Mass asymmetry is one of the factors that influence quasi-fission and true fusion competition. Generally speaking, adecrease in the mass asymmetry in the reaction entrancechannel leads to an increase in the quasi-fission and a de-crease in the fusion contributions into the capture crosssections. It appears that fusion is strongly hindered as thesize of the projectile relative to the target increases. There-fore, it is not at all surprising to see the failed attempt tomake even heavier element 120 using the Fe +
Pureaction [6]. Thus, a spherical neutron magic nucleus Ti(only two protons greater than Ca) seems to be a promis-ing candidate of projectile in the synthesis of superheavyelement heavier than Z = 118. Furthermore, it is worthmentioning that the lower limit of residue cross sectionwhich can be detected experimentally is in the magnitudeof 0.03 pb at present [7], so synthesis of superheavy ele-ments with projectile Ti is of great interest.According to the theory of compound nucleus reac-tions, the whole process of synthesizing the superheavynuclei is composed of fusion part and fission part. In theformer part the projectile is captured by the target anda amalgamated system is formed that then evolves intoa spherical compound nucleus, and then in latter part,besides being cracked into smaller fragments, few of thecompound nucleus may cool down by evaporating particlesand γ -rays and goes to its ground state. The evaporationresidue cross section is usually expressed as a sum over allpartial waves at a certain incident energy, σ res ( E c . m . ) = π ~ µE c . m . X J (2 J +1) P J fus ( E c . m . ) · P J surv ( E c . m . )(1) Ling Liu et al.: Residue cross sections of Ti-induced fusion reactions based on the two-step model where J is the total angular momentum quantum num-ber, E c . m . the incident energy in the center of mass sys-tem; P J fusion and P J surv denote the fusion and the survivalprobabilities, respectively.In the evaporation process, though there is a certainmargin of uncertainty in the estimations of evaporationresidue cross sections [8], the statistical evaporation modelis commonly accepted and used to calculate the evapora-tion probability P J surv . However in the fusion process, be-cause of the complexity of heavy ion reactions, there is stillno commonly accepted model to deal with this process.Several models are adopted to study the fusion reactions,such as fusion-by-diffusion model [9,10], DNS model [11,12], QMD-based model [13], etc. In this paper we adopttwo-step model to study the Ti-induced fusion reactionsleading to the synthesis of superheavy nuclei.The paper is arranged as follow: Sec. 2 gives a brief de-scription of the two-step model, and of the determinationof parameters fitting the experimental capture cross sec-tions; Sec. 3 shows the results of systematic calculationsand discussions; Sec. 4 gives a summary.
The two-step model was proposed to describe the fusionprocess in massive nuclear systems where fusion hindranceexists, as shown in Ref. [14,15,16]. In this model, the fu-sion process is divided into two stages: first, the stickingstage where projectile and target come to the touchingpoint over the Coulomb barrier from infinite distance, andsecond, the formation stage where the touched projectileand target evolve to form a spherical compound nucleus.Therefore, the fusion probability gets the form, P J fusion ( E c . m . ) = P J stick ( E c . m . ) · P J form ( E c . m . ) . (2)The energy dissipations in sticking stage and in formationstage are subtly considered in the model. It is worth toemphasize that the two-step model provides a method fora connection between a two-body collision process and thesubsequent one-body shape evolution. This is completelydifferent from the adiabatic, or the diabatic connection,and should be called as a “statistical connection’ [14].In principle, both of the sticking probability and for-mation probability need to be calculated via fluctuation-dissipation model, as shown in Ref. [14]. However, for sim-plicity, we also choose alternatively an empirical formula[17] to calculate the sticking probability, where the barrierheight is supposed to be Gaussian-distributed around theCoulomb barrier to simulate the energy dissipation in theapproaching phase. The P stick takes the form, P J stick ( E c . m . ) = 12 (cid:26) (cid:20) √ H ( E c . m . − B − ~ J ( J + 1)2 µR B ) (cid:21)(cid:27) , (3) -2 -1
10 20 30 40 50 10 20 30 40 50 60 Ti+ Pb Exp. Clerc C=0.095B=1.5 c ap ( m b ) Ti+ Bi Exp. Clerc C=0.095B=2.7
Exp. Itkis C=0.095 B=16.2
E* (MeV) Ti+ Pu Fig. 1.
Fitting (solid lines) the experimental capture crosssections [5,18] to get appropriate C and ∆B .
82 84 86 88 90 92 94 96 98 10004812162024 B ( M e V ) Z B = 1.225 Z - 98.95
Fig. 2.
The extrapolation of the shift of Coulomb barrier forthe nearby heavier targets. The solid circles correspond to thedata fitting to experimental data in Fig. 1, while the open onesare extrapolations for Am, Cm, Bk, Cf, and Es. where B is the barrier height of the Coulomb potentia, H the width of the Gaussian distribution of the barrierheight, µ the reduced mass, and R B the distance betweentwo centers of projectile and target at the Coulomb bar-rier. In a reasonable assumption B / ( √ H ) should bemuch greater than 1.To calculate sticking probability, the parameter C (afactor to calculate the width of the Gaussian distributionof the barrier height, see Ref. [15]) and the barrier heightof the Coulomb potential B = B + ∆B should be ad-justed as adequate as possible for very heavy systems. Thethree systems to be fitted are Ti+
Pb, Ti+
Bi [18]and Ti+
Pu [5]. The fitted results are shown in Fig.1. With a constant value of C = 0 .
095 and a linear in-crease of the barrier height shift ∆B with proton number Z , the available experimental capture cross sections arewell reproduced. The ∆B is extrapolated from the fittingformula ∆B = 1 . Z − .
95 for heavy targets havingvery close atomic numbers, namely Am, Cm, Bk, Cf, andEs, as is seen in Fig. 2.With Eq. (3) and the re-fitted parameter C and ∆B ,the sticking probability of Ti-induced fusion reactions iscalculated with confidence. Further more, for hot fusionreactions, we are interested in the residue cross sections ing Liu et al.: Residue cross sections of Ti-induced fusion reactions based on the two-step model 3 around E ∗ = 30 ∼
40 MeV, and the experimental dataare just located in the similar energy range, as shown inFig. 1, the calculated σ res around the same energy rangewill not sensitively depends on the determination of C and ∆B . In the excited compound nucleus, the de-excitation pro-cess includes usually light particle emissions, γ -ray emis-sions, and fission. However, because of existence of theCoulomb barrier for charged particle emissions, the prob-ability for the emission of light charged particles is muchsmaller than the one for the neutron emission. Therefore,most of the superheavy nuclei are obtained through theconsecutive neutron evaporations. In the calculation ofsurvival probability, the hivap code, based on the statisti-cal evaporation model, is adopted to evaluate the residuecross sections.The very important parameter in the evaporation pro-cess is the fission barrier B f . It is clear that for heavynuclei, the more stable one, which means having largershell correction energy E shell , usually has a heigher fissionbarrier. Thus, the classical way to calculate the fissionbarrier is B f = B LD − E shell , as did in Ref. [19]. Howeverthe microscopic calculations does not prove such so simplerelationship between B f and E shell . Since the microscopiccalculations do not give B f for so heavy compound nucleidiscussed in the present paper, we thus used the classicalform to calculate B f , but with an arbitrary factor f tothe shell correction energy, i.e., B f = B LD − f · E shell , (4)where E shell is taken from Ref. [20], and the factor f is de-termined by fitting the experimental data. In the presentcase, the inclusion of the factor f gives rise to reductionsof the fission barrier, and then the reduction of residuecross sections, but does not change general feature of theexcitation functions, i.e., peak positions etc., though de-creasing slopes in higher energies are a little affected. Theintroduction of the factor f , thus, is appropriate for pre-dictions of residue cross sections.Up to now, there are no experimental residue cross sec-tions for Ti-induced fusion reactions to synthesize super-heavy nuclei with Z ≥ Ca+
Bk had been measured by Dubnain 2010 [1], and together with fitting the experimentalresidue cross section of Ca+
Pb [21] and Ti+
Pb[18], the corresponding factor f for the three systems aredetermined to be 0 . .
72 and 0 .
77, respectively, using HIVAP code. Then, thefactor f for reaction Ti +
Bk can be approximatelyevaluated as 0 . × (0 . / .
72) = 0 .
48. Since the targetBk of the reaction have only one or two protons more orless than the targets such as Am, Cm, Cf and Es, thefactor f for Ti+Bk should also work with these target in Ti-induced reactions.
20 25 30 35 40 45 50 55 -3 -2 -1 -2 -1 Ti + Am
5n 4n 3n 2n 1n r e s ( pb ) E * (MeV) Ti + Am Fig. 3.
Predicted residue cross sections for producing super-heavy element Z = 117. The short dot line, short dash line,dash dot dot line, dash dot line, and dot line represent for 1 n ,2 n , 3 n , 4 n , and 5 n neutron evaporation channels, respectively. -2 -1 -2 -1
20 30 40 5010 -3 -2 -1
20 30 40 50
5n 4n 3n 2n 1n Ti+ Cm
5n 4n 3n 2n 1n Ti+ Cm
5n 4n 3n 2n 1n Ti+ Cm
5n 4n 3n 2n 1n Ti+ Cm
5n 4n 3n 2n 1n E * (MeV) r e s ( pb ) Ti+ Cm
5n 4n 3n 2n 1n Ti+ Cm Fig. 4.
The same as in Fig. 3, but for producing superheavyelement Z = 118. Now, all the ingredients in our calculations do not leaveany ambiguity. The target isotopes of Am, Cm, Bk, Cf,and Es with life times long enough for experiments arechosen, and then, the evaporation residue cross sectionswith Z = 117 −
121 in some selected reactions are calcu-lated systematically using the two-step model and HIVAPwith the purpose of searching the favorable reaction sys-tems and collision energies for the synthesis of superheavynuclei with even larger proton number. The correspondingresults are shown in Figs. 3-7.Fig. 3 shows very similar excitation functions of pro-ducing element Z = 117 in the Ti+ , Am reactions.It is reasonable since two targets differ with only two neu-
Ling Liu et al.: Residue cross sections of Ti-induced fusion reactions based on the two-step model trons. The maximum residue cross sections for Ti+
Am,is slightly larger with at in 3 n channels.The calculated residue cross sections for reactions Ti+ − Cm to produce elements Z = 118 are presented inFig. 4. According to the results, the optimum reactionto synthesize Z = 118 are Ti+
Cm with σ res (2 n ) =0 .
053 pb at E ∗ = 28 . Ti+
Cm with σ res (3 n )= 0.040 pb at E ∗ = 37 . n and 3 n evaporationchannels of the compound nuclei is around 8 MeV which isabout one neutron separation energy. Furthermore, it canbe seen from Fig. 4 that our results do not show strong iso-tope dependence of superheavy nucleus production. Gen-erally speaking, the formation of the superheavy nucleus isa complex dynamical process and depends on many phys-ical factors, such as Coulomb barrier, conditional saddlepoint, neutron separation energy, shell effect and so on. Itneeds more further explorations.The next superheavy element to be synthesized in ex-periment may be Z = 119 or 120, therefore, the inves-tigations of the synthesis of Z > Ti+ , Bk the maximum residuecross sections for producing superheavy element Z = 119are both in 3 n channels and are, respectively, 0.043 pband 0.033 pb, which are almost one order of magnitudesmaller than that of Ca+ , Bk in reference [15]. Thiscould be explained by the fact that for the same actinidetarget, the Coulomb potential for Ti-induced reactionis roughly 10% larger than that for Ca-induced reac-tions, and the fusion hindrance for the former one is alsostronger than the latter case [22]. Moreover, the results for Ti+
Bk from Ref.[9] with fusion-by-diffusion modeland from Ref.[23] with dinuclear system model are 0.57pb and 0.11 pb respectively, while our present calculationvalue is very close to 0.035 pb calculated with an ana-lytical expression for the description of the fusion proba-bility [24] and 0.05 pb in Ref. [25]. The different resultscan be attributed to the dependence of model and param-eter. However, although the cross sections are relativelysmall and are more than two orders of magnitudes lowerthan pico-barn, they are in the detection capability of thepresent experimental facilities.Fig. 6 and Fig. 7 gives the results for producing Z =120 and 121 in reactions Ti + − Cf and Ti + , Es, respectively. It can be seen from the figures thatthe residue cross sections are too small, approaching tothe order of femto-barn. As was noted in the figures, themaximum residue cross sections in the reactions Ti + , , Cf are, 9.7 fb, 7.5 fb, and 12.2 fb, respectively,which are obviously larger than those of the reaction Ti+
Cf (4.6 fb). Recently, Siwek-Wilczynska et al. predictthe cross section of Ti +
Cf to synthesize the element Z = 120 to be only 6 fb [10] which is more consistentwith our present result. In contrast, our result is smallerthan other several predictions with σ res ≈ ∼
200 fb [9,24,25]. In addition, it is worth arguing that the residuecross section with target
Cf is several times larger thanthose of the targets − Cf and hence is theoretically -2 -1
20 25 30 35 40 45 50 5510 -3 -2 -1 r e s ( pb ) E * (MeV) Ti+ Bk
5n 4n 3n 2n 1n Ti+ Bk Fig. 5.
The same as in Fig. 3, but for producing superheavyelement Z = 119.
20 30 40 5010 -4 -3 -2 -1
20 30 40 5010 -4 -3 -2 -1
5n 4n 3n 2n 1n Ti+ Cf
5n 4n 3n 2n 1n Ti+ Cf E * (MeV)
5n 4n 3n 2n 1n Ti+ Cf
5n 4n 3n 2n 1n Ti+ Cf r e s ( pb ) Fig. 6.
The same as in Fig. 3, but for producing superheavyelement Z = 120. the most favorable one for the synthesis of element 120.However, Cf may be difficult to be target because itsspontaneous fission would bring about serious backgroundin the experiment. Therefore, to produce element Z = 120, Ti + − Cf could be considered in the future exper-iments. As a further extension, we evaluate the residuecross sections of element 121 with targets , Es thoughEinsteinium is rather exotic and may be hardly preparedpresently. It shows that the maximum residue cross sec-tions for nucleus Z = 121 have comparable value of about3 fb and are far below the present experimental limit ofregistration (30 fb). Thus, the synthesis of elements with Z >
120 is rather problematic in the near future due toextremely low cross sections and short half-lives of theseelements.To illustrate the results clearly, the relatively largerresidue cross sections for different reactions are listed inTable 1. It should be mentioned that our reduction factor f of E shell of the compound system influence only on the ing Liu et al.: Residue cross sections of Ti-induced fusion reactions based on the two-step model 5
20 25 30 35 40 45 50 5510 -4 -3 -2 -3 -2
5n 4n 3n 2n 1n r e s ( pb ) E * (MeV) Ti+ Es
5n 4n 3n 2n 1n Ti+ Es Fig. 7.
The same as in Fig. 3, but for producing superheavyelement Z = 121. Table 1.
The relatively larger residue cross sections for the Ti-induced reactions to synthesize superheavy nuclei for dif-ferent target elements. The half-lives of the targets are takenfrom Ref. [26]. Z CN Target T / (target) E ∗ (MeV) σ res ( pb )117 Am 7370 y 37.3 0.044 (3n)118
Cm 18.10 y 28.0 0.053 (2n)118
Cm 4730 y 37.1 0.040 (3n)119
Bk 1380 y 37.0 0.043 (3n)120
Cf 2.645 y 35.3 0.012 (3n)121
Es 471.7 d 26.5 0.004 (2n) absolute values of residue cross sections, not on the shapesof the residue excitation functions.
In summary, Ti-induced fusion reactions to synthesizesuperheavy nucleus with Z = 117 ∼
121 are studied withthe two-step model and statistical evaporation model, wherefusion process is divided into two consecutive phases, i.e.,approaching phase and formation phase. The results showthat the reactions Ti + , Am, Ti + − Cm, Ti + , Bk to synthesize superheavy nucleus with Z = 117, 118 and 119 have smaller residue cross sectionsthan Ca-induced ones with nearly one order of magni-tude. However, the calculated residue cross sections arestill within the detection capability of experiment nowa-days. Whereas, Ti-induced fusion reactions with a tar-get − Cf, and , Es, to synthesize superheavy el-ements with Z = 120 and 121 respectively, have so smallresidue cross sections that the experiments can be per-formed only when the experimental facilities are developedin the future. Of course, for planning the experiments onthe synthesis of superheavy nuclei of up to Z = 122, new mechanism and more precise data obtained in the pro-cesses of fusion-fission and quasi-fission of these nuclei arerequired. This work was supported by the National Natural ScienceFoundation of China (Grant Nos. 11275068, 11547312,11375062, 11505057, 11505056 and 11305108), and theproject sponsored by SRF for ROCS, SEM. One of theauthors (L.L) is grateful to the hospitality and calcula-tion support of C3S2 Computing Center by Huzhou Uni-versity. The author (C.W.Shen) acknowledges the fruitfuldiscussions and suggestions from Prof. Y. Abe and D. Boil-ley, and the hospitality by RCNP Osaka University andGANIL.
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