Double folding analysis of 3 He elastic and inelastic scattering to low lying states on 90 Zr, 116 Sn and 208 Pb at 270 MeV

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1 Double folding analysis of 3 He elastic and inelastic scatteing to low lying states on 90 Z, 116 Sn and 08 Pb at 70 MeV Mawa N. El-Hammamy Physics Depatment, Faculty of Science, Damanhu Univesity, Egypt Abstact The expeimental data on elastic and inelastic scatteing of 70 MeV 3 He paticles to seveal low lying states in 90 Z, 116 Sn and 08 Pb ae analyzed within the double folding model (DFM). Femi density distibution (FDD) of taget nuclei is used to obtain eal potentials with diffeent powes. DF esults ae intoduced into modified DWUCK4 code to calculate the elastic and inelastic scatteing coss sections. Two choices of potentials fom factos; Woods Saxon (WS) and Woods Saxon Squaed (WS ) fo eal potential ae used, while the imaginay pat is taken as phenomenological Woods Saxon (PWS) and phenomenological Woods Saxon Squaed (PWS ). This compaison povides infomation about the similaities and diffeences of the models used in calculations. Key wods: elastic and inelastic scatteing, double folding, Squaed Femi density distibution, Woods Saxon Squaed potential, modified DWUCK4. PACS: 5.55.Ci 1. Intoduction When we bombad a nucleus with a nucleon o with light ions like d, 3 He, α-paticles, etc., vaious nuclea phenomena occu, including elastic scatteing, inelastic scatteing, nucleon tansfe eactions and pojectile fagmentation, depending on the pojectile species and the bombading enegy. The simplest among these phenomena is the elastic scatteing. Elastic scatteing can povide valuable infomation about the inteaction potential between two colliding nuclei [1]. Inelastic scatteing of 3 He paticles belongs to useful methods fo investigation of excited states of nuclei. Accoding to the impotance of these eactions, the analysis of expeimental data on elastic and inelastic scatteing of 3 He fom diffeent tagets is needed [-4]. The main poblem of investigating the light heavy ion eactions by using nuclea eaction models is to detemine the most suitable potential fom to explain the expeimental data. Optical (OM) and Folding models (FM) ae examples of simplified models that exist fo studying light heavy ion eactions [1,5-8]. The OM has a potential including the eal and the imaginay potentials. The eal potential descibes the elastic scatteing of the eaction. The imaginay potential expesses the loss of flux into non elastic channels. The eal and imaginay potentials can be detemined with eithe the phenomenological o the micoscopic model. In the micoscopic model, while the imaginay potential is taken PWS o PWS type potential, the eal potential can be defined using DFM. In DFM, the density distibutions (DD) of both pojectile and taget nuclei ae used. Theefoe, DD used in double folding calculations is vey impotant in examining of nuclea eactions. The pupose of this wok is to analyze angula distibutions of the elastic and inelastic scatteing of 3 He with an enegy of 70 MeV leading to the excitation of 90 Z levels,.18 MeV ( + ),.75 MeV (3 ), 116 Sn levels 1.9 MeV ( + ),.7 MeV (3 ), and 08 Pb levels 4.09 MeV

2 ( + ),.61 MeV (3 ) MeV [] in the famewok of DFM by using FDD of taget nuclei and WS potential foms with diffeent powes (n =1 o ) as case one and case two. The impotance of inelastic scatteing analysis to low lying states is to test the stength of these states within DFM. The method employed hee and discussions ae given in Section, and the conclusions ae pesented in Section 3.. Analysis and discussion To study the elastic scatteing fo the eactions of 3 He - paticles with 90 Z, 116 Sn and 08 Pb, the pogam code DFPOT [9] has been used. V() is the DF potential caied out by intoducing the effective nucleon-nucleon (NN) inteaction ove the gound state DD of the two colliding nuclei. It is evaluated fom the expession V ( ) 1( 1 ) ( ) V NN ( s) d1d. (1) 1 ( 1 ) and ( ) ae the nuclea matte density of the two colliding nuclei, and V NN (s) is the effective NN inteaction potential ( s 1 ). V NN (s) is taken to be a standad Reid- M3Y inteaction [10] in the fom, NN ( 00 e e V ) J ( E ) ( ). () The fist and second tems epesent the diect pat and the thid tem epesents the exchange pat of the inteaction potential. It plays an impotant ole in epoducing the expeimental esults fo elastic and inelastic scatteing [11,1]. The exchange pat can be witten to a good appoximation in the fom [10] E J 00 ( E) 76( ), (3) A whee E is the enegy in the cente of mass system and A is the mass numbe of the pojectile. In ou calculations, the nuclea matte DD of 3 He nucleus has the Gaussian fom exp( ), (4) 3 whee fm, 0.01 fm [13], and fo 90 Z [14], 116 Sn and 08 Pb [15] the following FDD fom is used n R 1 exp, (n =1 o ) (5) a The total potential must compise both the eal pat and the imaginay pat, the latte is being esponsible fo the absoption of the incident paticle in the inelastic channels. U ( ) N V ( ) iw ( ) (6) Since the M3Y inteaction is eal, the folding calculation gives the eal pat of the potential. In the model used hee, the volume eal pat has the folded fom with nomalization factos N.We have chosen this fom to be WS shape, while the imaginay pat is taken as PWS. The esulted folded fom factos, in addition with PWS potentials paametes in each case (n=1 o ) ae intoduced into the modified pogam code DWUCK4 [16] to compute the diffeential scatteing coss section, in which an additional fom facto

3 σ(θ)/σr(θ) Submitted to jounal (unde eview) fom (WS) is added. The calculations fo elastic scatteing wee calculated by DWUCK4. Fistly, we used (WS) fo eal and (PWS) fo imaginay pats of the potential as case one. Secondly, in case two, we used (WS) fo eal and (PWS) fo imaginay pats of the potential. Thus, the eal potentials ae epesented by n Rv V ( ) V 1 exp( ), av while the imaginay potentials ae epesented by n Rw W ( ) W 1 exp( ) (7) aw 3 3 in which Rv, w v, w ( A 1/ T A 1/ P ) and n 1o. V () is the DF potential of equation (1) and N W,, a ae vaiable paametes. Compaisons ae shown in figue (1) between the, o w w, pesent calculations and expeimental data. 1.E+09 1.E+07 x 10 8 x E Figue (1). Angula distibutions of 3 He elastically scatteed on 90 Z, 116 Sn and 08 Pb. The theoetical coss section obtained with DF model is epesented by dotted lines fo case one and solid lines fo case two. Expeimental points ae denoted by black symbols, fo 90 Z, fo 116 Sn and fo 08 Pb. The vaiable paametes of the two cases and DF-equivalent potential paametes ( V o, v, av ) ae listed in table (1).In ode to estimate the quality of the fit, one can calculate a elative eo

4 i N calc. exp. 1 ( ( i ) ( i ) R calc. exp, (8) N i 1 ( ( i ) ( i ) calc. whee N is the numbe of data points and ( ) is the i th calculated scatteing coss section and exp. ( ) is the coesponding expeimental scatteing coss section. i i Table (1): The eal and imaginay potentials paametes of 3 He elastic scatteing on diffeent nuclei Reaction n V o (MeV) R v (fm) a v (fm) N W 0 (MeV) w (fm) a w (fm) R 3 He+ 90 Z 3 He+ 116 Sn 3 He+ 08 Pb Fo fist case, the ageement of the theoetical angula distibution with the expeimental one is excellent at fowad angles θ c.m < o, then discepancy appeaed in lage angula egions. Thus, these esults should be impoved with anothe theoetical appoach. Theefoe secondly in case two, we inceased the powe of the potential fom to be squaed as it was successful in many othe analyses within OM [17,18]. Accoding to an incease of eal nomalization facto ( N 1 ) values by inceasing powe (n), the esults ae bette than in case one, with less elative eo R values. This is an impotant point in studying the inteaction of 3 He. Because, if it is investigated the inteaction of 3 He with diffeent taget nuclei within famewok of DFM, it is mostly needed nomalization to obtain a satisfied ageement esults with the expeimental data. Thus, we used the case two potential paametes to be used in inelastic scatteing analysis. The difficulty found in fitting elastic scatteing coss sections is eflected in the inelastic pedictions and indicated a deficiency in the pesent potential fom. The analysis of the inelastic scatteing of the 3 He paticles has been pefomed and the compaison of theoetical calculations and the expeimental data has been pesented in figue ().

5 σ(θ)mb/s σ(θ)mb/s σ(θ)mb/s Submitted to jounal (unde eview) 1.E+04 x 10 ( 3 ).75 MeV (a) 1.E+06 1.E+04 x 10 3 ( 3 ).7 MeV (b) 1.E+0 1.E+0 ( + ).18 MeV ( + ) 1.9 MeV 1.E+00 1.E E E+04 x 10 ( 3 ).61 MeV (c) 1.E+0 1.E+00 ( + ) 4.09 MeV Figue (). Angula distibutions of 3 He inelastically scatteed on 90 Z (a), 116 Sn (b) and 08 Pb (c). The theoetical coss section obtained with DF model is epesented by solid lines fo case two. Expeimental points ae denoted by black symbols In case of 90 Z and 08 Pb (3 ) state, thee is an oveestimation in fowad egions and poo ageement in case 08 Pb ( + ) state. The potentials fo elastic scatteing analysis ae subsequently used to calculate the inelastic scatteing coss sections in the modified DWUCK4. The inelastic potentials ae calculated accoding to the following fom U ( ) N V ( ) iw ( ), (9) whee λ is the multi-polaity [19]. V ( ) is the eal folded (tansition) inelastic potential multiplied by nomalization facto N and W ( ) is an imaginay defomed PWS potential.

6 The calculated eal folded inelastic potential nomalization facto N as well as the coesponding values of R ae shown in table (). R Table (): Real nomalization facto and values fom best fit to inelastic scatteing data fo diffeent levels of 90 Z, 116 Sn and 08 Pb Reaction n Level N R 3 He+ 90 Z 3 He+ 116 Sn 3 He+ 08 Pb.18( + ) (3 - ) ( + ) (3 - ) ( + ) (3 - ) Conclusion Although thee ae many detailed analyses concening the elastic and inelastic scatteing angula distibutions of these investigated systems studied in OM with vaious potential foms, just a few of them make an effot to evolve a systematization fo the folding potential paametes. So, we have eanalyzed elastic and inelastic scatteing of 3 He - paticles with 90 Z, 116 Sn and 08 Pb at 70 MeV with minimal 4-paamete nuclea potential sets having (WS +ipws ) and (WS+iPWS) foms. When the eal potential paametes ae used with diffeent nomalization factos given in this wok, FM analyses with these two folded potential sets have povided diffeent esults. Calculations with the squaed potential foms can epoduce the expeimental elastic angula distibutions in a good ageement, especially in case 116 Sn. The difficulty found in fitting elastic scatteing coss sections is eflected in the inelastic pedictions in case 90 Z and 08 Pb and indicated a deficiency in the pesent potential fom. Howeve, this appoach has shown that an incease of powe (n) fom 1 to is accompanied by an incease of eal nomalization facto ( N 1 fo all cases) values, i.e. it doesn't need nomalization to fit the data. The similaities and diffeences between the two cases used in ou analysis ae pleasantly visible in this compaison. Geneally on the basis of these esults, we conclude that that WS fom is moe suitable than the PWS fom fo eal potential. Acknowledgment Autho thanks the efeee fo valuable discussion and comments. The autho would like to thank the eseache Ahmed Fouad (Faculty of education, physics and chemisty depatment, Alexandia Univesity, Egypt) fo poviding an additional fom facto fom (WS) in DWUCK4 pogam.

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