α decay chains in superheavy nuclei

Transcription

1 PHYSICAL REVIEW C 84, 2469 (211 α decay chains in superheavy nuclei K. P. Santhosh, * B. Priyanka, Jayesh George Joseph, and Sabina Sahadevan School of Pure and Applied Physics, Kannur University, Payyanur Campus, Payyanur , India (Received 7 June 211; published 17 August 211 α decay of superheavy nuclei is studied using the Coulomb and proximity potential model for deformed nuclei (. The predicted α half-lives of and nuclei and their decay products are in good agreement with experimental values. Comparison of α and spontaneous fission half-lives predicts four-α chains and three-α chains, respectively, from and nuclei and are in agreement with experimental observation. Our study predicts two-α chains from 273,274,289 11, three-α chains from 27 11, and four-α chains consistently from 284,28, nuclei. These observations will be useful for further experimental investigation in this region. DOI:.13/PhysRevC PACS number(s: e, b I. INTRODUCTION Understanding nuclear stability-instability in the superheavy mass region is a long-standing question. Hence, during the past few decades, considerable attention has been given by the experimentalists to the investigation of the existence of superheavy nuclei (SHN beyond the valley of stability. Nuclei with larger Z (SHN are produced with accelerator-based experiments, most often via fusion of heavy ions. Several elements have been discovered using cold and hot fusion reactions 1,2 in the superheavy region. In heavy-ion-induced fusion reactions nuclei up to Z = 118 were synthesized during the last decade 1,3. There are mainly two types of reaction mechanisms that produce superheavy elements, namely, the cold fusion reactions 1 (isotopes of elements with Z up to 113 have been produced using Pb or Bi targets performed at GSI (Darmstadt, Germany and the hot fusion reactions 2 (isotopes of elements with Z = and 118 have been produced using 48 Ca projectile on actinide targets such as 233,238 U, 237 Np, 242,244 Pu, 243 Am, 24,248 Cm, and 249 Cf performed at JINR, FLNR (Dubna. The existence of long-lived superheavy nuclei is controlled mainly by spontaneous fission and the α decay process. Thus identifying and characterizing the α decay chains form a crucial part of nuclide identification in the synthesis of superheavy nuclei. α decay, one of the most common and important decay modes for heavy and superheavy nuclei, was first observed by Rutherford and co-workers 6,7 almost a century ago. By 1928 George Gamow had solved the theory of α decay via the quantum mechanical tunneling process 8. Gamow s theory gave a basis to successfully explain the experimental α decay half-lives using both phenomenological and microscopic models In order to predict the experimentally observed facts many theoretical models were also put forward in a similar fashion. The rapid progress in the experimental studies of the α decay of the superheavy nuclei has resulted in the observation of many alternative α emitters in the superheavy region. An * drkpsanthosh@gmail.com investigation of the α decay chains of the SHN is the main tool to obtain information regarding their degree of stability and existence in nature. During the past few years several experimental 1 4,22 28 and theoretical works 29 41have been devoted to understanding the formation of SHN and their α decay half-lives. Even though several theoretical works can be quoted to the α decay studies of superheavy nuclei, works on odd mass nuclei 36,37,42 46 are very rare. Of the various theoretical models just specified, α decay studies on Z = 11 have been done recently by Kumar et al. 38 and Ren and Xu 39 within the framework of preformed cluster model (PCM and Generalized Density Dependent Cluster Model (GDDCM, respectively. The purpose of this paper is to study the α decay and the spontaneous fission of the isotopes of the superheavy element Z = 11 and predict the α decay chains using Coulomb and proximity potential model for deformed nuclei ( 2. The success and applicability of formalism in predicting the α decay half-lives of heavy and SHN have been revealed in our previous works 47 49, by comparison with the experimental data. As we were successful in reproducing the experimental results in the case of and , which were synthesized in JINR, Dubna 4,22 through the fusion evaporation reaction 243 Am + 48 Ca, we have extended the work in predicting the α decay half-lives of a few more isotopes of the same element which gives scope for experimental works. The features of the Coulomb and proximity potential model for deformed nuclei are described in Sec. II, results and discussions on the α decay of the nuclei under study are given in Sec. III, and in Sec. IV we give our conclusions on the entire work. II. THE COULOMB AND PROXIMITY POTENTIAL MODEL FOR DEFORMED NUCLEI ( In the Coulomb and proximity potential model for deformed nuclei, the potential energy barrier is taken as the sum of deformed Coulomb potential, deformed two-term proximity potential, and centrifugal potential for the touching configuration and for the separated fragments. For the /211/84(2/2469( American Physical Society

2 SANTHOSH, PRIYANKA, JOSEPH, AND SAHADEVAN PHYSICAL REVIEW C 84, 2469 (211 1 (a 1 (b - log (T 1/ (c (d FIG. 1. (Color online Comparison of the calculated α decay half-lives with the spontaneous fission half-lives for the isotopes pre-scission (overlap region, simple power law interpolation as done by Shi and Swiatecki is used. The inclusion of proximity potential reduces the height of the potential barrier, which closely agrees with the experimental result. TABLE I. Comparison of α decay half-lives of and and their decay products with experimental values 23. The α half-life calculations are done for zero angular momentum transfers. Parent nuclei Q α (expt (MeV T av (s T α 1/2 Mode of decay Expt GLDM UMADAC ± ms ms ms ms 14.4 ms α ± ms ms ms ms ms α ± ms ms ms ms ms α3 27 Mt.48 ± ms ms ms ms 87. ms α Bh 9.6 a s.9612 s.499 s.912 s 267 Db 7.9 a min 3.83 min min 33 min min ± ms ± s ms ms ms ms α1 s 2.29 s α2 +.1 s s ± s s s s 1.38 s α3 276 Mt 9.8 ± s s s s.93 s 272 Bh 9.1 ± s. s 3. s s s Db 8.2 a h s.84 4 s 2.4 min s a Q values computed using experimental mass excess (Ref

3 α DECAY CHAINS IN PHYSICAL REVIEW C 84, 2469 ( (a (Expt. (b log (T 1/ (Expt. (c (Expt. (d FIG. 2. (Color online Comparison of the calculated α decay half-lives with the spontaneous fission half-lives for the isotopes The proximity potential was first used by Shi and Swiatecki in an empirical manner and has been quite extensively used over a decade by Gupta et al., 14 in the preformed cluster model (PCM. Dutt and Puri 1,2 have been using different versions of proximity potential for studying the fusion cross section of different target-projectile combinations. In our model the contribution of both the internal and external parts of the barrier is considered for the penetrability calculation. In the present model, assault frequency v is calculated for each parent-cluster combination that is associated with vibration energy. But Shi and Swiatecki 3 obtain v empirically, and find the unrealistic values of 22 for even-a parents and 2 for odd-a parents. The interacting potential barrier for two spherical nuclei is given by V = Z 1Z 2 e 2 r + V p (z + h2 l(l + 1 2μr 2, for z>. (1 Here Z 1 and Z 2 are the atomic numbers of the daughter and emitted cluster, z is the distance between the near surfaces of the fragments, r is the distance between fragment centers, l represents the angular momentum, μ the reduced mass, and V P is the proximity potential given by Blocki et al.,4 as V p (z = 4πγb C1 C 2 (C 1 + C 2 ( z b, ( with the nuclear surface tension coefficient, γ = (N Z 2 /A 2 MeV/fm 2, (3 where N, Z, and A represent neutron, proton, and mass number of the parent. represents the universal proximity potential, given as (ε = 4.41e ε/.7176, for ε 1.947, (4 (ε = ε +.169ε 2.148ε 3, for ε 1.947, ( with ε = z/b, where the width (diffuseness of the nuclear surface b 1. The Süsmann central radii C i of fragments related to sharp radii R i is ( b 2 C i = R i. (6 For R i we use the semiempirical formula in terms of mass number A i as 4 R i R i = 1.28A 1/3 i A 1/3 i. (7

4 SANTHOSH, PRIYANKA, JOSEPH, AND SAHADEVAN PHYSICAL REVIEW C 84, 2469 ( (Expt. (a (Expt. (b log ( T 1/ (Expt. (c (Expt. (d FIG. 3. (Color online Comparison of the calculated α decay half-lives with the spontaneous fission half-lives for the isotopes The potential for the internal part (overlap region of the barrier is given as V = a (L L n, for z <, (8 where L = z + 2C 1 + 2C 2 and L = 2C, the diameter of the parent nuclei. The constants a and n are determined by the smooth matching of the two potentials at the touching point. Using a one-dimensional Wentzel-Kramers-Brillouin (WKB approximation, the barrier penetrability P is given as P = exp 2 h b a 2μ(V Qdz. (9 Here the mass parameter is replaced by μ = ma 1 A 2 /A, where m is the nucleon mass and A 1, A 2 are the mass numbers of daughter and emitted cluster, respectively. The turning points a and b are determined from the equation V (a = V (b = Q. The integral in Eq. (9 can be evaluated numerically or analytically, and the half-life time is given by ( ( ln 2 ln 2 T 1/2 = =, ( λ υp where υ = ( ω 2π = ( 2E v represents the number of assaults h on the barrier per second and λ the decay constant. E v, the empirical vibration energy, is given as 6 { E v = Q exp (4 A2 2. }, for A 2 4. (11 In the classical method, the α particle is assumed to move back and forth in the nucleus and the usual way of determining the assault frequency is through the expression given by ν = velocity/(2r, where R is the radius of the parent nuclei. But the α particle has wave properties; therefore a quantum mechanical treatment is more accurate. Thus, assuming that the α particle vibrates in a harmonic oscillator potential with a frequency ω, which depends on the vibration energy E v,we can identify this frequency as the assault frequency ν given in Eqs. ( and (11. The Coulomb interaction between the two deformed and oriented nuclei taken from 7 with higher multipole deformation included 8,9 is given as V C = Z 1Z 2 e 2 + 3Z 1 Z 2 e 2 r λ,i=1,2 1 2λ + 1 Ri λ ( Y rλ+1 λ (α i β λi β2 λi Y ( λ (α iδ λ,2, (

5 α DECAY CHAINS IN PHYSICAL REVIEW C 84, 2469 ( (Expt. (a (Expt. (b log (T 1/ (Expt. (c (Expt. (d FIG. 4. (Color online Comparison of the calculated α decay half-lives with the spontaneous fission half-lives for the isotopes with R i (α i = R i 1 + λ β λi Yλ (α i, (13 where R i = 1.28A 1/3 i A 1/3 i. Here α i is the angle between the radius vector and symmetry axis of the ith nuclei (see Fig. 1 of Ref. 8. Note that the quadrupole interaction term proportional to β 21 β 22 is neglected because of its short range character. Nuclear interactions 6,61 can be divided into two variants: the proximity potential and the double folding potential. The latter is more effective in the description of interaction between two fragments. The proximity potential of Blocki et al.,4, has one term based on the first approximation of the folding procedure, which describes the interaction between two pure spherically symmetric fragments. The two-term proximity potential of Baltz et al., (Eq.(11 of62 includes the second component as the second approximation of the more accurate folding procedure. The authors have shown that the two-term proximity potential is in excellent agreement with the folding model for heavy ion reaction, not only in shape but also in absolute magnitude (see Fig. 3 of 62. The two-term proximity potential for interaction between a deformed and spherical nucleus is given by Baltz et al. 62as R 1 (αr 1/2 C V P 2 (R,θ = 2π R 1 (α + R C + S R 2 (αr 1/2 C R 2 (α + R C + S { ε (S + R 1(α + R C ε 1 (S 2R 1 (αr C ε (S + R } 2(α + R 1/2 C ε 1 (S. (14 2R 2 (αr C Here R 1 (α and R 2 (αare the principal radii of curvature of the daughter nuclei at the point where the polar angle is α, S is the distance between the surfaces along the straight line connecting the fragments, R C is the radius of the spherical cluster, and ε (S and ε 1 (S are the one-dimensional slab-onslab function. III. RESULTS AND DISCUSSIONS Using the Coulomb and proximity potential model for deformed nuclei, we have calculated the α decay half-lives of the nuclei in the range 271 A 294 with Z = 11, including 2469-

6 SANTHOSH, PRIYANKA, JOSEPH, AND SAHADEVAN PHYSICAL REVIEW C 84, 2469 ( Expt. 23 (a Expt. 23 (b log (T 1/ (c (d FIG.. (Color online Comparison of the calculated α decay half-lives with the spontaneous fission half-lives for the isotopes the recently synthesized superheavy elements, and In the external drifting potential barrier is obtained as the sum of deformed Coulomb potential, deformed two-term proximity potential, and centrifugal potential. The energy released in the α transitions between the ground state energy levels of the parent nuclei and the ground state energy levels of the daughter nuclei is given as Q gs gs = M p ( M α + M d + k ( Z ε p Zε d, (1 where M p, M d, M α are the mass excess of the parent, daughter, and α particle, respectively. The Q value is calculated using the experimental mass excess values taken from Audi et al.,63 and for those nuclei where experimental mass excess is unavailable, it is taken from Ref. 64. As the effect of atomic electrons on the energy of the α particle has not been included in the mass excess given in Ref. 63,64, for a more accurate calculation of the Q value, we have included the electron screening effect 6inEq.(1. The term k(z ε p Zε d represents this correction, where k = 8.7 ev and ε = 2.17 for nuclei with Z 6 and k = 13.6 ev and ε = 2.48 for nuclei with Z < 6. The input values necessary for our calculation of α half-lives are the Q value of the corresponding α decay and the quadrupole (β 2 and hexadecapole (β 4 deformation values of the both parent and daughter nuclei. As the experimental deformation values were not available in the case of odd mass nuclei, the theoretical values are taken from Ref. 66. In Table I we have incorporated the study on the α decay of the superheavy elements and and their α decay products. The first column gives the isotopes under study and their corresponding α decay chain. As these isotopes are already synthesized their experimental Q values were available 23 and column 2 gives the respective values. The α decay half-lives are evaluated using these experimental Q values in formalism, and are given in column. On comparison with the experimental half-life values 23 given in column 4, we can see that our calculated values are in good agreement with them. The half-life calculations are also done using the Viola-Seaborg semiempirical relationship ( with constants determined by Sobiczewski et al. 67 for α half-lives and are given as log (T 1/2 = (az + bq 1/2 + cz + d + h log, (16 where the half-life is in seconds, the Q value is in MeV, and Z is the atomic number of the parent nucleus. Instead of using the original set of constants by Viola and Seaborg 68, more recent values were determined in an adjustment taking account of new data for even-even nuclei 67 and are used here. The constants are a = , b = 8.166, c =.2228,

7 α DECAY CHAINS IN PHYSICAL REVIEW C 84, 2469 ( (a 2 1 (b log (T 1/ (c (d FIG. 6. (Color online Comparison of the calculated α decay half-lives with the spontaneous fission half-lives for the isotopes d = ; h log =, for Z, N even, h log =.772, for Z = odd, N = even, h log = 1.66, for Z = even, N = odd, h log = 1.114, for Z, N odd. The quantities a, b, c, and d are adjustable parameters and the quantity h log accounts for the hindrances associated with odd proton and odd neutron numbers given by Viola and Seaborg 68. The half-life values were computed using systematics and are shown in column 6. In Table I we can see that the values calculated using our formalism matches well with values, and values reported by the generalized liquid drop model (GLDM 46 and the unified model for α decay and α capture (UMADAC 69, with a few order differences in some cases. Now, to identify the mode of decay of the isotopes under study, the spontaneous fission ( half-lives are also calculated using the semiempirical relation given by Xu et al. 7: T 1/2 = exp { 2π C + C 1 A + C 2 Z 2 + C 3 Z 4 + C 4 (N Z 2 } ( Z , (17 A1/3 with constants C = , C 1 = 3.16, C 2 =.4386, C 3 = , and C 4 = As Eq. (17 was originally made to fit the even-even nuclei, and as we have considered only the odd mass (odd-even and odd-odd nuclei in this work, instead of taking spontaneous fission half-life T sf directly, we have taken the average of the fission half-life Tsf av of the corresponding neighboring even-even nuclei as the case may be. While dealing with the odd-even nuclei, we took the Tsf av of two neighboring even-even nuclei and for the case of odd-odd nuclei the Tsf avof four neighboring even-even nuclei was taken. Here we would like to mention that, in the case of the nuclei 27 Lr, T expt sf = s and Tsf av = s; in the case of 26 Md, T expt sf = 1.64 s and Tsf av = 9.7 s, which shows the agreement between experimental and computed average spontaneous fission half-lives. The spontaneous fission half-lives are calculated because isotopes with smaller α decay half-lives than spontaneous fission half-lives survive fission and can be detected through α decay in the laboratory. Now by comparing the α decay half-lives with the spontaneous fission half-lives given in column 3, we could identify the nuclei (both parent and decay products that will survive fission. Thus by comparing α half-lives with the corresponding spontaneous fission half-lives we predict four-α chains to be seen for and three-α chains for ; these predictions are tabulated

8 SANTHOSH, PRIYANKA, JOSEPH, AND SAHADEVAN PHYSICAL REVIEW C 84, 2469 (211 TABLE II. The α decay half-lives and spontaneous fission half-lives of The mode of decay is predicted by comparing the α decay half-lives with the spontaneous fission half-lives. The α half-life calculations are done for zero angular momentum transfers. Parent nuclei Q α (cal (MeV T av (s T 1/2 α (s Mode of decay α α α3 272 Mt α4 268 Bh α α α3 273 Mt α4 269 Bh Db Lr α α α3 274 Mt α4 27 Bh Db in column 9 of Table I. As one may notice, our prediction agrees well with the experimental observations 23,24. On the basis of the fact that formalism could successfully reproduce the experimental data for and , we could confidently extend our study to predict the α half-lives and the modes of decay of the isotopes of Z = 11 ranging from 271 A 294. The entire work is presented in Figs These figures give the plots for log (T 1/2 against the mass of the nuclei in the corresponding α chain. Here we have plotted α decay half-lives calculated using the Coulomb and proximity potential model ( formalism (without the ground state deformation of the both parent and daughter nuclei, in addition to the formalism (with the ground state deformation of both the parent and daughter nuclei. The α half-lives are found to be decreasing when the deformation values are included as it is clear from the plots. Along with these we have plotted the values for comparison. The spontaneous fission half-lives are computed for the parent nuclei and their decay products using the phenomenological formula of Xu et al. 7 and are given in these figures. The experimental spontaneous fission half-life values 71,72 are also displayed in these figures as scattered points for comparison. It can be seen that the computed Tsf av values match very well with experimental spontaneous fission half-life values. Figure 1 gives the plot of log (T 1/2 versus mass number for the nuclei By comparing the α half-life with the corresponding spontaneous fission half-life we can note that isotope will not survive fission, whereas isotopes , , and survive fission and hence one-α, two-α, and two-α chains can be predicted, respectively, for them. Figures 2 4 give the plots for the nuclei , , and , respectively. It is clear from the figures that all these nuclei will survive fission. Three-α chains can be predicted to be seen for the isotope from the comparison of its α half-lives with the spontaneous fission half-lives. But our calculations shows that the α half-lives for the isotopes of Z = 11 in the range 276 A 283 are below the millisecond region (e.g., T1/2 α = s for and for , T1/2 α = s and hence cannot be detected through α decay. In these figures we have also depicted the experimental spontaneous fission half-lives as scattered points. The nuclei , 28 11, and give four-α chains consistently and can be synthesized and detected experimentally via α decay. This observation may provide insight for the experimentalists. Figure gives the plot of log (T 1/2 versus mass number for the nuclei which includes the experimentally synthesized nuclei and The experimental α decay half-lives for these two isotopes and their decay products are represented as scattered points. Four-α and three-α chains are predicted to be given by these nuclei, respectively, as per our model, which agrees with the experimentally observed facts 23,24. In the case of two-α chains and in the case of one-α chain is predicted. Figure 6 gives the plot of log (T 1/2 versus mass number for the nuclei Of these four nuclei, survive fission and show a one-α chain, whereas the nuclei , , and will not survive fission and hence completely undergo spontaneous fission

9 α DECAY CHAINS IN PHYSICAL REVIEW C 84, 2469 (211 Table IIhas been included separately which may of interest to experimentalists; it contains the α decay half-lives and the average spontaneous fission half-lives of , 28 11, and isotopes, as they show four-α chains consistently. Here we have also incorporated the predictions of their α decay chain based completely on our model, the Coulomb and proximity potential model for deformed nuclei (. The sixth column contains the α decay half-lives calculated using the Coulomb and proximity potential model ( formalism which treat parent, daughter, and emitted α particle as a sphere. As no experimental Q values were available for these isotopes, we have calculated the Q values using Eq. (1; they are given in column 2. It can be seen that four-α chains can be seen consistently for all three isotopes and can be detected in the laboratory via α decay chain. We thus hope that the study on , 28 11, and will be a guide to future experiments. IV. CONCLUSION We present theoretical estimates for the α decay half-lives of 24 superheavy elements ranging from 271 A 294 with Z = 11, in the Coulomb and proximity potential model for deformed nuclei ( framework. This formalism is found to be quite reliable as the results could successfully reproduce the experimental α half-lives and decay chains observed for and Hence an extensive study was done confidently for 22 other isotopes of Z = 11, with a view to finding possible α decay chains which may open up another line of study in experimental investigations. The updated half-life estimations on reveal four-α chains from these isotopes which we hope to be useful for further experimental studies in this area. 1 S. Hofmann, and G. Munzenberg, Rev. Mod. Phys. 72, 733 (2. 2 Yu. Ts. Oganessian, J. Phys. G 34, R16 (27. 3 K. Morita et al., J. Phys. Soc. Jpn. 76, 41 (27. 4 Yu. Ts. Oganessian et al., Phys.Rev.C74, 4462 (26. S. Hofmann et al., Eur. Phys. J. A 32, 21 (27. 6 E. Rutherford and H. Geiger, Proc. R. Soc. London, Ser. A. 81, 162 ( E. Rutherford and T. Royds, Phil. Mag. 17, 281 ( G. Gamow, Z. Phys. 1, 24 ( D. S. Delion and A. Sandulescu, J. Phys. G: Nucl. Phys. 28, 617 (22. R. G. Lovas, R. J. Liotta, K. Varga, and D. S. Delion, Phys. Rep. 294, 26 ( B. Buck, A. C. Merchant, and S. M. Perez, Phys.Rev.C4, 2247 ( G. Royer, J. Phys. G: Nucl. Part. Phys. 26, 1149 (2. 13 H. F. Zhang, J. Q. Li, W. Zuo, B. Q. Chen, Z. Y. Ma, S. Im, and G. Royer, Chin. Phys. Lett. 23, 1734 ( S. S. Malik and R. K. Gupta, Phys. Rev. C 39, 1992 ( S. Peltonen, D. S. Delion, and J. Suhonen, Phys. Rev. C 78, 3468 ( D. Ni and Z. Ren, J. Phys. G: Nucl. Part. Phys. 37, 34 (2. 17 K. P. Santhosh and A. Joseph, Pramana J. Phys. 8, 611 ( J. Dong, H. Zhang, Y. Wang, W. Zuo, and J. Li, Nucl. Phys. A 832, 198 (2. 19 V. Yu. Denisov and A. A. Khudenko, Phys. Rev. C 8, 3463 (29. 2 K. P. Santhosh, Sabina Sahadevan, and Jayesh George Joseph, Nucl. Phys. A 8, 34 ( G. Royer, Nucl. Phys. A 848, 279 (2. 22 Yu. Ts. Oganessian et al., Phys.Rev.Lett.4, 1422 (2. 23 Yu. Ts. Oganessian et al., Phys.Rev.C69, 2161 (R ( Yu. Ts. Oganessian et al., Phys.Rev.C72, (2. 2 V. Yu. Denisov and S. Hofmann, Phys.Rev.C61, 3466 (2. 26 R. Smolanczuk, Phys. Rev. C 63, 4467 ( M. G. Itkis, Yu. Ts. Oganessian, and V. I. Zagrebaev, Phys. Rev. C 6, 4462 ( Y. K. Gambhir, A. Bhagwat, and M. Gupta, Phys. Rev. C 71, 3731 (2. 29 R. Smolanczuk, J. Skalski, and A. Sobiczewski, Phys. Rev. C 2, 1871 ( P. Moller and J. R. Nix, Nucl. Phys. A 49, 84 ( D. N. Poenaru, I.-H. Plonski, and Walter Greiner, Phys. Rev. C 74, ( D. S. Delion, R.J. Liotta,and R.Wyss,Phys. Rev. C76, 4431 ( M. Bhattacharya and G. Gangopadhyay, Phys. Rev. C 77, 4732 ( G. Gangopadhyay, J. Phys. G: Part. Nucl. Phys 36, 9 (29. 3 P. R. Chowdhury, C. Samanta, and D. N. Basu, Phys. Rev. C 77, 4463 ( A. Sobiczewski, Acta Phys. Pol., B 41, 17 (2. 37 P. Roy Chowdhury, G. Gangopadhyay, and A. Bhattacharyya, Phys.Rev.C83, 2761 ( S. Kumar, S. Thakur, and R. Kumar, J. Phys. G: Nucl. Part. Phys. 36, 4 ( Z. Ren and C. Xu, J. Phys.: Conf. Ser. 111, 124 (28. 4 V. Yu. Denisov and A. A. Khudenko, Phys. Rev. C 81, (2. 41 V. Yu. Denisov and A. A. Khudenko, Phys. Rev. 82, 993(E (2. 42 D. Ni and Z. Ren, J. Phys. G: Nucl. Part. Phys. 37, 7 (2. 43 L. S. Geng, H. Toki, and J. Meng, Phys.Rev.C68, 6133 ( D. N. Basu, J. Phys. G: Nucl. Part. Phys. 3, B3 (24. 4 S. Das and G. Gangopadhyay, J. Phys. G: Nucl. Part. Phys. 3, 97 ( G. Royer and H. F. Zhang, Phys. Rev. C 77, 3762 ( K. P. Santhosh, Sabina Sahadevan and R. K. Biju, Nucl. Phys. A 82, 19 ( K. P. Santhosh and R. K. Biju, J. Phys. G: Nucl. Part. Phys. 36, 17 ( K. P. Santhosh and R. K. Biju, Pramana J. Phys. 72, 689 (29. Y. J. Shi and W. J. Swiatecki, Nucl. Phys. A 438, 4 (

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