Effects of Isospin on Pre-scission Particle Multiplicity of Heavy Systems and Its Excitation Energy Dependence

Commun. Theor. Phys. (Beijing, China) 41 (2004) pp. 751 756 c International Academic Publishers Vol. 41, No. 5, May 15, 2004 Effects of Isospin on Pre-scission Particle Multiplicity of Heavy Systems and Its Excitation Energy Dependence YE Wei 1 and CHEN Na 2 1 Department of Physics, Southeast University, Nanjing 210096, China 2 Department of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing 210003, China (Received August 25, 2003; Revised October 15, 2003) Abstract Isospin effects on particle emission of fissioning isobaric sources 202 Fr, 202 Po, 202 Tl and isotopic sources 189,202,212 Po, and its dependence on the excitation energy are studied via Smoluchowski equations. It is shown that with increasing the isospin of fissioning systems, charged-particle emission is not sensitive to the strength of nuclear dissipation. In addition, we have found that increasing the excitation energy not only increases the influence of nuclear dissipation on particle emission but also greatly enhances the sensitivity of the emission of pre-scission neutrons or charged particles to the isospin of the system. Therefore, in order to extract dissipation strength more accurately by taking light particle multiplicities it is important to choose both a highly excited compound nucleus and a proper kind of particles for systems with different isospins. PACS numbers: 25.70.Jj, 25.85.Ge Key words: isospin effect, pre-scission particle multiplicity, excitation energy, fission diffusion model 1 Introduction The nature and magnitude of viscosity is a hot topic in nuclear physics. Particle multiplicity is a source to obtain the information of nuclear dissipation [1 9] because they are considered as the main indicators for the importance of dissipation effects in nuclear fission. The comparison between experimental data and theoretical simulations based on the diffusion models [10 14] or improved statistical models [1 9,15] allows us to derive the knowledge of nuclear dissipation. Nowadays, isospin influence on the formation and decay of hot nuclei is an important subject in heavy-ion collision physics. Its important roles in many phenomena have been discovered. [16 27] However, more experimental and theoretical studies are still needed in order to understand isospin physics better. The isospin effect on the emission of light particles in heavy-ion reactions has also been widely noted and extensively studied. This is both an easily measureable quantity experimentally and a very valuable observable for understanding nuclear reaction mechanisms. In this paper we investigate isospin effects on various light particles as probes of nuclear dissipation and its dependence on the excitation energy. The organization of this paper is as follows. In Sec. 2 we briefly describe fission-diffusion model. Section 3 presents our calculated results and discussions. A short summary and conclusion can be found in Sec. 4. 2 Brief Description of Theoretical Model In order to study particle emission in the diffusion process, we employ the following Smoluchowski equation [13] P (x, t) t = θ x ( U x P (x, t) ) P (x, t) + λ i P (x, t), (1) x i=n,p,α where P (x, t) represents the probability of the system at fission deformation coordinate x and time t. U = V/T is a dimensionless potential, V is fission potential and θ = T/(µβ), where T is the nuclear temperature, µ the reduced mass of the system and β the viscosity coefficient. Fission potential V is a function of coordinate x, consisting of a well and a barrier. The second term on the right-hand side of the equation represents light particle emissions. λ i = Γ i / h, Γ i (i = n, p, α) is the particle emission width. For an excited fissioning system, after it releases a particle the resulting daughter nucleus still has sufficient excitation energy to emit another particle. In this way, the decay chain is formed, and it ends by fission. This decay chain can be described by a set of the coupled equations as follows: d dt P s(t) = i=n,p,α [ λ i,s 1 Ps 1(t) i i=n,p,α ] λ i,s + λ f,s (t) P s (t), S = 1, 2,..., s m, (2) The project supported in part by the Teaching & Researching Foundation for Outstanding Teachers of Southeast University

752 YE Wei and CHEN Na Vol. 41 where P s is the probability of the s-th daughter nucleus. The first term on the right-hand side is the source term which results from the decay of the (s 1)-th nucleus through emission of particles. The second term represents its decay probability via fission or particle emission. The maximum number of times of evaporating particles for a decay chain is denoted by s m over which the produced nuclei is too cold to emit particles. Generally, the number of particles such as neutrons n s released by the s-th daughter nucleus is n s = Γ n,s h 0 P s (t)dt, (3) where Γ n,s is neutron emission width of the s-th daughter nucleus. The particle multiplicity N i (i=n,p,α) is defined as the total number of particles released from all the decay chains that are formed by emitting neutrons, protons, and α particles, N i = d m d=1 s m s=1 n ds. (4) The inner sum here is over the particle multiplicity for a single decay with the proper probability, and the outer sum here is over all possible decay chains. n ds represents particle multiplicity evaporated in the s-th decay on a decay chain denoted by d. The time-dependent fission width is defined as Γ f (t) = hλ f (t) = hj f (t)/π f (t), (5) where π(x, t) is the probability at the left-hand side of deformation at x and J(x, t) is the probability flow at this deformation. Physically, the reasonable fission point is the scission point, and therefore here we choose the scission point to compute the fission width. Via this set of extended Smoluchowski equations, survival probabilities, fission rates, and various particle multiplicities are calculated numerically with the initial conditions P s (0) = δ s,1, i.e. at the beginning, the probability of the mother nucleus is 1, while probabilities of daughters are zero. Concerning the detailed steps of deriving the above-mentioned quantities, we refer the reader to Ref. [13]. 3 Calculated Results and Discussions Several isotopes of Po are chosen as examples for the study of isospin effects in the framework of the diffusion model. The isospin asymmetry values (I = (N Z)/A) for 189 Po, 202 Po, and 212 Po are 0.111, 0.168, and 0.208, respectively. Moreover, in order to better study isospin effects on the emission of particles, a chain of isobaric nuclei, namely, 202 Fr, 202 Po, and 202 Tl, are also used to compare with the results in this work. Their values of I are 0.139, 0.168, and 0.198, respectively. Generally the value of nuclear dissipation strength can be extracted by comparing the calculated particle multiplicity with data. Also, we know that two essential inputs in the calculation of the diffusion model, i.e. particle separation energies and fission barriers, to a large extent, determine the change of particle multiplicity with dissipation strength. Considering that for systems with different isospins there has a large difference in fission barriers as well as in the neutron and charged particle separation energies, it can be expected that isospin has an influence on the sensitivity of the emission of different light particles to dissipation strength. Note that in the calculations we assume that a neutron or a proton carries away angular momentum of 1 h, whereas an α particle carries away angular momentum of 2 h. Particle emission widths are assumed to be independent of time and position and computed by a detailed balance principle method. [28] The emission barriers of protons and α particles are taken from Ref. [29]. The effects of angular momentum on the particle emission width are taken into account by removing the rotational energy from the excitation energy and modifying the angular momentum of daughter nuclei. The rotational energy due to angular momentum can be calculated by using a rigid body model. [30] The change of the fission barrier with angular momentum and isospin is evaluated with the code barfit. [30] Table 1 Comparison of model predictions with experiments. From left to right it is reaction system, bombarding energy, compound nuclei, experimental neutron multiplicity, and theoretical calculations. Errors in ν pre are shown in brackets. The unit of energy is MeV. Reaction E lab CN ν pre (error) cal. 20 Ne+ 168 Er [31] 148.0 188 Pt 3.53(0.30) 3.67 20 Ne+ 181 Ta [31] 148.0 201 Bi 3.80(0.30) 3.91 28 Si+ 164 Er [32] 167.7 192 Pb 2.00(0.25) 1.76 28 Si+ 170 Er [32] 132.4 198 Pb 1.61(0.18) 1.82 147.5 198 Pb 2.39(0.27) 2.53 162.6 198 Pb 3.00(0.37) 3.21 30 Si+ 170 Er [32] 157.6 200 Pb 2.44(0.30) 2.95 Before presenting the calculated results, we compare the model predictions with experiments. [31,32] Table 1 shows the comparison of the model calculations with data for neutron multiplicity of heavy systems, which correspond to the currently considered fissioning mass regions. In the calculations we chose the level density parameter value to be a = A/10, the viscosity coefficient value to be β = 5 10 21 s 1, and the average fission angular momentum to be 2/3 times the value of the fusion angular momentum predicted by the Bass model. [33] The excitation energy was taken from experiments. We see from

No. 5 Effects of Isospin on Pre-scission Particle Multiplicity of Heavy Systems and Its Excitation Energy Dependence 753 Table 1 that the agreement between calculation and data is rather satisfactory, except for the case of the 200 Pb nucleus. In that case, the theoretical result is a bit larger than the measured value. Table 2 Calculated pre-scission neutron (N n), proton (N p), and α particle (N α) multiplicity for Po isotopic chain 189 Po, 202 Po, and 212 Po as a function of viscosity coefficient (β) at angular momentum L = 60 h for three different excitation energies (E ). E = 80 MeV β 189 Po 202 Po 212 Po 2 0.06339 0.37419 0.11587 0.43323 0.05025 0.04512 1.67260 0.00935 0.01526 5 0.10873 0.61785 0.18757 0.84121 0.09303 0.08195 2.52894 0.01301 0.02102 10 0.15060 0.82826 0.24726 1.25860 0.13292 0.11485 3.31734 0.01510 0.02422 15 0.17767 0.95708 0.28257 1.52771 0.15654 0.13360 3.76154 0.01599 0.02557 20 0.2007 1.06018 0.30973 1.74409 0.17404 0.14703 4.05739 0.01651 0.02634 E = 100 MeV β 189 Po 202 Po 212 Po 2 0.15462 0.73956 0.27501 0.87133 0.15004 0.13721 2.61989 0.03505 0.05349 5 0.26015 1.16255 0.42456 1.55554 0.25059 0.22419 3.96728 0.04734 0.07139 10 0.36426 1.53246 0.54980 2.17243 0.33150 0.29063 4.92651 0.05417 0.08092 15 0.43240 1.75223 0.62136 2.53921 0.37519 0.32488 5.41805 0.05692 0.08478 20 0.48721 1.91440 0.67216 2.81473 0.40511 0.34737 5.75438 0.05854 0.08693 E = 120 MeV β 189 Po 202 Po 212 Po 2 0.28166 1.11866 0.46270 1.41166 0.30397 0.27623 3.55775 0.08521 0.12198 5 0.46192 1.65628 0.67568 2.35202 0.47120 0.41951 4.98473 0.11222 0.15887 10 0.62668 2.06585 0.83019 3.10059 0.59049 0.51608 5.98182 0.12598 0.17722 15 0.72578 2.29938 0.90690 3.51038 0.66995 0.56179 6.49994 0.15128 0.18418 20 0.79970 2.48076 0.96226 3.79941 0.74819 0.59683 6.87494 0.15408 0.19482 Table 2 lists the particle multiplicities as functions of isospin and viscosity coefficient for the Po isotopic sources 189 Po, 202 Po, and 212 Po at three excitation energies. We take the results at E = 100 MeV as a demonstrated example to discuss the isospin effects on different particle evaporations. One can see that N n is an increasing function of N/Z of isotopes, N p and N α are a decreasing function of N/Z of isotopes. This is obviously related to the systematics of the neutron number of fissioning sources. When β increases from 2 to 20, the values of N n of 189 Po and 212 Po change from 0.154 to 0.487 and from 2.61 to 5.75, respectively. For charged particles, N p and N α of 189 Po increase from 0.74 to 1.91 and from 0.27 to 0.67, respectively, whereas those for 212 Po are from 0.035 to 0.058 and from 0.053 to 0.086. This comparison indicates that the neutron multiplicity of the 212 Po nucleus is more sensitive to the dissipation strength than that of the 189 Po nucleus is, while the charged particle emission of 212 Po is almost independent of β, in contrast to the case of 189 Po. This means that as the isospin asymmetry of fissioning systems increases, protons and α particles do not provide sensitive observables with respect to dissipation strength. At other excitation energies such as E = 80 and 120 MeV, a similar picture was observed. In order to reveal the role played by the excitation energy in the isospin effects of particle emission we investigate the change of the differences of various particle multiplicities of two compound systems with β. It is more quantitative to define N 202 Po( 189 Po) i N i ( 202 Po) N i ( 189 Po), N 212 Po( 189 Po) i N i ( 212 Po) N i ( 189 Po), N 202 Po( 202 Fr) i N i ( 202 Po) N i ( 202 Fr), N 202 Tl( 202 Fr) i N i ( 202 Tl) N i ( 202 Fr), where i = n, p, α. The symbol N 202 Po( 189 Po) n denotes the difference of neutron multiplicity between 202 Po (N n ( 202 Po)) and that of 189 Po (N n ( 189 Po)). Other symbols are of similar meaning. Figure 1 shows the differences of various pre-scission particle multiplicity between 202 Po and 189 Po (circles),

754 YE Wei and CHEN Na Vol. 41 and between 212 Po and 189 Po (squares) as a function of viscosity coefficient at angular momentum L = 60 h and at the excitation energies E = 80 MeV (left panels), 100 MeV (middle panels) and 120 MeV (right panels). From this figure it is clear that at β = 2(20) the values of N 202 Po( 189 Po) n and N 212 Po( 189 Po) n are 0.37 (1.54) and 1.61 (3.85) at E = 80 MeV and they become 1.13 (3.00) and 3.27 (6.08) respectively at E = 120 MeV. For the case of charged particle emission, at β = 2 (20) and E = 80 MeV, N 202 Po( 189 Po) p = 0.32 ( 0.88), N 202 Po( 189 Po) α = 0.07 ( 0.16), and N 212 Po( 189 Po) p = 0.36 ( 1.04), N 212 Po( 189 Po) α = 0.10 ( 0.28). As E is increased up to 120 MeV, N 202 Po( 189 Po) p = 0.81 ( 1.73), N 202 Po( 189 Po) α = 0.186 ( 0.360), and N 212 Po( 189 Po) p = 1.03 ( 2.29), N 212 Po( 189 Po) α = 0.34 ( 0.77). From these results we see that the differences of particle multiplicity of two different nuclei increase with energies, and this enlargement is increased with an increase of isospin asymmetry difference between the two systems. This is because a higher excitation energy significantly increases neutron (charged particle) emission of high- (low-) isospin systems, but its influences are minor for the evaporation of neutrons (charged particles) of low- (high-) isospin systems. This explains the observations shown in Fig. 1. As is stated before, isospin has a pronounced effect on the different particles as probes of dissipation strength, the current calculation indicates that increasing the excitation energy can enhance the sensitivity of neutron or charged particle emission to the value of isospin asymmetry of the fissioning systems and to the variation of dissipation strength. This means that an accurate extraction of the value of dissipation strength by means of light particle emission needs to choose a fissioning system with a larger excitation energy. Fig. 1 The differences of the multiplicity of pre-scission neutrons ( N n), protons ( N p), and α particles ( N α) between 202 Po and 189 Po (circles), and between 212 Po and 189 Po (squares) as a function of viscosity coefficient at angular momentum L = 60 h and at the excitation energies E = 80 MeV (left panels), 100 MeV (middle panels), and 120 MeV (right panels). Solid points are the calculated results. The lines are for guiding eyes. We also use three isobaric sources of 202 Fr, 202 Po, and 202 Tl to investigate this isospin effect on particle emission and its variations with excitation energy. In Table 3 we see that the behavior of particle emission in this isobaric chain is analogous to the case of the Po isotopic chain. For instance, the emission of neutrons is enhanced but the emission of protons and α particles is suppressed by increasing the isospin asymmetry of the sources. The higher the isospin asymmetry of the fissioning system is, the more neutrons are evaporated. Increasing (Decreasing) isospin asymmetry of systems will increase the sensitivity of neutron (charged-particle) emission to dissipation strength. We show in Fig. 2 the variations of N n, N p, and N α for A = 202 system with β at three different excitation energies. As is seen, for neutrons (charged particles) all of dashed lines are always below (above) the solid lines, i.e. the larger the difference of isospin asymmetry of two systems is, the larger the difference of the emitted particle multiplicity is. Also, this difference gradually becomes larger with the increase of E. For example, at β = 2 (20), as E changes from 80 MeV to 120 MeV, N 202 Tl( 202 Fr) n rises from 2.03 (3.98) to 3.48 (5.37). In contrast, N 202 Tl( 202 Fr) p and N 202 Tl( 202 Fr) α at β = 2 (20)

No. 5 Effects of Isospin on Pre-scission Particle Multiplicity of Heavy Systems and Its Excitation Energy Dependence 755 fall from 0.131 ( 0.487) and 0.05 ( 0.215) to 0.455 ( 1.24) and 0.176 ( 0.585), respectively. This is also due to a consequence of the fact that particle multiplicity increases with increasing excitation energy and isospin effects on particle emission as mentioned before. Again, these results indicate that increasing the excitation energy can increase the effect of isospin on the particle emission sizeably. Fig. 2 The multiplicity differences of pre-scission neutrons, protons and α particles between 202 Po and 202 Fr (circles), and between 202 Tl and 202 Fr (squares) as a function of viscosity coefficient at angular momentum L = 60 h and at the excitation energies E = 80 MeV (left windows), 100 MeV (middle windows), and 120 MeV (right windows). Full points are the calculated results. The lines are to guide the eye. Table 3 Calculated pre-scission neutron (N n), proton (N p), and α particle (N α) multiplicity for isobaric sources 202 Fr, 202 Po and 202 Tl as a function of viscosity coefficient (β) at angular momentum L = 60 h for three different excitation energies (E ). E = 80 MeV β 202 Fr 202 Po 202 Tl 2 0.13708 0.15089 0.07830 0.43323 0.05025 0.04512 2.16254 0.02006 0.02851 5 0.26037 0.27713 0.14168 0.84121 0.09303 0.08195 3.16761 0.02578 0.03615 10 0.37940 0.38983 0.19692 1.25860 0.13292 0.11485 3.87021 0.02865 0.03986 15 0.45746 0.45953 0.23038 1.52771 0.15654 0.13360 4.24504 0.02980 0.04132 20 0.52642 0.51743 0.25742 1.74409 0.17404 0.14703 4.50418 0.03047 0.04214 E = 100 MeV β 202 Fr 202 Po 202 Tl 2 0.30254 0.35409 0.20377 0.87133 0.15004 0.13721 3.22157 0.06725 0.09049 5 0.53302 0.58454 0.33159 1.55554 0.25059 0.22419 4.51025 0.08492 0.11247 10 0.75447 0.78060 0.43709 2.17243 0.33150 0.29063 5.32711 0.09361 0.12283 15 0.89435 0.89325 0.49602 2.53921 0.37519 0.32488 5.72372 0.09702 0.12677 20 1.00833 0.97730 0.53853 2.81473 0.40511 0.34737 5.99589 0.09895 0.12893 E = 120 MeV β 202 Fr 202 Po 202 Tl 2 0.51742 0.60565 0.36762 1.41166 0.30397 0.27623 4.00257 0.15034 0.19158 5 0.87699 0.94088 0.56502 2.35202 0.47120 0.41951 5.35471 0.18563 0.23326 10 1.20872 1.20690 0.71654 3.10059 0.59049 0.51608 6.21216 0.20170 0.25173 15 1.40839 1.34955 0.79517 3.51038 0.64995 0.56179 6.63974 0.20758 0.25832 20 1.55844 1.44716 0.84709 3.79941 0.68819 0.58983 6.92951 0.21065 0.26173

756 YE Wei and CHEN Na Vol. 41 4 Summary and Conclusions In summary, we studied the effects of isospin on the particle emission in the fission process for heavy fissioning sources via the Smoluchowski equation. It is shown that as the isospin asymmetry of a fissioning compound nucleus becomes very high, the emission of protons and α particles becomes insensitive to the dissipation strength, indicating that for such fissioning systems with larger isospin asymmetries, light charged multiplicity is not a good observable for studying dissipation in fission. Moreover, the calculated results also show that decreasing the excitation energy will decrease the influence of nuclear dissipation on the isospin dependence of pre-scission particle emission. Therefore, in order to extract the value of dissipation strength more accurately by taking light particle multiplicities it is important to choose both a highly excited compound nucleus and a proper kind of particles for systems with different isospins. References [1] M. Thoennessen and G.F. Bertsch, Phys. Rev. Lett. 71 (1993) 4303. [2] D. Hilscher and H. Rossner, Ann. Phys. Fr. 17 (1992) 471; D.J. Hinde, et al., Phys. Rev. C45 (1992) 1229. [3] J.P. Lestone, et al., Phys. Rev. Lett. 67 (1991) 1078; A. Chatterjee, et al., Phys. Rev. C52 (1995) 3167. [4] T. Nakagawa, et al., Nucl. Phys. A583 (1995) 149; K. Yuasa-Nakagawa, et al., Phys. Rev. C53 (1996) 997. [5] P. Paul and M. Thoennessen, Ann. Rev. Nucl. Part. Sci. 44 (1994) 55; I. Diósezegi, et al., Phys. Rev. C61 (2000) 024613; N.P. Shaw, et al., Phys. Rev. C61 (2000) 044612. [6] B.B. Back, et al., Phys. Rev. C60 (1999) 044602. [7] A. Saxna, et al., Phys. Rev. C65 (2002) 064601; L.M. Pant, et al., Eur. Phys. J. A16 (2003) 43. [8] K. Mahata, et al., Nucl. Phys. A720 (2003) 209. [9] G. La Rana, et al., Eur. Phys. J. A16 (2003) 199; E. Vardaci, et al., Eur. Phys. J. A2 (1998) 55; M. Kaplan, et al., Nucl. Phys. A686 (2001) 109. [10] Y. Abe, et al., Phys. Rep. 275 (1996) 49. [11] P. Fröbrich and I.I. Gontchar, Phys. Rep. 292 (1998) 131. [12] P.N. Nadtochy, et al., Phys. Rev. C65 (2002) 064615; A.V. Karpov, et al., Phys. Rev. C63 (2001) 054610. [13] Z.D. Lu, et al., Z. Phys. A323 (1986) 477; Z.D. Lu, et al., Phys. Rev. C42 (1990) 707; W. Ye, et al., Z. Phys. A359 (1997) 385; W. Ye, et al., Eur. Phys. J. A18 (2003) 571. [14] G. Chaudhuri and S. Pal, Eur. Phys. J. A14 (2002) 287; A.K. Dhara, et al., Eur. Phys. J. A7 (2000) 209. [15] W. Ye, High Energy Phys. Nucl. Phys. 26 (2002) 52; ibid. 24 (2000) 945. [16] B.A. Li, C.M. Kuo, and W. Bauer, Int. J. Mod. Phys. E7 (1998) 147. [17] B.A. Li and W. Udo Schröder, Isospin Physics in Heavy- Ion Collisions at Intermediate Energies, Nova Science, New York (2001). [18] G.F. Bertsch, et al., Nucl. Phys. A490 (1988) 745; G.Q. Li and R. Machleidt, Phys. Rev. C48 (1993) 1702. [19] R. Pak, et al., Phys. Rev. Lett. 78 (1997) 1022; ibid. 78 (1997) 1026. [20] B.A. Li, et al., Phys. Rev. Lett. 76 (1996) 4492; ibid. 85 (2000) 4221. [21] W.Q. Shen, et al., Nucl. Phys. A491 (1989) 130; T. Suzuki, et al., Phys. Rev. Lett. 75 (1995) 3241. [22] V. Baran, et al., Nucl. Phys. A703 (2002) 603. [23] S.J. Yennello, et al., Phys. Lett. B321 (1994) 15. [24] J.Y. Liu, et al., Phys. Rev. C63 (2001) 054612. [25] Q.F. Li and Z.X. Li, Chin. Phys. Lett. 19 (2002) 321; Phys. Rev. C64 (2001) 064612. [26] Y.G. Ma, et al., Phys. Rev. C60 (1999) 024607. [27] Z.G. Xiao, et al., Chin. Phys. Lett. 18 (2001) 1037. [28] H. Delagrange, et al., Z. Phys. A323 (1990) 437. [29] L.C. Vaz and J.M. Alexander, Phys. Rep. 97 (1983) 1. [30] A.J. Sierk, Phys. Rev. C33 (1986) 2039. [31] D.J. Hinde, et al., Phys. Rev. C39 (1989) 2268. [32] J.O. Newton, et al., Nucl. Phys. A483 (1986) 126. [33] R. Bass, Nucl. Phys. A231 (1974) 45.