Influence of Shell on Pre-scission Particle Emission of a Doubly Magic Nucleus 208 Pb

Transcription

1 Commun. Theor. Phys. (Beijing, China) 41 (2004) pp c International Academic Publishers Vol. 41, No. 2, February 15, 2004 Influence of Shell on Pre-scission Particle Emission of a Doubly Magic Nucleus 208 Pb YE Wei Department of Physics, Southeast University, Nanjing , China (Received May 22, 2003; Revised July 8, 2003) Abstract Using Smoluchowski equation, we study the shell effects on the emission of light particles in the fission process of a doubly magic nucleus 208 Pb. Calculated results show that shell has a considerable effect on the neutron emission and that shell effect gradually becomes weak with increasing excitation energy. In addition, a dependence of shell effects in the neutron emission on the angular momentum has been found. PACS numbers: Jj, Ge Key words: shell effect, prescission particle multiplicity, diffusion model 1 Introduction Measurements of pre-scission light particle multiplicities in heavy-ion-induced fusion-fission reactions provide useful information on fission dynamics and the effects of nuclear viscosity on the fission process. [1 3] It is generally observed in many heavy- and light-ion-induced reactions that the measured pre-scission light particle multiplicities are much higher than expected on the basis of statistical model calculations. The excess particle emission from the composite fissioning system has been interpreted to be due to a time delay in the fission process arising due to dynamical effects in fission decays. Using diffusion models [4 6] many authors have studied some factors affecting the enhanced emission of pre-scission particles. As is well known, shell has an important effect on the determination of nuclear structure and the related information of the shell effects in the fission process is valuable in the context of production of superheavy elements. It is generally believed that the shell effects are washed out at high excitation energies. However, it has been recently suggested that neutron shell closure N = 126 might play a role in fission fragment anisotropies. [7,8] Back et al. [9] found that in order to reproduce the evaporation residue cross section of 224 Th (N = 134) and 216 Th (N = 126) nuclei, it is necessary to use a larger dissipation coefficient for the former nucleus, so they concluded that nuclear dissipation has a possible relation with neutron closure shell at N = 126. Karamian et al. [10] has also reached a similar conclusion. These works prompt us to investigate theoretically possible shell effects on the particle emission in the fission process. Since doubly magic nucleus 208 Pb (Z = 82, N = 126) has been known to have the largest shell correction, thus we use it to investigate the shell effects on the emission of light particles. The paper is organized in the following way. Section 2 briefly presents the fission diffusion model. The calculated results and discussions are given in Sec. 3. Section 4 is a short summary. 2 Model Generally, a Fokker Planck equation cannot be solved analytically. However, for a very large friction (β 2ω 1, where ω 1 is the frequency of the fission potential at the ground state), the Fokker Planck equation can be reduced to the Smoluchowski equation. In terms of van Kampen technique, [11,12] Smoluchowski equation can be solved analytically. Several authors have ever used this technique to calculate the fission rate. [13,14] Because we are interested in the particle emission in the diffusion process, the following extended Smoluchowski equation is adopted [14] P (x, t) t = θ ( U P (x, t) ) P (x, t) + x x x λ i P (x, t), (1) i=n,p,α where P (x, t) represents the probability of the system at fission deformation coordinate x (for simplicity, below called coordinate x) and time t. U is the potential function for the reaction system, which is a function of coordinate x, consisting of a well and a barrier. We have θ = T/(µβ), where T is the nuclear temperature, µ stands for the reduced mass of system and β stands for the viscosity coefficient. The second term on the right-hand side of the equation represents light particle emissions. λ i = Γ i / h, where Γ i (i = n, p, α) is the particle emission width. For high incident energy, the composite nucleus has high excitation energy or temperature. After it releases a particle, the resulting daughter nucleus still has sufficient excitation energy to emit another particle. In this way, a decay chain is formed, and it ends in 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 P i s 1(t) The project supported in part by the Foundation of Teaching & Researching of the Best Teacher of Southeast University

2 284 YE Wei Vol. 41 [ i=n,p,α ] λ i,s + λ f,s (t) P s (t), S = 1, 2,..., s m. (2) Here, 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 the emission of a particle. The second term represents its decay probability through 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 nucleus is cold enough not to emit particles. In this work, decay chains are defined as follows. Because a mother nucleus may evaporate neutrons or protons or α particles, correspondingly, three possible daughter nuclei are produced with different probabilities. Then each of the newly born daughter nuclei yields three possible granddaughter nuclei by evaporating these three light particles, and so on. In this way, various possible decay chains are formed. Generally, the number of particles such as neutrons, n s released by the s-th daughter nucleus, or the number of the (s + 1)-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. Concerning the detailed steps of deriving particle multiplicity, the reader may refer to Lu s work in Ref. [14]. The particle multiplicity M 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, M i = d m s m d=1 s=1 n ds, (4) where the inner sum is over the particle multiplicity for a single decay with the proper probability, and the outer sum is over all possible decay chains. n ds represents particle multiplicity evaporated in the s-th on a decay chain denoted by d. The time-dependent fission width in the Smoluchowski equation is defined as Γ f (t) = hλ f (t) = hj f (t)/π f (t). (5) Here, J f (t) is the probability flow passing over the saddle point, π f (t) is the surviving probability of system on the left-hand side of the saddle point. Existing probabilities of any nuclei, fission rates, and various particle multiplicities are calculated by numerically solving this set of extended Smoluchowski equations. Specific procedures are as follows. First, with Eqs. (1) and (5) one can compute existing probability of the mother nucleus P (t) as a function of time and its fission rates Γ f (t). Three possible daughter nuclei with proper existing probability arising from neutron, proton, and α particle emission of the mother nucleus and particle multiplicities stemming from the mother nucleus are calculated using Eqs. (2) and (3) with the initial conditions P s (0) = δ s,1, i.e. at the very beginning, the probability of the mother nucleus is 1, while probabilities of daughters are zero. Secondly, each of these three born daughter nuclei can produce another three granddaughter nuclei by evaporating these light particles. Again, via Eqs. (1) and (5) the fission rates of this daughter nucleus are obtained, then with the help of Eqs. (2) and (3) various particle multiplicities emitted by this daughter nucleus and the existing probability of these three granddaughter nuclei are worked out, and so on. It should be mentioned that according to Eq. (2), in order to evaluate particle multiplicity emitted by this daughter nucleus, the information on the existing probability P s 1 (t) of the source term that produces this daughter nucleus must be as an input quantity. Its value has been derived in the formal step. Total particle multiplicity is determined according to Eq. (4). The nuclear first fission probability P f is 1 substracting the sum of the emission probability of first three particles of the mother nucleus. 3 Results and Discussions In a statistical model, the fission width is determined by the ratio of level density at the saddle to that at the ground state, the nuclear temperature, and the fission barrier, while in a diffusion model, there is a transient process, in which fission rate increases continuously from zero to its stationary value. The delay time produced by this transient process increases with increasing the height of fission barrier. Thus particle emission is closely correlated to fission barrier. The fission barrier B f (T ) is composed of the part of the droplet model and temperature-dependent shell corrections, B f (T ) = B DM δu Φ(T ), (6) where B DM is the part of droplet model, and δu is the part of shell correction at T = 0, which is taken from Ref. [15]. For 208 Pb nucleus, its shell-correction value is MeV. Φ(T ) is the temperature-dependent factor of shell correction, which is parametrized as Φ(T ) = exp( at 2 /E d ), (7) where following the work by Ignatyuk et al., [16] a denotes the level density parameter. The shell-damping energy E d is chosen as 20 MeV. [17] Note that in calculations the shell corrections for fission barriers of various nuclei produced in decay processes are considered only in the ground state and they are assumed not to change with deformation. Angular momentum effect in the competition between fission and particle emission is contained in the present

3 No. 2 Influence of Shell on Pre-scission Particle Emission of a Doubly Magic Nucleus 208 Pb 285 work. The rotational energy due to angular momentum can be obtained by a rigid body model. [18] The dependence of the liquid-drop fission barrier on the angular momentum was evaluated with the code barfit, which is based on the finite range model of Sierk. [18] This model considers the finite-range effects in the nuclear surface energy by means of a Yukawa-plus-exponential potential and finite surface diffuseness effects in the Coulomb energy. The code has three input parameters, i.e. mass number A, atomic number Z, and angular momentum L. It gives a rather satisfactory description for the change of fission barriers of various nuclei with angular momentum. [18] Figure 1 displays the fission barrier corrected by shell correction for 208 Pb nucleus as a function of angular momentum at two excitation energies. As one sees that due to the effects of shell, fission barriers are not rather low at larger angular momenta. Meanwhile, we also notice that the values of barrier height decrease with increasing excitation energy. Here, it should be worth mentioning that shell correction is related to temperature. After a compound system evaporates particles, the system is cooled, which results in a restoration of shell correction to fission barrier. This is because the emitted particles carry away a part of excitation energy and that lowers the temperature of the compound system. By comparing the change of shell-correction fission barrier at two different excitation energies, i.e., 100 MeV and 200 MeV, one can easily find the lower the initial excitation energy of the compound nucleus, the larger the shell-correction barrier. Moreover, the speed of restoration of shell correction to fission barrier depends on the initial excitation energy of the compound system. Fig. 1 Fission barriers calculated according to Eq. (6) for 208 Pb nucleus as a function of angular momentum at two different excitation energies. In the present calculation, in a first approximation, a neutron or a proton is assumed to carry away angular momentum of 1 h, while an α particle is assumed to carry away angular momentum of 2 h. The particle emission width is calculated using the detailed balance principle method as [19] Γ i = (2I i + 1)m E B i i π 2 h 2 ɛσ ρ(e i (ɛ), A) V i ρ(e V i B i ɛ, A A i )dɛ, (8) where A is mass number of mother nucleus, ρ is the level density, E is excitation energy after substracting the rotational energy. I i, m i, B i, σ i, V i, and A i (i = n, p, α) are the emitted particle spin, mass, binding energy, inverse absorption cross section, emission barrier, and mass number, respectively. For neutrons, the emission barrier V n = 0. For protons and α particles their emission barriers are taken from Ref. [20] to be { 1.44Z2 /(1.18A 1/ ) for protons, V c = 2.88Z 2 /(1.18A 1/ ) for α particles. (9) Here, Z 2 and A 2 are the atomic number and mass number of the residue nucleus. In the calculation, the level density parameter is taken as A/10 with A being the mass number of the nucleus and particle emission width is assumed to be independent of time and position. In the decay process, the excitation energy of the s-th nucleus can approximately be calculated using the formula [21] E s = E s 1 B i,s 1 2T s 1 V i E(s, s 1), (10) where B i and V i (i = n, p, α) are particle binding energy and emission barrier, respectively. Also, 2T s 1 is the average kinetic energy of the particle emitted from the (s 1)-th nucleus. E(s, s 1) is the difference in the rotational yrast energy caused by angular momentum loss carried away through particle emission. Under the condition of large angular momentum, this becomes the difference in the rotational energy. The data of particle binding energies used here are taken from the estimations of Myers and Swiatecki. [15] It should be mentioned that the competition between particle emission and fission in each decay process is taken into account in this calculation. Using particle multiplicity to study shell effects, angular momentum is a key factor. It plays dual roles. On one hand, it reduces the liquid-drop fission barrier. On the other hand, because of its influence, the rotational energy occupies a part of excitation energy. This part of energy cannot be used as heat bath to raise the nuclear temperature. This indicates that angular momentum reduces the temperature of the fissioning system as well. The fall of temperature increases the shell-correction fission barrier and hence the shell effects on the emission of particles. So an introduction of large angular momenta will be important for studying shell effects on particle emission. However, considering the angular momentum

4 286 YE Wei Vol. 41 dependence of pre-scission particle emission, namely, with increasing the angular momentum, particle multiplicity decreases. It means that if the value of spin of the fissioning nucleus is very high, then the number of emitted particles may be very small, and this lowers the sensitivity of particle evaporation to the shell effects. This is unfavorable to the current research. As a compromise of the above-mentioned two requirements, a proper angular momentum range is necessary for the present investigation. In this study, we presented a detailed calculation of the shell effects on the particle emission as functions of excitation energy and angular momentum. Table 1 Pre-scission neutron (M n), proton (M p), and α particle (M α) multiplicities and the first fission probability (P f ) of the 208 Pb nucleus for the cases with and without shell correction at angular momentum L = 55 h and viscosity coefficient β = s 1 for different excitation energies. E (MeV) Without shell correction With shell correction M n M p M α P f (%) M n M p M α P f (%) Table2 The same as Table 1 but at angular momentum L = 60 h. E (MeV) Without shell correction With shell correction M n M p M α P f (%) M n M p M α P f (%) We first investigate the variation of the shell effects in particle emission with excitation energy for a doubly magic nucleus 208 Pb. We found that as the values of spin of the 208 Pb system lie between 55 h and 80 h, shell effects are evident. Tables 1 4 give the calculated results on the dependence of light particle multiplicities of the 208 Pb nucleus on the excitation energy at four angular momenta and viscosity coefficient β = s 1 for the cases with and without shell correction. In the following we take the results at angular momentum L = 60 h

5 No. 2 Influence of Shell on Pre-scission Particle Emission of a Doubly Magic Nucleus 208 Pb 287 presented in Table 2 as an example to study excitation energy dependence of shell effects on various light particle multiplicity. From Table 2 one can see the inclusion of shell correction does not alter the trend of the emission of light particles with excitation energy though the magnitude of particle multiplicity has a change. This change can be explained based on a complementary quantity, fission probability. Obviously, P f with shell correction is reduced as compared with that without this correction. In addition, calculations also indicate that for this system the charged-particle multiplicity is very small. At other angular momenta such as 55 h and 80 h (see Tables 1 and 4), a similar situation for the small multiplicity of protons and α particles is also observed. We think that lower M p and M α is a consequence of a higher N/Z of the 208 Pb system. It is interesting to study this phenomenon in detail in the future. Therefore in the following our attention focuses on the shell effects on the neutron emission. Table 3 The same as Table 1 but at angular momentum L = 70 h. E (MeV) Without shell correction With shell correction M n M p M α P f (%) M n M p M α P f (%) E (MeV) Table 4 The same as Table 1 but at angular momentum L = 80 h. Without shell correction With shell correction M n M p M α P f (%) M n M p M α P f (%) Figure 4 shows the differences of various light-particle multiplicity between the cases with and without shell correction as a function of excitation energy at L = 60 h. As seen, M p and M α are very small and they are almost not dependent on the excitation energy. This phenomenon can be ascribed to the smaller proton and α particle multiplicity of this system (see Table 2). Meanwhile M n monotonically decreases with increasing the excitation energy, indicating that at a higher energy the shell effects on the neutron emission become weak. When

6 288 YE Wei Vol. 41 excitation energy rises up to 230 MeV, M n approaches zero, suggesting that at this energy the effects of shell do not exist. This is because at E = 230 MeV shell correction to barrier height is negligible, which means that potential barrier mainly comes from liquid-drop contribution. Because only the variation of potential barrier affects transient time and hence the particle emission, a small shell-correction barrier implies a small shell effect in particle multiplicity. The lower the excitation energy is, the larger the shell correction to fission barrier is. At 140 MeV excitation energy, the contribution of shell correction to barrier is not negligible. As a result, M n increases by 0.36 relative to that without shell correction. Because a rise of fission barrier is favorable to the particle emission, this implies that shell will have a more stronger effect on the particle emission at a lower energy. From Fig. 4(a) it is seen that M n increases from 0.36 to 1.25 as excitation energy decreases from 140 MeV down to 100 MeV. At other angular momenta such as 55 h and 80 h, we also observed a similar picture (see Figs. 3 and 6). an increase of temperature of the fissioning system. These two factors reduce the shell effects on the neutron emission. In other words, the emission of neutrons is affected little by shell at this angular momentum. Fig. 3 The same as Fig. 2 but at angular momentum L = 55 h. Fig. 2 The differences of pre-scission neutron (a), proton (b), and α particle (c) multiplicity of the 208 Pb nucleus for the cases with and without shell correction at angular momentum L = 30 h and viscosity coefficient β = s 1 as a function of excitation energy. The solid circles are calculated results. The lines are for guiding eyes. Outside the sensitive range of the shell effects to the angular momentum, the influence of shell on the particle multiplicity is quite small. As an example, we show in Fig. 2 the M i (i=n,p,α) at an angular momentum of 30 h. One can see that the values of M n including shell correction are close to those without shell correction. This can be expected because low angular momenta lead to not only a higher macroscopic rotating nuclei barrier, but also Fig. 4 The same as Fig. 2 but at angular momentum L = 60 h. Because particle emission depends strongly on the angular momentum, so in this work we make a detailed computation concerning the shell effects on particle multiplicity as a function of angular momentum. The particle multiplicities at four angular momenta, i.e., 55 h, 60 h, 70 h, and 80 h, are listed in Tables 1 4 and the corresponding differences for the cases with and without shell

7 No. 2 Influence of Shell on Pre-scission Particle Emission of a Doubly Magic Nucleus 208 Pb 289 correction displayed in Figs. 3 6, respectively. By comparing these results we found that the shell effects on the neutron emission are sensitive to the angular momentum involved. These figures reveal the behavior of M n with angular momentum. Some interesting phenomena were observed. As L varies from 55 h to 80 h, M n does not change monotonically. This is different from the behavior of M n with L, where the higher the L is, the smaller the M n is. We notice that M n at L = 60 h is always larger than the ones at L = 55 h and 80 h for various energies and it is also larger than that at higher angular momenta 70 h for E < 130 MeV. Besides, one also notes that M n at a higher angular momentum L = 70 h is larger than M n at L = 55 h after E > 110 MeV. All these results suggest that shell effects on particle multiplicity depend on the angular momentum and excitation energy simultaneously. This conclusion can be understood as follows. Because M n in the present work is defined as a difference of neutron multiplicity including shell correction with that without the correction, so how particle emission containing shell correction changes with E and L is a key to understanding the behavior of M n at different excitation energies and angular momenta. Because a larger M n makes shell effects evident, this demands a high excitation energy and/or a small angular momentum. However particle emission is also closely connected with fission barrier. A higher fission barrier suppresses fission and enhances particle emission. Consider that the fission barrier including the contribution of shell correction depends on the temperature of a compound system. The two reasons imply that a high angular momentum and/or a low excitation energy is desirable for enhancing the shell effect. Therefore the complicated roles played by excitation energy and angular momentum in affecting the shell-correction barrier and particle evaporation produce some variations of M n with energy and nuclear spin. Moreover, if comparing M n at L = 70 h and that at 80 h shown in Figs. 5(a) and 6(a) respectively, we see that with increasing L, M n decreases. This means that shell effects become weak at L = 80 h. Although a high angular momentum increases the contribution of shell to fission barrier and hence the shell effects, it also greatly decreases pre-scission particle multiplicity and this lowers shell effects. As a competing result of the two factors, the shell effect at L = 80 h is smaller than that at a lower angular momentum 70 h. Fig. 5 The same as Fig. 2 but at angular momentum L = 70 h. Fig. 6 The same as Fig. 2 but at angular momentum L = 80 h. 4 Summary In conclusion, within the framework of the Smoluchowski equation, we investigate the shell effects on the emission of pre-scission particles of a doubly magic nucleus 208 Pb. Calculations show that shell has a significant effect on the pre-scission neutron multiplicity and that this effect gradually fades out with increasing excitation energy. Moreover, a dependence of this shell effect in neutron emission on the angular momentum has also been found.

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