Influence of the shell effects on evaporation residue cross section of superheavy nuclei

NUCLEAR PHYSICS AND INTERDISCIPLINARY RESEARCH

Influence of the shell effects on evaporation residue cross section of superheavy nuclei

D. Naderi，

S. A. Alavi

Nuclear Science and Techniques

Vol.29, No.11

Article number 161

Published in print 01 Nov 2018

Available online 28 Sep 2018

DOI：10.1007/s41365-018-0498-6

38500

In order to study the influence of the shell effects on the formation and fission of superheavy elements we applied multidimensional Langevin equations. The Evaporation residue cross sections have been calculated for 3n, 4n, and 5n evaporation channels using three (K=0) and four (K ≠ 0) dimensional Langevin equations. Calculations were done for ⁴⁸Ca+²³⁸U and ⁴⁸Ca +²⁴⁴Pu hot fusion reactions with 3n, 4n evaporation channels and ⁷⁰Zn +²⁰⁸Pb, and ⁵⁴Cr +²⁰⁹Bi cold fusion reactions with 1n and 2n evaporation channels. The calculations were performed for 4n and 5n evaporation channels of the ²⁶Mg+²³⁸U reaction, as well. Our results show that with increasing dimension of Langevin equations the residue cross section increases whereas the fission cross section decreases. The obtained results with four dimensional Langevin and considering shell effects are in better agreement with experimental data in comparison with three and four dimensional Langevin equations without shell effects.

SuperheavyLangevin equationsCross section

1 Introduction

The superheavy elements production is one of the important problems and outstanding research objects of recent nuclear physics. Some experimental and theoretical research has been done to produce and investigate the synthesis mechanism of superheavy elements. Cold fusion and hot fusion are two main classes of the heavy-ion fusion reactions for synthesis of superheavy nuclei. The superheavy nuclei with Z=102-118 has been produced by means of cold fusion reactions with the targets of ²⁰⁸Pb and ²⁰⁹Bi and ⁴⁸Ca-induced hot fusion reactions [1-5].

Various theoretical methods have been introduced for study of the superheavy nucleus production, such as the dynamical Langevin model [6-8], dinuclear system model [9-11], fluctuation dissipation model [12, 13], nuclear collectivization concept [14], macroscopic dynamical model [15], and multidimensional stochastic model [16]. In Refs. [17] the dynamical Langevin model has been used to estimate ⁵⁸Fe+²³⁸Pb cold-fusion reaction cross sections. In a similar manner, this method (one dimensional Langevin equation) has been used for ⁴⁸Ca+²³⁸U hot-fusion reaction in Refs. [18, 19]. In order to determine the cold and hot fusion reaction cross sections, the dinuclear system was used in Refs. [20-22].

In this paper, we applied multidimensional Langevin equations to evaluate the formation of a superheavy nucleus. The main topics of this paper are the analysis of the influence of the orientation degree of freedom and shell effects on synthesis of superheavy nucleus. In Sect. 2 the theoretical calculations based on Langevin equations are given. The obtained results are given in Sec. 3. Finally, the summary and conclusion remarks are presented in Sec. 4.

2 Model

The fusion and fission processes are considered with the same approach. The Langevin equations for the shape parameters of the ions give the time evolution of the system in both stages. The random force term gives the stochastic features of the process. For easiness, the shape of the system is considered in terms of two spheres with radii R₁ and R₂ gently connected with hyperboloidal neck. Let r stand for the total length of the system, s=r-2(R₁+R₂) for the surface separation between the two spheres, h is the radius of the neck, and α=[(R₁-R₂)/(R₁+R₂)]² is the asymmetry variable. The coupled Langevin equations of motion in multidimensional collective space are written as [17, 23-25]

\begin{array}{l} \frac{d q_{i}}{d t} = μ_{i j} p_{j}, \\ \frac{d p_{i}}{d t} = - \frac{1}{2} p_{j} p_{k} \frac{\partial μ_{j k}}{\partial q_{i}} - \frac{\partial V (q)}{\partial q_{i}} - γ_{i j} μ_{j k} p_{k} + θ_{i j} ξ_{j} (t), \end{array}

(1)

where qi= s, h, α represent the collective coordinates, pi gives conjugate momenta, and μjk denotes inverse matrix elements of the inertia tensor, mij [26, 27]. The strength θij of the random force is given by θikθkj=Tγij. T and γij are the temperature and friction tensor, respectively [23].

The potential energy of the system is defined as [28]

V (q, I, K, T) = V_{LD} (q) + \frac{[I (I + 1) - K^{2}] ℏ^{2}}{[0.8 M R_{0}^{2} J_{⊥} (q) + 8 M a^{2}]} + \frac{K^{2} ℏ^{2}}{[0.8 M R_{0}^{2} J_{∥} (q) + 8 M a^{2}]} + V_{SH} (q, T),

(2)

where V_LD(q) is the potential energy based on the liquid-drop model. R₀=1.2249A^1/3. A and M are the mass number and mass of the compound nucleus, respectively. a=0.6 fm. J_‖(q) and J_⊥(q) are the rigid body moments of inertia of the nucleus with respect to the symmetry axis and an axis orthogonal to it, respectively. K is the projection of the total spin of the compound nucleus (I) to the symmetry axis. The temperature dependent shell correction energy, V_SH, is given as

V_{SH} (q, T) = [Δ E_{Pair} (q) + Δ E_{Shell} (q)] Φ (T),

(3)

where ΔE_Pair(q)=E_Pair-〈E_Pair〉 is the pairing correlation energy which is determined by using BCS approximation [29]. Here 〈E_Pair〉 is the average of pairing energy at the ground state distortions. We can calculate the pairing correlation energy based on the method of Ref. [29]. Also ΔE_Shell(q) is the shell correction energy based on the Strutinsky method which can be defined as [30-32]

Δ E_{Shell} (q) = \sum ϵ_{k} - \int_{- \infty}^{μ} e g (e) d e,

(4)

ϵk, g(e), μ are the energy, single particle density of states, and chemical potential, respectively. The temperature dependent shell correction factor, Φ(T), in Eq. (3) is given as [33]

Φ (T) = \exp (- \frac{a T^{2}}{E_{d}}),

(5)

where E_d=18.5 MeV is the shell damping energy. a is the level density parameter [34]

a = {1 + \frac{V_{SH} (T = 0)}{E_{int}} [1 - \exp (- \frac{E_{int}}{E_{d}})]} \times (a_{1} A + a_{2} A^{2 / 3} B_{s} (q)),

(6)

with a₁=0.068 MeV^-1 and a₂=0.213 MeV^-1 [34]. The alternative form of these parameters as a₁=0.073 MeV^-1 and a₂=0.095 MeV^-1 has been given by Ignatyuk [35]. The nuclear temperature can be calculated as

T = \sqrt{\frac{E_{int}}{a}} .

(7)

The function B_s(q) is the dimensionless function of the surface energy in the liquid drop model with a sharp surface [34]. The intrinsic excitation energy is calculated as,

E_{int} = E_{c .m .} + Q - V (q, I, K, T = 0) - E_{Coll},

(8)

where the Q-value is the released energy of the reaction and E_Coll is the kinetic energy of the fusing system.

The variation of the orientation degree of freedom (K coordinate) is obtained as [36-38]

d K = - \frac{γ_{K}^{2} I^{2}}{2} \frac{\partial V}{\partial K} d t + γ_{K} I \sqrt{\frac{T d t}{2}} ξ (t) .

(9)

Here the spins of projectile and target nuclei have been neglected and I=l. ξ(t) is a random variable given as

\begin{array}{l} 〈 ξ_{i} 〉 = 0, \\ 〈 ξ_{i} (t_{1}) ξ_{j} (t_{2}) 〉 = 2 δ_{i j} δ (t_{1} - t_{2}) . \end{array}

(10)

The γK parameter controls the coupling between the orientation degree of freedom and heat bath [8, 37].

The evaporation residue cross section of a superheavy nucleus producion is calculated as [39]

σ_{ER} = \frac{π ℏ^{2}}{2 μ E_{cm}} \sum_{l = 0}^{\infty} (2 l + 1) P_{cap} (E_{cm}, l) \times P_{fus} (E_{cm}, l) P_{kn} (E_{cm}, l) .

(11)

P_cap is the capture probability of the colliding nuclei. This factor can be calculated by means of the semiphenomenological barrier distribution function as [14, 39]

P_{cap} = \int f (B) \frac{1}{1 + \exp (\frac{2 π}{ℏ ω_{B} (l)} (B + \frac{ℏ^{2}}{2 μ R_{B}^{2}} l (l + 1) - E_{cm}))} d B,

(12)

here $ℏ ω_{B} (l)$ is defined by the width of the parabolic barrier. f(B) is the barrier distribution function [39].

The modified fusion by diffusion model [40, 41] for evaluating the compound nucleus formation probability, P_fus, is given as

P_{fus} (E_{cm}, l) = \frac{1}{2} \int erfc (\sqrt{B (s_{inj}, l) / T}) \times f (s_{inj}) d s_{inj},

(13)

where s_inj is the distance between the surfaces of two approaching nuclei where injection into an asymmetric fission valley takes place and f(s_inj) is the probability distributions of s_inj. The dynamic equations were solved step by step and obtained the probability distribution of s_inj.

The survival probability is given as [7, 39]

\begin{array}{l} P_{kn} (E_{0}^{*}, l_{0}) = [\frac{Γ_{n}}{Γ_{tot}} (E_{0}^{*}, l_{0})] \int_{0}^{E_{0}^{*} - S n (1)} \frac{Γ_{n}}{Γ_{tot}} (E_{1}^{*}, l_{1}) \\ \times P_{n} (E_{0}^{*}, e_{1}) d e_{1} \\ \times \int_{0}^{E_{1}^{*} - S n (2)} \frac{Γ_{n}}{Γ_{tot}} (E_{2}^{*}, e_{2}) P_{n} (E_{1}^{*}, e_{2}) d e_{2} ... \\ \times \int_{K_{th}}^{E_{k - 1}^{*} - S n (k)} P_{n} (E_{k - 1}^{*}, e_{k}) d e_{k}, \end{array}

(14)

here Γ_tot≈Γ_n+Γ_fis is the total decay width. The details of the calculation of the P_cap, P_fus, and P_kn have been represented in Ref. [8]. The survival probability to fission strongly depend on the fission barrier, B_f, and how this barrier is damped with respect to the angular momentum, l, and nuclear temperature, T, of the Compound nucleus. The nuclear temperature is related to the excitation energy, E^*, and the level density parameter, a, by the expression $T = \sqrt{E^{*} / a}$ . The fission barrier was evaluated as [7]

B_{f} (E^{*}) = B_{LD} - Δ_{sh} e^{- E^{*} / E_{d}},

(15)

where B_LD is the liquid-drop model fission barrier [42], Δ_sh is the shell correction energy calculated for the nucleus in its ground state [43], and E_d=18.5 MeV [44] is the damping parameter which shows a decrease in the shell effects in an energy level density with increasing the excitation energy of the nucleus.

It is worthwhile to note that moreover P_CN, the P_cap can be calculated by using Langevin equations [45]. However we have used the approach proposed by Zagrebaev et al. [14, 39].

3 Results

In order to investigate the shell effects and orientation degree of freedom on residue cross section, we selected two hot fusion reactions, ⁴⁸Ca+²³⁸U and ⁴⁸Ca +²⁴⁴Pu with 3n and 4n evaporation channels and three cold fusion reactions, ⁷⁰Zn +²⁰⁸Pb, and ⁵⁴Cr +²⁰⁹Bi, with 1n and 2n evaporation channels. Also, we performed calculations for 4n and 5n evaporation residue cross section in the ²⁶Mg+²³⁸U reaction. The shell effects were investigated via introducing V_SH term in potential energy. The level density parameters of Ref. [34] have been used in calculations. Obtained results have been shown in Figs. 1 to 10.

Fig. 1.

Variations of the probability distributions of sinj for a) ⁴⁸Ca +²³⁸U reaction b) ⁴⁸Ca +²⁴⁴Pu reaction at E_cm=200 MeV. Dashed, solid and dotted curves are the results based on three dimensional Langevin equations without shell effects, four dimensional Langevin equations without shell effects, and four dimensional with shell effects, respectively.

Fig. 10.

Plot of excitation functions for 4n a) and 5n b) evaporation channels as a function of excitation energy for ⁴⁸Mg +²³⁸U reaction. Circles are experimental data [48]. Also, dashed, solid, and dotted curves are the results based on three dimensional Langevin equations without shell effects, four dimensional Langevin equations without shell effects and four dimensional with shell effects, respectively.

The distribution of the injection distance, f(s_inj), is calculated with the Langevin equations in three and four dimensional collective space. For this reason, we have numerically solved Eqs. (1) and (9), simultaneously. The injection point distribution is obtained in the simulation with the three and four dimensional Langevin equations. One of quantities which changes in three and four dimensional calculations is the potential energy. Consequently the numerical results for the distributions of the injection distance are different in three and four dimensional calculations. Obtained results based on the four dimensional model are higher than results of the three dimensional model. One can see the discrepancy of two models in Fig. 1 particularly in the position of picks. Figs. 1 a) and b) show variations of the probability distributions of s_inj for ⁴⁸Ca +²³⁸U and ⁴⁸Ca +²⁴⁴Pu reactions, respectively. Dashed, solid, and dotted curves are the results based on three Langevin without shell effects, four dimensional Langevin without shell effects and four dimensional Langevin equations with shell effects, respectively. When s is negative it means that nuclei are crossed. A shape describing the connection of two spheres has been depicted in Fig.1 of Ref. [25]. The probability distributions based on four dimensional model with shell effects show a noticeable difference and sharper peak in comparison with three and four dimensional models without shell effects. For the ⁴⁸Ca +²³⁸U reaction the four dimensional with shell effects model gives probability distributions larger than three and four dimensional models without shell effects particularly around the peak.

In Fig. 2 (a) and (b) the plot of the potential energy for ⁴⁸Ca +²³⁸U and ²⁸Mg +²³⁸U reactions have been depicted as a function of elongation. We can see for the case K ≠ 0 the fission barrier is higher than when comparison with K=0.

Fig. 2.

Variations of the potential energy as a function of elongation for a) ⁴⁸Ca +²³⁸U and b) ²⁸Mg +²³⁸U reactions. Dashed and solid curves are the results for I=60, K=60 and I=60, K=0, respectively. B_F is the fission barrier.

The variation of the survival probability as a function of the excitation energy for a) ⁴⁸Ca +²³⁸U, b) ⁴⁸Ca +²⁴⁴Pu, and c) ⁷⁰Zn +²⁰⁸Pb reactions have been displayed in Fig. 3. We can observe that by using three and four dimensional Langevin equations and with including the shell effects, the peak of the survival probability changes, significantly.

Fig. 3.

Plot of survival probability versus energy for a) ⁴⁸Ca +²³⁸U b) ⁴⁸Ca +²⁴⁴Pu c) ⁷⁰Zn +²⁰⁸Pb reactions. Dashed, solid and dotted curves are the results based on three dimensional Langevin equations without shell effects, four dimensional Langevin equations without shell effects and four dimensional with shell effects, respectively.

Using Eq. (10), the smooth value of the level density parameter has been modified due to the shell effects. The level density parameter of the daughter nucleus at the ground configuration increases as the excitation energy increases as a result of the damping of the shell correction energy of the ground state. Consequently, the damping of the shell effects directly influences the decay width of neutron emission rather than the fission width.

Fig. 4(a) and (b) show variations of fission cross section versus energy for ⁴⁸Ca +²³⁸U and ⁴⁸Ca +²⁴⁴Pu hot fusion reactions, respectively. Open squares, circles, and triangles show obtained results based on three dimensional Langevin equations without shell effects, four dimensional Langevin equations with shell effects, and four dimensional Langevin equations with shell effects, respectively. By increasing the excitation energy the fission cross section is increased. The obtained results show by increasing the dimension of the Langevin equations the fission cross section is decreased and obtained results based on four dimensional Langevin equations are in good agreement with the experimental data.

Fig. 4.

Variations of fission cross section versus energy for a) ⁴⁸Ca +²³⁸U reaction b) ⁴⁸Ca +²⁴⁴Pu reaction. Open squares, circles and triangles show obtained results based on three dimensional Langevin equations without shell effects, four dimensional Langevin equations without shell effects and four dimensional with shell effects, respectively. Solid squares are experimental data for ⁴⁸Ca +²³⁸U reaction [13] and ⁴⁸Ca +²⁴⁴Pu reaction [46].

Figs. 5 and 6 show the variation of the cross section of evaporation residue as a function of the excitation energy for a) 3n and b) 4n evaporation channels for ⁴⁸Ca +²³⁸U and ⁴⁸Ca +²⁴⁴Pu hot fusion reactions, respectively. The results show by increase of the dimension of the Langevin equations the larger values of evaporation residue cross section are obtained. The obtained results based on four dimensional Langevin equations with shell effects are in better agreement with the experimental data in comparison with other models. The evaporation residue cross section is related to dimensions of calculations via survival probability.

Fig. 5.

Plot of excitation functions for a) 3n and b) 4n evaporation channels as a function of energy for ⁴⁸Ca +²³⁸U reaction. Circles are experimental data [46]. Also, dashed, solid and dotted curves are the results based on three dimensional Langevin equations without shell effects, four dimensional Langevin equations without shell effects and four dimensional calculations with shell effects, respectively.

Fig. 6.

Same as Fig. 5 for ⁴⁸Ca +²⁴⁴Pu reaction.

In Fig 7 a) and b) the variation of excitation functions for a) 1n and b) 2n evaporation channels as a function of excitation energy have been depicted for the ⁵⁸Fe +²⁰⁸Pb cold fusion reaction. Similarly, Fig. 8 a) and b) have been plotted for the 1n evaporation channel for a) ⁷⁰Zn +²⁰⁸Pb and b) ⁵⁴Cr +²⁰⁹Bi cold fusion reactions. Similar results are deduced for these cold fusion reactions. Four dimensional Langevin equations with shell effects are in better agreement with experimental data in comparison with other models, as well.

Fig. 7.

Variation of excitation functions for a) 1n and b) 2n evaporation channels as a function of excitation energy for ⁵⁸Fe +²⁰⁸Pb reaction. Squares are experimental data [47]. Also, dashed, solid, and dotted curves are the results based on three dimensional Langevin equations without shell effects, four dimensional Langevin equations without shell effects and four dimensional with shell effects, respectively.

Fig. 8.

Variation of excitation functions for 1n, evaporation channel as a function of excitation energy for a) ⁷⁰Zn +²⁰⁸Pb and b) ⁵⁴Cr +²⁰⁹Bi reactions. Squares are experimental data [47]. Dashed, solid, and dotted curves are the results based on three dimensional Langevin equations without shell effects, four dimensional Langevin equations without shell effects and four dimensional with shell effects, respectively.

In order to investigate the effect of the level density parameter on the evaporation residue cross section, the variation of excitation functions for the ⁵⁸Fe +²⁰⁸Pb reaction in the 2n evaporation channel based on four dimensional Langevin equations with shell effects as a function of excitation energy for two set of level density parameters [34] and [35] have been depicted in Fig. 9. With the decreasing level density parameter the fission cross section decreases whereas evaporation residue cross section increases. These figures show by using the Ignatyuk parameters, larger values are obtained in comparison with Toke and Swiatecki parameters. For these reactions obtained data by using Ignatyuk parameters are closer to the experimental data, as well.

Fig. 9.

Variation of excitation functions based on four dimensional Langevin equations with shell effects for ⁵⁸Fe +²⁰⁸Pb reaction in 2n evaporation channel as a function of excitation energy for two set of level density parameters. Squares are experimental data [47]. Solid, and dotted curves are the calculated excitation functions by using Toke and Swiatecki level density parameter [34] and Ignatyuk parameter [35], respectively.

In Fig. 10 the calculated excitation functions have been compared with experimental data a) 4n and b) 5n evaporation channels as a function of excitation energy for the ⁴⁸Mg +²³⁸U reaction. One can see the same results from this figure in comparison with obtained results for ⁴⁸Ca +²⁴⁴Pu and ⁴⁸Ca +²³⁸U reactions. Also, in this figure one can see the shell effects clearly.

Although the calculated results of the evaporation residue cross section in Fig. 5-9 agree well with the available experimental results, the limits of this calculation are in the free parameter model dependence in the calculation of capture cross section and in the estimation of the survival probability to fission [49-52].

4 Summary and Conclusion

In this paper, the evaporation residue cross sections of two hot fusion reactions, ⁴⁸Ca+²³⁸U and ⁴⁸Ca +²⁴⁴Pu, with 3n and 4n evaporation channels and three cold fusion reactions, ⁷⁰Zn +²⁰⁸Pb, and ⁵⁴Cr +²⁰⁹Bi, with 1n and 2n evaporation channels were calculated by using three and four dimensional Langevin equations without shell correction, and four dimensional Langevin equations with shell correction. Also, the calculations were performed for 4n and 5n evaporation residue cross section in the ²⁶Mg+²³⁸U reaction. The difference between the results of the fission cross section and evaporation residue cross sections based on three and four dimensional Langevin equations is significant, and four dimensional Langevin equations with shell effects give better results in comparison with other models. Our results show that with increasing dimension of Langevin equations the evaporation residue cross section increases whereas the fission cross section decreases.

References

[1]

S. Hofmann and G. Münzenberg,

The discovery of the heaviest elements

, Rev. Mod. Phys. 72, 733 (2000). doi: 10.1103/RevModPhys.72.733