Department of Physics, Collaborative Innovation Center for Quantum Effects, and Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, Hunan Normal University, Changsha 410081, China Received Date:2020-04-03 Available Online:2020-09-01 Abstract:The survival probability of an excited compound nucleus was studied using two different approaches of the washing out of shell effects with excitation energy based on a superasymmetric reaction system. The estimated evaporation residue cross sections based on the two different methods are compared with the available experimental data. Both methods are in agreement with the experimental data to a certain extent for some specific reactions and $ xn$ emission channels.
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2.Theoretical frameworkThe ERCS of the compound nucleus can be defined as follows [6-8]:
where $ \sigma_{\rm{ERCS}} $ is the product of the capture cross section $ \sigma_{\rm cap} $, fusion probability $ P_{\rm{CN}} $, and survival probability $ W_{\rm{sur}} $, and $ E_{\rm c.m.} $ is the incident energy in the center of mass system. $ E_{\rm{CN}}^{*} $ and J separately represent the excitation energy and spin angular momentum of the compound nucleus, where $ E_{\rm{CN}}^{*} = E_{\rm{c.m.}}+Q $, $ Q = M(P) c^{2}+ M(T) c^{2}- M(C) c^{2} $, $ M(P) $, $ M(T) $, and $ M(C) $ represent the mass of the projectile nucleus, target nucleus, and compound nucleus, respectively. The capture cross section $ \sigma_{\rm cap}(E_{\rm c.m.},J) $ in Eq. (1) is determined by the penetration probability $ T\left(E_{\rm c . m .}, J\right) $ of the two colliding systems overcoming the Coulomb potential barrier in the entrance channel to form the DNS. The corresponding expression is defined as [31]:
where $ \hbar\omega(J) $ is the width of the parabolic Coulomb barrier at position $ R_{B}(J) $, and $ B(\theta_{1},\theta_{2}) $ is the orientation dependent barrier. The $ P_{\rm{CN}}\left(E^{*}, J\right) $ term in Eq. (1) is the probability that the dinuclear system evolves from a touching configuration to form a compound nucleus [6]. Furthermore, the fusion probability $ P_{\rm{CN}}\left(E^{*}, J\right) $ depends on the competition between the complete fusion and quasifission process. However, there is almost no effect of the dynamical fusion hindrance on the collision of superasymmetric reaction systems. Our calculations assume that the fusion probability $ P_{\rm CN}\approx1 $. However, if the asymmetry decreases and tends to a symmetrical system, then the quasifission channel will become increasingly important, i.e., $ P_{\rm CN}<1 $. The $ W_{\rm{sur}}\left(E^{*}, J\right) $ term in Eq. (1) represents the survival probability of the excited compound nucleus with a certain excitation energy $ E^{\ast} $ and angular momentum J [7]. Furthermore, the survival probabilities of excited compound nuclei in the process of deexcitation via neutron evaporation, $ \gamma $-ray emission, and light particle evaporation in competition with fission can be estimated within the statistical evaporation model. The most noteworthy point is that the excited compound nuclei are generally considered to be cooled mainly by neutron evaporation and fission in the medium excitation energy range ($ E^{*}_{\rm CN} = 10-60 $ MeV). The survival probability of neutron deficient compound nuclei is an interesting research topic because many fusion reactions of heavy nuclei lead to the formation of a neutron-deficient compound nucleus. In addition, the production of heavy [34-36] and superheavy nuclei [37-39] in fusion reactions accompanied by the evaporation of charged particles has been discussed previously in theoretical and experimental studies. Furthermore, there is experimental evidence that the evaporation of charged particles in the deexcitation of neutron deficient compound nuclei successfully competes with neutron emission [7]. In fact, it is necessary to take into consideration the evaporation channels of charged particles when calculating the survival probability at high excitation energies. We will study these effects in further studies. The survival probability of vaporizing $ xn $ neutrons can be expressed as:
where $ P_{xn} $ is the probability of vaporizing $ xn $ neutrons at a given excitation energy $ E_{\rm{CN}}^{*} $ [40]. $ \Gamma_{n} $ and $ \Gamma_{\rm f} $ are the neutron evaporation width and fission width, respectively. $ E_{i}^{*} $ is the compound nuclear excitation energy before vaporization of the $ i $th neutron. The partial width for the emission of a neutron from a compound nucleus with the excitation energy $ E_0 $ is given by the Weisskopf formula [41]
where $ m_{n} $ and g are the mass and spin degeneracy of the emitted neutron, and $ \sigma_{\rm inv} $ is the cross section of the decaying nucleus formed in the inverse process. The $ \rho_{0}(E_{0}-\delta_0) $ term is the level density of the parent nucleus at the thermal excitation energy of its corrected pairing energy $ \delta_0 $, and $ \rho_{n}(E_0-B_n-\delta_n-\varepsilon) $ is the corresponding level density of the daughter nucleus after emitting a neutron. $ B_n $ is the neutron separation energy, and $ \delta_n $ is the pairing energy of the daughter nucleus. The fission width can be expressed in terms of the transition state theory as [18]
where $ \rho_{\rm f}(E_0-B_{\rm f}-\delta_{\rm f}-\varepsilon) $ is the level density of the fissile nucleus at the saddle configuration. The back-shift Fermi-gas model is used to determine the level density [42],
where $ \sigma^2 = \dfrac{\Theta_{\rm rigid}}{\hbar^2}\sqrt{\frac{U}{a}} $, $ \Theta_{\rm rigid} = \dfrac{2}{5}m_{u}AR^2 $, and $ U = E-\delta_0 $. The back shifts $ \delta = -\Delta $ (odd-odd), 0 (odd A), and $ \Delta $ (even-even), respectively, are related to the neutron and proton pairing gap $ \Delta = 1/2[\Delta_n(Z,N)+\Delta_p(Z,N)] $ obtained from the mass differences of the neighboring nuclei. One way to consider that the washing out of shell effects with increasing excitation energy is introduced in the nuclear level density parameter is by proposing an exponential function. The dependence of the level density parameter a on the shell correction and excitation energy was initially proposed as
with $ \tilde{a}(A) = \alpha A+\beta A^{2/3} $ and $ f(U) = 1-\exp{(-\gamma_D U)} $. The values of the free parameters $ \alpha $, $ \beta $, and $ \gamma $ are determined by fitting to the experimental level density data [42]. The fission barrier $ B_{\rm f} = B_{\rm f}^{\rm L D}+B_{\rm f}^{\rm M}(E^{*} = 0) $ remains constant. The fission barrier height $ B_{\rm f} $ in Eq. (6) consists of the liquid drop $ B_{\rm{f}}^{\rm L D} $ and microscopic $ B_{\rm f}^{\rm M} $ parts in the macroscopic-microscopic approach. The liquid drop part was calculated according to the angular momentum dependent macroscopic fission barriers described by the finite range liquid model [43]. The microscopic value $ B_{\rm f}^{\rm M} = \delta W_{\rm shell}^{\rm saddle}-\delta W_{\rm shell}^{\rm{\rm gs}} $ is the difference between the shell correction energy of the saddle point and shell correction energy of the ground state [44]. In addition, the systematics analysis shows that the values of $ \delta W_{\rm shell}^{\rm saddle} $ are close to zero for the nuclei with $ 80\leqslant Z\leqslant 100 $ [24]. Therefore, the present theoretical calculations can be used to experimentally determine the measured values of the fission barrier when we assume the shell correction energy of the saddle point as $ \delta W_{\rm shell}^{\rm saddle} = 0 $ [24]. Another way to consider the washing out of shell effects with increasing excitation energy is to focus on the dependence of the fission barrier on the excitation energy. This energy dependence can be attributed to an effective compound nucleus ground state, where the ground state shell correction is washed out by the excitation energy. Therefore, the dependence of the fission barrier on the excitation energy is given by the following formula:
$ B_{\rm f}\left(E_{\rm C N}^{*}\right) = B_{\rm f}^{\rm L D}+B_{\rm f}^{\rm M}\left(E_{\rm C N}^{*} = 0\right) \exp \left[-E_{\rm C N}^{*} / E_{\rm D}\right] $
(9)
$ E_{\rm D} $ is the shell damping factor that describes the decrease in the shell effect with increase in the nuclear excitation energy. From a theoretical point of view, the washing out of microscopic effects with the excitation energy in the level density should be equivalent to smearing microscopic effects on the effective potential energy surface [28]. To ensure self-consistency in the calculation, we need to calculate the neutron separation energy through the effective compound nucleus ground state. For FRDM1995 [44], the dependence of the nuclear binding energies on the excitation energy can be expressed as