Dynamical cluster-decay model for hot and rotating light-mass nuclear systems applied to the low-energy ${}^{32}\mathrm{S}+{}^{24}\mathrm{Mg}\to {{}^{56}\mathrm{Ni}}^{\ast }$ reaction

The dynamical cluster-decay model (DCM) is developed further for the decay of hot and rotating compound nuclei (CN) formed in light heavy-ion reactions. The model is worked out in terms of only one parameter, namely the neck-length parameter, which is related to the total kinetic energy TKE($T$) or effective $Q$ value ${\mathit{Q}}_{\mathrm{eff}}(T)$ at temperature $T$ of the hot CN and is defined in terms of the CN binding energy and ground-state binding energies of the emitted fragments. The emission of both the light particles (LP), with $A\le 4,Z\le 2$, as well as the complex intermediate mass fragments (IMF), with $4<A<20,Z>2$, is considered as the dynamical collective mass motion of preformed clusters through the barrier. Within the same dynamical model treatment, the LPs are shown to have different characteristics compared to those of the IMFs. The systematic variations of the LP emission cross section ${\sigma }_{\mathrm{LP}}$ and IMF emission cross section ${\sigma }_{\text{IMF}}$ calculated from the present DCM match exactly the statistical fission model predictions. A nonstatistical dynamical description is developed for the first time for emission of light particles from hot and rotating CN. The model is applied to the decay of ${{}^{56}\mathrm{Ni}}^{\ast }$ formed in the ${}^{32}\mathrm{S}+{}^{24}\mathrm{Mg}$ reaction at two incident energies $E_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}=51.6$ and 60.5 MeV. Both the IMFs and average $\stackrel{̲}{\mathit{TKE}}$ spectra are found to compare resonably well with the experimental data, favoring asymmetric mass distributions. The LPs' emission cross section is shown to depend strongly on the type of emitted particles and their multiplicities.

Figure 1

Temperature- and angular-momentum-dependent scattering potentials, illustrated for ${}^{56}\mathrm{Ni}\to {}^{12}\mathrm{C}+{}^{44}\mathrm{Ti}$ at $T=3.39\text{MeV}$ (equivalently, $E_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}=51.6\text{MeV}$). The potential for each $\ell$ is calculated by using $V(R,T,\ell )=E_c(T)+V_P(T)+V_{\ell }(T)$, normalized to exit channel $T$-dependent binding energies $B_L(T)+B_H(T)$, each defined as $B(T)=V_{\mathrm{LDM}}(T)+\delta U(T)$. The decay path, defined by $V(R_a,\ell )=Q_{\mathrm{eff}}(T,\ell )$ for each $\ell$, is shown to begin at $R_a=C_t+\stackrel{̲}{\Delta R}$ for $\ell =0$ case, where $\stackrel{̲}{\Delta R}$ is the average over $\eta$ of the different neck-length parameters $\Delta R(=R_a-C_t)$ calculated for $V(R_a)=Q_{\mathrm{eff}}(T,\ell =0)$ for all possible fragmentations. The critical angular momentum ${\ell }_c$ value is determined from Eq. (23). For other details, see text.

Figure 2

Fragmentation potentials $V(A)$ for the compound system ${}^{56}\mathrm{Ni}\ast$ formed in the reaction ${}^{32}\mathrm{S}+{}^{24}\mathrm{Mg}$ at $E_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}=51.6\text{MeV}$, calculated for different $\ell$ values at a fixed $T=3.39\text{MeV}$ (corresponding to $E_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}=51.6\text{MeV}$) and $R_a=C_t+1.28\text{fm}$. The fragmentation potential $V(A)=V_{\mathrm{LDM}}(T)+\delta U(T)+E_c(T)+V_P(T)+V_{\ell }(T)$ for fixed $R=R_a$. The ${\ell }_c$ value is from Eq. (23).

Figure 3

Fragment preformation factor $P_0(\ell ,A_L)$, penetrability $P(\ell ,A_L)$, and decay cross section ${\sigma }_{\ell }(A_L)$, with $\ell$ summed over ${\ell }_{\max }={\ell }_c=32\hslash$ calculated from Eq. (23), for the decay of ${}^{56}\mathrm{Ni}\ast$ to various complex fragments (both LPs and IMFs), using the fragmentation potentials in Fig. 2 based on the DCM.

Figure 4

Variation of $P_0$ with $\ell$, for both the LPs (dashed lines) and even-$A,N=Z$ IMFs (solid lines), calculated with the DCM for the compound system ${}^{56}\mathrm{Ni}\ast$, using the fragmentation potentials in Fig. 2. For protons, the calculated $P_0$ values are three times those shown in the figure.

Figure 5

Variation of P with $\ell$, for both the LPs (dashed lines) and even-$A,N=Z$ IMFs (solid lines), calculated with the DCM for the compound system ${}^{56}\mathrm{Ni}\ast$, using the scattering potentials of Fig. 1. For protons, there is no barrier at any $\ell$ value and hence $P=1$.

Figure 6

Variation of evaporation residue cross sections due to the LPs' ${\sigma }_{\mathrm{LP}}$ (dotted line), the IMFs' ${\sigma }_{\text{IMF}}$ (dashed line), and their sum ${\sigma }_{\mathrm{Total}}$ (solid line) with $\ell$, for $\ell$ values up to ${\ell }_c$, calculated with the DCM for the compound system ${}^{56}\mathrm{Ni}$${}^{\ast }$ formed in a ${}^{32}\mathrm{S}+{}^{24}\mathrm{Mg}$ reaction at $E_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}=$ 51.6 MeV. The parameter $\stackrel{̲}{\Delta R}$ (=1.28 fm) is the same for both the LPs and IMFs. The cross sections given in parentheses are obtained by summing for ${\ell }_{\max }={\ell }_c=32\hslash$. Here the LPs consist of ${}^{1,2}$H and ${}^{3,4}$He.

Figure 7

Variation of $P_0(\ell ,A_L)$, $P(\ell ,A_L)$, and ${\sigma }_{\ell }(A_L)$ with $\ell$, for the emission of ${}^8\mathrm{Be}$ and ${}^{28}\mathrm{Si}$ fragments from ${}^{56}\mathrm{Ni}$${}^{\ast }$ formed in a ${}^{32}\mathrm{S}+{}^{24}\mathrm{Mg}$ reaction at $E_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}=51.6\text{MeV}$, calculated with DCM, as in Fig. 6. For ${}^{28}\mathrm{Si}$ decay, the calculated $P$ is ten times the plotted value and the calculated $P_0$ is one tenth of the plotted value.

Figure 8

The same as Fig. 6, but for use of different $\stackrel{̲}{\Delta R}$ values for LPs and IMFs.

Figure 9

The same as Fig. 8, but with proton emission replaced by neutron emission.

Figure 10

The same as Fig. 3, but for only the decay cross section $\sigma (A_L)$ and calculated for the case of Fig. 8. The DCM calculations are also made for another arbitrary higher ${\ell }_c$ value ($=36\hslash$) and are compared with two other model calculations, EHFM and TSM of Refs. 3 and 6, respectively, and the experimental data [6, 16] for $A_L\ge 12$ IMFs.

Figure 11

Variation of the first turning point $R_a$ with light fragment mass $A_L$, for $R_a=C_t,R_a=C_t+\stackrel{̲}{\Delta R}$(=1.28 fm), and the actually calculated $R_a$ from $V(R_a)=Q_{\mathrm{eff}}(T),\ell =0$, for the decay of ${}^{56}\mathrm{Ni}$ at $T=3.39\text{MeV}$. Note that the actually calculated $R_a$ from $V(R_a)=Q_{\mathrm{eff}}(T),\ell =0$, can be written as $R_a=C_t+\Delta R(\eta )$, where $\Delta R(\eta )$ is found to be, in general, positive. For some light fragments ($A_L=1\text{−}5,8$), we use $\ell >0$ since calculated $Q_{\mathrm{eff}}(T)$ is larger than the barrier for the $\ell =0$ case.

Figure 12

The same as Fig. 10, but for the DCM alone, calculated for different average $\stackrel{̲}{\Delta R}$ values and the actual $\Delta R(A_L)$ determined from $V(R_a)=Q_{\mathrm{eff}}(T,\ell =0)$ (Fig. 11). The DCM calculations are compared with the experimental data taken from Refs. 6, 16.

Figure 13

Histograms of the calculated IMF cross sections $\sigma (A_L)$ from the DCM, compared with the experimental data [6, 16] and two other model calculations, EHFM [3] and TSM [6], for even-$A_L$ IMFs at $E_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}=51.6\text{MeV}$ and for both the odd- and even-$A_L$ IMFs at $E_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}=60.5\text{MeV}$. The predicted $A_L=$14 fragment cross section from the DCM is very large, ten times of what is plotted here.

Figure 14

DCM excitation functions (i.e., cross sections at different incident c.m. energies) for emission of a ${}^{12}$C fragment from the excited ${}^{56}{\mathrm{Ni}}^{\ast }$ compound nucleus, calculated for a constant and an arbitrary $T$-dependent $\stackrel{̲}{\Delta R}$ value.

Figure 15

(a) Charge dispersion potential, (b) the corresponding (fractional) preformation probability $P_0$ in the DCM for a mass $A_L=14$ fragment, at different $\ell$ values, in the decay of ${{}^{56}\mathrm{Ni}}^{\ast }$.

Figure 16

The measured [6] and DCM-calculated average total kinetic energy ($\stackrel{̲}{\mathit{TKE}}$) for the reaction ${}^{32}\mathrm{S}+\hspace{1em}{}^{24}\mathrm{Mg}\to {}^{56}{\mathrm{Ni}}^{\mathrm{\ast }}\to A_L+A_H$, at two incident energies $E_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}=51.6\text{MeV}$ and 60.5 MeV. The total kinetic energy $\mathit{TKE}$ for the best fit to the $\ell$ value is also plotted. The average $\stackrel{̲}{\Delta R}=1.28\text{fm}$ and 1.29 fm, respectively, for the two energies.