Formation and decay of the compound nucleus ${}^{220}\mathbf{Th}^{*}$ within the dynamical cluster-decay model

Background: The radioactive ${}^{220}\mathrm{Th}^{*}$ compound nucleus (CN) is of interest since the evaporation residue (ER) cross sections are available for various entrance channels ${}^{16}\mathrm{O}+{}^{204}\mathrm{Pb}$, ${}^{40}\mathrm{Ar}+{}^{180}\mathrm{Hf}$, ${}^{48}\mathrm{Ca}+{}^{172}\mathrm{Yb}$, and ${}^{82}\mathrm{Se}+{}^{138}\mathrm{Ba}$ at near barrier energies. Within the dynamical cluster-decay model (DCM), the radioactive CNs ${}^{215}\mathrm{Fr}^{*}$, ${}^{242}\mathrm{Pu}^{*}$, ${}^{246}\mathrm{Bk}^{*}$, and ${}^{254}\mathrm{Fm}^{*}$ are studied where the main decay mode is fission, with very small predicted ER cross section. ${}^{220}\mathrm{Th}^{*}$ provides a first case with experimentally observed ER cross section instead of fission.

Purpose: To look for the optimum “hot-compact” target-projectile (t-p) combinations for the synthesis of “cold” ${}^{220}\mathrm{Th}^{*}$ and then its decay. For best fitting of the measured ER cross sections, with quasifission (qf) content, if any, the fusion-fission (ff) component is predicted. The magic-shell structure and entrance channel mass-asymmetry effects are analyzed, and the behavior of CN formation and survival probabilities $P_{\mathrm{CN}}$ and $P_{\mathrm{surv}}$ is studied.

Methods: The quantum-mechanical fragmentation theory (QMFT) is used to predict the possible cold t-p combinations for synthesizing ${}^{220}\mathrm{Th}^{*}$, and the QMFT-based DCM is used to analyze its decay channels for the experimentally studied entrance channels. The only parameter of the model, the neck length $\mathrm{\Delta }R$, varies smoothly with the excitation energy $E^{*}$ of CN and is used to best fit the ER data and predict qf and ff cross sections.

Results: The hot-compact and “cold-elongated” fragmentation paths show dissimilar results, whose comparisons with measured fission yields result in t-p combinations, the cold reaction valleys. For the decay process, the fixed $\mathrm{\Delta }R$ fit the measured ER cross section nicely, but not the individual decay-channel cross sections, which require the presence of qf effects, less so for asymmetric t-p combinations, and large (predicted) ff cross section.

Conclusions: The calculated yields for hot-compact fragmentation path compared favorably with the observed asymmetric fission-mass distribution, resulting in mostly the same t-p reactions as are used in experiments. The best fitted $\mathrm{\Delta }R$ are found to be independent of the entrance channels, despite their very different cross sections, in conformity with our earlier works. The entrance-channel effects lead to the largest decay cross section for the most asymmetric and doubly magic t-p combination, the magicity taking over asymmetry for both the total ER and channel cross section. Furthermore, the variation of both $P_{\mathrm{CN}}$ and $P_{\mathrm{surv}}$ with $E^{*}$ fit in with the known systematic of other radioactive CN studied so far, thereby giving credence to our DCM analysis of ${}^{220}\mathrm{Th}^{*}$.

Figure 1

Schematic configuration of two equal or unequal axially symmetric deformed, oriented nuclei, lying in the same plane (azimuthal angle $\mathrm{\Phi }=0^{\circ }$) for various ${\theta }_1$ and ${\theta }_2$ values in the range $0^{\circ }$ to ${180}^{\circ }$. The ${\theta }_i$ are measured anticlockwise from the colliding axis and angle ${\alpha }_i$ in clockwise from the symmetry axis.

Figure 2

The scattering potential $V\left(R\right)$ for the decay ${}^{220}\mathrm{Th}^{*}\to {}^{217}\mathrm{Th}+3n$ formed in ${}^{48}\mathrm{Ca}+{}^{172}\mathrm{Yb}$ at $E^{*}=39.9$ MeV, calculated at ${\ell }_{\min }$ and ${\ell }_{\max }$ with both fragments having quadrupole (${\beta }_{2i}$) deformations and optimum hot orientations ${\theta }_i^{\mathrm{opt}}$ of Table I in Ref. [17].

Figure 3

Mass-fragmentation potential minimized in charge coordinate ${\eta }_Z$ for the formation of ${}^{220}\mathrm{Th}^{*}$ at $E^{*}=39.9$ MeV having quadrupole (${\beta }_{2i}$) deformations and optimum orientations ${\theta }_i^{\mathrm{opt}}$ of Table I in Ref. [17] for (a) hot-fusion configurations, and (b) cold-fusion configurations, for ${\ell }_{\max }$ and ${\ell }_{\min }$ values, and at a fixed $\mathrm{\Delta }R=1.0$ fm.

Figure 4

Preformation yield $P_0\left(A_i\right)$, $i=1,2$, for the mass-fragmentation potentials of ${}^{220}\mathrm{Th}^{*}$ in Figs. 3 and 3, at ${\ell }_{\min }$ and ${\ell }_{\max }$ values.

Figure 5

Scattering potentials $V\left(R\right)$ for hot-fusion reactions, i.e., cold reaction valleys with hot-compact configurations of Fig. 3, listed in Table 1.

Figure 6

Variation of ${\sigma }_{\mathrm{ER}}$ as a function of $E^{*}$ for ${}^{16}\mathrm{O}+{}^{204}\mathrm{Pb}$ entrance channel, for all six measured $E^{*}$ (solid line with open squares) and two extrapolated energies (dashed line with open circles), compared with experimental data (filled circles) [10] and $\text{DNS}+\text{ASM}$ calculation of Kim et al. [12] (dashed line).

Figure 7

Variation of ER-fit neck-length parameter $\mathrm{\Delta }R$ with $E^{*}$ for the formation of ${}^{220}\mathrm{Th}^{*}$ via various entrance channels. Symbols give the $\mathrm{\Delta }R$ value, and the lines are to guide the eyes, except for ${}^{16}\mathrm{O}+{}^{204}\mathrm{Pb}$ where the solid line is a polynomial fit and the dotted line with open circles gives the extrapolated values.

Figure 8

Fragmentation potentials $V\left(A_2\right)$ as a function of light-fragment mass number $A_2$, for the decay of ${}^{220}\mathrm{Th}^{*}$, plotted at ${\ell }_{\min }$ and ${\ell }_{\max }$ values, for best-fit $\mathrm{\Delta }R$ values given in Table 2, Calc. 2 for $xn$ channels, at $E^{*}=39.9$ MeV, using quadrupole deformations (${\beta }_{2i}$). For $A_2=6-A/2$, $\mathrm{\Delta }R=1.0$ fm.

Figure 9

The preformation probability $P_0$ vs angular momentum $\ell$ for the fragmentation potential in Fig. 8.

Figure 10

The penetrability P vs angular momentum $\ell$ for $xn$ decay channels, as per details in Fig. 8.

Figure 11

The $xn$ decay channel cross sections ${\sigma }_{xn}$, $x=1–5$, vs angular momentum $\ell$ as per details in Fig. 8.

Figure 12

DCM calculated ER excitation functions of ${}^{220}\mathrm{Th}^{*}$ for best-fit $\mathrm{\Delta }R$ to individual $xn$ decay channels of ${}^{40}\mathrm{Ar}+{}^{180}\mathrm{Hf}$ entrance channel, composed of CN contribution ${\sigma }_{\mathrm{ER}}^{\mathrm{CN}}$ (solid lines with open up-triangle) and the noncompound quasifission component ${\sigma }_{qf}^{\mathrm{Calc}.}$ (solid lines with open down-triangle), and their sum ${\sigma }_{\mathrm{ER}}^{\mathrm{Calc}.}$ (solid line with open squares), compared with experimental data (filled circles) [11] and $\mathrm{DNS}+\mathrm{ASM}$ calculation of Kim et al. [12] (dashed line).

Figure 13

DCM-calculated excitation functions of individual $3n$, $4n$, and $5n$ decay channel cross sections ${\sigma }_{xn}\left(={\sigma }_{xn}^{\mathrm{CN}}+{\sigma }_{qf}^{\mathrm{Calc}.}\right)$, $x=3–5$ from ${}^{220}\mathrm{Th}^{*}$ formed due to the three entrance channels ${}^{48}\mathrm{Ca}+{}^{172}\mathrm{Yb}$, ${}^{40}\mathrm{Ar}+{}^{180}\mathrm{Hf}$, and ${}^{82}\mathrm{Se}+{}^{138}\mathrm{Ba}$ (solid lines with respective open symbols), compared with experimental data (solid symbols) [11, 13, 14] and hivap calculation [14] for ${}^{82}\mathrm{Se}+{}^{138}\mathrm{Ba}$ only (dashed line). The best-fit $\mathrm{\Delta }R$ values are from Tables 2 and 3, Calc. 2.

Figure 14

The $\ell$-summed $P_0$, $P$, and $\sigma$ for the decay of ${}^{220}\mathrm{Th}^{*}$ as a function of light-fragment mass $A_2$ at $E^{*}=39.9$ MeV for best-fit $\mathrm{\Delta }R$ given in Table 2, Calc. 2, and Fig. 8 caption.

Figure 15

Variation of DCM-calculated (a) $P_{\mathrm{CN}}$ and (b) $P_{\mathrm{surv}}$ as a function of excitation energy $E^{*}$ for the three entrance channels of ${}^{220}\mathrm{Th}^{*}$, compared with other DCM-studied reactions forming radioactive compound systems for $\mathrm{\Phi }=0^{\circ }$ case.

Figure 16

The best-fit neck-length parameters $\mathrm{\Delta }R$ for the CN- and nCN-like qf processes in $xn$ decay of ${}^{220}\mathrm{Th}^{*}\to {}^{220}\mathrm{Th}+xn$, where $x=3$, 4, 5 are the observed neutron channels, plotted as a function of $E^{*}$ for the three entrance channels.

Figure 17

Comparison of the experimental (solid symbols) [11, 13, 14] and DCM-calculated (solid lines) $xn$ cross sections for different channels at a fixed $E^{*}\approx 40$ MeV.