Non-coplanar compact configurations of nuclei and non-compound-nucleus contribution in the fusion cross section of the ${}^{12}\mathrm{C}+{}^{93}\mathrm{Nb}$ reaction

Background: In our earlier study of the ${}^{12}\mathrm{C}+{}^{93}\mathrm{Nb}\to {}^{105}\mathrm{Ag}^{*}$ reaction at three near- and below-barrier energies ($E_{\mathrm{c}.\mathrm{m}.}=41.097$, 47.828, and 54.205 MeV), using the dynamical cluster-decay model (DCM) with various nuclear interaction potentials (the Blocki et al. pocket formula and others derived from the Skyrme energy density formalism) for compact, coplanar $\left({\mathrm{\Phi }}_c=0^0\right)$ nuclei, we found a large non-compound-nucleus (nCN) contribution in the measured fusion cross section of this reaction.

Purpose: In the present work, we look for the effect of using non-coplanar, compact configurations $\left({\mathrm{\Phi }}_c\ne 0^0\right)$, in the Blocki et al. pocket formula of the nuclear proximity potential, on the non-compound-nucleus (nCN) contribution, using the DCM.

Methods: Allowing the $\mathrm{\Phi }$ degree of freedom in the DCM formalism, we calculate the compound-nucleus (CN) and nCN cross sections. The only parameter of the DCM is the neck-length parameter $\mathrm{\Delta }R$, which also fits the empirically determined nCN cross section nearly exactly, under the assumption of considering it like a quasifission process where the fragment preformation factor $P_0=1$.

Results: With the $\mathrm{\Phi }$ degree of freedom included, at the higher two energies the nCN cross section gets enhanced, and hence the pure CN cross section is decreased, since the calculated (total) fusion cross section is fitted to experimental data. The parameter $\mathrm{\Delta }R$ for the nCN contribution is smaller, and hence the reaction time larger, than for the CN decay process. Also, the contributing angular momentum ${\ell }_{\max }$ value increases in going from ${\mathrm{\Phi }}_c=0^0$ to ${\mathrm{\Phi }}_c\ne 0^0$ for both the CN and nCN processes. The intermediate mass fragments (IMFs), measured up to mass 13 in this reaction, are shown extended up to mass 16, and the fusion-fission $\left(ff\right)$ region is identified as $A/2\pm 16$, the same as for the ${\mathrm{\Phi }}_c=0^0$ case.

Conclusions: As a result of enhanced nCN cross section due to ${\mathrm{\Phi }}_c\ne 0^0$, the CN fusion probability $P_{\mathrm{CN}}$ for ${}^{105}\mathrm{Ag}^{*}$ changes its behavior from an increasing to a decreasing function of center-of-mass energy $E_{\mathrm{c}.\mathrm{m}.}$, and hence belongs to the group of weakly fissioning nuclei, instead of the strongly fissioning superheavy nuclei for ${\mathrm{\Phi }}_c=0^0$. On the other hand, with measured IMFs taken to represent the $ff$ component, non-coplanarity simply increases the magnitude of CN survival probability $P_{\mathrm{surv}}$, although its functional dependence on $E_{\mathrm{c}.\mathrm{m}.}$ remains the same as for weakly fissioning nuclei. In other words, on adding the $\mathrm{\Phi }$ degree of freedom, the inconsistent result of $P_{\mathrm{CN}}$ behaving like strongly fissioning superheavy nuclei and $P_{\mathrm{surv}}$ like the weakly fissioning nuclei for ${\mathrm{\Phi }}_c=0^0$ changes to both $P_{\mathrm{CN}}$ and $P_{\mathrm{surv}}$ behaving consistently like those of weakly fissioning nuclei. Thus, our calculations advocate for ${\mathrm{\Phi }}_c$ as an important degree of freedom, like deformations of nuclei themselves.

Figure 1

Two unequal nuclei, oriented at angles ${\theta }_1$ and ${\theta }_2$, with their principal planes $X^{\prime }{Z}^{\prime }$ and $XZ$ making an azimuthal angle $\mathrm{\Phi }$. The angle $\mathrm{\Phi }$ is shown by a dashed line, since it is meant to be an angle coming out of the plane $XZ$. Nucleus 2 is in the $XZ$ plane, and for the out-of-plane nucleus 1 another principal plane $Y^{\prime }{Z}^{\prime }$, perpendicular to $X^{\prime }{Z}^{\prime }$, is also shown. The orientation angels ${\theta }_i$ are measured counterclockwise from the collision $Z$ axis, and the angles ${\alpha }_i$ (and ${\delta }_i$) of radius vectors are measured in the clockwise direction from the nuclear symmetry axis.

Figure 2

The $\ell$-dependent scattering potential $V\left(R\right)$ for ${}^{103}\mathrm{Ag}+2n$, in the decay of ${}^{105}\mathrm{Ag}^{*}$ formed in the ${}^{12}\mathrm{C}+{}^{93}\mathrm{Nb}$ reaction at $E_{\mathrm{c}.\mathrm{m}.}=41.097$ MeV. The concept of barrier lowering $\mathrm{\Delta }{V}_B=V\left(R_a\right)-V_B$ is also shown in this figure for the ${\ell }_{\max }=80\hslash$ value. The first and second turning points $R_a$ and $R_b$ are also labeled.

Figure 3

Mass fragmentation potential minimized in charge-asymmetry coordinate ${\eta }_Z$ for the decay of ${}^{105}\mathrm{Ag}^{*}$ formed in the ${}^{12}\mathrm{C}+{}^{93}\mathrm{Nb}$ reaction at $E_{\mathrm{c}.\mathrm{m}.}=41.097$ MeV and at ${\ell }_{\min }$ and ${\ell }_{\max }$ values.

Figure 4

Preformation probability $P_0$ as a function of angular momentum $\ell$ for ER decays of ${}^{105}\mathrm{Ag}^{*}$ formed in the ${}^{12}\mathrm{C}+{}^{93}\mathrm{Nb}$ reaction at $E_{\mathrm{c}.\mathrm{m}.}=41.097$ MeV. $P_0\sim {10}^{-10}$ for ${\ell }_{\max }=80\hslash$.

Figure 5

Same as for Fig. 4, but for penetrability $P$. $P\sim {10}^{-10}$ for ${\ell }_{\min }=23\hslash$.

Figure 6

The DCM-calculated CN decay channel (ER and IMFs) cross sections for the two cases of coplanar $\left({\mathrm{\Phi }}_c=0^0\right)$ and non-coplanar $\left({\mathrm{\Phi }}_c\ne 0^0\right)$ degrees of freedom, compared with experimental data at $E_{\mathrm{c}.\mathrm{m}.}=41.917$ MeV, for ${}^{105}\mathrm{Ag}^{*}$ formed in the ${}^{12}\mathrm{C}+{}^{93}\mathrm{Nb}$ reaction.

Figure 7

The DCM-calculated (a) CN cross section ${\sigma }_{\mathrm{CN}}^{\mathrm{Cal}.}$, (b) the nCN component ${\sigma }_{\mathrm{nCN}}^{\mathrm{Cal}.}$, and (c) their sum, the fusion cross section ${\sigma }_{\mathrm{fusion}}^{\mathrm{Cal}.}$, for the ${}^{12}\mathrm{C}+{}^{93}\mathrm{Nb}\to {}^{105}\mathrm{Ag}^{*}$ reaction, ${\mathrm{\Phi }}_c\ne 0^0$ case, plotted as a function of $E_{\mathrm{c}.\mathrm{m}.}$, compared with earlier calculations of Ref. [1] for ${\mathrm{\Phi }}_c=0^0$. The experimental data [3] for ${\sigma }_{\mathrm{fusion}}^{\mathrm{Expt}.}$ is also shown in plot (c).

Figure 8

Variations of $\mathrm{\Delta }R$ values for best fitted CN and nCN cross sections as a function of center-of-mass energy $E_{\mathrm{c}.\mathrm{m}.}$ for the ${}^{12}\mathrm{C}+{}^{93}\mathrm{Nb}\to {}^{105}\mathrm{Ag}^{*}$ reaction, ${\mathrm{\Phi }}_c\ne 0^0$ case.

Figure 9

DCM-calculated (a) $P_{\mathrm{CN}}$ and (b) $P_{\mathrm{surv}}$ for ${}^{105}\mathrm{Ag}^{*}$ formed in the ${}^{12}\mathrm{C}+{}^{93}\mathrm{Nb}$ reaction for the ${\mathrm{\Phi }}_c\ne 0^0$ case, compared with the ${\mathrm{\Phi }}_c=0^0$ case of Refs. [5, 6] (solid lines), with ${\sigma }_{\mathrm{IMFs}}^{\mathrm{Cal}.}$ representing the fusion-fission component. In (b) $P_{\mathrm{surv}}^{\mathrm{predicted}}$ (star with dotted line) refers to the not-yet-measured $ff$ component ${\sigma }_{ff}^{\mathrm{Cal}.}$ taken as a sum of the cross sections due to the extended region of IMFs $\left(A_2=5\text{–}16\right)$, the HMFs $\left(A_2=30\text{–}36\right)$, and the near-symmetric and symmetric fission fragments $A/2\pm 16$.

Figure 10

The $\ell$-summed fragment preformation probability $P_0$, the penetrability $P$, and the decay channel cross section $\sigma$ as a function of the light fragment mass number $A_2$ for the compound system ${}^{105}\mathrm{Ag}^{*}$ formed in the ${}^{12}\mathrm{C}+{}^{93}\mathrm{Nb}$ reaction at three $E_{\mathrm{c}.\mathrm{m}.}$'s with their respective ${\ell }_{\max }$ values.