Systematic study of suppression of complete fusion in reactions involving weakly bound nuclei at energies above the Coulomb barrier

Complete fusion excitation functions of reactions involving breakup are studied by using the empirical coupled-channel (ECC) model with breakup effects considered. An exponential function with two parameters is adopted to describe the prompt-breakup probability in the ECC model. These two parameters are fixed by fitting the measured prompt-breakup probability or the complete fusion cross sections. The suppression of complete fusion at energies above the Coulomb barrier is studied by comparing the data with the predictions from the ECC model without the breakup channel considered. The results show that the suppression of complete fusion is roughly independent of the target for the reactions involving the same projectile.

Figure 1

The fitted results of $\nu$ shown as a function of $R_0$, i.e., $R_0=R_{\mathrm{P}}+R_{\mathrm{T}}$. The fitted results of $\nu$ taken from Ref. [34] are represented by the solid squares. The blue dotted line denotes the results obtained from the function of $\nu =a-\mu R_0$ with $a=0.557$ and $\mu =-0.884 {\mathrm{fm}}^{-1}$.

Figure 2

The experimental complete fusion excitation functions and calculated cross sections for reactions induced by ${}^9\mathrm{Be}$ on ${}^{89}\mathrm{Y}$ [57], ${}^{124}\mathrm{Sn}$ [58], ${}^{144}\mathrm{Sm}$ [6, 59], ${}^{169}\mathrm{Tm}$ [13], ${}^{181}\mathrm{Ta}$ [12], ${}^{187}\mathrm{Re}$ [13], ${}^{208}\mathrm{Pb}$ [4], and ${}^{209}\mathrm{Bi}$ [60, 61]. The black line (Fus) denotes the fusion cross sections obtained from the ECC model without the breakup channel considered, i.e., ${\sigma }_{\mathrm{Fus}}$. The pink dash-dotted line (CF) denotes the calculated complete fusion cross sections obtained from the ECCBU model with $\nu =0.557-\mu R_0$ and $\mu =-0.884 {\mathrm{fm}}^{-1}$. The blue dotted line denotes $F_{\mathrm{BU}}{\sigma }_{\mathrm{Fus}}$ with the suppression factor $F_{\mathrm{BU}}=0.68$ taken from Ref. [33].

Figure 3

The experimental complete fusion excitation functions and calculated cross sections for reactions induced by ${}^6\mathrm{Li}$ on ${}^{90}\mathrm{Zr}$ [27], ${}^{96}\mathrm{Zr}$ [14], ${}^{154}\mathrm{Sm}$ [16], ${}^{159}\mathrm{Tb}$ [29], ${}^{197}\mathrm{Au}$ [30], ${}^{198}\mathrm{Pt}$ [7], ${}^{208}\mathrm{Pb}$ [5], and ${}^{209}\mathrm{Bi}$ [4]. The black line (Fus) denotes the fusion cross sections obtained from the ECC model without the breakup channel considered, i.e., ${\sigma }_{\mathrm{Fus}}$. The pink dash-dotted line (CF) denotes the calculated complete fusion cross sections obtained from the ECCBU model with $\nu =0.557-\mu R_0$ and $\mu =-0.798 {\mathrm{fm}}^{-1}$. The blue dotted line denotes $F_{\mathrm{BU}}{\sigma }_{\mathrm{Fus}}$ with the suppression factor $F_{\mathrm{BU}}=0.60$ taken from Ref. [33].

Figure 4

The experimental complete fusion excitation functions and calculated cross sections for reactions induced by ${}^7\mathrm{Li}$ on ${}^{159}\mathrm{Tb}$ [65], ${}^{165}\mathrm{Ho}$ [9, 66], ${}^{197}\mathrm{Au}$ [30], ${}^{198}\mathrm{Pt}$ [67], and ${}^{209}\mathrm{Bi}$ [4]. The black line (Fus) denotes the fusion cross sections obtained from the ECC model without the breakup channel considered, i.e., ${\sigma }_{\mathrm{Fus}}$. The pink dash-dotted line (CF) denotes the calculated complete fusion cross sections obtained from the ECCBU model with $\nu =0.557-\mu R_0$ and $\mu =-0.964 {\mathrm{fm}}^{-1}$. The blue dotted line denotes $F_{\mathrm{BU}}{\sigma }_{\mathrm{Fus}}$ with the suppression factor $F_{\mathrm{BU}}=0.67$ taken from Ref. [33].

Figure 5

The experimental complete fusion excitation functions and calculated cross sections for reactions induced by ${}^{10,11}\mathrm{B}$ on ${}^{159}\mathrm{Tb}$ [68] and ${}^{209}\mathrm{Bi}$ [10]. The black line (Fus) denotes the fusion cross section obtained from the ECC model without the breakup channel considered, i.e., ${\sigma }_{\mathrm{Fus}}$. The pink dash-dotted line (CF) denotes the calculated complete fusion cross section using the ECCBU model with $\nu =0.557-\mu R_0$ and $\mu =-1.415{\mathrm{fm}}^{-1}$ for ${}^{10}\mathrm{B}$, while for ${}^{11}\mathrm{B},\mu =-1.900{\mathrm{fm}}^{-1}$. The blue dotted line denotes $F_{\mathrm{BU}}{\sigma }_{\mathrm{Fus}}$ with the suppression factor $F_{\mathrm{BU}}=0.80$ for ${}^{10}\mathrm{B}$ and $F_{\mathrm{BU}}=0.91$ for ${}^{11}\mathrm{B}$ taken from Ref. [33].

Figure 6

The values of $\mu$ for ${}^{6,7}\mathrm{Li},{}^9\mathrm{Be}$, and ${}^{10,11}\mathrm{B}$ as a function of the lowest breakup threshold energy $E_{\mathrm{BU}}$. The dotted line denotes the empirical formula (9).