Effects of projectile break-up on fusion cross sections at energies near and above the Coulomb barrier: A case of incomplete fusion

In the present work, the experimental studies of projectile break-up (incomplete fusion) at energies $\approx 4$–7 MeV/nucleon have been performed by using offline $\gamma$-ray spectroscopy. The excitation functions of reaction residues populated in the ${}^{19}\mathrm{F}+{}^{175}\mathrm{Lu}$ system via complete fusion and/or incomplete fusion processes were measured and analyzed within the framework of the statistical model code pace4. The measured excitation functions of $xn$ and $pxn$ channels are found to be well reproduced by the predictions of pace4, which clearly indicates the population of these residues, predominantly, via complete fusion processes. However, in the case of residues involving $\alpha$ particles in the exit channels, the experimentally measured cross sections are found to show a significant enhancement when compared with pace4 predictions. This enhancement points toward the onset of incomplete fusion reactions at the studied range of energies and is found to be projectile energy dependent. Further, an attempt was made to study the influence of projectile (strongly bound) break-up on fusion cross sections, above the Coulomb barrier, within the framework of the universal fusion function, which is a benchmark function that does not depend on the system parameters. The experimental fusion function was deduced for the complete fusion (CF) cross section $\left(\sum {\sigma }_{\text{CF}}=\sum {\sigma }_{xn+pxn}^{\text{expt}}\right)$ and total fusion (TF) cross section $\left(\sum {\sigma }_{\text{TF}}=\sum {\sigma }_{\text{CF}\text{+}\text{ICF}}^{\text{expt}}\right)$ for three strongly bound projectiles ${}^{13}\mathrm{C},{}^{16}\mathrm{O}$, and ${}^{19}\mathrm{F}$ (present work) on different target nuclei and compared with the universal fusion function. Analysis of data indicates 10–$35\%$ complete fusion suppression above the barrier, indicating that it is essentially due to the prompt break-up (incomplete fusion) of the strongly bound projectiles. Moreover, the deduced complete fusion suppression factor for the present work shows a conspicuous exponential relation with the break-up threshold of the projectile.

Figure 1

A typical $\gamma$-ray spectrum of the ${}^{19}\mathrm{F}+{}^{175}\mathrm{Lu}$ interaction at $103.64\pm 1.36$ MeV.

Figure 2

(a) Experimentally measured EFs of ${}^{190}\mathrm{Hg},{}^{189}\mathrm{Hg}$, and ${}^{188}\mathrm{Hg}$, residues populated via $4n,5n$, and $6n$, in the ${}^{19}\mathrm{F}+{}^{175}\mathrm{Lu}$ system (see text for details). (b) Experimentally measured and theoretically calculated ratios of given complete fusion $xn$ residues to the sum of all such residues for the ${}^{19}\mathrm{F}+{}^{175}\mathrm{Lu}$ system.

Figure 3

Experimentally measured EFs of ${}^{190}\mathrm{Au}$ and ${}^{189}\mathrm{Au}$ residues populated via $p3n$ and $p4n$ channels compared with pace4 calculations: (a) cumulative and (b) independent cross sections of ${}^{190}\mathrm{Au}$ residues, and (c) cumulative and (d) independent cross sections of ${}^{189}\mathrm{Au}$ residues (see text for details).

Figure 4

Excitation function of reaction residues ${}^{188}\mathrm{Au}$ populated via $p5n$ channel compared with pace4 predictions (see text for details).

Figure 5

Experimentally measured EFs of (a–d) ${}^{189,188,187,186}\mathrm{Pt}(\alpha xn$, where $x=1$, 2, 3, and 4) and (e, f) ${}^{183,182}\mathrm{Os}(2\alpha xn$, where $x=3$, 4) residues populated in the ${}^{19}\mathrm{F}+{}^{175}\mathrm{Lu}$ system and compared with those calculated by the pace4 model (see text for details).

Figure 6

Experimentally measured EFs of ${}^{187,186,185}\mathrm{Ir}(\alpha pxn$, where $x=2$, 3, and 4) and ${}^{181}\mathrm{Re}\left(2\alpha p4n\right)$ residues populated in the ${}^{19}\mathrm{F}+{}^{175}\mathrm{Lu}$ system and compared with those calculated by the pace4 model (see text for details). (d) The dashed line in the figure is to guide the eye.

Figure 7

The total fusion cross sections $\left({\sigma }_{\text{TF}}\right)$ and the sum of all CF channels $\left({\sigma }_{\text{CF}}\right)$, and $\left({\sigma }_{\text{ICF}}\right)$ cross sections (inset) are plotted as a function of incident projectile energy. Lines through the experimental data points are drawn to guide the eye.

Figure 8

The complete fusion function for strongly bound projectiles on different target nuclei. The solid black curve is the benchmark UFF. The dotted red line is the UFF multiplied by 0.75 (for details see text).

Figure 9

The complete fusion function plotted separately for the ${}^{19}\mathrm{F}$ projectile on different target nuclei, viz., ${}^{159}\mathrm{Tb},{}^{169}\mathrm{Tm}$, and ${}^{175}\mathrm{Lu}$. The solid black curve is the benchmark UFF. The dotted red line is the UFF multiplied by 0.66 (for details see text).

Figure 10

The deduced suppression factor for the ${}^{19}\mathrm{F}$ projectile is plotted as a function of the break-up threshold of the projectile. The dotted line represents the empirical Eq. (7) (for details see text).

Figure 11

The experimentally deduced total fusion functions for strongly bound projectiles on different target nuclei are compared with the UFF. The solid black curve is the UFF (for details see text).

Figure 12

Fusion $\ell$ distributions calculated using the code ccfull to understand the population of $\ell$ bins at each studied energy.