Empirical formulas for fragmentation production cross section

Abstract

Production cross sections of neutron-rich nuclei in projectile fragmentation reactions measured at zero degrees are systematically analyzed using published experimental data. By comparing the yields of O, F, Ne, Na, Mg, Al, Si, and P isotopes produced in $^{40}$Ar-induced fragmentation at beam energies of 35~MeV/nucleon, 57--120~MeV/nucleon, and 1~GeV/nucleon, we demonstrate that the production cross sections of neutron-rich fragments are, within experimental uncertainties, essentially independent of the projectile energy over this broad range. Motivated by this limiting-fragmentation behavior, we propose an empirical systematics for neutron-rich fragment production based on the binding energy per nucleon ($\mathrm{BE}/A$) of the final nuclei. The cross sections are found to follow a steep exponential dependence on $\mathrm{BE}/A$, spanning 8--9 orders of magnitude for small variations of $\mathrm{BE}/A$. The systematics is tested for reactions induced by $^{40}$Ar, $^{48}$Ca, $^{58,64,68}$Ni, $^{72}$Zn, $^{76}$Ge, $^{82}$Se, and $^{86}$Kr projectiles on Be and Ta targets, and the corresponding fit parameters are shown to correlate with the fragment charge $Z$ and the projectile neutron-to-proton ratio $(N/Z)_{\mathrm{projectile}}$. Additionally, a complementary analysis based on $Q_{gg}$ systematics (mass difference between projectile and detected fragments) is also presented, and the advantages and limitations of the $\mathrm{BE}/A$-based description for extrapolating yields toward the neutron drip line are discussed.