Breakup of a nucleus with two weakly bound neutrons on a proton target: Single-scattering approximation

Three- and four-cluster breakup reactions in the ${}^{16}$C-proton scattering are studied using three-body core-neutron-neutron model for ${}^{16}$C. Single-scattering approximation (SSA) of four-particle equations for transition operators is used to calculate three- and four-cluster breakup amplitudes at 200 and 300 MeV/nucleon energy near proton-neutron ($pn$) quasi free scattering (QFS) conditions. The differential cross section is sharply peaked at $pn$ QFS point and decreases rapidly whenever kinematical conditions deviate from $pn$ QFS. The accuracy of the SSA for the three-cluster breakup is estimated from a three-body model and is found to be as good as 6% at higher reaction energies in suitable angular regimes. Furthermore, under an additional approximation the three-cluster breakup amplitude factorizes into the $pn$ transition operator and the overlap integral of two- and three-particle bound states. That approximation usually reduces the cross section, in some cases even up to 10%.