Nuclear and Coulomb breakup of the Borromean nucleus $^{11}$Li

We have performed kinematically complete measurements of breakup reactions of $^{11}$Li with a $^{12}$C and with a Pb target at about 70~MeV/nucleon at the radioactive beam facility RIPS at RIKEN. Due to the loosly-bound nature of this nucleus, the breakup reactions will provide useful knowledge on the ground and low-lying states of $^{11}$Li, as well as that on the subsystems such as $^{10}$Li and di-neutron correlation. In the breakup with the C target, where nuclear interaction dominates over the Coulomb interaction, the knockout reaction becomes important. There, the $^{10}$Li nucleus is produced as an intermediate state. We show the relative energy spectrum of $^{10}$Li, which is important in understanding the structure of $^{11}$Li. In the breakup with the Pb target, where Coulomb breakup is dominant, we have observed strong E1 transition strengths at $E_{\rm rel}\sim$~0.3~MeV, which was missing in the previous experiments. In this presentation we also discuss the characteristics of reactions mechanisms of breakup reactions with halo nuclei.