Vortex Softening: Origin of the second peak effect in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$
Transverse ac permeability measurements in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta }$ single crystals at low fields and temperatures in a vortex configuration free of external forces show that the decrease of the critical current as measured by magnetization loops at the second peak effect is an artifact due to creep. On the other hand, the increase of critical current at the second peak is due to a genuine softening of the tilting elastic properties of vortices in the individual pinning regime that precedes the transition to a disorder state.