  By doing magnetization measurements during magnetic field sweeps on thin
films of the new superconductor $MgB_2$, it is found that in a low temperature
and low field region small flux jumps are taking place. This effect strongly
suppresses the central magnetization peak leading to reduced nominal
superconducting critical current density at low temperatures. A borderline for
this effect to occur is determined on the field-temperature (H-T) phase
diagram. It is suggested that the small size of the flux jumps in films is due
to the higher density of small defects and the relatively easy thermal
diffusion in thin films in comparison with bulk samples.
