Determination of the onset of bulk pinning and the low-temperature-irreversibility line in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$

We report the results of dc magnetization measurements of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ single crystals as a function of temperature. The evolution of the hysteresis loops with temperature is examined and the irreversibility line extracted. These loops can be scaled onto each other from 4.2 to 20 K, while the lower field part of the loops for the virgin curves continue to scale up to $T_c$. At intermediate temperatures, the scaling is broken by the appearance of the second peak or "arrowhead" which appears at around 20 K but gradually disappears above about 30 K. While the behavior is dominated by significant bulk pinning at low temperatures, the hysteresis and irreversibility fields at high temperatures are considerably modified by geometrical and surface barrier effects. We identify the onset, with increasing field, of bulk pinning and significant critical current from the emergence of the "arrowhead" feature from the background surface barrier-determined magnetization curve. Scaling in the low-temperature regime, and extrapolation, enables the irreversibility line at 0 K to be estimated at about 110 T. We propose a phase diagram up to high magnetic fields over a wide range of temperature.