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Measurements on magnetization and relaxation have been carried out on anoptimally doped Ba$_{0.59}$K$_{0.47}$BiO$_{3+\delta}$ single crystal with $T_c$= 31.3 K. Detailed analysis is undertaken on the data. Both the dynamicalrelaxation and conventional relaxation have been measured leading to theself-consistent determination of the magnetization relaxation rate. It is foundthat the data are well described by the collective pinning model leading to theglassy exponent of about $\mu\approx$ 1.64 to 1.68 with the magnetic fields of1 and 3 T. The analysis based on Maley's method combining with the conventionalrelaxation data allows us to determine the current dependent activation energy$U$ which yields a $\mu$ value of about 1.23 to 1.29 for the magnetic fields of1 and 3 T. The second magnetization peaks appear in wide temperature regionfrom 2 K to 24 K. The separation between the second peak field and theirreversibility field becomes narrow when temperature is increased. When thetwo fields are close to each other, we find that the second peak evolves into astep like transition of magnetization, suggesting a crossover from the secondorder to first order transition. Finally, we present a vortex phase diagram anddemonstrate that the vortex dynamics in Ba$_{1-x}$K$_x$BiO$_3$ can be used as amodel system for studying the collective vortex pining.

Collective vortex pinning and crossover between second order to first order transition in optimally doped Ba1-xKxBiO3 single crystals