  We have studied flux-pinning effects of MgB$_2$ superconductor by doping (Fe,
Ti) particles of which radius is 163 nm on average. 5 wt.\% (Fe, Ti) doped
MgB$_2$ among the specimens showed the best field dependence of magnetization
and 25 wt.\% one did the worst at 5 K . The difference of field dependence of
magnetization of the two increased as temperature increased. Here we show
experimental results of (Fe, Ti) particle-doped MgB$_2$ according to dopant
level and the causes of the behaviors. Flux-pinning effect of volume
defects-doped superconductor was modeled in ideal state. During the study, we
had to divide M-H curve of volume defect-dominating superconductor as three
discreet regions for analyzing flux pinning effects, which are diamagnetic
increase region, $\Delta$H=$\Delta$B region, and diamagnetic decrease region.
As a result, flux-pinning effects of volume defects decreased as dopant level
increased over the optimal dopant level, which was caused by decrease of
flux-pinning limit of a volume defect. And similar behaviors are obtained as
dopant level decreased below the optimal dopant level, which was caused by the
decreased number of volume defects. Comparing the theory with experimental
results, deviations increased as dopant level increased over the optimal dopant
level, whereas the two was well matched on less dopant level than the optimal
dopant level. The behavior is considered to be caused by segregation of volume
defects. On the other hand, the property of over-doped specimens dramatically
decrease as temperature increases, which is caused by double decreases of
flux-pinning limit of a volume defect and segregation effect.
