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Water-Quenched Effects of 5 wt.% (Fe, Ti) particle-doped MgB$_2$ Superconductor and Low Limit of Pinning Effect

\begin{abstract} We have studied magnetic properties of water-quenched 5 wt.% (Fe, Ti) particle-doped MgB$_2$ comparing with that of air-cooled one. Generally, grain refinement is achieved by increasing cooling rate, which implies an increase of grainboundaries in the superconductor. Here we show that increased grainboundaries influence what kinds of effects on the field dependence of magnetization and what is the mechanism. As a result, they are served as a pinning center at a high field whereas they are served as a pathway to facilitate the movement of fluxes pinned on volume defects at a low field. As modeling grainboundaries in a superconductor, we explained that they had a flux pinning effect as well as the flux-penetrating promotion effect. As temperature increases, the pinning ability of a grainboundaries decreases, which was caused by increased coherence length. Stacking fault planes and twin boundaries have also been considered by using the model. It explained the reason for that stacking fault planes of MgB$_2$ do not have any pinning effect and the twin boundary of HTSC have the strong pinning or strong flux-penetration effect depending on the direction of the applied field.