Equation of Motion for Pinned Fluxes at Volume Defects and Increases of a Diamagnetic Property by Flux Pinning in Superconductors

Whereas there are two critical fields that are H$_{c1}$ and H$_{c2}$ in the ideal type II superconductor, there is another critical field H$_{c1}'$ defined as the field showing the maximum diamagnetic property in the real type II superconductor. Here we present that H$_{c1}'$ is able to be proved theoretically and xperimentally. We have derived the equation based on the pinning effect of volume defects. MgB$_2$ bulks which were synthesized by Mg and B are similar to this model. The number of quantum fluxes pinned at a defect of radius r, a pinning penetration depth, the magnetic flux penetration method, and a magnetization at H$_{c1}'$ in the static state are suggested through the equation of the model. It is speculated that fluxes pinned on volume defects in the superconductor have to be picked out from the defect and move an inside of the superconductor when the pick-out forces of pinned fluxes is larger than the pinning force of the defect (F$_{pickout}$ $>$ F$_{pinning}$) or when the distance between quantum fluxes pinned at a volume defect is the same as that of H$_{c2}$. In reality, $\Delta$G$_{dynamic}$ is involved for movement of pinned fluxes. When volume defects are small and many, the experimental results are closer to the calculated ones because of a small $\Delta$G$_{dynamic}$. However, when volume defects are large and a few, the experimental results are much lower than the calculated ones because of a large $\Delta$G$_{dynamic}$.