Increases of a Diamagnetic Property by Flux-Pinning in Volume Defect-Dominating 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. We would present that H$_{c1}$ is able to be proved theoretically and experimentally. We have derived an equation based on flux-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, magnetic flux penetration method, and a magnetization at H$_{c1}$ in the static state are suggested through the equation of the model. It was speculated that pinned fluxes at a volume defect in the superconductor have to be pick-out depinned from the defect and move an inside of the superconductor when pick-out forces of pinned fluxes is larger than pinning force of the defect (F$_{pickout}$ $>$ F$_{pinning}$) or when the shortest distance between pinned fluxes at a volume defect is the same as that of H$_{c2}$. In reality, $Delta$G$_{dynamic}$ which is sum of fluxes movement energy and fluxes vibration energy is involved in movement of pinned fluxes. When volume defects are small and many, the number of pinned fluxes at a volume defect calculated by experimental results was closer to that of ideally calculated ones because of a small $Delta$G$_{dynamic}$. However, when volume defects are large and a few, the number of pinned fluxes at a volume defect calculated by experimental results were much fewer than that of ideally calculated ones because of a large $Delta$G$_{dynamic}$.