## 3 Results and Discussion
### AC Susceptibility as a Function of AC Field Amplitude: \(\chi_{1}^{\prime\prime}(h_{AC})\)
It is worth noting that the \(\mathrm{J_{c}}\) values  
Figure 1: \(\chi_{1}^{\prime\prime}\) as a function of the AC field amplitude at T = 77 K for different DC fields and AC field frequencies. decrease up to 3 T, while an increase in J\({}_{\rm c}\) is observed with further increasing the magnetic field. This can be associated with the intergranular and intragranular contributions already noted in Figure 1, respectively. The behavior of the experimental dependence of J\({}_{\rm c}\) as a function of ln(f\({}_{0}\)/f) provides insight into the suitability of different pinning models [42]: the Anderson-Kim model of pinning is represented by a downward curvature [48], the collective pinning model is represented by an upward curvature [49], and the suitability of a logarithmic dependence of the pinning potential on the current density is indicated by a straight line [50], which is the case with our data. For the latter case, as was shown in ref. [42], the pinning potentials for each DC applied field, U\({}_{0}=\) k\({}_{\rm B}\)T(1 + 1/b), can be estimated thanks to the slope b of the straight lines in Figure 2.  
The obtained values of U\({}_{0}\) in K (k\({}_{\rm B}\) = 1) as a function of the DC field are displayed in Figure 3. It is interesting to note that the two different behaviors of the pinning energy as the field are increased: up to about 4 Tesla, the slope of the U values is low and constant, but as the field further increases, the pinning energy values show a strong decreasing trend. Moreover, the U(H\({}_{\rm DC}\)) curve can be fitted by the power law \(U(H)\propto\) H\({}^{-\alpha}\) where the vortex pinning regime operating in the sample is indicated by the exponent \(\alpha\). Specifically, for a value of \(\alpha\approx 0\), we are considering a single vortex pinning regime [49], while for \(\alpha>0.5\), we are in a collective pinning regime [51]. In our case, we have \(\alpha=0.25\) for 0 T \(<\) H \(\leq\) 4 T, indicating a single vortex pinning regime with a very low vortex interaction. For H \(>\) 4 T, \(\alpha\) shows a much larger value equal to 3.6, indicating a collective pinning regime with very strong vortex interactions. This is in agreement with the inter- and intragranular hypothesis because, for lower magnetic fields, the interaction among the vortices is low (single vortex) because the number of vortices that can be accommodated in the normal links among the grains is low as well as their interaction, because each vortex is pinned by a defect. For higher fields, the number of vortices increases, forming bundles of vortices predicted in the collective pinning theory [49]. In this case, their interaction is much higher than the single vortex state and, together with the high values of the magnetic field, leads to a fast decrease in the pinning energy as evidenced in Figure 3.  
Figure 2: Critical current density as a function of frequency at 77 K at different DC fields. The slopes of the linear fits (their values are reported on the fitting lines) allow for estimating the values of the pinning potential U.