## 3 Results and Discussion
This behavior was already observed previously in [8] and may have indicated the possible presence of the so-called Mackie-Thompson (MT) regime of behavior of the temperature dependences of paraconductivity \(\Delta\)s in the system [14]. In the \(2D\) region, two-particle tunneling between layers is excluded, resulting in superconducting and the normal carriers being located directly in the planes of the leading layers. The dominant contribution to the FC in this regime is made by an additional contribution, justified by Mackie-Thompson [14] and determined as a result of the interaction of the fluctuation pairs with the normal charge carriers. This contribution depends on the lifetime of fluctuation pairs and is determined by the processes of pairing for each specific sample. Importantly, the degree of heterogeneity of the sample structure has to be taken into account. According to [8], for samples of a perfect structure:  
\[\rho(\text{T})=\text{AT}^{-1}+\text{BT} \tag{1}\]4  
Here,  
\[\rho(\text{T})=\text{AT}^{-1}+\text{BT} \tag{1}\]5  
are the communication and depairing parameters, respectively. Here, \(l\) is the mean free path, \(\xi_{\text{ab}}\) is the coherence length in the \(ab\)-plane and \(\tau_{\phi}\) is the existence time of fluctuation pairs. In the presence of inhomogeneities in the structure, the \(\Delta\sigma(T)\) dependence is determined by the Lawrence-Doniach (LD) model [13]. The dependence of the coherence length \(\xi_{c}(0)\) on the fluence of electron irradiation is shown in the inset (b) in Figure 5a. On the main panels and inserts (a) in Figure 5a,b shows the dependencies of excess conductivity in the \(ab\)-plane in In\(\Delta\sigma\)-In\(\varepsilon\) coordinates for YBa\({}_{2}\)Cu\({}_{3}\)O\({}_{7-8}\) (Figure 5a and insert (a)) and for Y\({}_{1-2}\)Pr\({}_{2}\)Ba\({}_{2}\)Cu\({}_{3}\)O\({}_{7-8}\) (Figure 5b and insert (a)). The dotted lines show the approximation of the experimental curves by straight lines with slopes \(\alpha_{1}\approx-0.5\) (\(3D\) regime) and \(\alpha_{2}\approx-1.0\) (\(2D\) regime). Arrows show \(2D\)-\(3D\) crossover points. Insets (b) show the values of \(\xi_{c}(0)\), calculated using Equation (5) for different \(\Phi\) or \(z\). It can be seen that \(\xi_{c}(0)\) passes through a maximum at \(\Phi_{m}\approx 80\times 10^{18}\) e/cm\({}^{2}\) (Figure 5a) or \(z_{m}\approx 0.43\) (Figure 5b). Figure 6 shows the dependencies of \(\xi_{c}(0)\) on \(T_{c}\) for all investigated samples. Dark squares show the data obtained previously for YBa\({}_{2}\)Cu\({}_{3}\)O\({}_{7-8}\) film samples at different volumes of \(\delta\)[8]. As is known from the general theory of superconductivity (Bogoliubov-de Gennes--BdG [32]), the relationship between \(\xi_{c}(0)\) and \(T_{c}\) in superconducting compounds obeys the following relation:  
\[\rho(\text{T})=\text{AT}^{-1}+\text{BT} \tag{1}\]6where \(\Delta(0)\) is the order parameter at \(T=0\) K. Since for YBa\({}_{2}\)Cu\({}_{3}\)O\({}_{7-8}\) the value \(2\Delta(0)/k_{\rm B}T_{c}\approx 5\), then, taking \(\xi_{0}=\xi_{\rm c}(0)\), this can be written as  
\[\rho(\text{T})=\text{AT}^{-1}+\text{BT} \tag{1}\]7  
where \(G=2K\hbar v_{F}/(5\pi k_{B})\) and the proportionality coefficient is \(K\approx 0.12\). The dependence \(\xi_{\rm c}(0)\) as a function of \(T_{c}\) is shown in Figure 6 with the red solid line, which indicates that the pairing mechanisms in HTSC films, in this range of temperatures and praseodymium concentrations, obey to a large degree the general theory of superconductivity. Qualitatively similar behavior of analogous dependencies was observed for YBa\({}_{2}\)Cu\({}_{3}\)O\({}_{7-8}\) (circles) and Y\({}_{1-x}\)Pr\({}_{x}\)Ba\({}_{2}\)Cu\({}_{3}\)O\({}_{7-8}\) (triangles) single-crystal samples at \(T_{c}\geq 65\) K and \(55\) K, respectively.