## 2 Thin Film Deposition and Characterization by X-ray Diffraction
### Characterization of Films by Optical Methods and Atomic Force Microscopy
(reported in the third column in Table 2), we reveal the values of the longitudinal acoustic velocities in these films, \(v_{S}~{}=~{}f_{0}\lambda_{0}/2n\), presented in the last column of Table 2.  
Optical and Acoustic Anomalies Accompanying Structural Transformation of BPFMO Films Deposited on Si  
According to our measurements (Table 2), the refractive index exhibits strong maximum at x = 0.15 Pr substitution, which correlates well with the expected position of the structural transition. The maximum measured value of \(n\) is larger by 25% and 20% than the values for the measured neighbor Pr substitutions of x = 0.10 and 0.20, respectively (Figure 6). Correspondingly, the dielectric constant, under the assumption of the magnetic constant (permeability) independent of Pr substitution, is larger in the phase transition point by 57% and 44% than for the neighbor measured Pr substitutions x = 0.10 and x = 0.20, respectively. The sharpness of the revealed anomaly in the dielectric constant is comparable to those measured earlier in these films for remnant magnetization and polarization (see Figure 5(b) in Ref. [13]). Thus, our measurements provide for the first time information on the optical anomaly in BPFMO thin films deposited on Si substrate. Additionally, the measured large values of the optical refractive index in BPFMO films (Table 2) provide an additional complexity/challenge in disentangling the refractive index and the sound velocity by only using the TDBS technique. For example, the magnitude of the refractive index at the structural transition, i.e., for 15% Pr substitution, is very large: \(n~{}=~{}3.22\). Thus the angle \(\phi\) between the directions of the propagations of the CAP and of the probe light inside the material, which controls the BF, even for \(\psi\) = 52\({}^{\circ}\) angle of probe incidence, is rather small (\(\phi~{}=~{}arcsin[(\sin\psi)/n]\approx 14^{\circ}\)). The measured acoustic velocities in the films with different Pr substitution concentrations (Table 2) demonstrate the highly non-monotonic velocity variation in the vicinity of the rhombohedral-tetragonal structural phase transition expected around the Pr substitution concentration of \begin{table}
\begin{tabular}{c c c c c c c} \hline
**1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline Pr substitution & \(H\) [nm] & \(f_{0}\) [GHz] & \(H_{\text{AFM }}\) [nm] & \((nH)_{\text{opt}}\) & \(n=~{}\frac{(nH)_{\text{opt}}}{H_{\text{AFM }}}\) & \(v_{S}=\frac{\xi_{3\alpha}}{2n}\) \\ concentration & nominal & \((\psi=0^{\circ})\) & \(H_{\text{AFM }}\) [nm] & \((nH)_{\text{opt}}\) & \(n=~{}\frac{(nH)_{\text{opt}}}{H_{\text{AFM }}}\) & \(v_{S}=\frac{\xi_{3\alpha}}{2n}\) \\
10\% & 600 & 17.0 \(\pm\) 0.7 & 679 \(\pm\) 18 & 1746 \(\pm\) 8 & 2.57 \(\pm\) 0.08 & 2672 \(\pm\) 193 \\
15\% & 600 & 17.7 \(\pm\) 0.7 & 529 \(\pm\) 11 & 1702 \(\pm\) 10 & 3.22 \(\pm\) 0.09 & 2227 \(\pm\) 151 \\
20\% & 600 & 25.1 \(\pm\) 0.3 & 649 \(\pm\) 18 & 1737 \(\pm\) 10 & 2.68 \(\pm\) 0.09 & 3784 \(\pm\) 172 \\ \hline \end{tabular}
\end{table} 
Table 2: Evaluation of the parameters of (Bi\({}_{1-x}\)Pr\({}_{x}\))(Fe\({}_{0.95}\)Mn\({}_{0.05}\))O\({}_{3}\) films deposited on Si by combining time-domain Brillouin scattering (TDBS), optical reflectivity, and atomic force microscope (AFM) measurements. Figure 6: Dependencies of the optical refractive index and of the longitudinal sound velocity on the Praseodymium substitution concentration in (Bi\({}_{1-x}\)Pr\({}_{x}\))(Fe\({}_{0.95}\)Mn\({}_{0.05}\))O\({}_{3}\) films deposited on Si substrate. x = 0.15. Although there are no published data for the direct comparison with our measurements, our observations in BPFMO films (Figure 6) follow the tendency revealed for BFO films in Ref. [33], i.e., substantial increase in the sound velocity in the transition from rhombohedral to tetragonal phase (see Introduction). More importantly and originally, the measurements in Table 2 reveal a very sharp minimum in the longitudinal sound velocity at the structural phase transition for x = 0.15.