## 2 Materials and Methods
### The Experimental Set-up for Dielectric Constant Measurements  
In this study, impedance measurements were performed to determine the dielectric constant of the device under test (DUT) as well as various other electrical properties. Special AC bridge circuits are used to accurately measure the complex impedance values for the DUT. More specifically, capacitance bridges such as the Schering Bridge (Figure 8) were used to assess the capacitance values of the DUT and subsequently calculate the dielectric constant as  
\[\kappa=\epsilon_{r}=\frac{\epsilon_{m}}{\epsilon_{0}}=\frac{C_{m}}{C_{0}} \tag{18}\]  
where \(C_{m}\) and \(C_{0}\) are the capacitances of the DUT and the air-gapped specimen cell, respectively. Implementation of the capacitance bridge is greatly simplified through the use of advanced LCR measurement devices. These meters are purpose-built to measure the inductance (L), capacitance (C), and resistance (R) of connected devices using a variable AC source. Many LCR meters utilize four BNC connections to independently measure current and voltage. These four connections can be configured as an auto-balancing bridge circuit which quickly and automatically determines the capacitance value for a DUT (Figure 9). For the calculations of the dielectric constant, a thin unpolished sample of PMN-PT (approximately 12 mm \(\times\) 10 mm \(\times\) 0.5 mm) was characterized using a capacitance bridge apparatus. Capacitance values of the PMN-PT sample at multiple temperatures were compared to \"open-cell\" capacitance to determine the dielectric constant as a function of temperature. Figure 8: The capacitance bridge circuit is a useful AC bridge circuit that allows for the precise measurement of impedance values through balancing (unifying) the voltages at nodes B and D.  
Figure 7: Schematic of the experimental set-up used for measuring the transmission of a 1.5 mm thick PMN-PT ceramic slab at three different wavelengths (532 nm, 632.8 nm, and 1550 nm). Those three measurements were used to verify the three points of the transmission spectrum calculated by Equation (17) using the refractive indices defined by the FTIR method. The PMN-PT samples for dielectric measurement require uniformity but can be unpolished. Beginning with one of the thin slices taken from the bulk sample segment, the pointed ends were removed. Additionally, it was necessary to remove the outer edge of the sample due to poor homogeneity in this area which is a result of the hot-press manufacturing process. Following the sample shaping, conductive silver paint was applied to the two parallel faces to serve as electrodes for impedance measurements (Figure 10). The conductive coating applied to either face should help to ensure that the electric field is distributed evenly across the area of the sample. The capacitance bridge can be implemented using a high-accuracy inductance capacitance resistance (LCR) meter operating at 1 kHz. Again, LiNbO\({}_{3}\) single crystal samples served as a control group to support the validity of this methodology. Due to the birefringent nature of LiNbO\({}_{3}\) single crystals, the dielectric constant of the material is orientation-dependent. For thoroughness, the capacitance bridge method was used to calculate the dielectric constant for both x-cut (11) and z-cut (33) LiNbO\({}_{3}\) samples at room temperature. Literature values for the dielectric constants of x-cut and z-cut LiNbO\({}_{3}\) crystals are given as 85.2 and 28.7, respectively [33]. If the dielectric constant of a material is known, an estimate of the capacitance can be calculated as  
\[C_{m}=e_{0}e_{r}\frac{A}{d} \tag{19}\]  
where \(A\) and \(d\) are the area and thickness of the material, respectively, the resulting calculations for the expected capacitance values of the two LiNbO\({}_{3}\) samples are summarized in Table 3, which also includes the physical parameters needed for Equation (19).