## 3 Results and Discussion
Then, the mechanical quality factor (Qm\({}_{\rm mt}\)) at the thickness vibration mode is high, and the impedance at the resonance frequency is low, which is advantageous when driven at low voltage. For the treatment of temporomandibular joint disorders, 3 MHz can be utilized for the deep penetration of tissue. Because 10 MHz has a high frequency, it can be used for facial skin massages with slightly less tissue penetration. As can be seen in Figure 4, the resonance peak was large at around 3 MHz, which is the first resonance, and the phase value showing the efficiency of polarization was also over 67\({}^{\circ}\), presenting excellent characteristics. When the side length of the piezoelectric device was 7.7 mm, the highest phase of 83.82\({}^{\circ}\) appeared. Figures 6 and 7 show electromechanical coupling factors k\({}_{\rm t1}\) and k\({}_{\rm t3}\) with the side length of the piezoelectric device, respectively. The maximum electromechanical coupling factors k\({}_{\rm t1}\) and k\({}_{\rm t3}\) were increased up to 0.555 and 0.1967 when the side length of the piezoelectric device was 7.7 mm and then decreased, while they slightly decreased at the side length = 6.3 mm. These phenomena are because the optimal condition was found at L/T (length/thickness) = 11,which is like the thickness vibration mode, which corresponds to a thickness-to-diameter ratio of about 10 in the IRE standard. Figure 3: The SEM micrographs of Pb\({}_{0.88}\)(La\({}_{0.6}\) Sm\({}_{0.4}\))\({}_{0.08}\)(Mn\({}_{1/3}\) Sb\({}_{2/3}\))\({}_{0.02}\) Ti\({}_{0.98}\) O\({}_{3}\) ceramics (a) sintered at 1200 \({}^{\circ}\)C and (b) sintered at 1230 \({}^{\circ}\)C.  
Figure 4: Resonant properties with varying sintering temperatures in the piezoelectric device. Figure 5: _Cont_. Figure 5: _Cont._  
Figure 8 and 9 show mechanical quality factors \(\mathrm{Qm_{t1}}\) and \(\mathrm{Qm_{3}}\) with the varying side length of the piezoelectric device, respectively. When the side length of the piezoelectric device was 7.7 mm (L/T = 11), excellent properties appeared. As the size of the ceramic substrate increases, its electrode area expands, which reduces resonance resistance at the resonant frequency and can increase the mechanical quality factor \(\mathrm{Q_{mt}}\). Figure 6: Electromechanical coupling factor (\(\mathrm{k_{t1,k_{t3}}}\)) with varying piezoelectric device side lengths. Figure 7: Electromechanical coupling factor (\(\mathrm{k_{t3}}\)) with varying piezoelectric device size lengths. Figure 10 shows the change in dynamic ratio (D.R) in the first and third vibration modes as the side length of the piezoelectric device changes. As the length of the piezoelectric device increased, the dynamic ratio (D.R) increased up to the side length = 7.7 mm and then decreased. Their highest values were 63.7 and 37.7 dB at 7.7 mm in the 1 st and 3 rd vibration modes, respectively, when the length of the piezoelectric device was 7.7 mm (L/T = 11). Figure 8: Mechanical quality factor (Q\({}_{\text{mt1}}\)) with varying piezoelectric device side lengths. Figure 9: Mechanical quality factor (Q\({}_{\text{mt3}}\)) with varying piezoelectric device side lengths. These results are also because the optimal condition in the thickness vibration mode is a thickness-to-diameter ratio of about 10. Here, it can be illustrated that as the high-order mode of the area vibration mode disappeared, unnecessary noise was removed. Finally, when the side length of the piezoelectric device was 7.7 mm, the parameters were suitable for device application as an ultrasonic transducer, with first and third overtone electromechanical coupling factors k\({}_{\text{t}}\) of 0.555 and k\({}_{\text{t3}}\) of 0.196, first and third overtone mechanical quality factors Qm\({}_{\text{tt1}}\) of 381 and Qm\({}_{\text{t3}}\) of 3938, and first and third overtone dynamic ratios (D.Rs) of 63.7 and 37.7 (dB), respectively.