# Study on Molybdenum-Rhenium Alloy Ultrasonic Resonance Temperature Sensor
## 2 Materials and Methods
### Material Selection and Parameter Design  
The performance of resonant ultrasonic temperature sensors for temperature testing in high-temperature environments depends mainly on the physical properties of the selected waveguide material, including a high melting point, low coefficient of thermal expansion, good acoustic properties, and temperature sensitivity. The physical properties of common waveguide materials are shown in the following Table 1. When sound waves propagate through waveguide materials [17], the energy of their resonant echoes [18] is influenced by the properties of the sensitive materials. This underscores the importance of selecting appropriate materials to ensure the stability and reliability of sensors under extreme conditions. Figure 2: Schematic diagram of resonance. Compared to single crystal waveguide materials such as sapphire [19] and spinel [20,21], metals and their alloys have good ductility and significant advantages in resonator processing, so metal alloys are usually chosen as waveguide materials. As a refractory metal, molybdenum has a melting point of about 2620 \({}^{\circ}\)C. However, pure molybdenum is susceptible to oxidation when exposed to high temperatures for a long period of time. To enhance its oxidation resistance, adding 5% rhenium to molybdenum effectively improves its mechanical properties and significantly enhances its stability in high-temperature environments. In addition, the molybdenum-rhenium alloy has good acoustic properties, and the acoustic-wave propagation process can effectively transmit ultrasonic signals [22]. Therefore, the molybdenum-rhenium alloy material was used to prepare ultrasonic temperature sensors. In order to ensure the accurate transmission of the signal, the end of the sensor away from the temperature field will be coupled with the magnetostrictive transducer when constructing the temperature-measurement system. Ultrasonic resonant temperature sensors reflect temperature changes through changes in the resonant frequency produced by the propagation of sound waves. They use echo signals crossing the zero point to determine the resonant state, thereby converting temperature measurements into frequency measurements. The characteristics of the ultrasonic guided wave are influenced by the dimensions of the sensor. Therefore, in designing the sensor, it is essential to select appropriate dimensions--including length, diameter and resonator size--to minimize the impact of size on the ultrasonic guided wave and to enhance the accuracy of the temperature measurements. Length of waveguide rod  
Energy is lost from sound waves as they propagate, including the attenuation of acoustic pressure [23] and intensity. These attenuation rates can be expressed as follows:  
\[P=P_{0}\mathrm{e}^{-\alpha x} \tag{4}\]  
\[I=I_{0}\mathrm{e}^{-2\alpha x} \tag{5}\]  
where \(\alpha\) represents the attenuation coefficient; \(P_{0}\) and \(I_{0}\) are the initial sound pressure and intensity, respectively; and \(P\) and \(I\), respectively, denote the sound pressure and intensity at position \(x\). Considering these attenuations, the length of the waveguide rods is appropriately controlled to avoid an increase in loss as the acoustic wave propagates through the rods. If the waveguide rod is too short, the transducer may be affected by external magnetic fields in high-temperature environments, compromising the stability of sound-wave transmission;  
Diameter size  
Changes in the waveguide rod's diameter and frequency-impact sound-wave propagation, led to variations in group velocity (\(C_{\mathrm{g}}\)) and phase velocity (\(C_{\mathrm{p}}\)) as well as frequency