# The Dependence of Ultrasonic Velocity in Ultra-Low Expansion Glass on Temperature
## 5 Results and Discussion
### Measurement and Analysis of \(c_{\mathrm{L}}\)-T Data
#### 5.2.2 Analysis of the Change in the \(c_{\mathrm{L}}\) with \(T\)  
The data from Figure 8 were used to directly plot the changes in \(c_{\mathrm{L}}\) as a function of \(T\) according to the exponential fitting in Equation (22). Table 2 displays the results of all six samples, with excellent exponential fits given for each case and with \(\alpha_{\mathrm{L}}\) mostly ranging from 0.000127 to 0.000129, which shows that for ULE materials with very small CTE differences, their \(\alpha_{\mathrm{L}}\) is almost constant. This means that the temperature effect pattern on ultrasonic velocity was essentially the same in ULE glasses with different CTE values. Figure 8: Changes in \(c_{\mathrm{L}}\)(m\(\cdot\)s\({}^{-1}\)) with temperature for all measured samples: (**a**) sample 1#; (**b**) sample 2#; (**c**) sample 3#; (**d**) sample 4#; (**e**) sample 5#; (**f**) sample 6#. Therefore, the average value of these \(\alpha_{\rm L}\) coefficients could be taken as the temperature correction coefficient of ultrasonic velocity in ULE glass. Of course, because of the small difference in the CTE, the \(c^{\prime}\) in Equation (22) showed relatively large fluctuations. In the physical sense, the \(c^{\prime}\) indicates that \(c_{\rm L}\) correposonds to 0 \({}^{\circ}\)C. It was found that as the CTE of the ULE sample increased, \(c^{\prime}\) increased correspondingly, which is consistent with the mechanism of the linear positive correlation between \(c_{\rm L}\) and the CTE. To represent the more pronounced difference in the ultrasonic velocity with increasing temperature for the ULE samples with different CTE values, a new coefficient is defined as follows:  
\[\beta_{\rm L}=\frac{\Delta c_{\rm L}}{c_{\rm L}\Delta T}. \tag{24}\]  
Here, we take the \(c_{\rm L}\) corresponding to 10 \({}^{\circ}\)C of the samples as the basis \(c_{\rm L}\), and calculate the \(\beta_{\rm L}\) via Equation (24) with temperature increments of 5 \({}^{\circ}\)C, i.e., at 15 \({}^{\circ}\)C, 20 \({}^{\circ}\)C, 25 \({}^{\circ}\)C, and 30 \({}^{\circ}\)C. The \(\beta_{\rm L}\) for samples 1#, 2#, and 3# is shown in Figure 9. We noted that for the different temperature ranges, the speed of the ultrasonic velocity changes with the temperature was inconsistent. This is possibly because the elastic modulus also has a temperature coefficient \(\beta_{E}\), which varies in different temperature ranges. The PV value of \(\beta_{\rm L}\) for the three samples was 0.000098, 0.000056, and 0.000101. Considering that the difference was minor, we conclude that the multi-data point fitting method that was used to obtain \(\alpha_{\rm L}\) in a given temperature range in this paper is a more feasible equivalent to an average function and has wider temperature applicability. Samples 4# and 5#, which had the same CTE, were also analyzed for differences in the \(c_{\rm L}\) as the temperature increased, and the results are shown in Figure 10. We found that the two samples had similar patterns of variation in \(\beta_{\rm L}\). This implies that the changing trends in the ultrasonic velocity with temperature was, to some extent, consistent with the \begin{table}
\begin{tabular}{c c c c} \hline
**No.** & **\(\alpha_{\rm L}\)/\(\times\)10\({}^{-6}\)** & **\(R^{2}\)** & **RMSE** \\ \hline
1\# & 127 & 0.977 & 0.7118 \\
2\# & 127 & 0.985 & 0.5718 \\
3\# & 129 & 0.970 & 0.8330 \\
4\# & 127 & 0.966 & 0.8734 \\
5\# & 128 & 0.974 & 0.7718 \\
6\# & 128 & 0.984 & 0.5944 \\ \hline \end{tabular} Note: No.—sample number; \(\alpha_{\rm L}\)—temperature coefficient of ultrasonic longitudinal wave velocity; \(R^{2}\)—\(R\)-square (a measure of goodness of fit).  
\end{table} 
Table 2: Data from the six samples shown in Figure 8 were taken to exponentially fit the relationship between \(c_{\rm L}\) and \(T\). Figure 9: The relationship between \(\beta_{\rm L}\) and temperature for ULE samples with different CTEs. changing trend of the CTE with temperature; however, this assumes that the thickness of the ULE glass sample did not change over a wide temperature range.