# The Dependence of Ultrasonic Velocity in Ultra-Low Expansion Glass on Temperature
## 3 Theoretical Model between Ultrasonic Velocity and Temperature  
Ultrasound is defined as an elastic wave of high frequency that propagates in a medium. Therefore, when an elastic wave propagates in an isotropic medium without being affected by volume stress, the ultrasonic longitudinal wave velocity, denoted as \(c_{\mathrm{L}}\), can be found from the following equation:  
\[c_{\mathrm{L}}=\sqrt{\frac{E(1-v)}{\rho(1+v)(1-2v)}}, \tag{7}\]  
where \(\rho\) is density, \(v\) is [PERSON]'s ratio, and \(E\) is [PERSON]'s modulus. Equation (7) indicates that the \(c_{\mathrm{L}}\) in the material is mainly related to [PERSON]'s modulus, density, and Poisson's ratio. For ULE glass, within the upper limit of 11.5 wt%, slight changes in the TiO\({}_{2}\) content will not change the density or Poisson's ratio but will cause changes to [PERSON]'s modulus [23]. Therefore, the difference in \(c_{\mathrm{L}}\) is mainly determined by the difference in [PERSON]'s modulus. Then, the effect of temperature on the ultrasonic velocity essentially reflects its effect on [PERSON]'s modulus. The microscopic analysis of [PERSON]'s modulus shows that the reaction of a solid to all forces comes from the potential  
Figure 2: The correlation calculation result of a simulated signal: (**a**) the waveform of the simulated signal; (**b**) the correlation coefficient distribution. energy of the interaction between atoms. The interaction potential \(U(r)\) of a pair of two atoms separated by \(r\) can be expressed as:  
\[U(r)=-\frac{A}{r^{n}}+\frac{B}{r^{m}}, \tag{8}\]  
where \(A\), \(B\), \(n\), and \(m\) are all positive constants. The first term represents the energy of attraction, and the second term represents the energy of repulsion. Assuming that the solid is stretched along the crystal axis when the tensile force changes \(\mathrm{d}f\), the interatomic distance \(r\) changes to \(\mathrm{d}r\). At this time, the cross-sectional area \(r^{2}\) of a unit cell is regarded as inconvenient. Then, [PERSON]'s elastic modulus can be expressed as:  
\[E=\frac{\sigma}{\varepsilon}=\frac{\frac{\mathrm{d}f}{r^{2}}}{\frac{\mathrm{d }r}{r}}=\frac{\mathrm{d}f}{r\mathrm{d}r}, \tag{9}\]  
where \(\sigma\) and \(\varepsilon\) represent the stress and strain, respectively. Since the binding force \(f\) of the tensioned solid is only related to the first term of Equation (8), its magnitude is:  
\[f(r)=-\frac{\mathrm{d}U(r)}{\mathrm{d}r}=-\frac{nA}{r^{n+1}}. \tag{10}\]  
Equation (10) is then substituted into Equation (9) by deriving the derivative for \(r\) to obtain  
\[E=\frac{n(n+1)A}{r^{n+3}}. \tag{11}\]  
Assume \(K=(n+1)\)\(A\) and \(Q=n+3\). Equation (11) can be changed to:  
\[E=\frac{nK}{r^{Q}}. \tag{12}\]  
Equation (12) takes the derivative of \(T\) and divides both sides by \(Er^{Q}\) at the same time, shifting the term to obtain:  
\[\frac{\mathrm{d}E}{E\mathrm{d}T}=Q\frac{\mathrm{d}r}{r\mathrm{d}T}. \tag{13}\]  
It is assumed that the distance between atoms still obeys the following rules when a solid is heated and expanded:  
\[r=r_{0}(1+\alpha T), \tag{14}\]  
where \(r_{0}\) is the atomic distance when the absolute temperature \(T_{0}=0\); \(\alpha\) is the linear expansion coefficient of the solid, and its differential definition is:  
\[\left\{\alpha=\frac{1}{r}\frac{dr}{dT},\eta=\frac{1}{E}\frac{dE}{dT}.\right. \tag{15}\]  
The \(\eta\) in Equation (15) is the temperature coefficient of the elastic modulus. Relevant studies have demonstrated that the elastic modulus of ULE glass increases with the increase in temperature, and the increment \(\mathrm{d}E\) of \(E\) has a positive value [23].