# Ultrasonic Attenuation of Ceramic and Inorganic Materials Using the Through-Transmission Method
## 4 Results and Discussion
### Hard Ceramics
This difficulty also extends to YSZ and polycrystalline alumina. Carbides and nitrides have fine microstructures with typical grain sizes of under 10 \(\upmu\)m [65; 66; 67; 68; 69; 70]. This appears to be the primary reason for their relatively low attenuation since Rayleigh scattering derived by Mason [50] is proportional to the third power of the size of scattering center. Converting [PERSON]'s equation to an attenuation coefficient, \(\upalpha\), one obtains  
\[\upalpha=7.51\times 10^{4}\,\mathrm{S}\,\mathrm{L}_{\mathrm{c}}^{3}f^{4}/\mathrm{v }^{4}, \tag{14}\]  
where S is anisotropy-induced scattering power of a grain and Lc is grain diameter. As noted above, dislocation damping is also expected to be low in materials with Vickers hardness exceeding 1000. The WC and Si\({}_{3}\)N\({}_{4}\) samples had Vickers hardness of 1755 (1 kg load) and 1170 (300 g load), while B\({}_{4}\)C was reported to have 3200 [71] and 1430-2300 for Si\({}_{3}\)N\({}_{4}\)[64; 72]. In B\({}_{4}\)C, sintering and hot pressing slightly decrease carbon content and also leave the intergranular amorphous phase [73]. Its anisotropy factor of 0.80 suggests low attenuation from grain boundaries, even with the presence of the intergranular glassy phase. In fact, attenuation due to Rayleigh scattering was absent, possibly due to the filling of remaining voids. The tungsten carbide sample is actually a cermet, containing a 16% Co binder. Typical microstructures of WC of the same density as the present test sample consisted of 0.3-5 \(\upmu\)m blocky WC grains in the Co matrix [65; 55]. The difference of their acoustic impedances is more than a factor of two (ZWC = 98.3 and ZCo = 42.0 Mrayl). Thus, the Co binder phase is expected to scatter propagating waves. However, low Rayleigh scattering effects were observed as listed in Table 3. Attenuation of WC with 16% Co was given as 17.9 dB/m at 10 MHz in [30], which is about 18 times lower than \(\upalpha\) = 325 dB/m at 10 MHz of Test No. 3 (cf. Table 3). The reported \(\upalpha\) for WC also had \(f^{2.3}\)-dependence [30], giving \(\upalpha\) of 0.082 dB/m \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Number** & **Material** & **E \({}^{*}\)** & **G \({}^{*}\)** & **v** & **C\({}_{11}\)** & **C\({}_{33}\)** & **C\({}_{44}\)** & **C\({}_{12}\)** & **C\({}_{13}\)** & **Thickness** & **Density** & **Notes** \\ \hline  & & **GPa** & **GPa** & & **GPa** & **GPa** & **GPa** & **GPa** & **mm** & **Mg/m\({}^{3}\)** & \\ \hline C46c & Sapphire (R-cut) & 386.0 & 170 & 0.248 & 497.6 & 498.1 & 147.2 & 162.6 & 117.2 & & 3.98 & [62] \\
9 & Bonded SiC & 58.3 & 25.3 & 0.154 & & & & & 8.87 & 2.32 & This study \\
10 & Bonded Emery & 10.6 & 4.38 & 0.206 & & & & 7.24 & 2.63 & This study \\ \hline  & & & & & & & & & & & \\ \hline \end{tabular} \({}^{*}\)E = (C\({}_{11}\) + 2C\({}_{12}\))/(C\({}_{11}\) + C\({}_{12}\)), G = C\({}_{44}\). \({}^{*}\)C\({}_{66}\) in parentheses.  
\end{table} 
Table 4: _Cont._at 1 MHz for WC if the same slope extends downward. The lower \(\alpha\) values in WC by the ASTM C1332 method were consistent as in the cases of SiC and Si\({}_{3}\)N\({}_{4}\), discussed in Section 1 [33; 34; 35; 36]. In fact, only a few metallic alloys (Al-2011, Al-2024 and Mg-AZ61) had that level of low attenuation of \(\sim\)18 dB/m at 10 MHz. Considering the absence of ringing of a hanging WC rod upon hitting, it is difficult to accept attenuation comparable to or lower than that of low-damping Al alloys. Silicon nitride showed two to three times higher longitudinal attenuation than the two carbides, B\({}_{4}\)C and WC. On the other hand, transverse attenuation was the lowest among this group of hard ceramics. With the addition of SiC, the longitudinal attenuation coefficient was reduced from 84 to 41 dB/m/MHz, but the transverse attenuation coefficient increased from 2.5 to 18.5 dB/m/MHz. Additionally, Rayleigh scattering effects were higher for Si\({}_{3}\)N\({}_{4}\) and Si\({}_{3}\)N\({}_{4}\) + 10%SiC in transverse mode.