## 3 Results and Discussion
### Resonant Ultrasound Spectroscopy
The resonance spectra were traced for the cylindrical samples noted corresponding to \(x=0.09\), \(0.13\) and \(0.17\) Ce concentrations in the samples (\(\pi 2\), \(\pi 3\), \(\pi 4\)), whose properties have been presented in Table 4. Due to the fact that the samples \(\pi 2\)-\(\pi 4\) are axisymmetric, isotropic, and homogeneous [25], we can conclude that every mode observed on samples must fall into the following three categories [33]:  
* Torsional axisymmetric pure share motion consisting of rigid rotation of rings of materials around the sample axis. The frequency of these modes depend strongly on the sample's shear velocity
* Extensional axisymmetric mixtures of compression and shear modulus
* Flexural modes along paths that are tilted with respect to the cylinder axis. These modes occur in pairs named doublets, with the same resonance frequency. Figure 8 shows the resonance spectra for the samples \(\pi 2\)-\(\pi 4\) in the frequency range \(80\) kHz to \(250\) kHz. The RUS spectrum describes a large amplitude response detected when the frequency corresponds to one of the samples eigenfrequencies. Figure 7: X-ray diffractograms corresponding to (**a**) \(x=0.09\); (**b**) \(x=0.13\); (**c**) \(x=0.17\); (**d**) distractograms presenting the dependence of intensity (I) on interplanar distances (**d**) and Ce concentration (\(x\)). The vertical bars correspond to the monoclinic (M) and tetragonal (T) phases (FullProf program was used for determination of parameters). SP represents the foreign phase. Figure 9 describes the typical response of one from the three tested samples, namely \(\pi 2\), in a frequency range comprised between 80 and 250 kHz chosen based on our preliminary analysis performed on similar materials. It is immediately observed that for the three samples, there is a pronounced amplitude response in intervals of frequency corresponding to the samples eigenfrequency. The amplitude responses in the swept frequency range are in tight connection to the sample's properties, especially the ones related to the density. The maximum displacements appear as red patches in the above representations, the intermediate ones appear in yellow and green and the minimum ones appear in blue. It is noticeable that for the analyzed frequencies, the maximum displacements take place at the edges of the cylinders in opposite directions, with a minima in the middle of the samples. The eigenfrequency intervals exhibit a slight shift towards smaller values as the density of the samples decrease (see Table 3 and Figure 8). Resonant ultrasound spectroscopy is a reliable technique for monitoring structural modifications related to samples density. Figure 8: Resonance ultrasound spectra for samples \(\pi 2\)–\(\pi 4\). Figure 9: Resonant modes for sample \(\pi 2\): (**a**) extensional mode, \(f\) = 163 kHz; (**b**) extensional mode, \(f\) = 185 kHz; (**c**) flexural mode \(f\) = 208 kHz; (**d**) flexural mode \(f\) = 214 kHz. In Figure 9, we are showing the simulation of deformations for two extensional modes (a,b) and two flexural modes (c,d) for sample \(\pi\)2. The simulations have been made using the finite element method using SolidWorks 2011 (Dassault Systemes SolidWorks Corporation, Waltham, MA, USA), Simulation-Frequency toolbox [36] using a solid mesh type, curvature based mesher, with a total number of 13145 nodes and 8777 total elements. The resonance frequencies obtained by simulations could give us information similarly to the ones experimentally obtained, that could be chosen at full width at half maximum of the amplitude [33]. It is important to specify that the simulated information is useful in order to determine which of the resonances are observables for an investigated spectrum.