## 2 Materials and Methods  
The Ce doped zirconia samples have been obtained through the standard ceramic technology [15], a mixture of ceria and zirconia oxides (Zr1-xCeO2 (x = 0-0.17)) being used in proportions established _a priori_. After grinding and pressing, the samples were shaped in form of cylinders. Finally we treated cylindrical samples at a temperature of 1500 \({}^{\circ}\)C for six hours in air. The phase composition, the lattice constants, space group, average size of coherent blocks, microstrains, and positions of cations and anions in the unit cell have been determined using XRD and ND data. For XRD analysis the samples surface were prepared by a standard metallographic technique [16]. XRD data were acquired with a X'pert Pro MPD PANalytical diffractometer (PANalytical Inc., Westborough, MA, USA) and a BRUKER AXS D8--Advance diffractometer (Bragg--Brentano geometry, CuK\(\alpha\), 20 range of 20\({}^{\circ}\)-80\({}^{\circ}\)) (BRUKER AXS Inc., Madison, WI, USA). A powder diffraction software package, which includes the standards of the Crystallography Open Database (COD) [17], was used in order to identify the phase composition of the samples. The microstrains and average size of crystalline blocks of the thin layer from the surface of the bulk samples were obtained using the XRD data, processed by the Rietveld method, PowderCell [18]. LaB\({}_{6}\) was used as standard for XRD measurements (NIST SRM-660--a lanthanum hexaboride powder LaB\({}_{6}\), for line position and line shape). The structural analysis of the Zr1-xCeO2 (x = 0.0, 0.09, 0.13, 0.17) ceramic samples was performed also using the time-of-flight (TOF) High Resolution Fourier Diffractometer (HRFD) at the IBR-2 pulsed reactor in Joint Institute for Nuclear Research, Dubna, Russia [19]. At the HFRD diffractometer, the correlation technique of data acquisition is used, providing a very high resolution of \(\Delta d/d\sim\) 0.001, practically constant in a wide interval of \(d_{\mathrm{hkl}}\) spacings [19]. We collected the high resolution diffraction patterns by using a detector placed at backscattering angles 20 = \(\pm\)152\({}^{\circ}\), \(d_{\mathrm{hkl}}=\) 0.6-3.6 A. It is important to mention that via X-ray diffraction, a layer of typically 10 \(\upmu\)m thickness has been investigated, whereas in the case of neutron diffraction, a thickness of tens of centimeters of a sample has been investigated. Information about microstructural parameters, such as crystalline size (average size of crystalline blocks) and microstrains were obtained from the lines broadening. From the dependence between the square of Full Width at Half Maximum (W) of neutron peaks and the square of the interplanar spacing, \(d^{2}\), described by the Equation (1) [20,21]  
\[W^{2}=C_{1}+\left(C_{2}+C_{3}\right)d^{2}+C_{4}d^{4} \tag{1}\]  
we extracted the microstrain values (\(\varepsilon\)). Here, C\({}_{1}\), C\({}_{2}\) are the refining constants of the resolution function of HRFD obtained from a standard Al\(\mathrm{O}_{3}\) (SRM-676 of NIST, USA); C\({}_{3}\)\(\sim\)\(\varepsilon^{2}\), where \(\varepsilon\) is the microstrain; C\({}_{4}\)\(\sim\) 1/L where L is the size of the coherently scattering blocks. The values of \(\varepsilon\) and L can be determined from the true physical characteristics of line broadening, whose value is defined as the difference between broadening of peaks of the experimental sample and the standard Al\(\mathrm{O}_{3}\). RUS is a complex method that allows the determination of elastic constant and the elements of elasticity matrix for samples with certain shapes [22; 23; 24; 25; 26]. Sample geometry affects data acquisition [26]. For cylinders with a high ratio of length to diameter we have few excitation modes and the spectra are simple [26; 27]. In the case of the studied samples the ratio is around the unit and the spectrum requires more analysis than a long bar. The increasing of spectral complexity supposes adjacent analyses before concluding about these samples.