## 3 Results and Discussion
The difference in the frequency of this line can be attributed to the choice of the calculation model, as well as the presence in our work of a lithium niobate crystal co-doped with magnesium and gadolinium. When excited by laser radiation in the  
visible range (532 nm), a line with a frequency of 732 cm\({}^{-1}\) was registered in the Raman spectrum. Taking into account the double doping of the studied crystal and the error of the used spectrometer, it can be attributed to second-order Raman scattering [42]. Figure 2: Raman spectra in the band of 50–2000 cm\({}^{-1}\) for the backscattering geometry of a LiNbO\({}_{3}\)Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%) crystal obtained upon excitation by a laser line with a wavelength of 532 nm. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{\(\lambda_{0}\) = 532 nm} & \multicolumn{4}{c}{\(\lambda_{0}\) = 785 nm} \\ \cline{2-7}
**Assignment** & \multicolumn{4}{c}{\(\
u_{\text{r}}\), cm\({}^{-1}\)} \\ \cline{2-7}  & \multicolumn{4}{c}{\(-\)} & \(-\) & \(-\) & \(-\) & \(-\) & \(-\) \\  & \(x(zz,zy)x\) & \(y(zz,zx)y\) & \(z(xx,yyy,xy)z\) & \(x(zz,zy)x\) & \(y(zz,zx)y\) & \(z(xx,yyy,xy)z\) \\ \hline
1E(TO) & 152 & 152 & 152 & 153 & 153 & 153 \\
1E(LO) & & & & 186 & 182 & 188 \\
2E(LO) & & & 238 & 237 & 237 & 237 \\
1A\({}_{1}\)(TO) & 250 & 250 & & 253 & 256 & \\
3E(TO) & & & 262 & & & 262 \\
3E(LO) & & & & & 296 \\
4E(TO) & 321 & 321 & 327 & 322 & 322 & 328 \\
5E(TO) & & & & 367 & 365 & 363 \\
6E(LO) & & & & & 296 & \\
6E(LO) & & & & & 367 & 365 & 363 \\
6(E(TO)) & 429 & 432 & 432 & 436 & 433 & 429 \\
8E(TO) & 581 & 581 & 578 & 578 & & \\
4A\({}_{1}\)(TO) & 629 & 632 & & 629 & 629 & \\
4A\({}_{1}\)(LO) & & & & & 2876 & 876 & 875 \\
(9E(LO)) & & & & & 1046 & 1036 & 1041 \\  & & & & & 1122 & & 1131 \\  & & & & & 1202 & & \\  & & & & 1281 & & 1291 & 1279 & 1291 \\  & 1319 & & & & 1351 & 1398 & \\  & & & & & 1411 & & \\  & & & & & 1475 & & \\
1548 & 1512 & 1539 & 1523 & 1525 & 1522 \\  & & & & & 1595 & & \\
1725 & 1734 & 1739 & 1709 & 1720 & 1720 & 1720 \\  & & & & & 1853 & & 1862 \\  & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The frequencies of transverse (TO) and longitudinal (LO) fundamental polar vibrations observed in the Raman spectrum of the LiNbO\({}_{3}\)Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%) crystal obtained in this work, as well as their assignment using [36; 37]. Below are the line frequencies corresponding to the second-order spectrum. Figure 3: _Cont._According to Figure 2, the registered Raman spectra contain intense lines with frequencies: 250, 629-632 cm\({}^{-1}\), corresponding to the A\({}_{1}\)(TO) fundamental vibrations of the symmetry type along the polar \(z\) axis; lines with frequencies 864-872 cm\({}^{-1}\), corresponding to fundamental vibrations A\({}_{1}\)(LO) of the symmetry type, and six lines with frequencies of 152, 262, 321-327, 359-362, 429-432, and 578-581 cm\({}^{-1}\), corresponding to fundamental doubly degenerate vibrations E(TO)-symmetries perpendicular to the polar \(z\) axis. It is important to note that in the spectra of the LiNbO\({}_{3}\):Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%) crystal in the range 1000-2000 cm\({}^{-1}\), upon excitation by a laser line in the visible range (532 nm), we found three broad bands in all the studied scattering geometries. Moreover, one of them has a close spectral position for different geometries. It is important to note that in the Raman spectra of the LiNbO\({}_{3}\):Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%) in the range of 900-2000 cm\({}^{-1}\), upon excitation by a laser line with a wavelength of 532 nm, we observed a number of lines of very low intensity in the involved scattering  
Figure 3: Raman spectra in the band of 50–2000 cm\({}^{-1}\) for the backscattering geometry of the LiNbO\({}_{3}\):Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%) crystal obtained upon excitation by a laser line with a wavelength of 785 nm. geometries (Figure 2).