## 3 Results and Discussion  
The composition of the LiNbO\({}_{3}\):Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%) crystal on the phase diagram of the Nb\({}_{2}\)O\({}_{5}\)-Li\({}_{2}\)O system is within the homogeneity region (solid solution region). Therefore, the unit cell of the LiNbO\({}_{3}\):Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%) crystal, such as the unit cell of the ferroelectric phase of the nominally pure LiNbO\({}_{3}\) crystal, is characterized by the space symmetry group C\({}_{3V}^{6}\) (R3c) and contains two formula units (10 atoms) [2]. In this case, the doping of the Gd\({}^{3+}\) and Mg\({}^{2+}\) cations in the structure of the LiNbO\({}_{3}\) crystalic the order of alternation of the Li\({}^{+}\) and Nb\({}^{5+}\) cations, and vacancies (V) along the polar axis, and it distorts the geometry of the oxygen-octahedral MeO\({}_{6}\) clusters, which is characteristic of a nominally pure crystal of congruent composition. The main Li\({}^{+}\) and Nb\({}^{5+}\) ions, as well as the doping of the Mg\({}^{2+}\) and Gd\({}^{3+}\) ions, occupy the C\({}_{3}\) position, while the O\({}^{2-}\) ions occupy the C\({}_{1}\) position. The phonon dispersion curve thus has 30 vibrational branches, of which 27 are optical and 3 are acoustic. The optical vibrational representation of a LiNbO\({}_{3}\) crystal has the following form:  
\[\Gamma=5\text{A}_{1}(\text{z})+5\text{A}_{2}+10\text{E}(\text{x},\text{y}).\]  
In Raman scattering and IR absorption at **k** = 0 (at the center of the Brillouin zone), active 4A\({}_{1}\)(z) + 9E(x, y) dipole-active fundamental vibrations occur, respectively, along and perpendicular to the polar axis. Due to the polar nature of all optical vibrations in a crystal, they are split into longitudinal (LO) and transverse (TO). Thus, in the Raman spectra, under the condition of the propagation of phonons along the main crystallographic axes, taking into account the LO-TO splitting, 26 lines corresponding to fundamental phonons should appear [2]. There are also A\({}_{1}\)(z) + E(x, y) acoustic and 5A\({}_{2}\) optically inactive fundamental vibrations that should not be detected in the Raman and IR absorption spectra. It follows from the form of the Raman tensors [16] that only nondegenerate phonons of A\({}_{1}\)(z) symmetry appear in the polarization (\(zz\)); in polarizations (\(xy\)), (\(xz\)), (\(yx\)), (\(yz\)), (\(zx\)), (\(zy\))--only doubly degenerate phonons of E(x, y)--symmetry types. In polarizations (\(xx\)) and (\(yy\)), phonons A\({}_{1}\)(z) and E(x, y) of symmetry types must be present simultaneously. Figures 2 and 3 show the Raman spectra of the LiNbO\({}_{3}\):Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%) crystal in some backscattering geometries, recorded upon excitation by laser lines with a wavelength of 532 and 785 nm, respectively. The experimentally observed frequencies and their assignments are presented in Table 2. The assignment of experimentally recorded frequencies was carried out on the basis of widely cited articles [36; 37], in which crystals of various stoichiometry, powders, and solid solutions were studied using RS and IR spectroscopy at various temperatures. For some registered Raman lines, there is an alternative assignment, shown in parentheses. As this table shows, we did not assign all registered Raman lines. Thus, upon the excitation of Raman radiation via laser radiation in the near-IR range (785 nm), a line with a frequency of 615 cm\({}^{-1}\) is observed. This line is close to the calculated value (610 cm\({}^{-1}\)) for the 4A\({}_{1}\)(z)TO mode [37]. In one of the first works on Raman spectroscopy in lithium niobate [38], a line with a frequency of 603 cm\({}^{-1}\) was observed, which the authors attributed to an A\({}_{1}\)(z)TO + E(x, y)TO vibration. Other theoretical calculations [39; 40; 41] refer to the 603 cm\({}^{-1}\) line for the fully symmetrical fundamental A\({}_{1}\)(z)TO-mode.