## 3 Results and Discussion
Namely, bands with maxima near the frequencies 1036-1046, 1122-1131, 1202, 1279, 1291, 1351, 1398, 1411, 1522-1525, 1595, 1709, 1720, 1853-1862, and 1963 cm\({}^{-1}\) correspond to the second-order Raman spectra, the frequencies of which are significantly higher than the frequencies corresponding to the fundamental modes of the crystal lattice, located in the range of 150-900 cm\({}^{-1}\). These lines correspond to overtone processes (bound states of optical phonons). Thus, we have found significant differences in the Raman spectra of the LiNbO\({}_{3}\):Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%) crystal obtained upon excitation by laser radiation at wavelengths of 532 and 785 nm. We managed to register the Raman spectra of the first and second orders of the crystal LiNbO\({}_{3}\):Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%). When excited by a near-IR laser line (785 nm), more lines are observed in the recorded second-order Raman spectrum. Differences in the Raman spectra of the LiNbO\({}_{3}\):Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%) obtained upon excitation by laser lines with wavelengths of 532 and 785 nm can be explained via a different mechanism of interaction, with visible and near-IR radiation of microstructural features and structural defects of the crystal, which determine the phononphonon interaction (bound states of phonons) and the anharmonicity of the vibrations of the crystal lattice. Methods for calculating the bound states of phonons in crystals were proposed earlier in the theoretical works [48; 49; 50]. In these papers, many-particle states of phonons are associated with the anharmonicity of vibrations of the crystal lattice. Using [PERSON]'s function method, one can calculate the density of two-phonon states \(\rho_{2}(\omega)\) as follows. In the simplest approximation, known as the quasi-Newtonian, the density of single-particle states \(\rho_{1}(\omega)\) can be written in the following form:  
\[\rho_{1}(\omega)=a\sqrt{\omega_{0}-\omega} \tag{1}\]  
Here \(a=-\frac{\omega_{0}V\sqrt{2\omega_{0}}}{2\pi^{2}g^{3}}\), \(V\)--unit of cell volume, \(\omega_{0}\) is the frequency near the center of the Brillouin zone, and \(s\)--speed of sound. The one-particle Green's function is written as:  
\[D_{1}\bigg{(}\overset{\rightarrow}{k},\omega\bigg{)}=\frac{\omega\bigg{(} \overset{\rightarrow}{k}\bigg{)}}{2}\Bigg{[}\frac{1}{\omega-\omega\bigg{(} \overset{\rightarrow}{k}\bigg{)}+\frac{1}{2}i\Gamma}-\frac{1}{\omega+ \omega\bigg{(}\overset{\rightarrow}{k}\bigg{)}-\frac{1}{2}i\Gamma}\Bigg{]} \tag{2}\]  
Then, the two-particle Green's function is written as follows:  
\[D_{2}\bigg{(}\overset{\rightarrow}{k},\omega\bigg{)}=\frac{2F(\omega)}{1- \frac{1}{2}g_{4}F(\omega)} \tag{3}\]  
where \(g_{4}\) is the anharmonicity constant and \(F(\omega)\) is defined as  
\[F(\omega)=\frac{i}{(2\pi)^{4}}\int d^{3}\overset{\rightarrow}{k}\int D_{1} \bigg{(}\overset{\rightarrow}{k},\omega-\omega^{\prime}\bigg{)}D_{1}\bigg{(} \overset{\rightarrow}{k},\omega\bigg{)}d\omega^{\prime} \tag{4}\]  
Integrating (4) once, we obtain:  
\[F(\omega)=\frac{1}{4}\omega_{0}^{2}a\int\limits_{0}^{\Delta}\frac{\sqrt{ \omega^{\prime}}}{\omega-2(\omega_{0}-\omega^{\prime})+i\Gamma}d\omega^{\prime} \tag{5}\]  
where \(\Delta\) is a small part of the dispersion curve of optical phonons. Finally, we can obtain the final expression for \(\rho_{2}(\omega)\):  
\[\rho_{2}(k,\omega)\approx-\frac{2}{\pi\omega_{0}^{2}}\frac{\text{Im}F(\omega )}{\left[1-\frac{1}{2}g_{4}\text{Re}F(\omega)\right]^{2}+\left[\frac{1}{2}g_{ 4}\text{Im}F(\omega)\right]^{2}} \tag{6}\]  
As can be seen from Figure 3, there are two second-order Raman lines (1853 and 1963 cm\({}^{-1}\)) of LiNbO\({}_{3}\):Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%), the frequencies of which are noticeably higher than the exact value overtone frequency of the mode 4A\({}_{1}\)(z)LO (1752 cm\({}^{-1}\)). Let us apply the methods for calculating the bound states of the optical phonons to describe these lines.