## 3 Results
### Numerical Study of the Loss Tangent of Dielectric Materials  
Using the experimental setup described in [73; 171], precision measurements provided frequency spectra of the dielectric loss tangent \(\text{tg}^{\text{S}(T)}(\
u)\) for Onot talc crystals \(\text{Mg}_{3}(\text{Si}_{4}\text{O}_{10})(\text{OH})_{2}\) (Figure 6) and gypsum \(\text{CaSO}_{4}\cdot 0.5\text{H}_{2}\text{O}\) (Figure 7). The Onot talc samples were pre-calcined at a temperature \(T_{\text{ac}}=923\,\text{K}\), while the calcination temperature for gypsum was set to \(T_{\text{ac}}=393\,\text{K}\), as in [4]. The frequency range of the alternating polarizing field varied from \(\
u_{\text{min}}=30\,\text{kHz}\) to \(\
u_{\text{max}}=300\,\text{MHz}\). The numerical values of the measured \(\text{tg}^{\text{S}(T)}(\
u)\) ranged within \(\text{tg}^{\text{S}(T)}(\
u)=\left(0.1-10\right)\times 10^{-3}\). The polarization temperature of the crystal was constant during the experiments. Figure 6: Frequency spectra of the dielectric loss tangent in Onot talc crystals. The frequency spectra of the real part of the complex dielectric permittivity \(\mathrm{Re}\Big{(}\epsilon^{(\mathrm{T})}(\
u)\Big{)}\) were also measured for Onot talc crystals (Figure 8) and gypsum (Figure 9). The frequency range in these spectra was the same as for \(\mathrm{tg}\delta^{(\mathrm{T})}(\
u)\), i.e., from \(\
u_{\mathrm{min}}=30\,\mathrm{kHz}\) to \(\
u_{\mathrm{max}}=300\,\mathrm{MHz}\). The numerical values of \(\mathrm{Re}\Big{(}\epsilon^{(\mathrm{T})}(\
u)\Big{)}\) were within the range \(\mathrm{Re}\Big{(}\epsilon^{(\mathrm{T})}(\
u)\Big{)}=4.5-5.5\). The polarization temperature of the crystal was constant during the experiments. Figure 8: Frequency spectra of the real component of the complex dielectric permittivity \(\mathrm{Re}\Big{[}\epsilon^{(\mathrm{T})}(\
u)\Big{]}\,\mathrm{B}\) in Onot talc crystals. Figure 7: Frequency spectra of the dielectric loss tangent in gypsum. Theoretical frequency graphs for the dielectric loss tangent \(\text{tg}\delta^{(\text{T})}_{\text{th}}(\
u)\) were constructed based on numerical calculations using expression (150) or its equivalent expression (151), which is the most general for the quasi-classical phenomenological model. This model is based on solutions of the nonlinear Fokker-Planck Equation (57) combined with Poisson's Equation (58), accounting for relaxation parameters (120), (121), (122). The functions \(\text{tg}\delta^{(\text{T})}_{\text{th}}(\
u)\) were calculated using Formula (151) around the frequencies \(\text{vexp}\),\(i\),\(\max\)vexp,\(i\),\(\max\) corresponding to the maxima of \(\text{tg}\delta^{(\text{T})}_{\text{exp},i}(\
u)\) on the experimental graphs 1-4. This was performed in conjunction with the method of comparison function minimization (MCF-method) [73; 171; 172], where \(i\) denotes the index of a maximum on one of the graphs 1-4. The domain restrictions for the measured \(\text{tg}\delta^{(\text{T})}_{\text{th}}(\
u)\) values were applied during calculations of \(\text{tg}\delta^{(\text{T})}_{\text{th},i}(\
u)\), as in the model described in [3], to enhance the accuracy of the MCF method when minimizing the comparison function between theory and experiment. The implementation of the MCF method for \(\text{tg}\delta^{(\text{T})}_{\text{exp},i}(\
u)\) and \(\text{tg}\delta^{(\text{T})}_{\text{th},i}(\
u)\) involves computational parameter optimization. These parameters include the activation energy of the relaxator at a chemical bond (in HBC, the activation energy of a proton at a hydrogen bond) \(\text{U}_{0,i}\); the natural oscillation frequency of the relaxator in the potential well \(\
u_{0,i}\); the width of the potential barrier \(\delta_{0,i}\); the equilibrium concentration of relaxators in the crystal \(n_{0,i}\); and the lattice constant \(a_{i}\). Optimization is performed using custom-developed computational programs operating around the field frequency \(\
u_{\text{exp},i,\max}\) (in a set of points localized near the experimental maximum).