## 3 Results and Discussion
### The Magnetoelectric Effect  
To investigate the magnetoelectric effect, a previously developed method was used, which was described in the publication [28]. The multiferroic magnetoelectric material can be treated as a system composed of two subsystems, namely, a magnetic subsystem and an electric subsystem. The coupling between these subsystems can be treated and described as a certain type of interaction energy. Figure 7: Temperature dependence of the real part of electric permittivity \(\varepsilon^{\prime}\) for the Bi\({}_{7-\chi}\)Gd\({}_{\chi}\)Fe\({}_{3}\)Ti\({}_{3}\)O\({}_{21}\) ceramic materials for x = (0, 0.2, 0.4, 0.6) in a measurement field with a frequency of f = \(\{\)10\({}^{3}\), 10\({}^{4}\), 10\({}^{5}\), 10\({}^{6}\)\(\}\) Hz. The Hamiltonian of the magnetoelectric system can be presented in the form proposed by [PERSON] and [PERSON] [29; 30; 31]. This Hamiltonian consists of three parts, namely (2)  
\[H=H^{m}+H^{e}+H^{me} \tag{2}\]  
where \(H^{m}\) is the Hamiltonian of the magnetic subsystem, \(H^{e}\) is the Hamiltonian of the ferroelectric subsystem, and \(H^{me}\) is the coupling interaction between the two subsystems. In the mean-field approximation, the electric permittivity can be written using the following relationship (3)  
\[\varepsilon(T)=\varepsilon_{0}\big{(}1+\alpha<S_{i}S_{j}>\big{)} \tag{3}\]  
where \(\varepsilon_{0}\) is the electric permittivity of free space, \(\alpha\) is the magnetoelectric coupling coefficient, and \(S_{i}S_{j}\) is the spin-pair correlation. The thermodynamic statistics can be used to receive the average value of the spin-pair correlation. It can be approximately obtained in the following (4):  
\[\left\langle S_{i}\right\rangle=\frac{4e^{4\beta D}\text{sinh}\left(2\beta \widetilde{h}\right)+2e^{\beta D}\text{sinh}\left(\beta\widetilde{h}\right) }{1+2e^{4\beta D}\cosh\left(2\beta\widetilde{h}\right)+2e^{\beta D}\cosh \left(\beta\widetilde{h}\right)}, \tag{4}\]  
where \(\left\langle S_{i}S_{j}\right\rangle\approx\left\langle S_{i}\right\rangle \left\langle S_{j}\right\rangle\), \(\beta=1/k_{B}T\), \(k_{B}\) is the Boltzmann constant, \(D\) is the uniaxial single-ion anisotropy constant, and \(\widetilde{h}\) is the effective mean field acting on the magnetic subsystem. Using Formulas (3) and (4), and the results of measurements of temperature dependences of electric permittivity, it is possible to calculate the values of the magnetoelectric coupling coefficient \(\alpha\), which is a measure of the magnetoelectric effect. The magnetoelectric coupling coefficient was calculated for undoped multiferroic Aurivillius-type material with chemical formula Bi\({}_{7}\)Fe\({}_{3}\)Ti\({}_{3}\)O\({}_{21}\) (Figure 8a) and for doped Aurivillius-type material with gadolinium ions (Gd\({}^{3+}\)) with chemical formula Bi\({}_{7-x}\)Gd\({}_{x}\)Fe\({}_{3}\)Ti\({}_{3}\)O\({}_{21}\) for x = 0.2; 0.4; 0.6 (Figure 8b-d respectively). Figure 8: Temperature dependences of electrical permittivity for the undoped material (**a**) and for the doped with gadolinium ions, materials: x = 0.2 (**b**), x = 0.4 (**c**), x = 0.6 (**d**). Solid lines (in all figures) indicate curves obtained from fitting. The calculated values of the magnetoelectric coupling coefficient \(\alpha\) are listed in Table 6. The weakest magnetoelectric effect is observed in the material doped with gadolinium ions in the amount of x = 0.6 at a frequency of f = 1 kHz. The value of the magnetoelectric coupling coefficient \(\alpha\) for this material is equal to 1.28\(\cdot\)10\({}^{-9}\) s/m. At the same time, the strongest magnetoelectric effect occurs in the material doped with gadolinium ions in the amount of x = 0.2 at a frequency of f = 1 MHz, for which the value of the magnetoelectric coupling coefficient \(\alpha\) is equal to 12.58\(\cdot\)10\({}^{-9}\) s/m. The obtained coupling coefficient values are comparable to the literature data for analogous Aurivilius-type materials, for which the magnetoelectric coupling coefficient is of the order of 10\({}^{-9}\div 10^{-7}\) s/m [6; 32].