## 3 Results and Discussion
The position of the Anderson line is defined by the resonance condition \(H_{0}=\frac{2\pi\,f}{\gamma}\times\frac{g_{\mathrm{e}}}{g_{\mathrm{R}}}\) (\(f\) is the radio-wave frequency, \(\gamma\) is the electron gyromagnetic ratio, \(g_{\mathrm{e}}\) is the electron \(g\)-value, \(g_{\mathrm{R}}\) is the sensor \(g\)-value) as well as by macroscopic demagnetizing fields \(H_{\mathrm{D}}\)[16] and by the fluctuation shift \(\delta_{\mathrm{d}}D_{\mathrm{A}}\)[17, 18]. The value and direction of the demagnetizing fields depend on the sample shape and orientation according to Equations (5.61)-(5.63) in [16]. For spherical and cubic samples the demagnetizing fields equal zero. The fluctuation shift is caused by the fluctuations in location of dipoles of the nearby sensor. These fluctuations result in the line shift \(\delta_{\mathrm{d}}D_{\mathrm{A}}\) to higher fields with  
\[\delta_{\mathrm{d}}=\frac{1}{\pi}\Bigg{(}\sqrt{3}+\ln\frac{\sqrt{3}+1}{\sqrt{3 }-1}\Bigg{)}=0.1321 \tag{1a}\]  
The Anderson line can be observed for the sensor located in the magnetoisotropic gel, where magnetic particles are uniformly distributed over the gel medium and isolated from each other. An experimental EPR spectrum of the radical in the magnetoisotropic gel (\(M_{\mathrm{G}}=0.45(5)\) G) is represented in Figure 3 together with the Anderson line. The Anderson line fits the experimental line when the line width \(D_{\mathrm{A}}=2.17(5)\) G and the line position \(H_{\mathrm{R}}=H_{0}+\delta_{\mathrm{d}}D_{\mathrm{A}}=3478.89(10)\) G. The spectrometer frequency \(f\)= 9.763559(1) GHz and \(g_{\mathrm{R}}=2.00532(5)\) correspond to \(H_{0}=3478.65(10)\) G. The value of \(D_{\mathrm{A}}\) is close to the theoretically predicted line width \(D_{\mathrm{A}}=2.27(25)\) G, and the difference of \(H_{\mathrm{R}}-H_{0}\) gives the fluctuation shift \(\delta_{\mathrm{d}}D_{\mathrm{A}}=0.24(10)\) G which is in agreement with Equation (1a). The macroscopic demagnetizing field was neglected according to the cubical symmetry of the sample (Section 2). The experimental spectrum of the sensor in magnetoanisotropic gels has an asymmetry and cannot be fitted by the Lorentzian (Figure 4). An attempt to fit the spectrum by the Lorentz function centered at \(H_{\mathrm{R}}=H_{0}+\delta_{\mathrm{d}}D_{\mathrm{A}}\) gives \(H_{0}=3479.28(1)\) G instead of the expected \(H_{0}=3478.65(10)\) G. The line shape transformation and line shift find a natural explanation in the model [9], which takes into account the formation of elongated aggregates of magnetic particles. The theoretical analysis presented in [9] is based on the calculation of free induction decay (FID). The FID function \(F(t)\) is related to the line shape function by the Fourier transformation. In the limit of short time FID coincides with the monopole one:  
\[F(t)=F_{\mathrm{m}}(t)=\exp\left(-\left|D_{\mathrm{m}}\right|^{3/2}-i\delta_{ \mathrm{m}}D_{\mathrm{A}}\,t\right) \tag{2}\]  
where \(D_{\mathrm{m}}=\frac{8\pi}{\left(15\right)^{2/3}}\gamma v_{\mathrm{c}}^{2/3}\) is the monopole line width (\(\
u\) is the linear density of the magnetic moment, \(c_{\mathrm{c}}\) is the number of aggregate centers in unit volume), and \(\delta_{\mathrm{m}}=\frac{3\sqrt{3}}{2\pi}=0.8270\) is such that the monopole shift \(\delta_{\mathrm{m}}D_{\mathrm{A}}\) is equal to the Lorentz field. In the opposite limit of long times \(t\), the structure of aggregates is unimportant, and \(F(t)\) corresponds to the dipole FID of the Anderson line (with consideration for the fluctuation shift):  
Figure 4: EPR spectrum of TEMPOL radical in magnetoanisotropic polyacrylamide gel (\(f=9.763559\) GHz). Dots: experimental spectrum. Solid line: theoretical spectrum calculated for elongated aggregates. Dashed line: theoretical Lorentz line. Figure 3: Electron magnetic resonance (EPR) spectra of TEMPOL radical in magnetoisotropic gel (\(f\)= 9.763559 GHz). Dots: experimental spectrum. Solid line: theoretical description by the Anderson line.