## 3 Results and Discussion
The study of the evolution of the magnetic structure of this sample with temperature [48] shows that the G-type antiferromagnetically ordered moments of Fe/Mn ions reorient from the \(c\) axis (this orientation forbids spin canting) [49] to the \(a\) axis (this orientation permits weak ferromagnetism) [49] as the temperature decreases. Stabilization of dissimilar domain structures upon zero field cooling, field-cooled cooling, and field-cooled warming should thus result in the formation of hysteresis between the corresponding curves (Figure 6a-e,g,h). The preceding analysis of the temperature dependencies of magnetization, aimed at clarifying the pattern of changes in the magnetic state of the Bi\({}_{1-y}\)Ca\({}_{y}\)Fe\({}_{1-x}\)Mn\({}_{y}\)O\({}_{3}\) samples below their Neel points, allows us now to return to the consideration of the metamagnetic behavior of the samples containing the \"\(R3c\)\" phase to formulate the most important findings/conclusions of this work. Figures 8 and 9 supplement the previously considered data (Figures 4 and 5) with measurements conducted at \(T\leq 100\) K. The main observations are summarized below. Metamagnetic behavior is not observed in the single-phase samples with the orthorhombic structure, which demonstrate weak ferromagnetic behavior below their Neel points (Figure 8f,i). The effect is likely due to the difference in the crystal electric field between the rhombohedral and orthorhombic samples, which dictates the character of the magnetic anisotropy. Considering that their low-temperature remnant magnetization (Figure 9f,i) is an order of magnitude higher than the spontaneous characteristic of the weak ferromagnetic state of BiFeO\({}_{3}\)-based perovskites [17; 18; 19; 27; 28; 29; 30; 50], the low-temperature magnetic properties of the samples are affected by the presence of the component, which, in the absence of long-range ferromagnetic order detected by neutron diffraction [34], should be characterized as glassy. Contrary to the field-driven transition from the cycloidal to the canted antiferromagnetic phase in pure BiFeO\({}_{3}\)[13], in the Bi\({}_{1-y}\)Ca\({}_{y}\)Fe\({}_{1-y}\)Mn\({}_{x}\)O\({}_{3}\) series, the decrease in temperature shifts the metamagnetic transition toward smaller fields (Figure 8a-e,g), indicating a difference in the origins of these magnetic field-induced transformations. An increase in the Mn concentration, \(x\), leads to a decrease in the magnetic field required to trigger the metamagnetic phase transition in the rhombohedral phase (Figure 8a-c or Figure 8d,e). In contrast, an increase in the Ca content, \(y\), seems to result in the opposite effect (Figure 8a,d,g or Figure 8b,e). Taking into account the compositional dependence of the critical field for the six samples suitable for such an analysis (Figure 8a-e,g), it can be assumed that the concentration of Jahn-Teller ions of Mn\({}^{3+}\), \(n_{Mn^{3+}}=x-y\), could be considered as one of the factors affecting the magnetic anisotropy and, accordingly, the stability of the collinear G\({}_{2}\)-type AFM structure [33; 48], which, as indicated by the  
Figure 9: Enlarged sections of the field dependencies of magnetization measured for Bi\({}_{1-y}\)Ca\({}_{y}\)Fe\({}_{1-x}\)Mn\({}_{x}\)O\({}_{3}\) samples at different temperatures ranging from 2.5 to 300 K: (**a**) \(y=0.1\), \(x=0.35\); (**b**) \(y=0.1\), \(x=0.4\); (**c**) \(y=0.1\), \(x=0.45\); (**d**) \(y=0.15\), \(x=0.35\); (**e**) \(y=0.15\), \(x=0.4\); (**f**) \(y=0.15\), \(x=0.45\); (**g**) \(y=0.175\), \(x=0.35\); (**h**) \(y=0.175\), \(x=0.4\); (**i**) \(y=0.175\), \(x=0.45\). Identical horizontal and vertical scales are used across all panels for direct comparability. vanishingly small values of the remnant magnetization in the single-phase rhombohedral samples (Figure 5a,b,d,e), should be formed near their Neel temperatures.