Understanding Complex Interplay among Different Instabilities in Multiferroic BiMn7O12 Using 57Fe Probe Mössbauer Spectroscopy

Here, we report the results of a Mössbauer study on hyperfine electrical and magnetic interactions in quadruple perovskite BiMn7O12 doped with 57Fe probes. Measurements were performed in the temperature range of 10 K < T < 670 K, wherein BiMn6.9657Fe0.04O12 undergoes a cascade of structural (T1 ≈ 590 K, T2 ≈ 442 K, and T3 ≈ 240 K) and magnetic (TN1 ≈ 57 K, TN2 ≈ 50 K, and TN3 ≈ 24 K) phase transitions. The analysis of the electric field gradient (EFG) parameters, including the dipole contribution from Bi3+ ions, confirmed the presence of the local dipole moments pBi, which are randomly oriented in the paraelectric cubic phase (T > T1). The unusual behavior of the parameters of hyperfine interactions between T1 and T2 was attributed to the dynamic Jahn–Teller effect that leads to the softening of the orbital mode of Mn3+ ions. The parameters of the hyperfine interactions of 57Fe in the phases with non-zero spontaneous electrical polarization (Ps), including the P1 ↔ Im transition at T3, were analyzed. On the basis of the structural data and the quadrupole splitting Δ(T) derived from the 57Fe Mössbauer spectra, the algorithm, based on the Born effective charge model, is proposed to describe Ps(T) dependence. The Ps(T) dependence around the Im ↔ I2/m phase transition at T2 is analyzed using the effective field approach. Possible reasons for the complex relaxation behavior of the spectra in the magnetically ordered states (T < TN1) are also discussed.


Introduction
The variety of structural and magnetic phase transitions in the so-called quadruple perovskite BiMn 7 O 12 [1][2][3][4] and its solid solutions, such as BiMn 7−x Cu x O 12 (0 < x ≤ 1.1) [5,6], has attracted strong interest from researchers.Numerous transitions of different origins are associated with the presence of Mn 3+ and Bi 3+ cations in the crystal lattice of these oxides, which promotes structural instability [7][8][9][10].High-spin Jahn-Teller (JT) cations Mn 3+ (d 4 ) in the non-distorted octahedral oxygen surrounding possess an energetically degenerate configuration, e g 1 , which provokes, along with a local distortion of polyhedra (MnO 6 ), a cooperative interaction in which the JT centers Mn 3+ proper, often called orbital ordering [7,8,11,12].The distortion driven by easily polarized Bi 3+ cations, containing the 6s 2 lone electron pair, results in off-centric cation displacements and the formation of local electric dipoles, which are responsible for the ferroelectric properties of many Bi-containing perovskites [7,10].
Non-zero magnetization (M) co-existing with electrical polarization (P s ) is characteristic of multiferroics, which can be grouped into two types [13].In the first group (type-I multiferroics), M and P s are independent of each other, i.e., magnetism and ferroelectricity have different origins.In the second group (type-II multiferroics), M and P s demonstrate a strong mutual influence, i.e., magnetism induces ferroelectricity [13].Such a magneto-electric coupling strongly correlates with local distortions, and, thus, the roles of local polar and magnetic clusters must be studied.Therefore, these compounds are increasingly studied not only by using diffraction and magnetic methods but also using local nuclear resonance techniques, such as NMR [14][15][16][17][18][19][20][21], NQR [22,23], muon spectroscopy [24][25][26], the spectroscopy of perturbed angular γ-γ correlations [27,28], and Mössbauer spectroscopy [9, 29,30].The temperature dependences of hyperfine magnetic fields (B hf ) and principal components {V ii } X,Y,Z of the tensor of the electrical field gradient (EFG) gained using these methods reproduce the corresponding dependences M(T) and P s (T).Meanwhile, the relationship B hf = αM is usually linear for 57 Fe, 55 Mn, and 53 Cr nuclei, and the dependencies of the EFG tensor parameters V ii = f(P s ) and η = f ′ (P s ) (where η = (V XX − V YY )/V ZZ is the asymme- try parameter) demonstrate nontrivial behavior.In some works, quadratic dependences V ii = a + bP s 2 and η ∝ P s 2 are used for approximation [31][32][33].However, such an approach is rather formal and does not allow for associating the hyperfine parameters of different resonant nuclei with the structural data and physical characteristics of the compounds under study.
In the present work, we present Mössbauer-based research on the hyperfine interactions of 57 Fe probes in perovskite BiMn 7 O 12 .This manganite demonstrates spontaneous electrical polarization in the temperature range of T < T C ≈ 440 K, whereas, at lower temperatures (T < T N1 ≈ 59 K), it acquires a magnetically ordered state and multiferroic properties [2].In contrast to perovskites ABO 3 , in their "quadruple" analogs (A ′ A ′′ 3 )B 4 O 12 , the sublattice A is divided into two sublattices formed by cations A ′ with a high coordination number (CN = 8-12) and by using JT cations A ′′ = Cu 2+ and Mn 3+ , which are located in the square planar oxygen coordination [34].In the case of (Bi 3+ Mn 3+ 3 )[Mn 3+  4 ]O 12 , these sublattices consist of the cations A ′ = Bi 3+ and A ′′ = Mn 3+ , whereas the sublattice with a distorted octahedral oxygen coordination (B) consists of the JT cations Mn 3+ that directly initiate orbital ordering (cooperative JT effect) [8].A combined effect of two cations (Mn 3+ ) B and (Bi 3+ ) A ′ , of which the latter contains the stereochemically active lone-pair electrons [7,9,10], results in a whole cascade of structural and magnetic phase transitions in BiMn 7 O 12 (Figure 1) [3].However, the mechanisms and driving forces of these phase transitions are still widely discussed despite abundant experimental data and theoretical studies [35].
According to the results of the earlier Mössbauer studies of perovskites AMn 6.96 57 Fe 0.04 O 12 (A = Ca, Sr, Cd, Pb) [35][36][37], the 57 Fe probes are localized in the structure solely in the formal oxidation state "3+", substituting manganese cations only in octahedral sublattice.Moreover, experimental and theoretical studies show that the hyperfine parameters of the 57 Fe spectra reflect the features of the local crystal structure of this class of manganites.It is essential to highlight that the studies involving macroscopic diagnostic methods found the influence of the 57 Fe probes on physical parameters to be insignificant, as well as the patterns of the structural and magnetic transitions in these oxides.Thus, utilizing Mössbauer spectroscopy to probe a more complicated BiMn 7 O 12 system is justified by the current experimental data, and the successful application of this technique is used to study other isostructural compounds of the AMn 7 O 12 family.
Our work aims to qualitatively obtain new information about the local structure of BiMn 7 O 12 and outline the features of the structural, electrical, and magnetic phase transitions.We describe a close interplay between the orbital and spin degrees of freedom, which is characteristic of systems with a strong electron correlation.
The Results and Discussion section of the manuscript is divided into several parts.The first part is devoted to analyzing the effect of 57 Fe probes on structural (T 1 , T 2 , and T 3 ) and magnetic (T N1 and T N2 ) transitions in BiMn 7 O 12 (Figure 1).Based on theoreti-cal calculations of the EFG parameters in the paraelectric range T > T 1 , the second part discusses the crystal chemistry of Bi 3+ cations and their influence on the transition of the bismuth sublattice into the ordered ferroelectric state.In the third part, we consider the dynamic Jahn-Teller effect of Mn 3+ cations in octahedral coordination at intermediate temperatures T 2 < T < T 1 .The fourth part presents an analysis of the temperature dependence of the spontaneous polarization in BiMn 6.94 Fe 0.04 O 12 at temperatures T N1 < T < T 3 and T 3 < T < T 2 .The final part describes the hyperfine magnetic interactions of 57 Fe probes in the magnetically ordered temperature range T < T N1 .Our work aims to qualitatively obtain new information about the local structure of BiMn7O12 and outline the features of the structural, electrical, and magnetic phase transitions.We describe a close interplay between the orbital and spin degrees of freedom, which is characteristic of systems with a strong electron correlation.
The Results and Discussion section of the manuscript is divided into several parts.The first part is devoted to analyzing the effect of 57 Fe probes on structural (T1, T2, and T3) and magnetic (TN1 and TN2) transitions in BiMn7O12 (Figure 1).Based on theoretical calculations of the EFG parameters in the paraelectric range T > T1, the second part discusses the crystal chemistry of Bi 3+ cations and their influence on the transition of the bismuth sublattice into the ordered ferroelectric state.In the third part, we consider the dynamic Jahn-Teller effect of Mn 3+ cations in octahedral coordination at intermediate temperatures T2 < T < T1.The fourth part presents an analysis of the temperature dependence of

Results and Discussion
2.1.Crystallographic, Magnetic, and Thermodynamic Data X-ray diffraction patterns show no additional reflections corresponding to impurity phases (Figure S1).Having been measured at different temperatures, they suggest that the studied sample retains all the crystal modifications characteristic of an undoped (without Fe) quadruple manganite BiMn 7 O 12 [3].The observed reflections at 615 K are associated with the cubic (Im3) BiMn 7 O 12 phase that is stable at T > T 1 (Figure 1a).A part of the reflections split upon transition below T 1 , which corresponds to the monoclinic symmetry I2/m (Figure 1b).The monoclinic phase reflection (242) splits upon the further cooling of the sample (Figure 1c).As noted in [3], the temperature T 3 of the phase transition monoclinic (Im) ↔ triclinic (P 1 ) (Figure 1d) was not detected in the thermodynamic measurements; however, it can be evaluated from the deviation of the α and γ monoclinic unit cell angles from 90 • (Figure S2).The estimated point T 3 ≈ 240 K for BiMn 6.96 Fe 0.04 O 12 is noticeably lower than ~290 K for the undoped BiMn 7 O 12 sample [3].
The BiMn 6.96 Fe 0.04 O 12 powder is characterized by a high degree of crystallinity according to the SEM data (Figure S3).It is inferred from particle agglomeration and the wide distribution of particle sizes ranging from 0.5 to 20 µm.Almost all crystallites have an irregular shape.
The peaks observed in the differential scanning calorimetry (DSC) curves of BiMn 6.96 Fe 0.04 O 12 (Figure 2a) correspond to the phase transitions at the temperatures associated with the cubic ( 3 Im ) BiMn7O12 phase that is stable at T > T1 (Figure 1a).A part of the reflections split upon transition below T1, which corresponds to the monoclinic symmetry I2/m (Figure 1b).The monoclinic phase reflection (242) splits upon the further cooling of the sample (Figure 1c).As noted in [3], the temperature T3 of the phase transition monoclinic (Im)  triclinic (P1) (Figure 1d) was not detected in the thermodynamic measurements; however, it can be evaluated from the deviation of the α and γ monoclinic unit cell angles from 90° (Figure S2).The estimated point T3 ≈ 240 K for BiMn6.96Fe0.04O12 is noticeably lower than ~290 K for the undoped BiMn7O12 sample [3].
The BiMn6.96Fe0.04O12powder is characterized by a high degree of crystallinity according to the SEM data (Figure S3).It is inferred from particle agglomeration and the wide distribution of particle sizes ranging from 0.5 to 20 μm.Almost all crystallites have an irregular shape.
The peaks observed in the differential scanning calorimetry (DSC) curves of BiMn6.96Fe0.04O12(Figure 2a) correspond to the phase transitions at the temperatures T1 ≈   By measuring the heat capacity C P /T(T) in BiMn 6.96 Fe 0.04 O 12 (Figure 2b), we obtained the values of the temperatures of transition to the magnetically ordered states T N2 and T N3 .T N2 reached ~50 K, and the temperature of the third magnetic transition T N3 ≈ 24 K was 3-5 K lower than that of the undoped manganite [4].The third phase transition at T N3 is also clearly seen in the temperature profile of the magnetic susceptibility χ(T) and is typical of antiferromagnets (Figure 2c).The parameters of the Curie-Weiss fit (Figure 2c) are in good agreement with the data obtained for the undoped manganite BiMn 7 O 12 [3].The temperature shift of structural and magnetic transitions upon iron-doping can be caused by the stabilization of a small quantity of iron in the manganite structure, rather than the precipitation of an impurity iron phase or its localization on the crystallite surface.

Mössbauer Data for T > T 1
Figure 3a represents the typical Mössbauer spectra of 57 Fe in BiMn 6.96 Fe 0.04 O 12 measured at high temperatures T > T 1 .At these temperatures, the spectra consist of a quadrupole doublet with small and virtually temperature-independent splitting ∆ ≈ 0.26 mm/s (Figure 4).The value of the isomer shift δ 633K ≈ 0.16 mm/s corresponds to Fe 3+ cations [38], isovalently substituting Mn 3+ in the octahedral positions of Mn2 (Figure 1a).Despite BiMn 6.96 Fe 0.04 O 12 having a cubic structure (Im3) at T > T 1 and the local octahedral anion environment of the Mn2 sites being formally considered to be undistorted, the local symmetry of the oxygen sites explains the non-zero quadrupole splitting of the spectrum (Table 1).Although all the Mn2 sites substituted by Fe 3+ probes are equivalent, the experimental spectra cannot be satisfactorily fitted using one doublet with unbroadened resonant lines.This suggests the presence of a distribution p(∆) of ∆ values (Figure 3a), i.e., that the crystalline environment of the 57 Fe probes is not homogenous.* When processing the spectra, the linewidth Γ was fixed in accordance with the "thin" absorber and properties of the source.<δ> is the mean isomer shift, <∆> is the mean quadrupole splitting, D exp p is the dispersion of the quadrupole splitting taken from the distribution reconstruction procedure.
The EFG parameters were calculated within the "ionic model" to support this assumption.It takes into account monopole (V mon ZZ ) and dipole (V dip ZZ ) contributions from ions that are located in the non-centrosymmetric sites in BiMn 7 O 12 .Having stereochemically active 6s 2 lone-pair electrons, Bi 3+ cations are considered to mainly contribute to V dip ZZ .The lone pair induces the displacement of Bi 3+ cations from their centrosymmetric positions, which is equivalent to inducing the electric dipole moment p Bi .Therefore, only dipole contributions (V dip ZZ,Bi ) from Bi 3+ were taken into account when calculating V dip ZZ using variable p Bi values in further calculations.Additionally, the dipole moments p Bi were considered to be randomly oriented in a crystal lattice since BiMn 7 O 12 is paraelectric at T > T 1 [3].See Appendix A for details.
The inclusion of the dipole contribution from Bi 3+ allows us to achieve a good agreement between the theoretical and experimental values of the quadrupole splitting.The calculated dipole moment p Bi ≈ 1.2 × 10 −29 C•m lays within the range of the corresponding values p Bi for other Bi 3+ oxide compounds [10].Most importantly, even with a random orientation of the p Bi moments, the Mn2 sites become non-equivalent in terms of the induced lattice contribution V dip ZZ,Bi .This is, in essence, the main cause of the observed broadening of the spectra, i.e., the appearance of the p(∆) distribution (Figure 3a).Using the calculated values of ∆ theor for each Mn2 site within the P1 pseudocell (see Appendices A-C), which is characterized by the peculiar relative orientation of the surrounding dipole moments p Bi , we calculated the mean value of the quadrupole splitting as well as the dispersion D theor p = 0.020 mm 2 /s 2 , which was found to be close to D exp p ≈ 0.017 mm 2 /s 2 of the experimental (Table 1) distribution p(∆).broadening of the spectra, i.e., the appearance of the p(Δ) distribution (Figure 3a).Using the calculated values of Δ theor for each Mn2 site within the P1 pseudocell (see Appendices A-C), which is characterized by the peculiar relative orientation of the surrounding dipole moments pBi, we calculated the mean value of the quadrupole splitting as well as the dispersion theor p D = 0.020 mm 2 /s 2 , which was found to be close to exp p D ≈ 0.017 mm 2 /s 2 of the experimental (Table 1) distribution p(Δ).Thus, the Mössbauer data indicate that, in the paraelectric cubic phase of BiMn 6.96 Fe 0.04 O 12 at T > T 1 , Bi 3+ cations exist in a locally distorted environment and retain their electric dipole moments p Bi that are randomly oriented in the cubic lattice.In this case, transitions to the anti-or ferroelectric states at lower temperatures T < T 2 should be accompanied by the ordering of the p Bi dipoles, i.e., they should represent the "order-disorder" phase transitions [39], as an alternative to the "displacive" phase transitions [40].Previously, in refs.[41,42], static dipole moments p Bi are assigned to the lone sp x -hybrid electron pairs of Bi 3+ cations, which are oriented parallel to the direction of the displacement of a bismuth cation from its conventional centrosymmetric site (Figure 5a).Such an approach can qualitatively explain the unusually large thermal ellipsoids of Bi 3+ cations reported in [1,3] for BiMn 7 O 12 at T > T 1 .These ellipsoids may form as a result of the superposition of multidirectional sp x -hybrid pairs, whose randomly oriented lobes create a sphere that manifests itself in the diffraction patterns as unusually large bismuth thermal ellipsoids (Figure 5b).However, it should be noted that this approach is a simplified, albeit illustrative, model that has not been experimentally confirmed for the majority of known Bi(III) phases [43][44][45].Thus, the Mössbauer data indicate that, in the paraelectric cubic phase of BiMn6.96Fe0.04O12at T > T1, Bi 3+ cations exist in a locally distorted environment and retain their electric dipole moments pBi that are randomly oriented in the cubic lattice.In this case, transitions to the anti-or ferroelectric states at lower temperatures T < T2 should be accompanied by the ordering of the pBi dipoles, i.e., they should represent the "order-disorder" phase transitions [39], as an alternative to the "displacive" phase transitions [40].Previously, in refs.[41,42], static dipole moments pBi are assigned to the lone sp x -hybrid electron pairs of Bi 3+ cations, which are oriented parallel to the direction of the displacement of a bismuth cation from its conventional centrosymmetric site (Figure 5a).Such an approach can qualitatively explain the unusually large thermal ellipsoids of Bi 3+ cations reported in [1,3] for BiMn7O12 at T > T1.These ellipsoids may form as a result of the superposition of multidirectional sp x -hybrid pairs, whose randomly oriented lobes create a sphere that manifests itself in the diffraction patterns as unusually large bismuth thermal ellipsoids (Figure 5b).However, it should be noted that this approach is a simplified, albeit illustrative, model that has not been experimentally confirmed for the majority of known Bi(III) phases [43][44][45].Thus, the Mössbauer data indicate that, in the paraelectric cubic phase of BiMn6.96Fe0.04O12at T > T1, Bi 3+ cations exist in a locally distorted environment and retain their electric dipole moments pBi that are randomly oriented in the cubic lattice.In this case, transitions to the anti-or ferroelectric states at lower temperatures T < T2 should be accompanied by the ordering of the pBi dipoles, i.e., they should represent the "order-disorder" phase transitions [39], as an alternative to the "displacive" phase transitions [40].Previously, in refs.[41,42], static dipole moments pBi are assigned to the lone sp x -hybrid electron pairs of Bi 3+ cations, which are oriented parallel to the direction of the displacement of a bismuth cation from its conventional centrosymmetric site (Figure 5a).Such an approach can qualitatively explain the unusually large thermal ellipsoids of Bi 3+ cations reported in [1,3] for BiMn7O12 at T > T1.These ellipsoids may form as a result of the superposition of multidirectional sp x -hybrid pairs, whose randomly oriented lobes create a sphere that manifests itself in the diffraction patterns as unusually large bismuth thermal ellipsoids (Figure 5b).However, it should be noted that this approach is a simplified, albeit illustrative, model that has not been experimentally confirmed for the majority of known Bi(III) phases [43][44][45].

Mössbauer Data for T 2 < T < T 1
BiMn 6.96 57 Fe 0.04 O 12 undergoes the structural transition at T 1 ≈ 590 K, transforming from cubic (Im3) to monoclinic (I2/m) lattice symmetry (Figure 1b) with decreasing temperature.Figure 3b illustrates a typical Mössbauer spectrum of 57 Fe probes in the monoclinic BiMn 7 O 12 , which has the shape of a symmetrically broadened quadrupole doublet.Despite the lowering of the manganite lattice symmetry, the obtained distributions p(∆) show a single maximum that corresponds to the average <∆> value which increases drastically upon decreasing temperature (Figures 3 and 4).Considering the fact that the main contribution to the EFG imposed on the spherical Fe 3+ cations is accounted for by the distortion of their crystalline surrounding (lattice contribution), it is difficult to explain the observed sharp change in the quadrupole splitting with temperature.
The results of the calculation of the EFG parameters for the different sites of Mn 3+ with the monopole contributions from all ions (Bi 3+ , Mn 3+ , and O 2− ), as well as additional dipole contributions from Bi 3+ and O 2− , show that the values V ZZ,Mn4 = 3.76 × 10 20 V/m 2 and V ZZ,Mn5 = 4.21 × 10 20 V/m 2 are close to each other, which is probably responsible for the presence of only one maximum in p(∆) (Figure 3b).As expected, the EFG parameters for both Mn sites are almost independent of temperature.Moreover, it was established that the calculated values ∆ theor Mni ≈ eQV doublet.Despite the lowering of the manganite lattice symmetry, the obtained d tions p(Δ) show a single maximum that corresponds to the average <Δ> value wh creases drastically upon decreasing temperature (Figures 3 and 4).Considering t that the main contribution to the EFG imposed on the spherical Fe 3+ cations is acc for by the distortion of their crystalline surrounding (lattice contribution), it is diff explain the observed sharp change in the quadrupole splitting with temperature.
The results of the calculation of the EFG parameters for the different sites o with the monopole contributions from all ions (Bi 3+ , Mn 3+ , and O 2− ), as well as add dipole contributions from Bi 3+ and O 2− , show that the values VZZ,Mn4 = 3.76 × 10 20 V/ VZZ,Mn5 = 4.21 × 10 20 V/m 2 are close to each other, which is probably responsible presence of only one maximum in p(Δ) (Figure 3b).As expected, the EFG parame both Mn sites are almost independent of temperature.Moreover, it was establish the calculated values Δ theor Mni ≈ eQ теор ZZ Mni , V for sites Mn4 and Mn5 (where Q is the rupole moment of 57 Fe nuclei) remarkably exceed the corresponding experimental Δ exp (Figure 4; Tables S2-S5).
We suppose that the abovementioned discrepancy between the calculated a perimental values of the quadrupole splitting (Δ theor > Δ exp ) and their unusually temperature dependences can be attributed to the dynamic JT effect of Mn 3+ cati curring in this temperature range [46].The JT interactions of the Mn 3+ cations in BiM can result in the so-called orbital ordering, or cooperative JT effect, which is a served in other perovskite-like Mn(III) oxide systems, namely, RMnO3 R1−xAxMnO3 [11], and AMn7O12 [37,49] (R = REE, A = Ca, Sr, Pb).All these syste exhibit a structural transition to a crystal lattice with enhanced symmetry in the t ature range T > TJT, which is ascribed to the dynamic JT effect, or the "melting" cooperative JT distortion [46].Similar phase transitions can occur through two m nisms: one involves increasing the symmetry of distorted (Mn 3+ O6) polyhedra un uniform population of Mn 3+ eg-orbitals is achieved, and the other entails the orient disordering of distorted (Mn 3+ O6) polyhedra while preserving the polarizat eg-orbitals even at high temperatures (T >> TJT) [50].As was noted in several refs.[ the local disordering of (Mn 3+ O6) polyhedra can start at a temperature (T*) that is icantly lower than the temperature of the structural phase transition TJT (>>T*).Ho there is still no reliable experimental data on the changes in the structure and ele state of manganite, which take place in this "intermediate" temperature range.
Local minima are known to appear on the adiabatic potential surface of p nuclear configurations of O 2− anions in the (MnO6) polyhedra if the anharmonicit bronic interactions is taken into account.These minima reflect the specific orthorh distortions of the corresponding polyhedra.With increasing temperature, the cry environment of the JT Mn 3+ cations stochastically relaxes between these minima thermally activated excitations or the tunnel effect [54].Although the Fe 3+ cations do not participate in the vibronic interactions, their local crystal environment als tuates dynamically due to the cooperative JT effect.Therefore, we suggest that served significant reduction in the Δ exp values compared to the theoretical calcu can be associated with the relaxation behavior of the Mössbauer spectra in the te ture range T2 < T < T1.In [54,55], it was shown that such spectra can be described w eop ZZ,Mni for sites Mn4 and Mn5 (where Q is the quadrupole moment of 57 Fe nuclei) remarkably exceed the corresponding experimental values ∆ exp (Figure 4; Tables S2-S5).
We suppose that the abovementioned discrepancy between the calculated and experimental values of the quadrupole splitting (∆ theor > ∆ exp ) and their unusually strong temperature dependences can be attributed to the dynamic JT effect of Mn 3+ cations occurring in this temperature range [46].The JT interactions of the Mn 3+ cations in BiMn 7 O 12 can result in the so-called orbital ordering, or cooperative JT effect, which is also observed in other perovskite-like Mn(III) oxide systems, namely, RMnO 3 [47,48], R 1−x A x MnO 3 [11], and AMn 7 O 12 [37,49] (R = REE, A = Ca, Sr, Pb).All these systems can exhibit a structural transition to a crystal lattice with enhanced symmetry in the temperature range T > T JT , which is ascribed to the dynamic JT effect, or the "melting" of the cooperative JT distortion [46].Similar phase transitions can occur through two mechanisms: one involves increasing the symmetry of distorted (Mn 3+ O 6 ) polyhedra until the uniform population of Mn 3+ e g -orbitals is achieved, and the other entails the orientational disordering of distorted (Mn 3+ O 6 ) polyhedra while preserving the polarization of e g -orbitals even at high temperatures (T >> T JT ) [50].As was noted in several refs.[51][52][53], the local disordering of (Mn 3+ O 6 ) polyhedra can start at a temperature (T*) that is significantly lower than the temperature of the structural phase transition T JT (>>T*).However, there is still no reliable experimental data on the changes in the structure and electronic state of manganite, which take place in this "intermediate" temperature range.
Local minima are known to appear on the adiabatic potential surface of possible nuclear configurations of O 2− anions in the (MnO 6 ) polyhedra if the anharmonicity of vibronic interactions is taken into account.These minima reflect the specific orthorhombic distortions of the corresponding polyhedra.With increasing temperature, the crystalline environment of the JT Mn 3+ cations stochastically relaxes between these minima due to thermally activated excitations or the tunnel effect [54].Although the Fe 3+ cations per se do not participate in the vibronic interactions, their local crystal environment also fluctuates dynamically due to the cooperative JT effect.Therefore, we suggest that the observed significant reduction in the ∆ exp values compared to the theoretical calculations can be associated with the relaxation behavior of the Mössbauer spectra in the temperature range T 2 < T < T 1 .In [54,55], it was shown that such spectra can be described with the "twolevel model" in the limit of "fast relaxation", i.e., when Ω R >> Ω 0 , where Ω R and Ω 0 are the frequencies of the relaxation of the oxygen environment and the precession of the 57 Fe quadrupole moment around the V ZZ direction, respectively.The model adopts the frequencies of forward (Ω 12 ) and reverse (Ω 21 ) transitions between states "1" and "2" as variables connected by the detailed equilibrium principle n 1 Ω 12 = n 2 Ω 21 , where n 1 and n 2 are the populations of states (Figure 6a) [55].
In the monoclinic structure (I2/m) of BiMn 7 O 12 , the distortion of (MnO 6 ) polyhedra corresponding to the energy minimum E 1 of the adiabatic potential, is described as a "bonding" Q (−) -the linear combination of the orthorhombic (Q 2 ) and tetragonal (Q 3 ) vibrational modes [7,56].In this case, the distortion with a higher energy E 2 is attributed to the "antibonding" vibration mode Q (+) .In a local approximation, when only the closest anion environment is considered, the two vibrational modes-Q (−) and Q (+) -correspond to the distortions that exhibit equal magnitudes but opposite signs in their EFG components (V ZZ ) imposed on the 57 Fe nuclei occupying the Mn4 and Mn5 sites [54].Consequently, when the population of the E 1 and E 2 levels equalizes with increasing temperature, quadrupole splitting sharply decreases, i.e., ∆(T) ∝ <V ZZ >, where <V ZZ > is averaged over the energy states E 1 and E 2 [54].On the other hand, the monotonous decrease in ∆(T) up to T 1 can suggest a gradual enhancement of the (FeO 6 ) symmetry upon approaching the temperature of the structural phase transition I2/m → Im3.This conclusion is consistent with the synchrotron X-ray diffraction studies of BiMn 7 O 12 , which also demonstrate the gradual decrease in the distortion parameter ∆ d of (MnO 6 ) polyhedra when T → T 1 [3].Thus, these parameters behave similarly when assessed by inherently different characterization techniques.These data suggest the second-order JT phase transition that can be referred to as the displacive structural transition, in contrast to the "order-disorder" transition mechanism."two-level model" in the limit of "fast relaxation", i.e., when ΩR >> Ω0, where ΩR and Ω0 are the frequencies of the relaxation of the oxygen environment and the precession of the 57 Fe quadrupole moment around the VZZ direction, respectively.The model adopts the frequencies of forward (Ω12) and reverse (Ω21) transitions between states "1" and "2" as variables connected by the detailed equilibrium principle n1Ω12 = n2Ω21, where n1 and n2 are the populations of states (Figure 6a) [55].In the monoclinic structure (I2/m) of BiMn7O12, the distortion of (MnO6) polyhedra corresponding to the energy minimum E1 of the adiabatic potential, is described as a "bonding" Q (−) -the linear combination of the orthorhombic (Q2) and tetragonal (Q3) vibrational modes [7,56].In this case, the distortion with a higher energy E2 is attributed to the "antibonding" vibration mode Q (+) .In a local approximation, when only the closest anion environment is considered, the two vibrational modes-Q (−) and Q (+) -correspond The fitting of the whole series of spectra within the framework of a two-level model allowed us to estimate the average relaxation frequency Ω R = Ω 12 Ω 21 /(Ω 12 + Ω 21 ) ≈ (2-7) × 10 7 Hz, which significantly exceeds the frequency of the quadrupole precession Ω 0 ≈ 8.5 × 10 6 Hz.Increasing temperature leads to a gradual equalization of the populations n 1 and n 2 .This should result in a sharp decrease in the quadrupole splitting and a slight broadening of the doublet components in the limit of fast relaxation Ω R >> Ω 0 .Indeed, this pattern describes the temperature-related changes of all the spectra at T 2 < T < T 1 (Figure S4).Using the linear approximation of ln(n 1 /n 2 ) = f(1/T) (obtained from the Arrhenius equation): where Ω 0 1(2) are temperature-independent parameters; E 12 (21) are the activation energies of "1" and "2" states, respectively; k B is the Boltzmann constant) we evaluated the energy difference between the relaxed states ∆E = 69(2) meV and the mean value of the activation energy E act = 220(9) meV (Figures 6a and S5) that closely corresponds to <E act > for other perovskite-like Mn(III) manganites [57].However, a deviation from linearity is observed at higher temperatures (T > 550 K) (Figure 6b).This is likely to result from the changes in the relative position of levels E 1 and E 2 between which the relaxation occurs.This explanation is indirectly supported by a similar temperature profile of the distortion parameters ∆ d of polyhedra (MnO 6 ) (Figure 6b), which govern the splitting of the 3d levels of Mn cations under the influence of the ligand field.
It is worth noting that the observed structural changes in BiMn 7 O 12 in the temperature range of the JT transition are similar to those in the isostructural phase of LaMn 7 O 12 [58] but differ crucially from the so-called "traditional" perovskites RMnO 3 (R = REE), in which the cooperative JT effect follows the "order-disorder" mechanism [39].In these oxides, polyhedra (Mn 3+ O 6 ) remain distorted even at temperatures significantly exceeding T JT .However, these distortions are randomly oriented in the crystal lattice, thus making the structure "macroscopically" more symmetrical compared to the low-temperature phase with orbital ordering.

Mössbauer Data for Temperature Ranges T 3 < T < T 2 and T N1 < T < T 3
In the temperature range T 3 < T < T 2 , the Mössbauer spectra of BiMn 6.96 Fe 0.04 O 12 consist of a broadened quadrupole doublet (Figure 3c).The observed bimodal profile of p(∆) (Figure 3c) suggests the stabilization of 57 Fe nuclei at two different crystallographic sites of the manganite.This conclusion agrees with the earlier structural data for BiMn 7 O 12 [3]: in the crystal lattice with the Im symmetry, the JT Mn 3+ cations occupy two sites, Mn1 and Mn2, in the very distorted octahedral oxygen environment (Figure 1c).Thus, the high value <∆> ≈ 0.62 mm/s observed in the spectra of Fe 3+ probe cations is conditioned by a low symmetry of the oxygen environment of the JT Mn 3+ cations.Based on the p(∆) distribution, we fitted the whole series of spectra measured in the range T 3 < T < T 2 as a superposition of two quadrupole doublets, Fe(1) and Fe(2), having close values of isomer shifts, δ 1 ≈ δ 2 (Figure 3c).
The analysis above yielded anomalously sharp temperature dependences ∆ i (T) for both Fe(1) and Fe(2) doublets (Figure 4).Such behavior can stem from the induction of spontaneous BiMn 7 O 12 polarization at T < T 2 [3].To support this assumption quantitatively, we obtained an expression relating the lattice contributions V lat ZZ with the values and mutual orientation of electric moments (p k ) in the lattice of BiMn 7 O 12 (for details, see SI).The values p k and their projections p ik were calculated using the Born model [59]: p k = Z k ∆r k or p ik = Z k ∆x ik , where Z k is the Born charge of the kth ion, which is an isotropic scalar value in our calculation; ∆r k (∆x ik ) is the vector of the displacement of the kth ion (and ith projection) from the centrosymmetric position.The values ∆r kk and ∆x ik were calculated using the crystallographic data for BiMn 7 O 12 obtained at 300 K [3].To estimate Born charges, we sequentially varied the charges {Z Bi , Z Mn(i) , Z O(i) } and their corresponding dipole moments p ik with the given displacement ∆x ik until the best agreement with the experimental splitting ∆ i was achieved (Table S1).The values approximated in such a manner that Z Bi = +3.30,Z Mn1 ≈ Z Mn2 = +3.30,and Z O = −2.20,and all lie within the range of the Born charges obtained earlier for the corresponding ions in other perovskite oxides [60].
Using the above approximations, we derived an equation that describes ∆(T) as a function of the ith projections {P i } i=x,y,z = ∑ k p ik of the spontaneous polarization P s = (P x 2 + P y 2 + P z 2 ) 1/2 : where p k (T 0 )/P s (T 0 ) is the ratio of the dipole moment of the kth ion (p k ) to the spontaneous polarization in the crystal, which is calculated based on the crystallographic data for BiMn 7 O 12 (Im).The first and the second terms in Equation ( 1) are the monopole and dipole contributions to the EFG.Using Equation ( 1), we plotted theoretical dependencies ∆ 1 (P s ) and ∆ 2 (P s ) (Figure 7).Using the numerical solution of the equations ∆ 1 (P s ) = ∆ 1 (T) and ∆ 2 (P s ) = ∆ 2 (T) and accounting for statistical errors allowed us to simulate the temperature dependence P s (T) (Figure 8).The same algorithm for constructing dependencies P s (T) using experimental data ∆ i (T) was applied to the triclinic phase (P1) of BiMn 7 O 12 .The distribution p(∆) has a trimodal profile for the given structure modification (Figure 3d), which indicates that Fe 3+ cations occupy at least three nonequivalent sites.According to the structural data [4], the Mn 3+ cations form four equally populated sites (Mn4, Mn5, Mn6, and Mn7) in an octahedral oxygen environment (Figure 1d).The calculated EFG parameters of Mn4 and Mn6 suggest that these atoms are located symmetrically near similar crystalline environments, making them indistinguishable in the 57 Fe Mossbauer spectra.Therefore, the spectra at T N1 ≤ T ≤ T 3 were fitted with three quadrupole doublets, namely, Fe(1), Fe(2), and Fe(3), with the Fe(3) component being two times more intense than Fe(1) and Fe(2) (Figure 3d).Using the structural data for the triclinic BiMn 7 O 12 phase at 10 K [4] and the described algorithm combining the theoretical dependencies ∆ i (P s ) with the experimental ∆ i (T) values, we were able to model P s (T) across the temperature range under investigation (Figure 8).
The dependences ∆ ∝ V ZZ = f(P s ) (Figure 7) agree with the results of [24][25][26][27], where, in a general case, the dependence of V ZZ (P s ) at low P s values was represented using a Taylor series expansion: It was shown in [12] that the linear term vanishes and the quadratic term in the expansion becomes significant (β >> γ ̸ = 0) for the centrosymmetrical crystal sites.At the same time, if the center of symmetry is absent, the linear term should be predominant (α >> β >> γ ̸ = 0).Since for both polymorphic modifications of the BiMn 7 O 12 octahedral Mn 3+ sites are not centrosymmetric, the above dependences V ZZ (P s ) can be described using an expansion in a series with nonzero parameters α and β, whose values (Table 2), in order of magnitude, are consistent with similar data from other perovskite-like ferroelectrics [24][25][26][27].When describing the dependences, P s (T) was obtained for two temperature ranges, and we used the model of the average effective field [61].Within the framework of this approach, it is assumed that every ion in the ferroelectric crystal is affected by an effective electric field (E eff ), which can be expressed as where E 0 is the external electrical field, and the following terms correspond to the dipolar, quadrupolar, and octupolar interactions, respectively.In our calculations, E 0 = 0.Moreover, only dipolar and quadrupolar interactions were taken into consideration, the latter (γ) serving to describe the phase transitions of both the first and second orders within the united approach.For statistical consideration, the P s (T) dependence of polarization can have a general form [61]: where P 0 is the spontaneous saturation polarization, and N is the number of elementary dipoles per unit cell.Using the relationship k B T C = β•P 0 2 /N (where T C is the Curie ferroelectric point) and designations of the normalized values σ s ≡ P s /P 0 , τ ≡ T/T C , g ≡ γP 0 2 /β, one can derive an expression that is convenient for analyzing the experimental data: where the parameter g is a quantitative criterion of the order of the transition to the ferroelectric state [61].The analysis of the experimental dependences P s (T) using Expression ( 5) is shown in Figure 8.The value g ≈ 0.48(3) in the temperature range T 3 < T < T 2 indicates a significant contribution of the quadrupolar interactions (γ ̸ = 0) in Expression (3), which, in turn, is a feature of the first-order transitions [61,62].A similar behavior was observed for many oxide systems, in particular, those demonstrating multiferroic properties [63].The transition point T C ≈ 437 K of the ferroelectric state, having been evaluated within the framework of this approach based on the description of P s (T), insignificantly differs from the temperature of the structural transition Im ↔ I2/m (T 2 ≈ 442 K) determined for the sample BiMn 6.96 Fe 0.04 O 12 from the thermodynamic data (Figure 2a).We suggest that the observed first-order transition at T 2 stems from an inevitable coupling between the electric polarization and the crystal lattice.There is a simultaneous structural change accompanying this transition as evidenced by the above finding.By using a simple phenomenological approach (see Appendix D), it can be shown that the coupling between spontaneous polarization (P s ) and strain (ε) can switch an otherwise second-order transition to a firstorder transition.Moreover, there is a relationship between the strength of the ferroelastic coupling and the size of the hysteresis ~20 K (Figure 2a) of the resultant first-order transition.It is known that the size of hysteresis is determined by the energy barrier at T C (Figure S4), which is largely dependent on the magnitude of the ferroelastic coupling coefficient λ in the fourth-order term of the Landau free energy (see Equation A8).On the other hand, a larger λ also leads to a larger spontaneous lattice distortion upon the first-order phase transition.Therefore, the magnitude of thermal hysteresis increases with an increase in lattice distortion.
In the range T N1 ≤ T ≤ T 3 , the dependence P s (T) demonstrates a kink at the point T 3 , and its course follows Expression (4) with the parameter g ≈ 0, which, upon extrapolation to ~294 K (see Figure 8), should correspond to a "gradual" second-order phase transition [61].A discussion of the nontrivial course of the dependencies P s (T) is beyond the scope of our work; however, it may motivate someone to study this unusual system using new independent local and macroscopic methods and theoretical approaches.

Mössbauer Study in the Temperature Range T < T N1
The quadrupole doublets at temperatures slightly below the Néel point (T < T N1 ≈ 50 K) contain the broadened components that reflect hyperfine magnetic fields B hf induced on the 57 Fe nuclei (Figure 9a).The spectra were fitted via a reconstruction of distributions p(B hf ) characterized by a certain dispersion D P (δ) at a given temperature.The kink on the temperature dependence D P (δ) = f(T) (Figure 9b; Table S6 for data) corresponds to the temperature 57(3) K, which, within the measurement error, coincides with T N1 ≈ 59 K for undoped manganite BiMn 7 O 12 (Figure 2b).This result gives independent confirmation of the stabilization of 57 Fe probes in the lattice of the BiMn 7 O 12 manganite under study.At low temperatures, T << TN1, the Mössbauer spectrum of BiMn6.96 57Fe0.04O12 has an asymmetric and slightly broadened Zeeman structure (Figure 10), which can be described by a superposition of four unequally broadened sextets in accordance with the structural data for the triclinic phase BiMn7O12 [4].With increasing temperature, the profiles of these sextets change noticeably, which is characteristic of the system exhibiting relaxation processes.Earlier studies showed [64] that this behavior could be attributed to the magnetic excitation of paramagnetic impurity ions within magnetically ordered matrices, where competing exchange interactions play a significant role.When embedded within the manganite matrix, impurity cations Fe 3+ with half-filled orbitals are surrounded by the JT Mn 3+ cations with anisotropic orbital occupation.This can lead to a noticeable weakening of the exchange interactions between the impurity cations and their magnetic environment.Essentially, this suggests that iron cations can undergo lower-energy magnetic excitations, influencing also the neighboring Mn 3+ cations rather than only Fe 3+ cations.An increase in temperature leads to the progressive occupation of magnetic Fe 3+ states characterized by different projections (SZ) of the total spin S = 5/2.As the magnetic interaction of iron with its surroundings is weakened, the relaxation of spin between the states |5/2, SZ> decelerates.If the relaxation period τR closely corresponds to At low temperatures, T << T N1, the Mössbauer spectrum of BiMn 6.96 57 Fe 0.04 O 12 has an asymmetric and slightly broadened Zeeman structure (Figure 10), which can be described by a superposition of four unequally broadened sextets in accordance with the structural data for the triclinic phase BiMn 7 O 12 [4].With increasing temperature, the profiles of these sextets change noticeably, which is characteristic of the system exhibiting relaxation processes.Earlier studies showed [64] that this behavior could be attributed to the magnetic excitation of paramagnetic impurity ions within magnetically ordered matrices, where competing exchange interactions play a significant role.When embedded within the manganite matrix, impurity cations Fe 3+ with half-filled orbitals are surrounded by the JT Mn 3+ cations with anisotropic orbital occupation.This can lead to a noticeable weakening of the exchange interactions between the impurity cations and their magnetic environment.Essentially, this suggests that iron cations can undergo lower-energy magnetic excitations, influencing also the neighboring Mn 3+ cations rather than only Fe 3+ cations.An increase in temperature leads to the progressive occupation of magnetic Fe 3+ states characterized by different projections (S Z ) of the total spin S = 5/2.As the magnetic interaction of iron with its surroundings is weakened, the relaxation of spin between the states |5/2, S Z > decelerates.If the relaxation period τ R closely corresponds to the period of Larmor precession (τ L ) of the 57 Fe nuclear spin around the hyperfine field B hf , the Mössbauer spectra typically have complex relaxation profiles [65].Additionally, it should be noted that the SEM data indicates that an average particle size exceeded 2 μm for the BiMn6.96Fe0.04O12sample (Figure S3).Consequently, the observed relaxation behavior of the Mössbauer spectra cannot be attributed to the superparamagnetic or superferromagnetic states of small particles [65,66].
Taking into account the considerations presented above, the spectra were fitted using a multilevel relaxation model [67].This model is based on the assumption that in the effective magnetic Weiss field, spin S = 5/2 of the Fe 3+ cation, stochastically relaxes between Zeeman states |5/2, SZ> [67].Along with the static parameters (δ, eQVZZ and Bhf), the model includes variable relaxation parameters, namely, the relaxation frequency ΩR (=1/τR) and the relative population (s) of the Zeeman sublevels between which the relaxation occurs.A detailed description of this model can be found in [67].This allowed us to process the whole series of spectra measured in the temperature range 10 K < T < TN1.It should be noted that the static and relaxation parameters of the Mössbauer spectra remain virtually unchanged, which indicates the stability of a complex model of spectrum processing.A smooth and continuous change in the hyperfine magnetic field near TN1 indicates the occurrence of a second-order magnetic phase transition.This conclusion is consistent with the results of a theoretical study of undoped manganite BiMn7O12, according to which the transition at TN1 = 59 K corresponds to the formation of a single E-type AFM structure [68].At the same time, the magnetic phase transitions at TN2 ≈ 50 K and TN3 ≈ 24 K cannot be seen in the Mössbauer spectra because the former does not Additionally, it should be noted that the SEM data indicates that an average particle size exceeded ~2 µm for the BiMn 6.96 Fe 0.04 O 12 sample (Figure S3).Consequently, the observed relaxation behavior of the Mössbauer spectra cannot be attributed to the superparamagnetic or superferromagnetic states of small particles [65,66].
Taking into account the considerations presented above, the spectra were fitted using a multilevel relaxation model [67].This model is based on the assumption that in the effective magnetic Weiss field, spin S = 5/2 of the Fe 3+ cation, stochastically relaxes between Zeeman states |5/2, S Z > [67].Along with the static parameters (δ, eQV ZZ and B hf ), the model includes variable relaxation parameters, namely, the relaxation frequency Ω R (=1/τ R ) and the relative population (s) of the Zeeman sublevels between which the relaxation occurs.A detailed description of this model can be found in [67].This allowed us to process the whole series of spectra measured in the temperature range 10 K < T < T N1 .It should be noted that the static and relaxation parameters of the Mössbauer spectra remain virtually unchanged, which indicates the stability of a complex model of spectrum processing.A smooth and continuous change in the hyperfine magnetic field near T N1 indicates the occurrence of a second-order magnetic phase transition.This conclusion is consistent with the results of a theoretical study of undoped manganite BiMn 7 O 12 , according to which the transition at T N1 = 59 K corresponds to the formation of a single E-type AFM structure [68].At the same time, the magnetic phase transitions at T N2 ≈ 50 K and T N3 ≈ 24 K cannot be seen in the Mössbauer spectra because the former does not change the local magnetic environment of manganese (probe iron) cations in B sites, and the second transition (T N3 ) leads to the magnetic ordering of Mn 3+ in the A ′′ sublattice.
Characteristic of spin-spin relaxation, there are no obvious changes in the frequency Ω R as a function of temperature.However, the values of Ω R ~(0.2-0.5)× 10 9 s −1 are found to be significantly lower than the characteristic frequencies of spin waves Ω S ≈ k B T/h = 10 11 -10 12 s −1 (T = 1-100 K) in conventional magnetic systems [65].This indicates that the spin fluctuations involving iron probes are predominantly local.Furthermore, the Ω R value is comparable to the Larmor frequency Ω R ≈ 10 8 s −1 of the 57 Fe nuclear spin in the hyperfine magnetic field <B hf > ≈ 50 T, thus supporting our assumption about the relaxed nature of the observed spectra.
Figure 11 shows the temperature dependence of the hyperfine field <B hf (T)> averaged over all partial spectra Fe(i), which was approximated for spin S = 5/2 using the parametric Brillouin function: where ξ = J FeMn /J MnMn is the ratio of exchange integrals that characterize the magnetic interactions of Fe 3+ probes with the surrounding manganese cations (J FeMn ) and the averaged interactions between the Mn 3+ cations themselves (J MnMn ).The value ξ = 0.67(3) obtained from the best fit of the experimental spectra evidences the weakening of the exchanged magnetic interactions of the iron cations with the manganese sublattice, which is equivalent to decreasing the effective Weiss field [69].This can result from a so-called "orbital dilution", a phenomenon characteristic of the impurity cations of transition metals with non-degenerate orbital electron states (Fe 3+ , Cr 3+ . ..: <L> = 0, where L is the total orbital momentum) if they are stabilized within the matrix of transition metals with degenerate orbital states (Rh 4+ , Mn 3+ . ..: <L> ̸ = 0) [70,71].It was shown previously that such impurity centers behave as peculiar "orbital defects" due to the absence of orbital degeneration, i.e., orbital degrees of freedom.Even at very low concentrations, they can significantly affect the magnetic structure of the compound.Experimental methods employed to study such systems are currently in their early stages of development.
change the local magnetic environment of manganese (probe iron) cations in B sites, and the second transition (TN3) leads to the magnetic ordering of Mn 3+ in the A′′ sublattice.Characteristic of spin-spin relaxation, there are no obvious changes in the frequency ΩR as a function of temperature.However, the values of ΩR ~ (0.2-0.5) × 10 9 s −1 are found to be significantly lower than the characteristic frequencies of spin waves ΩS ≈ kBT/ћ = 10 11 -10 12 s −1 (T = 1-100 K) in conventional magnetic systems [65].This indicates that the spin fluctuations involving iron probes are predominantly local.Furthermore, the ΩR value is comparable to the Larmor frequency ΩR ≈ 10 8 s −1 of the 57 Fe nuclear spin in the hyperfine magnetic field <Bhf> ≈ 50 T, thus supporting our assumption about the relaxed nature of the observed spectra.
Figure 11 shows the temperature dependence of the hyperfine field <Bhf(T)> averaged over all partial spectra Fe(i), which was approximated for spin S = 5/2 using the parametric Brillouin function: where ξ = JFeMn/JMnMn is the ratio of exchange integrals that characterize the magnetic interactions of Fe 3+ probes with the surrounding manganese cations (JFeMn) and the averaged interactions between the Mn 3+ cations themselves (JMnMn).The value ξ = 0.67(3) obtained from the best fit of the experimental spectra evidences the weakening of the exchanged magnetic interactions of the iron cations with the manganese sublattice, which is equivalent to decreasing the effective Weiss field [69].This can result from a so-called "orbital dilution", a phenomenon characteristic of the impurity cations of transition metals with non-degenerate orbital electron states (Fe 3+ , Cr 3+ …: <L> = 0, where L is the total orbital momentum) if they are stabilized within the matrix of transition metals with degenerate orbital states (Rh 4+ , Mn 3+ …: <L> 0) [70,71].It was shown previously that such impurity centers behave as peculiar "orbital defects" due to the absence of orbital degeneration, i.e., orbital degrees of freedom.Even at very low concentrations, they can significantly affect the magnetic structure of the compound.Experimental methods employed to study such systems are currently in their early stages of development.

Materials and Methods
The manganite BiMn 6.96 Fe 0.04 O 12 was synthesized in a high-pressure, "belt"-type chamber.A stoichiometric mixture of Mn 2 O 3 (99.9%,Rare Metallic Co., Tokyo, Japan) Bi 2 O 3 (99.9999%,Rare Metallic Co., Tokyo, Japan), and 57 Fe 2 O 3 (95.5% enriched with 57 Fe, Trace Sciences International, Richmond Hill, ON, Canada) was filled into a gold capsule in which it was subjected to a pressure of ~6 GPa followed by heating to 1323 K for 10 min.The sample was quenched to room temperature after holding for 120 min.The synthesis of undoped manganite BiMn 7 O 12 is described in more detail in [3].
The X-ray diffraction data were acquired using a synchrotron source of X-rays (SXRPD) in a large Debye-Sherrer chamber with the line BL15XU (SPring-8, Sayo, Japan) and the 2θ value ranging from 3 • to 60 • with a step of 0.003 • .The monochromatic radiation with the wavelength of λ = 0.65298 Å was used.Experiments were performed in a temperature range of between 100 and 670 K. Prior to measuring, the powder samples were tightly packed in a Lindemann glass capillary (for the 100-400 K range) and a quartz capillary (for the 350-670 K range) with an internal diameter of 0.1 mm.The capillaries were cooled using an N 2 flow when the low-temperature measurements were performed.The processing of the diffraction patterns and refinement of the crystal lattice parameters were performed using the Rietveld method, using the RIETAN-2000 software similar to the procedure described in [3].
Scanning electron microscopy (SEM) images were taken on a NVision 40 electron microscope (Carl Zeiss; Oberkochen, Germany) equipped with an Oxford Instruments X-Max analyzer.The accelerating voltage varied in the range from 3 to 20 kV.
For measuring differential scanning calorimetry (DSC) curves on a Mettler Toledo DSC1 STAR e calorimeter in the temperature range 125-673 K, samples were placed in Al crucibles, the rate of heating/cooling in the nitrogen flow being 10 K/min.
The heat capacity measurements were carried out on a PPMS calorimeter (Quantum Design, San Diego, CA, USA) in the temperature range of 2-300 K in the modes of heating and cooling in external magnetic fields ranging from 0 to 90 kOe.
Magnetic susceptibility was measured on a SQUID MPMS 1T magnetometer (Quantum Design, San Diego, CA, USA) in the temperature range of 2-350 K in the ZFC (cooling without external magnetic field) and FC (cooling in the external magnetic field 10 kOe in strength) modes.
Mössbauer spectra were measured with a conventional electrodynamic-type spectrometer in the constant acceleration mode with a 1450 MBq 57 Co(Rh) γ-ray source.The values of the isomer shift are given relative to α-Fe (298 K).Processing of the experimental spectra was performed with the use of the program package "SpectrRelax" [72].Computations of the EFG parameters were carried out using the "GradientNCMS" software (ver.8.3) designed by the authors and are represented in more detail in [73].

Conclusions
We explored the interplay between the local crystal structure of the multiferroic BiMn 7 O 12 manganite and the processes of its spontaneous polarization and magnetic ordering using 57 Fe-probe Mössbauer spectroscopy.It was shown that Fe 3+ probes statistically substitute isovalent Mn 3+ cations in the octahedral oxygen local environment.The parameters of the electric hyperfine interactions of 57 Fe nuclei reflect the symmetry of the crystalline environment of Mn 3+ cations in these sites.
The calculations of the EFG parameters, considering both monopole and dipole contributions, are in accordance with our experimental results, demonstrating that, in the paraelectric phase (at T > T 2 ), cations Bi 3+ , even while existing in locally distorted crystalline environments, maintain their electrical dipole moments p Bi , which are randomly oriented within the cubic lattice.As a result, the phase transitions into the ferroelectric state involve the ordering of p Bi dipoles, i.e., they may be considered the transitions of the "order-disorder" type.
It was determined that the monotonous decrease in ∆(T) from T 2 up to T 1 can indicate a gradual increase in the symmetry of (Fe 3+ O 6 ) polyhedra while approaching the temperature of the structural transition Im3 → I2/m, which is corroborated by the synchrotron diffraction studies of undoped BiMn 7 O 12 .This characteristic behavior has been independently registered through methods that are entirely different in their physical nature.Thus, it strongly indicates the occurrence of the second-order JT phase transition.Its mechanism can be classified as a structural transition of the "displacive" type, in contrast to "order-disorder" transitions.
Using the Born model, we calculated dynamic ion charges that indicate only a minor ion polarization and the predominance of ionic contributions in the spontaneous electrical polarization of the crystal.The observed robust temperature dependence of the quadrupole splitting ∆ i (T) of the partial spectra Fe(i) is governed by the temperature dependence P s (T).The dependence P s (T) on the opposite sides of the phase transition P1 ↔ Im (T 3 ≈ 240 K) significantly differs in its behavior.In the range T 3 < T < T 2 , P s (T) indicates that the ferroelectric-paraelectric phase transition is of the first order.The Curie point T C ≈ 437 K determined from the Mössbauer data closely coincides with the temperature of the structural transition Im ↔ I2/m.The proposed algorithm for finding the correlation between the experimental dependencies ∆ i (T) for the probe 57 Fe nuclei and the polarization P s (T) of the crystal can be applied to other systems with ferroelectric and multiferroic properties.
At low temperatures, T < T N1, and the 57 Fe Mössbauer spectra demonstrate relaxation behavior.This can result from a so-called "orbital dilution" characteristic of the impurity cations of transition metals with non-degenerate orbital electron states within the matrix of transition metals with degenerate orbital states.
where γ ∞ is the Sternheimer antishielding factor, q k is the effective charge of the kth ion in the lattice, r k is the distance between the kth ion and the 57 Fe nucleus, and r ik is the projection of the radius vector r k on the direction of the ith projection of the electric dipole moment of the kth ion p ik (where i = x, y, z).The dipole moments of the ions (p k ) and their projections (p ik ) were assumed to be calculated in terms of the Born model [59]: p k = Z k ∆p k or p ik = Z k ∆x ik , where Z k is the Born charge of the kth ion, which was taken to be an isotropic scalar quantity, and ∆p k (∆x ik ) is the displacement vector of the kth ion (and its ith projection) from its centrosymmetrical position.∆p k and ∆x ik were calculated using the crystallographic data for the BiMn 7 O 12 manganite obtained at 300 K [3].To estimate the Born charges, we used the following procedure: charges Z Bi , Z Mni , and Z Oi , and the corresponding dipole moments p ik were sequentially varied at displacement ∆x ik until the best agreement with the experimental quadrupole splittings ∆ i .At the first stage of this procedure, the desired Born charges of the ions Z k were taken to be equal to their formal oxidation levels in the BiMn 7 O 12 compound.As a result, we obtained the values Z Bi = +3.30,Z Mn1 ≈ Z Mn2 = +3.30,and Z O = −2.20,which fall within the Born charge range obtained for the corresponding ions in other perovskite-like oxides [60].Using the projections of the electric dipole moments p ik , we can calculate the spontaneous polarization of the manganite, The value obtained at T = 300 K (P s (T 0 ) ~9 µC/cm 2 ) agrees with the value (7 µC/cm 2 at 300 K) obtained earlier for 57 Fe-free manganite BiMn 7 O 12 (space group Im) [3].All these results demonstrate that, despite the initial assumptions, the proposed calculation scheme is quite self-consistent and can be extend to the temperature range for which structural data are absent.
The Born charges obtained for Bi 3+ , Mn 3+ , and O 2− ions (Z i ) can be compared to the effective charges (S) calculated using the Brown model by using the data on the crystal structure of the plain BiMn 7 O 12 manganite [3].
where r k is the metal-oxygen distance for the kth pair M 3+ -O 2− (where M = Bi, Mn), r 0 is a constant for the given metal grade (r 0 = 2.09 for Bi 3+ -O 2− and r 0 = 1.760 for Mn 3+ -O 2− ), and B = 0.37 [75].Using this equation, we calculated the effective charges S of the Bi 3+ , Mn 3+ , and O 2− ions occupying nonequivalent crystallographic positions for each of the two structural modifications of BiMn 7 O 12 .
Charges <S Bi >, <S Mn >, and <S O > that were averaged for each kind of ions are presented in the Table S1.As would expected, the obtained values of <S> are very close to the formal oxidation levels of the corresponding atoms but are lower than their Born charges.The high Born charges, which are several times higher than the formal oxidation levels for most perovskite-like metal oxides, are associated with the deformation (polarizability) of the electron shell of an ion when it is displaced from the centrosymmetrical position in a crystal [59].However, in our case, the discrepancy between the Born (Z) and effective (<S>) charges is not so significant (Table S1), which indirectly confirms the validity of the ion model used in this work.
The temperature dependences of the dipole moment p k (T) and the related spontaneous polarization P s (T), which are likely to be the main cause of the sharp temperature-induced change in the splitting ∆ 1 (T) and ∆ 2 (T), were taken into account using the approximate expressions p k (T) = ξ k (T)p k (T 0 ) and P s (T) = ξ k (T)P k (T 0 ), in which p k (T 0 ) and P k (T 0 ) are the

26 Figure 1 .
Figure 1.Crystal structures of BiMn7O12: (a) At T > T1 in the cubic 3 Im structure (without Bi splitting); (b) At T2 < T < T1 in the monoclinic I2/m structure; (c) At T3 < T < T2 in the monoclinic Im structure; (d) At T < T3 in the triclinic P1 structure (all viewed along the monoclinic b axis; Elongated Mn-O bonds due to the Jahn-Teller distortions in MnO6 octahedra are marked by red lines; The crystal cells are marked with black lined rectangles.The inset in the center depicts the accordance of the crystal structures and physical properties.

Figure 1 .
Figure 1.Crystal structures of BiMn 7 O 12 : (a) At T > T 1 in the cubic Im3 structure (without Bi splitting); (b) At T 2 < T < T 1 in the monoclinic I2/m structure; (c) At T 3 < T < T 2 in the monoclinic Im structure; (d) At T < T 3 in the triclinic P1 structure (all viewed along the monoclinic b axis; Elongated Mn-O bonds due to the Jahn-Teller distortions in MnO 6 octahedra are marked by red lines; The crystal cells are marked with black lined rectangles.The inset in the center depicts the accordance of the crystal structures and physical properties.

T 1 ≈
580-590 K and T 2 ≈ 420-440 K, i.e., structural transitions I2/m ↔ Im3 and Im ↔ I2/m, respectively, according to the literature data [3].It is worth noting that the undoped sample BiMn 7 O 12 demonstrated the same transitions at ~608 K and ~460 K, respectively, as was reported in the earlier experiments [3].Further lowering the lattice symmetry to P1 does not affect DSC curves.Measurements upon cooling and heating reveal a difference in the transition points ∆T 1 ~7 K and ∆T 2 ~20 K, both of which slightly exceed the corresponding values for the BiMn 7 O 12 sample [3].
580-590 K and T2 ≈ 420-440 K, i.e., structural transitions I2/m  3 Im and Im  I2/m, respectively, according to the literature data [3].It is worth noting that the undoped sample BiMn7O12 demonstrated the same transitions at ~ 608 K and ~ 460 K, respectively, as was reported in the earlier experiments [3].Further lowering the lattice symmetry to P1 does not affect DSC curves.Measurements upon cooling and heating reveal a difference in the transition points ΔT1 ~ 7 K and ΔT2 ~ 20 K, both of which slightly exceed the corresponding values for the BiMn7O12 sample [3].

Figure 2 .
Figure 2. (a) Differential scanning calorimetry (DSC) curves of BiMn6.96 57Fe0.04O12 upon heating and cooling (three runs were performed to check the reproducibility; since there were no peaks observed, data in the 125-300 K range are not shown); (b) Specific heat, plotted as CP/T versus T, of BiMn7O12 and BiMn6.96 57Fe0.04O12 at H = 0 (measurements were performed on cooling); (c) ZFC dc

Figure 2 .
Figure 2. (a) Differential scanning calorimetry (DSC) curves of BiMn 6.96 57 Fe 0.04 O 12 upon heating and cooling (three runs were performed to check the reproducibility; since there were no peaks observed, data in the 125-300 K range are not shown); (b) Specific heat, plotted as C P /T versus T, of BiMn 7 O 12 and BiMn 6.96 57 Fe 0.04 O 12 at H = 0 (measurements were performed on cooling); (c) ZFC dc (left scale) and reversed (right scale) magnetic susceptibility curves of BiMn 7 O 12 and BiMn 6.96 57 Fe 0.04 O 12(dashed vertical lines emphasize magnetic anomalies).

Figure 3 .
Figure 3. Left panel: typical Mössbauer spectra of the 57 Fe nuclei in BiMn6.96Fe0.04O12manganite measured at different temperatures (each at a specific range according to different crystal struc-Figure 3. Left panel: typical Mössbauer spectra of the 57 Fe nuclei in BiMn 6.96 Fe 0.04 O 12 manganite measured at different temperatures (each at a specific range according to different crystal structures).Right panel: the p(∆) distributions and their representation as the superposition of normal distributions corresponding to the crystal sites of 57 Fe probe nuclei within a manganite structure.(a-d) Correspond to particular temperature range (see text).
Int. J. Mol.Sci.2024, 25, x FOR PEER REVIEW 7 of 26 tures).Right panel: the p(Δ) distributions and their representation as the superposition of normal distributions corresponding to the crystal sites of 57 Fe probe nuclei within a manganite structure.(a-d) Correspond to particular temperature range (see text).

Figure 4 .
Figure 4.The temperature dependencies of the experimental values of the quadrupole splittings Δi exp (T) of the partial Fe(i) spectra at specific ranges according to different crystal structures of the BiMn6.96Fe0.04O12manganite (asterisks show the theoretical values).

Figure 5 .
Figure 5. Schematic representations: (a) the formation of the pBi dipole moment as a result of the displacement of Bi 3+ cations (brown balls) from their centrosymmetric positions (balls with a dotted line).The Bi 3+ center is shifted toward the lone pair; (b) the random orientation of the lone electron

Figure 4 .
Figure 4.The temperature dependencies of the experimental values of the quadrupole splittings ∆ i exp (T) of the partial Fe(i) spectra at specific ranges according to different crystal structures of the BiMn 6.96 Fe 0.04 O 12 manganite (asterisks show the theoretical values).

Figure 4 .
Figure 4.The temperature dependencies of the experimental values of the quadrupole splittings Δi exp (T) of the partial Fe(i) spectra at specific ranges according to different crystal structures of the BiMn6.96Fe0.04O12manganite (asterisks show the theoretical values).

Figure 5 .
Figure 5. Schematic representations: (a) the formation of the pBi dipole moment as a result of the displacement of Bi 3+ cations (brown balls) from their centrosymmetric positions (balls with a dotted line).The Bi 3+ center is shifted toward the lone pair; (b) the random orientation of the lone electron

Figure 5 .
Figure 5. Schematic representations: (a) the formation of the p Bi dipole moment as a result of the displacement of Bi 3+ cations (brown balls) from their centrosymmetric positions (balls with a dotted line).The Bi 3+ center is shifted toward the lone pair; (b) the random orientation of the lone electron pairs or displacements of Bi 3+ cations leads to the zero value of the total crystal polarization (<P Bi >) averaged over all directions (the large brown ball represents the ellipse of the thermal vibrations of bismuth).

Figure 6 .
Figure 6.(a) The explicative scheme of the two-level relaxation model: Ei-the energies of states "1" and "2", ni-the probabilities of states, Ωij-the frequencies of transitions between states.(b) The reciprocal temperature dependencies of the logarithm ln(n1/n2) of probabilities n1 and n2 ratio, and the distortion parameters Δd for Mn4O6 and Mn5O6 octahedra, calculated from structural data [3].Blue lines represent the linear approximation in the selected temperature range and are shown for visual convenience.The shaded part corresponds to the temperature range (T > 500 K) for which a change in the degree of the distortion (Δd) of the MniO6 polyhedra is expected and, as a consequence, so too is a change in the relative position of the energy levels E1 and E2 (see text) The black dots correspond to the left scale, and the circles correspond to the right scale.

Figure 6 .
Figure 6.(a) The explicative scheme of the two-level relaxation model: E i -the energies of states "1" and "2", n i -the probabilities of states, Ω ij -the frequencies of transitions between states.(b) The reciprocal temperature dependencies of the logarithm ln(n 1 /n 2 ) of probabilities n 1 and n 2 ratio, and the distortion parameters ∆ d for Mn 4 O 6 and Mn5O 6 octahedra, calculated from structural data [3].Blue lines represent the linear approximation in the selected temperature range and are shown for visual convenience.The shaded part corresponds to the temperature range (T > 500 K) for which a change in the degree of the distortion (∆ d ) of the MniO 6 polyhedra is expected and, as a consequence, so too is a change in the relative position of the energy levels E 1 and E 2 (see text) The black dots correspond to the left scale, and the circles correspond to the right scale.

Figure 7 .
Figure 7. Left panel: The dependencies of the theoretical Δi values versus spontaneous crystal larization Ps at (a) T = 300 K and (b) T = 10 K.The curves refer to the experimental values Δ exp i fr the Mössbauer spectroscopy data at (a) 300 K and (b) 101 K temperatures.Shaded areas correspo to evaluated Ps values when the theoretical Δi(Ps) values conform to the experimental Δ exp i ones the best way.The verticle lines in the shaded areas showed the approximate mean position for e of perception.Right panel: the dependencies of the theoretical VZZi values versus spontaneo crystal polarization Ps at corresponding temperatures fitted with quadratic functions (see text).

Figure 7 .
Figure 7. Left panel: The dependencies of the theoretical ∆ i values versus spontaneous crystal polarization P s at (a) T = 300 K and (b) T = 10 K.The curves refer to the experimental values ∆ exp i from the Mössbauer spectroscopy data at (a) 300 K and (b) 101 K temperatures.Shaded areas correspond to evaluated P s values when the theoretical ∆ i (P s ) values conform to the experimental ∆ exp i ones in the best way.The verticle lines in the shaded areas showed the approximate mean position for ease of perception.Right panel: the dependencies of the theoretical V ZZi values versus spontaneous crystal polarization P s at corresponding temperatures fitted with quadratic functions (see text).

Figure 7 .
Figure 7. Left panel: The dependencies of the theoretical Δi values versus spontaneous crystal polarization Ps at (a) T = 300 K and (b) T = 10 K.The curves refer to the experimental values Δ exp i from the Mössbauer spectroscopy data at (a) 300 K and (b) 101 K temperatures.Shaded areas correspond to evaluated Ps values when the theoretical Δi(Ps) values conform to the experimental Δ exp i ones in the best way.The verticle lines in the shaded areas showed the approximate mean position for ease of perception.Right panel: the dependencies of the theoretical VZZi values versus spontaneous crystal polarization Ps at corresponding temperatures fitted with quadratic functions (see text).

Figure 8 .
Figure 8.The temperature dependencies of the spontaneous polarization Ps(T) for the two crystal structures of the BiMn6.96Fe0.04O12manganite.The solid and dashed curves represent the fitting in order with theory explained in the text.The dotted line shows the temperature of the phase transition..The same algorithm for constructing dependencies Ps(T) using experimental data Δi(T) was applied to the triclinic phase (P1) of BiMn7O12.The distribution p(Δ) has a tri-

Figure 8 .
Figure 8.The temperature dependencies of the spontaneous polarization P s (T) for the two crystal structures of the BiMn 6.96 Fe 0.04 O 12 manganite.The solid and dashed curves represent the fitting in order with theory explained in the text.The dotted line the temperature of the phase transition.

Figure 9 .
Figure 9. (a) 57 Fe Mössbauer spectra of BiMn6.96Fe0.04O12near TN1 fitted as the distributions of the single Lorentz line; (b) the temperature dependence of the dispersion DP(δ) of the isomer shift δ.The kink was used to evaluate the magnetic phase transition point (see text).

Figure 9 .
Figure 9. (a) 57 Fe Mössbauer spectra of BiMn 6.96 Fe 0.04 O 12 near T N1 fitted as the distributions of the single Lorentz line; (b) the temperature dependence of the dispersion D P (δ) of the isomer shift δ.The kink was used to evaluate the magnetic phase transition point (see text).

26 Figure 10 .
Figure10.57Fe Mössbauer spectra of BiMn6.96Fe0.04O12at T < TN1, fitted with the multilevel magnetic spin relaxation (see text).Black curves are subspectra obtained during fitting procedure, red curves are summarized fitted spectra.The differecnces between experimental and fitted data are also shown in the bottom of each spectrum.

Figure 10 .
Figure10.57Fe Mössbauer spectra of BiMn 6.96 Fe 0.04 O 12 at T < T N1 , fitted with the multilevel magnetic spin relaxation (see text).Black curves are subspectra obtained during fitting procedure, red curves are summarized fitted spectra.The differecnces between experimental and fitted data are also shown in the bottom of each spectrum.

Figure 11 .
Figure 11.The temperature dependence Bhf(T) of the hyperfine magnetic field Bhf at 57 Fe nuclei approximated using a modified Brillouin function (see text).The dashed red line shows the "pure" Brillouin law for spin S = 5/2.Black dots are experimental mean values of the hyperfine fields.

Figure 11 .
Figure 11.The temperature dependence B hf (T) of the hyperfine magnetic field B hf at 57 Fe nuclei approximated using a modified Brillouin function (see text).The dashed red line shows the "pure" Brillouin law for spin S = 5/2.Black dots are experimental mean values of the hyperfine fields.

Table 2 .
Taylor expansion parameters V ZZ (0) , α, and β of the dependences V ZZ (P s ) calculated for both polymorphic modifications for all octahedral Mn 3+ sites.