Structural, Spectroscopic, Electric and Magnetic Properties of New Trigonal K5FeHf(MoO4)6 Orthomolybdate

A new multicationic structurally disordered K5FeHf(MoO4)6 crystal belonging to the molybdate family is synthesized by the two-stage solid state reaction method. The characterization of the electronic and vibrational properties of the K5FeHf(MoO4)6 was performed using density functional theory calculations, group theory, Raman and infrared spectroscopy. The vibrational spectra are dominated by vibrations of the MoO4 tetrahedra, while the lattice modes are observed in a low-wavenumber part of the spectra. The experimental gap in the phonon spectra between 450 and 700 cm−1 is in a good agreement with the simulated phonon density of the states. K5FeHf(MoO4)6 is a paramagnetic down to 4.2 K. The negative Curie–Weiss temperature of −6.7 K indicates dominant antiferromagnetic interactions in the compound. The direct and indirect optical bandgaps of K5FeHf(MoO4)6 are 2.97 and 3.21 eV, respectively. The K5FeHf(MoO4)6 bandgap narrowing, with respect to the variety of known molybdates and the ab initio calculations, is explained by the presence of Mott-Hubbard optical excitation in the system of Fe3+ ions.


Introduction
The design of complex molybdate crystals containing different cations attracts a lot of attention because it provides an opportunity to develop a new functionality of material properties. Over the past few decades, many molybdate compounds with various combinations of cations have been studied to see their structural and chemical properties and

Crystal Structure of K5FeHf(MoO4)6
Single crystals of K5FeHf(MoO4)6 suitable for a single crystal structure analysis were grown by the spontaneous flux crystallization with the use of superfluous reactant K2Mo2O7 as a flux. The K5FeHf(MoO4)6 powder was used as the charge. The weight ratio of the solvent to the charge was selected to be equal to 3:1. The starting mixture was ground in an agate mortar into a homogeneous powder for 15 min and put into a 20 mL quartz crucible with a smooth inner wall and a top cap. The batch was melted at 873 K and was held at this temperature for 7 h to reach a completely melting and transparent solution. After that, slow cooling was performed from 873 to 473 K at the rate of 2 K h −1 . Then, the furnace was switched off and naturally cooled to room temperature. The heating of the ground mixture, the isothermal holding, and the slow cooling of the melt were controlled and kept automatically with the precision of ±0.5 K. Transparent brown and regular shaped crystals sized 1-2 mm were separated from the flux by breaking up the solidified cake.

Crystal Structure of K 5 FeHf(MoO 4 ) 6
Single crystals of K 5 FeHf(MoO 4 ) 6 suitable for a single crystal structure analysis were grown by the spontaneous flux crystallization with the use of superfluous reactant K 2 Mo 2 O 7 as a flux. The K 5 FeHf(MoO 4 ) 6 powder was used as the charge. The weight ratio of the solvent to the charge was selected to be equal to 3:1. The starting mixture was ground in an agate mortar into a homogeneous powder for 15 min and put into a 20 mL quartz crucible with a smooth inner wall and a top cap. The batch was melted at 873 K and was held at this temperature for 7 h to reach a completely melting and transparent solution. After that, slow cooling was performed from 873 to 473 K at the rate of 2 K h −1 . Then, the furnace was switched off and naturally cooled to room temperature. The heating of the ground mixture, the isothermal holding, and the slow cooling of the melt were controlled and kept automatically with the precision of ±0.5 K. Transparent brown and regular shaped crystals sized 1-2 mm were separated from the flux by breaking up the solidified cake.
The crystal structure of K 5 FeHf(MoO 4 ) 6 was determined by X-ray single crystal diffraction analysis. Centrosymmetrical space group R3c (167) was chosen based on the analysis of the absences in the data array. The structure refinement showed the cooperative occupation of the Hf and Fe atom positions. The SOFs for the Hf and Fe atoms in positions M1 and M2 were refined from the difference electron density maps and fixed with ratio 0.732/0.268 for M1 (opposite for M2) in the final refinement to reduce the shifts. Position M1 is located on the inversion axis 3 and is occupied mainly by Hf atoms. Position M2 is located at the intersection point of axes 2 and 3 and is occupied mainly by Fe atoms.

Mo Tetrahedron (Fe,Hf) Octahedra
Bond 3.278(2)-3.292 (2) All of the A 5 MT(MoO 4 ) 6 compounds, the unit cell parameters of which are known, are listed in Table S1 and shown in Figure 3. As is evident from Table S1, the giant cell volume variation, by~17%, is possible among the known members of the A 5 MT(MoO 4 ) 6 crystal family, and this indicates the high stability of this structure type. As is seen in Figure 3, in the a-c-V space, all known crystals can be captured by the extremely narrow elongated inclined ellipsoid, which apexes are determined by Cs 5 BiZr(MoO 4 ) 6   The physical properties of the K5FeHf(MoO4) 6 were measured with the use of the powder sample. The XRD pattern recorded for the synthesized powder sample of K5FeHf(MoO4)6 is presented in Figure 4. All of the peaks were indexed by trigonal cell (R 3c) with the parameters close to the single crystal data of K5FeHf(MoO4)6, as obtained in the present study. Therefore, this structure was taken as the starting model for the Rietveld refinement, which was performed using TOPAS 4.2 [60]. There are two Hf/Fe sites in the asymmetric part of the unit cell. The Hf:Fe ratio in each of these sites was refined in the assumption of 100% site occupation and the total Hf:Fe = 1:1 ratio in the final chemical formula. The refinement was stable and gave low R-factors (Table S2, Figure 4). A digital photo of the K5FeHf(MoO4)6 powder sample is presented in Figure 5. The sample is characterized by a light brown tint attributed to the presence of Fe 3+ ions. The physical properties of the K 5 FeHf(MoO 4 ) 6 were measured with the use of the powder sample. The XRD pattern recorded for the synthesized powder sample of K 5 FeHf(MoO 4 ) 6 is presented in Figure 4. All of the peaks were indexed by trigonal cell (R3c) with the parameters close to the single crystal data of K 5 FeHf(MoO 4 ) 6 , as obtained in the present study. Therefore, this structure was taken as the starting model for the Rietveld refinement, which was performed using TOPAS 4.2 [60]. There are two Hf/Fe sites in the asymmetric part of the unit cell. The Hf:Fe ratio in each of these sites was refined in the assumption of 100% site occupation and the total Hf:Fe = 1:1 ratio in the final chemical formula. The refinement was stable and gave low R-factors (Table S2, Figure 4). A digital photo of the K 5 FeHf(MoO 4 ) 6 powder sample is presented in Figure 5. The sample is characterized by a light brown tint attributed to the presence of Fe 3+ ions.   . Measured (red), calculated (black) and differential (gray) X-ray diffraction patterns of the K 5 FeHf(MoO 4 ) 6 powder sample. Figure 4. Measured (red), calculated (black) and differential (gray) X-ray diffraction patterns of the K5FeHf(MoO4)6 powder sample.

Thermochemical Properties
The thermal stability of K5FeHf(MoO4)6 was characterized using DSC, as shown in Figure 6. From the DSC curve, there are two endothermal peaks, located at 835 and 957 K.

Thermochemical Properties
The thermal stability of K 5 FeHf(MoO 4 ) 6 was characterized using DSC, as shown in Figure 6. From the DSC curve, there are two endothermal peaks, located at 835 and 957 K. The small endothermic peak at 835 K has a phase transition enthalpy of ∆H = −0.45 J/g. A sharp peak at the temperature of 957 K corresponds to the incongruent melting of the sample (∆H = −101 J/g). For comparison, the known thermal parameters of the A 5 MT(MoO 4 ) 6 crystals are listed in Table S3 [21,31,[33][34][35]61]. As it is seen in Table S3, the K 5 FeHf(MoO 4 ) 6 melting point is among the typical values observed for A 5 MT(MoO 4 ) 6 crystals. The analysis of the powder XRD pattern of the solidified melt revealed that K 5 FeHf(MoO 4 ) 6 is decomposed to double and simple molybdates, and it indicates that K 5 FeHf(MoO 4 ) 6 is an incongruent melting compound.
Molecules 2023, 28, x FOR PEER REVIEW 9 of 25 The small endothermic peak at 835 K has a phase transition enthalpy of ΔH = −0.45 J/g. A sharp peak at the temperature of 957 K corresponds to the incongruent melting of the sample (ΔH = −101 J/g). For comparison, the known thermal parameters of the A5MT(MoO4)6 crystals are listed in Table S3 [21,31,[33][34][35]61]. As it is seen in Table S3, the K5FeHf(MoO4)6 melting point is among the typical values observed for A5MT(MoO4)6 crystals. The analysis of the powder XRD pattern of the solidified melt revealed that K5FeHf(MoO4)6 is decomposed to double and simple molybdates, and it indicates that K5FeHf(MoO4)6 is an incongruent melting compound.

Vibration Dynamics
The Raman and infrared spectra collected for K5FeHf(MoO4)6 at normal conditions are shown in Figure 7. In both cases, the phonon modes are observed up to 1000 cm −1 , with a gap from approximately 450 to 700 cm −1 . The existence of such a phonon gap is typical of molybdate crystals with MoO4 tetrahedra [11,21,[62][63][64][65]. According to the group theory, the vibrational modes of K5FeHf(MoO4)6 in the center of the Brillouin zone can be listed as Γvibr = 17A1g + 19A2g + 36Eg + 18A1u + 20A2u + 38Eu, where A1g and Eg modes are Ramanactive, A1u and A2g modes are silent, A2u and Eu modes are infrared-active. The acoustic modes are Γacoustic = A2u + Eu. According to the structural data presented above, the K5FeHf(MoO4)6 structure contains one crystallographically independent MoO4 tetrahedron. The [MoO4] 2− ion is of Td symmetry and has four internal vibration modes: ν1-symmetric stretching, ν3-asymmetric stretching, ν2 and ν4-symmetric and asymmetric

Vibration Dynamics
The Raman and infrared spectra collected for K 5 FeHf(MoO 4 ) 6 at normal conditions are shown in Figure 7. In both cases, the phonon modes are observed up to 1000 cm −1 , with a gap from approximately 450 to 700 cm −1 . The existence of such a phonon gap is typical of molybdate crystals with MoO 4 tetrahedra [11,21,[62][63][64][65]. According to the group theory, the vibrational modes of K 5 FeHf(MoO 4 ) 6 in the center of the Brillouin zone can be listed as Γ vibr = 17A 1g + 19A 2g + 36E g + 18A 1u + 20A 2u + 38E u , where A 1g and E g modes are Ramanactive, A 1u and A 2g modes are silent, A 2u and E u modes are infrared-active. The acoustic modes are Γ acoustic = A 2u + E u . According to the structural data presented above, the K 5 FeHf(MoO 4 ) 6 structure contains one crystallographically independent MoO 4 tetrahedron. The [MoO 4 ] 2− ion is of T d symmetry and has four internal vibration modes: ν 1 -symmetric stretching, ν 3 -asymmetric stretching, ν 2 and ν 4 -symmetric and asymmetric bending modes, respectively. The correlation diagram for the internal MoO 4 vibrations in the K 5 FeHf(MoO 4 ) 6 crystal is given in Table 3. According to Table 3, it can be concluded that three Raman-active and three infrared-active modes exist for MoO 4 ν 1 vibrations. In the Raman spectrum, the ν 3 modes are 3A 1g + 6E g , and, in the infrared spectra, they are 3A 2u + 6E u . Thus, 12 vibrational bands should be present in the Raman, as in infrared spectra, between 700 and 1000 cm −1 . The classification of the spectral bands according to irreducible representations is extremely difficult in the case of unpolarized spectra and, as can be seen in Figure 7, these bands are strongly overlapped. For comparison, the calculated DFT wavenumbers of the Raman and infrared modes are shown in Figure 7, with vertical tick marks with the corresponding irreducible representations labels. The existence of the phonon gap between 450 and 700 cm −1 is confirmed by the calculations and a large number of modes below 450 cm −1 are observed. Fifteen of these modes are related to ν 2 and ν 4 types and observed in the Raman and infrared spectra from 270 to 450 cm −1 . The group of lines are rotation-like vibrations of MoO 4 tetrahedra. The bands below 175 cm −1 are the lattice modes of K 5 FeHf(MoO 4 ) 6 . Table 3. Correlation diagram of MoO 4 internal vibrations in K 5 FeHf(MoO 4 ) 6 .

Molecular Group
Symmetry. Site Symmetry. Space Group Symmetry Molecules 2023, 28, x FOR PEER REVIEW 10 of 25

Molecular Group
Symmetry.   In Figure 8, we show the partial phonon density of states for K 5 FeHf(MoO 4 ) 6 . The high-wavenumber part is dominated by the vibrations of oxygen atoms in the MoO 4 tetrahedra (ν 3 and ν 1 modes). It is interesting to note that the O ions are involved in almost all of the modes that appeared in the low-wavenumber region. As was mentioned above, all types of structural units yield vibrations in the region of spectra below 175 cm −1 . Several peaks related to the vibrations of Fe atoms are observed at 228, 435 and 450 cm −1 .

Symmetry. Space Group Symmetry
Molecules 2023, 28, x FOR PEER REVIEW 11 of 25 In Figure 8, we show the partial phonon density of states for K5FeHf(MoO4)6. The high-wavenumber part is dominated by the vibrations of oxygen atoms in the MoO4 tetrahedra (ν3 and ν1 modes). It is interesting to note that the O ions are involved in almost all of the modes that appeared in the low-wavenumber region. As was mentioned above, all types of structural units yield vibrations in the region of spectra below 175 cm −1 . Several peaks related to the vibrations of Fe atoms are observed at 228, 435 and 450 cm −1 .

Calculated Electronic Structure
The Brillouin zone (BZ) and calculated electronic band structure of K5FeHf(MoO4)6 are shown in Figure 9a and

Calculated Electronic Structure
The Brillouin zone (BZ) and calculated electronic band structure of K 5 FeHf(MoO 4 ) 6 are shown in Figure 9a,b, respectively. The list of k-points defining the path along the Brillouin zone should be written as Γ-T-H 2 |H 0 -L-Γ-S 0 |S 2 -F-Γ. The coordinates of these points are Γ(0,0,0), T(0.5,0.5,0.5), H 2 (0.794,0.206,0.5), H 0 (0.5,−0.206,0.206), L(0.5,0,0), S 0 (0.353,−0.353,0), S 2 (0.647,0,0.353). The conduction band minimum and the valence band maximum are located in the center of the BZ, and the calculated direct band gap is equal to 3.73 eV. The obtained band gap value is typical of crystals with MoO 4 tetrahedra [65]. As can be seen in Figure 10, the valence band maximum is dominated by the O-2p states, while the conduction band minimum is formed by Mo-4d states. Thus, we can say that the optical transitions in K 5 FeHf(MoO 4 ) 6 are associated with the charge transfer from oxygen to molybdenum ions. Some weak bands are observed in the electronic density of states (DOS), shown in Figure 10, for Fe and Hf ions, and these peaks are related to the narrow flat branches that appeared in the band structure at 1.48 and 3.14 eV above the valence band top (Figure 9b). The low-lying unoccupied levels predicted for Fe are, in fact, avatars of excited Fe 3d states that cannot be correctly calculated because the software used does not allow a switch on of the interaction between electrons within the 3d shell. However, the spectral measurements described in the next section show that the d-d transitions are really the lowest electronic excitations in the crystal under study, and the corresponding spectra look very similar to those for the isolated Fe 3+ ion, which is in agreement with the band structure calculations.

Reflection Spectrum and Experimental Bandgap
The crystal structure of K5FeHf(MoO4)6 belongs to the trigonal symmetry class and is of the R3c space symmetry group. The Fe 3+ ions are positioned within the oxygen octahedra and occupy two inequivalent positions with local symmetry C3i (occupancy 0.268) and D3 (occupancy 0.732), respectively, whilst the rest of the occupancies belong to hafnium.

Reflection Spectrum and Experimental Bandgap
The crystal structure of K5FeHf(MoO4)6 belongs to the trigonal symmetry class and is of the R3c space symmetry group. The Fe 3+ ions are positioned within the oxygen octahedra and occupy two inequivalent positions with local symmetry C3i (occupancy 0.268) and D3 (occupancy 0.732), respectively, whilst the rest of the occupancies belong to hafnium. The deviation from the undistorted Oh symmetry in both cases is rather minor, and the

Reflection Spectrum and Experimental Bandgap
The crystal structure of K 5 FeHf(MoO 4 ) 6 belongs to the trigonal symmetry class and is of the R3c space symmetry group. The Fe 3+ ions are positioned within the oxygen octahedra and occupy two inequivalent positions with local symmetry C 3i (occupancy 0.268) and D 3 (occupancy 0.732), respectively, whilst the rest of the occupancies belong to hafnium. The deviation from the undistorted O h symmetry in both cases is rather minor, and the lower crystal field symmetry components are expected to be rather weak.
In Figure 11a, the diffuse reflection spectrum of K 5 FeHf(MoO 4 ) 6 is shown; the low energy part of the Kubelka-Munk function F(R) = (1 − R) 2 /(2R), where R is the reflection coefficient, is shown in Figure 11b. The spectral features observed in the 1-3 eV range are characteristic for the d 5 electronic system and are well assignable to the transitions from the Fe 3+ ground state 6 A 1 to 4 T 1 (G) 4 T 2 (G), and 4 A 1 , 4 E(G) states. The corresponding energy for the 4 T 1 (G) state is 1.46 eV, and the narrow peak corresponding to the degenerate 4 A 1 and 4 E states is observed at 2.71 eV. In the intermediate energy range, three subbands can be located at 1.83, 2.13 and 2.25 eV; these subbands must be ascribed to the 4 T 2 (G) states at two inequivalent sites within the crystal structure, rather than to the splitting of the 4 T 2 (G) state by the low symmetry crystal field components.
The direct and indirect bandgaps were determined using the modified Kubelka-Munk functions. The fundamental absorption edge for the indirect transition in K 5 FeHf(MoO 4 ) 6 ( Figure 11c, red line) is found to be more pronounced than that of the direct transition. The indirect bandgap value is found to be 2.97 eV, while the direct bandgap in K 5 FeHf(MoO 4 ) 6 is equal to 3.21 eV. Remarkably, these bandgap values are noticeably lower than those known for a variety of molybdates. The latter are typically assigned to the transitions between the bands originating from the energy levels of the MoO 4 group. The corresponding bandgap in molybdates may be due to a charge transfer band in the system of Mo 6+ -O 2− and is expected to be observed at 3.5-4 eV; additionally, this was admitted by the calculations presented in Section 2.5. of the present paper. The bandgap narrowing in K 5 FeHf(MoO 4 ) 6 , with respect to the common molybdates, cannot be ascribed to the contribution of subbands originating from the Hf 4+ ions because the transitions originating from the Hf subsystem in the oxides are expected in the range of 5 eV.
To clarify the origin of the bandgap narrowing, we synthesized the reference crystal with the K 5 GaHf(MoO 4 ) 6 content in which Fe 3+ ions are replaced by Ga 3+ ions, featured by the closest ionic radius to Fe 3+ and by the closed shell electron configuration 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 , which is different from the opened shell 3d 5 configuration of Fe 3+ . The optical transitions starting from the closed d shell are known to be substantially higher in energy than those from the opened d shell. The obtained K 5 GaHf(MoO 4 ) 6 compound, as expected, is characterized by a crystal structure very similar to K 5 FeHf(MoO 4 ) 6 and belongs to the same space group R3c, with the cell parameters coinciding those of K 5 FeHf(MoO 4 ) 6 within 0.1%. The Kubelka-Munk function built using the diffuse reflectance data modified for the case of indirect transition is plotted in Figure 11c by the blue line. The indirect bandgap is equal to 3.24 eV, while the direct bandgap for K 5 GaHf(MoO 4 ) 6 is as wide as 3.95 eV. Therefore, both the direct and indirect K 5 GaHf(MoO 4 ) 6 bandgaps are substantially larger than those of K 5 FeHf(MoO 4 ) 6 and narrowing the bandgap of K 5 FeHf(MoO 4 ) 6 must be associated with an additional absorption in the Fe 3+ system. Three of the five forbidden d-d transitions of Fe 3+ are observed in the range below 3 eV, and the next two d-d transitions (to 4 T 2 (D) and 2 E(D) states) can lie in the region above 5 eV; however, their absorption must be of the same order of magnitude, such as that of lower states, and 4 T 2 (D) and 2 E(D) cannot contribute to the onset of the fundamental absorption edge.
The most direct hypothesis for the additional strong absorption of K 5 FeHf(MoO 4 ) 6 in the range between 3 and 4 eV is the charge transfer transition O 2− 2p-Fe 3+ 3d. However, the known charge transfer band positions of the Fe 3+ ions in oxides are typically in the range 4.4-4.8 eV (see, e.g., references in [66]). In the present situation, the only eligible explanation for the bandgap narrowing is the absorption at the Mott-Hubbard optical excitation channel (see, e.g., [67]). Mott-Hubbard optical transitions are interatomic d-d excitations. These excitations are shown to be the reason for the bandgap narrowing of gadolinium iron huntite [66]. The factor that favors interatomic excitations in the Fe 3+ system is the antiferromagnetic interaction that is known in huntites. We showed (Section 2.8) that the antiferromagnetic interaction is also present in K 5 FeHf(MoO 4 ) 6 . Surprisingly, in K 5 FeHf(MoO 4 ) 6 , Mott-Hubbard excitation in the Fe 3+ system is observed at the rather large distance of 7 Å between the Fe 3+ sites within the crystal lattice, as well as in the presence of structural Fe/Hf disorder. The direct and indirect bandgaps were determined using the modified Kubelka-Munk functions. The fundamental absorption edge for the indirect transition in K5FeHf(MoO4)6 ( Figure 11c, red line) is found to be more pronounced than that of the direct transition. The indirect bandgap value is found to be 2.97 eV, while the direct bandgap in K5FeHf(MoO4)6 is equal to 3.21 eV. Remarkably, these bandgap values are noticeably lower than those known for a variety of molybdates. The latter are typically assigned to the transitions between the bands originating from the energy levels of the MoO4 group. The corresponding bandgap in molybdates may be due to a charge transfer band in the system of Mo 6 + -O 2-and is expected to be observed at 3.5-4 eV; additionally, this was admitted by the calculations presented in Section 2.5. of the present paper. The bandgap narrowing in K5FeHf(MoO4)6, with respect to the common molybdates, cannot be ascribed to the contribution of subbands originating from the Hf 4+ ions because the transitions originating from the Hf subsystem in the oxides are expected in the range of 5 eV.
To clarify the origin of the bandgap narrowing, we synthesized the reference crystal with the K5GaHf(MoO4)6 content in which Fe 3+ ions are replaced by Ga 3+ ions, featured by the closest ionic radius to Fe 3+ and by the closed shell electron configuration 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 , which is different from the opened shell 3d 5 configuration of Fe 3+ . The optical transitions starting from the closed d shell are known to be substantially higher in energy than those from the opened d shell. The obtained K5GaHf(MoO4)6 compound, as expected, is characterized by a crystal structure very similar to K5FeHf(MoO4)6 and belongs to the same space group R 3c, with the cell parameters coinciding those of K5FeHf(MoO4)6 within 0.1%. The Kubelka-Munk function built using the diffuse reflectance data modified for the case of indirect transition is plotted in Figure 11c by the blue line. The indirect bandgap is equal to 3.24 eV, while the direct bandgap for

Electrical Properties
The electrical properties of K 5 FeHf(MoO 4 ) 6 were investigated by impedance spectroscopy. This is a reliable technique to study the electrical properties of the oxide compounds [21,[67][68][69][70][71][72][73][74]. The Nyquist plots (Z vs. Z ), recorded for the K 5 FeHf(MoO 4 ) 6 sample at different temperatures, are shown in Figure 12. The spectra were recorded in the frequency range between 1 Hz and 1 MHz. Commonly, the impedance diagram of a typical electrolyte consists of three semicircles, and each semicircle represents a distinct process. The high frequency semicircle and medium frequency semicircle represent the contributions of the grain bulk and grain boundary, while the low frequency semicircle represents the electrode contribution (C = 10 −4 −10 −5 F). In practice, not all characteristic impedance arcs can be observed in the experimental spectrum. In our experiment, the capacitance values at the level of 10 −10 F were observed for the first semicircle, which can be considered as the average values of the capacitances for the bulk and grain boundary conductivities (10 −12 and 10 −8 F). Thus, the semicircle in the −Z vs. Z plots is associated with the sum of these two contributions. Moreover, at a certain temperature, a straight line appears just after the depressed semicircle. This tail is due to the effect of the electrode processes. It can be seen that the feature of impedance is varied at different temperatures. The resistance decreases with the increasing temperature and the conductivity increases accordingly. This behavior is a characteristic of solid electrolytes, and it is in a good agreement with the results reported previously for selected oxides [21,[72][73][74].
In Figure 13, the variation in the ac conductivity (σ) with 1000/T at various frequencies is presented for K 5 FeHf(MoO 4 ) 6 . The conductivity dependences have two linear parts, with different slopes. A change of the graph in the slope is observed near the transition temperature of approximately 750 K. Below and above the phase transition, the dependences can be described by the Arrhenius-Frenkel law, i.e., the observed processes are thermally activated. This behavior is typical of conductivity in most solid electrolytes, and it may be considered as an indication of the ionic transport in K 5 FeHf(MoO 4 ) 6 . Molybdate K 5 FeHf(MoO 4 ) 6 has a comparatively low conductivity (2.9 × 10 −5 S cm −1 at 550 K), and, as a result of phase transition, it passes into the superionic state (about 10 −3 S cm −1 at 850 K). In the high-temperature range (740-893 K, cooling mode), the activation energy for the electrical conduction is~0.8 eV and the electrical conductivity reaches 4 × 10 −4 S/cm at 773 K. Then, at 893 K, it reaches a value as high as 1.1 × 10 −3 S cm −1 . The K 5 FeHf(MoO 4 ) 6 electrical conductivity values are slightly lower than those of K 5 CrHf(MoO 4 ) 6 , but higher than those of K 5 ScHf(MoO 4 ) 6 and K 5 InHf(MoO 4 ) 6 (Table 4). It is seen in Table 4 that the replacement of the trivalent element with iron did not significantly improve the electrical conductivity of the compound, but, at the same time, the influence of the trivalent cation nature is observed. In Figure 13, the variation in the ac conductivity (σ) with 1000/T at various frequencies is presented for K5FeHf(MoO4)6. The conductivity dependences have two linear parts, with different slopes. A change of the graph in the slope is observed near the transition temperature of approximately 750 K. Below and above the phase transition, the dependences can be described by the Arrhenius-Frenkel law, i.e., the observed processes are thermally activated. This behavior is typical of conductivity in most solid electrolytes, and it may be considered as an indication of the ionic transport in K5FeHf(MoO4)6. Molybdate K5FeHf(MoO4)6 has a comparatively low conductivity (2.9 × 10 −5 S cm −1 at 550 K), and, as a electrical conduction is ~0.8 eV and the electrical conductivity reaches 4 × 10 S/cm at 7 K. Then, at 893 K, it reaches a value as high as 1.1 × 10 −3 S cm −1 . The K5FeHf(MoO4)6 el trical conductivity values are slightly lower than those of K5CrHf(MoO4)6, but higher th those of K5ScHf(MoO4)6 and K5InHf(MoO4)6 ( Table 4). It is seen in Table 4 that the repla ment of the trivalent element with iron did not significantly improve the electrical co ductivity of the compound, but, at the same time, the influence of the trivalent cation ture is observed.   6 1.59 × 10 −4 0.85 (750-920 K) [35] At high temperatures, the conductivity does not significantly depend on the f quency. At the temperatures below 700 K (Figure 13), the conductivity data show the havior is strongly dependent on the frequency. It is seen that the K5FeHf(MoO4)6 cond tivity gradually increases with the increase in the applied electric field frequency. T high K5FeHf(MoO4)6 conductivity is possible due to sufficiently large sizes of voids in structure, which appreciably decrease the steric hindrances to the ion transport. The cr tal-chemistry analysis of K5FeHf(MoO4)6 shows that the Fe 3+ , Hf 4+ and Mo 6+ cations, wh are part of the crystal framework, form stronger chemical bonds with oxygen than   [35] At high temperatures, the conductivity does not significantly depend on the frequency. At the temperatures below 700 K (Figure 13), the conductivity data show the behavior is strongly dependent on the frequency. It is seen that the K 5 FeHf(MoO 4 ) 6 conductivity gradually increases with the increase in the applied electric field frequency. The high K 5 FeHf(MoO 4 ) 6 conductivity is possible due to sufficiently large sizes of voids in the structure, which appreciably decrease the steric hindrances to the ion transport. The crystalchemistry analysis of K 5 FeHf(MoO 4 ) 6 shows that the Fe 3+ , Hf 4+ and Mo 6+ cations, which are part of the crystal framework, form stronger chemical bonds with oxygen than the singly charged K + cations located in structural channels. This indicates a higher mobility of K + cations in comparison with the "framework" cations, Fe 3+ , Hf 4+ and Mo 6+ .

Magnetic Properties
The temperature dependence of K 5 FeHf(MoO 4 ) 6 magnetization in the 500 Oe magnetic field is presented in Figure 14; this dependence evidences the paramagnetic state of K 5 FeHf(MoO 4 ) 6 down to 4.2 K. At low temperatures, in the vicinity of 4.2 K, the tendency of magnetic moment value saturation can be noticed, and that is a sequence of the orientation nature of K 5 FeHf(MoO 4 ) 6 paramagnetism.
The temperature dependence of K5FeHf(MoO4)6 magnetization in the 500 Oe magnetic field is presented in Figure 14; this dependence evidences the paramagnetic state of K5FeHf(MoO4)6 down to 4.2 K. At low temperatures, in the vicinity of 4.2 K, the tendency of magnetic moment value saturation can be noticed, and that is a sequence of the orientation nature of K5FeHf(MoO4)6 paramagnetism. In Figure 15, the field dependences of magnetization are presented, as recorded at different temperatures. These curves evidence that, at all temperatures, including the liquid helium temperature (T = 4.2 K), the coercive force is equal to zero. This indicates the absence of the magnetic domains that are present in magnetically ordered crystals. Therefore, the field dependences of M(H) also confirm the statement that the crystal under study is a paramagnet in a good approximation. The preservation of the paramagnetic state is associated with large distances between the magnetic ions of iron, which leads to a small exchange integral value comparable to the dipole-dipole interaction. In this case, the magnetic ordering can be expected only at temperatures below 1 K. In Figure 15, the field dependences of magnetization are presented, as recorded at different temperatures. These curves evidence that, at all temperatures, including the liquid helium temperature (T = 4.2 K), the coercive force is equal to zero. This indicates the absence of the magnetic domains that are present in magnetically ordered crystals. Therefore, the field dependences of M(H) also confirm the statement that the crystal under study is a paramagnet in a good approximation. The preservation of the paramagnetic state is associated with large distances between the magnetic ions of iron, which leads to a small exchange integral value comparable to the dipole-dipole interaction. In this case, the magnetic ordering can be expected only at temperatures below 1 K. The paramagnetic Curie temperature θ determined from the temperature dependence of the inverse susceptibility ( Figure 16) is equal to θ = 6.7 K. This Curie temperature value indicates the presence of a weak antiferromagnetic interaction in the Fe subsystem of K5FeHf(MoO4)6. The effective magnetic moment corresponding to the Curie temperature value is equal to μef = 5.97 μB. Assuming that only Fe 3+ ions possess a magnetic moment among all of the ions in a crystal, the theoretical value of the magnetic moment per formula unit equals μ theor = gs•μB•(S(S + 1)) 1/2 , where gs = 2 is the degeneracy factor accounting only the spin contribution to the magnetic moment, and S = 5/2 is the Fe 3+ ion spin. The corresponding theoretical value μ theor = 5.48 μB is in good agreement with the experi- The paramagnetic Curie temperature θ determined from the temperature dependence of the inverse susceptibility ( Figure 16) is equal to θ = −6.7 K. This Curie temperature value indicates the presence of a weak antiferromagnetic interaction in the Fe subsystem of K 5 FeHf(MoO 4 ) 6 . The effective magnetic moment corresponding to the Curie temperature value is equal to µ ef = 5.97 µ B . Assuming that only Fe 3+ ions possess a magnetic moment among all of the ions in a crystal, the theoretical value of the magnetic moment per formula unit equals µ theor = g s ·µ B ·(S(S + 1)) 1/2 , where g s = 2 is the degeneracy factor accounting only the spin contribution to the magnetic moment, and S = 5/2 is the Fe 3+ ion spin. The corresponding theoretical value µ theor = 5.48 µ B is in good agreement with the experimental one.

Materials and Methods
Commercially available Fe2O3 (chemically pure, Ural Plant of Chemical Reagents, Russia), HfO2 (chemically pure, IGIC RAS, Russia) and MoO3 (chemically pure, Red Chemist, Russia) were used as the starting materials for the synthesis of simple molybdates. Potassium molybdate K2MoO4 (pure) was supplied by Red Chemist, Russia. Iron molybdate Fe2(MoO4)3 and hafnium molybdate Hf(MoO4)2 were prepared at 673-1023 K for 120 h. First, the simple oxides were well mixed according to the nominal compositions of Fe2(MoO4)3 and Hf(MoO4)2 and ground in an agate mortar. For better reactivity, the reaction mixtures were then progressively calcined at 673-1023 K, and intervened by mixing and grinding every 24 h of annealing. In the synthesis, after every 24 h of annealing, the reaction cakes were cooled, grinded and mixed for better reactivity. The phase purity of the final products was verified by the XRD analysis in reference to the known structures of Fe2(MoO4)3 (01-083-1701 [76]) and Hf(MoO4)2 (00-038-1467 [77]).
Simple K2MoO4, Fe2(MoO4)3 and Hf(MoO4)2 molybdates were used in the study of the phase formation in the quasi-ternary system K2MoO4-Fe2(MoO4)3-Hf(MoO4)2. In addition, these simple molybdates were applied for the preparation of powder samples and the single crystal growth of K5FeHf(MoO4)6.
Powder X-ray diffraction measurements were carried out with the use of a Bruker D8 ADVANCE diffractometer (CuK radiation, Vantec-1 detector, maximum angle 2 = 100°). The step size of 2θ was 0.0208°, and the counting time was 2 s per step. The XRD patterns were recorded at room temperature and normal atmospheric pressure. To identify the sample phase composition, the ICDD database was used. As we see, K 5 FeHf(MoO 4 ) 6 crystals preserve their paramagnetic state in the temperature range down to 4.2 K, in contrast to some other double molybdates, such as CsFe 5 (MoO 4 ) 7 [75] or RbFe 5 (MoO 4 ) 7 [22], which exhibit their magnetic ordering below 20 K. The paramagnetic Curie temperatures in those two compounds are two times larger than in K 5 FeHf(MoO 4 ) 6 , and, hypothetically, the magnetic ordering in K 5 FeHf(MoO 4 ) 6 could be expected in the vicinity of 10 K, and that was not found. Most likely, this is due to the peculiarities of the magnetic structure in our crystal.

Materials and Methods
Commercially available Fe 2 O 3 (chemically pure, Ural Plant of Chemical Reagents, Russia), HfO 2 (chemically pure, IGIC RAS, Russia) and MoO 3 (chemically pure, Red Chemist, Russia) were used as the starting materials for the synthesis of simple molybdates. Potassium molybdate K 2 MoO 4 (pure) was supplied by Red Chemist, Russia. Iron molybdate Fe 2 (MoO 4 ) 3 and hafnium molybdate Hf(MoO 4 ) 2 were prepared at 673-1023 K for 120 h. First, the simple oxides were well mixed according to the nominal compositions of Fe 2 (MoO 4 ) 3 and Hf(MoO 4 ) 2 and ground in an agate mortar. For better reactivity, the reaction mixtures were then progressively calcined at 673-1023 K, and intervened by mixing and grinding every 24 h of annealing. In the synthesis, after every 24 h of annealing, the reaction cakes were cooled, grinded and mixed for better reactivity. The phase purity of the final products was verified by the XRD analysis in reference to the known structures of Fe 2 (MoO4) 3 (01-083-1701 [76]) and Hf(MoO 4 ) 2 (00-038-1467 [77]).
Simple K 2 MoO 4 , Fe 2 (MoO 4 ) 3 and Hf(MoO 4 ) 2 molybdates were used in the study of the phase formation in the quasi-ternary system K 2 MoO 4 -Fe 2 (MoO 4 ) 3 -Hf(MoO 4 ) 2 . In addition, these simple molybdates were applied for the preparation of powder samples and the single crystal growth of K 5 FeHf(MoO 4 ) 6 .
Powder X-ray diffraction measurements were carried out with the use of a Bruker D8 ADVANCE diffractometer (CuK α radiation, Vantec-1 detector, maximum angle 2θ = 100 • ). The step size of 2θ was 0.0208 • , and the counting time was 2 s per step. The XRD patterns were recorded at room temperature and normal atmospheric pressure. To identify the sample phase composition, the ICDD database was used.
The single-crystal X-ray diffraction data for K 5 FeHf(MoO 4 ) 6 (I) were collected on a Bruker Nonius X8 Apex diffractometer equipped with the graphite monochromated MoK α (λ = 0.71073 Å) radiation at 293(2) K. The ϕ-and ω-scan technique was employed to measure the intensities. Absorption corrections were applied empirically using the SADABS program [78]. The structure was solved by the direct methods of the difference Fourier synthesis and further refined by the full-matrix least squares method using the SHELXTL package [79]. The atomic thermal parameters for all of the atoms were refined anisotropically.
Differential scanning calorimetric (DSC) measurements were performed on an NET-ZSCH STA 449C TG/DSC/DTA (Jupiter) thermoanalyzer over the temperature range of 373-1000 K with the heating/cooling rates of 10 K min −1 Ar flow. Pt crucibles were used as vessels. Pt-PtRh thermocouples were applied for the temperature control. The temperature measurement precision was ±1 K. The DSC curves were calculated using a specially developed program from Netzsch. The Raman spectra were recorded with the help of the Horiba Jobin Yvon T64000 spectrophotometer using the excitation from the Spectra Physics Excelsior 532 nm laser. The spectral resolution of the Raman spectra was 2.7 cm −1 . The infrared spectra were recorded with the help of the Bruker VERTEX 80 FTIR spectrometer.
The electrical conductivity was measured by the two-contact impedance spectroscopy method as a function of the temperature and frequency exploring an impedance meter Z-1500J. The measurements were performed in the heating/cooling mode in the frequency range of 1 Hz-1 MHz. The pure phase powder of K 5 FeHf(MoO 4 ) 6 was used for the tablet fabrication. The tablets with the thickness of 2 mm and 10 mm in diameter were prepared with the use of a hydraulic laboratory press PLG-12 at 100 bar and then sintered at 773 K for 2 h. For the making of electrodes, large disk surfaces were covered with a paste-type mixture of hexachloroplatinate (IV) ammonium (NH 4 ) 2 [PtCl 6 ] in toluene. Then, the tablet with the paste layers was annealed at 773 K for 1 h. The electrical conductivity σ total for each fixed temperature was calculated by relation: where σ total , L, S and R total are the total conductivity, thickness of specimen, area of the big tablet surface and the total resistance, respectively. The magnetic properties of K 5 FeHf(MoO 4 ) 6 were investigated using vibrating magnetometer Quantum Design PPMS (Krasnoyarsk Regional Center of Research Equipment of Federal Research Center "Krasnoyarsk Science Center SB RAS") over the 4.2-300 K temperature range at magnetic fields up to 9 T.

Computation Details
The ab initio calculations were carried out using the density function theory (DFT) approach implemented within the CASTEP code [80]. The Vanderbilt-type ultra-soft pseudopotentials were used, and the energy cutoff was taken as 450 eV. The 3s 2 3p 6 4s 1 , 3d 5 4s 2 , 5d 2 6s 2 , 4s 2 4p 6 4d 5 5s 1 , 2s 2 2p 4 valence electron configurations were used for K, Fe, Hf, Mo and O, respectively. The model structure of K 5 FeHf(MoO 4 ) 6 was fully optimized using the generalized gradient approximation (GGA) with the PBESol (Perdew-Burke-Ernzerhof for solids) exchange-correlation functional [81]. As a starting point, the structural parameters were used, as obtained in the present work. The convergence tolerance was set to 0.005 eV/Å and 0.02 GPa for the maximum force and maximum stress, correspondingly. The 3 × 3 × 3 Monkhorst-Pack mesh [82] was used for the Brillouin zone sampling.

Conclusions
A new, structurally disordered molybdate K 5 FeHf(MoO 4 ) 6 was synthesized and found to crystallize in the trigonal space group R3c. The direct band gap value obtained from the first-principles calculations is 3.73 eV. According to the results of the lattice dynamics density functional theory calculations and the group theory analysis, most of the spectral lines above 175 cm −1 in the vibrational spectra of K 5 FeHf(MoO 4 ) 6 are associated with the MoO 4 vibrations. The calculations of the partial phonon density of states revealed that the translations of oxygen ions involved almost all of the vibrational modes. K 5 FeHf(MoO 4 ) 6 is a paramagnetic down to 4.2 K. The negative Curie-Weiss temperature of −6.7 K indicates dominant antiferromagnetic interactions in the compound. The direct and indirect optical bandgaps of K 5 FeHf(MoO 4 ) 6 are 3.21 and 2.97 eV. The bandgap narrowing, with respect to a variety of molybdates and to the ab initio calculations, is explained by the Mott-Hubbard interatomic optical excitation channel.