Incommensurate structures of the [CH3NH3][Co(COOH)3] compound

The temperature evolution of [CH3NH3][Co(COOH)3] on a single crystal and the crystal structure determination of intermediate incommensurate phases of [CH3NH3][Co(COOH)3] by means of neutron diffraction are presented.


Introduction
The development and characterization of new materials are key challenges in condensed matter chemistry and physics. The ability of metal-organic frameworks (MOFs) to combine within the same framework different physical properties has attracted much interest in recent years (Zhu & Xu, 2014;Cui et al., 2016;Lin et al., 2014;Coronado & Espallargas, 2013;Liu et al., 2014Liu et al., , 2016Li et al., 2016). These compounds constitute a promising approach for combining paradielectric, ferroelectric or antiferroelectric behaviours with long-range magnetic order. This new generation of materials are the multiferroic metal-organic frameworks (Jain et al., 2009;Rogez et al., 2010;Tian et al., 2014). To date, most of the known multiferroic materials are purely inorganic perovskites. The most abundant representative compounds exhibiting such behaviour are, certainly, perovskite oxides with the ABO 3 formula (Shang et al., 2013;Van Aken et al., 2004;Vrejoiu et al., 2008;Rout et al., 2009;Khomchenko et al., 2011;Catalan & Scott, 2009). One example is the BiFeO 3 compound, which represents a rare case with magnetic and ferroelectric ordering coexisting at room temperature (Lebeugle et al., 2008). Another important material that marked the beginning of the success of this family of compounds is YMnO 3 , characterized by a multiferroic behaviour where the ferroelectric order derives from geometrical effects (Van Aken et al., 2004). A more recent system in which a different magnetoelectric coupling can be found is TbMnO 3 , whereby ferroelectricity occurs as a consequence of a special kind of magnetic order (Kenzelmann et al., 2005). All the multiferroic materials cited above are only inorganic-based perovskites, but a first exception was introduced in the literature by Jain et al. (2008) with a hybrid inorganic-organic framework material of the general formula ABX 3 , in which A is an organic cation, B is the metal centre and X is an organic bridging ligand. The presence of organic molecules contributes to the formation of hydrogen bonds that are often responsible for structural phase transitions, giving ferroelectric behaviour to the compound (Ramesh, 2009). In this context, formate-based metal-organic compounds have been revealed to exhibit a combination of both dielectric and magnetic orders (Lawler et al., 2015;Qin et al., 2015;Mą czka, Gą gor et al., 2017;Mą czka, Janczak et al., 2017). These compounds are typically synthesized by reaction of the formate ligand with a metal salt under solvothermal conditions or by slow evaporation or diffusion techniques. Typically, these compounds present a three-dimensional framework constructed from the formate ligand and metal ions, where a counter-ion is located in the cavities. The formate ligand has the ability to mediate ferro-or antiferromagnetic interactions between the connected metal ions, depending on its coordination mode, promoting long-range magnetic order in the framework. The paradielectric, ferroelectric or antiferrolecric order is normally achieved due to order-disorder of the counter-ion within the cavities, triggered by slight differences in the hydrogen-bonded network. Then, the adequate combination of a well known framework with different ammonium-based counter-ions could be used to favour the occurrence of structural transitions and change the weak interaction network of the sample, promoting remarkable changes in its physical properties. Hence, the possible combination of long-range magnetic order with electric order makes these compounds excellent candidates for the development of multiferroic behaviour (Wang et al., 2007;Xu et al., 2011;Cañ adillas-Delgado et al., 2012;Mą czka et al., 2016).
The occurrence of an order-disorder transition can involve the blocking of the libration of the counter-ion or the reorganization of these molecules in the cavities. These effects produce a symmetry breaking of the system. Sometimes a doubling of the crystallographic axes is needed in order to explain the physical nature of the phase transition (Cañ adillas-Delgado et al., 2012). However, more complex order-disorder transitions are also possible. The competition between atomic interactions in pure inorganic ABO 3 perovskites is responsible for the occurrence of unusual instabilities, which occasionally lead to incommensurate structures (Arakcheeva et al., 2017;Khalyavin et al., 2015;Lin et al., 2015;Aré valo-Ló pez et al., 2015;Du et al., 2014;Szczecinski et al., 2014;Magdysyuk et al., 2013). The atomic interactions responsible for the incommensurate distortions involve a counter-ion displacement, as well as antiferrodistorsive motions, mainly the tilting of the oxygen octahedron centred in the B-site.
Among the low number of incommensurate structures reported in the Bilbao Crystallographic Server database (Aroyo et al., 2011) (149 entries of which 24 are composites),  there are no examples of organic-inorganic perovskite-like  systems, although a similar scenario to the pure perovskite  compounds is expected (see, for example, Fü tterer et al., 1995). To the best of our knowledge, the closest example is the (C 6 H 11 NH 3 ) 2 [PbI 4 ] compound, which presents a hybrid layered perovskite-like structure (K 2 NiF 4 -type). This compound crystallizes in the orthorhombic space group Pbca at room temperature (RT), and below 128 K it presents a structural phase transition to the superspace group Pca2 1 ( 1 2 0), disturbing its physical properties (Yangui et al., 2015). Furthermore, other metal-organic layer-based examples, such as n-propylammonium manganese chloride, have also been studied previously (Depmeier, 1981).
The refinement of modulated structures requires the use of the superspace formalism in which every structural parameter p ð Þ, (e.g. atom positions, displacement parameters, occupation factors etc.) is described in terms of the average parameter p 0 ð Þ and a periodic modulation function of the internal coordinate x 4 . The internal coordinate is defined as Þþt, where q is the modulation vector, g is a phase reference point that, in our case, is the atomic position of the average structure, n defines the position of the current unit cell and t is the phase factor. The actual value of the parameter p for a particular atom in the incommensurate phase is given by where p 0 is the value of the parameter in the average structure, p sn and p cn are the amplitudes of the displacement modulation and n is the order of harmonics in the Fourier series. Then, for each structural parameter affected by modulation, it is necessary to refine p 0 , p sn and p cn in order to fully determine the incommensurate structure (Petřiček et al., 2016). Formate compounds have a predilection for the 4 12 Á6 3 -cpu topology, where the formate group acts as bis-monodentate in an anti-anti coordination mode. This coordination gives rise to structures with medium-sized cavities, where an adequate guest molecule can be located in order to promote electric order. The occurrence of an intricate hydrogen-bonded network is mainly due to the ability of the formate anion to act as a proton acceptor. As in the case of pure inorganic compounds, the interaction between the guest molecule and the framework can be the driving force of unusual structural instabilities, promoting incommensurate structures.
The present article is devoted to the study of a series of phase transitions of the [CH 3 NH 3 ][Co(HCOO) 3 ] (1) formate compound at temperatures below the well known Pnma orthorhombic phase. In previous studies, a phase transition between the space group Pnma (at 135 K) and the monoclinic space group P2 1 /n (at 45 K) has been reported and related with changes in the electrical behaviour of compound 1 (Mazzuca et al., 2018). Here, we will describe the occurrence of orthorhombic incommensurate structures upon cooling from 135 to 45 K. It should be noted that modulation parameters research papers 106 Laura Canadillas-Delgado et al. Incommensurate structures are usually very sensitive to defects in the basic structure model. Moreover, determination of the atomic position in each phase is fundamental to understanding the mechanism of these phase transitions. In order to consider all atoms, including the hydrogen atoms, and to establish the hydrogenbond network, single-crystal neutron diffraction measurements at different temperatures were carried out to fully determine the modulated crystal structures.

Single-crystal neutron Laue diffraction measurements
The Laue diffraction measurements were collected on the multiple CCD diffractomer CYCLOPS (Cylindrical CCD Laue Octagonal Photo Scintillator) at ILL (Grenoble, France) (Ouladdiaf et al., 2011). The Laue pattern permits us to perform a fast exploration of the reciprocal space as a function of an external parameter. A single crystal of about 36 mm 3 was mounted on a vanadium pin and placed in a standard orange cryostat, the diffraction patterns were recorded in the temperature range from 140 to 65 K, following a ramp of 0.1 K every 3 min. The sample was centred on the neutron beam by maximization of the intensity of several strong reflections in the x, y and z directions, after which, a specific orientation was selected and the temperature evolution was collected. Each Laue diffraction pattern was collected over a period of 15 min with a difference of temperature of 0.5 K. From the temperature evolution of the Laue diffraction pattern, the occurrence of several unknown phases was observed. The graphical visualization of the Laue patterns, as well as the indexing of commensurate phases, was performed using the ESMERALDA software developed at ILL (Fig. 1) .
Although there is a notable change in the Laue diffraction pattern (Fig. 1) as a function of temperature, the preliminary specific heat measurement did not show any signal in this temperature range. Above 128 K, the Laue pattern can be indexed using the Pnma orthorhombic unit cell (Table 1 and Fig. 1). Between 128 and 78 K, the occurrence of satellite reflections suggests the presence of incommensurate structures. The evolution of these satellites as a function of temperature indicates a variation in the wavevector, while the change in intensity of the main and satellite reflections points to a change in the modulation amplitude and therefore a structural evolution (Fig. 2). As a result of this crystal structure evolution, below 78 K the crystal can be indexed using a commensurate monoclinic unit cell, although the presence of two twin domains is observed in the Laue diffraction pattern, in agreement with the results reported previously from singlecrystal data obtained at 45 K (Mazzuca et al., 2018). It should be noted that there is a clear correlation between the different phases, probably due to a group-subgroup relation.
The temperature evolution of the Laue diffraction patterns shows that the orthorhombic reflections remain almost at the same positions in the incommensurate phases; however, the intensity of these main reflections diminishes as the temperature decreases, particularly in the vicinity of the monoclinic phase transition. As shown in Fig. 2, the behaviour of the first-order satellites is the opposite; as the temperature decreases, the intensity of the satellites increases. Up to second-order satellites are observed for the strongest reflections. However, below a critical temperature, the main reflections from the orthorhombic phase and the first and higher order satellites abruptly disappear and new reflections belonging to the monoclinic phase are then observed. Although the monoclinic reflections appear close to the firstorder satellites, the non-coexistence of these phases in the single-crystal measurements, together with the abrupt change Temperature evolution of the Laue patterns: (a) corresponds to the orthorhombic phase collected at 138 K, (b)-(i) correspond to an enlarged area, highlighted in (a) and (j) with a white rectangle. The (b)-(i) patterns were collected at approximately 135.5, 127.5, 122.5, 114, 105, 94, 81 and from an incommensurate to a monoclinic phase, which takes place in less than 0.5 K, preclude the unambiguous definition of this last phase transition. Although different thermal treatments have been used (fast or slow cooling), at low temperature the compound always becomes a two-domain monoclinic twinned crystal. The observed twin law fð100Þð010Þð001Þg corresponds to a rotation of 180 around the crystallographic a* axis (in the monoclinic setting). It should be noted that after thermal treatment, and above 135 K, the orthorhombic commensurate phase (Pnma) is recovered without any evidence of damage to the sample.

Monochromatic single-crystal neutron diffraction measurements
Monochromatic diffraction data were collected on the fourcircle D19 diffractometer at ILL (Grenoble, France) with Cu(220)-monochromated radiation (take-off angle 2 M = 69.91 ), providing neutrons with a wavelength of 1.456 Å , which is a good compromise between instrumental resolution, data completeness and the overlapping of neighbouring reflections in the incommensurate phases. The same sample used for Laue diffraction was used on D19. The sample was placed on a closed-circuit displex device, which was operated following a ramp of 2 K min À1 . The sample and the beam stability were checked by collecting a short scan around the ð513Þ reflection. The measurement strategy consists of several ! scans with steps of 0.07 at different and positions. The collected data set on the incommensurate phases consists of 25 long ! scans at 122 (2), 106 (2) and 86 (2) K and 21 scans at 90 (2) K.
In previous work, two extra data sets were collected at 135 (2) and 45 (2) K in the orthorhombic and monoclinic phases, above and below the incommensurate phases. At 135 K, we collected 25 ! scans at 0.949 Å , while at 45 K, we  Table 1 Experimental and crystallographic data of compound 1, measured on the single-crystal neutron diffractometer D19 and refined with JANA2006.  Temperature evolution of the integrated intensities of the ð161Þ reflection in the orthorhombic phase above 128 K. Below 128 K, the first-order satellites with index ð1; 6; 1:14Þ and ð1; 6; 0:86Þ, corresponding to (hkl) AE q, are observed. Below 90 K, a shift in the position of the satellites together with an increase in intensity of the second-order satellites is noticeable. This shift in the position of the reflections corresponds to the change of the wavevector from q = [0, 0, 0.1430 (2)] to q = [0, 0, 0.1247 (2)], while the change in intensity of these satellites is related to changes in the modulation amplitudes. Below 78 K, two single reflections are observed. These two reflections were indexed with the indices ð161Þ and ð161Þ, corresponding to the same diffraction plane belonging to two different twin domains. The inset highlights the region of interest used during the integration. acquired 35 ! scans at 1.454 Å (Mazzuca et al., 2018). Although the crystallographic studies on these phases are not the objective of this work, a brief description of these phases will be included for completeness. The multi-detector acquisition data software (MAD) from ILL was used for data collection. Unit-cell determinations were performed using PFIND and DIRAX programs, and processing of the raw data was applied using RETREAT and RAFD19 programs (Duisenberg, 1992;Wilkinson et al., 1988).
The calculation of possible wavevectors was carried out using the DIRAX program (Duisenberg, 1992) and the full data set was indexed with a single wavevector in the form q = c*. Second-or third-order satellites were observed depending on the temperature range. At each temperature, the indexed wavevector was used to obtain a supercell. With this supercell, all reflections, main and satellites, were successfully integrated. The decomposition into main and satellite reflections following the superspace formalism was carried out using the new D19 software SATELLITE. An absorption correction was applied using D19ABS (Matthewman et al., 1982). The structures were solved with SUPERFLIP (Palatinus & Chapuis, 2007) using a chargeflipping algorithm.

Structural determination and refinement details
Full-matrix least-squares refinement on |F 2 | using SHELXL2014/76 as implemented in the program WinGX was used for structure refinement of the high-temperature phase (commensurate orthorhombic phase), while for the lowtemperature data, the crystal structures were solved using the SUPERFLIP program (Palatinus & Chapuis, 2007). SUPER-FLIP was used to determine the non-hydrogen-atom positions, while the hydrogen atoms were located using difference Fourier maps. The incommensurate phases were refined using the superspace formalism included in the JANA2006 program (Petřiček et al., 2016), which is currently the only available program able to handle this formalism. Exploration of the three-dimensional +1 Fourier density maps clearly indicates a displacive character close to harmonicity for both framework and counter-ion. Therefore, the displacement parameters of the different atoms were included in the refinement. After the convergence of the model, all atoms -including hydrogen atoms -were refined with anisotropic displacement parameters (ADPs). Then, the first-and second-order harmonic waves of the ADPs were introduced into the model, which take into account the changes in the crystal structure modulation. The ratios between main and satellite reflections are 0.26 for the 122 (2) and 106 (2) K data sets, and 0.11 and 0.18 for the 90 (2) and 86 (2) K data sets, respectively. The decrease of the ratio between the main and satellite reflections can be attributed to the occurrence of third-order satellite reflections in the 90 (2) and 86 (2) K data sets. There are 333 refined parameters for all incommensurate structures; a summary of the experimental and crystallographic data is given in Table 1.
The commensurate monoclinic phase collected at 45 K was refined using JANA2006 (Petřiček et al., 2016) against F o 2 data using the full-matrix least-squares algorithm. The occurrence of a twin at this temperature was taken into account during the data refinement. The contribution of each twin domain was close to 50%. In the final refinements, all atoms, including the hydrogen atoms, were refined with ADPs. Graphical representations of all phases were produced using the program DIAMOND (Version 4.4.0; Brandenburg & Putz, 1999). Crystallographic data, in CIF format, for the structures of phases II and III have been deposited at the Bilbao Crystallographic Server Database with reference number 13542El8AS4.

Results and discussion
Although the structures of compound 1 in the orthorhombic space group Pnma at high temperature and in the monoclinic phase (P2 1 /n space group) at low temperature have already been reported (Boča et al., 2004;Gó mez-Aguirre et al., 2016;Mazzuca et al., 2018), for the sake of clarity in the comparison of these two phases with the incommensurate phases, we will give some details about them. The description of the different phases will be given as a function of the decreasing temperature, although a similar behaviour is observed on heating.

Commensurate orthorhombic phase
Compound 1 crystallizes in the orthorhombic space group Pnma between RT and $128 K (phase I). The crystal structure consists of a three-dimensional anionic [Co(HCOO) 3 ] À framework with a 4 12 Á6 3 -cpu perovskite-like topology (Schlä fli notation), where the crystallographically independent cobalt(II) ion is located in an inversion centre and is sixcoordinated in an almost ideal CoO 6 octahedron. The cobalt(II) atoms are connected through formate ligands in an anti-anti conformation along the ½101, ½010 and ½101 directions (BO 3 sites of the Perovskite structure). In order to achieve electroneutrality, methylammonium [CH 3 NH 3 ] + counter-ions fill the cavities of the framework (A sites of the Perovskite structure). This molecule sits in a mirror plane that crosses the molecule through the nitrogen and carbon atoms, parallel to the ac plane. At room temperature, the three hydrogen atoms connected to the carbon atom in the methylammonium molecule are disordered over two different positions with occupancy factors of 0.5. However, at 135 K, before the first phase transition takes place, all three hydrogen atoms sit in single positions. There are two crystallographically independent hydrogen atoms, one is crossed by the mirror plane and the other generates the third hydrogen atom by symmetry. Note that these hydrogen atoms are not involved in any hydrogen bonding, nevertheless the hydrogen atoms connected to the nitrogen atom of the counter-ion establish two different hydrogen bonds (Fig. 3). These hydrogen bonds between the counter-ion and the three-dimensional network contribute to the stabilization of the whole structure.

Incommensurate structures
The temperature dependence of the Laue diffraction patterns reveals the occurrence of two different incommensurate phases (Fig. 1). Below 128 K, the occurrence of weak new reflections close to the high-temperature orthorhombic reflections suggest a first phase transition from the orthorhombic commensurate phase (I), crystallized in the space group Pnma, to an incommensurate phase (II). The indexing of both main and satellite reflections gives rise to an incommensurate unit cell with the wavevector q, with a unique component along the c axis. The full data set collected at 122 (2) K suggests a wavevector of the form q = 0.1430 (2)c*. The same result was obtained for the full data set collected at 106 (2) K. However, for the data collected at 90 (2) and 86 (2) K, the modulation is clearly longer with wavevector q = 0.1247 (2)c* (Table 1). This change in modulation length, with values of [c/q z ] of 56.98 (1), 57.02 (2), 65.46 (2) and 65.37 (2) Å for 122 (2), 106 (2), 90 (2) and 86 (2) K, respectively, suggest the presence of two different incommensurate phases. Herein, the crystal structures refined in the tempera-ture range from 128 to 96 K will be called phase II and those refined between 96 and 78 K will be named phase III.
The refined wavevectors are close to being commensurate. The length of the c axis, as well as the unit-cell volume in the incommensurate phases (from 128 to 78 K), is close to seven or eight times bigger than in the commensurate orthorhombic phase (Pnma phase I). In a preliminary refinement, a supercell (assuming a strictly commensurate unit cell) was used to determine a crystal structure model. The model was solved in the three-dimensional space group P2 1 2 1 2 1 . However, the refined model, using the three-dimensional space group, gives rise to an unstable refinement, which can only converge after applying constraints. Therefore, even if we assume that phases II and III are almost commensurate, the quality of the data refinements is significantly better if we use the superspace group formalism. In the superspace formalism, each atom follows a curve forming the so-called 'atomic domain'. This curve, defined as a modulation function, can be described by a periodic function characterized by a set of refined parameters. After each refinement, a four-dimensional density map can be    View of the possible hydrogen bonds, denoted in discontinuous blue and green lines, involving the methylammonium cation in (a) the orthorhombic commensurate phase, (b) the incommensurate phases and (c) the commensurate monoclinic phase. The distances denoted in grey are too large to be considered as hydrogen bonds. In the incommensurate phases, the distances H1NÁ Á ÁO3d and H1NÁ Á ÁO3e are in the ranges 2.075-2.133, 2.009-2.132, 1.973-2.113 and 1.988-2.123 Å at 122, 106, 90 and 86 K, respectively (see details in Tables 4 and 5). Symmetry codes: ðdÞ x þ 1 2 , Ày þ 1 2 , Àz À 1 2 ; ðeÞ x þ 1 2 , y, Àz À 1 2 ; ðfÞ À x þ 3 2 , Ày, z À 1 2 ; ðgÞ x, Ày þ 1 2 , z; ðhÞ À x þ 3 2 , y þ 1 2 , z þ 1 2 ; ðiÞ À x þ 3 2 , y þ 1 2 , Àz þ 1 2 ; ðjÞ À x þ 1 2 , y þ 1 2 , Àz þ 1 2 .
calculated and different two-dimensional sections through a specific atom can be calculated. The shape of the modulation function must reproduce the modulation of the atomic domain in the Fourier maps, as it occurs (Fig. 4). The determination of the superspace group was carried out with the help of the SUPERFLIP program for each data set from 122 (2) to 86 (2) K. Although a change in the modulation length occurs between 106 (2) and 90 (2) K, there is no breaking of symmetry and the Pnma(00)0s0 superspace group remains invariant in these temperature ranges (see details in Table 1).
Based on the determined superspace group, the average structure is described in the space group Pnma. Each independent atom in the average structure is modulated by the application of a modulation function. This modulation function should be defined for each data set; however, in the current case, all exhibit a sinusoidal behaviour, as shown in the sections of the four-dimensional Fourier maps (Fig. 4). The refined modulation functions for the cobalt atom at 86 (2), 90 (2), 106 (2) and 122 (2) K, represented as a blue curve in Fig. 4, are in good agreement with the experimental (F obs ) Fourier maps. The shape of this modulation suggests a continuous character, with an increase in the displacive modulation with decreasing temperature. The refined model shows that the amplitudes of the displacive modulation have the main components along the b axis. A summary of the refined amplitude displacements for cobalt(II) and the carbon and nitrogen atoms of the methylammonium counter-ion, as representative of the framework and guest molecule, can be found in Tables 2 and 3. Note that, due to symmetry restrictions, the modulation of the cobalt(II) ion presents only the sine term of the Fourier coefficients (see Table 2), while for the other atoms, both sine and cosine terms are present (see Table 3).
Based on the atomic positions present in the average structure (8d, 4c or 4b Wyckoff positions of the Pnma average structure), symmetry constraints are applied to the sine or cosine terms of the Fourier coefficients. This implies that slight tilts or distortions of the CoO 6 octahedron are allowed by symmetry. However, these terms are notably smaller than those responsible for the modulation along the b axis. Therefore, the final model presents small differences between the modulation of the [Co(COOH) 3 ] À framework and the [CH 3 NH 3 ] + counter-ions. The distortion of the CoO 6 octahedron [the maximum variation in the Co-O bond distances is ca 0.05 (1) Å ] is much smaller than the variation of the cobalt(II) atom position due to the modulated displacement [maximum displacement = 0.375 (6) Å at 86 K] (Fig. 5).
A graphical representation of the bond-distance modulation for each Co-O bond in phases II and III is shown in Fig.  5. It is interesting to observe that at any t value [t = x 4 (mod 1)], all the Co-O bond distances are in the range from 2.117 (5) to 2.075 (6) Å for phase II and from 2.125 (6) to 2.080 (7) Å for phase III, which denotes that the CoO 6 octahedron remains an almost ideal octahedral environment in the whole range of t. The modulation function (Figs. 5b-5e) shows temperature dependence, since the amplitude functions are also dependent on temperature. Furthermore, the change in the modulation wavelength from q = 0.1430 (2)c* to q = 0.1247 (2)c* produces a drastic change in the shape of the modulation function, which is in agreement with the two phases description. Finally, we would like to mention that the average displacement value, which is maximal along the b axis, is much larger than the variation of the bond distances due to the modulation function (Fig. 5). The same effect is observed in the [CH 3 NH 3 ] + counter-ions: the variation in the individual N-C, C-H and N-H bond lengths, as well as the variation in the C-N-H or H-C-N angles are much smaller than the average displacement values.
A graphical description of the modulated structure for each phase, i.e. II at 122 K and III at 86 K, is represented in Fig. 6. The graphical representation has been carried out considering a supercell (ten times the average unit cell along the c axis), in order to include at least a complete period. Tables 4 and 5 show the possible hydrogen bonds defined along the incommensurate structure using a fraction of the t parameter for phase II (122 and 106 K) and III (90 and 86 K), respectively (Fig. 3). Along t, not all DÁ Á ÁA distances correspond to the expected range for an ideal hydrogen bond. The distances and angles not considered to be hydrogen bonds are shown in italics. As shown, the distances elongate as a function of increasing temperature, which is compatible with an increase from thermal expansion. However, close to 106 K, we observed a slight alteration of this tendency, which is related to the proximity to the phase transition from II to III.
Slight variations in the hydrogen-bond network, which are in agreement with the subtle changes in the angles and distances, contribute significantly to the explanation of the occurrence of two incommensurate phase transitions. In Tables 4 and 5 Table 3 Amplitude displacement for the sine and cosine terms of the first order of the harmonics in the Fourier series corresponding to the N and C atoms from the dimethylammonium cation.  Table 2 Amplitude displacements of the cobalt atom (sine term).

Commensurate monoclinic phase
Below 78 K, the satellite reflections are completely absent and the patterns can be indexed with a twinned monoclinic unit cell (phase IV) in the space group P2 1 /n. The orthorhombic (I) and monoclinic (IV) phases are related by a group-subgroup relation. The transition from phase I to phase IV involves an overall permutation of the crystallographic axes (a RT = Àb LT , b RT = Àc LT and c RT = a LT ) following the next transformation matrix fð010Þð001Þð100Þg, together with a change in the angle of 1.89 . The relation between the two twin domains is a rotation of ca 180 about the a* axis (in the monoclinic setting). Although these two phases are separated by two orthorhombic incommensurate phases, the unit-cell volume of each phase remains almost invariable, with a slight decrease of the unit-cell volume due to the thermal contraction. It should be noted that this thermal contraction is notably   (b)-(e) Modulation of the bond lengths between the cobalt and oxygen atoms at 86 and 90 K in phase III, and at 106 and 122 K in phase II. The distances Co1-O1, Co1-O1a, Co1-O2b, Co1-O2c, Co1-O3 and Co1-O3a are represented in light-green, dark-green, orange, red, black and grey continuous lines, respectively. The distances Co1-O1, Co1-O2c and Co1-O3 in the commensurate phase I are represented in green, brown and blue discontinuous lines, respectively. [Symmetry codes: ðaÞ À x þ 1, Ày, Àz; ðbÞ x À 1 2 , y, Àz þ 1 2 ; ðcÞ x À 3 2 , Ày, z À 1 2 .]

Figure 6
View along the a axis of the superstructure obtained at (a) 122 K and (b) 86 K; the solid lines are included to highlight the structural modulation, green and blue lines for the framework modulation at 122 and 86 K and orange and red for the modulation of the methylammonium cations. View along the wavevector direction of the model refined at (c) 122 K and (d) 86 K, in order to emphasize the increase of the amplitudes of the displacive modulation with decreasing temperature; all atoms have been represented as sticks except for cobalt (pink) and nitrogen and carbon atoms (blue and black, respectively) of methylammonium. Hydrogen atoms are represented by light-grey sticks. The graphical representations were carried out considering a supercell (ten times the average unit cell along the c axis), in order to include at least a complete period.
anisotropic. While the a and b axes (defined in the orthorhombic phase) decrease continuously with temperature, with a compression of 1.5 and 0.5%, respectively, the c axis, which agrees with the direction of the incommensurate wavevector, increases in length by approximately 0.8%.
Although the topology of compound 1 in phase IV remains invariable, at low temperature there are two crystallographically independent cobalt(II) atoms (Co1 and Co2) because of the loss of symmetry operations in the transition from the orthorhombic to the monoclinic space group. Both  Table 4 Selected distances (Å ) and angles ( ) involving the ammonium group of compound 1 at 122 and 106 K (phase II).

Table 5
Selected distances (Å ) and angles ( ) involving the ammonium group of compound 1 at 90 and 86 K (phase III).
Distances and angles further from the ideal hydrogen-bond geometry are emphasized in italics. Symmetry codes: ðdÞ x þ 1 2 , Ày þ 1 2 Àz À 1 2 ; ðeÞ x þ 1 cobalt atoms sit on inversion centres and are six-coordinated in an almost ideal CoO 6 octahedron. Each Co1 atom is bonded to six Co2 atoms and every Co2 atom is also surrounded by six Co1 atoms, all of them connected through formate ligands in an anti-anti manner along the ½110, ½110 and ½001 directions, building an octahedral perovskite-like framework. In the cavities of the three-dimensional structure, the methylammonium counter-ion is no longer located in a mirror plane, which means that six crystallographically independent hydrogen atoms are observed in this low-temperature phase. These variations in the structure with respect to the orthorhombic phase, together with the thermal contraction, imply changes in the hydrogen-bonded network. While in phase I there exists two hydrogen bonds between the guest molecule and the host framework, at low temperature the three hydrogen atoms connected to the nitrogen atom of the counter-ion establish three hydrogen bonds (Fig. 3).

Conclusions
The crystal structure analysis of compound 1 has revealed three different phase transitions between RT and 45 K. At RT, it crystallizes in the orthorhombic space group Pnma. Upon cooling, at around 128 K, compound 1 undergoes a phase transition from the commensurate orthorhombic phase (I) to the orthorhombic incommensurate phase (II), crystallized in the Pnma(00)0s0 space group with q = 0.1430 (2)c*. Below 96 K, a second orthorhombic incommensurate phase was observed. The change towards the wavevector q = 0.1247 (2)c* involves an elongation of the modulation length. Moreover, the amplitudes of the displacive modulation increase with decreasing temperature, having the main components along the b axis. The evolution from the commensurate hightemperature phase to the incommensurate phases is 'continuous', which is compatible with a 'displacive' phase transition. Moreover, the modulation of the incommensurate waves is not frozen within each incommensurate phase, as the intensity of the satellites changes with temperature. This suggests a complex scenario with a 'sluggish' or 'partly first-order' transition, which can also explain the lack of signal in specific heat measurements. Furthermore, the shape of the previously reported relative permittivity curve, which shows a continuous decrease, suggests that the structural phase transition occurs in a broad temperature range. The unexpected shape of this curve prompted us to study the temperature evolution of this compound. The order-disorder phase transition, due to the hydrogen-bond reorientation, shows a jump in the permittivity curves similar to those observed for the [NH 4 ][Zn(HCOO) 3 ] compound (Xu et al., 2010) or [NH 2 (CH 3 ) 2 ] n [Fe III-Fe II (HCOO) 6 ] n (Cañ adillas-Delgado et al., 2012). However, this curve reminds us of the occurrence of a slight reorientation of the methylammonium ions into the cavities or a small distortion of the framework. After the temperature evolution studies, this scenario has been confirmed due to the existence of incommensurate structures. The occurrence of modulated structures produces a reorganization of the existing electric dipoles, due to the continuous variation in the amplitude of the displacive modulation. Therefore, the shape in the relative permittivity curve is compatible with the slight variations in the electric dipoles due to the temperature evolution of the crystal structure. This compound presents an intricate hydrogen-bonded network, mainly due to the ability of the formate anion to act as a proton acceptor. This fact, together with the presence of the methylammonium counter-ion in the cavities, which acts as an excellent proton donor, gives rise to a system where the competition between weak interactions produces crystal structures very close in energy. Small changes in the hydrogenbonded network can trigger structural phase transitions, giving rise to incommensurate phases.
Below 78 K, the orthorhombic incommensurate phase becomes a monoclinic phase, resulting in a twinned crystal. The two twin domains are related by a rotation of 180 along the a* axis (in the monoclinic setting). This axis corresponds with the c axis in the orthorhombic phase, hence the one that becomes incommensurate between 128 and 78 K. As observed in the Laue patterns, the phase transition between the incommensurate phase [Pnma(00)0s0 space group with q = 0.1247 (2)c*] and the monoclinic phase is abrupt and therefore compatible with a first-order character. The indexing of the pattern in both phases suggests that the separation into two domains is caused by the enhancement of the amplitude of the displacement modulation in the incommensurate phase. When the amplitudes of modulation are large enough, the lowerenergy structure is no longer the incommensurate structure, and a breaking of symmetry is needed to decrease the energy, giving rise to a monoclinic structure. The breaking of symmetry gives rise to a twinned crystal, with both twin domains related by the lost symmetry elements that are no longer present in the monoclinic space group P2 1 /n.
The hydrogen-bonded network between the methylammonium and the carboxylate oxygen atoms in the incommensurate structure shows a different behaviour than in orthorhombic phase I. Two of the three hydrogen atoms of the NH 3 group establish hydrogen bonds, while the third fluctuates between two oxygen atoms from the same formate ligand (Fig. 3), giving rise to a 'flip-flop' behaviour. The modulation in the hydrogen-bond interactions suggests that competition between these interactions is responsible for the change in the modulation vector. The analysis of the hydrogen-bond interactions in the monoclinic phase shows that the three hydrogen atoms of the NH 3 group are involved in hydrogen bonding, therefore the methylammonium counter-ions are better anchored into the framework cavities, giving rise to a more stable structure.

Funding information
Partial funding for this work is provided by the Ministerio Españ ol de Ciencia e Innovació n through project MAT2015-68200-C02-2-P. We are grateful to the Institut Laue Langevin for the allocated neutron beam-time through project 5-15-617