research papers
Structure of incommensurate ammonium tetrafluoroberyllate studied by structure refinements and the maximum
method^{a}Laboratory of Crystallography, University of Bayreuth, D95440 Bayreuth, Germany
^{*}Correspondence email: smash@unibayreuth.de
Incommensurately modulated ammonium tetrafluoroberyllate (AFB) occurs in a narrow temperature interval between the paraelectric roomtemperature phase with Pnma (T_{i} = 178 K) and the ferroelectric lowtemperature phase with Pna2_{1} (T_{c} = 173 K). The structure is determined from accurate singlecrystal Xray diffraction data collected with synchrotron radiation at 175 K. The of the structure is Pnma(α00)0ss with α = 0.4796 (4). Both structure refinements and the maximum method lead to the same structure model, which involves only single harmonic modulations. The building units of the structure are BeF and NH complex ions with approximately tetrahedral They are relatively rigid and the modulations consist mainly of translations of the tetrahedra and their rotations around a fixed axis. The modulation is related to changes in the network of the hydrogen bonds. The lowtemperature can be described as a commensurately modulated structure with the same symmetry. The first harmonic modulations of the lowtemperature and incommensurate phases are related by a scale factor with a value of approximately two. In addition, the lowtemperature phase exhibits a second harmonic modulation that is responsible for shifts along c and the ferroelectricity in this phase. The experimental data of the incommensurate phase do not contain any evidence for the presence of a second harmonic in the modulation functions. This suggests that the development of the second harmonic, i.e. the development of the spontaneous polarization, is responsible for the lockin transition.
Keywords: incommensurate modulation; superspace; maximum entropy method.
1. Introduction
Ammonium tetrafluoroberyllate (AFB) is centrosymmetric with the Pnma at room temperature. It undergoes two phase transitions at lower temperatures. After the first transition at T_{i} = 182.9 K (Strukov et al., 1973; Makita & Yamauchi, 1974) the structure becomes incommensurate with the modulation wavevector close to 0.5a^{*} (Iizumi & Gesi, 1977). The second leads to a noncentrosymmetric ferroelectric phase (Pepinsky & Jona, 1957). The reported values of T_{c} vary between 174.2 K (Pepinsky & Jona, 1957) and 177.2 K (Strukov et al., 1973). The crystal structures of the paraelectric and ferroelectric phases have been studied by Xray diffraction (Onodera & Shiozaki, 1979; Garg & Srivastava, 1979) and neutron diffraction (Srivastava et al., 1999). O'Reilly et al. (1967) and Onodera & Shiozaki (1979) suggest that one of the ammonium ions is orientationally disordered in the paraelectric phase. O'Reilly et al. (1967) propose that the is of the order–disorder type. However, the structures by Garg & Srivastava (1979) and the most recent work by Srivastava et al. (1999) do not indicate any disorder in the roomtemperature structure. These authors conclude that the is a result of changes in the hydrogenbonding scheme.
Iizumi & Gesi (1977) have proposed a model of the phase transitions similar to that proposed for potassium selenate (Iizumi et al., 1977). In this model, the spontaneous polarization would already be present in the modulated phase on a microscopic scale, but the incommensurateness of the modulation would cause the average spontaneous polarization to be zero. The incommensuratetocommensurate would then correspond to ordering of the directions of the microscopic polarizations, resulting in a macroscopic spontaneous polarization.
None of the previous works have studied the ) are insignificant in the incommensurate phase within the experimental resolution.
of the intermediate incommensurate phase. Knowledge of the structure of this phase can give additional insight into the mechanism of the phase transitions in AFB and into the origin of the ferroelectricity of the lowtemperature phase. In particular, we will show that the microscopic polarizations supposed by the model of Iizumi & Gesi (1977Singlecrystal diffraction data were collected with synchrotron radiation. The structural model was found by refinements in ). The maximum formalism can be extended to (Steurer, 1991; van Smaalen et al., 2003). The MEM in can give a nonparametric estimate of the shapes of the modulation functions. Thus, the MEM can be used to either independently confirm the refined model or to find a shape of modulation function that differs from the parametrized model (van Smaalen et al., 2003; Palatinus et al., 2004).
The structure was further studied by means of the Maximum Method (MEM). The Maximum Method (MEM) is a general tool for a modelfree reconstruction of positive additive distributions. One of many applications in crystallography is a reconstruction of the electron density from phased structure factors (Gilmore, 19962. Experimental
Crystals of AFB were grown by slow evaporation at 278 K from an aqueous solution of a stoichiometric mixture of NH_{4}F (ACROS, ACSReagenz, purity ≥ 98%) and BeF_{2} (Alfa Aesar, 99.9% purity of the metals basis). Several crystals were tested on a Nonius Mach3 diffractometer with a rotating anode generator and Mo K radiation. A thick plate of the dimensions 0.20 ×0.13 ×0.08 mm^{3} was selected for the data collection. The diffraction data were collected on a Huber fourcircle diffractometer at beamline D3 at Hasylab, DESY, Hamburg. The beam was monochromated by a Si(111) doublecrystal monochromator. The wavelength was set to 0.7100 Å. An Oxford Cryojet was used to cool the sample by a cold nitrogen gas flow. Incommensurate satellite reflections were observed in the temperature range between 173 and 178 K; the data were collected at 175 K. Three standard reflections were measured every 2 h as a check of the stability of the experimental setup. Experimental details are given in Table 1.
The modulation wavevector was determined from the positions of 32 satellites. The value q = 0.4976 (4)a* is very close to . Thus, reflections of the type h k l m are very close to reflections . This can result in overlaps of these reflections in the scans. The value of the angle (rotation of the crystal around the scattering vector) was therefore optimized so that the distance between the positions of the reflections in the scans was maximal. Even with this procedure the occurrence of the two neighboring reflections in one scan could not always be avoided, but the distances between the peaks became large enough to allow intensities of the individual reflections to be determined.
The critical problem in the determination of the integrated intensities was the determination of the background. The standard procedures for background determination could not be used because of the presence of more than one peak in some scans. Therefore, the following procedure was used. Every profile was first smoothed by calculating a 13point sliding average. Then a leastsquares line was fitted to every 16 points of the smoothed profile. As a result, a smooth first derivative of the scan profile was obtained. Starting from the expected peak position, the derivative was scanned to the left. The first negative point, that was not followed by any significantly positive point, was considered to be the border of the
On the right side, the procedure was repeated symmetrically. The was determined for each peak in the profile and the points lying outside the peak areas were considered to be the background points. At least ten points were always assigned to the background at each side of the peak. The parameters of the method were selected empirically to produce the best results. In the case of the present data the method proved to be very robust and produced reliable estimates of the background.The raw counts were corrected for the deadtime of the detector, and the integrated intensities were corrected for the Lp effects and for variation in the intensity of the primary beam. The absorption correction was not applied because of the very small ^{1}).
( 0.215 mmThe classes of reflections (0 k l 0) with k+l = 2n+1, (h k 0 m) with h+m = 2n+1 and (h 0 l m) with m = 2n+1 were systematically extinct. No other extinction rule was observed. The unique corresponding to these extinction rules is Pnma(00)0ss.
Secondorder satellites (2) were not observed in preliminary measurements. Therefore, these satellites were not measured during the data collection. However, due to the almost commensurate value of the qvector the secondorder satellites occur close to the main reflections. Analysis of the collected data set showed that profiles of 1131 main reflections (out of 1460) contained positions of the secondorder satellites. All these 1131 profiles were visually checked and no peak was observed at the positions of the secondorder satellite reflections.
3. Structure refinements
The structure was refined using the et al., 1981; van Smaalen, 1995). All refinements were performed using the computer program JANA2000 (Petříček & Dušek, 2000). The coordinates of the roomtemperature structure (Srivastava et al., 1999) were used as a starting point for the of the average structure. The positional parameters of all the atoms were refined together with the isotropic temperature parameters of H atoms and anisotropic harmonic temperature parameters of all the other atoms. The fit to main reflections converged to R_{main}(obs) = 0.043. In the next step the modulation was introduced. The displacive modulations in the directions i = 1,2,3 were described by the Fourier series
approach (de WolffFunctions with n = 1 (firstorder harmonic modulation) were refined for all atoms, starting from arbitrary but small values. The with 97 parameters converged to R_{main}(obs) = 0.042 and R_{sat}(obs) = 0.131. The differenceFourier maps showed significant structure around the positions of the F atoms (Fig. 1a). The first harmonic modulation of the harmonic displacement parameters improved the fit only marginally.
It turned out that the fit can be significantly improved by introducing anharmonic displacement parameters (ADP) of the third order (Kuhs, 1992) for the F atoms as well as their firstorder harmonic modulation. The R values dropped to R_{main}(obs) = 0.028 and R_{sat}(obs) = 0.066 after the introduction of the modulated ADP. This decrease is significant, although the number of refinable parameters increased to 175. The maxima in the difference electron density of all the F atoms decreased by a factor larger than three (Fig. 1b). There is no significant negative region of the probability density function at any of the three atoms. Moreover, the BeF_{4} tetrahedron becomes more regular in comparison with the without ADPs. All these observations support the conclusion that the ADP parameters are the appropriate way to describe the structure. The positional parameters of the basic structure are listed in Table 2; the parameters of the modulation functions are listed in Table 3.^{1}


Another possibility to improve the fit is to introduce the second harmonic modulation of the harmonic displacement parameters. However, R values [R_{main}(obs) = 0.039, R_{sat}(obs) = 0.076] than the of the thirdorder ADPs and predicts unrealistically high intensities of the secondorder satellites. The of the second harmonic modulation is also very difficult without any observed secondorder satellites. For these reasons this model was not adopted.
of these parameters leads to higherAs mentioned in §2, we did not observe any secondorder satellites. This indicates that the harmonic displacement modulation of second order is very weak or absent. Indeed, the introduction of the secondorder harmonic displacement modulation did not improve the fit and did not lead to significant amplitudes of the secondorder modulation functions.
The final structure model (firstorder harmonic modulation only) produced calculated structure factors of the secondorder satellites that were small, but with some of them strong enough to be observable in our experiment. The intensities of the secondorder satellites can be effectively decreased by introducing the second harmonic modulation of the temperature factors and refining this modulation against a data set, which contains the experimentally measured reflections together with the secondorder satellites with zero intensity and small
If the structural model is refined against this combined data set, the fit to the main reflections and firstorder satellites remains essentially unchanged, while the calculated intensities of all the secondorder satellites become unobservable, in agreement with the experiment. This can be regarded as proof of the presence of some second harmonic component in the modulation of the temperature factors, although the exact form of this modulation cannot be determined from the present data set, which does not contain any observed secondorder satellites.4. The ferroelectric structure as a commensurately modulated structure
The lowtemperature phase can be described as a twofold q_{c} = 0.5a* and with the same as the incommensurate phase. For a given and modulation wavevector, the symmetry and structure of the depend on the t_{0} section of (Yamamoto, 1982; van Smaalen, 1995). For , Hogervorst (1986) has listed all the possible space groups of the for different values of t_{0} and different commensurate modulation wavevectors. The Pna2_{1} of the twofold corresponds to the sections t_{0} = 1/8+n/4, n = 0,1,2,3. Alternative values of n correspond to a shift in the phase of the modulation or a shift in the origin of the threedimensional The actual value of t_{0} for the transformation between the lowtemperature structure reported by Srivastava et al. (1999) and the model used in this work is t_{0} = 7/8. Using this information, we have derived the modulation amplitudes of the description for the lowtemperature structure (Tables 4 and 5). Both first and secondorder harmonics were necessary to fit the coordinates. Because of the commensurateness of the modulation (q_{c} = 0.5a*), the phase and amplitude of the secondorder harmonic are correlated. We have chosen the phases so that the modulation amplitudes are minimal.
of the roomtemperature phase. Alternatively, it can be described in as a commensurately modulated structure with


The average N—H bond length in the lowtemperature structure is 1.005 Å, while the presently determined incommensurate structure gives 0.900 Å. This difference is because of the different experimental techniques (neutron versus Xray scattering). In order to facilitate the comparison of the modulation amplitudes in the two structures, we have modified the positions of the H atoms in the lowtemperature structure in such a way that the N—H bonds are shorter than the published values by a factor of 0.9. This procedure is used only in Tables 4 and 5 and does not have any impact on the discussions about the hydrogenbonding scheme in §6.2.
5. Maximum method
The 128 ×100 ×162 ×32 pixels. This corresponds to a resolution of approximately 0.06 Å in each realspace direction. The modulation functions were sampled at 32 points, allowing, in principle, the determination of up to eight harmonics. The experimental amplitudes of the structure factors, corrected for and extinction, were combined with the phases of the best refined model to produce the input for the MEM calculation (Bagautdinov et al., 1998). This input is called `observed data'. For checking purposes, the MEM calculations were performed with the structure factors derived directly from the refined model (`calculated data'). The computer program BayMEM (van Smaalen et al., 2003) was used for two MaxEnt calculations, with different algorithms and different constraints for each data set (Table 6). Convergence was obtained for all four calculations.
electron density of AFB was discretized on a grid of

The MaxEnt calculations result in optimized ). The positions of the atoms as a function of the fourth coordinate have been determined by the computation of the centers of charge around each local maximum for different realspace sections (different values of t). The positions of the H atoms cannot be determined with this method, because these atoms do not form separate maxima in the electron density. The determination of the hydrogen positions and the investigation of the bonding electron density will be a topic of future research.
electron densities . exhibits local maxima in the form of strings parallel to the fourth dimension of at the positions of the atoms (Fig. 2The agreement of the modulation functions derived from with the refined functions of the model is excellent (Fig. 3). The differences between the results by the MEM on the observed data and the refined functions are similar to or even smaller than the differences between the results by the MEM on calculated data and the refined functions. All differences in atomic positions are found to be below 10% of the pixel size. Thus, within the accuracy of the MEM calculations, the MEM shows perfect agreement with the refined model and the model is confirmed (van Smaalen et al., 2003). Inspection of also provides an independent indication of the modulated oddorder anharmonic temperature factors of the F atoms (Fig. 2).
A possible displacement along c of the atoms in special positions is of particular importance, because they are responsible for the spontaneous polarization in the lowtemperature phase. In the approach the z displacements are described by the second harmonics. Any evidence for the presence of second harmonics in the incommensurate structure has not been found in the MaxEnt calculations. We have tested the sensitivity of the MEM for second harmonic modulations by two additional calculations on simulated data. The structure model of the incommensurate phase (Table 3) is combined with secondorder harmonic modulation functions, as they were obtained by dividing the second harmonics of the lowtemperature structure by a factor of 2 (Table 5). The structure factors calculated from this model were used as input in the MEM calculations. The first calculation contained main reflections, and first and secondorder satellites; the second calculation only the main reflections and the firstorder satellites. The results of the first calculation clearly reproduced the weak secondorder modulations along x and z. However, the results of the second calculation do not show any such modulations (Fig. 4). It can be concluded that it is not possible to detect the weak secondorder modulations if the corresponding satellites have not been measured and included in the dataset, because the main reflections and firstorder satellites do not contain enough information about this modulation.
6. Discussion
6.1. Description of the modulated structure
Among the seven structures of AFB published previously (see §1), the work by Srivastava et al. (1999) is the most recent and it involves the most extensive data sets. The structures by Srivastava et al. (1999) basically confirm earlier refinements of the paraelectric and ferroelectric structures, but they are more accurate. Therefore, we compare the present structure of the incommensurate phase with the structures from Srivastava et al. (1999). If not otherwise stated, the expression `RT structure' refers to the roomtemperature structure, while the `LT structure' indicates the twofold at 163 K, as they were reported by Srivastava et al. (1999).
The BeF complex anion has nearly perfect tetrahedral geometry. The small deviations from this symmetry are of equal sizes for the RT and incommensurate structures, while the LT structure exhibits slightly larger distortions (Table 7). The modulation has little effect on the geometry (Table 8). This implies that the modulation of the BeF complex anion can be described as rigidbody translations and rotations in very good approximation.


A quantitative analysis shows that the modulation of BeF can be described as the combined effect of small displacements along b and rotations around an axis in the mirror plane. While the angle of rotation varies with the phase of the modulation, the direction of this axis is fixed and is not affected by the modulation (Table 9, Fig. 5).

The deviations from tetrahedral symmetry are larger for the NH cations than they are for the BeF complex anions (Table 7). However, the variations of 0.04 Å due to the modulations of the individual N—H bond lengths (Table 8) are much smaller than the displacements up to 0.35 Å of the H atoms due to the modulation. Similarly, the variations of the H—N—H angles due to the modulation (up to 4.6°) are much smaller than the rotations of the whole cations (Tables 8 and 9). The result is that the largest part of the modulations of the NH ions is described by rigidbody modulations. The quantitative analysis again shows that the modulation is a combination of small displacements along b and rotations about a single axis in the mirror plane (Table 9, Fig. 5).
The modulation of the NH cations affects the N1—H13 and N2—H23 bond lengths only, as well as a few bond angles involving either H13 or H23. The other bond lengths and angles are almost independent of the modulation (Table 8). The variation of deformations in the LT structure is slightly smaller for the N1H cation, while it is much larger for the N2H cation. For example, the N2—H21 bond length does not vary in the incommensurate structure, while it results in two distances different by 0.059 Å in the LT structure. These differences are significant with respect to the standard uncertainties of the positions in both structures. We believe that the differences in distortions between the incommensurate and LT structures are related to the development of the spontaneous polarization in the latter phase.
The structure can be considered to consist of two alternating layers stacked along c (Fig. 6). Layer I is centered on z = 0, while layer II is centered on z = 0.5. The translations and rotations of the complex ions within the same layer are correlated so that the neighboring ions have approximately the same deviations from an average position and orientation at the same value of t. The modulations of the ions of the second layer are shifted by approximately 0.25 in t. The result is such that at the places of the structure, where the ions of one layer reach their maximal deviations, the ions of the other layer are in their average positions and vice versa. The ions in layer I have the largest deviations approximately at t = 0.25 and t = 0.75; the ions in layer II are most displaced around t = 0.0 and t = 0.5. Because of the commensurateness of the LT structure, each ion adopts only four different orientations. The value t_{0} = 0.875 corresponds to intermediate deviations of the ions in both layers.
As we will discuss below, the dissection of the structure into layers is strongly correlated with the modulations of the hydrogen bonds.
6.2. Hydrogenbonding scheme
The interactions between the complex anions and cations are governed by ionic interactions and hydrogen bonds. Changes in the pattern of hydrogen bonds are believed to be responsible for the phase transitions in AFB (Onodera & Shiozaki, 1979; Garg & Srivastava, 1979; Srivastava et al., 1999).
The H⋯F distances as a function of t are shown in Fig. 7. We take 2.6 Å as the limit for the effective bonding interaction between H and F in the present structure. There is a gap between the distances below and above this limit. Inclusion of all the distances below this limit is necessary and sufficient to fully connect the structural units in a threedimensional network of hydrogen bonds.
The H⋯F interactions can be roughly separated into two classes: `stable' and `unstable' interactions. The `stable' interactions change only a little with the phase of the modulation. They involve the atoms H11, H12, H21 and H22, which lie in the mirror plane. H11 and H22 have only one very short distance to a neighboring F atom. This distance is not influenced by the modulation at all. These distances represent the strongest hydrogenbond interactions in the structure and they remain almost unchanged in all three phases of AFB. The atoms H12 and H21 have three almost equally long distances to F atoms. These distances are considerably longer than the H11⋯F and H22⋯F distances and they can be considered weak, but the simultaneous occurrence of three such interactions in different directions stabilizes the position of the H atoms and is probably responsible for the special behavior of these atoms. Note that the H12 atom rotates considerably less around the rotational axis of the N1H_{4} tetrahedron than the other atoms and H21 lies almost exactly in the rotational axis of the N2H_{4} tetrahedron (Table 9).
The `unstable' interactions involving the H13 and H23 atoms are subject to large changes of the H⋯F distance with the phase of the modulation. The result is that the H13 and H23 atoms of one NH ion are alternatively bonded to F2 or F3 atoms of different BeF anions, depending on the phase of the modulation. The formation of one very short and one very long H⋯F distance for each of the H13 and H23 atoms will have lower energy than all distances being of intermediate length, as found in the RT structure. The modulation of these bonds thus appears to be the driving force for the phase transition.
All the interlayer hydrogen bonds belong to the `stable' interactions. The changes in the hydrogenbonding scheme occur only within the layers and the bonds between the layers are not influenced by the modulation.
The infinite number of different H⋯F distances described by each curve in Fig. 7 reduces to four different distances in the LT structure. In each `unstable' interaction, two of the four distances represent the stronger H⋯F interaction and two the weak or broken interaction. However, no general distance limit can be defined that separates the stronger and weaker interactions within the modulated distances. The limit 2.1 Å used by Srivastava et al. (1999) yields 13 strong hydrogen bonds involving four crystallographically independent ammonium ions, but these bonds involve both stronger and weaker H⋯F bonds on the H23⋯F3 curve, and neglect one of the shorter distances on the H23⋯F1 curve. Obviously, it is inappropriate to reduce the description of the hydrogenbond interactions in the LT structure of AFB to a simple categorization of the bonds to strong and weak based only on the distance limit.
6.3. The ferroelectric phase transition
The and 5). The LT structure also contains secondorder harmonic displacements that are responsible for the ferroelectricity, as is demonstrated by the following argument: All central atoms of the complex ions lie on special positions in mirror planes. Symmetry restricts the firstorder harmonic modulations of these atoms to shifts along the y axis. The vectors representing the dipole moments of the NH and BeF complex ions also lie in the mirror planes and they are subject to the same symmetry restrictions. As a consequence, the firstorder modulation cannot change the z components of the dipole moments of the individual complex ions, nor can it create a z component of the by relative shifts of the NH and BeF ions. However, the small spontaneous polarization in the LT structure is along c, because this is the polar axis of the of the LT structure. Consequently, secondorder harmonic modulations, which include displacements along a and c, are necessary to describe the spontaneous polarization.
description reveals a striking similarity between the incommensurate and the LT structures. The phases of the first harmonic modulation functions in the two structures are almost equal. The amplitudes of these modulation functions in the LT structure are approximately two times larger (Tables 3Secondorder harmonic modulation amplitudes have not been found in the incommensurate structure, although neither the MEM nor the refinements could disprove the possibility of small secondorder amplitudes. To analyze this further we have performed a series of calculations to estimate the upper boundary on the amplitude of the second harmonic that is consistent with our experiment. The following points of concern were raised:

The experimentally measured reflections were analyzed to obtain an estimate of the limit of observability. A fraction of 95% of the 2291 unobserved reflections [] have an intensity lower than I_{limit} = 0.412 ( with F scaled to correspond to Fourier transform of one unit cell). Therefore, this value was taken as the lowest intensity that could have been observed in the experiment. The standard uncertainties of the secondorder satellites were estimated as 0.09 on the absolute scale, which corresponds to the lower limit of the experimental standard uncertainties of all reflections. The results are not influenced by the exact value of the standard uncertainties.
Several hypothetical structure models were created by combining the final structure model with the second harmonic modulation derived from the LT structure with amplitudes multiplied by a constant factor f_{2nd}. Each model was refined against the experimental data set combined with the secondorder satellites with zero intensity and a equal to 0.09. All parameters of the model were refined; only the second harmonic positional modulation was kept constant. After the the intensities of the secondorder satellites corresponding to the refined model were calculated. The number of observed intensities of the secondorder satellites was 0,0,0,2,8,17 for f_{2nd} = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, respectively. The lowest f_{2nd} that leads to observable intensities is 0.4 of the original amplitudes (Table 5). Because the procedure described above was designed to suppress the intensities as much as possible, the occurrence of observable intensities indicates that this amount of modulation is inconsistent with the experiment. Thus, the amplitudes of the secondorder harmonic positional modulation in the modulated structure are smaller than 0.4 times the amplitudes derived from the LT structure.
Any spontaneous polarization of the incommensurate phase is forbidden by the centrosymmetric et al. (1977) for the in potassium selenate and applied to AFB by Iizumi & Gesi (1977). These authors propose that the incommensurate phase contains a modulated local that, at the lockin transition, orders to form the spontaneous polarization in the commensurate phase. Our analysis of the incommensurate structure indicates that the local dipole moments are very small and might even be absent in the modulated structure. This suggests that the responsible for the observed spontaneous polarization might be formed at the lockin transition.
although a local of the basic structure would be possible if a secondorder harmonic displacement modulation is present. Because we did not obtain any evidence for such a secondorder harmonic, our experiment does not support the model proposed by IizumiIf this conclusion is considered and if it is further considered that the first harmonics are sufficient to stabilize the pattern of hydrogen bonds (see §6.2), we obtain the following possibilities for the mechanisms of the phase transitions: The transition at T_{i} is most probably governed by changes in the hydrogen bonding, in accordance with previous suggestions (Onodera & Shiozaki, 1979; Garg & Srivastava, 1979; Srivastava et al., 1999). The formation of the spontaneous polarization (forbidden in the incommensurate structure) might be the driving force for the transition at T_{c}. Alternatively, the rearrangements of the hydrogen bonds might also be responsible for the lockin transition at T_{c} and the spontaneous polarization would be a side effect of this rearrangement. However, the absence of significant local dipole moments in the incommensurate structure and the similarity of the overall pattern of the hydrogenbonding scheme in the incommensurate and LT structures suggest that the spontaneous polarization is important for the stabilization of the LT lockin phase, thus making the first mechanism the most probable one.
7. Conclusions
The atomic structure of the incommensurately modulated phase of ammonium tetrafluoroberyllate has been determined at T = 175 K. The transition from the paraelectric phase to the incommensurate phase is found to be due to rearrangements in the hydrogenbonding scheme. The structure of AFB can be described as an alternate stacking of two layers along c. In the incommensurate structure H⋯F distances between the layers remain constant at their values in the RT phase. Within the layers, some of the H⋯F distances strongly vary between values corresponding to strong and very weak hydrogen bonds. This change is the driving force for the at T_{i}.
A microscopic polarization is found to be very small or absent in the modulated structure. The presence of this microscopic polarization in the incommensurate structure is supposed in the mechanism proposed by Iizumi & Gesi (1977). At T_{c} the local polarization would rearrange to form the LT structure with a spontaneous polarization. Instead, we do not find evidence for a local polarization in the incommensurate structure within the sensitivity of our experiment. Further specialized experiments are necessary that will determine the amount of the local polarization in the incommensurate structure with higher precision and possibly support the hypothesis of zero local polarization.
Two mechanisms can be envisaged for the transition at T_{c}. The first is further rearrangements in the hydrogenbonding scheme, with the spontaneous polarization as an `accidental' corollary. The second, more likely mechanism is that the development of the spontaneous polarization is the driving force for the lockin transition.
Interesting questions pertaining to the mechanisms of the transitions remain. For example, it could be possible that the secondorder harmonic modulation (and consequently local dipole moments) develops in the incommensurate phase close to T_{c} or that the secondorder harmonics develop as critical fluctuations. Whether this is true or not can be investigated by highresolution diffraction experiments measuring the temperature dependence of the secondorder satellites, as is possible at the thirdgeneration synchrotron sources.
Supporting information
10.1107/S0108768104000874/ck5001sup1.cif
contains datablocks global, (I). DOI:Structure factors: contains datablock . DOI: 10.1107/S0108768104000874/ck5001sup2.hkl
Supporting information file. DOI: 10.1107/S0108768104000874/ck5001sup3.txt
Data collection: DIF4 (Eichhorn, 1993); cell
DIF4 (Eichhorn, 1993); program(s) used to refine structure: (Jana2000; Petricek and Dusek, 2000); molecular graphics: (Diamond 2.1c, Brandenburg, 1999); software used to prepare material for publication: (Jana2000; Petricek and Dusek, 2000).H_{8}BeF_{4}N_{2}  F(000) = 248 
M_{r} = 121.1  D_{x} = 1.739 (1) Mg m^{−}^{3} 
Orthorhombic, Pnma(α00)0ss†  Synchrotron radiation, λ = 0.71 Å 
q = 0.47956a*  Cell parameters from 24 reflections 
a = 7.5284 (12) Å  θ = 14.7–25.1° 
b = 5.8848 (9) Å  T = 175 K 
c = 10.436 (2) Å  Thick plate, colourless 
V = 462.35 (14) Å^{3}  0.20 × 0.13 × 0.08 mm 
Z = 4 
† Symmetry operations: (1) x_{1}, x_{2}, x_{3}, x_{4}; (2) x_{1}, 1/2−x_{2}, x_{3}, 1/2+x_{4}; (3) 1/2+x_{1}, x_{2}, 1/2−x_{3}, 1/2+x_{4}; (4) 1/2+x_{1}, 1/2−x_{2}, 1/2−x_{3}, x_{4}; (5) −x_{1}, −x_{2}, −x_{3}, −x_{4}; (6) −x_{1}, 1/2+x_{2}, −x_{3}, 1/2−x_{4}; (7) 1/2−x_{1}, −x_{2}, 1/2+x_{3}, 1/2−x_{4}; (8) 1/2−x_{1}, 1/2+x_{2}, 1/2+x_{3}, −x_{4}. 
Huber fourcircle diffractometer  θ_{max} = 40.6°, θ_{min} = 3.3° 
Radiation source: Hasylab, DESY, Hamburg  h = −13→0 
Si 111 monochromator  k = 0→10 
profile data with ω–scans  l = 0→18 
No. of measured, independent and observed reflections: 4109, 3969, 2262 [I > 3σ(I)]; ?, 1460, 1256 (m=0 main reflections); ?, 2509, 1006 (m=1 main reflections)  3 standard reflections every 120 min 
R_{int} = 0.051  intensity decay: none 
Refinement on F  Weighting scheme based on measured s.u.'s w = 1/(σ^{2}(F) + 0.000036F^{2}) 
R[F^{2} > 2σ(F^{2})] = 0.034, wR(F^{2}) = 0.040, S = 1.99; R[F^{2} > 2σ(F^{2})] = 0.0276, R(F^{2}) = 0.0368 (m=0 main reflections); R[F^{2} > 2σ(F^{2})] = 0.0662, R(F^{2}) = 0.0646 (m=1 main reflections)  (Δ/σ)_{max} = 0.001 
3969 reflections  Δρ_{max} = 0.40 e Å^{−}^{3} 
175 parameters  Δρ_{min} = −0.34 e Å^{−}^{3} 
0 restraints  Extinction correction: BC type 1 Gaussian isotropic (Becker & Coppens, 1974) 
0 constraints  Extinction coefficient: 0.05 (3) 
All Hatom parameters refined 
H_{8}BeF_{4}N_{2}  c = 10.436 (2) Å 
M_{r} = 121.1  V = 462.35 (14) Å^{3} 
Orthorhombic, Pnma(α00)0ss†  Z = 4 
q = 0.47956a*  Synchrotron radiation, λ = 0.71 Å 
a = 7.5284 (12) Å  T = 175 K 
b = 5.8848 (9) Å  0.20 × 0.13 × 0.08 mm 
† Symmetry operations: (1) x_{1}, x_{2}, x_{3}, x_{4}; (2) x_{1}, 1/2−x_{2}, x_{3}, 1/2+x_{4}; (3) 1/2+x_{1}, x_{2}, 1/2−x_{3}, 1/2+x_{4}; (4) 1/2+x_{1}, 1/2−x_{2}, 1/2−x_{3}, x_{4}; (5) −x_{1}, −x_{2}, −x_{3}, −x_{4}; (6) −x_{1}, 1/2+x_{2}, −x_{3}, 1/2−x_{4}; (7) 1/2−x_{1}, −x_{2}, 1/2+x_{3}, 1/2−x_{4}; (8) 1/2−x_{1}, 1/2+x_{2}, 1/2+x_{3}, −x_{4}. 
Huber fourcircle diffractometer  3 standard reflections every 120 min 
No. of measured, independent and observed reflections: 4109, 3969, 2262 [I > 3σ(I)]; ?, 1460, 1256 (m=0 main reflections); ?, 2509, 1006 (m=1 main reflections)  intensity decay: none 
R_{int} = 0.051 
R[F^{2} > 2σ(F^{2})] = 0.034, wR(F^{2}) = 0.040, S = 1.99; R[F^{2} > 2σ(F^{2})] = 0.0276, R(F^{2}) = 0.0368 (m=0 main reflections); R[F^{2} > 2σ(F^{2})] = 0.0662, R(F^{2}) = 0.0646 (m=1 main reflections)  All Hatom parameters refined 
3969 reflections  Δρ_{max} = 0.40 e Å^{−}^{3} 
175 parameters  Δρ_{min} = −0.34 e Å^{−}^{3} 
0 restraints 
x  y  z  U_{iso}*/U_{eq}  
Be  0.25141 (8)  0.25  0.41847 (6)  0.01427 (12)  
F1  0.05442 (9)  0.25  0.38015 (8)  0.03386 (12)  
F2  0.27030 (10)  0.25  0.56419 (6)  0.03064 (12)  
F3  0.33820 (7)  0.03489 (9)  0.36575 (5)  0.02881 (7)  
N1  0.18709 (6)  0.25  0.09990 (4)  0.01742 (9)  
N2  0.45778 (6)  0.25  0.80326 (4)  0.01731 (9)  
H1  0.3059 (12)  0.25  0.1228 (10)  0.041 (3)*  
H2  0.1075 (16)  0.25  0.1586 (12)  0.061 (3)*  
H3  0.1718 (10)  0.1298 (15)  0.0498 (8)  0.058 (2)*  
H4  0.4876 (13)  0.25  0.7213 (10)  0.048 (3)*  
H5  0.5606 (18)  0.25  0.8468 (13)  0.075 (4)*  
H6  0.3946 (12)  0.1227 (17)  0.8194 (8)  0.062 (2)* 
U^{11}  U^{22}  U^{33}  U^{12}  U^{13}  U^{23}  
Be  0.0136 (2)  0.0146 (2)  0.0147 (2)  0  0.00028 (17)  0 
F1  0.01412 (13)  0.0449 (3)  0.0426 (2)  0  −0.00767 (13)  0 
F2  0.03108 (18)  0.0469 (3)  0.01393 (13)  0  −0.00103 (12)  0 
F3  0.02590 (12)  0.01822 (11)  0.04229 (16)  0.00019 (9)  0.01060 (11)  −0.01041 (11) 
N1  0.01510 (15)  0.01725 (16)  0.01991 (16)  0  0.00066 (12)  0 
N2  0.01748 (16)  0.01793 (17)  0.01651 (15)  0  0.00124 (12)  0 
Average  Minimum  Maximum  
Be—F1  1.5370 (9)  1.5359 (9)  1.5381 (9) 
Be—F2  1.5302 (9)  1.5273 (9)  1.5331 (9) 
Be—F3  1.5294 (14)  1.5251 (14)  1.5334 (14) 
Be—F3^{i}  1.5293 (14)  1.5251 (14)  1.5334 (14) 
N1—H1  0.928 (10)  0.926 (10)  0.930 (10) 
N1—H2  0.860 (12)  0.857 (12)  0.864 (12) 
N1—H3  0.901 (16)  0.878 (16)  0.919 (16) 
N1—H3^{i}  0.901 (16)  0.878 (16)  0.919 (16) 
N2—H4  0.884 (11)  0.884 (11)  0.884 (11) 
N2—H5  0.901 (14)  0.898 (14)  0.905 (14) 
N2—H6  0.909 (17)  0.895 (17)  0.926 (16) 
N2—H6^{i}  0.909 (17)  0.895 (17)  0.926 (16) 
F1—Be—F2  110.52 (5)  110.43 (5)  110.61 (5) 
F1—Be—F3  108.55 (8)  108.19 (8)  108.81 (8) 
F1—Be—F3^{i}  108.55 (8)  108.19 (8)  108.81 (8) 
F2—Be—F3  108.58 (8)  108.46 (8)  108.73 (8) 
F2—Be—F3^{i}  108.58 (8)  108.46 (8)  108.73 (8) 
F3—Be—F3^{i}  112.06 (6)  111.98 (6)  112.15 (6) 
F3^{i}—Be—F3  112.06 (6)  111.98 (6)  112.15 (6) 
H1—N1—H2  118.9 (11)  118.4 (11)  119.5 (10) 
H1—N1—H3  106.2 (12)  104.2 (12)  108.1 (12) 
H1—N1—H3^{i}  106.2 (12)  104.2 (12)  108.1 (12) 
H2—N1—H3  109.4 (14)  107.0 (14)  111.6 (14) 
H2—N1—H3^{i}  109.5 (14)  107.0 (14)  111.6 (14) 
H3—N1—H3^{i}  105.7 (14)  105.5 (14)  105.8 (14) 
H3^{i}—N1—H3  105.7 (14)  105.5 (14)  105.8 (14) 
H4—N2—H5  105.6 (11)  105.5 (11)  105.7 (11) 
H4—N2—H6  108.2 (13)  107.6 (12)  108.8 (13) 
H4—N2—H6^{i}  108.2 (13)  107.6 (12)  108.8 (13) 
H5—N2—H6  111.1 (14)  109.1 (14)  113.1 (14) 
H5—N2—H6^{i}  111.1 (14)  109.1 (14)  113.1 (14) 
H6—N2—H6^{i}  112.3 (14)  112.1 (14)  112.5 (14) 
H6^{i}—N2—H6  112.3 (14)  112.1 (14)  112.5 (14) 
Symmetry code: (i) x_{1}, −x_{2}+1/2, x_{3}, x_{1}/2. 
Experimental details
Crystal data  
Chemical formula  H_{8}BeF_{4}N_{2} 
M_{r}  121.1 
Crystal system, space group  Orthorhombic, Pnma(α00)0ss† 
Temperature (K)  175 
Wave vectors  q = 0.47956a* 
a, b, c (Å)  7.5284 (12), 5.8848 (9), 10.436 (2) 
V (Å^{3})  462.35 (14) 
Z  4 
Radiation type  Synchrotron, λ = 0.71 Å 
µ (mm^{−}^{1})  ? 
Crystal size (mm)  0.20 × 0.13 × 0.08 
Data collection  
Diffractometer  Huber fourcircle diffractometer 
Absorption correction  – 
No. of measured, independent and observed [I > 3σ(I)] reflections  4109, 3969, 2262 ; ?, 1460, 1256 (m=0 main reflections); ?, 2509, 1006 (m=1 main reflections) 
R_{int}  0.051 
(sin θ/λ)_{max} (Å^{−}^{1})  0.917 
Refinement  
R factors and goodness of fit  R[F^{2} > 2σ(F^{2})] = 0.034, wR(F^{2}) = 0.040, S = 1.99; R[F^{2} > 2σ(F^{2})] = 0.0276, R(F^{2}) = 0.0368 (m=0 main reflections); R[F^{2} > 2σ(F^{2})] = 0.0662, R(F^{2}) = 0.0646 (m=1 main reflections) 
No. of reflections  3969 
No. of parameters  175 
Hatom treatment  All Hatom parameters refined 
Δρ_{max}, Δρ_{min} (e Å^{−}^{3})  0.40, −0.34 
† Symmetry operations: (1) x_{1}, x_{2}, x_{3}, x_{4}; (2) x_{1}, 1/2−x_{2}, x_{3}, 1/2+x_{4}; (3) 1/2+x_{1}, x_{2}, 1/2−x_{3}, 1/2+x_{4}; (4) 1/2+x_{1}, 1/2−x_{2}, 1/2−x_{3}, x_{4}; (5) −x_{1}, −x_{2}, −x_{3}, −x_{4}; (6) −x_{1}, 1/2+x_{2}, −x_{3}, 1/2−x_{4}; (7) 1/2−x_{1}, −x_{2}, 1/2+x_{3}, 1/2−x_{4}; (8) 1/2−x_{1}, 1/2+x_{2}, 1/2+x_{3}, −x_{4}.
Computer programs: DIF4 (Eichhorn, 1993), (Jana2000; Petricek and Dusek, 2000), (Diamond 2.1c, Brandenburg, 1999).
Average  Minimum  Maximum  
Be—F1  1.5370 (9)  1.5359 (9)  1.5381 (9) 
Be—F2  1.5302 (9)  1.5273 (9)  1.5331 (9) 
Be—F3  1.5294 (14)  1.5251 (14)  1.5334 (14) 
Be—F3^{i}  1.5293 (14)  1.5251 (14)  1.5334 (14) 
N1—H1  0.928 (10)  0.926 (10)  0.930 (10) 
N1—H2  0.860 (12)  0.857 (12)  0.864 (12) 
N1—H3  0.901 (16)  0.878 (16)  0.919 (16) 
N1—H3^{i}  0.901 (16)  0.878 (16)  0.919 (16) 
N2—H4  0.884 (11)  0.884 (11)  0.884 (11) 
N2—H5  0.901 (14)  0.898 (14)  0.905 (14) 
N2—H6  0.909 (17)  0.895 (17)  0.926 (16) 
N2—H6^{i}  0.909 (17)  0.895 (17)  0.926 (16) 
F1—Be—F2  110.52 (5)  110.43 (5)  110.61 (5) 
F1—Be—F3  108.55 (8)  108.19 (8)  108.81 (8) 
F1—Be—F3^{i}  108.55 (8)  108.19 (8)  108.81 (8) 
F2—Be—F3  108.58 (8)  108.46 (8)  108.73 (8) 
F2—Be—F3^{i}  108.58 (8)  108.46 (8)  108.73 (8) 
F3—Be—F3^{i}  112.06 (6)  111.98 (6)  112.15 (6) 
F3^{i}—Be—F3  112.06 (6)  111.98 (6)  112.15 (6) 
H1—N1—H2  118.9 (11)  118.4 (11)  119.5 (10) 
H1—N1—H3  106.2 (12)  104.2 (12)  108.1 (12) 
H1—N1—H3^{i}  106.2 (12)  104.2 (12)  108.1 (12) 
H2—N1—H3  109.4 (14)  107.0 (14)  111.6 (14) 
H2—N1—H3^{i}  109.5 (14)  107.0 (14)  111.6 (14) 
H3—N1—H3^{i}  105.7 (14)  105.5 (14)  105.8 (14) 
H3^{i}—N1—H3  105.7 (14)  105.5 (14)  105.8 (14) 
H4—N2—H5  105.6 (11)  105.5 (11)  105.7 (11) 
H4—N2—H6  108.2 (13)  107.6 (12)  108.8 (13) 
H4—N2—H6^{i}  108.2 (13)  107.6 (12)  108.8 (13) 
H5—N2—H6  111.1 (14)  109.1 (14)  113.1 (14) 
H5—N2—H6^{i}  111.1 (14)  109.1 (14)  113.1 (14) 
H6—N2—H6^{i}  112.3 (14)  112.1 (14)  112.5 (14) 
H6^{i}—N2—H6  112.3 (14)  112.1 (14)  112.5 (14) 
Symmetry code: (i) x_{1}, −x_{2}+1/2, x_{3}, x_{1}/2. 
Acknowledgements
The authors thank K.L. Stork (Laboratory of Crystallography, University of Bayreuth) for growing the crystals, and P. Daniels and W. Morgenroth for assistance with the synchrotron experiment. The financial support by the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged. The Xray diffraction experiments were performed at beamline D3 of Hasylab at DESY.
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