Introduction

Diffuse scattering from a crystalline material is the coherent scattered intensity that is not localised on the reciprocal lattice points. It comes about due to short-range order (SRO) in the crystal, and is sensitive to two-body correlations. Its analysis can determine details about the nature of the SRO in the crystal [32]. As systems undergo phase transitions from disorder to order (for example on cooling) the correlations between atoms and molecules grow, and eventually the correlation length diverges and a new phase is obtained. Hence, the study of the diffuse scattering from a system close to it phase transition(s) can shed light on the nature and mechanism of the transition [12].

Triglycine sulphate, (TGS), (NH2CH2COOH)3H2SO4, undergoes a ferroelectric phase transition at T C  = 322 K (49 °C) (∼334 K in the case of fully deuterated TGS, FDTGS [15]), and the phase transition is of second-order, order-disorder type [11, 23]. Above T C , in the paraelectric phase, one of the glycine molecules (labelled G1 in Fig. 1), and particularly the NH +3 group within it, is disordered across a mirror plane [3]. The G2 and G3 molecules are then equivalent and the structure has the space group symmetry P21/m. On cooling through T C , the orientation of G1 goes from being disordered about the mirror plane to being long-range ordered, with all of the NH +3 groups on one side, and the mirror plane disappears. The ordered orientation of G1 molecules lowers the space group symmetry to P21 and produces a spontaneous polarisation along the ferroelectric b axis.

Fig. 1
figure 1

Part of the unit cell of TGS in a the ferroelectric phase (T < T C ) and b the paraelectric phase (T > T C )

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While, the hydrogen bonds are generally believed to play a significant role in the phase transition of hydrogen containing ferroelectrics [6, 17], the role of the hydrogen bonds in TGS is unclear. The role of the short hydrogen bond between G2 and G3 is particularly poorly understood, with some studies suggesting that it may play a crucial role in triggering the phase transition [4, 14, 19], and others suggesting that it may not be involved [8, 9]. Kikuta et al. [21, 22] refined the structure of TGS with both positive and negative polarity by poling the crystal. They found that the position of the H3 atom in the G2–G3 bond is correlated with the G1 orientation. This supports the G2–G3 hydrogen bond being involved in the phase transition, although, it does not mean that it triggers it.

An argument against the G2–G3 hydrogen bond playing a dominant role in the phase transition is the fact that the increase in the phase transition temperature of TGS on deuteration is relatively small.

The change in the G2–G3 hydrogen bond geometry in TGS on deuteration is similar to that of the short hydrogen bond in KDP [5], but the increase in T C is only 12 K. This indicates that the G2–G3 short hydrogen bond is unlikely to be driving the transition. An alternative to the G2–G3 bond driving the phase transition is that the N1–D13⋯O22 and N1–D11⋯O3 hydrogen bonds (see Fig. 1) weaken with increasing temperature, allowing the G1 orientation to become disordered [16, 18].

It has been suggested that it is the dipole–dipole interactions between G1 molecules that drive the phase transition, rather than the hydrogen bonds. These interactions have been modelled in a number of ways. These include an Ising-type model, in which the pseudo Ising spin variables are the two orientational states of G1 along the ferroelectric b axis [10], and a unified oscillator model in which the NH +3 groups on G1 are considered to be interacting oscillators in a double-well potential [5, 6].

Critical diffuse scattering measurements show an anomalous rise in the diffuse scattering around some of the Bragg peaks in TGS for T ∼ T C [10, 25, 29, 30]. The intensity of this critical diffuse scattering is a maximum at T C and decreases with |T − T C |; Further, its distribution is elongated perpendicular to b * and narrow parallel to b *, which suggested that the interactions between dipoles is strongest parallel to the ferroelectric b axis and that there are short-range correlations along b.

Here, analysis of X-ray and neutron diffuse scattering using Monte Carlo (MC) modelling is used to examine the SRO in TGS, and through that determine likely drivers for the phase transition. This is practicable because diffuse scattering is sensitive to two-body correlations [32], and therefore, allows us to explore the interactions between atoms and molecules.

Experimental

X-ray diffuse scattering

X-ray diffuse scattering data from TGS were collected on the 11-ID-B beamline at the Advanced Photon Source (APS) using the approach described in [34]. The crystal was mounted in front of a PerkinElmer amorphous silicon two-dimensional area detector. To collect a full three-dimensional volume of reciprocal space, the sample was rotated through 184° in ω with a step size of 0.25°. For each crystal orientation, 20 × 1 s exposures were recorded and summed together. A dark file (no beam) for the detector was collected every 4°.

The photon energy was 58.26 keV [\(\lambda =0.2127(1)\,\AA\)] and the size of the detector was 2048 × 2048 pixels, with a pixel size of 200 × 200 μm. The sample to detector distance was determined using a CeO2 powder standard and found to be 720.405 mm. This gave data out to a maximum q of approximately 8 \(\AA^{-1}\).

Data were collected at 103, 223, 295, 322 (T C ), 373 and 423 K. The temperature was controlled using an Oxford Cryosystems Cryostream 700 Plus.

The program XCAVATE was used to reconstruct a three-dimensional volume of reciprocal space from the raw data and to extract two-dimensional slices of reciprocal space of interest [7, 26, 31, 34]. The 2/m Laue symmetry of the crystal was applied to generate data in the unrecorded regions, and to improve counting statistics in regions that were recorded more than once.

Neutron diffuse scattering

Neutron diffuse scattering data from FDTGS were measured on the SXD beamline at the ISIS spallation neutron source [20]. SXD combines the use of a time-of-flight measurement technique, and an array of 11 position-sensitive area detectors (covering a total of 49.4 % of the solid angle [20]) to access large volumes of reciprocal space for a given orientation of the crystal [20, 33]. Each detector has 64 × 64 pixels, with a pixel size of 3 × 3 mm . For each neutron pulse, a time of flight spectrum is recorded for each pixel of each detector.

Close to full three-dimensional volumes of reciprocal space were obtained by recording diffuse scattering data at 6 different crystal orientations for each temperature. Data were collected at each crystal orientation until the integrated beam current reached 1500 μA (∼10 h).

Data were collected at 300, 334 (T C ) and 373 K. The temperature was controlled using a closed-cycle refrigerator (CCR).

The raw data were processed using the program SXD2001 [20]. The reciprocal volume was calculated out to \(q = 15\,\AA^{-1}\) with a step size of \(0.1\,\AA^{-1}\) and two-dimensional slices of reciprocal space were then extracted. The 2/m Laue symmetry of the crystal was applied to improve reciprocal space coverage.

Diffuse scattering data

Figure 2a shows neutron diffuse scattering in the 0 k l reciprocal layer of FDTGS, centred on the origin. These data were measured at 334 K (T∼T C ). The Bragg peaks show irregular shapes due to the open geometry of the SXD instrument, which effectively images the crystal onto the detector. However, the intense neutron beam and large area detectors make the instrument very useful [33]. The most crucial features revealed in these data are the short streaks of scattering perpendicular to b *; Fig. 2b shows these abstracted from (a). By taking further cuts at small increments of h (i.e. cuts parallel and close to 0 k l) it can be shown that the streaks are cross sections through small pancake-like regions. This indicates that they are the results of small one-dimensional (1-d) domains parallel to b *; this shows good agreement with earlier results [10]; however, these more extensive data allow a model of the correlation structure of the polarising molecules.

Fig. 2
figure 2

The 0 k l slice of reciprocal space from the SXD neutron data at 334 K (T∼T C ). a Shows a detailed view of the region in the centre of the slice, and b shows a schematic of the positions at which the diffuse streaks occur

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This is especially true when X-ray diffuse scattering is incorporated, since the 11-ID-B beamline gave high-quality, low-noise data over a wide region of reciprocal space. As it is the NH +3 group that displaces most pronouncedly, and X-ray scattering is not sensitive to the H atoms, or D atom in the case of FDTGS, such data (see for example Fig. 3) is less sensitive to the polarised G1 molecule than is the neutron data, but shows thermal diffuse scattering very clearly, and does indeed show the domain-related streaking, although in fewer places (see Fig. 3b).

Fig. 3
figure 3

Part of the 0 k l slice of reciprocal space at a 223, b 322 (T ∼T C ), c 373 and d 423 K from the X-ray diffuse scattering data from HTGS. The positions of two of the diffuse streaks associated with the phase transition are indicated by the yellow arrows in (b). The powder rings in the 423 K data, indicated by the red arrow in (d) were caused by the crystal starting to decompose, but did not affect the rest of the diffuse scattering pattern (Color figure online)

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Although not as sensitive to the polarised G1 molecules, the high resolution of the X-ray data made it suitable for investigating the evolution of the diffuse streaks with temperature. Figure 3 shows part of the 0 k l slice of reciprocal space from the HTGS X-ray diffuse scattering data at 223, 322 (T∼T C ), 373 and 423 K. The region around the (041) and (042) Bragg peaks is shown in more detail in Fig. 4, along with the plot profiles, measured in ImageJ [27], of the diffuse intensity between the peaks (Fig. 5).

Fig. 4
figure 4

The region around the (041) and (042) Bragg peaks in the 0 k l slice of reciprocal space from the X-ray data at a 223, b 322 (T∼T C ), c 373 and d 423 K. Plot profiles of the intensity in the region between the (041) and (042) Bragg peaks (indicated by the rectangles) are shown in (e). The plot profiles were measured in ImageJ and have been vertically offset for clarity

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Fig. 5
figure 5

The change in the FWHM and area with temperature of the diffuse streak between the (041) and (042) Bragg peaks observed in the 0 k l slice of the X-ray data. The intensities of the peaks were measured using the plot profile tool in ImageJ, as shown in Fig. 4e, and fitted with a Gaussian

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Well below T C , at 223 K (Fig. 3a) all the diffuse scattering intensity is thermal diffuse scattering, which is as expected since the system should be long-range ordered and all the G1 molecules in a given domain should have the same orientation. At 322 K, T∼T C (Fig. 3b), the intensity of the thermal diffuse scattering is stronger than at 223 K, and the G1 SRO diffuse streaks perpendicular to the b * (k) axis are clearly visible, as indicated in the figure. Above T C , at 373 and 423 K (Fig. 3c, d), the streaks are still visible but appear weaker and more diffuse.

Although, the large temperature gaps between the measurements makes it impossible to determine exactly when the diffuse streaks (and therefore the onset of SRO) first occur, they most likely begin to appear between 40 and 30 K below T C . No diffuse streaks were observed in the neutron data for FDTGS at 295 K (T − T C  ≈ 40 K), but diffuse streaks were visible in the X-ray diffuse scattering from HTGS at 295 K (T − T C  ≈ 25 K).

In order to gain a more quantitative understanding of the evolution of the diffuse streaks with temperature, the plot profiles of the intensities of the diffuse streak in the X-ray data between the (041) and (042) Bragg peaks were measured in ImageJ (Fig. 4e). It should be noted that no diffuse streaks were observed in the 223 K data, so the small peak in the 223 K plot profile is most likely caused by the tail of the thermal diffuse scattering around the (041) Bragg peak. The plot profile at each temperature was fitted with a Gaussian.

Figure 5 shows that the FWHM decreases to a minimum at the phase transition temperature and then increases with increasing temperature. Similarly, the the area increases to a maximum at the phase transition temperature and then decreases with increasing temperature. This indicates that the short-range correlations causing the diffuse streaks are strongest around the phase transition temperature and then weaken with increasing temperature.

Model description

For a two state system, occupancy can be described using a model analogous to the Ising model in magnetism [1, 24, 28]. The site occupancies are assigned a value of S i  =  ±1, where S =  +1 implies the NH +3 is displaced one way, and S =  −1 implies it is displaced the other. If only nearest neighbour (NN) interactions are considered, the energy of the spin is given by

$$ E = \omega_{\rm{NN}} \sum_{ij}S_iS_j $$
(1)

where ωNN is the interaction constant and the sum is over NNs. The sign of ωNN determines whether the occupancies tend to be positively or negatively correlated. In these simulations, a correlation magnitude was specified, and ωNN was periodically adjusted to obtain the desired correlation. Once the correlation structure was generated, a model crystal of 32 × 32 × 32 unit cells was constructed and its diffraction pattern calculated, using the program ZMC [13].

While, the principal correlation direction for the G1 molecules must clearly be parallel to the ferroelectric b axis, the G1 first neighbours directly along b (see the chain dotted line in Fig. 6), which is to say those G1 neighbours related by the full b unit cell vector, rather than the nearest G1 in the next cell along in b, are over 12 \(\AA\) apart. This distance is not unreasonable for dipole–dipole interactions, but is too large for there to be any direct intermolecular interactions.

Fig. 6
figure 6

Schematic of the ordering between b1 first neighbour type G1 molecules being mediated by the short hydrogen bond between G2 and G3. Depending on the orientation of G1, the N1 atom can either be hydrogen bonded to O22 (as shown here) or to O32. If N1 is hydrogen bonded to O22 via N1–H13⋯O22 (indicated by the solid arrows), then H3 is covalently bonded with O32. This prevents the N1 atom on the next G1 molecule from being able to form a hydrogen bond with O32 via N1–H11⋯O32 (indicated by the dashed arrows). This interaction propagates along the chain of b1 first neighbours and leads to ordering of the G1 orientations. The chain dotted line indicates a direct G1 to G1 vector parallel to b

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Inspection of the average crystal structure suggests a mechanism, whereby, the molecules might be linked such that the G1 displacements will correlate as the data suggests. This is illustrated in Fig. 6. While the correlations appear to propagate along b, the interaction must be indirect. The short H-bond between G2 and G3 may act as intermediary, giving the zigzag network of interactions as illustrated (G1 pairs connected by these interactions are denoted to be of type b1 to differentiate them from neighbours indirectly along b). Both correlations between b1 type neighbouring G1 molecules and G1 molecules directly along b were tested in the MC modelling.

The diffuse scattering due to the G1 SRO will be modulated by the thermal motions of the molecules. This was modelled following a recently established approach [2].

Results and discussion

Figure 7 shows the calculated distribution of diffuse streaks in 0 k l for a correlation coefficient (cc) of +0.95 between the b1 neighbours (Fig. 7b) and between neighbours directly along b (Fig. 7c). The different correlation structures lead to significantly different distributions of the diffuse streaks in reciprocal space. The distribution extracted from the neutron data at 334 K is also shown for comparison purposes. No other disorder is present in the model, but it is clear that the diffuse streaks occur in the correct places for correlations between b1 neighbours, whereas, the diffuse streaks for correlations between neighbours in b give a very poor match to the data.

Fig. 7
figure 7

The distribution of the diffuse streaks in the 0 k l slice of reciprocal space a from the SXD neutron diffuse scattering data at 334 K (Fig. 2) and calculated from the orientational short-range order model with strong positive correlations (cc ≈ 0.95) of the G1 orientations between b b1 first neighbour types (zigzag chain as per Fig. 6), and c first neighbours directly along b (chain dotted line in Fig. 6). The positions of the Bragg peaks, indicated by the grey dots, have been superimposed on the calculated patterns and the blue crosses indicate the origin of reciprocal space. Two different regions of reciprocal space have been circled for ease of comparison between the patterns (Color figure online)

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The G1 orientations being correlated for b1 first neighbour types suggests that the interactions are driven by the hydrogen bonds, rather than the dipole–dipole interactions since, as discussed above, the interactions between b1 first neighbour types could be mediated by the short hydrogen bond between G2 and G3 (see Fig. 6).

The thermal diffuse scattering also needs to be included in the model since the diffuse streaks due to the orientational SRO will be modulated by the thermal motions of the molecules. The calculated X-ray diffuse scattering pattern for the 0 k l slice of reciprocal space from the combined SRO and thermal diffuse scattering model is shown in the lower half of Fig. 8. The correlation between b1 neighbours is 0.6. The agreement with the X-ray data at 322 K both in terms of the broad thermal diffuse blobs centred on the Bragg peaks and the short occupancy-related streaks between Bragg peaks, is excellent.

Fig. 8
figure 8

The 0 k l slice of reciprocal space from the X-ray diffuse scattering data (top half) at 322 K and the calculated diffuse scattering pattern (bottom half) from the combined orientational and displacive disorder model with a correlation coefficient of 0.6 for the G1 orientations between b1 first neighbour types. The insets show the area around the diffuse streak between the (041) and (042) Bragg peaks

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Correlation coefficient for b1 first neighbour types

To investigate the effect on the diffuse streaks of the strength of the correlations between b1 neighbours, simulations were run with different values for the correlation coefficient. The real space distribution of the G1 orientations in the bc plane of the model crystal, as well as the corresponding calculated X-ray diffuse scattering in the 0 k l slice of reciprocal space, are shown in Fig. 9. In real space as the correlation coefficient increases it causes the formation of linear domains of increasing length parallel to the ferroelectric b axis. In reciprocal space, increasing the correlation coefficient causes the intensity of the diffuse streaks to become stronger.

Fig. 9
figure 9

The distribution of the G1 orientations (represented by the blue and orange circles) in the bc plane with a correlation coefficient (cc) between b1 first neighbour types of a 0.2, b 0.6 and c 0.9. The corresponding calculated X-ray diffuse scattering patterns for the 0 k l slice of reciprocal space are shown in parts (d) to (f), respectively

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While, it is difficult to determine exactly what the correlation coefficient is between the G1 molecules in TGS from the model, a correlation coefficient of somewhere between 0.5 and 0.7 seems reasonable at T C . Below 0.5, the diffuse streaks in the calculated patterns are barely visible, and above 0.7, they are unreasonably sharp (Fig. 10).

Dipole–dipole interactions between chains

These results strongly suggest that at temperatures above but close to T C the G1 displacements couple via the short hydrogen bond between the G2 and G3 molecules. However, the ferroelectric state shows three-dimensional (3-d) ordering, so the question remains: is this 3-d ordering driven by hydrogen bonding or dipolar forces? It is likely that both contribute; as the 1-d chains lengthen, the weak lateral H-bonds will compound in their effect, and the local electrical polarisation will increase, resulting in a cooperative switch to 3-d order. The dipole energy was explored by testing which of four possible inter-chain configurations it preferred. Four different correlation structures between neighbouring chains along a and c were tested, as shown schematically in Fig. 11.

Fig. 10
figure 10

The area around the diffuse streak between the (041) and (042) Bragg peaks in the 0 k l slice of reciprocal space calculated from the combined orientational and displacive disorder model with a correlation coefficient (cc) between b1 first neighbour types of a 0.2, b 0.6 and c 0.9. The plot profiles through the diffuse streak are shown in part (d) and have been vertically offset for clarity

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Fig. 11
figure 11

Schematic of the different correlation structures along a and c between neighbouring chains of G1 molecules for which the dipole interaction energy (Eq. 2) was calculated: a parallel along a and c, b antiparallel along a and parallel along c, c parallel along a and antiparallel along c and d antiparallel along a and c. The view is looking down the b axis, so the zigzag chains of G1 molecules correlated via b1 neighbours, are coming out of the page. Chains with the G1 molecules positively and negatively oriented along the ferroelectric b axis are represented by the + and − signs, respectively

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The potential energy of a pair of electric dipoles, p i and p j , is given by

$$ U = \frac{{\bf p}_i \cdot {\bf p}_j}{ {\bf r}_{ij}^3} - \frac{3({\bf p}_i \cdot {\bf r}_{ij})({\bf p}_j \cdot {\bf r}_{ij})}{ {\bf r}_{ij}^5} $$
(2)

where r ij are the displacement vectors between the dipoles p i and p j [5]. The dipole interaction energy can be obtained by summing over all such interactions out to some cutoff radius.

The distribution of the dipole moments in the model was taken to be the same as the distribution of the disordered nitrogen atoms on G1. The dipole moments of the NH +3 groups were taken to be ±1, since it is the relative energies of the different correlation structures that is of interest. The orientations of the dipole moments within chains were assumed to be always parallel, which meant that the length of the chains along b in the model was infinite. The convergence of dipole interaction energies is known to be problematic [25], so the energy was calculated as a function of cutoff radius for all cases. The summation showed a strong preference for a 3-d parallel ordering of all moments.

Conclusions

This study shows how the analysis of diffuse scattering can be used to build up a real-space picture of how molecular interactions lead to molecular ordering. As TGS is cooled the NH +3 groups form into chains which run in a zigzag parallel to the b axis. The growth of these chains appears related to the H-bonding network, and particularly interactions via the short hydrogen bond between the G2 and G3 molecules; that this initial ordering is driven by H-bonding and not dipolar interactions is supported by its strong anisotropy. It gives a 1-d correlation along b even though the dipolar interaction is 3-d in nature and neighbours in c are closer than those along the b and b1, and neighbours in a show a separation similar to neighbours along b1. Once the 1-d chains form, giving local regions of high polarisation, the dipolar interaction favours neighbouring chains to stack in a parallel fashion, such that when the ferroelectric state is obtained it shows full 3-d ordering. Hence, we suggest that this study gives a very reasonable real-space picture of the ferroelectric phase transition in TGS, including insight into the roles played by the different interactions. It also delivers a structural model, in the form of a model crystal, which successfully reproduces all the key features observed in the X-ray diffuse scattering, giving a comprehensive model of structure and mechanism in TGS.