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Journal of

AppliedCrystallography

ISSN 1600-5767

Approaches to modelling thermal diffuse scattering intriglycine sulfate, (NH2CH2COOH)3·H2SO4

J. M. Hudspeth, D. J. Goossens and T. R. Welberry

J. Appl. Cryst. (2014). 47, 544–551

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J. Appl. Cryst. (2014). 47, 544–551 J. M. Hudspeth et al. · Thermal diffuse scattering in triglycine sulfate

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544 doi:10.1107/S1600576713034547 J. Appl. Cryst. (2014). 47, 544–551

Journal of

AppliedCrystallography

ISSN 1600-5767

Received 5 August 2013

Accepted 23 December 2013

# 2014 International Union of Crystallography

Approaches to modelling thermal diffuse scatteringin triglycine sulfate, (NH2CH2COOH)3!H2SO4

J. M. Hudspeth,*‡ D. J. Goossens and T. R. Welberry

Research School of Chemistry, The Australian National University, Canberra, ACT 0200, Australia.

Correspondence e-mail: jessica.hudspeth@esrf.fr

The thermal diffuse scattering in triglycine sulfate, (NH2CH2COOH)3!H2SO4,has been modelled by treating the intermolecular interactions that give rise tothe correlated atomic displacements like Hooke’s law springs. To limit thenumber of variables in the model, the force constants for the interactions wereparameterized in a number of ways, the most successful of which was anempirical interaction potential using an exponential function of the interatomicseparations.

1. Introduction

Diffuse scattering from a crystalline material is the coherentscattered intensity that is not localized on the reciprocal latticepoints. It is sensitive to two-body correlations (Welberry, 2004;Egami & Billinge, 2003; Nield & Keen, 2001) and providesinformation about the local structure and short-range order inthe system. Displacive disorder due to the thermal motions ofthe molecules produces thermal diffuse scattering. Ratherthan being completely random, the interactions betweenatoms or molecules cause the atomic displacements to becorrelated on a local scale. These correlations determine thedistribution of the thermal diffuse scattering in reciprocalspace.

It is noted that, in an experiment using X-rays and thequasistatic approximation, the displacements may not be trulythermal in origin. However, by being able to model displace-ments that are distributed as if thermal in origin (that is,essentially, in a symmetrical distribution that accords with aharmonic like potential as a function of displacement awayfrom the average) then this component of the diffuse scat-tering can be modelled, interpreted where it appears in thedata, and separated out from the scattering due to other,possibly more interesting, effects such as strain, chemicalorder and so on.

One of the difficulties in modelling the short-range orderthat gives rise to diffuse scattering is that there are relativelyfew widely applicable methods. Being able to reliably modelthe thermal diffuse scattering is particularly useful since it isalways present and can be difficult to distinguish from diffusescattering due to other types of disorder. Whether the thermaldiffuse scattering is what is of interest, or its effects must befactored out to reveal what is of interest, modelling it can beuseful.

Monte Carlo (MC) simulations (Metropolis & Ulam, 1949)have been highly useful for studying order–disorder

phenomena (Neder & Proffen, 2008; Welberry et al., 2010;Nield et al., 1995) in a wide variety of systems (Barabash et al.,2009; Schweika, 1998; Zinkin et al., 1997). For molecularcrystal systems, the intermolecular interactions can bemodelled by setting up a model crystal in which the atoms onadjacent molecules are connected by a series of contactvectors which are treated like Hooke’s law springs (Goossenset al., 2011). The Monte Carlo energy of the system, EMC, isthen given by the sum over all of the contact vectors,

EMC "Pi

Ki di # di0! "2

; $1%

where di0 is the length of the ith contact vector from theaverage structure, di is the length of the ith contact vector inthe model and Ki is the force constant. Once the system hasbeen brought to thermal equilibrium through the MC simu-lation, the diffuse scattering patterns from slices of reciprocalspace of interest are calculated and compared with the data.

For triglycine sulfate (TGS), (NH2CH2COOH)3!H2SO4,there are 74 atoms in the unit cell, and the cell is of relativelylow symmetry. Further, a large ensemble of unit cells must bemodelled to capture the local structural correlations withsufficiently good statistics to allow high-quality diffractionpatterns to be calculated from the model. Such a system isdifficult to model using complex and realistic potentials, asused for example in ab initio methods. An extension of thework discussed here is to use such methods to determinepotentials which can then be simply parametrized such thatthe calculation becomes fast enough for use in a large simu-lation.

One of the challenges of using a network of contact vectorsto model the intermolecular interactions is choosing whichinteractions should be included in the model and determiningappropriate force constants. Including all the contact vectorsup to some cutoff length (typically 3–4 A) helps ensure that allthe important interactions are included in the model.However, this leads to too many contact vectors for it to bepracticable to determine suitable values for their force

‡ Current address: European Synchrotron Radiation Facility, BP 220, 38043Grenoble Cedex, France.

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constants individually. It is therefore necessary to para-meterize the force constant values in some way.

Here this approach is used to model the thermal diffusescattering in triglycine sulfate. Different methods of para-meterizing the force constants have been tested and thecalculated diffuse scattering patterns compared. The compar-ison is qualitative, discussing the abilities of the differentparameterizations to reproduce the shapes of key features.Many features are equally well modelled by the threeapproaches, so the qualitative comparison focuses on thediffraction features that are most sensitive to the differentmodelling approaches. To better explore the qualities of thedifferent approaches, the modelling was undertaken using theMC method at all temperatures, with the parameters of the

occupancy model (Hudspeth, Goossens, Gutmann & Studer,2013) changed to reflect the changes in the ordering close toand far away from the phase transition.

TGS is a hydrogen-bonded ferroelectric that undergoes asecond-order order–disorder phase transition at TC = 322 K(49&C) (Gonzalo, 1966; Lines & Glass, 2001). The transition ischaracterized by one of the glycine molecules, G1 (see Fig. 1),going from being disordered about a statistical mirror plane inthe paraelectric phase (T>TC) to being long-range ordered,with all of the NH3

+ groups on one side, in the ferroelectricphase (T<TC).

The fact that the transition is second order suggests that theG1 orientations will go from long-range ordered, to short-range ordered, to disordered as TGS is heated through theferroelectric phase transition. Being able to accurately modelthe thermal diffuse scattering in TGS is important because anydiffuse intensity arising from short-range order of the G1orientations will be modulated by the thermal motions of themolecules. Hence, to understand the short-range order, andthus gain insight into the mechanism of the phase transition,modelling the thermal diffuse scattering is useful (Hudspeth,Goossens, Gutmann & Studer, 2013).

2. Experimental

2.1. Diffuse scattering data collection

X-ray diffuse scattering data from TGS were collected onthe 11-ID-B beamline at the Advanced Photon Source (APS)using the approach described by Welberry et al. (2005). Thecrystal was mounted in front of a PerkinElmer amorphoussilicon two-dimensional area detector. To collect a full three-dimensional volume of reciprocal space, the sample wasrotated through 184& in ! with a step size of 0.25&. For eachcrystal orientation, 20' 1 s exposures were recorded andsummed together. A dark file (no beam) for the detector wascollected every 4&.

The photon energy was 58.26 keV [! = 0.2127 (1) A] andthe size of the detector was 2048' 2048 pixels, with a pixelsize of 200' 200 mm. The sample-to-detector distance wasdetermined using a CeO2 powder standard and found to be720.405 mm. This gave data out to a maximum q of approxi-mately 8 A#1.

Data were collected at 103, 223, 295, 322 (TC), 373 and423 K. The temperature was controlled using an OxfordCryosystems Cryostream 700 Plus.

The program XCAVATE was used to reconstruct a three-dimensional volume of reciprocal space from the raw data andto extract two-dimensional slices of reciprocal space ofinterest (Estermann & Steurer, 1998; Scheidegger et al., 2000;Weber et al., 2001; Welberry et al., 2005).

2.2. Observed diffuse scattering data

The (0kl) and (h0l) slices of reciprocal space at 295 K areshown in Fig. 2. These slices were of the most interest sincethey proved useful for distinguishing between the differentforce constant models.

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J. Appl. Cryst. (2014). 47, 544–551 J. M. Hudspeth et al. ( Thermal diffuse scattering in triglycine sulfate 545

Figure 1Part of the unit cell of TGS in (a) the ferroelectric phase (T <TC) and (b)the paraelectric phase (T >TC).

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Fig. 3 shows part of the (0kl) slice of reciprocal space at 223,322 (T ’ TC) and 423 K. Well below TC, the system isexpected to be long-range ordered and all the G1 molecules ina given domain should have the same orientation. This meansthat the diffuse scattering intensity in the 223 K data (Fig. 3a)should all be thermal diffuse scattering due to correlations inthe thermal displacements of the atoms.

At 322 K, T ’ TC (Fig. 3b), the intensity of the thermaldiffuse scattering is stronger than at 223 K, but there is alsoadditional diffuse intensity in the form of streaks perpendi-cular to the b) (k) axis, as indicated in the figure. Above TC, at423 K (Fig. 3c), the streaks still appear to be visible but aremuch weaker and more diffuse. These diffuse streaks werefound to be due to short-range correlations in the G1 orien-

tations parallel to the ferroelectric b axis developing as TGSpasses through the ferroelectric phase transition (Hudspeth,

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546 J. M. Hudspeth et al. ( Thermal diffuse scattering in triglycine sulfate J. Appl. Cryst. (2014). 47, 544–551

Figure 3Part of the (0kl) slice of reciprocal space at (a) 223, (b) 322 (T ’ TC) and(c) 423 K from the X-ray diffuse scattering data from TGS. The positionsof two of the diffuse streaks associated with the phase transition areindicated by the yellow arrows in (b). The powder rings in the 423 K data,indicated by the red arrow in (c), were caused by the crystal starting todecompose but did not affect the rest of the diffuse scattering pattern.

Figure 2The (a) (0kl) and (b) (h0l) slices of reciprocal space for TGS from theX-ray diffuse scattering data collected on the 11-1D-B beamline at APSat 295 K.

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2012; Hudspeth, Goossens, Gutmann & Studer, 2013). Theonset of significant short-range order appears to occurbetween 40 and 30 K below TC (Hudspeth, Goossens,Gutmann & Studer, 2013; Hudspeth, Goossens, Welberry &Gutmann, 2013).

The diffuse scattering for the (h0l) slice of reciprocal spaceis not affected by the short-range order of the G1 orientations,since in projection down b) the G1 ordering is not apparent.The only change is that the thermal diffuse scattering intensitygets stronger with increasing temperature, so this has not beenshown.

3. Model description

3.1. Displacive disorder model

The correlated atomic displacements that give rise to thethermal diffuse scattering were modelled using the programZMC (Goossens et al., 2011). A model crystal of 32' 32' 32units cells was populated, based on the average structure ofTGS. Atoms on adjacent molecules were connected by anetwork of contact vectors treated like Hooke’s law springs[equation (1)].

Contact vectors up to about 3 A in length were included inthe model, which gave about 100 different contact vectortypes. Symmetry-related contact vectors were considered to bethe same type. The network of contact vectors in TGS is shownin Fig. 4.

Three different approaches to parameterizing the forceconstants were investigated. The first was simply to make all ofthe contact vectors of equal strength. This is not expected tobe a highly realistic model, since some interactions should bemore important than others and should therefore havestronger force constants. However, it is still of interest to seehow well this model reproduces the observed diffuse scat-tering and to compare it with more complex parameterizationmethods that use more free parameters.

The second method was to use the Buckingham potential(Jensen, 1999) as a basis for parameterizing the forceconstants. The Buckingham potential describes the interactionpotential between two non-bonded atoms as a function ofdistance, r, and is given by

!$r% " Ajk exp #Bjkr! "

# Cjk=r6; $2%

where Ajk, Bjk and Cjk are constants that depend on whichtypes of atoms are interacting [here, taken from Filippini &Gavezzotti (1993)]. The Buckingham potential between ahydrogen and an oxygen atom is shown in Fig. 5. The valuesfor the force constants in the model, as a function of equili-brium separation, di0 , are given by the derivative of equation(2),

Ki " AjkBjk exp #Bjkdi0! "

# 6Cjk=d7i0: $3%

The force constant values for interactions between a hydrogenand an oxygen atom are shown in Fig. 6(a). For short contactvectors, the force constants are very large, and as the averagedistance between the atoms increases, the force constants goto zero. This means that in the model there will be a largerenergy penalty for deviations in the length of short contactvectors from their average value, compared to longer contactvectors. This seems physically reasonable since atoms that arecloser together are expected to be interacting more stronglythan those further apart. Once the atoms are far enough apartthey will essentially be independent of each other and can beconsidered as not interacting at all.

The only modification that was made to the force constantvalues from equation (3) was to set a maximum value. This waschosen to be 50 (in the arbitrary units used – all energies areultimately relative to the temperature used in the MC simu-lation), which corresponds to a contact vector length of about2 A. The reason for setting this maximum is that the values ofthe force constants from equation (3) increase very rapidlybelow about 2 A, but only about ten of the contact vectortypes are in this length range. Giving them very large forceconstants therefore risks overweighting their importance inthe model.

The third method of parameterizing the force constants wasdeveloped by Chan, Welberry, Goossens & Heerdegen (2010)

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J. Appl. Cryst. (2014). 47, 544–551 J. M. Hudspeth et al. ( Thermal diffuse scattering in triglycine sulfate 547

Figure 4A subset of the network of contact vectors in TGS up to 3 A in length inthe plane perpendicular to c).

Figure 5The Buckingham potential between a hydrogen and an oxygen atom.

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and is also based on the idea that the spring constants shouldbe largest for short contact vectors and successively smallerfor longer ones. The force constants are parameterized usingthe empirical formula

Ki " A exp #B di0 # Rjk

! "# $* C; $4%

where Rjk is the sum of the Van der Waals radii of the twoatoms [taken from Bondi (1964)], di0 is the average length ofthe contact vector, and A, B and C are empirically determinedconstants. The values of A, B and C used in the simulationswere A " 11, B " 0:4 and C " #8 since similar values havepreviously been refined using data from a range of molecularcrystal systems (Chan, Welberry, Goossens & Heerdegen,2010; Chan, Welberry, Heerdegen & Goossens, 2010; Chan &Goossens, 2012). The force constant values as a function ofaverage separation for contact vectors between a hydrogenand an oxygen atom are shown in Fig. 6(b).

Like the force constants from the Buckingham potentialmodel, this assigns the largest force constants to the shortestcontact vectors and the force constant value decreases to zerowith increasing contact vector length. However, the decreasein the force constant value with distance is far more gradualthan for the Buckingham potential model. This means thatshort contact vectors are not as strongly weighted as in theBuckingham model, since changing the length of a short

contact vector will cause a smaller relative increase in theenergy of the system.

It should be noted that, while the values of the forceconstants relative to each other are important, their absolutevalues are not. The relative strengths of the contact vectorsremain constant during the Monte Carlo simulation, but theabsolute values are globally scaled to achieve the requiredaverage atomic displacements at thermal equilibrium, asdescribed below. Once the model crystal is set up, ZMC uses aforward Monte Carlo algorithm.

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548 J. M. Hudspeth et al. ( Thermal diffuse scattering in triglycine sulfate J. Appl. Cryst. (2014). 47, 544–551

Figure 6The force constant values, as a function of average separation, for acontact vector between a hydrogen and an oxygen atom from (a) theBuckingham potential model [equation (3)] with a maximum value of 50and (b) the model developed by Chan et al. [equation(4)] with A " 11,B " 0:4 and C " #8.

Figure 7The (a) (0kl) and (b) (h0l) slices of reciprocal space from the X-raydiffuse scattering data (top half) at 223 K and the calculated diffusescattering patterns (bottom half) from the purely displacive disordermodel. The force constants were parameterized from the Buckinghampotential model. The region enclosed by the rectangle in the (h0l) slice in(b) is shown in more detail in Fig. 8, for the different force constantmodels.

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3.2. Orientational disorder model

The short-range order in the G1 orientations close to TC

was modelled using an Ising-type model in which the two G1orientations were treated as spin up and spin down (Hudspeth,2012; Hudspeth, Goossens, Gutmann & Studer, 2013). Tomodel both the orientational and displacive short-range order,the occupancy file from the Ising-type MC simulation was readinto ZMC, which was then used to model the correlatedatomic displacements.

4. Results and discussion

4.1. Displacive disorder only

Well below TC, the G1 orientations should be long-rangeordered and only thermal diffuse scattering is expected to bepresent. The purely displacive disorder model was thereforeused to model the diffuse scattering observed in TGS at 223 K.

The observed X-ray diffuse scattering at 223 K for the (0kl)and (h0l) slices of reciprocal space along with the calculateddiffuse scattering patterns from the Buckingham potentialforce constant model are shown in Fig. 7.

The model successfully reproduces the size and relativeintensities of the thermal diffuse scattering around the Braggpeaks. This indicates both that the Buckingham potentialmodel gives reasonable values for the force constants and thatall of the important intermolecular interactions have beenincluded in the model.

The slices of reciprocal space calculated from the otherforce constant models have not been shown since they lookedvery similar to the slices calculated from the Buckinghammodel in terms of the size and relative intensity of the diffusescattering. There was, however, some variation in the shape ofthe diffuse scattering around some of the Bragg peaks. Thiscan be seen in Fig. 8, which shows the diffuse scattering arounda few of the Bragg peaks in the (h0l) plane for the differentforce constant models and the data.

Both the Buckingham force constant parameterization andthe parameterization by Chan et al. [equation (4)] are quitesuccessful at reproducing the shapes of the diffuse scatteringseen in the data. For the model with all the force constants ofequal value, the agreement is less good since the shapes of thediffuse scattering around the Bragg peaks are too elliptical

compared to the data. This indicates that parameterizing theforce constants so that shorter contact vectors have largerforce constants than longer ones is a good approach. Howmuch stronger the short contact vectors need to be is unclear,since the two models have quite different relative forceconstant strengths but give quite similar results.

4.2. Combined orientational and displacive disorder model

In the vicinity of the phase transition temperature, there isexpected to be short-range order of the G1 orientations, so the

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J. Appl. Cryst. (2014). 47, 544–551 J. M. Hudspeth et al. ( Thermal diffuse scattering in triglycine sulfate 549

Figure 8The shape of the diffuse scattering around selected Bragg peaks in the(h0l) plane from the X-ray data at 223 K (a) and calculated from thedisplacive disorder model with the three different parameterizations ofthe force constants: (b) Buckingham, (c) Chan et al. and (d) all equal.

Figure 9The (a) (0kl) and (b) (h0l) slices of reciprocal space from the X-raydiffuse scattering data (top half) at 322 K and the calculated diffusescattering patterns (bottom half) from the combined orientational anddisplacive disorder model with a correlation coefficient of 0.6 for the G1orientations between b1 first neighbour types. The force constants werecalculated from the parameterization method of Chan et al. [equation(4)]. The region enclosed by the rectangle in the (0kl) slice in (a) is shownin more detail in Fig. 10, for the different force constant models.

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combined orientational and displacive disorder model wasused to model the diffuse scattering at 322 K, T ’ TC.

The observed X-ray diffuse scattering data at 322 K forTGS and the calculated diffuse scattering patterns with acorrelation coefficient of 0.6 for the G1 orientations and theforce constant parameterization by Chan et al. [equation (4)]are shown in Fig. 9. The calculated diffuse scattering patternsgenerally look very similar to those for the purely thermalmodel shown in Fig. 7, since most of the diffuse intensity is stillthermal diffuse scattering.

The force constant model used for the calculated patternsshown in Fig. 9 was the parameterization by Chan et al.[equation (4)]. As for the purely displacive model, the calcu-lated diffuse scattering patterns with the different forceconstant models globally look very similar, so the calculatedpatterns with the Buckingham parameterization of the forceconstants and with all the force constants of equal value havenot been shown. However, the different force constant modelsdo affect the intensity of the diffuse streaks due to the G1short-range order, as shown in Fig. 10. For the parameteriza-tion by Chan et al. and the model with all the force constants ofequal value, there is a clear diffuse streak in the calculatedpattern between the 041 and 042 Bragg peaks. However, forthe Buckingham parameterization, the intensity of the diffusestreak is barely above the background. In addition, theBuckingham parameterization causes significant distortion of

the shapes of the diffuse scattering around the Bragg peakscompared to the experimental data.

The difference between the force constant parameterizationby Chan et al. and the Buckingham parameterization is thatthe decrease in the force constant values with distance is farmore gradual for the parameterization by Chan et al. (seeFig. 6). The fact that the Buckingham parameterization is lesssuccessful at reproducing the intensity of the diffuse streakstherefore suggests that the Buckingham potential model isoverweighting the importance of the short contact vectorscompared to the larger ones.

While the Buckingham potential model should in theorygive accurate force constants for the interactions that areincluded in the model, all the contact vectors of greater thanabout 3 A in length have been excluded. While this is also thecase for the parameterization by Chan et al., the fact that theparameters of Chan et al.’s formula are determined empiricallyusing data from a range of molecular crystals means that theforce constants on the contact vectors that are included in themodel have been implicitly adjusted to compensate for the factthat not all of the contact vectors are included. This is not thecase for the Buckingham parameterization.

While both the force constant model by Chan et al. and themodel with all of the force constants of equal value success-fully reproduce the diffuse streak between the 041 and 042Bragg peaks, the model with all of the force constants of equalvalue has already been shown not to reproduce the shape ofthe thermal diffuse scattering around some of the Bragg peaksfor the purely displacive model (see Fig. 8). The para-meterization by Chan et al. is therefore the most successfulmethod for modelling the relative strengths of the inter-molecular interactions.

The fact that the force constant parameterization methodby Chan et al. has previously been used to model othermolecular crystal systems (Chan, Welberry, Goossens &Heerdegen, 2010; Chan, Welberry, Heerdegen & Goossens,2010; Chan & Goossens, 2012; Welberry et al., 2012) and alsoworks well for TGS is a useful result because it suggests thatthis method for modelling the intermolecular interactionsshould be applicable in a whole range of molecular crystalsystems.

While on the one hand this may seem to limit the insightthat can be gained from analysis of thermal diffuse scatteringfrom molecular crystals, it has the great advantage ofproviding a means of helping to determine which effects arethermal in nature and which have other causes – occupancies,for example.

5. Conclusions

The thermal diffuse scattering in TGS was successfullyreproduced by modelling the intermolecular interactions bytreating them like Hooke’s law springs. The most successfulmethod of parameterizing the force constants was theempirical formula developed by Chan, Welberry, Goossens &Heerdegen (2010). The reason why this method was moresuccessful than the Buckingham potential parameterization

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550 J. M. Hudspeth et al. ( Thermal diffuse scattering in triglycine sulfate J. Appl. Cryst. (2014). 47, 544–551

Figure 10The area around the diffuse streak between the 041 and 042 Bragg peaksin the (0kl) slice of reciprocal space from the X-ray data (a), as well asthat calculated from the combined orientational and displacive disordermodel with the different force constant parameterizations: (b) Buck-ingham, (c) Chan et al. and (d) all equal. The models for the calculatedpatterns all had a correlation coefficient of 0.6 between b1 first neighbourtypes. The plot profiles of the integrated intensities through the diffusestreak are shown in part (e) and have been vertically offset for clarity.

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appears to be that the force constants for the interactions thatare included in the model have been implicitly adjusted tocompensate for the fact that not all of the interactions areincluded. The fact that the parameterization method by Chanet al., which has previously been used to model the inter-molecular interactions in other molecular crystal systems, wasalso successful for TGS suggests that it should be applicable ina wide range of molecular crystal systems.

We acknowledge travel funding provided by the Interna-tional Synchrotron Access Program (ISAP) managed by theAustralian Synchrotron and funded by the AustralianGovernment. JMH thanks the Australian Institute of NuclearScience and Engineering (AINSE) for support through apostgraduate award. DJG thanks the Australian ResearchCouncil and the NCI National Facility for support. We thankDr Kevin Beyer, Dr Karena Chapman and Dr Peter Chupas ofthe Advanced Photon Source (APS) as well as Dr RossWhitfield and Dr Marek Pasciak for assistance with the diffusescattering data collection. The support of the APS for beamtime grant GUP-24581 is appreciated. Use of the APS wassupported by the US Department of Energy, Office of Science,Office of Basic Energy Sciences, under contract No. DE-AC02-06CH11357.

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J. Appl. Cryst. (2014). 47, 544–551 J. M. Hudspeth et al. ( Thermal diffuse scattering in triglycine sulfate 551electronic reprint