use of monte carlo simulation for the interpretation and analysis of diffuse scattering

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Use of Monte Carlo simulation for theinterpretation and analysis of diffusescatteringT.R. Welberry a , E.J. Chan a , D.J. Goossens a & A.P. Heerdegen aa Research School of Chemistry, College of Physical Sciences, TheAustralian National University, Canberra, ACT 0200, AustraliaPublished online: 13 Jan 2010.

To cite this article: T.R. Welberry , E.J. Chan , D.J. Goossens & A.P. Heerdegen (2010) Use ofMonte Carlo simulation for the interpretation and analysis of diffuse scattering, Phase Transitions: AMultinational Journal, 83:2, 80-98, DOI: 10.1080/01411590903491752

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Phase TransitionsVol. 83, No. 2, February 2010, 80–98

RESEARCH ARTICLE

Use of Monte Carlo simulation for the interpretation and analysis of

diffuse scattering

T.R. Welberry*, E.J. Chan, D.J. Goossens and A.P. Heerdegen

Research School of Chemistry, College of Physical Sciences, The Australian NationalUniversity, Canberra, ACT 0200, Australia

(Received 12 October 2009; final version received 17 November 2009)

With the development of computer simulation methods there is, for the first time,the possibility of having a single general method that can be used for any diffusescattering problem in any type of system. As computers get ever faster it isexpected that current methods will become increasingly powerful and applicableto a wider and wider range of problems and materials and provide results inincreasingly fine detail. In this article we discuss two contrasting recent examples.

The first is concerned with the two polymorphic forms of the pharmaceuticalcompound benzocaine. The strong and highly structured diffuse scattering inthese is shown to be symptomatic of the presence of highly correlated molecularmotions. The second concerns Agþ fast ion conduction in the pearceite/polybasitefamily of mineral solid electrolytes. Here Monte-Carlo simulation is used tomodel the diffuse scattering and gain insight into how the ionic conduction arises.

Keywords: diffuse scattering; Monte-Carlo simulation; polymorphism;benzocaine; fast ion conductors; pearceite

1. Introduction

In the nearly 100 years since the structure of NaCl was determined by Bragg [1],conventional crystallography (Bragg diffraction) has developed to the point where thestructures of complex materials and huge molecules, such as proteins can often bedetermined in a matter of hours by fully automated systems. However, this progressovershadows the fact that conventional crystallography is a strictly limited techniquebecause it is only sensitive to the average structure within a crystal. Because conventionalcrystallography only measures the intensities of the Bragg reflections, it can only be usedto reconstruct the average unit cell. However, in a real crystal, individual atoms all deviateto some extent from their ideal positions in the average unit cell. Local structure anddynamics within the crystal contribute intensity to regions of the diffraction pattern thatlie far from any Bragg reflection, and are therefore completely ignored by conventionalcrystallography. This intensity is known as diffuse scattering. As it is weak and difficult tointerpret, only a few specialist groups around the world practise the analysis of diffusescattering [2–5]. The vast majority of crystallographers ignore (or simply do not notice) thediffuse scattering that is always present in their experiments. However, with the increasing

*Corresponding author. Email: welberry@rsc.anu.edu.au

ISSN 0141–1594 print/ISSN 1029–0338 online

� 2010 Taylor & Francis

DOI: 10.1080/01411590903491752

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use of synchrotron X-ray sources and area detectors, more and more observations ofdiffuse scattering are being made, and there is increasing recognition that it containsimportant information over and above that contained in Bragg reflections [5].

While much of the early work on diffuse scattering was confined to the area of metalalloys and simple oxides [6–8], our aim has been to develop methods to make the techniqueapplicable to a wide range of different materials. To this end we have pioneered the use oflarge-scale Monte Carlo (MC) computer simulations of disordered structures [9–12].In this method a computer model is set-up based on known physical and chemicalprinciples with adjustable parameters describing the basic interatomic or inter- andintra-molecular interactions and possible mechanisms for disorder. Diffraction patternsare then calculated from the results of the simulations and these are compared to theobserved data. As a result of the comparison adjustments are made to the modelparameters and the process is repeated until satisfactory agreement between observed andcalculated patterns is obtained. The final outcome is a model defined in terms of arelatively small number of parameters that describe real physical and chemical interactionsand readily gives insight into the fundamental reasons for the local structural order anddisorder. Use of such MC computer simulations of a model structure has become apowerful and well-accepted technique for aiding the interpretation and analysis of diffusescattering patterns [9,10].

The advantage of the method is that it can be applied generally to all systems regardlessof their complexity or the size of the atomic displacements that might be present. The onlylimitation is the extent to which the MC energy can be made to represent realistically thereal system energy. At one extreme a very simplified model may be useful in demonstratingparticular qualitative effects [13] while at the other a quantitative and very detaileddescription of a disordered structure can be obtained [14]. This latter type of study hasbeen enhanced by the development of quantitative automatic methods of making thecomparison between observed and calculated diffraction patterns and adjusting the modelparameters accordingly [15–18].

As computers get ever faster it is expected that the methodology will becomeincreasingly powerful and so applicable to a wider and wider range of materials. The mostquantitative studies to date have achieved an agreement (R-factor) between observed andcalculated diffraction patterns (on a pixel by pixel basis), comparable to values achievedfor conventional crystallographic studies using Bragg reflections (see e.g. [17]). Once aviable MC model has been established it can be probed to reveal details of the localstructure.

In this article, we describe two contrasting examples where MC simulation has beenused to give insights into the behaviour of the respective systems. The first example dealswith polymorphism in the pharmaceutical compound benzocaine, C9H11O2N. The twoknown room temperature forms of benzocaine exhibit strong and highly structured diffusescattering which is symptomatic of the presence of highly correlated molecular motions.A quantitative study of the diffuse scattering in this and other pharmaceutical systems isbeing carried out in order to gain insight into how and why polymorphism occurs.

The second example concerns Agþ fast ion conduction in the pearceite/polybasitefamily of mineral solid electrolytes of general composition [M6T2S7][Ag9CuS4] whereM¼Agþ, Cuþ and T¼As3þ, Sb3þ. Electron diffraction patterns of various members ofthis family show highly structured diffuse scattering that includes incommensurate diffusepeaks. Conventional average crystal structure determination shows the Agþ ions as acontinuous distribution of electron density in channels within CuS4 layers of theframework. We here describe the use of MC simulations to model the diffuse scattering in

Phase Transitions 81

order to help determine the exact nature of the Agþ ion distribution and so gain insightinto how the ionic conduction arises. Since, because of multiple scattering effects, theobserved electron diffraction patterns do not provide a quantitative measure of the diffusescattering the aim in this study is to obtain a qualitative fit to the observations.

2. Analysis of diffuse scattering in benzocaine, C9H11O2N

At room temperature benzocaine occurs in two different polymorphic forms – form I ismonoclinic and form II is orthorhombic. The cell dimensions and the intermolecularstacking of the two polymorphs are very similar (see Table 1 and Figure 1). In bothstructures linear chains of molecules involving strong end-on N–H� � �O hydrogen bonds(dotted lines in Figure 1(a) and (b) are generated by translations along the a-axis.Coplanar pairs of these linear chains are interlocked sequentially via the in-planeinteractions of the ethyl groups (dashed lines). These substructural ribbon-motifs areidentified by the light grey shaded envelopes in Figure 1(a)–(d). The ribbons areinterconnected through quasi-perpendicular N–H� � �N hydrogen bonds (thick solid lines)via a 21-screw axis along b. The N–H� � �N–H bonding network generates the herring-bonearrangement of layers of ribbons which can be seen when the structure is viewed down thea-axis (Figure 1(c) and (d)). The stacking difference between the two polymorphs is asymmetry operation which describes how the ethyl groups of the two halves of each ribbonslot into each other, and how the layering of a stack of ribbons is generated along theb-axis. That is, this is either an inversion centre (form I) or a 21-screw (form II).

2.1. Diffuse scattering data collection

Complete three-dimensional (3D) diffuse scattering data for forms I and II were recordedat 300K on the 11-ID-B beamline of the advanced photon source (APS) using 60 keVphotons (�¼ 0.2128(1)A). Each data set consisted of 500 individual frames of datarecorded in steps of 0.36� of the crystal rotation angle ! using a Mar 345 image plate (IP)area detector with a pixel resolution of 0.15mm. The sample to detector distance was543.866mm.

Reciprocal-space sections for use in the analyses were reconstructed from the framesusing the programs MarIndex for indexing of Bragg peaks and XCAVATE [19] forreconstruction of specified two-dimensional (2D) reciprocal sections. Figure 2 showsexamples of such 2D sections of data. It is still not feasible to use full 3D data in analysesand for the present study 10 planar regions of the data were selected from the principalsections h0l, hk0 and 0kl. These regions were selected to include the strongest and mosthighly structured features of the scattering as well as regions of low scatttering to establish

Table 1. Cell data for the three polymorphs of benzocaine.

Polymorph Space group a(A) b(A) c(A) �(�) �(�) �(�) T (K)

I P21/c 8.257 5.501 19.956 90.0 91.699 90.0 300II P212121 8.242 5.311 20.904 90.0 90.0 90.0 300IIIa P21 8.188 10.637 20.476 90.0 90.0 99.370 150

Note: aPolymorph III was discovered subsequent to the initial analysis.

82 T.R. Welberry et al.

Figure 1. (a), (b) Plots of the structure of polymorphs I and II of benzocaine viewed down b.(c), (d) Plots of the structures viewed down a. The grey shaded regions enclose themolecular ribbons referred to in the text. Note the different symmetry of the ribbons in the twoforms.

background levels and both high and low-angle regions to establish magnitudes of atomicdisplacements.

2.2. MC models

2.2.1. Model crystals

For each polymorph a model crystal consisted of 48� 48� 48 unit cells, each containingfour molecules. The atomic positions for each molecule were defined using a z-matrixdescription. This method, commonly used for example in ab initio molecular-orbitalcalculations [20], allows the geometry of the molecule to be specified in terms of

Figure 2. Examples of (a) form II, 0kl observed and (c) form II, hk0 observed and (b) form II, 0klcalculated and (d) form II, hk0 calculated diffraction patterns for benzocaine. It should be noted thatthe calculated patterns contain no Bragg peaks as the average lattice has been subtracted during thecalculation.

bond lengths, bond angles and dihedral angles. This meant that the molecules could betreated as a number of rigid fragments with the only flexibility allowed arising fromvariation of the dihedral angles associated with single C–C, C–N and C–O bonds.The basic molecular orientation was defined by a set of four quaternion components andthe position of the molecule within the unit cell by centre of mass coordinates.

2.2.2. MC energy

The intermolecular interactions were modelled using sets of harmonic (Hooke’s law)springs, with force constants Ki, placed along a carefully selected subset of the largenumber of atom–atom vectors that typically make up a given intermolecular interaction.This subset of atom–atom interactions comprise the effective intermolecular interactionsthat are used in the simulations. Use of such effective interactions is necessary toreduce the computational task to a tractable level. In choosing these vectors the aim isto reduce the number to a manageable level while retaining a sufficient number to definerobustly the separation and mutual orientation of neighbouring molecular fragments.Two or three such atom–atom vectors are usually sufficient for each pair of neighbouringmolecular fragments. Provided this criterion is met the actual choice of vectors is lessimportant. The aim is to arrive at a set of springs that are able to mimic the vibrationalproperties of the real crystal. Torsional springs, with force constants Li, may be used toprovide restoring forces on variable internal dihedral angles.

The form of the MC energy that is used when there is no occupancy disorder and thediffuse scattering is purely thermal in origin may be written:

E ¼Xall linearsprings

Kiðd� d0Þ2þX

torsionalsprings

Lið�� 0Þ2:

Here d0 is the equilibrium length of the given spring and d its instantaneous length.Similarly �0 is the equilibrium value of the particular torsional angle and � itsinstantaneous value. d0 is assumed to be equal to the interatomic distance and �0 to thevalue of the particular torsion angle, observed in the average crystal structure.

Both analyses used 15 independent types of spring each with its associated value of Ki.In addition there were four independent torsional force constants Li, making in total 19independent parameters that were used in the fitting process. Of the 15 linear springs used,12 connected the same pairs of atomic sites in the two models and only three weresubstantially different – these latter being involved in the symmetrically different interfacesjoining the two halves of the molecular ribbons.

2.2.3. MC simulation and model refinement

Initial values were assigned to the Ki and Li force constants and then MC simulation wascarried for 2000MC cycles. At each MC step a molecule was chosen at random and thevariables specifying its conformation, orientation and position were given randomincrements. The system energy was calculated before and after the shift using Equation (1)and moves were accepted or rejected using the normal Metropolis algorithm [21] withBoltzmann partition P¼ exp(�DE/kBT). The simulation temperature was maintained at aconstant value of T¼ 1/kB, where kB is Boltzmann’s constant. For further detailssee ref. [22].

From the initial MC model the system parameters were then adjusted to obtain the bestfit to the observed data. This was carried out, first by manual adjustment of the Ki and Li

force constant parameters and subsequently using least-squares refinement [15]. This lattermethod is very computer intensive as it requires two complete MC simulations to obtaineach of the partial differentials that make up the least-squares matrix, but is ideally suitedto parallel computer processing.

2.3. Results

2.3.1. Fits to the data

The final agreement factor, R ¼ ½P!ðDI Þ2=

P!I2obs�

1=2, was 12.6% for form I and 17.2%for form II calculated on a pixel by pixel comparison over the selected regions of the threereciprocal sections of data, hk0, h0l and 0kl. In total 21,795 data points were used for formI and 30,522 for form II, with weights given by !¼ 1/Iobs. The quality of the fits was alsoreflected in the general appearance, in comparison to the observed data, of calculatedcomplete reciprocal sections. Two examples are shown in Figure 2 but a morecomprehensive comparison of all three sections for both polymorphs is given in Chanet al. [22]. As a stand alone study each of the analyses would have been considered asatisfactory result. However, though both fits were good, that for form II was rather lessgood and despite attempts to improve the model, by better selection of the spring vectors,the difference remained unaccounted for and it was concluded that something must behappening in this system that was not being captured by the simple harmonic springmodel. It was found that one feature of the observed data in particular was notbeing modelled well – a strong and rather elongated diffuse peak in the vicinity of the 200Bragg position (Figure 2(c)).

2.3.2. Correlated molecular motions

The analysis revealed two main types of molecular motion in both polymorphs. The firstof these is that the molecular ribbons which run along the a-axis tend to slide as semi-rigidunits along their length so producing high correlations of the molecular displacements inthis direction (i.e. the correlations are longitudinal correlations since the displacementdirection is parallel to the correlation direction). This produces planes of scatteringperpendicular to a and these may be seen in both the hk0 section (Figure 2(c)) and the h0lsection (not shown) as well as other non-zero level sections that are intersected by theplanes.

The second type of motion involves the lateral motion of the molecules making up themolecular ribbons. The ribbons are stacked together in a herring-bone pattern as seenwhen viewed down the a-axis (see Figure 1(c) and (d)). The very strong and highlystructured scattering in the 0kl section indicates that the molecules are also undergoinghighly correlated motions in directions within the 0kl planes. The displacements that arestrongly correlated are ones that are parallel to the planes of the ribbons. Figure 3 shows adrawing of the molecular packing together with (inset) a small magnified region of thediffuse streaking. The strong correlation direction is normal to the length of the streaksand is approximately in the [0 4 1] direction as indicated by the arrow. The narrowness ofthe streaks that can be seen in Figure 2(a) and in Figure 3 (inset) indicates that thedisplacements are significantly correlated over distances �50 A. It should be noted that,although the correlated displacement direction is in the plane of the ribbons,

the correlations are propagated through the intermediary layers of molecules which areinclined almost at right angles. The rather broad profile along the length of these elongatedstreaks is indicative of the fact that there is very little correlation between thedisplacements in neighbouring ribbons.

Both of the above types of motion involve longitudinal correlations: i.e. ones in whichthe correlation direction and the displacement direction are parallel. The direction of thedisplacement is revealed by the presence of a plane of zero intensity normal to thedisplacements and passing through the origin. In Figure 2(a) and (b), the streaks that runfrom top left to bottom right only occur in the top right and bottom left quadrants andhave no intensity along a line running from top left to bottom right through the origin.Similarly in Figure 2(c) and (d) the vertical streaks running parallel to b* have zerointensity along the central h¼ 0 line.

2.3.3. Phase transition

As the modelling of form II was not as satisfactory as that for form I it was surmised inSection 2.3.1 that the model was lacking some particular feature that was present in thereal system. That the diffuse peak near 200 was observed to be very strong while theharmonic model gave only weak scattering seemed to indicate that some rather substantialchange in the molecular structure factors was required. It was thought possible that somestrong anharmonic effects might be present or even some static disorder where theobserved average structure was the average of two quite different local structures.

In order to investigate this further some low temperature experiments were carried out.It was found that form II undergoes a reversible phase transition only just below roomtemperature (�270K) to a new form which we have designated form III (Table 1).On transformation the single crystal of form II results in a twinned crystal of form III.The average structure of this has been determined and its relationship to form IIdiscussed [23]. Form I does not undergo a phase transition.

Figure 3. A view of the structure down the a-axis. The arrow indicates the direction of stronglongitudinal displacement correlation. This is perpendicular to the narrow streaking (see inset) that isobserved in the 0kl diffraction patterns. See also Figure 2.

Figure 4(c) and (d) shows plots of a single molecular layer of the structure viewed down

the c-axis. This shows that in the low temperature form every second molecular chain has

slipped in the a-direction and now forms a different interface with its neighbour (indicated

by the dotted intermolecular vectors). The interface involving vectors indicated by the dark

solid lines remain virtually unchanged. In the room temperature form the solid and dotted

vectors are translationally equivalent. Figure 4(a) and (b) shows corresponding small

regions of the hk0 diffraction patterns at temperatures either side of the phase transition.

The low temperature pattern contains Bragg peaks for two orientations of the form III

structure. Although much of the diffuse scattering in the diffuse sheets normal to a has

Figure 4. (a), (b) Small regions of the hk0 section of the diffraction patterns of a crystal of form IIrecorded above and below the phase transition at 270K and 260K, respectively. The rectangleshighlight the elongated diffuse peak around 200 which resolves into strong Bragg peaks in thetwinned low temperature structure. (c), (d) Drawings of a single molecular layer of the structures offorms II and III of benzocaine viewed down the c-axis. The two types of vector shown as a heavyblack line and a dotted line are identical in form II but different in form III. Note that form III is in anon-standard monoclinic setting with c as the unique axis (see Table 1).

condensed into the form III Bragg peaks there is still some residual diffuse scatteringconnecting these. This is not unusual. The pattern clearly shows that in the region of theelongated diffuse peak near 200 in form II, form III shows two strong Bragg peaks (onefrom each twin). This adds weight to the original conjecture that at room temperature thestructure may, on a local scale, contain two different configurations that are precursors ofthe different twin components. Further work is in progress to incorporate this possibilityinto a more sophisticated model for the form II structure.

3. Diffuse scattering from Agþ fast ion conduction in solid electrolytes

3.1. Average structure

The average structure of the pearceite/polybasite family of mineral solid electrolytes ofgeneral stoichiometry [M6T2S7][Ag9CuS4] has been described by Bindi et al. [24].Figure 5(a) shows a drawing of the structure viewed in projection down a* andFigure 5(b) down c. The structure is seen to be made of alternate layers of the [M6T2S7]and [Ag9CuS4] substructures. The [M6T2S7] substructure is found to be ordered but the[Ag9CuS4] substructure is highly disordered. Figure 5(c) shows this substructure on its ownand it is seen that the Agþ ions have been modelled by distributing them over a number ofdifferent possible crystallographic sites. In fact a Fourier synthesis of the observedcrystallographic structure factor data shows a more-or-less continuous distribution ofelectron density around the hexagons whose edges occur between each Cu and S site inFigure 5(c). It should be noted that in each Agþ layer the stoichiometry requires that thereare 4.5 Agþ ions and one CuS2 unit within each unit cell.

3.2. Observed electron diffuse scattering patterns

Figure 6 shows examples of the diffuse scattering that has been observed in these materials[25]. The two patterns shown are of the same sample but with different small tilts awayfrom the c zone-axis position. For electron diffraction it is difficult to obtain quantitativemeasurements of diffuse scattering because of multiple scattering and tilting away fromthe ideal zone-axis in this way is a means by which such multiple scattering can at least bepartially eliminated to reveal the true form of the scattering.

In the patterns shown it is seen that there is a motif of six diffuse peaks surroundingsome of the Bragg peaks corresponding to the average crystal lattice. Close inspectionreveals that these diffuse peaks occur at positions that are incommensurate with thelattice of Bragg peaks. Despite the effects of multiple scattering it is also clear that there isstrong asymmetry in the intensities of the six peaks with those on the low-angle side ofthe particular Bragg peak being generally weaker than those on the high-angle side.Such asymmetry is characteristic of local size-effect distortions.

3.3. Development of a model for the disordered structure

3.3.1. Ag ion substructure

First, we consider that the diffuse peaks that are observed arise because the Agþ ions lie ona primitive hexagonal lattice which is incommensurate with the framework substructure.The lattice spacing of the framework is aF¼ 7.4805 A while that measured for the Agþ ionsfrom the positions of the diffuse peaks is aAg¼ 3.13 A. The relative areas of these two

lattices is 5.71:1, implying that if the Ag ion lattice is fully occupied there would be 5.71 Agions in the framework substructure unit cell instead of the 4.5 required by thestoichiometry. This implies that �21.2% of the potential Ag sites must be vacant.

3.3.2. Vacancies

It might be supposed that these vacancies arise because the Ag ions cannot get too close tothe Cu and the two S ions that are in the adjacent CuS2 layer. If the Ag lattice is overlaidwith the CuS2 lattice then, because the two lattices are incommensurate, some Ag ions willnaturally fall on or very close to Cu or S sites. Suppose a criterion is used that any Ag ionfalling within DCu of a Cu site or DS of a S site is to be removed from the lattice. The valuesof DCu and DS can be adjusted until the required 21.2% vacancies is achieved.For simplicity it was assumed that DCu¼DS.

Figure 5. (a) The structure of pearceite viewed down a* showing the different layers. (b) View downc showing the disordered Ag sites. (c) View of the CuS2 and Ag layers alone. (d) View down cshowing the exclusion zones (larger circles) around the Cu and S sites.

3.3.3. Bond-valence requirements

Removing the required 21.2% of Ag ions in this way satisfies the stoichiometry but manyof the remaining ions are still much closer to Cu or S sites than is ideal. A second step indeveloping a plausible model for the Ag ion arrangement is thus to try to imposereasonable local crystal chemical constraints. It is known from the literature that theCu–Cu distance in metallic Cu is 2.5561 A while the Ag–Ag distance in metallic Ag is2.8895 A [26], thus a Cuþ–Agþ distance shorter than �2.72 A is rather unlikely. Likewise,S2þ–Agþ distances shorter than 2.3 A are almost unknown in silver sulfide mineralogy.Thus a S2þ–Agþ distance shorter than �2.3 A was also deemed to be rather unlikely.The remaining 78.8% Ag ions must therefore undergo some additional local relaxation toensure that they satisfy these constraints. In Figure 5(d) the larger shaded circlessurrounding each Cu and S position represent the regions that should be avoided owing tothese chemical constraints.

3.3.4. Highly distorted lattices – paracrystals

The above arguments have been found on the basis that the Ag ions tend to form a simplehexagonal lattice. However, the fact that the scattering does not show sharp Bragg peaksbut rather broad diffuse peaks due to the Ag ion sublattice is an indicator that this lattice isfar from perfect. In fact the 6-peak motif of scattering that appears in the observeddiffaction patterns of Figure 6 is very reminiscent of patterns that we have obtainedpreviously for a dense liquid close to the freezing point [27]. The 6-peak motifs ofscattering that appear in Figure 6 are also reminiscent of the motifs of scattering that wereobserved around Bragg reflections in the diffraction patterns of wustite, Fe1�xO [28].These were modelled using a class of highly distorted lattices called paracrystals [29,30].In that case the models used were defined on a square lattice. In the present case we use acomparable model defined on a hexagonal lattice. Such hexagonal paracrystals havepreviously not been reported but they are a simple extension of the square lattice examples.

Figure 6. Diffuse (electron) scattering patterns of pearceite viewed aproximately down c* from thesame sample with slightly differing degrees of incident beam tilt. The arrows indicate positions of thediffuse peaks that define a hexagonal reciprocal array that is incommensurate with the reciprocallattice of sharp peaks due to the parent structure.

Further details of these will be published elsewhere. For the present purposes we use an

example which is shown in Figure 7. This distribution is used instead of the perfecthexagonal lattice and then the process described in Section 3.3.2 is carried out as before toremove 21.2% of the Ag ions that conflict with the Cu and S positions. Similarly thesubsequent relaxation as described in Section 3.3.3 is also carried out.

3.4. MC simulation

MC simulation was used in two different stages of the modelling – in the production of theparacrystal distribution used for the initial Ag array and in the later relaxation of theAg ions away from the Cu and S positions.

3.4.1. Paracrystal distribution

Gaussian random variables �i, j with zero mean and unit variance are assigned to the i, jsites of a hexagonal lattice (Figure 8). It is assumed that the value of �i, j depends on thevalues of its neighbours along all three equivalent directions. �i, j is used to define thedisplacement of a site along the [0 1] direction. We define a MC energy as,

E½01� ¼X

all sites

�i, j½RLð�i, jþ1 þ �i, j�1Þ þ RTð�iþ1, j þ �i�1, j þ �iþ1, jþ1 þ �i�1, j�1Þ�: ð2Þ

Here RT and RL are interaction energies used to induce longitudinal and transversecorrelations between the variables (Figure 8).

If additional sets of random variables �i, j and i, j are used to describe displacements

along the [10] and [11] directions, respectively, similar equations to Equation (2) may beobtained for E[1 0] and E[1 1] involving these random variables. The same values for RT andRL are used in the expressions for E[0 1], E[1 0] and E[1 1] in order to maintain trigonalsymmetry.

Figure 7. (a) Part of the paracrystal distribution used for the initial Ag lattice distribution.(b) Diffraction pattern corresponding to the distribution in (a).

In carrying out MC simulations using Equation (2) and its symmetry equivalents the

aim is to produce realisations with particular chosen values of the transverse and

longitudinal correlations, CT and CL. Since the required values of RT and RL to achieve

this are initially unknown a feed-back procedure is adopted. After each cycle of iteration

the values of the correlations present in the arrays are computed and compared to the

target values. As a result of the comparison the values of RT and RL are adjusted by an

amount proportional to the difference between the current and target correlation values.

For the high values of correlations required to produce the paracrystal distributions used

here many cycles of iteration are required (�2000).Once the three sets of random variables, �i, j, �i, j and i, j have been produced these

are combined to produce just two variables representing orthogonal Xi, j and Yi, j

displacements.

Xi, j ¼

ffiffiffi3p

2ð�i, j þ i, jÞ

Yi, j ¼ �i, j þ1

2ð�i, j � i, jÞ: ð4Þ

Since the sum of two Gaussian variables is itself Gaussian the variables Xi, j and Yi, j are

also Gaussian. They already have a zero mean but must first be renormalised to have unit

variance (2¼ 1). Then, by further scaling to have a particular chosen variance, these

random variables may be used to produce a paracrystal realisation like that shown in

Figure 7(a). At each point on the regular hexagonal lattice the lattice point is displaced

away from its average position by an amount Xi, j along the x-cartesian direction and Yi, j

along the y-cartesian direction. The value of is used to control the magnitude of the

displacements relative to the cell spacing. In the example in Figure 7(a) ¼ 0.375. The

values of the longitudinal and transverse correlations for this example are CL¼ 0.99 and

CT¼ 0.965. The high value of the correlation means that, although there is a large

variation of an individual site away from the underlying average lattice, the variation of

Figure 8. Relationship between the random variables �i, j which are used to represent a vertical(along the b-axis) displacement on a hexagonal lattice. RT and RL are interactions governingtransverse and longitudinal correlations, respectively.

nearest-neighbour distances are quite small. This may be calculated using the formula [30],

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� CL

q: ð5Þ

Using the values ¼ 0.375 and CL¼ 0.99 d is found to be 0.053, i.e. only �5% of the cellspacing. d is the standard deviation of the neigbouring bond distance. By using CT inEquation (5) instead of CL the amount of lateral variation is found to be higher at �10%.It should be noted that for values of 4�0.3 the highly distorted lattice produces noBragg peaks – only the diffuse peaks.

3.4.2. Ag array relaxation

MC simulation is also used to carry out the local relaxation of the Ag ion array asdescribed in Section 3.3.3. For simplicity it has been assumed that the Cu and S sitesremain fixed and that the Ag ions move to try to avoid these while at the same timemaintaining chemically realistic distances between themselves. Each of these terms hasbeen modelled using a Lennard Jones potential and the MC energy thus has three terms:

Erelax ¼ CuECu þ SES þ AgEAg ð6Þ

where,

ECu ¼Xr

� �12

�RCu

� �6" #( )

ES ¼Xr

� �12

� �6" #( )

EAg ¼Xr

� �12

�RAg

� �6" #( )

: ð9Þ

In these equations the summation is over all neighbours within a distance r¼ 10.8 A.RCu, RS, RAg are parameters defining the position of the start of the steep repulsive wall ofthe potential for each type of ion and Cu, S and Ag are relative magnitudes for the threeterms. For the simulation shown here, it was assumed that Cu¼ S¼ 2Ag; RCu¼ 2.36 A;RS¼ 1.83 A and RAg¼ 3.88 A. The values for RCu and RS were based on the bond-valencecalculations described in Section 3.3.3 but taking into account that those distancesincluded out-of-plane components. The value of RAg was rather higher than the observedaverage Ag–Ag distance but was used to force the Ag ions to move into the space made bythe initial creation of the 21% of vacancies. The relaxation process was carried out for 20cycles of MC simulation, where a cycle is defined as that number of individual MC stepsrequired to visit each Ag ion once on average.

3.5. Results and discussion

Figure 9 shows the results of the different stages of the modelling process. In Figure 9(a)the initial paracrystal distribution is seen superposed on the underlying CuS2 array.Circles have been drawn on the figure where the Ag ions conflict with the Cu sites.

Figure 9. The three stages of modelling of the distribution of Ag ions. (a) shows the initial paracrystaldistribution superimposed on the CuS2 layer. (c) shows the distribution when conflicting sites havebeen removed to produce 21% vacancies. (e) shows the final distribution after further relaxation usingthe potential given in Equation (6). Corresponding calculated diffraction patterns are given in (b), (d)and (f), respectively. In all three cases the scattering is calculated only from the Ag ions.

These, together with similar ones where the conflict is with the S sites, are the sites wherethe vacancies are assumed to occur. The diffraction pattern (Figure 9(b)), calculated solelyfrom the Ag array, shows the diffraction pattern of the paracrystal distribution unaffectedby the CuS2 lattice.

Figure 9(c) shows the distribution once the 21% vacancies have been made. These areseen to cluster in small islands and these islands have a paracrystal-like distributionthemselves. It is also noticable that many of the Ag ions still fall within the chemicallyforbidden exclusion zones. The diffraction pattern (Figure 9(d)) shows the modifiedparacrystal pattern and now there is evidence of the interference with the CuS2 lattice sinceBragg peaks corresponding to this lattice have appeared even though the calculation is stillsolely from the Ag ions.

Finally, Figure 9(e) shows the distribution after relaxation. It is now seen that the Agions have largely been excluded from the chemically forbidden zones. The diffractionpattern (Figure 9(f )) now shows remarkable similarity to the observed electron diffractionpatterns of Figure 6.

The diffraction patterns shown in Figures 7 and 9 were calculated using the programDISCUS [31].

4. Conclusion

In this article, we have shown how MC computer simulation can be used to interpret andanalyse diffuse scattering. At one extreme, as illustrated by the example in Section 2, themodelling may be sufficiently quantitative to allow definite predictions to be made fromthe results. In this example, the fitting process indicated the presence of anharmonic effectsthat turned out to be precursors of a phase transition that was subsequently discovered.At the other extreme MC simulation is an effective tool to allow the qualitativeexploration of observed disorder phenomena in a diverse range of situations. This has beenillustrated by the second example described in Section 3, where a convincing explanationfor the observed scattering in a fast ion conductor was obtained.

Acknowledgements

The support of the Australian Research Council, the Australian Partnership for AdvancedComputing and the Australian Synchrotron Research Program are gratefully acknowledged. DJGgratefully acknowledges the support of the Australian Institute of Nuclear Science and Engineering.Use of the Advanced Photon Source was supported by the U.S. Department of Energy, Office ofScience, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.

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use of monte carlo simulation for the interpretation and analysis of diffuse scattering

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