Neutron total scattering method: simultaneous ...chemb191.chem.ucl.ac.uk/.../EJM_RMC_2002.pdf · in disordered crystalline materials, there will be large atomic displacement parameters.

Introduction

Powder diffraction measurements have traditionallyfocused on accurate measurements of the positions andintensities of Bragg peaks The former can give a gooddetermination of the lattice parameters and the lattercontain information about the average positions of atomsWith the development of the Rietveld method more thanthree decades ago neutron powder diffraction has maturedinto a powerful tool for structure determination and refine-ment as discussed elsewhere in this issue (Redfern 2002)In the Rietveld method the background scattering ismodelled using polynomials with fitted coefficientsHowever it is frequently found that the backgroundcontains characteristic oscillations and structure as notedfor example in the case of the high-temperature cubicb-phase of cristobalite SiO2 (Schmahl et al 1992) Thisstructured background is the diffuse scattering whichmany single-crystal experiments reveal to have a strongvariation across reciprocal space Diffuse scatteringcontains information about fluctuations of the atomicstructure from the arrangement of the average positions(see discussion in the opening paper of this issue Dove2002) These fluctuations contribute to the short-rangestructural order of the material

The case of b-cristobalite shows clearly the type ofshort-range order or fluctuations from the average struc-ture that can give rise to diffuse scattering The average

position of each oxygen atom lies exactly half-way betweentwo silicon atoms (Schmahl et al 1992) If this is takenliterally it implies that the SindashOndashSi bond is linear which isknown to be a high-energy configuration of these atomsCrystallographic analysis more properly gives the distribu-tion of positions of the atoms Usually the average positionof an atom is taken to be the mid-point of its distribution ofpositions and the overall distribution is often representedby a three-dimensional Gaussian function The width ofthis distribution is known as the atomic displacementparameter In the case of b-cristobalite the spread of distri-butions of the oxygen atoms has significant elongationnormal to the SihellipSi vector as shown in Fig 1a Thisimplies that there is considerable bending of the OndashSindashObond but still with the mid-point of the distribution of posi-tions of the oxygen atoms corresponding to a linear bondIt is reasonable to ask whether the OndashSindashO bond is everlinear and whether instead the midpoint of the SihellipSiseparation is really a minimum in the distribution ofoxygen positions This would correspond to the OndashSindashObond always being bent with the position of the oxygenatom being randomly distributed in an annulus around themidpoint of the SihellipSi vector This can be modelled in astructure refinement by using a set of equivalent positionsfor each oxygen atom with a partial occupancy (Schmahl etal 1992) Figure 1b shows the case of refining six sites foreach oxygen atom and this leads to an improved level ofagreement with the diffraction data In crystal structure

Eur J Mineral2002 14 331-348

Neutron total scattering method simultaneous determination oflong-range and short-range order in disordered materials

MARTIN T DOVE(1) MATTHEW G TUCKER(1) and DAVID A KEEN(2)

(1)Department of Earth Sciences University of Cambridge Downing Street Cambridge CB2 3EQ UK(2)Physics Department Oxford University Clarendon Laboratory Parks Road Oxford OX1 3PU

and ISIS Facility Rutherford Appleton Laboratory Chilton Didcot Oxfordshire OX11 0QX

Abstract Neutron total scattering provides simultaneous information about long-range order through the Bragg peaks and short-range order through the diffuse scattering We review recent progress in applying total scattering methods for the study of short-range structural order in silicates and other materials focussing on changes in short-range order that accompany changes inlong-range order at structural phase transitions

Keywords Neutron total scattering Reverse Monte Carlo pair distribution function cristobalite quartz silica diffuse scatteringsulphur hexafluoride

0935-1221020014-0331 $ 810copy 2002 E Schweizerbartrsquosche Verlagsbuchhandlung D-70176 StuttgartDOI 1011270935-122120020014-0331

Author for correspondence email martinesccamacuk

MT Dove MG Tucker DA Keen

analysis the distance between mean positions is oftentaken to be approximately equal to the bond length Whenthe mean position of the oxygen atom is exactly half waybetween its neighbouring silicon atoms the distancebetween the Si and O mean positions is significantlyshorter than the normal SindashO bond length When the meanposition of the oxygen atom is taken to be one of the sixsites in the annulus about the SihellipSi separation thedistance between the Si and O mean positions is closer tothe values of the SindashO bond length in many other tetrahe-drally-coordinated silicates This point is shown in Fig 2

There are however limits to how far one can developthis analysis The first limitation arises from the veryformalism of Bragg diffraction Structure is properlydefined by the relative positions of atoms On the otherhand the intensities of Bragg peaks contain no informationabout the relative positions of atoms only about the distri-butions of positions of individual atoms Put into formallanguage the relative positions of atoms are describedusing two particle and higher-order correlation functionswhereas Bragg diffraction is only a probe of single-atomdistribution functions The second limitation arises fromthe low resolution of a typical diffraction analysis Thespatial resolution of a structure is given by

Dr =2pmdashQmax

(1)

Qmax is the largest value of the scattering vector Q used inthe measurement For scattering processes with no changein wavelength l

Q = 4p sinql (2)

In X-ray diffraction with CuKa radiation (wavelengthl = 154 Aring) the maximum value of Q is 4pl so that theresolution is l2 = 077 Aring this resolution is improved withMoKa radiation which gives minimum resolution of035 Aring In a typical 6-site refinement of b-cristobalite thedistances between two closest sites in an annulus is ca

05 Aring Clearly the 6-site model is pushing against the limitsof the resolution and at best it can only be said to be butone possible representation of the spread of positions of theoxygen atoms Whilst it is possible to extend the range ofQ in a typical X-ray powder diffraction experiment withCuKa radiation the fall-off of the atomic scattering factorswith Q means that it is hard to identify Bragg peaks abovethe normal background before reaching Qmax This is not aproblem with neutron diffraction but usually for values ofQ typically above 10ndash15 Aringndash1 there is too significant anoverlap of Bragg peaks to be able to distinguish them abovethe background This will give a best possible resolution of04 Aring in a structure refinement It is better than with X-raydiffraction but may still not be good enough Furthermorein disordered crystalline materials there will be largeatomic displacement parameters These also will cause theBragg peaks at higher values of Q to become too weak tobe measurable both in X-ray and neutron diffraction andthis will again limit the structural information that can beobtained from analysis of the Bragg diffraction alone

These limitations can be effectively addressed using themethod of lsquototal scatteringrsquo This involves collecting thecomplete diffraction pattern to high values of Q perhaps to50ndash60 Aringndash1 when possible and performing an analysis ofthe total diffraction pattern rather than restricting the anal-ysis to the Bragg peaks alone As we have noted the totalscattering will contain the diffuse scattering and we willshow below that this contains information about atomicpair distribution functions Thus a total scattering experi-ment on a crystalline material is able to give simultaneousinformation about both the long-range crystallographicorder and the short-range fluctuations It is interesting tocontrast the behaviour over both length scales in disorderedcrystalline materials including some of the phases ofsilica and when heating a material through a phase transi-tion

Total scattering methods have long been the only meansof obtaining structural information on glasses or liquids(eg Wright 1993 1994 1997) They are now increasinglybeing applied to the study of crystalline phases (egBillinge amp Thorpe 1998) including more recently to thestudy of mineralogical materials (Dove amp Keen 1999) Thefirst study of this type was of the a and b phases of cristo-balite (Dove et al 1997) and this work has been extended

332

a) b)Fig 1 The refined crystal structure of b-cristobalite with theatoms drawn as ellipses that represent the Gaussian spread ofdistributions of positions (from Tucker et al 2001a) In a) thedistribution is shown about a single site half way along the SihellipSivector and in b) the distribution is based on six mean sites for eachoxygen atom

Actual bondlength

Apparentbond length

Averageposition

Distributionof positions

Fig 2 The apparent short-ening of the SindashO bondwhen the midpoint istaken to be directlybetween the SihellipSi vectoras compared to an actualposition displaced fromthis midpoint

to cover several other silica phases (Keen amp Dove 19992000 Dove et al 2000 Tucker et al 2000a 2001a and b)In this paper we will describe the basics of methodologyand illustrate the ideas with examples of our work on thecrystalline phases of silica and on the molecular crystalSF6

The formalism of total scattering

We start from the equation for the interference term thatdefines scattering of a beam of radiation from an ensembleof atoms for a given scattering vector Q

S(Q) = 1ndashN S

jkltbjbk exp(iQtimes[rj ndash rk]) (3)

In a powder diffraction experiment we need to averageover all orientations of Q relative to the vector rj ndash rkbecause we have the powder average of random grainorientations Moreover on instruments on time-of-flightneutron sources (such as the GEM diffractometer describedbelow) data are collected from all around the diffractioncone and then summed The average over all orientations ofQ leads to the Debye result (derived for example in theintroductory paper Dove 2002)

S(Q) = 1ndashN S

jkbjbk

sin(Qmdash

Qccedilrj ndash rk ecircmdashccedilrj ndash rk ecirc)

(4)

In fact the individual instantaneous atomic separations arenot useful quantities because the average is over time andhence over very many instantaneous configurationsInstead it is better to express this function using distribu-tion functions for the interatomic separations We thereforeneed to separate the components that arise from differentatoms from those that arise when j = k We therefore write

S(Q) = i(Q) + Sm

cmb2m

ndash(5)

The second term accounts for the terms in equation (4)with j = k and is equal to the total scattering cross-sectionof the material (it is known as the self-scattering term) Thesummation is over all atom types and cm is the proportionof atom type m i(Q) is related to the total radial distribu-tion function G(r) by the pair of Fourier transforms

i(Q) = r0ograveyen

0

4pr2G(r) sin Qrmdash

Qr dr (6)

G(r) = (2p)mdash1mdash

3r0ograveyen

0

4pQ2i(Q) sin Qrmdash

Qr dQ (7)

with average atom number density r0 = NV (in atomsAring3)G(r) may also be defined in terms of the partial radialdistribution functions gij(r)

G(r) = Sm n

cmcnbndash

mbndash

n(gmn(r) ndash 1) (8)

where

gmn(r) = 4pr2mdashnmnmdashrm

(r)mdashdr

(9)

nmn(r) is the number of particles of type n betweendistances r and r + dr from a particle of type m andrm = cmr0 Two other versions of correlation functions arecommonly used First the differential correlation functionD(r) is defined as

D(r) = 4prr0G(r) (10)

Neutron total scattering method 333

0 4 8 1 2 1 6 2 0

r (Aring)

G(r)

D(r)

T(r)

Neu

tron

-wei

ghte

d ov

eral

l pdf

Indi

vidu

al p

artia

l pdf

SindashSi

SindashO

OndashO

Fig 3 Example of the representation of the pair distribution func-tions (pdf) obtained for b-cristobalite at 300degC Top shows theSindashO SindashSi and OndashO partial pdf functions (origin displaced) eachof which tend to values of unity at large r by definition Bottomshows the merging of the three partial distribution functions intothe neutron-weighted overall functions G(r) is the weighted sum ofthe partial pdf functions (equation 8) which tends to the value

Sm n

cmcnbndash

mbndash

n at r = 0 and to a value of zero at large r D(r) obtained

from G(r) via equation (10) oscillates around zero for largedistances and approaches zero linearly from negative values as rtends to zero T(r) obtained from D(r) via equation (13) has zerovalue at low r and oscillates around a line of constant slope atlarger values of r D(r) shows the structure in the pair distributionfunction most clearly at intermediate and large values of r whereasT(r) shows most clearly the pair distribution functions for the near-ests-neighbour bonds

MT Dove MG Tucker DA Keen

Thus we can write equations (6) and (7) as

Qi(Q) = ograveyen

0

D(r) sin Qr dr (11)

D(r) = 2ndashp ograve

yen

0

Qi(Q) sin Qr dQ (12)

Second the total correlation function T(r) is defined as

T(r) = D(r) + 4prr0 Sm

cmbndash

m

2

(13)

These different functions are illustrated schematically inFig 3 Technically the function rG(r) is the transform ofthe experimentally derived quantity Qi(Q) and thus bestreflects the direct analysis of experimental data T(r) isuseful because it has a value of zero below the first fewpeaks and D(r) is useful because at larger distances itoscillates around zero

A number of alternative representations of the nomen-clature of total scattering are in common use Keen (2001)has compared several of these ndash we follow closely therecommendations given in that paper The formalism isalso reviewed by Wright (1993 1994 1997)

Experimental methods

Basic requirements

Neither the experimental procedure necessary forperforming total scattering measurements nor the treat-ment of data prior to detailed analysis are trivial There arethree main experimental constraints The first of these isthat it is necessary to perform measurements to relatively

large values of Q following the earlier discussionconcerning the fact that the resolution is given by theinverse of Qmax For an experiment performed with CuKaX-radiation and collecting data to 2q = 180deg the resolutionis around half of the length of a SindashO bond Ideally itwould be useful to aim for a resolution that is a smallpercentage of a bond length With time-of-flight neutronsit is possible to obtain good data to values of Q of around60 Aringndash1 and even higher if necessary (albeit with the caveatthat the data quality is lower at higher values of Q andtherefore more experimental effort would be required)This will give a resolution of around 01 Aring which is ca 6of the SindashO bond length

The second essential experimental condition arisesfrom the fact that the total scattering needs to be a goodintegration over all possible changes in energy Theneutrons can be scattered elastically with no change inenergy (as in Bragg scattering) or scattering inelasti-cally with either a gain in energy due to the absorptionof one or more phonons or a loss in energy due tocreation of one or more phonons For an accurate totalscattering experiment all these processes must beallowed to occur The last condition is the hardest Itimplies that the energy of the incident neutron beammust be higher than the energy scale of the phonons inthe material

The third experimental requirement is that backgroundscattering (ie scattering from sources other than from thesample) needs to be minimised and that it must be possibleto measure to high values of Q with good statistical accu-racy Variations in S(Q) are much weaker at higher valuesof Q which means that data at these values need to be goodif the ripples are to be used to provide information aboutstructure over short distances

334

Fig 4 Schematic diagram of the GEM diffrac-tometer at ISIS (Williams et al 1998)showing the banks of detectors that cover mostof the scattering angles This instrument hasbeen optimised for both pure diffraction (highresolution and high intensity) and total scat-tering (detectors covering a wide range of Qwith high intensity and high stability) Thehigher angle banks have the higher resolutionin Q and allow measurements to higher valuesof Q whereas the lower-angle banks allowmeasurements to low values of Q

As a result of these three requirements the best data areobtained at spallation time-of-flight neutron sources Thereis a rich flux of high-energy neutrons and with appropriateinstrument design there are no geometric constraints on therange of Q accessible for measurements Certainly it is quitepossible to obtain good data for values of Q in excess of 60Aringndash1 beyond which it is often found that the total scatteringsignal has reached a constant value (ie i(Q) = 0) Theinstrument GEM at ISIS is ideal for total scattering experi-ments giving large values of Qmax low intrinsic back-ground and high resolution for measurement of Braggpeaks in addition to having a high coverage of the range ofscattering angles with banks of detectors This instrument isshown in Fig 4 A reactor source of neutrons has its largestdistribution of neutron energies at lower energies so thatdata can typically be obtained to Qmax ~ 20 Aringndash1Synchrotron sources can also produce high-energy beamsof X-rays to permit measurements to Qmax ~ 30ndash40 Aringndash1 weexpect to see an increasing use of synchrotron X-ray beamsfor total scattering measurements

Basic data reduction

The formalism and interpretation of total scatteringexperiments are both firmly grounded in the possibility ofthe measurements giving absolute values of the intensity ofthe scattered beam This is quite different from some otherneutron scattering techniques For example in the Rietveldmethod the scale factor is treated as an adjustable param-eter a number of adjustable parameters are used to define abackground function (which is often treated as a polyno-mial) and other factors such as absorption and extinctioncoefficients can also be treated using adjustable parametersThe values of all these parameters can be varied in the leastsquares refinements and finally should have little effect onthe quantitative values of the important refined structuralparameters On the other hand there is no scope for the useof adjustable parameters in the analysis of total scatteringThe scale factors background and absorption correctionsneed to be known absolutely This means that all correctionsneed to be measured separately or it must be possible tocalculate them The set of corrections (discussed in Howe etal 1989 Wright 1993 1994 1997) is1 One set of corrections accounts for background scat-

tering from the components of the instrument the equip-ment used to control the sample environment (furnacesor cryostats) and the sample container These threeexperimental components together with the samplealso give an attenuation of the signal which needs to beaccounted for The procedure for performing thesecorrections is outlined in the appendix

2 The data also need to be properly normalised Accountneeds to be taken of multiple scattering (ie processesin which the beam is scattered more than once withinthe sample) and of factors such as the energy spectrumof the incident beam solid angles of the detectors anddetector efficiencies Multiple scattering can be calcu-lated for a sample that does not scatter an appreciablefraction of the incident beam (typically up to around20) Its contribution tends to be constant with Q

3 The energy spectrum of the incident beam is easilymeasured using a special detector called the monitorpositioned just in front of the instrument All measuredspectra must be scaled by the spectrum recorded in themonitor The factors concerned with the detectors canbe taken into account by performing a measurement ofthe incoherent scattering from a sample of vanadiumthat is ideally of the same size as the sample Thecoherent Bragg scattering from vanadium is extremelyweak (which is why it is so useful in this context)However the Bragg peaks can still be observed in themeasurement and are taken account of by fitting thevanadium scattering with a smooth function that liesbelow the Bragg peaks The intensity of the incoherentscattering from vanadium is known theoretically (it isindependent of Q and given by the inelastic crosssection) and therefore the normalisation of themeasurements is straightforward Multiple scatteringcorrections need to be applied also to the scatteringfrom the vanadium used for normalisation

4 The other important correction is known as the Placzekcorrection In an X-ray experiment the changes inenergy of the scattered X-ray beams are tiny comparedto the energy of the incident beam This means that thescattered beam has almost the same wavelength as theincident beam and when scattering at a fixed angle thevalue of Q can be established from the equation (2)However in neutron scattering the change in energygives rise to a significant change in wavelength In atotal scattering experiment there is no measurement ofthe energy of the neutrons when they reach the detec-tors and hence the wavelength is unknown In effectthe integration over energy implicit in a total scatteringexperiment is performed at constant scattering anglerather than constant Q The correction required to bringthe integral back to constant Q is the Placzek or inelas-ticity correction It also needs to account for the factthat the efficiency of a detector typically scales as theinverse of the neutron velocity The Placzek correctioncan be calculated for simple atomic systems (see forexample Bacon 1975 Chieux 1978) and thesecorrections can be adapted for more complex systemsor for different experimental arrangements

Data reduction and analysis of G(r)

Once the data have been corrected as outlined abovethe task is to obtain G(r) or its alternative forms With thedata properly normalised the first stage is to subtract theself-scattering term from S(Q) to give i(Q) (equation 5)and then to generate Qi(Q) for the Fourier transform ofequation (12) The functions i(Q) and Qi(Q) are comparedin Fig 5 showing the range of detail within the dataincluding the oscillations in Qi(Q) at large Q that reflect thestructure of polyhedral units within the material

If data are collected in a single measuring process suchas from one set of detectors a single function Qi(Q) can beconstructed for Fourier transform There are two issues thatneed to be considered The first is that the Fourier trans-

Neutron total scattering method 335

MT Dove MG Tucker DA Keen

form will contain lsquotruncation ripplesrsquo due to the finiterange of Q being used These are reduced if Qmax isincreased particularly if Qi(Q) is close to zero at Qmax It iscommon to multiply Qi(Q) by a modification functionM(Q) that falls smoothly to zero at Qmax such as

M(Q) = sin(pmdashpQ

mdashQmdashQmax

mdashQmax) (14)

(Wright 1994) Whilst this reduces the termination ripplesin the Fourier transform it does mean that the resultantG(r) is convoluted with the Fourier transform of M(Q)This leads to a broadening of the peaks in G(r) which isparticularly significant for low values of Qmax

The second issue in the analysis of i(Q) is that themeasurements will contain the effects of the experimentalresolution If the resolution is not taken into account itseffect will be that the resultant G(r) will be multiplied bythe Fourier transform of the resolution function This willlead to a reduction in the size of G(r) on increasing r Theissue of resolution is not trivial since the resolution func-tion is actually a function of Q

On instruments such as GEM (Fig 4) different sets ofdetectors will measure i(Q) for different ranges of Q andthe data will need to be combined in some way to performthe Fourier transform of equation (12) One approachmight be to paste the measurements of i(Q) for differentranges of Q into one single overall i(Q) function Howevereach set of detectors will be subject to a different resolutionfunction and it is not possible to properly account for thisin the subsequent Fourier transform One solution to theproblem is to construct G(r) using inverse Fourier methodsIn this approach a trial form of G(r) is adjusted until itsFourier transform is in close agreement with the experi-mental measurements of i(Q) It is relatively straightfor-ward to account for resolution in this approach and thereare no termination ripples The inverse Fourier transformcan be compared with any number of sets of data each withdifferent ranges of Q and resolution We use a Monte Carlomethod to adjust the trial form of G(r) pointwise using theMCGR program of Pusztai amp McGreevy (1997) modifiedto account for the resolution from time-of-flight neutron

336

- 1 0

-0 5

00

05

10

15

20

Qi(Q

)

0 5 1 0 1 5 2 0 2 5

Q (Aringndash1)

3 0

00

01

02

03

04

05

06

07

S(Q

)

08

Fig 5 Representation of the S(Q) and Qi(Q) functions obtained forb-cristobalite at 300degC where i(Q) is simply obtained from S(Q)by sutracting the constant value to which S(Q) tends at large Q(Tucker et al 2001a) The Qi(Q) function highlights the oscilla-tions out to large values of Q

0 5 10 15 20 25

0

10

20

30

40

50

r (raquo)

D (

r)

1073K

973K

863K

857K

843K

833K

823K

793K

673K

473K

293K

290K

150K

20K

Fig 6 D(r) functions for quartz across a wide range of tempera-tures showing progressive broadening of the main features astemperature is increased The left-hand near-vertical dashed lineshows the small variation of the SindashO peak with temperature Theright-hand near-vertical solid line shows how the cube root of thecrystal volume varies with temperature highlighting how the mainstructural features at intermediate distances scale with the thermalexpansion of the crystal by comparison with the right-hand dashedline that is exactly vertical

r (Aring)

instruments This modified program is called MCGRtofand is described in detail elsewhere (Tucker et al 2002a)

The analysis of G(r) or its alternative representationscan be considered with two aspects The first is the analysisof the distinct peaks at low r These contain informationabout the true distribution of interatomic distances whichmay be different from the distances between the mean posi-tions determined by crystal structure refinement from theintensities of the Bragg peaks This is illustrated by ourmeasurements on quartz (Tucker et al 2000a 2001b) TheD(r) functions over a wide range of temperatures encom-passing the andashb displacive phase transition are shown inFig 6 The position of the first peak gives the mean instan-taneous SindashO distance which is denoted as daacuteSindashOntilde andshown as a function of temperature in Fig 7 This iscompared with the temperature-dependence of the distancebetween the mean positions as obtained from analysis ofthe Bragg peaks denoted as daacuteSintildendashaacuteOntilde These distances wereobtained by Rietveld refinement of the same data used toobtain G(r) Clearly the temperature dependence of daacuteSindashOntildeis different from that of daacuteSintildendashaacuteOntilde with the former showingonly a weak positive variation with temperature and thelatter having a significant variation that reflects the phasetransition In particular daacuteSintildendashaacuteOntilde decreases on heatingparticularly on heating in the high-temperature b-phaseand it is probable that this can be understood as a result ofincreased rotational vibrations of the SiO4 tetrahedra whichgive the appearance of bond shortening This differencebetween daacuteSindashOntilde and daacuteSintildendashaacuteOntilde is even more acute in b-cristo-balite if the average position of the oxygen atom is taken tobe half way between two silicon atoms (Tucker et al2001a) Even when using the split-site model the distancebetween the mean postions of the silicon and oxygen atomsis still lower than the mean instantaneous SindashO distance

The analysis of the pair distribution functions has beencarried out on a number of silica phases The overalltemperature dependence of the SindashO bonds in all phases isshown in Fig 8 (Tucker et al 2000b) From the analysiswe obtained a value for the coefficient of thermal expan-sion of the SindashO bond of 22 (plusmn 04) acute 10ndash6 Kndash1 Otherattempts to determine the intrinsic temperature dependenceof the bond have been indirect through applying correc-tions to the crystal structure from detailed analysis of thethermal displacements parameters (Downs et al 1992)The coefficient of thermal expansion obtained from G(r) islower than that obtained by indirect analysis from thecrystal structure (see discussion of Tucker et al 2000b)

The second aspect of the analysis of G(r) concerns itsform for distances beyond the first few peaks There will betoo many overlapping peaks to be able to identify specificneighbour distances but this region of G(r) can provideinformation about mid-range order Consider the D(r)functions for quartz shown in Fig 6 The main featuresacross all distances broaden on heating which shows theincrease in thermal disorder Many features vary smoothlythough the andashb phase transition without significantchange In particular the positions of the lower-r peaksvary only slightly with temperature However the positionsof some of the features at larger r have a variation withtemperature that reflects the variation of the volume of the

unit cell which in turn has a variation with temperaturethat is strongly correlated with the displacive phase transi-tion The case of cristobalite is particularly interesting forthe mid-range distances (Dove et al 1997 Tucker et al2001a) For distances greater than 5 Aring the features in G(r)are quite different which can be traced to changes in theoxygenndashoxygen partial distribution functions (Tucker etal 2001a) This shows that the structure of b-cristobalite is

Neutron total scattering method 337

158

159

160

161

162

258

259

260

261

262

263

264

265

T (K)0 200 400 600 800 1000

305

306

307

308

309

310

Si-O

dis

tanc

e (Aring

)O

O

dis

tanc

e (Aring

)Si

Si

dis

tanc

e (Aring

)

(a)

(b)

(c)

Fig 7 Comparison of the temperature dependence of the averageinstantaneous SindashO OndashO and SindashSi shortest interatomic distancesin quartz obtained from the pair distribution functions (opencircles) and compared with the distances between the mean posi-tions obtained from crystal structure refinements (filled squares)and the RMC analysis (filled circles) The crystal structure has twodistinct distances between the Si and O distances which are shownseparately The instantaneous SindashO and OndashO distances vary onlyweakly with temperature whereas the distances between the meanpositions are clearly affected by the andashb displacive phase transi-tion The variations of both the instantaneous SindashSi distance andthe distance between the mean positions of neighbouring Si atomsreflect the phase transition on heating

MT Dove MG Tucker DA Keen

significantly different from that of a-cristobalite overdistances as small as one unit cell length A similar analysishas been carried out for the high-temperature phase oftridymite (Dove et al 2000)

Reverse Monte Carlo modelling

The obvious question posed by the preceding discussionis how the crystal structure of a material such as quartz orcristobalite can accommodate a significant differencebetween the instantaneous SindashO bond length and thedistance between the mean positions of the two atoms Thefact that there have been various proposals in the literaturefor the structures of the high-temperature disordered phasesin the cases of quartz and cristobalite (and also to a lesserextent in tridymite where the issues are similar) suggeststhat this issue is not trivial What is required is a data-basedmodel for the whole structure that goes beyond considera-tion only of nearest-neighbour distances The ReverseMonte Carlo (RMC) method (McGreevy amp Pusztai 1988McGreevy 1995) provides one useful tool in this direction

The basis of the RMC method is straightforward Theatomic coordinates in a configuration are adjusted using aMonte Carlo algorithm to improve agreement betweencalculated functions and experimental data For total scat-tering measurements the important data are G(r) or i(Q)(or their variants) An energy function can be defined withthe following form based on the differences between thecalculated (subscript lsquocalcrsquo) and experimental (subscriptlsquoexprsquo) values of i(Q) and G(r)

c2RMC = S

m

c2m

c2i(Q) = S

kS

j[icalc(Qj)k ndash iexp(Qj)k]2

s 2k(Qj) (15)

c2G(r) = S

j[Gcalc(rj) ndash Gexp(rj)]2

s 2(rj)

c2f = S[f calc ndash f req]2

s 2

The s variables give specific weightings and can be related toexperimental standard deviations or set to favour one type ofdata over another The last term in equation (15) matches anyquantity calculated in the RMC configuration which wedenote as f calc against a pre-determined (or required) valuef req and acts as a set of constraints The most common form ofconstraint is on bond lengths or bond angles (Keen 19971998) These constraints need not be artificial and can bebased on the same experimental data For example if a bondlength constraint is used the value of the bond length can beset to equal the position of the corresponding peak in G(r) andthe spread of bond lengths as controlled by the value of s2 canbe equated to the width of the corresponding peak in G(r)

The starting point is a configuration of atoms based ona model structure which will be the average crystal struc-ture when the analysis is being carried out on crystallinematerials The Monte Carlo process involves a series ofsteps in which an atom is chosen at random and then movedby a random amount This will lead to a change in the valueof c2

RMC which we denote as Dc2RMC If Dc2

RMC is negativethe change is accepted and the process repeated If Dc2

RMCis positive the move is accepted with probabilityexp(ndashDc2

RMC 2) The process is repeated for many steps until c2

RMC oscillates around a stable mean valueThe RMC method was developed by McGreevy amp

Pusztai (1988) initially to use total scattering data for thedevelopment of structural models of fluids and glasses forwhich there is no equivalent of an average crystal structureMore recently the RMC method has been used for the studyof crystalline materials (Mellergaringrd amp McGreevy 19992000 Tucker et al 2001c) In principle the use of the RMCmethod for crystalline materials could be carried out inexactly the same way as for liquids and amorphous mate-rials The main difference between the two types of data isthat there are sharp Bragg peaks in the crystalline case thatare absent in data from liquids and amorphous materialsHowever in the basic methodology of RMC this differenceis not significant Sharp Bragg peaks imply structural orderover effectively infinite distances On the other hand the

338

1605

1610

1615

1620

0 200 400 600 800 1000 1200

quartzcristobalitetridymitezeolite Yzsm5

Si-

O d

ista

nce

(Aringcopy)

Temperature (K)

Fig 8 Overall thermal expansion of the SindashObond for a range of silica polymorphs obtainedfrom the corresponding peak in the pair distri-bution function obtained by neutron total scat-tering (Tucker et al 2000b)

RMC configuration is of finite size of the order of 1000 unitcells and the G(r) function can only be calculated up to thedistances of size L2 where L is the length of the shortestside of the RMC configuration The Fourier transform willtherefore contain truncation ripples In order to make a validcomparison between the i(Q) from an RMC configurationwith experimental data the i(Q) must first be convolutedwith the Fourier transform of a box function of size L2

irsquo(Q) = 1ndashp ograve

yen

ndashyeni(Qrsquo) sin (Lmdash

Qndashmdash2(Qmdash

QrsquondashQrsquo ))dQrsquo (16)

Thus the sharp Bragg peaks of the data are artificiallybroadened before comparing with the calculated i(Q) Thisimplies some degradation of data used in the RMC methodand particularly for studies of crystalline materials itwould be helpful to avoid this situation

Mellergaringrd amp McGreevy (1999 2000) have developed anew version of the RMC program (called RMCPOW) inwhich the calculated i(Q) is calculated for a three-dimen-sional grid of scattering vectors Q and then mapped onto theone-dimensional representation of the powder measurementThe values of Q that can be used are determined by thedimensions of the RMC sample The individual calculationsare broadened before being added so that the resultant i(Q)is a continuous function and not a discreet set of spikes Inprinciple this broadening can be related to the experimentalresolution The main problem with this approach is that itbecomes computationally demanding when measurements ofQ are taken to ideally large values because the size of thegrid scales with the cube power of the maximum value of Q

We have used a different approach in which we combine thedata for i(Q) and G(r) with data for the explicit intensities of theBragg peaks Initially we used the Pawley (1981) method toextract the intensities of Bragg peaks from the diffraction data(Tucker et al 2001c) The Bragg peak intensities are incorpo-rated into the basic RMC method by adding the following term

c2Bragg = S

hk(Icalc(hk ) ndash Iexp(hk ))2

s 2hk (17)

The program is called RMCBragg and has been used forstudies of quartz and cristobalite It has been described inTucker et al (2001c)

More recently we have incorporated a full profile fittingof the diffraction data as in Rietveld refinement using thefull resolution function for the Bragg peaks (Tucker et alsubmitted) Thus we write down the equation for thediffraction pattern as

Iprofile (tj) = B(tj) + SjShk

R(tj ndash thk )IBragg(hk ) (18)

where B(tj) is the background at tj R(tj ndash thk ) is the resolu-tion function associated with an hk reflection andIBragg(hk ) is the integrated intensity of the hk reflectionwhich we can write as

IBragg(hk ) = L(Qhk ) ecircF(hk ) ecirc2 (19)

L(Qhk ) is the Lorentz factor and |F(hk )|2 is the square ofthe structure factor of the hk reflection Multiplicity isaccounted for by explicit calculations for all combinations

of hk The elastic scattering profile is incorporated into theRMC model through the new residual

c2profile = S

kS

j(Icalc

profile(tj)k ndash Iexpprofile(tj)k)2

s 2k(tj) (20)

where the sum over k denotes the inclusion of diffractionpatterns from different banks of detectors (each with adifferent range of Q and different resolution) The back-ground function B(t) in equation (18) arises from thediffuse scattering and it is treated as a fitted functionbecause it is not directly given by the computed G(r) Theprogram for this work is called RMCprofile and will bedescribed in detail elsewhere (Tucker et al submitted)

Inclusion of the Bragg peak intensities has the merit ofensuring that the RMC method is giving both single-atomand pair distribution functions that are consistent with thefull range of data or equivalently giving both the long-rangeand short-range order implied by the data RMC is clearlybased in statistical mechanics and like nature will lead to amaximisation of the entropy consistent with fulfilling theconstraints of the energy function defined by c2 This meansthat the configurations produced by the RMC method willhave the maximum amount of disorder possible whilst beingconsistent with the experimental data What is not known is

Neutron total scattering method 339

Scattering vector Q (Aringndash1)

-1 0

00

10

20

30

40

50

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0

-0 5

00

05

10

15

20

25

30

Qi( Q

)

50 K ordered phase

190 K disordered phase

Fig 9 Qi(Q) functions for the two phases of SF6 measured usingthree different banks of detectors (data points) and compared withthe RMC calculations taking account of instrument resolution(continuous curves)

MT Dove MG Tucker DA Keen

whether there is a wide range of configurations with equaldata consistency but with varying degrees of disorder Thisis in fact one of the criticisms levelled at the RMC methodand is known as the uniqueness problem We believe that thisproblem is rather over-stated ndash we would expect there to bemany configurations that are consistent with the databecause nature in an experiment produces many configura-tions that contribute to the same data set By using as widea set of data as possible together with data-basedconstraints the variation between different configurations offactors such as degree of order can be minimised We haveshown in Tucker et al (2001c) that using different subsetcombinations of the data lead to very similar final configu-rations as measured by the c2 functions

The other main advantage in using Bragg peaks is thatthey give to the simulation some of the three-dimensionalnature of the problem The total scattering data are strictlyone-dimensional ie the measurements are only dependenton Q = |Q| rather than on the truly three-dimensional Q It isalso true that the Bragg peaks are measured in a one-dimen-sional sense However if it is possible to extract reliablevalues for the intensities of the Bragg peaks in the diffrac-tion profile the fact that each Bragg peak can be associatedwith a three-dimensional Q = ha + kb + c means that we

do recover some of the three-dimensional nature of theproblem in the data The outcome is that we would expectthe three-dimensional distribution of atom positions to bereproduced reasonably well in the RMC simulation

Example studies

1 Example of RMC data fitting sulphur hexafluoride

Although not a mineralogical example recent work onthe molecular crystal SF6 highlights several of the aspects ofthe analysis described in this paper There are two crystallinephases Between 90ndash230 K the crystal structure is body-centred cubic with one molecular per lattice point (Dollinget al 1979) The SndashF bonds lie along the aacute100ntilde directionsbut with considerable orientational disorder This disorderhas been studied in detail using molecular dynamics simula-tion techniques (Dove amp Pawley 1983 1984) The origin ofthe orientational disorder seems to arise from the fact thatwhen the molecules are ideally aligned in their average posi-tions the shortest distance between the closest F atoms ofneighbouring molecules with the interatomic vector lyingalong the unit cell edge would be too short As a result the

340

10

15

20

25

30

35

10

15

20

25

6 8 1 0 1 2 1 4 1 6 1 8

Flight time (ms)

Inte

nsity

(ar

bitr

ary

units

)

50 K ordered phase

190 K disordered phase

Fig 10 Bragg diffraction profiles for the two phases of SF6

obtained on one bank of detectors (points) and compared with theRMC calculations taking account of instrument resolution (contin-uous curves)

- 1 0

00

10

20

30

40

0 4 8 1 2 1 6 2 0

r (Aring)

- 0 5

00

05

10

15

20

25

30

D( r

)

190 K disordered phase

50 K ordered phase

Fig 11 Pair distribution functions D(r) for the two phases of SF6

as obtained using the MCGRtof program (points) and comparedwith the RMC calculations

molecules are constantly pushing each other out of the wayand this results in tumbling motions of the molecules Thisproblem is resolved at low temperatures by a phase transi-tion to an ordered structure with monoclinic symmetry(Powell et al 1987 Dove et al 1988)

Figure 9 shows the set of Qi(Q) data from differentbanks of detectors on GEM for both phases of SF6 whichhave been fitted by the MCGRtof program Figure 10shows the normal diffraction pattern I(t) for both phasesfitted by the profile fitting part of RMCprofile The D(r)functions are shown in Fig 11 Examples of layers ofmolecules from the resultant RMC configurations of bothphases are shown in Fig 12

The intermolecular FhellipF pair distribution functions forthe disordered phase highlight how it is possible to extract

information from the RMC simulations The main interestconcerns the shortest FhellipF contacts along along aacute100ntildeGiven the SndashF distance of 1565 Aring obtained from the totalT(r) and the unit cell parameter of 589 Aring the shortestFhellipF distance to be 276 Aring if the molecules were in orderedorientations However it was argued on the basis of mol-ecular dynamics simulations (Dove amp Pawley 1983 1984)that this contact distance would lead to too close an overlapof the electron distribution in the two atoms and that themolecules would reorient in a disordered manner in orderto allow the shortest contact distances to expand Thisprocess should be seen in the FhellipF distribution functionsUnfortunately these are complicated functions because fortwo molecules there will be 36 distances In Fig 13 weshow the g(r) function for all FhellipF contacts including bothinter-molecular and intra-molecular It is not possible todraw any conclusions from this function In Fig 13 we alsoshow the components for two distinct sets of inter-mol-ecular contacts The most important component is that forneighbouring molecules along aacute100ntilde The small peak at the

Neutron total scattering method 341

Fig 12 RMC configurations for the disordered (top) and ordered(below) phases of SF6 These show considerable orientationaldisorder of the molecules in the high-temperature phase andordered orientations in the low-temperature phase

00

10

20

30

40

50

60

70

0 2 4 6 8 1 0 1 2

r (Aring)

total FndashF g(r)

aacute121212ntilde neighbourFndashF g(r)

aacute100ntilde neighbourFndashF g(r)

Fig 13 Top FhellipF distribution function of the disordered phase ofSF6 obtained from the RMC simulation and containing both inter-molecular and intra-molecular distances Bottom Correspondinginter-molecular FhellipF distribution function for aacute121212ntilde andaacute100ntilde neighbouring molecules

MT Dove MG Tucker DA Keen

low-r side of the distribution function is that correspondingto the shortest contact along aacute100ntilde It can be seen that theposition of the peak is increased from the value of 276 Aringcited above to a value of 3 Aring This is consistent with thevalues expected from the molecular dynamics simulationanalysis (Dove amp Pawley 1983 1984) The important pointillustrated by this analysis however is that it is possible toobtain detailed information about specific aspects of the

sample information that may otherwise be hidden inoverall distribution functions

2 Changes in structure arising from phase transitions in quartz and cristobalite

The RMC study of quartz (Tucker et al 2000a 2001b)is interesting as giving an illustration of the changes in bothshort-range and long-range order that can accompany adisplacive phase transition The D(r) data for quartz shownin Fig 6 highlight several aspects of the change in structurethrough the displacive phase transition It can be seen thatthe lower-r peaks have very little temperature dependenceThe variation of the instantaneous SindashO bond length shown

342

160

Fig 15 Top SindashSindashSi angle distribution function of quartzobtained by analysis of the RMC configurations for all tempera-tures indicated in Fig 6 The lower temperatures give the sharperpeaks Note that on heating the two peaks in the range 120ndash150degmerge to give a single peak at 1325deg Bottom left shows thetemperature dependence of the midpoints of the two peaks thatmerge and bottom right shows the temperature dependence of thewidths of the peaks that merge The positions of the peaks clearlyshow a dependence on the andashb phase transition whereas thewidths of the peaks are virtually insensitive to the phase transition(from Tucker et al 2000a)

B

Fig 14 (100) layers from RMC atomic configurations of quartzfor two temperatures in the a-phase and one in the b-phase SiO4

units are represented by tetrahedra The insets show the averagestructures obtained from the same configurations In this projec-tion the small parallelopiped spaces between tetrahedra becomerectangles in the b-phase giving a clear representation of thesymmetry change associated with the phase transition (fromTucker et al 2000a 2001b)

in Fig 7 shows no clear change at the transition tempera-ture unlike the distance between the mean Si and O posi-tions The same is also true for cristobalite (data shown inFig 8) On the other hand the positions of some of thepeaks at higher values of r in the quartz D(r) data of Fig 6do show a temperature dependence that reflects the temper-ature dependence of the crystal volume which is itselfstrongly affected by the phase transition Thus we see thatdata for different length scales are affected by the phasetransition in different ways The other important point tonote is that with increasing temperature the features in D(r)for quartz are broadened on heating showing the effects ofthermal disorder

(110) layers of atomic configurations from the RMCsimulations of quartz at three temperatures are shown inFig 14 one for the a-phase at very low temperature one ata temperature just below the andashb phase transition and onein the b-phase This particular projection was chosen

because the shear of the structure can be seen relativelyeasily (note the inserts showing the average structureparticularly the small channel between SiO4 tetrahedrawith a rectangular projection in b-quartz that is sheared ina-quartz) There are two important points to note First isthat the RMC simulation has produced a very orderedstructure at low temperature This highlights that a carefulRMC simulation does not necessarily give an exaggerateddegree of disorder The second point is that there is aconsiderable degree of disorder of the structure with large-amplitude rotations of the SiO4 tetrahedra at high tempera-tures in both the a and b phases

The change in short-range and long-range order inquartz on heating through the phase transition can be seenin three-atom distribution functions calculated from theRMC configurations Figure 15 shows the distributionfunctions for the SindashSindashSi angles At low temperatures thedistribution function has four peaks On heating towards

Neutron total scattering method 343

575 K (b) 700 K (b)

825 K (b)

475 K (a)

900 K (b)

Idealised a

Fig 16 Polyhedral representationsof the RMC configurations of thehigh and low temperature phases ofcristobalite viewed down acommon direction that correspondsto [111] in b-cristobalite (Tuckeret al 2001a)

MT Dove MG Tucker DA Keen

the andashb phase transition two of the peaks broaden andtheir midpoints become closer In the b-phase the two

peaks merge into a broad single peak The midpoints andwidths of these peaks are shown as functions of tempera-ture in Fig 15 The temperature-dependence of themidpoints clearly reflects the change in long-range orderassociated with the phase transition The fact that theSindashSindashSi angle distribution function clearly reflects thephase transition means that the changes in long-range orderare felt down to the length scale of two neighbouring tetra-hedra (the second-neighbour SihellipSi distance is ca 6 Aring) Itis sometimes argued that the structure of b-quartz consistsof small domains of a-quartz but the analysis of theSindashSindashSi angle distribution function shows that the struc-tural changes associated with the phase transition areclearly felt at the length scale of the unit cell On the otherhand the widths of the peaks in the SindashSindashSi angle distri-bution function shown in Fig 15 increase significantly onheating without any change associated with the phase tran-sition These results shows that there are considerable fluc-tuations of the SindashSindashSi angle as reflected in theconfiguration plots of Fig 14 The changes in the long-range order occur on a significant background of large-amplitudes short-range disorder

344

cos q02

q)

Fig 17 Distribution functions for the OndashSindashO (top) SindashOndashSi(middle) and SindashSindashSi (bottom) angles in the two phase of cristo-balite obtained from the RMC analysis (Tucker et al 2001a) Notethat only the OndashSindashO distribution function has any clear variationwith temperature in the b-phase but that the all the distributionfunctions for the a-phase are markedly different from the corre-sponding functions in the b-phase

Fig 18 Comparison of the diffuse scattering in the (hk0) layerreciprocal lattice of quartz measured by single-crystal neutrondiffraction and calculated from the RMC configurations at atemperature in each of the two phases (Tucker et al 2001b) TheRMC data show the Bragg peaks as single pixels whereas theBragg peaks in the experimental data are broadening by experi-mental resolution and therefore appear as more prominent features

Figure 16 shows layers of the RMC configurations ofcristobalite in both a and b phases (Tucker et al 2001a)We have shown elsewhere (Dove et al 1997 Tucker et al2001a) that there is little variation in the structure of b-cristobalite with temperature The configuration of b-cristobalite shows a considerable degree of disorder of theSiO4 tetrahedra and there is also a relatively high degree ofdisorder in the structure of a-cristobalite (albeit less than inthe b-phase) The changes of structure are also shown bythe SindashSindashSi angle distribution function shown in Fig 17This function shows a large central peak with two sidepeaks in a-cristobalite but only one peak in b-cristobaliteAs for quartz this shows that the structure of the b-phaseis distinct from that of the a-phase over the length scale ofone unit cell (this is also clearly seen in the OndashO pair distri-bution functions Tucker et al 2001a) but that there is aconsiderable degree of short-range disorder For complete-ness we note that similar results have been obtained froman RMC analysis of the high-temperature phase oftridymite (Dove et al 2000)

3 Calculations of three-dimensional diffuse scattering

An interesting challenge is to use the RMC configura-tions produced by the one-dimensional total scatteringand some three-dimensional Bragg peaks to generatethree-dimensional patterns of diffuse scattering These canthen be compared with single-crystal diffuse scatteringexperiments Such comparisons for quartz (Tucker et al2001b) and cristobalite (Tucker et al 2001a) are shownin Fig 18 and 19 respectively The reader should note that

the Bragg scattering in the RMC maps is presented assingle-pixel points whereas the Bragg scattering in theexperimental data is broadened by the instrumental resolu-tion and hence appears more prominent than in the RMCmaps The experimental data for a- and b-quartz are fromneutron scattering measurements on the PRISMA spec-trometer at ISIS (Tucker et al 2001b) The experimentaldata for b-cristobalite are from TEM measurements (Huaet al 1988) The agreement between the experimentaldata and the RMC reconstruction is very good in eachcase That may not be too surprising since the streaks ofdiffuse scattering are associated with rigid unit modes andthese arise from the pattern of constraints imposed by thethree-dimensional network of corner-linked SiO4 tetra-hedra On the other hand the encouraging point about thequality of the comparisons is that the RMC simulationshave reproduced the physics of the dynamics of thesesystems

In fact it is possible to quantify the agreementbetween the calculated diffuse scattering and the experi-mental data The calculated and measured temperaturedependence of the diffuse scattering in quartz (Tucker etal 2001b) are compared in Fig 20 This is a much strictertest because it relies on being able to calculate the inten-sity of the diffuse scattering and its variation through aphase transition rather than simply calculating a patternof diffuse scattering It can be seen that the RMC hasreproduced the experimental data in both phases We aretherefore able to conclude that the RMC method is able toproperly capture the three-dimensional physics of thesystem under study

Neutron total scattering method 345

10

Fig 19 Calculated [001] zone diffuse scatteringfor the two phases of cristobalite (including twotemperatures in the high-temperature disorderedphase) compared with TEM measurements (Huaet al 1988) of the diffuse scattering in the high-temperature phase (Tucker et al 2001a) TheRMC data show the Bragg peaks as single pixelswhereas the Bragg peaks in the experimental dataare broadening by experimental resolution andtherefore appear as more prominent features

MT Dove MG Tucker DA Keen

Summary

The central point of this review has been to highlightthe way in which neutron total scattering measurements bycombining both Bragg diffraction and diffuse scattering ina single experiment can provide information about long-range and short-range order simultaneously This is particu-larly useful for the study of disordered crystallinematerials when the local structure fluctuates strongly fromthe structure averaged over all unit cells Key examples ofthis are the crystalline silica phases The Reverse MonteCarlo method facilitates the development of structuralmodels that are consistent with the data in all aspects andacross all length scales through fitting to the total scat-tering intensity across a wide range of Q to the pair distri-bution function and to the intensities of the Bragg peaksOur work in this area has mostly focussed on disorderedcrystalline materials but the RMC method can also beapplied to glasses (Keen 1997) magnetic materials (Keenamp McGreevy 1991 Keen et al 1995 1996 Karlsson etal 2000 Mellergaringrd amp McGreevy 2000) and crystallinematerials with site disorder (Mellergaringrd amp McGreevy2000) The RMC method has been applied to neutronsingle crystal diffuse scattering data (Nield et al 1995 seealso Proffen amp Welberry 1997ab and Welberry amp Proffen1998 for the application of the RMC method to the anal-

ysis of single-crystal X-ray diffuse scattering data) but therestricted range of Q that can be obtained in such measure-ments as compared to the range of Q accessible in totalscattering measurements means that it is not possible toachieve the same resolution in real space

Appendix correction of total scattering data for background scattering and beam attenuation

A total scattering experiment requires a number ofmeasurements in addition to that involving the sampleThese are a measurement with an empty instrument ameasurement with empty sample environment equipmentand a measurement with an empty sample container withinthe sample environment equipment We label the measuredintensity from each of these as II (empty instrument) IE(sample environment equipment) and IC (samplecontainer) together with IS for the measured intensity fromthe sample We denote the expected intensity afteraccounting for all corrections with a prime as Icent with thesame subscripts The relationships between the measuredintensities and the expected intensities can be can bewritten as the following set of equations

IE = aEEIrsquoE + IrsquoI

IC = aCCEIrsquoC + aE

CEIrsquoE + IrsquoI (21)

IS = aSSCEIrsquoS + aC

SCEIrsquoC + aESCEIrsquoE + IrsquoI

The a coefficients give the corrections for each contribu-tion The component in the superscript of each a coeffi-cient denotes the source of scattering and the subscriptdenotes the source of attenuation The values of all the acoefficients can be calculated from knowledge of thecomponents of the experiment It is assumed that IrsquoI = II Forexample the measured intensity from the empty sampleenvironment is given by the intensity from the emptyinstrument together with scattering from the sample envi-ronment that is also attenuated by the sample environmentThe measured intensity of scattering from the emptycontainer consists of the background from the instrumenta component of scattering from the sample environmentequipment attenuated by both the sample container and thesample environment equipment and a component of scat-tering from the sample container which is also attenuatedby the sample container and the sample environment equip-ment

In practice the measurements from both the emptyinstrument and the instrument containing the empty sampleenvironment II and IE respectively can be combined toyield IrsquoE This is then combined with the measurements ofthe empty sample environment and the empty sample canwithin the sample environment II and IC respectively togive IrsquoC Finally all these separate components of the scat-tering are combined with the measurement from the sampleto recover the true scattering from the sample alone IrsquoSThese corrections are usually applied in a single programsuch as ATLAS (Hannon et al 1990) into which is read

346

T(K)

400 600 800 10000

2

4

6

8

10

12

Dif

fuse

Int

ensi

ty

Fig 20 Temperature dependence of the diffuse intensity fromquartz in the (z00) directions The circles correspond to valuesfrom the diffuse scattering obtained from RMC configurationsaround (45350) whereas the other data points correspond to theexperimental diffuse intensity at (25150) and (45350) (greyand black squares respectively) The line is a guide to the eyethrough the circles and black squares All plots have been scaled togive a maximum of 10 at the transition temperature (Tucker et al2001b)

T(K)

details such as the geometry of the instrument sampleenvironment and sample can

Acknowledgements We are grateful to EPSRC forsupport

References

Bacon GE (1975) Neutron diffraction (third edition) ClarendonPress (Oxford) 636 p

Billinge SJL amp Thorpe MF (ed) (1998) Local structure fromdiffraction Plenum New York 412 p

Chieux P (1978) Liquid structure investigation by neutron scat-tering in ldquoNeutron Diffractionrdquo H Dachs ed Springer-VerlagBerlin 271ndash302

Dolling G Powell BM Sears VF (1979) Neutron diffractionstudy of the plastic phases of polycrystalline SF6 and CBr4Molecular Physics 37 1859ndash1883

Dove MT (2002) An introduction to the use of neutron scatteringmethods in mineral sciences Eur J Mineral 14 000ndash000

Dove MT amp Keen DA (1999) Atomic structure of disorderedmaterials in ldquoMicroscopic properties and processes inmineralsrdquo CRA Catlow K Wright ed pp 371ndash387

Dove MT amp Pawley GS (1983) A molecular dynamics simula-tion study of the plastic crystalline phase of sulphur hexafluo-ride J Phys C Solid State Physics 16 5969ndash5983

mdash mdash (1984) A molecular dynamics simulation study of the orien-tationally disordered phase of sulphur hexafluoride J Phys CSolid State Physics 17 6581ndash6599

Dove MT Keen DA Hannon AC Swainson IP (1997)Direct measurement of the SindashO bond length and orientationaldisorder in b-cristobalite Phys Chem Minerals 24 311ndash317

Dove MT Powell BM Pawley GS Bartell LS (1988)Monoclinic phase of SF6 and the orientational ordering transi-tion Molecular Physics 65 353ndash358

Dove MT Pryde AKA Keen DA (2000) Phase transitions intridymite studied using ldquoRigid Unit Moderdquo theory ReverseMonte Carlo methods and molecular dynamics simulationsMin Mag 64 267ndash283

Downs RT Gibbs GV Bartelmehs KL Boisen M (1992)Variations of bond lengths and volumes of silicate tetrahedrawith temperature Am Mineral 77 751ndash757

Hannon AC Howells WS Soper AK (1990) Institute ofPhysics Conference Series 107 193ndash211

Howe MA McGreevy RL Howells WS (1989) The analysisof liquid structure data from time-of-flight neutron diffracto-metry J Phys Cond Matter 1 3433ndash51

Hua GL Welberry TR Withers RL Thompson JG (1988)An electron-diffraction and lattice-dynamical study of thediffuse scattering in b-cristobalite SiO2 J Appl Cryst 21458ndash465

Karlsson L Wannberg A McGreevy RL Keen DA (2000)Modeling the magnetic structure of Dy7Fe3 metallic glassPhys Rev B 61 487ndash491

Keen DA (1997) Refining disordered structural models usingreverse Monte Carlo methods Application to vitreous silicaPhase Transitions 61 109ndash24

mdash (1998) in ldquoLocal Structure from Diffractionrdquo SJL Billingeand MF Thorpe eds Plenum Press New York 101ndash119

Keen DA (2001) A comparison of various commonly used corre-lation functions for describing total scattering J Appl Cryst34 172ndash177

Keen DA Bewley RI Cywinski R McGreevy RL (1996)Spin configurations in an amorphous random-anisotropymagnet Phys Rev B 54 1036ndash1042

Keen DA amp Dove MT (1999) Comparing the local structures ofamorphous and crystalline polymorphs of silica J PhysCondensed Matter 11 9263ndash9273

mdash mdash (2000) Total scattering studies of silica polymorphs simi-larities in glass and disordered crystalline local structure MinMag 64 447ndash457

Keen DA amp McGreevy RL (1991) Determination of disorderedmagnetic-structures by RMC modeling of neutron-diffractiondata J Phys Condensed Matter 3 7383ndash7394

Keen DA McGreevy RL Bewley RI Cywinski R (1995)Magnetic-structure determination of amorphous materialsusing RMC modeling of neutron-diffraction data Nucl InstrMethods Phys Research A 354 48ndash52

McGreevy RL amp Pusztai L (1988) Reverse Monte Carlo simula-tion A new technique for the determination of disordered struc-tures Molecular Simulations 1 359ndash67

McGreevy RL (1995) RMC ndash progress problems and prospectsNucl Inst Methods Phys Research A 354 1ndash16

Mellergaringrd A amp McGreevy RL (1999) Reverse Monte Carlomodelling of neutron powder diffraction data Acta Cryst A 55783ndash789

mdash mdash (2000) Recent developments of the RMCPOW method forstructural modelling Chem Phys 261 267ndash274

Nield VM Keen DA McGreevy RL (1995) The interpretationof single-crystal diffuse-scattering using Reverse Monte Carlomodeling Acta Cryst A 51 763ndash771

Pawley GS (1981) Unit cell refinement from powder diffractionscans J Appl Cryst 14 357ndash61

Powell BM Dove MT Pawley GS Bartell L (1987)Orientational ordering phase transition and the low temperaturestructure of SF6 Molecular Physics 62 1127ndash1141

Proffen T amp Welberry TR (1997a) An improved methodfor analysing single crystal diffuse scattering using theReverse Monte Carlo technique Zeit Kristall 212764ndash767

mdash mdash (1997b) Analysis of diffuse scattering via the reverse MonteCarlo technique A systematic investigation Acta Cryst A 53202ndash216

Pusztai L amp McGreevy RL (1997) MCGR An inverse methodfor deriving the pair correlation function from the structurefactor Physica B 234ndash6 357ndash358

Redfern SAT (2002) Neutron powder diffraction of minerals athigh pressures and temperatures some recent technical devel-opments and scientific applications Eur J Mineral 14251ndash261

Schmahl WW Swainson IP Dove MT Graeme-Barber A(1992) Landau free energy and order parameter behaviour ofthe andashb phase transition in cristobalite Zeit Kristall 201125ndash145

Tucker MG Dove MT Keen DA (2000a) Simultaneous ana-lyses of changes in long-range and short-range structural orderat the displacive phase transition in quartz J Phys CondensedMatter 12 L723ndashL730

mdash mdash mdash (2000b) Direct measurement of the thermal expansionof the SindashO bond by neutron total scattering J PhysCondensed Matter 12 L425ndashL430

Tucker MG Squires MD Dove MT Keen DA (2001a)Dynamic structural disorder in cristobalite Neutron total scat-tering measurement and Reverse Monte Carlo modelling JPhys Condensed Matter 13 403ndash423

Neutron total scattering method 347

MT Dove MG Tucker DA Keen

Tucker MG Keen DA Dove MT (2001b) A detailed struc-tural characterisation of quartz on heating through the andashbphase transition Min Mag 65 489ndash507

Tucker MG Dove MT Keen DA (2001c) Application of theReverse Monte Carlo method to crystalline materials J ApplCryst 34 780-782

mdash mdash mdash (2002a) MCGRtof Monte Carlo G(r) with resolutioncorrections for time-of-flight neutron diffractometers J ApplCryst (in press)

Welberry TR amp Proffen T (1998) Analysis of diffuse scatteringfrom single crystals via the reverse Monte Carlo technique IComparison with direct Monte Carlo J Appl Cryst 31309ndash317

Williams WG Ibberson RM Day P Enderby JE (1997)GEM General materials diffractometer at ISIS Physica B 241234ndash236

Wright AC (1993) Neutron and X-ray amorphography inldquoExperimental techniques of glass sciencerdquo ed Simmons CJand El-Bayoumi (Ceramic Transactions American CeramicSociety Westerville) pp 205ndash314

mdash (1994) Neutron scattering from vitreous silica V The structureof vitreous silica What have we learned from 60 years ofdiffraction studies J of Non-Crystalline Solids 179 84ndash115

mdash (1997) X-ray and neutron diffraction in ldquoAmorphous Insulatorsand Semiconductorsrdquo ed MF Thorpe and MI MitkovaNATO ASI series 3 High Technology (Kluwer Amsterdam)23 83ndash131

Received 18 July 2001Modified version received 12 September 2001Accepted 6 November 2001

348