Crystal Structure Determination From Powder

7/30/2019 Crystal Structure Determination From Powder

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Reviews

Crystal Structure Determination from PowderDiffraction Data

K e nneth D . M. Ha r r is*

School of Chemi st ry, Un iversi t y of Bir mi ngham, Edgbast on, Birm ingham B15 2TT, U . K.

Maryjane Tremayne

Department of Ch emi str y, Un iversity College L ondon, 20 Gordon St reet,London WC1H 0AJ, U. K.

Receiv ed A pr il 8, 1996. Revi sed M anu scr i pt Recei ved A ugu st 15, 1996X

A wide r a n g e of imp ort a n t cryst a l lin e so lids ca n n o t be p re pa re d in t h e fo rm o f s in g lecrystals of suitable size and quality for structural characterizat ion by conventional single-crysta l X-ra y diffra ction meth ods. The development of techniques for crysta l structuredeterminat ion from powder diffraction dat a is clear ly importa nt for allowing t he structura lchar acteriza tion of such ma teria ls. Although the str ucture refinement stage of the structuredetermination process can now be carried out fairly routinely using the Rietveld profilerefinement technique, structure solution directly from powder diffraction data is associatedw i t h s ev er a l i n t r in s ic d if fi cu lt i es . Th e a r t i cl e s u r ve ys t h e f ie ld of cr y s t a l s t r u ct u r edetermination from powder diffraction da ta . P ar t icular emphasis is given to the challengingstr ucture solution sta ge of the structur e determina tion process, wit h illustra tive ca se studieshighlighting the features of each of the main methods tha t a re currently used for str ucturesolution from powd er diffraction da ta . The current scope and future potentia l of pow derdiffraction as an approach for crystal structure determination are discussed, and contem-porary applications of this approach across several disciplines within materials chemistryare reviewed.

1. Introduction

X-ray diffraction is undoubtedly the most importantand powerful technique for char acterizing th e structura lproperties of crysta l line sol ids. Single-crystal X-ra yd if fr a c t ion , i n p a r t icu la r , i s n ow u s ed w i d el y a n droutinely for crystal structure determinat ion. However,many important crystal l ine solids cannot be preparedas s ingle cryst a l s of suff ici ent s iz e a nd qual i t y forconventional single-crysta l X-ra y diffra ction st udies, andin such cases it is essential that structural informationcan be determined from powder diffraction da ta . In thisreview, we focus on t he a pplicat ion of powd er diffra ctionto determine informa tion on th e str uctural propertiesof crysta lline solids.

Cryst a l structure determina tion from diffraction datacan be divided into t hree sta ges: (1) determina tion oflat t ice par ameters a nd a ssignment of crysta l symmetrya nd space group, (2) str ucture solution, and (3) str ucturerefinement. In structure solution, an ini t ial structura lmodel is derived directly from the experimental diffrac-tion da ta . If this initial structural model is a sufficientlygood representa t ion of the true structure, refinementof this model against the experimental diffraction datacan b e c arr i ed out t o ob t ai n a good-qual i t y c ryst a ls t ruct ure. For s ingl e-cryst a l di ff rac t ion dat a , b ot h

structure solution and structure refinement calculations

can now general ly be carried out in a straightforwardm anner. For pow der di f frac t i on da t a , on t he ot herhand, refinement of crystal structures (usually carriedout using t he Rietveld profile refinement technique1) c a nnow be ca rried out fairly r outinely, whereas solution ofcrysta l structures directly from powd er diffraction datais a significa ntly grea ter challenge. The field of cryst a lstructure solution from powder diffraction data is cur-rently a n a ct ive ar ea of research, both in the develop-m ent of new and m ore pow erful m et hodol ogies andimproved instrumentation and in the applicat ion ofexist ing techniques to tackle problems of increasingcomplexity. As described below, significant progress hasbeen m ade in recent years in a l l aspects of th is f ield.

We now consider in more detai l the di fficult ies as-sociated with solving crystal structures directly frompowder di ffract ion da ta . Essential ly the sa me informa-tion is contained in single-crystal and powder diffractionpat t erns , b ut i n t he form er c ase t hi s i nform at i on i sdistributed in three-dimensional space whereas in thelat ter case the three-dimensional di ffract ion data arecompressed into one dimension. As a consequence,there is generally considerable overlap of peaks in thepowder di ffract ion patt ern, leading t o severe a mbigu-it ies in extracting the intensi t ies I(h k l) of individualdiffra ct ion maxima. Despite this fact , the tra di t iona lapproach for crystal structure solution from powder

* To wh om correspondence should be ad dressed.X Abstract published in Advance ACS Abstracts, October 1, 1996.

2554 Chem. Mater. 1996, 8, 2554-2570

S0897-4756(96)00218-9 CCC : $12.00 1996 America n Ch emical S ociety


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diffra ct ion data is to at t empt t o extra ct the intensi t iesI(h k l) of individual reflections directly from the powderdiffra ct ion pat tern a nd then t o use these I(h k l) dat a inthe types of crystal structure solution technique (e.g.,direct methods and t he Pa tterson method) tha t a re usedfor single-crysta l diffra ction da ta . The major difficultywith t his approach a rises in extracting values of I(h k l)t hat are suff ici ent l y reli ab le t o l ead t o a successfulstru cture solution calculat ion. This problem of exten-

si v e peak ov erl ap i n t he pow der di f f rac t i on pat t ernlimits t he complexity of structures tha t can be solvedsuccessfully by su ch methods. As a consequence, so-phisticated m ethods a re now being developed to improvethe rel iabi l i ty of extracting intensi ty information forsuch overlapping peaks. Virtua l ly a l l the techniquesfor s t ruct ure sol ut ion from pow der di ff rac t ion dat a(including the P at terson method, direct methods, a ndthe method of entropy maximization and l ikelihoodran king, discussed in sections 5.1-5.3, respectively)consider values of I(h k l) ext rac t ed from t he pow derdiffraction pat tern, a nd, despite the difficulties that thisentai ls, these approaches are nevertheless the mostwidely used a t t he present t ime.

An alternative philosophy for crystal structure solu-tion from powder diffraction data is to postulate struc-tural models independently of the powder di ffract iondata, with the suitabi l i ty of these models assessed bydirect comparison of the powder di ffract ion patternscalculated for these models against the experimentalpowder diffra ction patt ern. This comparison is qua nti-fied using the profi le R factor (as used in Rietveldrefinement), wh ich considers t he w hole digitized int en-sity profile (not the integrated intensities of individualdiffraction maxima) and therefore implicitly takes careof t he ov erl ap of peaks. Thi s a pproac h av oids t heproblemat ic step of extra cting va lues of I(h k l) from th epowder di ffract ion pattern; no part i t ioning of the ex-

peri m ent al pow der di ff rac t ion prof il e i nt o a set ofsingle-cryst a l-like inten sities I(h kl) is carr ied out, a ndt he av oi danc e of suc h part i t i oni ng of t he dat a i s animportant strength of this a pproach. This philosophyi s em b odi ed w i t hi n t he Mont e C arl o and s i m ul at edan nea ling methods, which a re discussed in section 5.4.

Although considerable progress has been made inrecent years2-8 (see Figure 1), the solution of crystalstructures from powder diffraction data is still far fromroutine, and there is still considerable potent ial for thecontinued development and improvement of the meth-odologies in this f ield. The a im of this review is tosurvey t he current scope a nd l im i t a t i ons of cryst a lstructure determination from powder di ffract ion d at a,

with pa rticular emphasis on th e structure solution sta geof the str ucture determina tion process. Here we focuson the use of the P a tt erson method, direct meth ods, themethod of entr opy ma ximization a nd likelihood ra nking,and Monte Ca rlo an d simulated an neal ing methods forst ruct ure solut i on, a l t hough w e not e t hat ot her ap-proaches (including gr id sea rches, energy minimizat ioncalculat ions, a nd other computer simula tion techniquesfor structure predict ion) have also been used. Aftergiving a n overview of the methods presently ava i lable

for crystal structure determination from powder di f-fra ction da ta , the applica tion of these methods across awide range of disciplines within materials chemistry isreviewed.

2. Radiation Sources

Although a great deal of work on crystal structuredetermination from powder di ffract ion data has madeuse of conventional labora tory X-ra y di ffra ctometers,there are significant advantages in using synchrotronX-ra y di ffract ion da ta over labora tory da ta . The com-binat ion of high bright ness a nd good vert ica l collimat ionof synchrotron X-ra diat ion ca n be fully exploited in the

const ruct i on of di ff rac t om et ers t hat g iv e da t a w i t hsubstant ial ly improved signal/noise rat io a nd higherresolution. With h igh resolution, the problem of peakoverlap is substantial ly al leviated, al lowing a greateram ount of unam b i guous i nt ensi t y i nform at i on t o b eext rac t ed from t he pow der di ff rac t ion pat t ern andsometimes enabling successful determination of th elattice parameters in cases for which this is not possiblew i t h l ab orat ory X -ray pow der di f frac t i on da t a . Fur-thermore, the tuna bility of synchrotron radia tion sourcesallows the X-ra y wa velength t o be cha nged read ily, a ndthis can be exploi ted t o diminish the effects of X-ra yabsorption or t o give insights into th e positions and siteoccupa ncies of a toms of a specific type (by stud ying t he

differences between powder di ffra ct ion pa tterns re-corded for wavelengths on either side of an absorptionedge for the selected type of atom 5).

Although it is clearly preferable to use synchrotronX-ra y powder di ffract ion d at a for st ructure determina -t i on , t h e u s e o f s y n ch r ot r on d a t a i s g en er a l ly n otessential ; this is a consequence of the continued im-provements in the resolution and sensitivity of labora-tory X-ra y powder di ffra ctometers (al lowing da ta ofsufficiently high quality to be recorded 9) and adva ncesin th e methodology for structure determina tion. Clear lythe opportunity t o determine crysta l structures reliablyfrom X-ra y powder diffra ction da ta collected on labora-tory diffractometers is opening up the field to a much

wider community of users.The vast ma jority of cryst al st ructure determina tions

are carr i ed out using X -ray di ff rac t ion dat a , b ut i ncertain circumstances i t may be advantageous to useneut ron di f frac t i on dat a . One i m port ant di fferenceb et w e en n eu t r on a n d X-r a y d if fr a c t ion i s t h a t t h erelat ive scattering powers of atoms for neutrons andX-ra ys a re significant ly di fferent . In t he case of X-ra ydiffraction, the scat tering power increases monotonica llyw i t h a t om i c num b er and henc e l i ght a t om s, suc h ashydrogen, are w eak X-ra y scatterers. Neutrons, on theother han d, ar e scat tered by a tomic nuclei, an d neutron-scattering power varies irregularly across the periodicta ble (hence a toms w ith similar at omic number, even

Figure 1. Number of previously unknown crystal st ructuressolved since 1977 from X-ray and neutron powder diffractiond a t a .

R evi ew s Ch em . M at er ., V ol . 8, N o. 11, 1996 2555


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isotopes of the same element, may have significantlydifferent neutron-scat tering properties). Another im-porta nt di fference is tha t interference effects causeX-ra y scatt ering by an a tom to diminish with increasingscat tering angle, whereas t he scat tering of neutrons byan at om is essential ly isotropic. As a consequence, aneutron powder di ffra ct ion pat tern conta ins signifi-cantly more intensity informa tion, part icularly a t highdiffract ion angle, than the X-ray powder di ffract ionpat t ern of t he sam e m a t eria l .

A small number of structures have been determineddirectly from neutron powder diffraction data (examplesare given in refs 10-14), although in most cases it iseasier to solve a structure from X-ray powder diffractiondat a rat her tha n neutron powder diffra ction data . WithX-ra y da ta , i t is often sufficient to locate only a smallsubset of the atoms (usual ly the strongest scatterers)during the structur e solution sta ge, with t he remainderof t he a t om s l ocat ed subsequent ly i n t he s t ruct urerefinement stage (by application of difference Fouriertechniques). With neutron dat a, on the other ha nd, i tis usually necessary to locate the majority of the atomsduring the structure solution stage, in order for subse-quent st ructure refinement to be successful. For these

reasons, a good approac h i s t o sol v e as m uc h of t hestructure as possible from X-ray data and to use neutrondat a subsequently t o locate light a toms or to distinguishb et w e en a t o ms t h a t h a v e s im il a r X-r a y s ca t t e r in gpower s. A joint Rietveld refinement ca n then be carr iedou t u s in g t h e X-r a y a n d n eu t r on d if fr a c t ion d a t a ,lead ing to a crystal st ructure determination of improvedqua lity. Illustra tive examples of such joint refinementsare given in refs 15-18 and emphasize th e complemen-ta ry na ture of X-ra y a nd neutron diffraction techniques.

The subsequent discussion in this article is focusedon the use of X-ray powder diffraction data (both fromsynchrotron and conventional labora tory sources) forc ryst a l s t ruc t ure sol ut i on, a l t hough al l t he m et hods

di sc ussed are appl i c ab l e t o b ot h X - ray and neut ronpowder di ffract ion da ta . In section 5, the a pplicat ionof each structure solution technique is illustrated by adeta iled example; all of these exa mples refer t o the useof X -ray pow der di ff rac t ion dat a m easured using aconventional laboratory diffractometer.

3. Phase Problem in Crystal Structure Solution

In t he diffraction pa tt ern from a crysta lline solid, thediffraction maximum with Miller indexes (h k l) is cha r-acterized by the scat tering vector h in reciprocal space,w i t h h ) ha* + kb* + lc* (a*, b* , and c* denote thereciproca l lat tice vectors). The scat tering for reflectionh

is completely defined by the structure factor F(h

)w hi c h has am pl i t ude |F(h)| a n d p h a s e R(h) a n d i srelated to the electron density F(r) within the unit cellby

w here r is th e vector r ) xa + yb + zc in direct space(a, b, a n d c denot e t he di rect l a t t i ce v ect ors) andintegration is over all vectors r in the unit cell . Fromeq 1, it follows that

w here V denotes the volume of the unit cel l , and thesummation is over all vectors h wit h int eger coefficientsh, k, a n d l. I f the values of both |F(h)| a nd R(h) couldbe measured directly from the diffraction pattern, thenF(r) (i .e. , the crystal structure) could be determineddi rect l y f rom eq 2 b y sum m ing ov er t he m easuredreflections h (n ot e t h a t t h is w ou ld on ly b e a n a p -proximation to F(r), a s only a finite set of reflections his a ctually measured experimentally). However, wh ilethe values of |F(h)| can be obta ined experimenta lly (they

ar e rela ted to the measured diffra ction intensities I(h)),the va lues of R(h) cannot be determined directly fromthe experimental di ffract ion pa ttern. This consti tutesthe so-cal led pha se problem in crystal logra phy. Tosolve the crystal structure, it is clearly necessary to havemethods that provide an est imate of the values of thephases R(h) tha t should be combined with the experi-mentally derived values of |F(h)|.

4. Preliminary Stages of StructureDetermination

4.1. Determination of L attice Parameters andSpace Group Assignment. An essentia l prerequisite

for crystal structure determination is that the lat t icepara meters and the space group are known. Determi-nation of the lat t ice parameters from the powder di f-fraction pa ttern requires a ccurat e determination of thepeak positions (i.e., accurate d-spacing da ta ), wh ich cannormally be achieved using a peak-search process,provided al l systematic errors have been el iminated(e.g., by careful measurement of the zero-point error inthe posi t ion of the detector). Although in fa vora blecases the lat t ice parameters can be determined fromfirst principles (generally feasible only for high-sym-m et ry s t ruc t ures) , i t i s usual l y nec essary t o use anautoindexing program such as ITO,19 TREOR,20 orD I C V OL.21. Th es e p rog r a m s a d op t d if fe re nt a p -

proaches,22

and i t i s v al uab l e t o hav e m ore t han oneprogram av ai l ab le s ince experi ence show s t hat t herelative successes of different autoindexing programscan di ffer from one set of powder di ffract ion data toan other. In general, the autoindexing progra ms gener-at e several possible sets of lat t ice para meters tha t a reconsistent, to a greater or lesser degree, with the set ofmeasur ed peak positions; a var iety of figures of merit23,24

can be used to rank the proposed sets of lattice param-eters.

The spa ce group is assigned by identifying t he condi-t ions for systematic absences in the indexed powderdiffraction dat a. If i t is not possible to assign the spacegroup uniquely, i t may be necessary to carry out the

structure solution calculat ion in paral lel for severaldifferent plausible space groups.

4.2. Extraction of Diffraction Intensities fromthePowder Diffraction Pattern. In the conventionalapproac h (adopt ed i n t he P a t t erson m et hod, di rectmethods an d the ma ximum entropy method) for cryst a lstructure solution from powder di ffract ion data, theintensities (I(h)) of individual di ffract ion maxima ar eextra cted from the experimenta l powder di ffra ct ionpat t ern. I n t he i deal c ase, each peak i n t he pow derdiffractogram would be individual ly resolved, and i tw ou ld t h en b e s t r a i g h t for w a r d t o e xt r a c t a c cu r a t evalues of I(h). However, extra ction of intensities fromthe powder diffra ctogram is complica ted by th e overlap

F(h) ) |F(h)| exp(iR(h)) ) F(r) exp[2ihr] d r (1)

F(r) ) (1/V)h

|F(h)| exp[iR(h) - 2ihr] (2)

2556 Ch em . M at er ., V ol . 8, N o. 11, 1996 R evi ew s


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of nonequivalent reflections due to (1) accidenta lequa lity (or n ear-equa lity) of d spacings for nonequiva -lent reflections, especially a t high scat tering a ngle (thiscan be particularly severe for low-symmetry structures),or (2) symmetr y-imposed eq ua lity of d spacings for well-defined groups of nonequivalent reflections [this occursonly for high-symmetry stru ctures; e.g. , for reflections{h k l} a nd {kh l} in a t etragonal system with La ue group4/m, d(h k l) ) d(kh l) but I(h k l) * I(k h l)].

In ea rly work in th is field, peaks t ha t overlap signifi-

cantly were commonly ignored or assigned arbi trary(often equal) contributions to the total intensity of theov erl appi ng set . How ev er, t h i s approac h i s c l earl yunsat isfactory, and more sophist icat ed a pproaches fordetermining reliable relative intensities from overlap-ping peaks ha ve been developed. The most commonlyused of t hese patt ern decomposit ion techniques a rebased on the use of Rietveld profile fitting procedures(see section 6.2) in w hich th e w hole powder diffractionpat tern is d ecomposed in one step. These techniquesadopt a least-squares approach to fit a calculated profileto the experimenta l powder diffraction pa tt ern (withoutt he use of a s t ruct ural m odel) b y refi nem ent of t helat t ice para meters, t he zero-point error, peak-shape

parameters, and parameters defining the background.Integrated peak intensities can then be determined fromthe fi t ted profi le. The seminal work in this f ield w asc arri ed out b y P aw l ey,25 with part icular reference toneutron powder diffra ction da ta . Specific developmentsof the method for use with X-ray powder diffraction data(for which the 2 dependence and asymmetry of thepeak shape require more detai led considera tion) ha veb een addressed b y Toraya .26 Another widely usedprofile-fitting procedure is that of Le Bail, 27 in w hichproblems arising from negative intensities in the Pawleymethod ar e overcome. These pat tern decomposit iontechniques a re incorpora ted in a number of progra ms,including ALLHK L,25 WP P F,26 GSA S,28 F U L L P R O F ,29

LSQP ROF,30 P ROFI L, 31 a nd EXTRA.32

As most a pproaches for st ructure solution from pow-der di ffract ion dat a depend heavily on extra cting r eli-able intensity information, pattern decomposition con-stitutes an important step dictating the overall successof these a pproaches. Inevitably, however, the intensi-t ies determined for overlapping peaks by pattern de-composition contain inherent uncertainties, particularlywhen the positions of the peaks in the overlapping setare separat ed b y l ess t han hal f t he hal f- w i dt h of t hepeaks. Approaches are currently being developed toallow th e relat ive intensit ies of such overla pping peaksto be determined accurat ely and include the a pplicat ion

of relations between the structure factors derived fromdirect methods an d the P at terson function (DORE ES 33),an i terat ive procedure involving the calculat ion of asquar ed Pa tterson map a nd subsequent back-tra nsfor-m at i on gi v i ng a new set o f s t ruc t ure fac t ors for t heoverlapping reflections (FIPS 34) , a m et hod b ased onentropy ma ximization of a Pa tterson function,35,36 a nda Ba yesian fi t t ing procedure.37

5. Crystal Structure Solution from PowderDiffraction Data

5.1. Patterson Method. 5.1.1. M ethod. The P at ter-son function 38

uses only th e observed structur e factor am plitudes |F(h)|(determined from the measured diffraction intensitiesI(h)) a nd does not require informa tion on the pha ses ofr ef le ct i on s . E a c h p ea k i n t h e P a t t e rs on m a p P(r)corresponds to a n int erat omic vector (ri - rj) within t heunit cell , with the height of each peak proportional tothe product of the scattering powers of the atoms i a nd

j. In principle, the posi t ions of a toms in th e unit cellcan b e deduced from t he set of i n t erat om ic v ect orsr ep re se nt e d i n t h e P a t t e r son m a p . I f t h e s t r u ct u r econtains a small number of atoms that scatter signifi-cantly more strongly than the others, the interatomicvectors between these atoms dominate the Pattersonmap, al lowing the posi t ions (rj) of t hese a t om s t o b ed ed u ce d r e a d il y. F or m a n y ca s e s o f t h i s t y pe , t h ePatterson method has been used successfully for struc-ture solution from X-ra y powder di ffra ct ion da ta (ex-amples are given in refs 16, 18, and 39 -44). Un fortu-na tely, if the str ucture does not conta in a sm all numberof dominant scatterers, the Patterson map is denselypacked with pea ks of compar a ble intensity , from which

it may be di fficult or impossible to derive a rel iableinterpretation of the positions of the atoms in the unitcell.

The Patterson function may also be used to determinethe posi t ion of a structural fragment of well-definedgeometry (e.g., a r igid group), even if th is fra gment doesnot contain a domina nt scatt erer. This interpretat ionof t he P at t erson funct i on ar i ses b ecause a s t ruct uralfra gment of well-defined geometry gives rise to a well-defined set of interatomic vectors which will be repre-sented by a cha ra cterist ic set of peaks in the P at tersonm ap; t hus, a know l edge of m ol ecul ar geom et ry i sexploited in this structure solution approach. In ma nycases, the fragment search process is applied in Patter-

son spac e and i s usual l y di v i ded i nt o t w o part s , asfollows.

(1) A rotation search to determine the orientation ofthe fra gment. A vector model is constructed from theknown geometry of the fragment, superimposed on theP at terson map, a nd rota ted into all possible orientat ionsunti l the optimum fit w ith the P at terson map is found.

(2) A translation search to determine the position ofthe oriented fragm ent w ithin th e unit cell. This involvestranslat ion of the correctly oriented fragment into al lpossible posi t ions in th e a symmetric unit . The inter-atomic vectors corresponding to each posi t ion of thefragm ent are t hen c al culat ed and com pared w i t h t hePatterson map derived from the experimental data to

find the position corresponding to the best fit .Similar procedures ha ve also been applied to fra gment

searches operating in reciprocal space (PATMET45) andin a combination of direct space an d reciprocal space(ROTSEARCH 46). Pa t terson sear ch routines have alsobeen incorporat ed int o direct meth ods program s (PAT-S E E 47). These model-based P at terson methods havebeen us ed (PATME T,48,49 ROTSEARCH, 50,51 PATSEE 52)to solve the crystal structures of molecular materialsfrom X-ra y powder di ffract ion dat a.

The number of intensity m easur ements I(h) requiredfor successful structure solution by the P at terson methodis usual ly somewhat smaller than for direct methods(see section 5.2), an d i t may therefore be possible to

P(r) ) (1/V)h

|F(h)|2 exp[-2ihr] (3)



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reduce the number of overlapping peaks considered inthe P at terson calculation (by tr uncating t he experimen-t al dat a a t a m axi m um v al ue of 2 t hat i s low er t ha ntha t required for direct methods). Furthermore, theP at t erson m et hod i s o f t en ab l e t o deri v e reasonab l est ruc t ural i nform at i on from di f f rac t i on dat a t hat areinferior in quality to that required for successful struc-tur e solution by direct methods. These fea tures ma kethe P at terson method a n a t tra ct ive choice for str ucturesolution from powder di ffra ct ion da ta , provided t he

s t r u ct u r e i s k n ow n t o con t a i n a s m a ll n u m ber ofdominant scatterers or a structural fragment of well-defined geometry.

5.1.2. E xampl e. We now i llustrat e t he a pplicat ionof the Patterson method to determine the previouslyunknown structure of anhydrous l i thium perchlorate(LiClO4) from X-ray powder diffraction data (step size2 ) 0.02 ; t otal dat a col lection t ime ) 3 h).95 Thepowder di ffract ion pa ttern wa s indexed using t he pro-gram TREOR, on the basis of the first 20 observablereflections, producing the following unit cell: a ) 8.651, b ) 6.913 , c ) 4.829 , R ) ) ) 90 (withfigures of merit M20 ) 44, F20 ) 39 (0.007 843, 66)). The

system wa s assigned as orthorhombic, with systema ticabsences consistent with space groups P n m a (cen-trosymmetric) an d Pn21a (noncentrosymmet ric). S tr uc-ture solution was initially attempted in the centrosym-metric space group P n m a . Integra ted intensit ies wereextracted from the powder di ffract ion pattern in twora nges (10 < 2 < 40 and 35 < 2 < 75 ) using theLe Ba il profile-fitting procedure. The dat a fr om theset w o regi ons w ere t hen c om b i ned t o gi v e a set o f 84reflections a nd w ere then used to genera te a P at tersonma p, from w hich t he position of the Cl a tom wa s clearlyevident. After Rietveld refinement of the position of th eCl atom, a di fference Fourier synthesis revealed theposit ions of a l l three O a toms in t he a symmetric unit .

Subsequent refinement of these posi t ions an d furtherdifference Fourier synthesis identified the position oft he Li a t om . The f i nal Ri et veld refi nem ent of t hecomplete structure converged to Rw p ) 10.60%, Rp )8.32%, RF ) 11.10%, RB ) 9.21%, a nd 2 ) 1.53 for 28var iables a nd 3250 profile points in th e ra nge 10 < 2< 75 (84 reflections) [note: a genera l discussion ofRietveld refinement is given in section 6.2 and includesdefinitions of these agr eement factors]. The experimen-tal and calculated X-ray powder di ffract ion patterns,and the corresponding difference profile, are shown inFigure 2. The posi t ion of the Cl a tom found from t heP a tt erson ma p wa s close (0.54 ) to the position of t hisat om in t he final refined str ucture.

The final r efined fra ctiona l coordinat es an d isotropicat omic displacement para meters for LiClO4 are givenin Table 1. As shown in Figure 3, there is a distortedoctahedral arrangement of O atoms surrounding eachL i a t o m . Al l b on d l en g t h s a n d b on d a n g le s i n t h erefined structure are within acceptable limits consistentw i t h t he preci s ion of t he da t a [C l-O b ond l engt hs1.43(1)-1.46(1) ; O-Cl-O bond a ngles 108(1)-111(1) ;Li -O distances 1.98(1)-2.40(1) ]. It is relevan t to notet h a t b ot h t h e a c cu r a cy a n d t h e p re ci si on of t h es est ruct ural param et ers w ould b e i m prov ed w i t h t hecombined use of X-ray and neutron powder diffractiondat a in a joint refinement.

5.2. Direct Methods. 5.2.1. M et hod. The termdirect met hods describes a class of sta tistical methodst hat a t t em pt t o deri ve know ledge of t he phases R(h)directly from the measured diffraction intensities I(h).The direct methods approach is ba sed on the fa ct tha tthe observed structure factor amplitudes |F(h)| (deter-mined from I(h)), together w ith t he correct (but initia llyunknown) va lues of R(h), must correspond (via eq 2) toan electron densi ty tha t is posi t ive everywh ere withinthe unit cell . This imposes constr a ints on R(h), an d by

a pplying probability r elationships, the probable pha ses(Rp(h)) of well-defined groups of reflections can bededuced. The positions of atoms in the initial str uctura lm od el a r e t h en u s ua l ly ob t a i ne d f r om a n E -m a p ,constructed using the trial phases Rp(h) as follows:

w here |E (h)| are normalized structure factors (calcu-lated from |F(h)|). All aspects of the direct methodsprocedure for crystal structure solution are discussedin refs 53 and 54.

The direct methods a pproach ha s become the mostw i del y used t echni que for s t ruct ure sol ut ion from

Figure 2. E xp e r ime nt al (+), c a lcu lat e d (solid l ine), anddifference (below) powder diffraction profiles for the Rietveldrefinement of LiClO4. Reflection posit ions ar e ma rked.

Table 1. Final Atomic Coordinates and Isotropic AtomicDisplacement Parameters Obtained from the Rietveld

Refinement Calculation for LiClO4a

a t om s it e t y pe x/a y/b z/c Uiso /2

C l 4c 0.318(4) 1/4 0.543(8) 0.025(2)O1 4c 0.149(1) 1/4 0.546(2) 0.020(3)

O 2 8d 0.370(1) 0. 079(1) 0.681(1) 0.021(2)O3 4c 0.374(1) 1/4 0.261(2) 0.031(3)Li 4b 1/2 0 0 0.069(9)

a P n m a ; a ) 8.6522(4) , b ) 6.9152(3) , c ) 4.8299(2) .

Figure 3. Final refined crystal structure of LiClO 4 showingthe octahedra l coordination of ClO4- t e t r ahe d r a a r ound e ac hL i a t om.

(r) ) (1/V)h

|E (h)| exp[iRp(h) - 2ihr] (4)

2558 Ch em . M at er ., V ol . 8, N o. 11, 1996 R evi ew s


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powder di ffra ct ion dat a. Un like the Pa tterson method,t he di rect m et hods approac h does not rely on t hepresence of a domina nt s catt erer or prior knowledge ofthe geometry of a w ell-defined structura l fragm ent, a nd,in principle, it can be applied to a much great er var ietyof str uctura l problems. Although wea k reflections ca noften be omitted from the da ta used in structur e solutionby Patterson methods, this is not advisable for directmethods. These reflections play an important role inthe direct methods calcula tion, and their a bsence would

l ead t o errors i n t he norm al i zat i on a nd phasing pro-cesses and in the calculation of some figures of meritused t o di scri m inat e b et w een t he correct s t ruct uresolution a nd incorrect stru cture solutions. A significantnumber of such wea k reflections in a powder diffracto-gram ar e present as overlapping peaks, a nd the fact th atthe intensi t ies assigned to these reflections are oftenunreliable may lead t o problems in t he pha sing process.

Unti l recently, the applicat ion of direct methods topowder diffraction dat a w a s ca rried out using progra msdeveloped for single-crysta l diffra ction da ta (for exampleMULTAN,55 S H E L X S ,56 MITHRIL,57,58 S I R ,59,60 SI M-P E L61). How ever, direct methods procedures optimizedfor powder di ffract ion data have now been developed

(SIRPOW,60,62 S I M P E L63). For example, in th e ca se ofSI RP OW, knowledge of the number of reflections in ea chgroup of overla pping reflections an d th e tota l intensityof each group a re requi red. I ni t ia l l y , t he i ndi vi dualreflections within each group are assigned equal inten-si t ies, and these intensi t ies are then modified duringthe calculation as more phase information is obtained.The use of rel iable intensi t ies (importantly for weakreflections) strengthens the process of phase determi-nat ion and improves t he a bi li ty of the figures of meri tto discriminate the correct structure solution.

Exam pl es of t he successful appli cat i on of di rectmethods in structure solution from powder diffraction

data are given in refs 44 and 64-71.5.2.2. E xampl e. As a n exam ple of th e applica tion of

direct methods to crysta l structur e solution from powd erdiffra ct ion dat a, w e highlight the st ructure determina-tion of p-met hoxybenzoic acid (CH 3OC 6H 4C O2H). Thisi s an equal -at om syst em (see sect i on 8.2.2), t hest ruc t ure of w hi c h w as a l ready know n from si ngl e-crystal X-ray diffraction 72 (monoclinic, P21/a; a ) 16.968, b ) 10.962 , c ) 3.968 , ) 98.13). For thestructure solution from X-ray powder diffraction data,individual intensi t ies were extra cted from the di ffra c-togram in two ranges (7.5 < 2 < 35 and 35 < 2 70 , so a subsequent directm et hods c al c ul at i on w as c arr i ed out usi ng onl y t hei nt e ns it y d a t a i n t h e r a n g e 7 .5 < 2 < 70 (306reflections; 162 n onoverlapping an d 144 overlapping).Although no molecular fragment could be recognizedreadily from the E-map, six peaks slightly higher than

t he rest w ere used as an i ni t i a l m odel for s t ruc t urerefinement. The remaining non-hydr ogen at oms werel ocat ed using di fference Fouri er ana l ysi s , a nd t hecomplete structure was refined, with the application ofgeometric restraints, by the Rietveld refinement tech-nique. No at tempt wa s made to determine the positionsof the H atoms.

In Figure 4, the a tomic positions established from thedirect methods calculation are compared with the cor-responding positions in the final refined crystal struc-ture (the distances between corresponding atoms werein the ra nge 0.44-0.89 ). Although t he model takenfrom the structure solution stage comprised only six

atoms, i t was clear in retrospect (in comparison withthe final structure) that the direct methods calculationhad actual ly located a further O atom.

5.3. Method of Entropy Maximization and Like-lihood Ranking. 5.3.1. M ethod. The maximum en-tropy technique is a powerful method of ima ge recon-struction and has been applied successful ly in a widerange of scientific fields.73-75 In crystal lography, themaximum entropy criterion has been used to examinethe crystallographic inversion problem,76 t o generat ehigh-quality electron density maps,77,78 a s a m ea n s o fpartitioning the intensities of completely overlappingreflections in powder diffraction pat terns, 35,36 a n d a s a napproach for direct phase determina tion.79-82 Recently,

a t echni que for cryst a l s t ruct ure sol ut ion, i n w hi chphases are determined on the basis of entropy maximi-zat ion a nd likelihood ran king,83 has been developed anda pplied successfully t o single-crysta l X-ra y diffractiond a t a ,84 protein di ffract ion d at a, 85 and electron micros-copy data .86,87 As now described, this method has alsob een used i n cryst a l s t ruct ure sol ut ion from X -raypowder di ffract ion da ta .15,88-91

The ma ximum entr opy a nd likelihood meth od ad optsa similar approach to conventional direct methods forphase det erm i nat i on. Bot h m et hods c onsider a n un-known crystal structure to be composed of atoms ofknown identity but unknown positions. Initia lly, theseposit i ons are considered as random w i t h a uniform

di st r i but i on i n t he asym m et ric uni t . The s t ruct uresolution process consists of the gra dua l removal of thisra ndomness. In i ts a pplicat ion to powder di ffract iondat a , t he m axi mum ent ropy and l ikel ihood m et hodhandl es groups of ov erl appi ng peaks i n a ra t i onalma nner, enabling intensity informat ion for these peaksto be used productively in th e str ucture solution process.The procedure for structure solution from powder dif-fraction data using the maximum entropy and likelihoodmethod (implemented in the program MICE 84) is nowsummarized.

First, the peaks in the powder diffraction pattern aredi vi ded i nt o t w o set s, ac cordi ng t o w het her t hey arenonoverlapping or overlapping. For each overla pping

Figure 4. Atomic positions (filled circles) determined fromthe direct methods st ructure solution ca lculat ion for p-meth-oxybenzoic acid compared with the molecular structure (opencircles) in the fina l refined crystal structure.



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group of reflections, the intensi t ies of the individualreflections are summed to give a combined intensity forthe group. B oth sets of reflections are then norma lizedto give unita ry st ructure factors.

The ori gi n i s def ined b y f ix ing t he phases of anappropriat e number of reflections, chosen from t he setof n onoverlapping reflections a nd sat isfying t he usua lrules and cri teria.54 These reflections constitute theinit ial ba sis set . As described below, t he basis set issubsequently extended in the course of the structure

solution process by incorporat ing new r eflections. Thus,a phasing tree is constructed, with each addit ion ofnew reflections defining a successive level of this t ree.

The known phases of the reflections in the basis setare used a s the constra ints in a n entropy maximizat ionprocedure tha t gives rise to pha se extra polat ion. Specif-ically, a maximum entropy map is constructed, and theFourier tr ansform of this map produces new intensityand phase information for reflections not in the basisset (while also reproducing the amplitudes and phasesof the ba sis set reflections used to generat e th e ma p).However, wh en the ba sis set comprises only th e origin-defining reflections, the extrapolated intensi t ies andphases are not reli ab le . Thi s s i t uat i on i s i m prov edsuccessively at each level of the phasing tree by incor-porat ing new pha se informa tion into the ba sis set withthe a ddition of selected strong reflections. The phasesof these reflections are ta ken a s 0 or for centrosym-metric structures and +/4, -/4, +3/4 or -3/4 fornoncentrosymmetric structures (thus gua ra nteeing tha teach trial pha se must be within /4 of th e correct pha se).In a dding new reflections to the basis set , al l permuta-tions of the pha ses of these new r eflections a re consid-ered. The a ddit ion of new reflections w ith permutedphases represents the next level of the phasing tree,with each possible permutation of phases representinga di fferent node within th is level. Ea ch node is sub-jected to constrained entropy maximization, thus updat-ing the corresponding maximum entropy map.

At ea ch level of the pha sing tr ee, the most promisingnodes ar e identified using t he likelihood function, wh ichevaluates the agreement between the structure factorampli tudes extrapolated from the maximum entropyma p for the node and t hose from the experimenta l dat a.Thus, the l ikel ihood function indicates the extent towhich the experimental ly determined (unphased) in-tensities a re rendered more likely by t he pha se choicesm a d e f or t h e b a s is s et r ef le ct i on s f or t h e n od e i nquestion, tha n under t he initial assumption of a uniformdistribution. The use of overlapping reflections in thecalculation of the likelihood greatly increases the ability

t o di scri m inat e t he opt im um node. The nodes areranked in order of log-likelihood gain (LLG)88 a n d t h eLLG v al ues are t hen anal yz ed for phase i ndi c at i onsusing the Student t test .90 Applica tion of the S tudent ttest removes any inherent subjectivi ty that would beintroduced in selecting nodes on the basis of likelihooda lone. The most promising nodes a re then reta ined, andt he next l ev el o f t he phasi ng t ree i s c onst ruc t ed b yadding new reflections with permuted phases.

The pha sing procedure a nd d evelopment of successivel ev el s of t he phasi ng t ree are c ont i nued unt i l m oststructure factors ha ve significan t phase indicat ions orunti l the centroid map92 reveals a recognizable struc-tura l model . The maximum entropy distribution a s-

sociated w ith a node is not a tra ditiona l electron densityma p. To genera te a ma p from which positions of at omscan be determined, the maximum entropy map is usedto calculate a centroid map in which the weights involvenormalized structure factors but the coefficients areunitary st ructure factors. B oth basis set and non-basisset reflections a re used to calculat e th e centroid ma p,and i t is importa nt to emphasize that the inclusion ofoverlapping reflections in this calculation generates ac ent roi d m ap t hat i s s i gni f i c ant l y c l earer t han t hat

generated with the overlapping reflections excluded.Recent developments in the maximum entropy and

likelihood method include the use of a more efficientscheme for sa mpling permuta tions of tria l phases basedon error-correcting codes,93 the use of envelopes inforbidden zones,94 an d th e use of a fragment recyclingprocedure92 in which a known fra gment position is usedactively in the structure solution calculat ion 95 (as il-lustra ted in s ection 5.3.3).

5.3.2. Examp le 1. The combined maximum entropyand l ikelihood ra nking technique has been applied todet erm i ne a prev iously unknow n m ol ecul ar cryst a lstructure, p-toluenesulfonhydra zide (CH 3C 6H 4S O2NH -

NH 2), from X-ra y powder diffra ction da ta .

89

The X-raypowder di ffractogram was indexed using the programTREOR on the basis of the first 26 observable reflec-tions, producing the following unit cell : a ) 18.568 ,b ) 5.630 , c) 8.524 , R ) ) 90, ) 106.2 (withfigures of merit M26 ) 26, F26 ) 56 (0.009 830, 46)). Thesyst em w as ass i gned as m onoc l i ni c , and syst em at i cabsences identified the space group unambiguously asP21/n. Integra ted intensit ies were extra cted from thepowder di ffra ct ion patt ern in tw o ra nges (7.5 < 2

Crystal Structure Determination From Powder

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