1. IntroductionNumerous compounds are known, whose dif-

fraction patterns require four (or more) integerMiller indexes to be indexed completely. Suchcompounds are said to have modulated struc-tures, which can be interpreted as perturbationswith respect to an average conventional struc-ture. Reflections, which cannot be indexed withthe usual three-dimensional unit cell, are calledsatellites. Fourth vector, namely the modulationvector, is introduced as a combination of recip-rocal vectors (see Formalism, Eq. (1)), thus cre-ating special cell set of four vectors, called vec-tor modulo of order 4 in three dimensionalspace. The first indications that crystal struc-tures may not exhibit normal lattice periodicitywere found late in 19th century. Physically itmeans, that neighboring unit cells are not alike,but the deviation of each unit cell from an aver-age structure can be described by a periodicfunction with its own period. The presence ofsuch periodicity is proved by the existence of adiffraction pattern itself. The formalism of thedescription was suggested by P. M. de Wolff in1974 [1]. Reflections that are predominantlystrong are consistent with a regular 3-dimen-sional reciprocal lattice and are called main re-flections, but others, called satellites do not fitthe same lattice. On the other hand, while thesatellites do not fully match the main lattice, the whole pattern can be described by vectormodulo of order 4 in three dimensional space.Hence, it can be imagined that the 3-dimen-sional diffraction pattern is a projection of a3�d-dimensional reciprocal lattice, in which the

main (R) and the satellite (S) reflections are allregularly situated at the lattice nodes (Fig. 1a).From the properties of the Fourier transform theincommensurate modulated structures aretreated as 3�d dimensional and real structurescan be depicted as three-dimensional sectionsR3 of a multispace (Fig. 1b, c). Such 3�d unitcells retain the atomic periodicity with atomsrepresented as strings, while the three-dimen-sional structure is not periodic. On the otherhand, diffraction methods reflect the averageatomic distribution over a structure. Thus, theaverage three-dimensional structure can betreated as the summation of an electron densityover additional dimensions (x4 in the case of3�1 dimensional structure) and its projectionon R3. Obviously, the refinement of this type ofstructure will result in unreal atomic and/orstructural parameters, like atomic displacementparameter(s) (ADP), interatomic distances or co-ordination polyhedra. If the behaviour of atomsalong additional dimensions is described by aperiodic function, it means, that exact atomicparameters for each unit cell can be calculated.If the ratio between the components of themodulation vector and cell parameter(s) is ratio-nal, a structure is called commensurate and canbe described in terms of a superstructure, oth-erwise—this structure is referred to as incom-mensurate.

It should be stressed, that a structure remainsthree-dimensional and additional dimensionsreflect any structural peculiarity, which the radi-ation source used is sensitive to. For neutrons itcan be orientation of magnetic moments. For X-

THE RIGAKU JOURNALVOL. 19 / NO. 2 & VOL. 20 / NO. 1 / 2003

Vol. 19 No. 2 & Vol. 20 No. 1 2003 23

POWDER DIFFRACTION OF MODULATED AND COMPOSITESTRUCTURES

A. V. MIRONOV, A. M. ABAKUMOV AND E. V. ANTIPOV

Department of Chemistry, Moscow State University, Moscow 119899, Russia.e-mail: mironov@icr.chem.msu.ru, abakumov@icr.chem.msu.ru, antipov@icr.chem.msu.ru

In the past two decades intensive development of the theory of 3�d dimensional crystal-lography, experimental techniques and corresponding software makes it possible to performphase analysis and even structure refinement from powder diffraction data for modulated and composite structures. In the present paper the X-ray powder diffraction from such struc-tures is considered. Two examples of multispace structure refinement of the modulated(Bi2.3Sr1.7CuO6.23) and composite (Sr1.32Mn0.83Cu0.17O3) structures from conventional X-ray sourcedata are presented.

rays only positional, occupancy and ADP modu-lation of atoms are relevant.

Over the next years after P. M. de Wolff publi-cation more stones were put in the foundationof multidimensional crystallography. Practicalapplication of this method came to life. Twoprograms most commonly used for this pur-pose were developed: JANA* by V. Petricek andP. Coppens and REMOS† by A.Yamamoto, fol-lowed by other software‡. Late in 1980 s, firststructures were refined from single crystal datawith modulation parameters for heavy atomsonly. Later light elements were included in thislist. Nevertheless, many compounds are noteasily grown as single crystals or the interestingphases present complex polydomain structures.Therefore, the study of a structure by singlecrystal techniques becomes very complicated, if

possible. Rietveld refinement from neutron and/or X-ray data is the important step in the devel-opment of the multidimensional approach. X-ray powder diffraction from conventional sourcesis widely used in scientific studies. The correctinterpretation of diffraction patterns is impor-tant for phase analysis and successive structurerefinement. If unindexed peaks remain on apowder XRD pattern of a newly prepared com-pound, and other preparation routes producematerials having these peaks with similar rela-tive intensities, the following problem shouldbe considered: do extra lines correspond to anadmixture, either the superstructure was notcorrectly taken into account or could they reflectthe modulated or composite structure of a pure phase? If these extra lines correspond to satellite reflections of a modulated structure,then their intensities are relatively low. This isnot the case for composite structures. Their dif-fraction patterns include at least two subsets ofmain reflections with intensities of one order ofmagnitude and thus, two sets of cell parameterscan be refined. Does such a pattern belong to atwo phase sample or the sample contains a sin-gle phase with the composite structure? In thisreview we would like to focus on such prob-lems, using our own experience.

2. Short Notes on FormalismThe overwhelming majority of refined struc-

tures are described in four-dimensional space,so we will take it as an example. The theory of3�d dimensional crystallography is describedin details so far in many papers including thelatest edition of International Tables for Crystal-lography [2]. We present here a brief descrip-tion of the four-dimensional case.

The fourth basic vector a4, which is perpen-dicular to the a1, a2, a3 basic vectors of the R3 di-rect space, is defined through a projection ofcorresponding reciprocal vector a4* (calledmodulation vector q) in R3* reciprocal space(Fig. 1a):

(1)

where at least one of the coefficients a , b and gis irrational in general and possibly one or twoare rational coefficients. Then, reciprocal cellvectors in R4* space are defined as: ai�*�ai* for i�1–3 and a4*�q�e4, and from orthogonal-ity condition:

ai��ai�kie4, ki�a , b , or g , i�1–3 a4�e4 (2)

q a a b c� � � ��

ki ii

∗ ∗ ∗ ∗∑ α β γ1

3

24 The Rigaku Journal

Fig. 1. Multispace presentation by P. M. de Wolff[1]: (a) Diffraction pattern, reciprocal spaceR3* – three-dimensional section of 3�1 reciprocalspace, a3* and a4* – reciprocal cell vectors, R andS – main and satellite reflections respectively; (b) Astructure with displacive and occupational modula-tions; (c) A structure with no modulations.

* The present development of the JANA2000 package iscontinued by V. Petricek and M. Dusek. Find the package athttp://www-xray.fzu.cz/jana/jana.html† Find the package at http://quasi.nims.go.jp/yamamoto/index.html‡ For more information on the software go to http://www-xray.fzu.cz/sgip/aphome.html or http://www.ccp14.ac.uk/so-lution/incomm.htm

The modulation vector is presented by q�qi�qr, where qi and qr are irrational and rationalparts of a modulation vector. The positions ofreflections are given by a diffraction vector:

H�ha*�kb*�lb*�mq (3)

where hklm are four Miller indices of a reflec-tion. Main reflections (R) correspond to m=0,satellite (S) are others (Fig. 1a).

Functions that describe modulation waves areusually expanded in Fourier series:

(4)

where f v is the actual x, y or z coordinate, occu-pancy or corresponding ADP, A0

v – correspond-ing parameter of the average structure, Av

s,n andAv

c,n – modulation parameters. For some specialcases the other functions were used, namelylinear or saw-tooth function for displacive mod-ulation [3]:

(5)

and the step function for occupancy modulation[4]:

p(x4)�1

p(x4)�0 (6)

where Ux is the amplitude of modulation, D isthe length of the interval, where the function isdefined, and x4

0 is the parameter of the functionrepresenting the center of the step.

Interatomic distances and coordination poly-hedra are the most important information forchemists. Two aspects should be taken into ac-count. First, neighboring atoms in the averagestructure may have different x4 coordinates.Thus, the comparison of any atoms includinginteratomic distance calculation should be per-formed for the same R3 section defined by thephase shift parameter t :

t�x4�(q · r) (7)

where q is a modulation vector and r is a vectordefining the position of an atom in the averagestructure. The modulated structure itself shouldbe considered as a function of any structural pa-rameter (positions, ADPs, distances) vs. t. Sec-

ond, two neighbouring unit cells on the modu-lation curves do not correspond to nearestpoints but differ by some value Dt�t2�t1 (t1 andt2 at Fig. 1b). If we shift the structure along anycell vector by its length, Dt becomes equal to anappropriate component of the modulation vec-tor.

In the case of a composite structure two ormore subsystems can be identified, each havingits own set of cell parameters and space groupsymmetry. Let us consider the case with twosubsystems, when both of them have commona and b parameters and different c1 and c2. Eachsubsystem can be defined from common latticevectors and modulation vector by interlatticematrices:

(8)

where a*, b*, c*, q – are reciprocal cell vectorsand modulation vector of the composite struc-ture, W v and av*, bv*, cv*, qv – interlattice matrix(W 1 is usually unit and a*, b*, c*, q are recipro-cal vectors of the first subcell), reciprocal cellvectors and modulation vector of the v subsys-tem. If W 2 is equal

(9)

then c2*�q1. Indices of a reflection in respect toa certain subsystem are defined by:

(10)

For the example given above the main reflec-tions of the first subsystem have hkl0 indices,while hk0m reflections are the main reflectionsfor the second one. Reflections with hklm in-dices are satellites for both subsystems. Thus,cell parameters of each subsystem can be re-fined using conventional approach, if necessary,but satellite reflections still can not be indexedin any subsystem using three-dimensional ap-proach.

Symmetry of the 3�1 dimensional modulated

h

k

l

m

W

h

k

l

m

v

v

v

v

v

⋅

� �[( ) ]1 T

W 2

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

�

abcq

abcq

v

v

v

v

vW

∗∗∗

⋅

∗∗∗

�

x x x4 40

40

2 2∉ � �

∆ ∆,

x x x4 40

40

2 2∈ � �

∆ ∆,

x xx x

x� ��

04 4

0

2∗ ∗U( )

∆

f nx nxv vn

v

n

m

nv

n

m

� � �� �

A A sin A coss c0 41

41

2 2, ,( ) ( )π π∑ ∑

Vol. 19 No. 2 & Vol. 20 No. 1 2003 25

structures is described by four-dimensional spacegroup [2, 5]. In general, each symmetry opera-tion (R, e | t,t) in the superspace implies (R – 3�3 rotational matrix, t – translational vector of asymmetry operation, e and t refer to the addi-tional dimension):

(1) Rotational and translational parts (R | t)of the symmetry operation belong to the three-dimensional space group of the average struc-ture;

(2) Point group operation R must keep theorientation of the modulation vector invariant.Thus, the value of e is defined by the equation,often called de Wolff equation: Rqi�eqi�0, oreqi�Rqi (index i means irrational part of themodulation vector);

(3) If two atoms n , h are related by a sym-metry operation (R | t) in the average structure,their modulation parameters are related as fol-lows:

uh(x4)�Run[e(x4�t)] (11)

ph(x4)�pn[e(x4�t)] (12)

where u(x4) is the modulation function of coor-dinates, p(x4) is the occupancy modulation func-tion. Non-zero translational part t is associatedwith e�1.

Equations (11) and (12) should be satisfied ifatoms n and h coincide, that is for atoms in spe-cial positions, thus putting restrictions on theform of a modulation function. For example, inthe case of one harmonic function either sin orcos waves (if any) should be used, dependingon e .

For composite structures the symmetries ofboth subsystems are defined by:

Riv�W vRi (W

v)�1 (13)

tiv�W vti (14)

where Riv is the 4�4 rotational matrix of a sym-

metry operation and tiv is the translational part.

The subsystems interact, giving rise to modula-tions of atoms, which are described in respectto their own bases. In addition to main reflec-tions of undistorted subsystems, then satellitereflections occur. In general, such a structurecan be described in one subsystem like a modu-lated structure, but in this case modulations ofatoms from the second subsystem will be toolarge and nonharmonic. A detailed descriptionof a composite structure approach is given inreviews (e.g. S. van Smaalen [6]).

3. Powder X-ray Diffraction PatternsPowder XRD patterns of incommensurate

modulated phases have several specific fea-tures, which have to be taken into account forsuccessful extraction of information on the unitcell of the average structure and components ofmodulation vector. The hklm reflections onpowder XRD patterns of incommensurate mod-ulated phase form a dense set being projectedon the 2q axis, whereas for non-modulated orcommensurate crystals this set becomes dis-crete. A profile decomposition and Rietveld re-finement in the incommensurate case becomepossible because in most cases the intensitiesof satellite reflections quickly vanish with in-creasing order (i.e. with increasing | m |) that al-lows to consider the set of reflections as finite.In the majority of cases only first order satellitesare visible on conventional powder XRD pat-terns. The restriction of maximal value of | m |allows to assign indexes for satellite reflections.On the other hand, frequent overlapping ofweak satellites with intense main reflectionsand generally low peak to background noiseratio for satellite reflections strongly restrict theaccuracy in the positions of the satellites thatcan be determined from the experimental pro-file. Unfortunately poor quality data are usuallyavailable from conventional powder XRD pat-tern for determination and subsequent refine-ment of the components of the modulation vec-tor. Up to now there is no reliable method,which can help to distinguish a modulatedstructure and determine a modulation vectorfrom powder data. “...Powder indexing worksbeautifully on good data, but with poor data itwill usually not work at all...” (“Data accuracyfor powder indexing” – R. Shirley – NBS Spec.Publ. 567 (1980)), and, in agreement with thisstatement, a knowledge on three-dimensionalreciprocal lattice is still needed for unambigu-ous determination of length and orientation ofthe modulation vector in reciprocal space. Se-lected area electron diffraction in a transmissionelectron microscope appears to be particularlyuseful for this purpose since it allows one to ob-serve reciprocal space sections produced by dif-fraction from small areas of tens nanometers,i.e. from single crystallites from which the pow-der sample is formed. From the set of such elec-tron diffraction patterns with known mutual ori-entations a full reconstruction of the three-di-mensional reciprocal lattice can be performed.Let us consider several examples of indexing ofsuch patterns. In Fig. 2 the [010] electron diffrac-tion pattern of Bi2.3Sr1.7CuO6.23 is shown. Therectangle marks the reciprocal unit cell corre-sponding to the average structure with the

26 The Rigaku Journal

monoclinic A2/a space symmetry. The satellitereflections form rows assigned to each basicspot; the satellites can be indexed with themodulation vector q�0.22a*�0.61c*. The aand g components of the modulation vectorwere used then for indexing of powder XRDpattern based on peak positions calculated ac-cording to the formula (the monoclinic angle bis equal to 90°):

1/d 2hklm�(h�ma)2/a 2�k 2/b2�(l�mg)2/c 2

Cell parameters and components of the modu-lation vector could be refined with standardleast-squares procedure, as it is done, for example, in the CSD package [7]. For theBi2.3Sr1.7CuO6.23 phase it gives q�0.217(1)a*�0.608(1)c*, in a good agreement with the elec-tron diffraction data. The indexation of the XRD pattern for Bi2.3Sr1.7CuO6.23 (PDF 45-0315)with four hklm indexes is present in Table 1.

For composite structures two sets of cell pa-rameters should be determined, each belongsto its own subsystem. A separation of reflec-tions belonging to the two main sets becomeseasier if the XRD or electron diffraction patternsof both incommensurate and commensuratecomposite structures are available. A compari-son of electron diffraction patterns of Sr4/3Mn2/3-Cu1/3O3 and Sr1.32Mn0.83Cu0.17O3 compounds isshown in Fig. 3. This structures consist of twomodulated subsystems, namely [(Mn,Cu)O3]�

and [Sr]�, which have trigonal lattices with com-mon parameter a�9.6 Å and different parame-ters c1�2.6 Å and c2�3.9 Å. The modulation vec-tor can be chosen as q�c1/c2c1*. The structure of Sr4/3Mn2/3Cu1/3O3 is commensurate (that is q�2/3c*), then it can be considered as a super-structure with c�3�2.60 Å�2�3.90 Å�7.80 Å. Forthe Sr1.32Mn0.83Cu0.17O3 the c1/c2�0.662 value re-quires a large denominator to be approximatedby a rational number, and the structure is con-

Vol. 19 No. 2 & Vol. 20 No. 1 2003 27

Table 1. X-ray powder diffraction pattern of Bi2.3Sr1.7CuO6.23 (PDF 45-0315). The deviationsDq�qexp�qcalc do not exceed 0.05°.

d I/I100 hklm d I/I100 hklm

12.279 6 0 0 2 0 2.4092 7 2 0 2 1, 1 2 0 05.262 7 0 1 1 0 2.3947 1 0 2 4 14.4773 1 0 0 6 1 2.3654 1 1 2 2 04.0911 13 0 0 6 0 2.3532 <1 1 1 7 13.7678 3 1 1 1 0 2.3118 <1 2 1 3 03.6271 <1 0 1 5 0 2.3017 <1 0 1 9 13.6021 2 1 1 3 1 2.2512 1 2 0 6 03.4546 90 1 1 3 0 2.2447 1 1 2 4 03.3387 2 1 1 1 1 2.0773 2 1 2 6 0, 2 0 8 13.2444 2 1 1 3 1 2.0459 3 0 0 12 03.0669 22 0 0 8 0 2.0244 18 2 0 8 03.0098 94 1 1 5 0 2.0086 2 2 2 0 12.9780 5 2 0 2 1 1.9982 2 2 2 2 12.9175 4 1 1 5 1 1.9617 1 2 0 8 12.8440 1 1 1 7 1 1.9252 7 1 1 11 02.6947 100 2 0 0 0 1.9058 26 2 2 0 02.6312 2 2 0 2 0 1.8136 1 2 0 10 02.5808 5 1 1 7 0 1.8037 2 2 2 0 12.4675 1 2 0 4 0 1.7957 3 2 2 2 12.4547 3 0 0 10 0 1.7611 1 3 1 3 12.4270 6 2 0 0 1

Fig. 2. [010] electron diffraction pattern of Bi2.3Sr1.7CuO6.23.

sidered as incommensurate. Deviation frombeing commensurate causes clear splitting ofthe satellite hklm reflections on the [120] elec-tron diffraction pattern of Sr1.32Mn0.83Cu0.17O3phase, while the hkl0 (from the [(Mn, Cu)O3]�)and hk0m reflections (from the [Sr]�) remainunsplit on the [120] patterns of both Sr4/3Mn2/3-Cu1/3O3 and Sr1.32Mn0.83Cu0.17O3 phases. The in-dexation scheme of the electron diffraction pattern of Sr1.32Mn0.83Cu0.17O3 phase is shown in Fig. 4, the indexation of powder XRD pat-terns is given in Table 2. This indexation can be compared with that of the commensurateSr4/3Mn2/3Cu1/3O3 phase (a�9.5928(8) Å, c�7.8167(6) Å, S.G. P321, Table 3). The details onthe Sr1.32Mn0.83Cu0.17O3 structure are describedbelow.

After indexation the set of hklm reflectionscan be analyzed for the presence of systematicextinctions, and a (3�1)d superspace symmetrygroup (or several possible symmetry groups)can be proposed based on results of this analy-sis. This step-by-step procedure in the (3�1)dcase is given in of International Tables for Crys-tallography [2] together with a list of (3�1)d su-perspace symmetry groups.

The correctness of the unit cell, modulationvector and superspace symmetry can be veri-fied by the structure refinement. As well as for

28 The Rigaku Journal

Fig. 3. Electron diffraction patterns of Sr4/3Mn2/3Cu1/3O3 (left column) and Sr1.32Mn0.83Cu0.17O3

(right column).

Fig. 4. Indexation of [120] (a) and [110] (b) ED pat-terns of the incommensurate Sr1.32Mn0.83Cu0.17O3

phase. Solid and dashed lines outline the unit cells ofthe [(Mn, Cu)O3]� and [Sr]� subsystems, respectively.Black circles, black squares, open circles, gray circlesand open squares mark the reflections of the [Sr]�

subsystem, the [(Mn, Cu)O3]� subsystem, commonfor both subsystems, satellites and reflections causedby double diffraction, respectively.

the Rietveld refinement of conventional struc-tures, the initial model for modulated structurecomprising atomic coordinates of the averagestructure and the starting parameters of modu-lation waves should be defined prior to the re-finement. The initial model can be taken from a modulated structure of a similar compound,which was determined earlier from single crys-tal data, or derived from general structural andchemical considerations, which also can imposeseveral sensible restrictions on the modulatedfunctions thus decreasing a number of refine-able parameters. Two such examples of Riet-veld refinement of modulated structures arediscussed below.

4. Incommensurate Modulated Supercon-ducting Bismuth and Copper Mixed Oxides

Bismuth and copper mixed oxides were firstsynthesized by B. Raveau with co-workers [8].Some of these substances were observed to exhibit superconducting properties, which re-

sulted in extensive study of these compounds.The problem of the structure determination andrefinement occurred at once due to significantmodulations in their structures.

Bi-containing layered copper oxides Bi2Sr2Can�1CunO2n�4�d (n=1–3) have an inter-growth structure with perovskite and rock saltfragments alternating along the c axis (Fig. 5).They have an incommensurate modulatedstructure, and the appearance of modulation isassociated with a mismatch between the BiOlayer and perovskite-like fragment. This mis-match is reduced via strong shifts of atoms inall layers from ideal sites and insertion of addi-tional oxygen atoms in the bismuth-containinglayer. This model was suggested by H. Zandber-gen et al. [9] and later proved by numerousstudies.

An influence of modulations on a distributionof scattering density in Bi-2201 (n�1) phases is illustrated by a comparison of two high-resolution electron microscopy images. These

Vol. 19 No. 2 & Vol. 20 No. 1 2003 29

Table 2. X-ray powder diffraction pattern of Sr1.32Mn0.83Cu0.17O3. Thedeviations Dq�qexp�qcalc do not exceed 0.05°. Reflections with indicesin two subsets are common for both subsystems. Only reflections withI/I100�1% are shown.

d I/I100[(Mn, Cu)O3]� [Sr]� Satellitessubsystem subsystem

5.7762 3 1 0 1 2

4.7961 3 2 1 0 0 2 1 0 0

3.6904 1 0 2 1 2

3.0397 84 1 1 0 1

2.7696 100 3 0 0 0 3 0 0 0

2.4811 1 1 0 1 0

2.3986 2 4 2 0 0 4 2 0 0

2.3041 2 1 3 2 3

2.2141 3 1 3 1 2

2.2049 15 2 2 1 0

2.0475 57 2 2 0 1

2.0201 1 1 3 1 3

2.0105 1 4 0 1 2

2.0026 14 3 1 1 0

1.9643 11 0 0 0 2

1.8553 1 2 3 1 2

1.8505 2 2 3 1 1

1.8181 6 1 1 0 2

1.7246 4 4 3 1 0

1.6466 18 1 4 0 1

1.6033 10 3 0 0 2

1.5996 18 6 3 0 0 6 3 0 0

Table 3. X-ray powder diffraction patternof Sr4/3Mn2/3Cu1/3O3. The deviations Dq�

qexp�qcalc do not exceed 0.05°. Only reflec-tions with I/I100�1% are shown.

d I/I100 Hkl

5.7008 5 1 0 1

4.7985 3 1 1 0

3.6692 2 2 0 1

3.5351 4 1 0 2

3.0305 100 1 1 2

2.7693 80 3 0 0

2.4846 2 1 0 3

2.3981 2 2 2 0

2.2926 1 3 1 0

2.2082 18 2 0 3

2.0442 50 2 2 2

2.0052 15 2 1 3

1.9847 1 3 1 2

1.9547 17 0 0 4

1.8970 5 3 0 3

1.8516 2 3 2 1

1.8099 14 1 1 4

1.7260 3 3 1 3

1.6447 15 4 1 2

1.6242 2 5 0 1, 4 0 3

1.5977 21 3 3 0

images show the BiO layers in the modulatedBi2Sr1.6La0.4CuO6.33 phase (Fig. 6, top) and in the same phase after strong fluorination, wherethe modulations are suppressed by fluorine insertion into the BiO slabs (Fig. 6, bottom) [10].Apparently, the structure refinement of Bi-2201phase without modulations should result in arough model with very large atomic displace-ment parameters especially for Bi atoms andunrealistic long Bi–O distances. The location of extra oxygen cannot be revealed within such model. The refinement of incommensuratemodulated structures was successfully madefrom single crystal data [11–13]. However, it isvery difficult to grow single crystals of Bi con-taining cuprates with desired composition. Sev-eral studies were performed using powerfulsources such as neutron TOF or/and synchro-tron radiation to obtain powder data and struc-tures were refined [14, 15]. Although satellite re-flections are much weaker than main reflec-tions, their maximum intensities may be as highas 5–10% (relative scale). The significance is,that structure refinement from powder data

without taking modulations into account can re-sult in poor agreement factors. Now we con-sider the refinement of modulated bismuthcuprates from powder data and compare resultswith single crystal refinement. Sample prepara-tions and data collections are described previ-ously [16, 17]. The structures were refined usingJANA2000 program package [18, 19].

A powder sample with the composition Bi2.3Sr1.7CuO6.23 was used in this study. The 2201phases crystallize in A2/a(a 0g) monoclinic su-perspace group with pseudotetragonal cell pa-rameters: a�5.4 Å, c�24.5 Å. First, the averagestructure was refined with the initial parameterstaken from Leligny et al. [20]. Strontium sitewas supposed to be occupied by Sr and Bi with

30 The Rigaku Journal

Fig. 5. Average structure of Bi2.3Sr1.7CuO6.23 (2201phase). Small dark-gray circles–Bi, large gray cir-cles–Sr, large white circles–oxygen, extra oxygenatoms are not shown. Cu atoms are situated in theoctahedra.

Fig. 6. HREM images of modulated Bi2Sr1.6La0.4-CuO6.33 phase (top) and the same phase after fluori-nation (bottom). The corresponding electron diffrac-tion patterns are shown as an inset. Note absence ofsatellite reflections on the electron diffraction patternof the fluorinated phase.

a 0.85 : 0.15 ratio. The refinement convergedonly with a very low damping factor, fixed x co-ordinates for Bi and Sr atoms, and resulted topoor agreement with the experimental and cal-culated data (RF�0.096, Rp�0.112). Such resultwas expected, because there are about 20–25satellite reflections below 2q�50° (for CuKa1 ra-diation), which can be detected by profile de-composition of the pattern.

In incommensurate structure refinements sev-eral reliability factors are calculated accordingto the separation of the reflections into thegroup of main reflections and the groups ofsatellites of different order. RF or RI factorsshould be defined for each set of reflections. Ingeneral, satellite peaks are characterized by lowpeak to background ratios. This characterizationleads to a significant effect on integral intensi-ties of the satellites by the background noiseand to larger reliability factors for these sets ofreflections, even if the structure model is cor-rect. Only first order satellites were used in therefinement of modulation parameters, higherorder satellites have too low an intensity in thepowder pattern and their intensities were notdetected with profile analysis.

Displacive modulation parameters (one har-monic) for Bi atom were introduced in the re-finement, resulting in a significant reduction ofresiduals: RF0�0.072, RFm�0.052, RF1�0.088,Rp�0.087 (here and further on index 0 standsfor overall, m – for main reflections, 1 – for first

order satellite reflections). Starting parametersfor modulation were also taken from Leligny et al. [20]. Modulation parameters for othercations were then refined. Introduction of oneharmonic for displacive modulation of all cationsresulted in RF0�0.049, RFm�0.036, RF1�0.060,Rp�0.073. The refinement of the second har-monic revealed significance over 3s only for Bimodulation parameters and along x3 for Sr. Atthis stage R-factors were reduced to RF0�0.039,RFm�0.027, RF1�0.049, Rp�0.068. ADPs for alloxygen atoms were fixed to Uiso�0.025 andwere not refined. For oxygen modulation refine-ment the model suggested in [21] was used.Modulations of oxygen atom in CuO2 layer wererefined with a harmonic function only. For otheroxygen atoms (in the BiO and SrO layers) morecomplicated scheme was chosen. Modulationsalong x1 were refined by a linear function withD�1 , while along x2 and x3 were left harmonic.Additional oxygen atom in BiO layer was not in-cluded into the refinement since its impact inpowder XRD pattern is too small to be properlytaken into account. The refinement convergedonly with low damping factor and reduced R-factors to RF0�0.036, RFm�0.026, RF1�0.045,Rp�0.065. Experimental, calculated and differ-ence powder patterns are presented at Fig. 7.

The results of the refinement of the modula-tion functions for heavy scattering cations arevery similar to those obtained from single crys-tal data for Bi2.26Sr1.74CuO6.19 [21]. Amplitude of

Vol. 19 No. 2 & Vol. 20 No. 1 2003 31

Fig. 7. Experimental, calculated and difference powder XRD patterns for Bi2.3Sr1.7CuO6.23.Ticks mark the position of the main hkl0 reflections.

modulations and t regions with maximal shiftare essentially the same, some examples arepresented at Fig. 8. As a result we may state,that the refinement of modulated structure fromX-ray powder data produces a rough model ofstructural modulation. It reflects the generaltendency and may be used for indexing of pow-der data, but exact atomic parameters and dis-tances are the rough approximation of the realstructure. The importance of such refinement isconnected with full indexing of a diffraction pat-tern. The situation can be significantly im-proved if structural constrains are used in therefinement. In comparison with a single crystalX-ray diffraction experiment, X-ray powder pat-terns provide only information on a muchsmaller amount of structure amplitudes. There-fore, the number of refineable structure para-meters should be restricted to those, which arevery essential for the structure description. Nowwe will consider such approach for the compos-ite Sr1.32Mn0.83Cu0.17O3 structure.

5. Sr1�x(Mn, Cu)O3 Hexagonal PerovskiteBased Oxides with Chain Structures

Sr1.32Mn0.83Cu0.17O3 belongs to the family ofhexagonal perovskite based oxides with the

common formula A3m�3nA’nB3m�nO9m�6n. Thestructure of these oxides is generally consid-ered as a derivative of the 2H hexagonal per-ovskite ABO3 (Fig. 9a) that is based on the (hh)close packing of the (A3O9) layers. This structureconsists of infinite chains of face-shared BO6 oc-tahedra with columns of A cations in betweenforming a hexagonal array in the a�aper√ 2

—,

c�(2/√ 3—

)aper unit cell. One third of the oxygenatoms can be removed from the (A3O9) layers inan ordered manner, so that one trigonal pris-matic cavity is created instead of two face sharing octahedral polyhedra (Fig. 9b). Addi-tional A’ cations can be located inside these trigonal prisms resulting in a (A3A’O6) layercomposition. The ordered alternation of m(A3O9) and n (A3A’O6) layers leads to numer-ous A3m�3nA’nB3m�nO9m�6n polytypes describedby J. M. Perez-Mato et al. [22]. A comprehen-sive list of the A3m�3nA’nB3m�nO9m�6n compoundsis given in [23]. The different stacking se-quences of the (A3O9) and (A3A’O6) layers resultsin different values of c parameter and oftenleads to an enormous increase, which makesthe structure refinement impossible. The com-posite structure model was first suggested byM. Evain et al. [24] and generalized by J. M.Perez-Mato et al. [22]. These structures are con-

32 The Rigaku Journal

Fig. 8. Displacive modulation along x for Bi (a)and along z for Cu (b) in Bi2.3Sr1.7CuO6.23.

Fig. 9. Crystal structures of two A3m�3nA’nB3m�n

O9m�6n parent compounds: a) m�1, n�0; b) m�0,n�1.

sidered as consisting of [(B, A’)O3]� and [A]�

subsystems with a common a parameter, differ-ent c1 and c2 parameters and a modulation vec-tor defined as q�gc1*. In general they are com-mensurate with g=3(n�m)/2(2n�3m), but formembers with large denominators it is better toconsider the modulation as incommensurate.

Intensities of satellite reflections for modu-lated structure are rather weak and sometimescould be ignored. Composite structures providereflections from both sublattices with intensitiesof the same order of magnitude that should dedefinitely taken into account. Commensuratestructures can be treated as the superstructure,while in the incommensurate case 3�d ap-proach is the only way out.

The synthesis and data collection for Sr1.32-Mn0.83Cu0.17O3 are described elsewhere [25]. Thecrystal structure of Sr1.32Mn0.83Cu0.17O3 was re-fined from X-ray powder diffraction data inR 3m(00g)0s (3�1)d superspace group with twosets of cell parameters obtained from powderpattern profile analysis and successive least-squares refinement: a1�a2�9.5971(8) Å, c1�2.60008(17) Å, c2�3.9312(3) Å, thus producingmodulation vector q�0.6616(1)c1* with interlat-tice matrices:

The incommensurate composite model forthe A3n�3A’nBn�3O6n�9 compounds has been previously discussed, and will be briefly sum-

marized here. The average structure is pre-sented as [Mn0.83Cu0.17O3]� (Mn, Cu site – 0, 0, 0;O site* – 0.16, 0.16, 1/2) in the first subcell and[Sr]� (Sr site – 1/3, 0, 1/4) in the second one. TheSr content is determined by a volume ratioV1/V2�0.662, where V1 and V2 are cell volumesof the first and second subsystems, respectively.The amount of the A cations determines g�(1�x)/2, where x is taken from the A1�x(A’xB1-x)O3formula. The occupancy modulations for Sr,(Mn, Cu) and O atoms are modeled with thesaw-tooth functions.

The saw-tooth function with D�1/2 and x40�

1/4 for oxygen atoms with the average coordi-nates (0.16, 0.16, 1/2) separates them into twosets of atoms related by a mirror plane ms, Oaand Ob, so that each set forms the (001) trian-gular face of an octahedron or a trigonal prism.Depending upon the t coordinate, the trigonalprismatic sites are created by either Oa–Oa orOb–Ob combinations, whereas the Oa–Ob com-bination corresponds to an octahedral site (Fig.10, top). In the idealized structure model ofA3m�3nA’nB3m�nO9m�6n hexagonal perovskites thewidth of the octahedron DO measured along the c axis is one half the width of the trigonalprism DP, however in reality the DP�2DO condi-tion is not fulfilled. To overcome this differencea displacement of oxygen atoms along the caxis is introduced, which is characterized by the u0, 3(O)�(DO�DP)/2 component (u0, 1(O) andu0, 2(O) were assumed to be small and fixed tozero). The (Mn, Cu) cations at (0, 0, 0) are sepa-rated into two sites, one situated at the B octa-hedral sites and the other situated at the A’

W W1 2

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

� �

Vol. 19 No. 2 & Vol. 20 No. 1 2003 33

Fig. 10. Top: saw-tooth function domains limiting t regions where either octahedral (O) ortrigonal prismatic coordination exists; bottom: domains of existence of B and A’ cations.

* The site is half occupied; see the further explanation.

trigonal prismatic sites. Windows of the saw-tooth functions for B and A’ positions should coincide with the t regions where either octa-hedral or prismatic oxygen environment oc-curs, that requires DB�(1�x)/2 and DA’�x/2 and x 0

4,B�0, x 04,A’�1/4 (Fig. 10, bottom). By sym-

metry u0,1(B, A’)�u0,2(B, A’)�0, whereas u0,3(B)and u0,3(A’) should result in the same slope ofthe saw-tooth functions as for oxygen atoms.Hence, u0,3(B)�(1�x)u0,3(O) and u0,3(A’)�xu0,3(O).Thus, the distribution of the cations betweenoctahedral and prismatic sites is determined bya single refineable parameter u0,3(O). In the second subsystem, the Sr atoms at (1/3, 0, 1/4)can be located along the c axis either at thelevel of the centers of the prisms or at the levelof the triangular faces. These two subsets aredescribed by saw-tooth functions with DSr1�x/(1�x), x 0

4,Sr1�0 and DSr2�(1/3)�[x/(1�x)], x 04,Sr2

�1/2, u0,3(Sr1)�[3x/(1�2x)]u0,3(Sr2).This model was introduced to the structure

refinement taking into account satellite reflec-tions up to second order. The preferred orienta-tion was refined with the March-Dollase func-tion. The initial starting model gave reasonablevalues of reliability factors (Rall�0.057, RP�0.017). Due to the small difference in scatteringpower of Mn and Cu we did not refine the occu-pancy factors for the B and A’ sites to determinethe distribution of these cations. The B site wasassumed to be fully occupied by Mn, whereas53.9% Cu and 46.1% Mn were placed at the A’

site to achieve the nominal phase composition.After the initial refinement step large value ofatomic displacement parameter for the A’ cation(Uiso�0.024 Å2) was observed. The A’ cationswere displaced from their ideal position (0, 0, 0)to a (x, x, 0) position, that means a shift of thesecations towards one of the rectangular faces ofthe trigonal prism. This shift results in a twotimes decrease of the atomic displacement pa-rameter for these atoms, followed by a de-crease of the reliability factor to Rall�0.049. Fur-ther improvement of the fit was achieved byadding one harmonic component of the posi-tional modulation function to the saw-tooth dis-placement of the B cations. Prior to that, the or-thogonalization procedure was performed forthe modulation functions of the B cations [26].The final values of the reliability factors Rall�0.044, RP�0.016 reflect the good agreement be-tween calculated and experimental profiles (Fig.11).

Let us consider the chemical formula A1�xA’x-B1�xO3, where x�p/q, where p and q are co-prime numbers. If Nprism and Noct are respec-tively the numbers of trigonal prismatic and oc-tahedral cavities in [(B, A’)O3]� chains, then theratio Nprism/Noct�x/(1�x)�p/(q�p). BaNiO3 [27](Nprism�0, Noct�1, consequently p1=0, q1=1) (Fig.9a) and Sr4PtO6 [28] (Nprism�1, Noct�1, conse-quently p2�1, q2�2) (Fig. 9b) can be taken asparent structures. The combination of thesestructures will produce a new structure with

34 The Rigaku Journal

Fig. 11. Experimental, calculated and difference X-ray diffraction profilesfor Sr1.32Mn0.83Cu0.17O3.

p/q�(p1�p2)/(q1�q2)�1/3. This is the structureof the commensurate Sr4/3Mn2/3Cu1/3O3 phase(Nprism�1, Noct�2, consequently p3�1, q3�3).The process is continued to infinity, giving riseto enormous number of phases as described byFarey tree [22]. For Sr1.32Mn0.83Cu0.17O3 the re-fined value of g�0.6616(1) corresponds to x�0.3232. It is close to p/q�1/3, where p is thenumber of prisms in the chains and (q�p) is thenumber of octahedra. Using Farey tree [22], thefractional value p/q�32/99�0.3232 for x can befound. Expressing p/q through the preceding“generators” p1/q1 and p2/q2, so that p/q�(p1�p2)/(q1�q2) the sequence of octahedra andprisms can be found. Thus, the repeat periodalong the chains comprises 99 polyhedra ac-cording to the sequence 1�(3 oct�1 prism)�9�(2 oct�1 prism)�1�(3 oct�1 prism)�10�(2 oct�1 prism)�1�(3 oct�1 prism)�10�(2 oct�1 prism). Such polyhedral arrangement was con-firmed by high resolution electron microscopy[25].

6. Concluding RemarksThe intrinsic one-dimensional character of

powder XRD pattern results in a significant lossof structural information if compared with thesingle crystal X-ray diffraction technique. A lackof data seems to be crucial for possibility to de-termine ab initio the structure model for an in-commensurate modulated compound usingpowder XRD data as a unique source of infor-mation. However, compared with other tech-niques such as electron diffraction and high-res-olution electron microscopy, powder XRD pat-terns of incommensurate phases become ap-plicable not only for phase analysis, but also fordetermination and precise refinement of theunit cell metrics, modulation vector and possi-ble superspace group. If an appropriate modelis available from a similar known structure orfrom chemical considerations, Rietveld refine-ment appears to be able to provide sensible in-formation on modulation parameters and shedlight on peculiarities of the structural organiza-tion of composite structures.

Acknowledgement This work was partially supported by Russian

Foundation of Basic Researches (grant No.03-03-32610) and International Centre of DiffractionData (grant APS 91-05). The authors thanks Dr.V. Petricek for very fruitful discussion and P.Chizhov for help in the paper preparation. Ac-knowledgement is made to the donors of theAmerican Chemical Society Petroleum Research

Fund for the partial support of the research.

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