direct methods in electron crystallography: image processing and solving incommensurate structures

Direct Methods in Electron Crystallography: Image Processingand Solving Incommensurate StructuresH.F. FAN*Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China

KEY WORDS direct methods; image processing; incommensurate structures; dynamical diffrac-tion effect

ABSTRACT Theory and practice of direct methods are given in detail for application in imageprocessing of high-resolution electron microscopy and for solving incommensurate crystal struc-tures. The effect of dynamical electron diffraction is also discussed. Microsc. Res. Tech. 46:104–116,1999. r 1999 Wiley-Liss, Inc.

INTRODUCTIONFor the structure analysis of crystalline materials,

electron-crystallographic methods are in some casessuperior to X-ray methods. Firstly, many crystallinematerials important in science and technology, such ashigh Tc superconductors, are too small in grain size andtoo imperfect in periodicity for an X-ray single crystalanalysis to be carried out, but they are suitable forelectron-microscopic observation. Secondly the atomicscattering factors for electrons differ greatly from thosefor X-rays and it is easier for electron diffraction toobserve light atoms in the presence of much heavieratoms. Finally, the electron microscope is the onlyinstrument that can produce simultaneously for acrystalline sample a microscope image and a diffractionpattern at atomic resolution. In principle, either theelectron microscopic (EM) image or the electron diffrac-tion (ED) pattern could lead to a structure image of thesample. However, the combination of the two will makethe procedure much more efficient and powerful. Solv-ing crystal structures from ED data based on thekinematical-diffraction approximation was pioneeredby Vainshtein and his colleagues (Vainshtein, 1964)and by Cowley (1953). The first application of directmethods in ED analysis was presented by Dorset andHauptman (1976). High-resolution EM observation ofcrystal structures appeared in the early 1970s forinorganic compounds (Allpress et al., 1972; Cowley andIijima, 1972) and for the organic compound, chlorinatedcopper phthalocyanine (Uyeda et al., 1972). The idea ofcombining the information from an EM and the corre-sponding ED pattern was proposed by Gerchberg andSaxton (1971). The first attempt to enhance the resolu-tion of an EM by making use of the corresponding EDwas reported by Ishizuka et al., (1982) using the phasecorrection method (summarized by Gassmann, 1976).Direct methods in X-ray crystallography were intro-duced into high-resolution EM as a tool of imageprocessing combining the information from an EM andthe corresponding ED (Fan and Li, 1987; Li and Fan,1979). The procedure is in two stages, i.e., imagedeconvolution and phase extension. The former aims ateliminating the blurring effect of the contrast transferfunction, while the later is for enhancing the resolutionof the image. Multidimensional direct methods (Fan et

al., 1993; Hao et al., 1987) have been developed forsolving crystal structures containing periodic defects.They have been used to reveal the incommensuratestructure modulation in high Tc superconductors usingelectron crystallographic data. The focus of this paperwill be on direct methods applied to electron crystallog-raphy for image processing and for solving incommensu-rate crystal structures. While kinematical ED is as-sumed for all examples in this paper, the effect ofdynamical ED on the results will be discussed.

IMAGE PROCESSINGCOMBINING EM AND ED

Simulation calculations with the sample, chlorinatedcopper phthalocyanine, have shown that the combina-tion of EM and ED can be a much more powerful toolthan either of them alone (Fan et al., 1985; Han et al.,1986). The results are summarized in Figure 1. Theobject, chlorinated copper phthalocyanine (C32N8Cl16Cu),crystallizes in the space group C2/c with unit cellparameters a 5 19.62 Å, b 5 26.08 Å, c 5 3.76 Å, andb 5 116.5°. Figure 1a is the expected image of the unitcell at 0.1-nm resolution projected down the c axis.Figure 1b is the theoretical EM at 0.2-nm resolutiontaken with the photographic conditions: acceleratingvoltage 5 500 kV, Cs 5 1 mm, D 5 15 nm and Df 5 2100nm. Figure 1c is the corresponding ED pattern at0.1-nm resolution. It is seen from Figure 1b that, evenin the case of kinematical diffraction, an EM will notnecessary directly reflect structure details of the object.However a direct-method image deconvolution (Han etal., 1986) could bring Figure 1b to 1d. Here the shape ofthe molecule can be recognized but individual atomsother than copper are not visible. On the other hand, aset of structure-factor magnitudes at 0.1-nm resolutioncan be measured from the ED pattern. Ab initio direct-method phasing of these data resulted in the bestE-map shown in Figure 1e. Although the chlorine atomsare on the map, the large number of ghost peaks makesthe E-map impossible to interpret. With the same set of

Contract grant sponsor: National Natural Science Foundation of China.*Correspondence to: Fan Hai-Fu, Institute of Physics, Chinese Academy of

Sciences, Beijing 100080, China.Received ; accepted in revised form

MICROSCOPY RESEARCH AND TECHNIQUE 46:104–116 (1999)

r 1999 WILEY-LISS, INC.

structure-factor magnitudes, by incorporating the phaseinformation at 0.2-nm resolution obtained from theFourier transform of Figure 1d, a direct-method phaseextension (Fan et al., 1985) led to Figure 1f, which isnearly the same as Figure 1a, the theoretical structureimage. While the advantage of combining EM and ED isevident, it does not mean that to solve the structurefrom an ED alone is impossible. Dorset et al., (1991)and Tivol et al., (1993) reported the determination ofthe structure of chlorinated copper phthalocyanine.They started from an experimental ED pattern at0.091-nm resolution obtained with the acceleratingvoltage of 1,200 kV. Direct-methods phasing of the EDrevealed all the heavy atoms. The Fourier refinementbased on which led to the complete structure.

Image DeconvolutionThe goal of image deconvolution is to retrieve the

structure image from one or a series of blurred EMs, orequivalently, to extract a set of structure factors fromthem. Different procedures have been proposed. Most ofthem use a series of EMs with different defocus values.Uyeda and Ishizuka (1974, 1975) first proposed amethod for the deconvolution of a single EM under theweak-phase-object approximation (WPOA). Inspired bythis work, direct methods in X-ray crystallographywere introduced into high-resolution electron micros-copy for the image deconvolution using a single EM(Han et al., 1986; Li and Fan, 1979; Liu et al., 1990).

With the WPOA, in which dynamical diffractioneffects are neglected, the Fourier transform of an EMcan be expressed as

T(h) 5 d(h) 1 2sF(h) sin x1(h) exp [2x2(h)], (1)

which can be rearranged to give

F(h)hÞ0

5 T(h)/2s sin x1(h) exp [2x2(h)]. (2)

Here s 5 p/lU, where l is the electron wavelength andU the accelerating voltage. h is the reciprocal-latticevector within the resolution limit. F(h) is the structurefactor of ED, which is the Fourier transform of thepotential distribution w(r) of the object, andsin x1(h) exp [2x2(h)] is the contrast transfer function,in which

x1(h) 5 pDflh2 1 1⁄2(pCsl3h4),

and

x2(h) 5 1⁄2(p2l2h4D2).

Here Df is the defocus value, CS is the spherical-aberration coefficient, and D is the standard deviationof the Gaussian distribution of defocus due to thechromatic aberration (Fijes, 1977). The values of Df, CS,and D should be found by image deconvolution. Of thesethree factors, CS and D can be determined experimen-tally without much difficulty. Furthermore, in contrastto Df, CS and D do not change much from one image toanother. This means that the main problem is theevaluation of Df. With the estimated values of CS and D,a set of F(h) can be calculated from Equation (2) for agiven value of Df. If the Df value is correct then the

corresponding set of F(h) should obey the Sayre equa-tion (Sayre, 1952)

Fh 5u

V oh8

Fh8Fh2h8, (3)

where u is the atomic form factor and V is the volume ofthe unit cell. Hence the true Df can be found by asystemic change of the trial Df so as to improve thevalidity of the Sayre equation. For the evaluation of thequality of each trial, figures of merit used for directmethods in X-ray crystallography (see Woolfson andFan, 1995) were introduced. The procedure has beenautomated in the program VEC (Wan et al., 1997).

An example K2O · 7Nb2O5 is given here. The spacegroup is P4bm and unit cell parameters are a 5 b 527.5 Å and c 5 3.94 Å. Figure 2a is the expected imageat 0.3 nm resolution of the unit cell projected down the caxis. Two experimental EMs, taken in the [001] direc-tion, one near to and the other far from the optimumdefocus, are shown in Figures 2b and c, respectively. Itis seen that Figure 2b and c are quite different androughly inverse to each other. Starting with the images2b and c, respectively, the program VEC (Wan et al.,1997) produced the deconvoluted images 2d and e. Boththe deconvoluted images have similar features to thoseof the expected image 2a; i.e., they all have fourheptagonal channels surrounded by hexagonal, pentago-nal, tetragonal, and trigonal channels.

As has been shown by simulation calculations (Hanet al., 1986), ED data can help improve the deconvolu-tion result. An example using real experimental data isgiven on the left of Figure 3. By comparing Figures 3a,b, and d, we see that, even for an EM taken under theoptimum defocus condition, the image quality can besignificantly improved by deconvolution making use ofthe corresponding ED data.

Phase ExtensionAn ED pattern usually contains observable reflec-

tions at a resolution equal to or better than 0.1 nm. Inaddition, the intensities of the ED pattern from acrystalline specimen are independent of defocus andthe spherical aberration of the electron-optical system.Accordingly, under the WPOA, better-quality imagescould be obtained from ED. However, structure analysisby ED alone is subject to the well-known difficulty of thecrystallographic phase problem (but see Dorset et al.,1991, 1992). On the other hand, since an EM, afterdeconvolution, can provide phase information at about0.2-nm resolution, the complexity of the solution of thephase problem can be greatly reduced. An improvedhigh-resolution image can be obtained by a phase-extension procedure using the magnitudes of the struc-ture factors from ED and starting phases from thecorresponding EM. The tangent formula (Karle andHauptman, 1956) is used as the basis of the phaseextension:

tan ah 5

oh8

0Eh8Eh2h8 0 sin (ah8 1 ah2h8)

oh8

0Eh8Eh2h8 0 cos (ah8 1 ah2h8), (4)

105DIRECT METHODS IN ELECTRON CRYSTALLOGRAPHY

Fig. 1. Simulation calculations on crystalline chlorinated copper phthalocyanine. a: Expectedstructure image. b: Theoretical EM at 0.2 nm resolution. c: Corresponding ED pattern. d: Direct-methoddeconvolution result of b. e: Result of direct-method phasing with theoretical ED data at 0.1 nmresolution. f: Result of image processing combining EM and ED.

106 F. HAI-FU

where ah is the phase associated with the normalizedstructure factor Eh. The latter is in turn expressed interms of the conventional structure factor Fh as

Eh 5 Fh/(eS)1/2.

S is the average value of 0Fh 02 for the scattering anglecorresponding to h. The factor e usually equals unitybut is, in general, a small integer that depends on thespace group and the type of reflection and gives theeffect that for all the reflections of that particular type70Eh 028 5 1.

By substituting the normalized structure factorswith known phases into the right-hand side of thetangent formula, originally unknown phases can bepredicted according to the outcome on the left-hand

side. In the program VEC, a procedure similar to that ofRANTAN (Yao, 1981) is used for phase extension. Allreflections with their E value greater than a certainlimit are put into the right-hand side of Equation (4).Trial sets of random values are given to the unknownphases. The random phases in each trial set are thenchanged gradually during the recycling process untilthey converge. Figures of merit (see Woolfson and Fan,1995) are used to pick out the best solution from theresulting phase sets.

An example of phase extension using the experimen-tal EM and ED of chlorinated copper phthalocyanine isgiven below. Structure-factor magnitudes out to 0.1-nmresolution were measured from the ED pattern (Fig.3c). Phases out to 0.2-nm resolution were obtained fromthe Fourier transform of the deconvoluted image (Fig.

Fig. 2. Image deconvolution ofhigh resolution EMs of K2O · 7Nb2O5.a: Expected structure image at 0.3nm resolution. b,c: ExperimentalEMs with defocus value at about290 nm and 240 nm, respectively.d,e: Deconvoluted images of b and c,respectively.


Fig. 3. Image processing of the high-resolution EM of chlorinatedcopper phthalocyanine. a: Expected structure image of the objectprojected along the c axis. b: Experimental EM of the object at 0.2 nmresolution taken near the optimum defocus value. c: Correspondingexperimental ED pattern. d: Direct-method deconvoluted image of b.e: Electron-potential projection of the object at 0.1 nm resolution

obtained from the direct-method phase extension based on the decon-voluted image at 0.2 nm resolution and experimental ED data at 0.1nm resolution. f: Electron-potential projection after 4 cycles of Fourierrefinement. The experimental EM and ED were provided by ProfessorN. Uyeda.

108 F. HAI-FU

3d). The phase extension with the program VEC re-sulted in Figure 3e, which shows the essential featuresof the structure at atomic resolution. Four cycles ofFourier iteration based on this gave significant improve-ment as is shown in Figure 3f. This work was originallydone by Fan et al. (1991). The results shown here werereproduced with improvements by using the programVEC (Wan et al., 1997).

Fan and Zheng (1975) have shown that it is possibleto extrapolate not only the phases but also the magni-tudes of structure factors beyond the resolution limit ofthe original diffraction data. An example of the struc-ture-factor extrapolation from the theoretical EM ofchlorinated copper phthalocyanine at 0.2-nm resolutionto the structure image at 0.1-nm resolution has beenreported by Liu et al. (1988). Applications to real EMsstill have to be explored.

MULTIDIMENSIONAL DIRECT METHODSAND THE DETERMINATION OF

INCOMMENSURATE CRYSTAL STRUCTURESIn diffraction analysis, crystal structures are as-

sumed to be ideal 3-dimensional periodic objects. Sincereal crystals are never perfect, what we obtain is just animage of the real structure averaged over a largenumber of unit cells. However, a knowledge of theaverage structure is not enough for understanding theproperties of many solid-state materials. Therefore, animportant task for methods of solving crystal struc-tures is to extend their scope to include real crystals.Modulated crystal structures belong to that kind ofcrystal structures in which atoms suffer from certainoccupational and/or positional fluctuation. If the periodof fluctuation is commensurate with that of the three-dimensional unit cell, then a superstructure results;otherwise an incommensurate modulated structure isobtained. Incommensurate modulated phases can befound in many important solid-state materials. Inmany cases, the transition to the incommensuratemodulated structure corresponds to a change of certainphysical properties. Hence, it is important to know thestructure of incommensurate modulated phases in or-der to understand the mechanism of the transition andproperties in the modulated state. Up to the present,many incommensurate modulated structures weresolved by trial-and-error methods. With these methods,it is necessary to make assumptions about the modula-tion in advance. This leads often to difficulties orerroneous results. In view of diffraction analysis, it ispossible to phase the reflections directly and solve thestructure objectively without relying on any assump-tion about the modulation wave. Multidimensionaldirect methods have been developed for this purpose(Fan et al., 1993; Hao et al., 1987). A brief description ofthe theory involved is given below.

An incommensurate modulated structure produces a3-dimensional diffraction pattern, which contains satel-lites around the main reflections. An example of asection of such a 3-dimensional diffraction pattern isshown schematically in Figure 4. The main reflectionsare consistent with a regular 3-dimensional reciprocallattice but the satellites do not fit the same lattice. Onthe other hand, while the satellites are not commensu-rate with the main reflections, they have their ownperiodicity. Hence, it can be imagined that the 3-dimen-

sional diffraction pattern is a projection of a 4-dimen-sional reciprocal lattice, in which the main and thesatellite reflections are all regularly situated at thelattice nodes. From the properties of the Fourier trans-form, the incommensurate modulated structure hereconsidered can be regarded as a 3-dimensional ‘‘section’’of a 4-dimensional periodic structure. This representa-tion was first proposed by De Wolff (1974) to simplifythe structure analysis of the incommensurate modu-lated structure of g-Na2CO3. The above example corre-sponds to a one-dimensional modulation. For an n-dimensional (n 5 1, 2, . . .) modulation, it needs a 3 1 n)-dimensional description. A (3 1 n)-dimensionalreciprocal vector is expressed as

h 5 h1b1 1 h2b2 1 h3b3

1 · · · 1 h31n b31n(n 5 1, 2, . . . ),(5)

where bi is the ith translation vector defining thereciprocal unit cell. The structure factor formula iswritten as

F(h) 5 oj51

N

fj(mod)(h) exp [i2p(h1xj11 h2xj2

1 h3xj3)], (6)

where

fj(mod)(h) 5 fj(h) e0

1dx4 . . . e

0

1dx31nPj(x4, . . . , x31n)

3 exp 5i2p[(h1Uj11 h2Uj2

1 h3Uj3)

1 (h4xj41 · · · 1 h31nxj31n

)]6.

(7)

The fj(h) on the right-hand side of (7) is the ordinaryatomic scattering factor, Pj is the occupational modula-tion function and Uj describes the deviation of the jth

atom from its average position (xj1, xj2, xj3). For moredetails on (6) and (7) the reader is referred to the papersby De Wolff (1974), Yamamoto (1982), and Hao et al.

Fig. 4. Schematic diffraction pattern of an incommensurate modu-lated structure. The vertical line segments indicate projected latticelines parallel to the fourth dimension.


(1987). What should be emphasised here is that, accord-ing to (6) a modulated structure can be regarded as a setof ‘‘modulated atoms’’ situated at their average posi-tions in 3-dimensional space. The modulated atom, inturn, is defined by a ‘‘modulated atomic scatteringfactor’’ expressed as (7). A special kind of incommensu-rate modulated structure is called a composite struc-ture, the characteristic of which is the coexistence oftwo or more mutually incommensurate 3-dimensionallattices. Owing to the interaction of coexisting lattices,composite structures are also incommensurate modu-lated structures. Unlike ordinary incommensuratemodulated structures, composite structures do not havea 3-dimensional average (basic) structure. The basicstructure of a composite structure corresponds to a 4- orhigher-dimensional periodic structure. For a detaileddescription of composite structures, the reader is re-ferred to the paper by Van Smaalen (1992). Obviously,crystal-structure analysis of incommensurate modu-lated structures would better be implemented in multi-dimensional space. For this purpose we need, firstly, atheory on multi-dimensional symmetry and, secondly, amethod to solve directly the multi-dimensional phaseproblem. The work of Janner and co-workers (seeJanssen et al., 1992) has been aiming at the firstproblem, while the multidimensional direct methodsare trying to solve the second.

Hao et al. (1987) showed that the Sayre (1952) can beextended into multidimensional space. We have

F(h) 5u

V oh8

F(h8)F(h 2 h8); (8)

here h is a multi-dimensional reciprocal vector definedas (5). The right-hand side of (8) can be split into threeparts, i.e.,

F(h) 5u

V 5oh8

Fm(h8)Fm(h 2 h8)

1 2 oh8

Fm(h8)Fs(h 2 h8) 1 oh8

Fs(h8)Fs(h 2 h8)6,(9)

where subscript m stands for main reflections whilesubscript s stands for satellites. Since the intensities ofsatellites are on average much weaker than those ofmain reflections, the last summation on the right-handside of (9) is negligible in comparison with the second,while the last two summations on the right-hand side of(9) are negligible in comparison with the first. LettingF(h) on the left-hand side of (9) represent only thestructure factor of main reflections, we have, to a firstapproximation,

Fm(h) <u

V oh8

Fm(h8)Fm(h 2 h8). (10)

On the other hand, if F(h) on the left-hand side of (9)corresponds only to satellites, it follows that

Fs(h) <u

V oh8

Fm(h8)Fm(h 2 h8)

12u

V oh8

Fm(h8)Fs(h 2 h8).

(11)

For ordinary incommensurate modulated structures,the first summation on the right-hand side of (11)vanishes, because any three-dimensional reciprocallattice vector corresponding to a main reflection willhave zero components in the extra dimensions so thatthe sum of two such lattice vectors could never give riseto a lattice vector corresponding to a satellite. We thenhave

Fs(h) <2u

V oh8

Fm(h8)Fs(h 2 h8). (12)

For composite structures on the other hand, since theaverage structure itself is a 4- or higher-dimensionalperiodic structure, the first summation on the right-hand side of (11) does not vanish. We have instead of(12) the following equation:

Fs(h) <u

V oh8

Fm(h8)Fm(h 2 h8). (13)

Equation (10) indicates that the phases of main reflec-tions can be derived by a conventional direct methodneglecting the satellites. Equation (12) or (13), respec-tively, can be used to extend phases from the mainreflections to the satellites for ordinary incommensu-rate modulated structures or composite structures.This provides a way to determine the modulationfunctions objectively. The procedure comprises the fol-lowing stages:

1. derive the phases of main reflections using Equation(10);

2. derive the phases of satellite reflections using Equa-tion (12) or (13);

3. calculate a multi-dimensional Fourier map usingthe observed structure factor magnitudes and thephases from 1 and 2;

4. cut the resulting Fourier map with a 3-dimensional‘‘hyperplane’’ to obtain an ‘‘image’’ of the incommen-surate modulated structure in the 3-dimensionalphysical space;

5. measure the parameters of the modulation functionsdirectly on the multi-dimensional Fourier map result-ing from 3.

A program package, DIMS (Direct methods forIncommensurate Modulated Structures), has beenwritten in Fortran for the implementation of steps 1and 2 (Fu and Fan, 1994, 1997).

The widespread occurrence of incommensurate modu-lations in the high-Tc superconducting phases andrelated compounds requires the multi-dimensional crys-tallographic methods for their structure analysis. Onthe other hand, the modulation in the high-Tc supercon-ductors involves both the metal and the oxygen atoms.The modulation of the latter in the Bi-O layer isimportant for understanding the mechanism of super-conductivity, since it plays an important role in theincorporation of extra 0 atoms in the Bi-O layer andhence contributes to the hole concentration in the Cu-Olayer. Owing to the dominating effect of heavy atoms inX-ray diffraction, ED can be a better technique to studythe oxygen modulation. In the following, there areexamples of studying structural details of high-Tc super-

110 F. HAI-FU

conductors by multidimensional direct methods withelectron crystallographic data.

Incommensurate Modulationof the Pb-Doped Bi-2223 Superconductor

The analysis was based on the preliminary ED studyof Li et al. (1989) and the average structure obtained bySequeira et al. (1990). The diffraction intensities usedin the study were measured from the ED patternnormal to the a axis. Since the a axis is rather short(5.49 Å) and is perpendicular to the modulation vectorq, no attempt was made to use 3-dimensional diffrac-tion data. The symmetry of the sample belongs to thesuperspace group P [B bmb] 111 (see Janssen et al.,1992, for a discussion of superspace groups and theirdesignations.) with the 3-dimensional unit cell a 5 5.49Å, b 5 5.41 Å, c 5 37.1 Å; a 5 b 5 g 5 90° and themodulation vector q 5 0.117b*. Five photographs weretaken with different exposure times for the same OklmED pattern. This is an analogue of the multifilmmethod in X-ray crystallography for collecting diffrac-tion intensities. A microdensitometer was used to mea-sure the integrated intensities. Structure factor magni-tudes were obtained as the square root of diffractionintensities. The R factor for the discrepancy of symmetri-cally related reflections is 0.12 for the 42 main reflec-tions and 0.13 for the 70 first-order satellite reflections.A few second-order satellites were also observed; how-ever, they are much weaker than the first-order onesand were neglected in the structure analysis. Thestructure analysis was carried out in 4-dimensionalspace, in which the real and the reciprocal unit cells aredefined respectively as

a1 5 a, a2 5 b 2 0.117d, a3 5 c, a4 5 d

and

b1 5 a*, b2 5 b*, b3 5 c*, b4 5 0.117 b* 1 d

where d is the unit vector normal to the 3-dimensionalspace, i.e., a unit vector simultaneously perpendicularto the vectors a, b, c, a*, b*, and c*. An atom in the4-dimensional space without modulation will be some-thing like an infinite straight bar parallel to the fourthdimension a4. Occupational modulation will periodi-

cally change the width, while positional modulationwill periodically change the direction of the bar. Ourtask is to find out such a periodic change. This can beaccomplished by solving the phase problem and calculat-ing the 4-dimensional potential distribution, the 4-di-mensional Fourier map. The incommensurate modu-lated structure in the 3-dimensional physical space canbe obtained by cutting the 4-dimensional Fourier mapwith a 3-dimensional hyperplane perpendicular to thedirection a4. The modulation wave of all the atoms canbe measured directly on the 4-dimensional Fouriermap.

The phases of the main reflections Ok10 were calcu-lated from the known average structure, while phasesof the satellites Oklm were derived by the multidimen-sional direct method. A Fourier map in multidimen-sional space was then calculated, from which modula-tion waves of all metal atoms were measured directly.On this basis, least-squares refinement including alsomodulation parameters of oxygen atoms ended at R-factors of 0.16 for the main and 0.17 for the first-ordersatellite reflections. Figure 5 shows the modulationwaves of the metal atoms. Figure 6 shows the incommen-surate modulation of the Pb-doped Bi-2223 superconduc-tor in the 3-dimensional physical space projected alongthe a axis. As is seen, both occupational and positionalmodulations are evident for most metal atoms. Anotherprominent feature is that the oxygen atoms on the Cu-Olayers move towards the Ca layer, forming a disorderedoxygen bridge across the layers of Cu(2)-Ca-Cu(1)-Ca-Cu(2). This work was originally done by Mo et al.(1992). The figures shown here were reproduced byusing the program VEC.

Incommensurate Modulation of Bi-2212The superconducting phase Bi-2212 has been exten-

sively studied (Fu et al., 1995; Gao et al., 1988, 1993;Kan and Moss, 1992; Matsui and Horiuchi, 1988;Petricek et al., 1990; Yamamoto et al., 1990). Moststudies were done using X-ray and/or neutron diffrac-tion data. We provide here an example of using electroncrystallographic data and show how the combination ofEM and ED can be used to determine the incommensu-rate structure even when the average structure isunknown (Fu et al., 1994). The sample belongs to the

Fig. 5. Modulation waves in the Pb-dopedBi-2223 superconductor, revealed on the map, a3-dimensional ‘‘section’’ of the 4-dimensionalFourier map at x2 5 0, projected along the aaxis.


superspace group N [B bmb] 111 with a 5 5.42 Å, b 55.44 Å, c 5 30.5 Å, a 5 b 5 g 5 90° and the modulationwave vector q 5 0.22 b* 1 c*. We started with an EM at0.2 nm resolution (Fig. 7a) and the corresponding ED(Fig. 7b) at 0.1-nm resolution. A set of structure-factormagnitudes with indices Oklm was measured from theED. The phases of the main reflections within 0.2 nmresolution were obtained from the Fourier transform ofthe deconvoluted EM. Phases of the satellite reflectionsand of the main reflections beyond 0.2 nm resolutionwere derived by the direct-method phase extensionusing the program DIMS (Fu and Fan, 1994). Theresulting Fourier map is shown in Figure 7c, whichreveals much more structure detail in comparison withthe original EM.

Incommensurate Modulation of Bi-2201Crystals of Bi-2201 belong to the superspace group P

[B 2/b] 11 with a 5 5.41 Å, b 5 5.43 Å, c 5 24.6 Å, a 5b 5 g 5 90° and the modulation wave vector q 50.217b* 1 0.62c*. The structure analysis was based onthe known average structure (Gao et al., 1989). OnlyED intensities of the Oklm reflections were used. Sinceinconsistent results have been reported on the oxygenatoms (Gao et al., 1989; Yamamoto et al., 1992), specialcare was taken for their determination. The 4-dimen-sional Fourier map projected along the a axis wascalculated with phases from the average structure andthe direct-method phase extension. Apart from twooxygen atoms, which are overlapped with metal atomsrespectively on the Bi-O and Sr-O layers, the modula-tion waves of all symmetrically independent atomswere measured directly from this Fourier map. Up tothe fourth-order harmonics were included in the expres-sion of the modulation function. Fourier recycling andleast-squares refinement led to the R factors: RT 5 0.32,RM 5 0.29, RS1 5 0.29, RS2 5 0.36, and RS3 5 0.52. Herethe R factor is defined as R 5 S\Fo 0 2 0Fc \ /S 0 Fo 0, RTdenotes the R factor for the total reflections, RM for themain reflections, RS1, RS2, and RS3, respectively, for thefirst-order, second-order, and third-order satellite reflec-tions. The resulting Fourier map clearly shows the

main features of the modulation. However, it contains anumber of small additional peaks near some of the Biand Sr sites. After failed to eliminate them, they weretreated as extra oxygen atoms. By including extraoxygen atoms in the Bi-O layer, a few cycles of refine-ment brought the R factors to 0.23, 0.20, 0.24, 0.27, and0.30 for RT, RM, RS1, RS2, and RS3, respectively. Furtherinclusion of extra oxygen atoms in the Sr-O layer led tothe R factors of 0.18, 0.13, 0.19, 0.25 and 0.26 for RT,RM, RS1, RS2 and RS3 respectively. The final Fourier mapis shown in Figure 8.

Solving Composite StructuresThere have been no examples of solving composite

structures by multidimensional direct methods usingED data. We provide here an example from X-raycrystallography since the subject is of interest and themethod can be used with ED data as well. For compos-ite structures it is much more difficult to solve theaverage structure than to determine the modulation.The reason is that the average structure of a compositecrystal corresponds to a four- or higher-dimensionalperiodic object, and ab initio phasing rather than phaseextension should be used for solving the phase problem.However, by using multidimensional direct methodsthe problem can be much simpler. The composite crys-tal structure (PbS)1.18TiS2 (Van Smaalen et al., 1991)belongs to the space group C2/m (a, 0, 0) s 2 1. Itconsists of two subsystems: the subsystem TiS2 witha1 5 3.409 Å, b 5 5.880 Å, c 5 11.760 Å, a 5 95.29° andthe subsystem PbS with a2 5 5.800 Å, b 5 5.881 Å, c 511.759 Å, a 5 95.27°. A default run of DIMS in a test(Mo et al., 1996) using Equation (10) was able to revealdirectly the basic structure (see Fig. 9). In Figure 9 (topleft) is the resulting direct-method electron densitymap, which is the 3-dimensional hypersection with x4 50 projected along the b axis. The map shows thealternate stacking of TiS2 and PbS layers along the caxis. This results in a ‘‘chimney-ladder’’ structure alongthe a axis with two incommensurate periodicities a1and a2. The other three direct-method maps in Figure 9are 2-dimensional sections of the 4-dimensional elec-

Fig. 6. The 3-dimensional poten-tial distribution function of the Pb-doped Bi-2223 superconductor pro-jected along the a axis on an area of10b 3 1c.

112 F. HAI-FU

tron density function showing clearly all the atoms inthe unit cell. Determination of the modulation bymulti-dimensional direct methods was also successful(Fan et al., 1993; Sha et al., 1994).

ON THE EFFECT OF DYNAMICALELECTRON DIFFRACTION

It is well known that the dynamical diffraction effectcauses serious problems in crystal-structure analysisusing ED data. However, a recent study (Li et al., 1999)showed that this effect is often much weaker than whatpeople would usually predict. The thermal motion ofatoms, the incommensurate modulation of the struc-

ture, and any deviation from the ideal periodicity cansignificantly weaken the dynamical diffraction effect. Inpractice, many structural defects, which break the idealperiodicity, are either unrecognized or difficult to handlein simulation calculations. Hence, the dynamical EDeffect is often overestimated in theoretical consider-ations.

It is believed that, with kinematical electron-diffrac-tion analysis, a reasonable result is unlikely to beobtained from a sample much thicker than 10 nm,especially when the sample contains heavy atoms.However, the following simulation calculation showsthat, under the kinematical-diffraction approximation,

Fig. 7. Image processing for the Bi-2212 superconductor. a: Experi-mental EM taken with the incident electron beam parallel to the aaxis. b: Corresponding ED pattern. c: 3-dimensional potential distribu-tion projected along the a axis on an area of 8b 3 1c. The structure-

factor magnitudes used for calculating c were from the ED, while thephases were from the direct-method phase extension based on thedeconvolution of a. The experimental EM is provided by Dr. S.Horiuchi.


the incommensurate modulation of Bi-2201 could bedetermined by using ED data from a sample as thick as30 nm. Details on the determination of the incommensu-rate modulation in Bi-2201 from experimental ED datahave been given in the section Incommensurate Modu-lation of Bi-2201. The resulting potential map is shownin Figure 8. Here we start from the resulting structuremodel, atoms that are expressed by their scattering

factor, positional, and thermal parameters, and theirmodulation parameters. Four different sets of dynami-cal diffraction intensities were calculated assumingrespectively the sample thickness equal to 1 3 0.541nm (the length of the a axis), 20 3 0.541 nm, 40 3 0.541nm, and 60 3 0.541 nm. Potential maps (shown in Fig.10) were then obtained by assigning the correct phasesto the square roots of dynamical diffraction intensities.

Fig. 8. The final Fourier map (three-dimensionalpotential distribution) of the Bi-2201 superconductorprojected down the a axis on to an area of 7b 3 3c.

Fig. 9. Direct-method phased Fou-rier maps of (PbS)1.18TiS2. Top left: er(x1,x2, x3, 0)dx2, the 3-dimensional hypersec-tion of the electron density function pro-jected down the b axis; Bottom left:r(x1, 0, x3 0); Upper-right: r(0.25, x2, x3,0.25); Lower-right: r(0, x2, x3, 0).

114 F. HAI-FU

Figure 10a, which can be considered as nearly the truestructure image, is close to the experimental map inFigure 8. The other three maps all have their essentialfeatures similar to that of Figure 10a. Furthermore,even some oxygen atoms are visible on the map corre-sponding to the sample thickness of 30 nm (Fig. 10d). Itturns out that, based on the kinematical-diffractionapproximation, the modulation of all metal atoms inBi-2201 could be determined by using ED data from asample as thick as 30 nm, provided we could find thecorrect phases by whatever method. The completestructure may then be solved by Fourier recycling.Besides, given a rough structure model or just a partialstructure, there are some techniques to compensate forthe dynamical diffraction effect (Huang et al., 1996; Shaet al., 1993). This would further improve the results.

The calculation, measurement, and display of imagesshown in this paper, except for those in Figures 1 and 7,were done by the program VEC (Wan et al., 1997)running on an IBM-compatible PC.

ACKNOWLEDGMENTSThe author thanks Z. H. Wan, Y. D. Liu, and Y. Z. He

for their assistance in preparing the manuscript. Thework was supported in part by the National NaturalScience Foundation of China.

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