Introduction to Electron CrystallographyWILLIAM F. TIVOL*Wadsworth Center, Albany, New York 12201-0509

KEY WORDS phases, direct methods, inverse scattering, unitary

ABSTRACT This introduction describes the contributions to this topical issue, and in somecases gives observations on the importance and implications of these articles. Subsequently, there isa discussion of the phase problem and its solution using the unitary equation. This equation, whichis true for all known scattering processes, provides a firm theoretical basis for the direct methods ofphasing. Microsc. Res. Tech. 46:69–74, 1999. r 1999 Wiley-Liss, Inc.

Many years ago when I was in graduate school, aprofessor pointed out that if one had a complete knowl-edge of the scattering states of a system, one coulddetermine its structure. My experience since then hasonly confirmed that statement, and the contributions tothis topical issue demonstrate that determination ofstructures from the scattering of electrons is possible inpractice as well as in theory.

In his article in this issue, J. M. Cowley (Cowley,1999) states ‘‘the range of significant [electron] scatter-ing is seen to extend at least to the range of 20 or 30nm21.’’Figure 1, a zone-axis pattern from copper perchlo-rophthalocyanine, shows diffraction spots to 20 nm21.This illustrates that such high-resolution informationcan, indeed, be obtained in favorable circumstances.

The ability to acquire this high-resolution informa-tion combined with the capability of forming a beamless than 1 nm in diameter using a strong objective lensto image a field-emission electron source implies thatstructures of nanometer-scale arrangements of atomscan be investigated at sub-Angstrom resolution. Cow-ley shows how a dedicated scanning transmission elec-tron microscope can be used to produce the informationrequired to determine these structures, discusses theeffects of the coherence of the incident beam in bothcomplicating the interpretation of the data and addingto the information that can be obtained, describescurrent instruments and procedures, and enumeratesexamples of what can be done in practice.

The examples include studies of small particles,which often have significant effects on the properties ofthe bulk materials within which they reside, localvariations in the structures of carbon nanotubes, disor-der and defects in thin films, including the difficulties ofinterpreting the data obtainable from partially disor-dered materials, and changes in structures duringchemical or physical processes. He concludes that thereis much to be done in the future to explore the applica-tions of nanodiffraction and to develop improved instru-ments and techniques.

D.L. Dorset shows how amplitude and phase informa-tion not directly observable due to experimental limita-tions—in particular, limitations that cannot generallybe overcome for membrane protein crystals—can befilled in using the Sayre equation. Not only can thisequation be used to extend phase data to the unob-served orientations, it can also predict the magnitudesfor the unmeasured reflections.

A model protein, rubredoxin, was analyzed, startingfrom the X-ray data, but eliminating those parts ofreciprocal space for which data could not be obtained inan electron microscope due to tilt limitations of thespecimen stage. Amplitudes and phases for the missingdata were calculated, and improvements in the recon-struction of the rubredoxin structure are shown.

The ability to fill in such missing information raisesthe question of what the minimum amount of datanecessary for structure determination really is. (Thisminimum will be a function of the accuracy of the data.)This unusual and creative use of a crystallographictechnique will, I’m sure, find wide application in struc-ture analysis.

H.-F. Fan describes how direct methods can be usedto combine the information in images with that fromdiffraction patterns to deconvolute the images andextend the information to the higher resolution avail-able in the diffracted intensities. The Sayre equation isused to refine the defocus value, and the contrasttransfer function can then be deconvoluted from theimage data. This process results in a ‘‘structure image.’’He then uses the phase information from the deconvo-luted image and the higher-resolution intensities froma diffraction pattern to determine the phases, to thelimit of the diffraction data, by means of the tangentformula.

Next, he applies the 4-dimensional Sayre equation tosolve the structures of crystals that are incommensu-rately modulated. The average lattice gives rise to themain peaks, and the occupational and/or positionalfluctuations—modulations—give rise to satellite peaks.When the periodicities of the modulations are incom-mensurate with those of the average lattice, the three-dimensional diffraction pattern is treated as a projec-tion of a four-dimensional reciprocal lattice. Heconcludes with a brief discussion of dynamical effects.

C. Gilmore describes the maximum-entropy methodfor finding phase information. This method is gainingin popularity both for direct phasing and for otherforms of data analysis. The basis of the method is that,

Contract grant sponsor: National Center for Research Resources, Departmentof Health and Human Services/Public Health Service; Contract grant number:RR01219.

*Correspondence to: Dr. William F. Tivol, Wadsworth Center, Empire StatePlaza, PO Box 509, Albany NY 12201-0509. E-mail: william.tivol@wadsworth.org.

MICROSCOPY RESEARCH AND TECHNIQUE 46:69–74 (1999)

r 1999 WILEY-LISS, INC.

within the constraints imposed by the data, the solutionshould be as unbiased as possible. This leads to theinclusion of only those features demanded by theexperimental measurements and the smoothing out offeatures that arise only from the process of obtaining aparticular fit to the data. A likely reason for the increasein the use of this technique is its suitability for theanalysis of data sets that have regions where experimen-tal limitations preclude measurements or introducelarge errors into the data.

From a small set of amplitudes—the basis set—forwhich the phases are known or can be assigned a priori,

and from another set of large amplitudes, for each ofwhich one of two values of the phases are specified,maps are constructed, the entropies of these maps arecalculated and maximized, and the several most likelyof these solutions are retained for recycling through theprocedure. The process continues until the structureemerges or the likelihood values calculated for thephase sets, as compared to that for a null hypothesis,start to decrease.

Examples where maximum entropy was used todetermine structures are discussed for (1) small organicmolecules, which can consist of many atoms, but for

Fig. 1. Composite of prints produced from one ED pattern of copperperchlorophthalocyanine. The pattern was obtained using an AEIEM7 high-voltage electron microscope at an incident beam energy of1.2 MeV. The prints were made using different exposure times andwere otherwise unmodified. The outermost region was produced from

a short exposure to highlight the faint, but clearly present, reflectionsthat extend to a resolution of 0.05 nm; the inner region was producedat an exposure that was appropriate to show the lower-order reflec-tions. Note that there are variations in intensity even at the highestresolution, which indicate that structural information is present.

70 W.F. TIVOL

which dynamical scattering is minimal since the atomsare light, (2) inorganic molecules, which have fewer butheavier atoms and can be somewhat disordered, (3)membrane proteins, where the basis set can come froma Fourier transform of the image, and (4) surfaces,where the data are incomplete and have large errors.

R. Holmestad et al., address the use of convergent-beam electron diffraction (CBED) in materials science.The instrumentation needed for this kind of workincludes an electron microscope with a small, conver-gent beam—best realized by having a field-emissionsource—a double-tilt cryo-stage, energy-filtering, and adigital recording device.

Measurements for which CBED is useful includecrystal thickness, lattice parameters, and Debye-Waller factors. These are reviewed briefly, and then theprimary advantages of CBED in the determinations ofcharge-density distributions and the use of dynamicaleffects to measure phases and absolute polarities areexplored. These kinds of measurements take structureinvestigations beyond the location of atoms in the unitcell and are necessary to understand many importantproperties of materials.

S. Hovmoller and X. Zou consider parameters affect-ing structure determination from high-resolution trans-mission electron microscopy (HRTEM) images of crys-tals, which include the thickness and orientation of thespecimen. Whereas it is obviously correct to align themicroscope precisely when obtaining the images, it isnot always possible to control the thickness of thespecimen. Furthermore, multiple scattering may bereduced if the specimen is tilted a few degrees awayfrom a zone axis. Thus, it can be optimal not to align thespecimen along a zone axis, but to measure thicknessand tilt and correct for their effects.

The theory for such measurements and correctionshas been worked out, and the authors show thatexperimental data confirm the theory and can be usedin practice to derive structural information fromHRTEM images including those for which a directinterpretation of the image in terms of atomic positionsis not possible. They show that in the latter case imageprocessing can be used both to measure the thicknessand tilt and to provide corrections so that an interpret-able image can be generated. Some limitations of thisapproach are also given.

The focus of the paper is on metal oxides, which arerelatively radiation resistant and mechanically strong.For such a specimen, there are techniques to produceareas having the desired range of thickness, and theorientation can be optimized by tilting the specimenwhile observing the image or diffraction pattern untilthe desired orientation is achieved, then observing anadjacent area for which the orientation and thicknessare the same. For radiolabile specimens and for thosethat deform readily—such as organic compounds—itmay not be possible to produce a particular thickness ororientation. Those parameters can vary rapidly enoughthat an adjacent area can have sufficiently differentvalues that optimization on one area and observation ofa nearby unirradiated area is not useful. In this case,measurement of thickness and tilt in the observed areaand correction for their effects is the only way to obtaingood structural information. For these specimens, the

techniques discussed by Hovmoller and Zou could proveespecially valuable.

Another unusual use of electron crystallography, theanalysis of the structures of surfaces, is dealt with inthe paper by C. Leslie et al. These structures are oftenquite different from those of bulk materials. Surfaceatoms have high energies due to their having fewerneighbors and, therefore, unsatisfied chemical bonds.This makes surfaces more likely to react, and manyheterogeneous catalysts owe their useful properties tothis feature. This increased reactivity causes surfacesto accumulate adsorbed material rapidly, so specialultra-high-vacuum techniques must be used.

In order to study the surface layer and the near-surface layers—which exhibit the transition from thebulk structure to that of the surface—techniques thatare sensitive both to atomic types and positions must beused. These include reflection diffraction—both low-energy, where the electrons are coherently backscat-tered, and high-energy, where the electrons are inci-dent at a small angle to the surface plane—photoelectronholography, where the coherent electrons are producedby absorbed photons, and transmission electron micros-copy and diffraction.

Analysis of the data is accomplished by maximumentropy techniques and by a technique of iteration inboth real and reciprocal spaces using a ‘‘sharpening’’operator. This operator enhances strong features anddamps out weak ones, so that the background will beminimized and the information from the true structurewill be strengthened. The relative entropy of the real-space potential is a suitable operator, and the techniqueis, thus, called minimum relative entropy. A geneticalgorithm is used to select optimized solutions. Severalexamples of successful surface structure determina-tions are presented.

I.G. Voigt-Martin describes the use of electron crystal-lography in conjunction with quantum-mechanical mod-eling to design and test materials having desirednon-linear optical properties. This again demonstratesthat electron crystallography goes beyond the mere (!)location of atoms within the unit cell.

Non-linear optical properties arise from polarizationinduced by an electric field, which, in turn, can becalculated from properties of both the crystal and itsconstituent molecules. Thus, one can design materialspredicted to have desirable optical properties usingquantum-mechanical modelling. After these materialshave been engineered, the actual structure and proper-ties can be measured and compared to what wasdesired.

The experiment proceeds by synthesizing the mol-ecule, measuring the optical properties, calculating thestructure, designing the appropriate crystal, checkingthe optical properties of that crystal, determining thecrystal structure using both X-ray and electron diffrac-tion (ED) as well as imaging, doing ab initio analysis ofthe diffraction data using maximum entropy and com-paring the images to simulations, then relating themolecular properties to those of the macroscopic mate-rial. This is demonstrated for three compounds wherethe calculated optical properties are compared with theexperimental measurements.

X. Zou discusses the relationships among the phasesof structure factors, exit waves, and images. She points

71INTRODUCTION

out that the term ‘‘phase’’ can have different meanings,so one must define it clearly whenever the term is used.The point is made that the structure factors are aproperty of the crystal, whereas the exit wave andimage are determined also by experimental param-eters. In order to derive the structure factor phasesfrom an experiment, one must know how the experimen-tal conditions shift the phases. Although this paper isconcerned with determining structures from images,the results are also applicable to diffraction. Since,given precise enough data, the phases of the scatteringamplitude can be determined (see below), knowledge ofthe relationship between these phases and those of thestructure factors is sufficient to allow the determina-tion of structures from diffraction experiments.

There is a brief theoretical discussion of the processesof the formation of an image of a thin crystal. Theinteractions of electrons with both the crystal and themicroscope are described, as are the effects of theseinteractions on the phase of the electron wave and theFourier transform of the image. The weak phase objectapproximation is generally used, but there is alsodiscussion of the case where this approximation is nolonger applicable. The method of determining the crys-tal structure factor phases from images of weak phaseobjects is given.

The theory is then compared with data from a realcrystal at various thicknesses. Images were obtainedand compared with multislice calculations, which incor-porate dynamical effects and are not limited to weakphase objects. Simulations of images at two defocusvalues were compared with the experimental images.Accurate atomic positions were derived from the experi-mental images by crystallographic image processing,both for the smallest thickness and for a larger thick-ness where multiple scattering effects are significant.The fact that the structure factor phases derived fromthe images were correct for the large-amplitude dif-fracted beams accounts for the ability to determinecorrect structures even in the presence of multiplescattering.

J.M. Zuo describes the use of energy-filtered CBEDfor accurate structure refinement. He points out that,whatever the difficulties inherent in deriving an initialstructural model from ED data, refinement can be doneaccurately, since dynamical theory is well developed.He lists several possibilities for ED that cannot beaccomplished by either X-ray or neutron diffraction.Due to the possibility of forming a sub-nanometerelectron beam, small regions of a specimen can bestudied, and—a major subject of the paper—ED is quitesensitive to the charge redistribution due to bonding.

He then outlines the Bloch-wave theory, uses it tocalculate the intensity at each point in a CBED disk,and derives the expression for temperature effects fromthe observation that each fast electron sees only aninstantaneous configuration of a crystal.

The equipment and experimental parameters foraccurate collection of CBED data are described in thenext section as are the effects of energy filtering andimage deconvolution on the accuracy of that data.

Refinement of a structure model to achieve the bestfit to the experiment is discussed. Possible goodness-of-fit (GOF) measures are described as are their effectson the refinement. The theoretical expression for the

intensity distribution within a CBED spot is presented.Possible structural parameters and estimation of theerrors in refined parameters are addressed. And refine-ment algorithms are listed, with the author’s beingbriefly described.

Examples are presented to illustrate points made inthe preceding sections. The refinements of two low-order silicon structure factors, one strong and oneweak, are performed using several GOF criteria, show-ing that the results depend on the criteria used andconcluding that there is no single best criterion for allcases. The charge density distribution in MgO is deter-mined and shown to have differences from theoreticalpredictions, which are ascribed to polarization by ther-mal lattice vibrations.

One of the longstanding beliefs in crystallography isthat when one measures the diffracted intensities, theinformation concerning the phases is lost. Although it istrue that the intensities are equal to the absolutesquares of the diffraction amplitudes, and that thephase of a particular amplitude cannot be determinedfrom the value of the corresponding intensity, it is nottrue that the phase information is therefore irretriev-able. Methods for recovering the phases from thetotality of the intensity data have been known foralmost 50 years (Hauptman and Karle, 1953), and workon the ‘‘phase problem’’—both recent and not-so-recent—has shown that within broad limitations the scatteringamplitude can be determined from the differentialcross-section (Gerber and Karplus, 1970, 1972, 1973;Martin, 1969; Mishnev, 1991; Mishnev and Belyakov,1992; Mishnev and Shvets, 1979; Newton, 1968).

Sayre (1952) published a method to determine phasesfrom intensities. This method was derived from theproperties of the Fourier transform, and treated thecase of identical, non-overlapping atoms. Subsequently,it was assumed that the conditions of positive densityeverywhere and of discrete, identical atoms were neces-sary for the applicability of the Sayre equation. Thisand other methods of direct phasing, it was assumed,could only be successfully used with molecules com-posed of point-like atoms, which were not too differentin atomic number, and would succeed, in general, onlywith additional chemical knowledge (Cochran, 1953).

Other methods of direct phasing were developedstarting from the premise that a correct assignment ofphases gives a better fit among the diffraction ampli-tudes than does assigning random phases. These meth-ods begin with known phases for a set of amplitudes,5H6, and another set of unphased amplitudes, 5K6. Proba-bilistic equations, constrained by the phases in 5H6,predict phases for 5K6. These are used to calculate alikelihood gain, which provides a figure of merit for thephase solution. Some phases may be assigned one of aset of values and for each such assignment a figure ofmerit is calculated. The best few of these sets areretained. The newly-phased amplitudes are then addedto 5H6 and a new set 5K6 is chosen and the process isiterated.

Electron microscopy offers an excellent way to obtainthe phases for the set 5H6; they can be readily calculatedfrom a Fourier transform of an image. Although thereare some complications introduced by the contrasttransfer function, phases from images have proved tobe quite useful, when combined with intensity data

72 W.F. TIVOL

from ED, and several of the direct methods can be usedto extend the phases to the maximum resolution of thediffraction data.

All the direct methods are consistent in that they willfind the same phase solution. That is not unexpected ifeach possible structure gives rise to a different set ofdiffracted intensities; however, enantiomorphs, for ex-ample, will give rise to the same set of intensities. Suchambiguities as those for enantiomorphs or other closelyrelated structures (i.e., homometric structures) are notbothersome, since they can be resolved by independentknowledge of the crystal whose structure is to bedetermined. If there are more serious ambiguities—substances with unrelated structures giving the samediffracted intensities—then phasing methods would beexpected to be inconsistent or to fail.

It should be noted that data of atomic resolution mustbe available in order to interpret a proposed structureusing chemical bonding criteria. If not, then it is notpossible to identify a promising phase set for refine-ment in a multisolution environment, since there are nosufficiently robust figures of merit available. This is dueto the impossibility to determine a priori if the arrange-ment of low-resolution densities in a map is correct(Dorset, 1997; Podjarny and Urzhumetsev, 1998). Thepreceeding must be modified somewhat for macromol-ecules, where the presence of secondary structures(e.g., a-helices or b-sheets) could allow the identifica-tion of a phase set at a resolution corresponding to themolecular envelope.

Physicists have been trying for a long time to solvethe inverse scattering problem; i.e., given a set ofscattering data, to determine the potential that pro-duced those data. Solution of this problem would obvi-ously imply that structures could be determined solelyfrom diffraction intensity data, although such a determi-nation might have ambiguities, such as those fromenantiomorphs.

About 30 years ago, there was a flurry of activityinvestigating one aspect of the problem—if one hascomplete knowledge of the differential cross-section,can the scattering amplitude be determined. A series ofpapers answered this question in the affirmative, withinbroad limits for the form of the differential cross-section(Gerber and Karplus, 1970, 1972; Martin, 1969; New-ton, 1968).

The determination of the scattering amplitude fromthe differential cross-section is the same for the continu-ous case as determining phases from diffracted intensi-ties is for the discrete case, and, in fact, the methodselaborated in the papers mentioned above were shownto apply to X-ray diffraction by Mishnev and co-workers(Mishnev, 1991; Mishnev and Belyakov, 1992; Mishnevand Shvets, 1979), who derived the Sayre equationfrom the unitary equation. In addition, Gerber andKarplus (1973) derived the tangent formula from theunitary equation for the case of X-ray diffraction. Thederivation of these methods of phase determinationfrom the fundamental physical principle of unitarityboth gives a theoretical foundation for direct phasingand extends its range of applicability (Tivol, 1995).

As long as the scattering is unitary, the phases can bedetermined—no assumptions of everywhere-positivedensity or of point-like atoms are necessary—and thephase problem becomes a problem of collecting suffi-

cient high-quality data to solve the equations. Theconsistency of direct methods is a logical consequence ofthis foundation.

The unitary equation is a statement of the conserva-tion of probability and applies to all known scatteringprocesses. What is not yet fully appreciated is that theunitary equation also constrains the phase (and themodulus) of the scattering amplitude. There are twoambiguities relating to the unitary equation and phases:one can add 2pn to any phase, and the phase for thetime-reversed scattering process (w8 5 p 2 w) will alsosatisfy the unitary equation. The first of these ambigu-ities is trivial, and the second leads to two possible setsof phases (n.b., it is not the case that each phase canindependently have either of two values).

Other than these ambiguities the unitary equationwas shown (Gerber and Karplus, 1970) to convergeuniformly to a phase solution. This powerful resultmeans that if one starts with any set of phases anditerates the equation (Martin, 1969)

sin [wn11(k)] 51

4pe 0Fk80 0Fk90

0Fk0cos [wn(k8) 2 wn(k9)] dVk9,

where 0Fk 0 and 0Fk8 0 are the moduli of the scatteringamplitudes in the directions of the momentum vectors kand k8, 0Fk9 0 is the modulus of the scattering amplitudefrom the direction of k8 to that of k and the correspond-ing w’s are the phases, the sequence 5wn(k)6 will convergeto one of two solutions, for each of which sin [wn(k)] willhave the same value. For the discrete case, the integralis replaced by a sum and the factor of 4p (5e dV)disappears.

The fact that only two related phase sets are consis-tent with the intensities and that there is a path ofconvergence to one of these sets from any startingvalues provides a theoretical basis for probabilisticmulti-solution methods that start from a random distri-bution of the atoms in the unit cell. This does not,however, give any direction as to how to recognize thatone is converging on the correct phase set.

Once the phased diffraction amplitudes are in hand,the only remaining problem for structure determina-tion is relating these amplitudes to the structure fac-tors. For the case of kinematical scattering, the relation-ship is trivial, so if dynamical and secondary scatteringare small enough to be neglected, the structure can besolved (Dorset, 1995). Even in the presence of multiplescattering, if the diffraction amplitudes are sufficientlyclosely related to the structure factors to provide areasonable starting structure, refinement using a modelthat includes the effects of multiple scattering can beused to determine the structure (Jansen et al., 1998).

Although it would be preferable to arrange the experi-mental conditions so that multiple scattering is negli-gible, this cannot always be done in practice. In order toobtain sufficiently accurate data, one must use a certainnumber of electrons. Since these will cause some dam-age to the crystal, one must have enough unit cells sothat the dose to any part of the crystal will be limited.These must be appropriately well ordered, so thecrystal has to have sufficient mechanical rigidity. Thus,some particular minimum thickness—determined by

73INTRODUCTION

both scattering and mechanical properties of the sub-stance—will be necessary. If this minimum thickness issufficiently small, as it will be for a relatively simplestructure that forms rigid crystals, one can obtain EDdata that can be treated kinematically, but for sub-stances whose structures are more complicated—requiring more reflections for their determination—and/or that form crystals that can easily be distorted,the minimum required thickness can be so large thatmultiple scattering cannot be neglected.

Multiple scattering can be minimized by the use ofhigh-voltage electrons. ED intensity data obtained withan incident electron energy of 1.2 MeV were success-fully phased and the structures of copper perchloroph-thalocyanine (Dorset, et al., 1991) and copper perbro-mophthalocyanine (Dorset, et al., 1992) weredetermined. It was also noted that ED data obtained at0.4 MeV for copper perchlorophthalocyanine could notbe phased due to the effects of dynamical scattering(Tivol et al., 1993). In fact, data for copper perbromoph-thalocyanine could not be phased for any voltage below1.2 MV, and even at 1.2 MV only three atoms could belocated on the initial map; however, this minimalsuccess was sufficient for the phasing process to pro-ceed. The final step in the refinement process requiredexplicit inclusion of dynamical effects.

In light of these results, a study of the influence ofvoltage on the success of structure determination forcopper perchlorophthalocyanine was undertaken (Tivolet al., 1993). It was shown that dynamical scatteringaccounted for the major part of the differences amongdata sets in the voltage range from 100 kV to 1.2 MV.Furthermore, the effect of the curvature the Ewaldsphere on the resolution of the data was demonstrated.

Another difficulty arises when a substance formscrystals that are not well ordered. In these cases, acrystal can consist of many regions that are eithermisoriented or displaced from one another. Each ofthese regions will diffract coherently within itself, butwill not be appropriately located with respect to otherregions, to give coherence between regions. The scatter-ing from the crystal will consist of within-region scatter-ing for which the amplitudes add and contributionsfrom many regions for which the intensities add. Forsufficient thickness, this can lead to both dynamicaland secondary scattering, and the derivation of struc-tural information from ED data will be very compli-cated.

Not too long ago it was widely believed that suchcomplications made determination of structures fromED impossible. However, as the papers in this specialtopical issue demonstrate, structure determination fromED is not only possible, but is achievable in practice andis almost routine for the more ideal cases. For thedifficult cases, there are still challenges to be overcome,but ED can be advantageous for nanoscale regions,crystals with periodic disorder, situations where onlyincomplete data sets are obtainable, surfaces, andsurely other cases where no other technique is useful.Furthermore, the use of ED to determine charge distri-

butions and to predict optical properties of materialsshows that the range of usefulness of this techniquegoes well beyond determination of atomic positions, andI expect that the range of problems to which electronbeam methods will be successfully applied will expandgreatly in the coming years.

ACKNOWLEDGMENTSThis work was supported by Biotechnological Re-

source grant RR01219, awarded by the National Centerfor Research Resources, Department of Health andHuman Services/Public Health Service, to support theWadsworth Center’s Biological Microscopy and ImageReconstruction Facility as a National BiotechnologicalResource.

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