Electron nanodiffraction

$Page 1: Electron nanodiffraction$
Electron NanodiffractionJ.M. COWLEY*Department of Physics and Astronomy, Arizona State University, Tempe, Arizona 85287-1504

KEY WORDS electron nanodiffraction; STEM imaging; coherence effects; nanoparticles; thinfilms

ABSTRACT Electron nanodiffraction, particularly as performed in a dedicated STEM instru-ment with coherent illumination, provides, in principle, a means for obtaining information onstructural detail in the range between that of STEM image resolution, about 0.2 nm, and the limitsof information possible from elastic scattering from atoms, about 0.03 nm. In practice, mostnanodiffraction work has been concerned with finding the crystal structure, crystal defects, andsometimes crystal shapes for nanoparticles in the size range of 1–2 nm or for regions of thincrystalline films of about these dimensions. Available equipment allows for the recording ofnanodiffraction patterns at the rate of 30 per second, or faster, providing means for detailed study ofextended areas or of dynamical processes. Microsc. Res. Tech. 46:75–97, 1999. r 1999 Wiley-Liss, Inc.

INTRODUCTIONHigh-resolution transmission electron microscopy

(HRTEM) offers the possibility of determining thearrangements of atoms in solids with a resolution of0.15 to 0.2 nm for medium-energy electrons (100 to 400keV), with the possibility of extending this range toapproach 0.1 nm at the expense of using million-voltinstruments (Dehm et al., 1996) or by special tech-niques of electron holography (Van Dyck et al., 1996).However, the information obtainable by such imagingmethods falls short of that derivable in principle fromthe elastic scattering of electrons by solids.

As suggested by Figure 1, the atomic scatteringfactors for electrons extend out to 20 or 30 nm21 inreciprocal space before they are reduced to less than10% of their values for the small-angle scattering, inthe range of less than 10 nm21, which is the rangeutilized in HRTEM. It may be considered that, becausethe contribution to the image intensity comes from alldirections of scattering, one should consider not thevalue of the atomic scattering factor, f(u), where u 52(sinu)/l, and u is half the scattering angle, but theintegration around the circumference of the scatteringdiagram, 2puf(u), as a better indication of the impor-tance of the electrons scattered to high angles. Thisfunction (divided by p) is also plotted in Figure 1. Theapplication of the Debye-Waller factor, arising from thethermal vibration of the atoms, reduces the value ofthis function for high scattering angles, as shown, but,even so, the range of significant scattering is seen toextend at least to the range of 20 to 30 nm21.

The information available from HRTEM correspondsto a direct projection of the structure in the beamdirection only under the very restricted conditions forwhich the weak-phase-object approximation is appli-cable, namely for very thin objects consisting of lightatoms. For thicknesses of a few nm, especially formedium-weight or heavy atoms, dynamical diffractioneffects give serious non-linearities of the dependence ofimage contrast on scattering power of the atoms pre-sent. A recent publication (Malm and O’Keefe, 1997),for example, reports detailed simulations of images of

very small crystallites, which reveal that, for manyorientations of the crystals, the images appear to giveincorrect indications of faulting and incorrect values forthe inter-planar spacings. The images may be calcu-lated using the standard computer programs for mul-tiple-beam dynamical diffraction and electron-opticalimaging, but the image interpretation is then not directand unambiguous but involves fitting of the observa-tions with calculations from model structures.

The alternative to an imaging process is the collec-tion of data concerning the specimen structure inreciprocal space, i.e., from diffraction patterns. Thestandard method of selected-area electron diffraction(SAED) as performed in a transmission electron micro-scope for very thin specimens gives the intensities ofdiffraction features for the whole available range ofscattering angles out to 20–30 nm21 and beyond. In thecase of thick specimens, multiple-scattering effects areprominent and can affect the angular range of scatter-ing. Especially in the presence of inelastic scattering,the range of scattering can extend to much higherangles, but these cases will not be considered in thecontext of this review.

The SAED patterns, even under ideal conditions ofweak-phase objects or kinematical scattering, necessar-ily represent information concerning structures aver-aged over a relatively large specimen area, usually 100nm or more in diameter. The periodicities of crystallineregions and the statistics of interatomic distances maybe derived. Considerable progress has been made inusing such patterns for crystal structure analysis (Dor-set, 1995). The possibility of deriving information con-cerning individual, local configurations of atoms isremote. For such a purpose, it is necessary to obtaindiffraction data from very small areas of the specimen,not much bigger than the local arrangements of atomsto be investigated.

Received 20 October 1997; accepted in revised form 17 April 1998.*Correspondence to: J.M. Cowley, Arizona State University, Box 871504,

Department of Physics and Astronomy, Tempe, AZ 85287-1504. E-mail:[email protected]

MICROSCOPY RESEARCH AND TECHNIQUE 46:75–97 (1999)

r 1999 WILEY-LISS, INC.

Fortunately, the possibility of obtaining diffractionpatterns from such regions, as small as 1 nm or less indiameter, exists because the strong objective lensesused to form high-resolution TEM images may be usedalso to form electron beams of very small diameter. Thediameters of such electron probes may be comparable tothe HRTEM resolution limit. The use of electron beamsof such diameter provides the basis for scanning trans-mission electron microscopy (STEM) instruments inwhich the probe formed by the objective lens, demagni-fying a very bright small source such as that producedby a field-emission gun, is scanned over the specimen.The reciprocity relationship (Cowley, 1969) suggeststhat STEM and TEM resolutions and image contrastsare equivalent (see Fig. 2).

When the incident beam scan in a STEM instrumentis stopped so that the beam illuminates a small regionof the specimen, which can be chosen from a STEMimage, the diffraction pattern of the chosen regionappears on a plane of observation distant from thespecimen. Since the chosen area may be of a diameterconsiderably less than 1 nm, the technique for observ-ing and recording such diffraction patterns has beennamed ‘‘nanodiffraction.’’ This term emphasizes thedistinction from ‘‘microdiffraction,’’ which has beenused rather loosely to refer to diffraction from verysmall regions of poorly defined dimensions, usuallygreater than about 10 nm. Observation of nanodiffrac-tion patterns is usually made by means of a fluorescentscreen. Recording of the patterns may be made byimaging the fluorescent screen by means of a light-optical system, outside the vacuum of the microscopeand using a low-light-level television (TV) camera plusa video-cassette recorder (VCR), or else by use of acharge-coupled-device (CCD) camera and associateddigital recording system, as suggest in Figure 3.

The interpretation of electron-diffraction pattern in-tensities is well known to present considerable difficul-

ties because of the limitations of the single-scattering,kinematical approximation (Dorset, 1995) related tothe coherent multiple scattering, or dynamical scatter-ing, which becomes important for all but very thin,light-atom specimens. The kinematical approximationbecomes equivalent to the weak-phase object approxima-tion used for image-contrast theory in the small-angleapproximation for very thin specimens, so that thelimitations due to the scattering strength of atoms arecomparable for TEM, STEM, and diffraction patterninterpretation.An essential aspect of the imaging modes,however, is that the images are formed by coherentinteraction of the transmitted and scattered radiationso that both the amplitude and phase of the scatteredradiation contribute to the observations. By contrast, indiffraction patterns, such as those obtained in theSAED mode, it is only the scattered beam intensitiesthat are recorded and the phase information is lost sothat, even though the information extends further outin reciprocal space than for imaging, the interpretationof the information is limited, even for kinematicalscattering, by the loss of this phase information. This isthe well-known ‘‘Phase Problem’’ of X-ray crystallogra-phy.

In nanodiffraction, however, the possibility exists forobserving interference effects between the incident andscattered radiation and hence for extracting informa-tion concerning the relative phases of the scattering sothat the phase problem no longer exists. This raises thepossibility for complete unambiguous structure analy-sis of small specimen areas to a level of structural detaillimited only by the observable range of electron scatter-ing, i.e., to 20–30 nm21 in reciprocal space or toresolutions of better than 0.05 nm in real space.

The preferred method for taking advantage of thispossibility is to use a high-resolution STEM instrumentto scan a very fine incident beam over a thin specimen,preferably with a digital scan, and to record the nanodif-fraction pattern for each beam position, with an overlapof adjacent illuminated areas. Then the bright-field ordark-field STEM image, recorded by detecting eitherthe paraxial or the integrated high-angle scattering,may be used to give an image with a resolution of theorder of 0.2 nm, and the information from the nanodif-fraction patterns may be used to extend the resolutionlimit to 0.05 nm or better.

The reciprocity relationship (Fig. 2) suggests that anequivalent procedure, based on a HRTEM instrument,would be to record HRTEM images of a specimen for allangles of incidence within a cone of half angle equal tothe range of 20–30 nm21 in reciprocal space, andcorrelate the intensity distributions in all the recordedimages.

The scheme for recording nanodiffraction patternsfor all image points of a high-resolution STEM imagehas been analyzed and attempted in two cases usingdifferent bases for the analysis of the data. In the caseof Konnert et al. (1989), the beam of diameter about 0.3nm in an HB-5 STEM microscope was scanned over asmall area, 1.7 3 1.3 nm, of a thin silicon crystal in[110] orientation and a total of 714 nanodiffractionpatterns were recorded using a TV-VCR system. Ex-amples of the recorded patterns, compared with pat-terns computed for the known structure of Si, areshown in Figure 4. Fourier transforms of the intensity

Fig. 1. The atomic scattering factors, f(u), and the functions 2uf(u),for Au and Si plotted against the angular variable u, in Å21. For Au,the latter plot multiplied by the Debye-Waller factor, e2M, is also given.

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distributions of the nanodiffraction patterns outsidethe central-beam disk gave auto-correlation functionsshowing interatomic vector maps. In particular, a maxi-mum of an interatomic vector peak occurs when thecenter of the incident beam coincides with the center ofthe interatomic vector. Correlation of the maps foradjacent probe positions can then lead to a mapping ofthe atomic positions in the [110] projection of the siliconstructure. This mapping showed a very clear resolutionof the atom peaks separated by 1.36 Å in the projection.

The experiments of Rodenburg et al. (1994) werebased on a more formal analysis of the scatteringprocess in terms of a four-dimensional function (Nellistand Rodenburg, 1994) involving the two dimensions ofthe STEM image space and the two dimensions of thenanodiffraction space, including the central beam disk.It was shown, in particular, that a two-dimensionalsection of this space gives directly an image with aresolution better than that of the STEM instrument bya factor of two.

In the nanodiffraction pattern for a weak phaseobject, two regions may be distinguished. Outside of thecentral beam disk, the diffraction pattern intensitiesare given by the square of the Fourier transform of thescattering function and may be interpreted in terms ofthe auto-correlation function, as was done by Konnertet al. (1989). Within the central disk of the pattern,given by the incident beam, there is interference be-tween the incident and diffracted beam amplitudes.This region is then an in-line hologram of the specimenstructure. As suggested by Gabor (1949), it is possible tomake a reconstruction from this hologram to producean image of the specimen without the perturbation dueto the spherical aberration of the objective lens and,hence, with greatly improved resolution. This processcan be carried out by digital recording of the hologramand reconstruction of the image in a computer (Cowleyand Walker, 1981; Lin and Cowley, 1986a). The maindifficulty by this process is that the reconstructed,high-resolution image is accompanied by a defocused,aberrated ‘‘conjugate image.’’ Removal of this conjugateimage represents a fundamental problem and severalmeans for its removal have been suggested (Lin andCowley, 1986a; Cowley and Gribelyuk, 1994). It is notedthat one such method involves the correlation of holo-

grams obtained from series of overlapping regions,obtained as the incident beam is scanned over thespecimen.

It is suggested that an extension of these methodsmay be possible involving the use of the much finerelectron probes that may be produced by ‘‘atomic focus-ers’’ (Cowley et al., 1997b). It has been pointed out thatsingle heavy atoms, or columns of atoms extendingthrough a thin crystal, may act as atomic focusers toform beams having diameters of less than 0.05 nmwhen illuminated by a plane wave or by the focusedprobe of a normal high-resolution STEM instrument(Fig. 5). There are several TEM and STEM modes thathave been proposed whereby such atomic focusers maybe incorporated in TEM or STEM instruments toproduce images having a resolution of better than 0.05nm (Cowley et al., 1997b, 1998; Sanchez and Cowley,1998). For atomic focusers, the nanodiffraction patternhas an incident beam spot that covers the entireangular range of scattering by an atom with no clearlydefined boundary and, hence, the whole pattern is anin-line hologram. Since the images formed have theresolution of better than 0.05 nm, corresponding to thecoherent imaging of all parts of the pattern out to 20 or30 nm21, the observation and interpretation of thenanodiffraction patterns may not be needed, except forsome special purposes.

Until such time as atomic focuser systems are readilyand conveniently available, however, and even beyondthat time, there will be many cases for which the use ofnanodiffraction with beam diameters of the order of 1nm represents the most rapid and convenient way ofderiving the required information on particular types ofsamples. We, therefore, devote the remainder of thisreview to the practical methods of nanodiffraction withcurrently available instruments and examples of thetypes of results that have been obtained.

COHERENCE AND INCOHERENCEIN NANODIFFRACTION

The electron-beam geometry used for nanodiffractionis essentially the same as for convergent-beam electrondiffraction (CBED), which is practiced widely withTEM instruments. The assumption is usually made inCBED that the formation of the diffraction pattern is

Fig. 2. Diagram suggesting the reciprocity relationship for the essential elements (objective lens only)for STEM (electrons going from left to right) and conventional TEM (electron direction reversed).

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incoherent, i.e., that the intensity in the diffractionpattern can be calculated on the basis that the radia-tion on the specimen coming from any one direction isunrelated in phase to that from any other direction, sothat the diffraction-pattern intensity distribution is thesum of the intensity distributions for all incident beamdirections. This contrasts with the assumption made inthe above discussion of nanodiffraction in relation toimaging that the incident beam on the specimen iscompletely coherent so that the complex amplitudes ofscattering from all directions of incidence are added. Inpractice, the coherence of the radiation used for theformation of nanodiffraction patterns is sometimessignificant and sometimes not, so that the question ofcoherence needs to be discussed in some detail.

The cone of radiation incident on the specimen,formed by the objective lens of a STEM instrument, orthe condenser lens of a TEM/STEM instrument, islimited by an aperture. If that aperture is illuminatedcoherently by the electron source, it may be assumed

that the radiation on the specimen from all directions iscoherent. The coherence of the radiation at the apertureposition is determined by the dimensions and distanceof the electron source, which may be regarded as ideallyincoherent in that all atom-sized regions of the sourcemay be considered to emit electrons independently (seeFig. 6).

According to the Zernike-Van Cittart Theorem, formonochromatic electrons, the coherence function, defin-ing the degree of coherence between the wave functionsat any two points in a plane distant from the source, isgiven by the Fourier transform of the source intensityfunction. Assuming the source to be a circular disk ofuniform intensity, which subtends an angle b at theaperture position, the coherence function at the posi-tion of the aperture has the form J1(pbr/l)/(pbr/l)where J1 is the first-order Bessel function and r is theradial coordinate in the plane of observation. Thus thecoherence function is unity for r 5 0 and decreases tozero for r 5 1.2 l/b. This value of r may be called the

Fig. 3. Diagram of a dedicated STEM instrument with one (or more) post-specimen lenses and adetector system arranged for observation of nanodiffraction patterns and a thin annular detector (lowerpart of figure).

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‘‘coherence width’’ and provides a convenient singlenumerical measure of the distance between two pointsfor which the incident radiation is coherent. If thecoherence width is much greater than the aperturediameter, the radiation on the specimen may be consid-ered to be coherent. If it is much smaller, the radiationon the specimen is assumed to be incoherent. For athermal-emission source such as a LaB6 point, com-monly used in TEM instruments, the effective diametermay be around 1 µm, so that, for a distance to theaperture of 10 cm, b is approximately 1025 radians andthe coherence width at the aperture is of the order of 0.3µm, which is much smaller than the normal aperturesizes of approximately 50 µm. For a cold field-emissiongun, the effective source size is around 4 nm, so that thecoherence width at the aperture is about 250 timeslarger and is greater than the usual aperture size.Hence, the convergent illumination on the specimenmay be assumed to be coherent when a field-emissiongun is used but incoherent for a thermal emissionsource.

An alternative way of approaching this matter ofcoherence is to consider that, for a source of diameter 4nm at a distance of 10 cm, the spread of the probeformed by an objective lens of focal length 1 mm due tothe source size is 0.04 nm, corresponding to an angularwidth of 20–30 nm21 in the diffraction pattern. Thus,

the scattering of electrons may be considered as coher-ent within this angular range.

For current electron sources, the energy spread of theelectron beam is usually so small that the chromatic, orlongitudinal, coherence of the illumination of the speci-men is rarely a factor in the consideration of interfer-ence effects in the specimen. The spread in the values ofthe objective lens focal length, which can limit theeffective angular range of diffracted beams contributingto the image, is a somewhat different form of incoherentimaging factor, limiting the resolution of bright-fieldTEM and STEM images but not affecting the informa-tion content of the diffraction patterns appreciably.Other incoherent resolution-limiting factors, such asmechanical vibrations and electrical or magnetic inter-ference in the microscope also have less influence on thedetails of the diffraction pattern than on the imageresolution.

For a STEM or TEM/STEM instrument with a field-emission gun, therefore, the coherence of the incidentbeam must be taken into account. This coherence haslittle effect in some special cases, as when the specimenis a thin, perfect crystal, but in many cases it introducescomplications for the interpretation of the diffractionintensities but may add considerably to the informationobtainable from the interpretation of the patterns. Its

Fig. 4. Comparison of calculated (a) and observed (b) nanodiffraction patterns for an incident beam ofdiameter 0.3 nm at one position within the unit cell for a thin Si crystal in [110] orientation (Konnert et al.,1989).


implication must, therefore, be considered in somedetail.

For a two-dimensional perfectly periodic object, whichcan be approximated by a very thin perfect crystal, theradiation from each direction of incidence gives adiffraction pattern consisting of a set of regularlyspaced delta-functions. For different directions of inci-dence, the delta-functions occur at different positions.For an incident beam cone limited by an objective-lensaperture, the diffraction pattern consists of a central

beam disk, an image of the objective aperture, and onesimilar disk for each diffracted beam. If the objectiveaperture is so small that the diffracted beam disks donot overlap, as suggested in Figure 7a, then no point inthe diffraction pattern can receive radiation from twodifferent incident beam directions. Hence, the diffrac-tion pattern intensity is independent of whether theincident radiation from different directions is coherentor not. The diffraction patterns can be interpreted onthe assumption of complete incoherence, as for normal

Fig. 5. Diagram of an ‘‘atomic focuser’’ system in which a single heavy atom, or a row of atoms within acrystal, is used to demagnify a STEM beam to give a cross-over of diameter 0.05 nm or less (Cowley et al.,1997b).

Fig. 6. Diagram for the estimation of the coherence width of radiation at the position of the objectiveaperture of a STEM instrument.

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CBED patterns obtained in a conventional TEM instru-ment.

If, for the perfect crystal specimen, adjacent diffrac-tion-beam disks overlap, as suggested in Figure 7b,radiation diffracted in one direction for one incidentbeam direction can coincide with the radiation dif-fracted in another direction from a different incidentbeam direction, and interference effects can be pro-duced if the radiation in the two incident beam direc-tions is coherent. Thus, the regions of overlap of adja-cent diffraction disks are crossed by fringe patterns.This has been demonstrated very effectively in experi-ments made in TEM instruments fitted with field-emission guns (Tanaka et al., 1994; Vine et al., 1992).The theoretical derivation of the form of these fringes isrelatively simple for the one-dimensional case of adja-cent reflections with indices h and h 1 1. The ampli-tudes and phases of the diffracted beams from thecrystal are given by the complex amplitudes, Fh, theFourier coefficients for the Fourier series representingthe exit wave from the crystal, equal to the kinematical‘‘structure factors’’ in the case of a weak phase object.The phases of these diffracted beams are modified bythe transfer function of the objective lens, which re-flects the influence of the aberrations and defocus of thelens. For a defocus D and a third-order sphericalaberration coefficient CS, which are the only factorsusually considered in electron microscopy, the transferfunction multiplies the diffracted waves by exp 5ix(u)6,where x(u) 5 pDlu2 1 (p/2)CSl3u4. For the two beamscoming together at a distance e from the center of theregion of overlap of the disks, the u-values are h/a 1 eand (h 1 1)/a 2 e. The intensity at that point is

I(e) 5 0Fh exp 5ix(h/a 1 e)6

1 Fh11 exp 5ix((h 1 1)/a 2 e)6 02

and, in the limiting case of large defocus and small e,this becomes

I(e) 5 0Fh 02 1 0Fh11 0

2 1 2 0Fh 0

· 0Fh11 0 cos 5[2pDl(2h 1 1)(1/2a2 1 e/a)] 1 a6,(1)

where a is the difference in phase between the h 1 1and h reflections. Thus there are regular fringes havinga periodicity which decreases with defocus and a phase,which depends on the relative phases of the two reflec-tions.

On the basis of the equation (1), it has been proposedthat, by considering the phases of the cosine intensitydistributions in all regions of overlap of adjacent diffrac-tion disks, the relative phases of all diffracted beamsfrom a crystal could be determined. In this way, the‘‘phase problem’’ of kinematical diffraction could besolved and a complete, unambiguous structure analysisof the crystal could be made. However, since the phasesof the structure factors depend on the position in theunit cell chosen as origin, it is the relative phases of theoverlap cosine fringes as they move with the incidentbeam position that must be observed. Also, the restric-tion to the weak-phase object approximation, whichimplies the use of very thin crystals of light atoms, hasprevented any practical application of this method to

real specimens. However, recent methods proposed forthe reconstruction of the potential distribution from theamplitudes and phases of the exit waves from quitethick crystals (Spence, 1997) may change this situation.

For the case that a large defect of focus does notdominate the phase factor in (1), the presence of thespherical-aberration terms of x(u) can give strong distor-tions of the fringes in the overlap regions, as seen inFigure 8a, obtained from a thin MgO crystal close to thein-focus position. These fringes are similar to thedistorted ‘‘electron Ronchi fringes’’ observed with aSTEM instrument when no objective aperture is pres-ent so that there is a very wide range of incident beamdirections (Cowley and Disko, 1980; Cowley, 1981a), asin Figure 8b,c. Such fringes are analogous to the Ronchifringes of light optics, (Ronchi, 1964) used for testinglarge telescope mirrors for their aberrations. It can beshown that the observation of electron Ronchi fringes isa very effective way for measuring the astigmatism,spherical aberration, and defocus of the objective lensesused for STEM (Lin and Cowley, 1986b).

When a large objective-lens aperture is used in aSTEM system to illuminate a thin-crystal specimen,the diffraction-spot disks may become so large thatmany diffraction disks may overlap at any point of thepattern. Then, given sufficient coherence, there may beinterference effects for large numbers of incident beamdirections. The intensity distribution of the diffractionpattern becomes very complicated and there may nolonger be any apparent separation into diffraction-spotdisks. This corresponds to the situation that the diam-eter of the incident beam probe is smaller than thedimensions of the unit cell of the projected crystalstructure so that no indication is given of the presenceof periodicity in the specimen. The intensity distribu-tion in the pattern depends on the relative positions ofthe atoms in the small part of the unit cell that isilluminated by the incident beam. The symmetry of thepattern depends on the symmetry of the arrangementof atoms relative to the center of the incident beamprobe (Cowley and Ou, 1989). The intensity distribu-tion of the pattern varies strongly with movements ofthe probe by distances of 0.1 nm or less, as differentgroups of atoms are illuminated (Cowley, 1981b). Thisis the situation appropriate for the schemes discussedin the Introduction, above, for the achievement of theultimate resolution by combining information fromnanodiffraction and imaging modes.

One practical application of this approach was to thestudy of the form of the planar defects in diamond thathave been studied by X-ray diffraction and electronmicroscopy for many years with no clear indication oftheir structure (Cowley et al., 1984). A typical defect ina thin part of a diamond crystal was imaged in STEMand then the incident beam, of diameter about 0.3 nm,was scanned along a line passing over the defect andnanodiffraction patterns were recorded by means of aTV system with VCR at a rate of 30 per second so thatone pattern was recorded for each 0.02-nm movement ofthe beam. The observed patterns were then comparedwith patterns computed using many-beam dynamicaldiffraction programs based on the various models thathad been proposed for the defect. Moderate agreementwas found for the model that had been proposed byHumble (1982) and that, unlike the other proposed


Fig. 7. Diagram suggesting the formation of a convergent-beamelectron diffraction pattern with (a) no overlap and (b) overlap andinterference of neighboring diffracted-beam disks.

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models, was not based on the assumption that thedefect contained nitrogen impurity atoms.

As stated above, for a perfect thin crystal there is nodifference between the nanodiffraction patterns forcoherent and incoherent radiation, provided that thediffraction disks in the nanodiffraction pattern do notoverlap. If there are any interruptions of the periodicity,for example at crystal edges, grain boundaries, orfaults, or if the crystal is imperfect, with local defects orimpurities, the plane-wave diffraction pattern of thecrystal no longer consists of a set of delta-functionspots. There is a continuous distribution of scattering togive diffuse lines or spots or a continuous background(Cowley, 1995). Then with a convergent, coherent inci-dent beam, interference is possible between beamscoming from different directions but scattered into thesame direction. These interference effects may give riseto splitting or distortion of the diffraction disks in a waythat is characteristic of the nature of the crystal defectspresent and will be described later in relation to thevarious applications of the nanodiffraction technique.

CURRENT PRACTICEFigure 3 illustrates a representative dedicated STEM

instrument configured for the recording of nanodiffrac-tion patterns as well as for the usual forms of bright-field, dark-field, and energy-loss imaging (Cowley, 1993).The cold field-emission gun, operating at 100 keV in thevacuum of better than 10210 torr that is required for thestability of its emission, provides an electron source ofeffective diameter about 4 nm. One or more condenserlenses are used to adjust the intensity and size of thefine electron probe formed by the objective lens at thespecimen level. The specimen may be surrounded bydetectors to collect the low-energy electrons or charac-teristic X rays emitted by the specimen for secondary-electron microscopy (SEM) or for X-ray energy-disper-sive spectroscopy (EDS) or element-distributionmapping.

Two or more post-specimen lenses (one of which maybe formed by the post-specimen magnetic field of theobjective lens) allow a flexible manipulation of the

electrons that have passed through the specimen sothat the size of the diffraction pattern on the observa-tion screen or the effective acceptance angle of theEELS spectrometer may be varied readily. A phosphoror fluorescent screen turns the diffraction pattern into adistribution of visible light, which is then observed witha wide-aperture optical lens placed outside the vacuumvessel and a low-light-level TV camera equipped with avideo-cassette recorder. In this way, a series of nanodif-fraction patterns may be recorded at the rate of 30 persecond while the beam is scanned along a line in theSTEM image, as in Cowley et al. (1984), or over aselected area. Patterns so recorded are not obviouslylimited by noise due to the finite number of contributingelectrons.

Alternatively, for digital recording, the light may bedirected to a CCD camera with an associated digitalrecording system. The more common CCD systems,with an integration time of the order of one second perframe, have the advantages of higher sensitivity to lowlight levels and more quantitative recording, but CCDsystems having a much shorter frame time are avail-able to give faster recording at the expense of morenoisy patterns.

In the original method for recording diffraction pat-terns, provided in the commercial instruments from VGMicroscopes, Ltd. U.K. the post-specimen deflector coilsare used to scan the diffraction pattern over the en-trance aperture to the electron energy-loss spectrom-eter, while the diffraction pattern is displayed on thesame cathode-ray tube as the scanned images. Thismethod is very inefficient in its use of the availableelectrons in that only a very small proportion of thediffraction pattern is detected at any one time, but itdoes have the advantages that a quantitative measureof the electron intensities may be made and also theelectron energies may be filtered. Thus, the diffractionpattern recorded may show only the elastically scat-tered electrons or else only those electrons that havesuffered a particular energy loss.

Other special-purpose recording systems may beused alternatively, or else in parallel, with the observa-

Fig. 8. a: Nanodiffraction pattern from a thin MgO crystal showing interference fringes for the 200reflections (d 5 2.04 Å) distorted by spherical aberration. b,c: Electron Ronchi fringes for MgO, in thesame orientation, for two different values of the defocus.


tion of the diffraction pattern on the fluorescent screen.For example, annular detectors having various accep-tance angles, variable by use of the post-specimenlenses, may record signals from electrons scattered tohigher angles than those observed on the main fluores-cent screen. Also, the electrons in the diffraction pat-tern, or the light from the fluorescent screen, may bedirected to detector systems of various special shapessuch as, for example, the split or quadrant detectorsused for differential imaging modes (Chapman et al.,1992).

It is convenient to have a variety of annular detectorsavailable for dark-field imaging. In the original schemeintroduced by Crewe and colleagues (Crewe and Wall,1970), an annular detector collects almost all theelectrons scattered outside the central beam cone. Thisis a highly efficient way to form a dark-field image and,to a first approximation (Cowley and Au, 1978), thesignal for weak-phase objects is proportional to thesquare of the deviation of the projected potential fromthe local average value and the resolution is better thanfor bright-field by a factor of approximately 1.4.

The heavy dependence of the contrast on Z, theatomic number of the elements present, implies thatthis annular dark-field (ADF) imaging mode can behighly effective for the imaging of heavy-metal particleson light-atom supports, as in the case of supportedmetal catalysts. It was pointed out by Howie (1979),however, that the oxides that are often used for thesupport material for catalysts tend to be micro-crystalline rather than amorphous and diffraction ef-fects from the micro-crystallites tend to give maxima inthe ADF images, which can be confused with the metalparticle images. Then it is preferable to collect onlythose electrons that have been scattered to high angles,beyond the range of the stronger diffraction spots.Thus, the high-angle annular dark-field (HAADF) modehas been preferred for catalyst studies. Also, it has beenshown that the HAADF mode has important advan-tages for the imaging of relatively thick crystal struc-tures in axial orientations (Pennycook and Jesson,1991). With this mode, the strong, non-linear dependen-cies of the BF or DF TEM image contrasts on crystalthickness and diffraction conditions no longer apply,and the contrasts of atom rows in crystals are given, toa good approximation, just by the numbers and atomicnumbers of the atoms present (Jesson and Pennycook,1995). For this purpose, a 300-keV STEM with adark-field resolution of the order of 0.2 nm (Pennycooket al., 1996) has been used very effectively for the studyon an atomic scale of interfaces of semiconductor crys-tals.

It has recently been shown that, if a thin annulardetector, as suggested in the lower part of Figure 3, isused for dark-field imaging (i.e., TADDF), it is possibleto gain further useful information for particular typesof specimen, for example, by forming the image withelectrons scattered to particular diffraction angles toimage particular phases separately (Liu and Cowley,1996). It was shown, for example, that a thin film ofamorphous carbon can be imaged preferentially in thepresence of a greater thickness of amorphous silica if aTADDF detector is used to select the electrons scatteredto the diffraction angle for the peak at about d 5 0.12

nm in the diffuse ring pattern given by the amorphouscarbon.

If such a thin annular detector is used to form abright-field image, by selecting the outer edge of thecentral beam spot of the diffraction pattern, it ispossible to improve the BF resolution, by a factor ofabout 1.7, relative to that for the conventional BFimage obtained with an axial detector (Cowley et al.,1995).

In many current commercial TEM instruments, somepossibility for producing nanodiffraction patterns isavailable. Particularly for instruments fitted with field-emission guns, the condenser lenses, used in conjunc-tion with the pre-specimen field of the objective lens,and an added mini-lens can be adjusted to provide asmall probe focused on the specimen with a diameter assmall as 1 nm. The nanodiffraction pattern is focusedon the final viewing screen of the microscope, as in thecase of SAED patterns, and can be viewed in relationwith normal TEM images or STEM images formed withpre-specimen deflection coils and a suitable detector. Tosome extent, the same sort of detectors for elastically orinelastically scattered electrons and secondary radia-tions can be incorporated in the microscope column, butthe design of the microscope, directed primarily to-wards the attainment of high-resolution TEM imaging,usually precludes the flexibility and performance of thededicated STEM instrument for these purposes. Thereare, however, reports demonstrating that nanodiffrac-tion patterns have been recorded from regions of diam-eter 1–2 nm (Matsushita et al., 1996; Tanaka andMihama, 1988). Also EELS or X-ray EDS analyses ofregions of comparable diameter with such instrumentshave been reported (Cook et al., 1996).

The TEM instruments are well suited for producingconvergent-beam electron diffraction (CBED) fromsomewhat larger areas of relatively thick crystals andhave been applied extensively to the analysis of thedetailed structure of near-perfect crystal regions (Spenceand Zuo, 1992). The detailed intensity distributions ofdiffraction lines in patterns from regions of diameter 10nm or more are interpreted to give highly accuratedeterminations of crystal structure amplitudes, includ-ing phase information, leading to studies of bonding incrystals, as well as determinations of crystal thickness,distortion, and defect structures.

The interpretation of nanodiffraction patterns mayhave all the complications of interpreting SAED pat-terns, taking into account the many-beam dynamicalinteractions of diffracted beams. However, since nano-diffraction is frequently applied to the study of speci-men regions that are very small in the beam directionas well as laterally, it is usually possible to make thesimplifying assumption of a projection approximation.When the relatively weak inelastic scattering processesare neglected, this approximation becomes the phase-object approximation in which the transmission func-tion of the object, multiplying the incident beam wave-function, is written as q(x,y) 5 exp52isw(x,y)6, where sis the interaction constant (5l/2E, where l is theelectron wavelength and E is the accelerating voltage)and w(x,y) is the projection in the incident beam direc-tion, the z-axis, of the electrostatic potential distribu-tion of the object.

84 J.M. COWLEY

The complex amplitude distribution of the coherentbeam focused by the objective lens on the specimen ist(x,y), the spread-function of the lens, given by theFourier transform of the lens transfer function, T(u,v) 5A(u,v) · exp5ix(u,v)6, where A(u,v) is the aperture func-tion, equal to unity within the objective aperture andzero elsewhere, and x(u,v) is the phase factor due to thedefocus, D, and aberrations of the objective lens. Here uand v are the angular variables corresponding to the xand y components of 2sin(f/2)/l, where f is the scatter-ing angle. Usually, for electron-microscope lenses, it isconsidered adequate to include only the third-orderspherical aberration, assuming that astigmatism iscorrected, so that x(u,v) is written

x(u,v) 5 pDl(u2 1 v2) 1 (p/2)Csl3(u2 1 v2)2. (2)

The wave leaving the specimen is then q(x,y) · t(x,y),and the diffraction pattern intensity is given by themodulus squared of the Fourier transform of thisproduct. For many purposes, it is desirable to distin-guish the central beam spot of the diffraction pattern.This is done by writing q(x,y) 5 1 2 p(x,y), so that thediffraction pattern intensity is written

I(u,v) 5 0FT[t(x,y) 2 p(x,y) · t(x,y)] 02

5 0T(u,v) 02 2 T*(u,v) · 5P(u,v) p T(u,v)6

2 T(u,v) · 5P*(u,v) p T*(u,v)6

1 0P(u,v) p T(u,v) 02,

(3)

where P(u,v) is the Fourier transform of p(x,y) and thesymbol *, on line, represents a convolution integral.

The first term of this expression is just A2(u,v) 5A(u,v), the aperture function, representing the centralbeam spot in the absence of an object. The last term isthe Fraunhofer diffraction pattern of the portion of theobject illuminated by the incident beam. The secondand third terms represent the interference of the dif-fracted beams with the incident beam. Since theseterms are multiplied by T(u,v) or its complex conjugate,they are zero outside the central beam, as defined bythe central aperture. Within the central beam spot,they provide the modulation of the intensity thatrepresents the in-line hologram of the object. Outsidethe central beam, especially if there is no overlapping ofthe diffracted beam disks, one sees a diffraction patternin which the ‘‘normal’’ diffraction pattern of the object(as in SAED) is spread out by the transfer function ofthe lens. However, this ‘‘spreading-out’’ is not straight-forward, since both P(u,v) and T(u,v) are, in general,complex functions.

The case is simpler when the phase-change in theobject is very small and the weak-phase-object approxi-mation can be made so that p(x,y) 5 isw(x,y), and for aperfect thin crystal with non-overlapping spots, thefinal term of (3) can be written

I(u,v) 5 s2F2(u,v) p A(u,v), (4)

which implies that each diffraction spot from a crystalbecomes a disk of uniform intensity. On this basis, thediffraction pattern dimensions and intensities can be

interpreted as for a SAED pattern. However, for any-thing but a single crystal with a suitably small aper-ture, the more general expression

I(u,v) 5 0sF(u,v) p T(u,v) 02 (5)

must be maintained and this includes the interferencefringes for overlapping disks in the diffraction patternsfrom perfect crystals and the interference effects forimperfect or disordered crystals, mentioned in Coher-ence and Incoherence in Nanodiffraction.

APPLICATIONSNanoparticles

Small particles of various materials having dimen-sions in the range of a few nanometers have been theobjects of intense research for many years. In particu-lar, there has been much interest in the metal particlessuspended in various poorly crystalline materials suchas the oxides of aluminum, titanium, and magnesiumbecause these form the active elements of supportedmetal catalysts. Recently a wide range of nanoparticlesand materials composed of nanometer-size particleshave been studied intensively because their optical,electrical, and other physical properties are very differ-ent from those of the bulk materials. The informationrequired regarding the nanoparticles usually includestheir sizes and size distributions, their crystal struc-tures and crystal defects, their external shapes, and thecrystallographic nature of their boundary surfaces andedges, as well as their compositions and impuritycontent. Clearly, nanodiffraction and associated tech-niques are well suited to such investigations.

In many cases, nanodiffraction studies have shownthat the small metal particles in supported metalcatalyst systems are single crystals with no obviousdefects, as in the case of Pt particles supported onalumina, for example. For these samples, the moreinteresting questions concern the nature of the aluminasupport and the relationship of the Pt particles to thesupport. The support is usually said to be g-aluminaand assumed to be amorphous. However, the crystallin-ity and texture of this material is found to vary verywidely with its source. Nanodiffraction from the alu-mina and associated STEM imaging suggests that it iscrystalline with an average crystal size varying from1–2 nm up to about ten times that size, and with avariable amount of ordering of the aluminum ions inthe basic oxygen network structure. Because there areindications that the catalytic activity of the metal maydepend to some extent on metal-support interactions,the relative orientations of the crystallites of the metaland oxide are significant. Some studies have shownthat there is a clearly defined epitaxial relationship,with the cubic axes of the Pt crystallites lying parallelto those of the g-alumina (Pan et al., 1987) (see Fig. 9).

In other Pt-alumina catalyst systems, however, amore complicated arrangement exists, with crystals ofthe oxide, a-PtO2, appearing for both the calcined andthe reduced forms of the catalyst (Pan et al., 1987). Inthese oxide particles, the unit cell dimensions appear tobe about 6% smaller than for the bulk material. Thisdifference may be a result of the small particle size or,possibly, it may result from non-stoichiometry.


In recent years, attention has been concentrated oncatalyst systems involving the zeolite structures (Pan,1996). Small metal particles, caught in the network oflarge intersecting channels in these structures, canhave interesting catalytic properties, and it is of greatinterest to find out the forms of these particles and theirrelationship to the zeolite lattice. Severe difficultiesarise for the electron microscopy of such systems,however, because the zeolite structures are very sensi-tive to irradiation by high-energy electrons. Nanodiffrac-tion studies of nanometer-size platinum particles heldin zeolites have depended on the possibility of recordingtheir diffraction patterns rapidly with minimum expo-sure of the zeolite to the incident beam. Patterns showthat the Pt particles are single crystals of nanometersize (Pan et al., 1990), but it is difficult to find therelationship of the crystallite orientations to the zeolitelattice. The crystallite orientations appear to be ran-dom.

For some other metals, the very small particles maytend to be twinned, rather than single crystals. Forgold, in fact, theoretical considerations and TEM obser-vation on larger particles, in the 10-to-100-nm range,have suggested that the most stable forms for smallparticles may be multiply-twinned, with either a deca-hedral form, consisting of ten tetrahedral regions fittedtogether by twinning on 51116 planes, or icosahedralform with twenty tetrahedral regions fitted together(Allpress and Sanders, 1970; Marks and Smith, 1981).Nanodiffraction studies of gold particles dispersed in apolyester film showed that, for particles of about 5 nmdiameter, such multiply twinned forms do occur, alongwith a few single-crystal particles and some singly- ordoubly-twinned. However, the nanodiffraction patternsfrom particles in the 2–3-nm size range suggested thatthe proportion of multiply-twinned forms decreases forthe smaller sizes (Cowley and Roy, 1982). This result isnot necessarily inconsistent with the theoretical predic-tions, because these predictions were made for isolatedgold particles. For gold particles surrounded by polyes-ter, the surface energies may be expected to differconsiderably from those for a vacuum environment, sothat the balance of surface energy to the energies

associated with the distortions of the gold lattice due tothe twinning may be very different.

Other studies of metal catalyst particles on oxidesupports suggest that, for very small particles, thestructure of the metal may not be the same as for largerparticles. For example, a ruthenium-gold on MgO cata-lyst system was investigated in an effort to determinewhy the catalytic activity of the ruthenium is enhancedby the addition of the supposedly inert gold (Cowleyand Plano, 1987). For the particles of 5-nm diameter orgreater, it could be determined that the gold occurs inits normal face-centered cubic structure and the ruthe-nium occurs in its normal hexagonal form, with noapparent formation of any alloyed or mixed-crystalforms. Nanodiffraction shows clearly that, for these andsmaller particles, both the Au and Ru form singlecrystals sitting on the MgO crystallites in epitaxialarrangements. Other nanodiffraction patterns occur,however, which can be interpreted only in terms of abody-centered-cubic (BCC) form of ruthenium, un-known in bulk. Evidence from the strong diffuse streak-ing visible in some of these patterns suggests that theBCC crystallites are heavily faulted on 51106-type planes.

For studies of particle-support interactions, it issometimes preferable to have the support in the form ofthe well-characterized face of a bulk single crystal, andto examine the form and orientations of the crystallitesdeposited on this face. For this purpose, the STEM andnanodiffraction techniques may be used in the reflec-tion mode. As in the case of reflection high-energyelectron diffraction (RHEED) and reflection electronmicroscopy (REM), performed in a TEM instrument,the incident beam hits the flat surface of the crystal at agrazing angle of a few degrees, equal to the Bragg anglefor the fifth or sixth order of reflection from the princi-pal lattice planes parallel to the surface. If the incidentbeam is parallel to a principal set of planes perpendicu-lar to the surface, the RHEED pattern contains theregular pattern of spots corresponding to the zero layerof spots (the zero-order Laue zone) for the zone axislying in the surface almost parallel to the beam. Thenthere are outer rings of spots corresponding to thehigher-order Laue zones. In addition to these spot

Fig. 9. a–c: Three typical nanodiffraction patterns from small Ptcrystals on a support of nominally ‘‘amorphous’’ alumina, which isclearly microcrystalline with a well-defined Pt crystal orientation

relative to the alumina. In each case, the regular pattern of widelyspaced strong spots comes from a Pt crystallite. The more closelyspaced patterns of weaker spots come from the alumna.

86 J.M. COWLEY

arrays, there is a strong background due to inelastic orelastic diffuse scattering, taking the form of Kikuchilines, parabolas, and circles (Larsen and Dobson, 1987).The REM image is obtained by aligning the crystal andthe incident beam so that a prominent diffracted spot isdirected along the axis of the objective lens of the TEMinstrument. Usually, it is arranged that the crystalorientation is such that the intensity of the selectedreflection is enhanced by the so-called ‘‘surface channel-ing’’ effect (Gajdardziska-Josifovska and Cowley, 1991).

The equivalent geometry is used to give the equiva-lent surface diffraction and imaging techniques in aSTEM instrument. The convergence of the incidentbeam on the specimen has the same effect as intransmission in that the diffraction spots are spreadinto disks, but the pattern of Kikuchi lines and curves isnot affected because these features act as if firmlyattached to the crystal. The scanning reflection electron-microscopy (SREM) images are much the same as forREM images if formed by selecting a strong diffractedbeam to form the image, but possibilities also exist forthe use of annular detectors to give images withcontrast generated by different scattering mechanisms(Liu and Cowley, 1993).

In this way, studies have been made of the formationof nanocrystals of Au, Pt, and Pd on the near-perfect50016 faces of large crystals of MgO smoke. In each case,a parallel epitaxial orientation was found (Cowley andOu, 1989). For Au, it was found by nanodiffraction andSREM that the nanocrystals tend to grow on theinfrequent surface steps of the crystal face (Ou andCowley, 1987) (see Fig. 10).

A further application of nanodiffraction and SREMhas been for the study of the oxidation of the faces of acopper single crystal (Milne, 1987).

In the cases treated so far, the emphasis has been ondetermining the crystal structure, orientation, or fault-ing of the nanocrystallites, and for these purposes it issufficient to consider the positions and intensities of thediffraction spots. Further information is available, how-ever, from the shapes of the diffraction spots when wetake into account the coherent nature of the diffractionprocess.

Initial observations were made of the nanodiffractionpatterns obtained when a nanobeam was translatedover the edge of a cubic crystal of MgO smoke, with thebeam parallel to one of the flat faces of the crystal. Itwas noted that, when the beam included the crystaledge, the diffraction spots were transformed into hollowbright rings (Cowley, 1986) (see Fig. 11). This effect wasconfirmed by calculations, based on the assumptions ofcoherent diffraction (Cowley and Spence, 1981). Moredetailed calculations dealt with the specific case of a1-nm-diameter beam illuminating a cuboctahedral FCCgold crystal containing 55 atoms (Pan et al., 1989). Itwas shown that the diffraction spots could be split ordistorted in various ways, depending on the beamposition. The general conclusion of such calculations isthat, when the beam is intersected by a long, straightcrystal edge, the diffraction spots take the form ofcontinuous bright rings, as in Figure 11. When thecrystal edge extends only partway across the beamdiameter, however, the bright rings are not continuousbut take the form of two bright arcs, centered on the

Fig. 10. Scanning reflection electron-microscopy image. a: Distribution of small Au crystallites on the[001] surface of a large MgO smoke crystal (Marker 5 10 nm) and the nanodiffraction pattern. b: Parallelorientation of an Au crystal on the surface.


line perpendicular to the crystal edge. A distinction canbe made between the cases that the crystal edge iswedge-shaped and when it is abrupt, with a flat faceparallel to the incident beam direction (Pan, 1990). Onthis basis, an analysis can be made of the three-dimensional shape of a nanometer-size particle.

Carbon NanotubesCarbon nanotubes were first observed by Iijima (1991)

while examining preparations of fullerenes in HRTEM.He interpreted the long, thin objects as being made upof sheets of carbon atoms, as in the ‘‘graphene’’ sheetsthat constitute graphite, wrapped into the form ofcylinders. The initial observations were of multi-walledtubes, with a number of concentric cylindrical graphenelayers, separated by the spacing of 3.4 Å, which isslightly bigger than the inter-layer spacing of graphite(See Fig. 12a). Later (Iijima and Ichihashi, 1993),single-walled tubes with diameters in the range of 1–3nm were observed. Selected-area diffraction patternsmay be obtained from the tubes, although with consid-erable difficulty because the scattering from carbon isweak and the selected area of the image is very muchgreater than the area of the tube. It is necessary toselect a part of the tube that is uniform and unbent.Nanodiffraction is a much more efficient and flexiblemeans for studying the tube structure.

From his SAED patterns, however, Iijima (1991) wasable to deduce that the tubes could occur with varioushelix angles in that, when a graphene sheet is wrappedinto a cylinder, it is not necessary that one of the maincrystallographic directions of the sheet structure shouldbe perpendicular, or parallel, to the axis of the cylinder.For a helix angle of zero, the diffraction pattern showsthe symmetrical set of six sharp spots, corresponding tothe 51006 reflections of graphite or of the graphene sheet.For a helix angle of other than zero, each of these spotsis split, circumferentially, into two spots with a separa-tion of twice the helix angle, since the top and bottomlayers of the tube are, in effect, twisted in oppositedirections. It has been shown subsequently that, formulti-walled tubes of large diameter, containing ten to

twenty or more layers, there may be several differenthelix angles present. The helix angle appears to changewith the addition of approximately four layers, so thatthe number of helix angles present is approximately thenumber of layers divided by four (Liu and Cowley,1994a). An explanation for this observation in terms ofgeometric factors has been given.

From HRTEM, the amount of information regardingthe structures of the tubes is limited since, in general,appreciable contrast is given only by those parts of thetube where the planes of the graphene layers areparallel to the beam. Thus, an HREM image of a tubenormally consists of two sets of parallel lines, 3.4 Åapart, as in Figure 13a. It was difficult to interpret theobservation, therefore, that, particularly for some prepa-rations, many of the images of multi-walled nanotubesshow a different spacing for the two sets of parallellines. As in Figure 13b, the spacing of the lines on oneside of the tube is 3.4 Å, while on the other side, thespacing is more irregular and averages about 4 Å. Theinterpretation proposed for this is that the cross-sections of such tubes are not circular, as had beenassumed, but instead are polygonal, presumably pen-tagonal, as suggested in Figure 12b. This interpreta-tion is strongly supported by nanodiffraction (Liu andCowley, 1994b).

If an incident beam of diameter 0.7–1 nm is trans-lated across a multi-walled tube of circular cross-section, it first illuminates one wall of the tube. Therethe parallel stacking of the layers gives rise to a strongline of diffraction spots corresponding to the 3.4 Åinter-layer spacing (Fig 14a). For an ordered array ofgraphene layers, as in graphite, this line of 001 spotswould be accompanied by parallel layers of spots, thehk1 reflections. However, in a multi-walled tube therecan be no regular stacking of the sheets because thesuccessive layers have different radii and so rapidly get‘‘out-of-step’’ with each other around their circumfer-ences. This essential disorder implies that there are nosharp reflections, but only diffuse streaks, along thesehk1 lines.

Fig. 11. Nanodiffraction patterns obtained with the 1 nm diameter beam firstly transmitted through asmall MgO smoke crystal (a), and then moved across the crystal edge, showing the split diffraction spots(b) and ring-shaped spots (c).

88 J.M. COWLEY

When the incident beam is in the middle of the tubeimage, it is perpendicular to the graphene layers. Ifthere is only one helix angle present and that is zero,the pattern is that of a single graphene layer, namely ahexagonal array of spots with the same geometry, butnot the same intensity distribution, as the hk0 patternfrom graphite (Fig. 14a). For a single helix angle otherthan zero, the spots of this pattern are all split,circumferentially, into a pair of spots with a separationthat is twice the helix angle. For multiple helix angles,each spot is replaced by a multiple array of spots and,because the individual diffraction spots are ratherlarge, this may give the appearance of continuous arcsor complete rings of scattering.

If the incident beam is displaced a few tenths of annm away from the center of the tube, it may be possible,in favorable cases, to observe the effect on the diffrac-tion pattern of the tilting of the graphene layersrelative to the beam. The rings of spots, or continuousrings, of the hk0 pattern become elliptical, with axialratios of the ellipses depending on the cosine of theangle of tilt. This effect has been observed earlier for thecase of disordered planar graphene sheets (Schiffmakeret al., 1980) having the ‘‘turbostratic’’ structure postu-lated by Biscoe and Warren (1942) on the basis of X-raydiffraction observations.

When the incident beam reaches the other side of thetube of circular cross-section, the pattern again showsthe strong line of 001 spots, corresponding to theinter-layer spacing (Fig. 14a). Simple kinematical orweak-phase-object theory suggests that the diffractionpattern from the two sides of the tube should be thesame.

For the multiple-walled tubes of polygonal cross-section, as suggested in Figure 12b, the sequence ofnanodiffraction patterns is quite different. For theincident beam passing through the left-hand wall of thetube of Figure 12b, the line of 001 reflections is verystrong and the parallel lines of hk1 reflections showsharp spots (Fig. 14b). Thus, this flat part of the tubewall actually approximates a single crystal of graphite.It seems that the relatively small ordering energy thatproduces the ordered graphite structure can be effectivein this case since the misfitting of the successive layersthat prevents the ordering for the tubes of circularcross-section can presumably be accommodated in thedisordered regions around the sharp bends of thelayers.

For the beam on the right-hand side of the tube ofFigure 12b, the pattern is given mainly by the stronglybent and disordered graphene sheets on the corner. Onesees only a weak row of 001 spots, fading rapidly withincreasing diffraction angle (Fig. 14b). This is consis-tent with the model of layers of variable spacing and nolateral ordering. For the beam near the middle of thetube of Figure 12b, one can see a variety of patternsdepending on the exact form and orientation of the tube(Liu and Cowley, 1994b). With the beam almost perpen-dicular to a flat part of the tube wall and in the absenceof one or more helix angles, one sees the hexagonal hk0spot pattern, but this time, the spot intensities arethose typical of the ordered graphite lattice, with 110spots considerably stronger than the 100 spots.

For the most part, one can interpret such nanodiffrac-tion patterns on the basis of the simple kinematical or

Fig. 12. Diagrams of a multi-walled carbon nanotubes of circularcross-section (a) and of pentagonal cross-section (b).


weak-phase-object approximation (WPOA), with theusual assumption of kinematical theory that the diffrac-tion-pattern intensities show a center of symmetry,whether or not there is a center of symmetry in thestructure of the sample. However, if the projectedpotential distribution for a nanotube of circular cross-section is plotted, as in Figure 15, it is seen that thepeaks of projected potential corresponding to the indi-vidual layers produce relatively large phase changes sothat the WPOA is not applicable. The phase-object

approximation, valid for thin-enough regions, remainsa good approximation because the peak of the projectedpotential for each layer is given almost entirely by aregion only about 2 nm thick. These tubes of circularcross-section thus form excellent test objects for thetheoretical result that for phase objects, as for anythicker object for which appreciable dynamical scatter-ing exists, a lack of a center of symmetry in thediffraction pattern is expected to reflect a lack ofsymmetry of the illuminated area of the specimen

Fig. 13. HREM images of multi-walled carbonnanotubes of circular cross-section (fringes of spac-ing 3.4 Å) (a) and polygonal cross-section (fringes onone side average 4 Å) (b).

Fig. 14. Sets of nanodiffraction patterns obtained as the beam is moved from one side to the other formulti-walled tubes. a: Circular cross-section. (b) Polygonal cross-section (Liu and Cowley, 1994b).

90 J.M. COWLEY

(Cowley, 1992). The structures of these objects areknown and there is little or no dependence of theprojected potential distribution on the orientation.

From Figure 15, it is seen that the projected potentialfor one side of the circular tube can be described interms of a regular array of strongly asymmetric unitswith a periodicity of 0.34 nm. Each of these units maybe expected to act as a prism, deflecting the electrons tothe right. The periodicity of 0.34 nm implies that thisdeflection will affect the first-order reflections of the 001row of spots. Calculations show that these reflectionsshould show an asymmetry of about 10%, as measuredby the ratio of the 0,0,1 and 0,0,21 reflections. Suchasymmetries have been observed of about that magni-tude (Cowley and Packard, 1996) (Fig. 16). As expected,the asymmetry is in opposite directions for the twosides of a tube. It has been demonstrated that theobservation of such asymmetries may be used as a toolfor the elucidation of the three-dimensional forms ofsome tube-ends and other features for which the inter-pretation of the HRTEM images is ambiguous.

Considerable interest has been shown in the possibil-ity of encapsulating metals or metal compounds withinthe central spaces of multi-walled nanotubes as apossible basis for experiments to explore the propertiesof conductors of very small, well-specified dimensions.To fill the tubes by capillary action (Ajayan and Iijima,1993), it is normally necessary to remove the caps thatclose off the tube ends. More simply, the tubes may befilled during their formation in a carbon arc if aquantity of a metal or its oxide is placed in a recess inone of the carbon arc electrodes (Seraphin et al., 1993).In the latter case, the material that fills the tube isfound to be one form or other of metal carbide. Thecrystallinity of the carbide and the relationship withthe nanotube walls has been shown, by a combination ofHRTEM, SAED, and nanodiffraction, to vary widelywith the nature of the metal concerned.

At one extreme, as in the case of manganese (Liu andCowley, 1995), crystals of carbides of various composi-tions grow in rather irregular shapes and the walls ofthe nanotubes appear to grow around them, followingthe contours of the crystals, although the outlines of thecrystals are sufficiently elongated along the axes of thenanotubes to suggest that the carbides and the nano-tubes grow simultaneously and interactively. Anotherextreme example is that of yttrium (Cowley and Liu,1994), in which the nanotube shapes appear to be quiteunaffected by the presence of the carbide, as if thecarbide solidifies within the completely formed tube,without distorting it. In this case, it is interesting toinvestigate how the tetragonal carbide crystals cangrow in relationship to the smoothly curved inner wallof the tube. Nanodiffraction patterns reveal an epitax-ial relationship in which the (110) planes of the carbidelattice are parallel to the planes of the graphene layers.But such a relationship cannot be maintained over anyappreciable distance around the curved walls of thetube except by frequent faulting and some distortion ofthe carbide lattice.

In the case of tubes filed by diffusion through openedends, the original tube formats appear to be unaffectedby the intercalates. The structures of some intercalatedmaterials have been investigated in a TEM/STEMinstrument fitted with a field-emission gun using nano-diffraction with a beam of diameter 2–3 nm (Cook et al.,1996). Also, in this case, the compositions of the interca-lated materials were determined by energy-dispersiveX-ray spectroscopy from the same small areas.

For the single-walled nanotubes (SWnT), as firstobserved by Iijima and Ichihashi (1993), nanodiffrac-tion in a dedicated STEM instrument is possible forindividual tubes and is being applied to the determina-tion of the distributions of diameters and helix anglesresulting from various processes of tube formation.There is considerable interest in the structural param-eters because of the theoretical predictions that theywill strongly affect the electrical properties of the tubes.

The possibility that the SWnT may be bound togetherto form ropes is also of considerable interest because itis foreseen that a rope of SWnT will have a strength-to-

Fig. 15. The projected potential variation as a function of radius,calculated (with simplifying assumptions) for one side of a multi-walled carbon nanotube of circular cross-section (Cowley and Packard,1996).

Fig. 16. The measured intensity profile for the line of 001 diffrac-tion spots from a multi-walled carbon nanotube of circular cross-section (Cowley and Packard, 1996).


weight ratio that is far greater than for any knownmaterial. Progress towards the manufacture of suchropes has been reported, for example, by Thess et al.(1996), who have shown that ropes of aligned SWnT ofuniform diameter may be grown to diameters of 4 to 30nm and lengths of 100 nm by use of a process of doublelaser irradiation. Studies using X-ray diffraction andelectron microscopy revealed that the SWnT within theropes all have the same diameter of 1.37 nm and arepacked together in a near-perfect hexagonal array withperiodicity 1.69 nm. A theoretical model for their growthprocess suggests that SWnT should all have the samediameter and should, in addition, all have the samehelix angle, with the structure designated as (10,10),which has the symmetry C5V and metallic electricalconductivity.

Nanodiffraction studies of such ropes have helped toelucidate the details of their structures. Patterns, suchas those of Figure 17, were obtained in several ways.Firstly, with a stationary beam, a pattern could beobtained, which reveals the structure averaged along aline through the rope in the beam direction. Secondly, ifa rapid two-dimensional scan is applied to the beamwith an amplitude of perhaps 10 nm, a pattern can beobtained that gives the average structure for a shortsection of the rope. Thirdly, sets of nanodiffraction

patterns obtained with a slow linear scan of the beamcan reveal the way in which the structure of the ropevaries across its diameter or along its length.

The strong azimuthal line of reflections in this case,does not reveal the 0.34 nm layer stacking of themulti-walled tubes, but reflects the lateral hexagonalstacking of the SWnT with 1.69 nm periodicity. Al-though with a beam of diameter 7 nm it is not possibleto resolve a 1.69-nm periodicity, the intensity distribu-tions along this line may be interpreted in terms ofvarious recognizable orientations of the hexagonal lat-tice. In this way, it is found that the orientation of thehexagonal lattice tends to vary considerably both acrossand along the individual ropes of SWnT (Cowley andSundell, 1997). Portions of the ropes, usually around 10nm in diameter, tend to be composed of an almostperfect hexagonal packing of SWnT of identical diam-eter and helix angle, but a rope of larger diameter thanabout 10 nm may be made up of several such regionswith hexagonal arrays in different orientations andwith SWnT of the same diameter but different helixangles. Apparently as a result of the fitting together ofsuch differently-oriented regions, the whole ropes arenormally strongly twisted and bent.

The theoretical prediction that all the SWnT of thepreparation should have identical helix angles, with

Fig. 17. Nanodiffraction patterns from ropes of single-walled car-bon nanotubes. a: Pattern averaged over a (10,10) rope, helix angle 0°.b: Pattern from one spot on rope (a). c: Pattern from a similar rope but

with the rope tilted by 30°. d: Pattern from one spot on a rope withhelix angle 5°. e: Pattern averaged over a rope with spread of helixangles up to 10°. f: Pattern from one spot on rope (e).

92 J.M. COWLEY

the (10,10) structure, was found to be supported by thesummary of the determinations from the nanodiffrac-tion patterns that the helix angles are concentrated atthis value, with a spread of only about 65° (Cowley etal., 1997a).

Thin Films: Defects and DisorderThe extreme case of disorder in thin films is that of

amorphous or near-amorphous materials for which theSAED patterns show only diffuse haloes. Such diffusehalo patterns are given by averaging the diffractioncontributions from all possible orientations of the inter-atomic vectors, and an analysis of the patterns can bemade in order to derive the distributions of lengths ofthe interatomic vectors present, as is done in thediffraction analysis of gas molecules (Hargittai, 1992).For nanodiffraction, however, such an averaging cannottake place because the illuminated area of the specimenmay be less than 1 nm2 and for a specimen only a fewnm thick, the number of atoms contributing to thediffraction pattern may be only a few hundred. Thus,the diffuse haloes cannot be generated. The patterncontains distributions of maxima and minima, appar-ently at random positions, and changing appreciablywith a movement of the beam by as little as 0.1 nm.

It may be considered that the nanodiffraction pat-terns from individual regions of near-amorphous mate-rials may be an excellent means for investigating themost important aspects of the structures of thesematerials, namely, the preferred local configurations ofthe atoms on the scale of 0.5 to 2 nm. The nearest-neighbor preferences of atoms (the short-range order-ing) are given effectively by analysis of the diffuse halodiffraction patterns, or by a number of other tech-niques. The structures of ordered regions 2 nm orgreater in diameter are treated as the structures ofsmall crystallites for diffraction purposes and also forHRTEM imaging. It is in the intermediate range ofordering over distances of 0.5 to 2 nm that many of theinteresting properties of the materials reside and thisrange is the one least accessible to conventional diffrac-tion or imaging techniques. However, although a greatamount of information regarding structure on thisintermediate scale is obviously present in nanodiffrac-tion patterns, no realistic process for deriving meaning-ful conclusions from the observations has as yet beendevised.

From any one nanodiffraction pattern, one could, inprinciple, making use of coherent nanodiffraction pro-cesses to overcome the ‘‘phase-problem’’ of kinematicalscattering, derive a model of the two-dimensional projec-tion of the arrangement of the one particular set ofatoms selected by the beam position. From even thethree-dimensional configuration of that one set of at-oms, one could not make any significant conclusions.The information required is likely to be the relativefrequency with which various types of local configura-tions occur, and for this purpose it seems that therecording and correlation of an enormous amount ofdata would be necessary (Cowley, 1981c), especiallysince only the information of two-dimensional projec-tions seem likely to be available.

The same sort of considerations have, to much thesame extent, inhibited efforts to use nanodiffractionexperiments to investigate the local ordering of atoms

in crystals. For the relatively simple systems of binaryalloys, the local ordering may be described in terms ofshort-range-order parameters, and an accepted model-ling of the disordered structures in terms of small localregions of the known ordered phases, separated byout-of-phase domain boundaries, is available (Gragg etal., 1971). For other disordered crystal systems, how-ever, the situation is far from clear. Some oxides, suchas TiOX, VOX, LiFeO2, etc., show complicated distribu-tions of diffuse scattering in their diffraction patterns,with lines, loops, and arcs, and the interpretation ofthese distributions in terms of nearest-neighbor cluster-ing, while suggestive, appears to be far from adequate(Sauvage and Parthe, 1974).

Again, in this case, nanodiffraction patterns fromindividual regions are of little help in solving theproblem of what types of medium-range ordered struc-tures are likely to occur. The accumulation of resultsfrom many nanodiffraction patterns from randomlyselected areas may be suggestive (Tanaka and Cowley,1985), but a great amount of statistical informationwould be required for significant results, even thoughin this case the presence of a crystal lattice as areference point for orientations simplifies the consider-ations.

One approach to the gathering and interpretation ofstatistical data from nanodiffraction patterns is tomake use of pattern-recognition techniques. In therelatively simple case of identifying small regions in apartially ordered binary alloy in which the out-of-phasedomain structure has the same orientation, this ap-proach has proved an effective means for detecting thepositions in the specimen that give nanodiffractionpatterns having the same array of super-lattice spots(Monosmith and Cowley, 1983). The method may beextended to include the possibility of recognizing re-gions giving the same diffraction pattern even when thepattern axes are not parallel, i.e., when a rotationalcorrelation of pattern intensities is required.

For application to cases of thin single crystals havingisolated individual faults, the use of the nanodiffractiontechniques is more straight-forward, even though, ex-cept for exceptionally thin samples, the presence ofdynamical diffraction conditions is inevitable. The caseof the planar faults in diamond (Cowley et al., 1984)discussed previously, is an extreme one in which thebeam diameter was smaller than the unit-cell dimen-sions and the consideration of the continuous intensitydistribution of the whole pattern had to be made on thebasis of dynamical diffraction calculations. For manypurposes, the use of the larger beam sizes, approaching1 nm, gives valuable results that are much easier toassess. Very little attempt has been made to makedetailed nanodiffraction studies of local defects. Thepossibility was demonstrated early (Cowley and Spence,1979) of using nanodiffraction to explore the latticedistortions around the dislocations in iron, viewedend-on, but no such experiments have been reported.

Several studies have dealt with planar defects incrystals. As in the case of small particles, a planardiscontinuity in the scattering power of the specimencan result in the deformation of diffraction spots to formbright rings or spot-pairs (as in Fig. 11). For a fault in acrystal, the discontinuity may not be the discontinuous


cessation of the whole structure, but a discontinuity ofonly one part of the structure.

For example, it is well known that faults or twins onthe [111] planes of face-centered cubic metals or on thebasal plane of hexagonal close-packed metal structureshave no effect on the (001) reflections (in the commonhexagonal indices) or on reflections for which h 1 k 53n, since these reflections are insensitive to lateralshifts of the close-packed layers of atoms such as theshifts with x,y fractional coordinates (1/3, 2/3) or (2/3,1/3), which separate successive layers. The reflectionswith indices for h 1 k 5 3n 6 1, however, are highlysensitive to the stacking order of the close-packedlayers. For these reflections, a stacking fault such as atwinning plane represents a discontinuity of the struc-ture. When a nanodiffraction beam is centered on sucha fault plane, the corresponding diffraction spots ap-pear to be split whereas the diffraction spots for h 1 k 53n are not. Observations of this preferential splittinghave been made for twin boundaries and other faults inthin films of stainless steel (Zhu and Cowley, 1983).

Partially ordered binary alloys may be described asbeing made up of out-of-phase domains of the orderedstructure. At the domain boundaries, it may be consid-ered that the origin of the unit cell of the ordered cellshifts from one equivalent site to another of the aver-age, disordered cell. For some of the reflections, theso-called ‘‘fundamental reflections’’ given by both theordered and disordered structures, the amplitudes andphases of diffraction are unaffected by the domainboundaries. For the ‘‘superstructure reflections’’ givenby the ordered structure but not by the averageddisordered lattice structure, the shift of origin at anout-of-phase boundary represents a change of phaseand/or amplitude. The corresponding diffraction spotsof the nanodiffraction pattern show a splitting when theincident beam lies on the boundary.

For the case of thin crystal films of the partiallyordered Cu3Au alloy, it was observed that the severaldifferent classes of superstructure reflections can showdifferent types of splitting behavior, depending on thenature of the out-of-phase domain boundary, or bound-aries, illuminated by a nanodiffraction beam (Zhu andCowley, 1982). Thus, it is possible to distinguish readilybetween the ‘‘good’’ and ‘‘bad’’ types of boundaries, forwhich the usual avoidance of nearest-neighbor goldatoms is maintained or not. This is information noteasily derived from HRTEM images.

In polycrystalline thin films, the main defects may beboundaries between crystals of the same structure butdifferent orientations, or between crystals of differentstructures. Nanodiffraction from the boundaries maythen be treated in many cases as are the boundaries ofisolated crystals. However, there are cases of interest,particularly when crystals of one type are grown epitaxi-ally on another, when the most important factor to beinvestigated is the change at the boundary of the totalprojected potential, resulting from a sudden change inthe mean inner potential of the material. The abrupt-ness of the change is to be measured as an indication ofthe sharpness of the boundary on an atomic scale, andvarious methods of HRTEM, HAADF, STEM, etc., havebeen applied to this problem. Nanodiffraction offers analternative approach in that the change of projectedpotential between the two materials may be considered

to act like a prism to deflect the incident beam. A linearvariation of potential gives a single, easily-calculateddeflection. A non-linear change gives a streak. In thecase of alternate layers of Mo and Si in the cross-sectional film cut from a multi-layer sample, streaks inthe nanodiffraction pattern are seen to be consistentlylonger for the Mo/Si interface than for the Si/Mointerface, suggesting a more abrupt boundary in theformer case (Gajdardziska-Josifovska et al., 1995).

Dynamical Effects: Radiation DamageThe possibility of recording nanodiffraction patterns,

not appreciably degraded by electron noise, at a rate of30 per second by means of a TV and VCR system, maybe a valuable asset for many purposes. It allows therecording of large numbers of nanodiffraction patternsin a short time during one- or two-dimensional scans ofthe specimen. It suggests the possibility of recordingpatterns while the specimen structure is being modifiedduring chemical reactions or physical changes. How-ever, to date, few applications have been made to thestudy of transient structures. The most common casehas been in the recording of patterns from materialsthat are very sensitive to irradiation damage by theincident electron beam.

It is well known that for most biological and organicmaterials, and for many inorganic materials, the struc-ture of the specimen is rapidly degraded by irradiationin a normal TEM or STEM mode in some small fractionof a second under high-resolution imaging conditions.With the focused STEM beam used for nanodiffraction,it may be possible to glimpse some suggestion of adiffraction pattern before it is replaced by the diffuse,fluctuating intensity distribution characteristic of mo-bile, amorphous material. But if a recording is madewith the TV-VCR system from the time that the firstelectrons pass through a given part of the specimen, theoriginal, decaying form of the diffraction pattern maybe recorded on two or three frames. The desired speci-men area may be identified in a low-resolution STEMimage for which the beam is greatly defocused to reducethe intensity of specimen irradiation or for which thedistance between the scan lines for a focused beam isvery much greater than the beam diameter so that theirradiation of the specimen is only along well-separatedthin lines. Then the TV-VCR recording of the nanodif-fraction patterns is started and the beam is stopped at aselected position.

In this manner, nanodiffraction patterns have beenrecorded showing crystalline order in clay minerals (seeFig. 18) and in the lipid tubules of diameter 0.3 to 1.0µm that have been formed from a polymerizable leci-thin (Yager et al., 1985).

For many inorganic compounds, the effect of theelectron irradiation may be a change from one crystal-line phase to another. For thin films of Al2O3 and AlF3,reduction to metallic Al has been observed (Scheinfeinet al., 1985). Applications in solid-state device technol-ogy have been envisaged. Similarly, reductions of fluo-rides or oxides to form very small dots or wires offerromagnetic metals have been explored in relation toresearch on micromagnetics. Nanodiffraction is an es-sential adjunct to this research.

While the reduction of oxides as a result of electronirradiation is well known, there are other cases in

94 J.M. COWLEY

which oxidation processes have been observed. Thus, inconjunction with HRTEM and STEM, nanodiffractionstudies have shown that thin films of the oxide Ag2Omay grow under electron irradiation at the interfaces ofsmall Ag particles sitting on MgO crystals (Lodge andCowley, 1984), and small crystals of Pd on MgO facesmay be transformed to single crystals of PdO (Ou andCowley, 1988).

For some compounds, the loss of materials, presum-ably by dispersion into the vacuum, leads to the possibil-ity of using a STEM beam for the drilling of very smallholes in thin films, with possible technological applica-tions. It has been shown, for example, that by drillingholes of 2-nm diameter in a thin AlF3 film it is possibleto produce lettering on such a scale that the whole ofthe Encyclopedia Britannica could be written on apinhead (Humphreys et al., 1990). The drilling of holesin MgO crystals is of particular interest in that, with abeam of circular cross-section, the holes that are drilledhave square cross-section, defined by the prominentlattice planes of the cubic crystal (Turner et al., 1990).Nanodiffraction patterns observed during the drillingoperation and from the walls of the holes produced inthis way may be an effective means for studying thenature of the hole-drilling process.

CONCLUSIONSOf the multitude of possible applications of nanodif-

fraction in even its simplest, incoherent form, only afew have been explored. This is presumably a result ofthe fact that the number of dedicated STEM instru-ments in operation is very small as compared with thenumber of TEM instruments, and, of the dedicatedSTEM instruments currently in use, most are special-ized for the purposes of microanalysis using EELS orX-ray EDS analysis. Only a very few have provision forthe efficient observation and recording of nanodiffrac-tion patterns. Since the one commercial manufacturerof STEM instruments has ceased operation, the avail-ability of suitable instruments will undoubtedly de-crease in the future. The increased availability of TEMinstruments fitted with field-emission guns may, to

some extent, compensate for this loss, but such instru-ments are not usually equipped for the same flexibilityof recording of nanodiffraction patterns, shadow im-ages, and holograms, or for the same convenient inter-action of nanodiffraction with microanalysis and imag-ing modes.

The possibilities for observation and application ofcoherent nanodiffraction have scarcely been touched.Sufficient indications have been given of the extremesensitivity of the diffraction-pattern intensity distribu-tions to the structures, shapes, and sizes of crystallitesor defect regions containing only a few hundred atoms,to indicate that the systematic application of the nano-diffraction techniques in many areas could add greatlyto the knowledge of the detailed structures in terms ofthe positions and interactions of the atoms.

One difficulty in the wider application of the coherentnanodiffraction is that of instrumental stability. TheSTEM instruments now being applied for these pur-poses are old ones, designed for operation on a nanom-eter scale of resolution, but not for operation at an 0.1nm level. For adequate exploitation of the coherentnanodiffraction effects, it is necessary that the mechani-cal and electrical instabilities of the tilted specimen andof the microscope column should be controlled at leastas well as in modern TEM instruments, and preferablybetter. Clearly there is an urgent need for a newgeneration of STEM instruments, engineered accordingto the most advanced principles.

The possibility of using atomic focusers to produceelectron beams of diameter 0.05 nm or less suggeststhat, at some future time, it will be necessary toreassess the relationship of nanodiffraction and imag-ing. Both the theoretical basis and the experimentalprocedures will need to be reexamined and instrumentswill need to be redesigned in such a way that theoptimum combination of diffraction and imaging infor-mation can be made available for any specific purpose.

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Fig. 18. Nanodiffraction patterns from the radiation-sensitive clay mineral, kaolinite. a: Pattern fromthe wall of a rolled-up tube. b: Pattern with the beam perpendicular to a flat sheet. c: Degraded patternafter exposure to the beam for a fraction of one second.


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Electron nanodiffraction