Review

Applications of electron nanodiffraction

J.M. Cowley*

Department of Physics and Astronomy, Arizona State University, P.O. Box 871504, Tempe, AZ 85187-1504, USA

Received 12 November 2003; revised 15 December 2003; accepted 16 December 2003

Abstract

Diffraction patterns from regions 1 nm or less in diameter may be recorded in scanning transmission electron microscopy instruments, and

have been applied to the investigation of the structures of various nanoparticles, including catalysts, ferrihydrite and ferritins. Applications to

nanotubes and related materials and near-amorphous thin films are reported. The coherence of the incident beams may be exploited in studies

of crystals and their defects. Several schemes are outlined whereby the information from sequences of nanodiffraction patterns may be

combined to provide ultra-high resolution in electron microscope imaging.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Electron; Nanodiffraction; Nanoparticles; Scanning transmission electron microscopy

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

2. Experimental procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

3. Nanoparticles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

4. Nanotubes, nanoshells and nanobelts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

5. Ferrihydrite and ferritin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

6. Amorphous and disordered systems: quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

7. Coherent nanodiffraction: crystal defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

8. Nanodiffraction and holography: atomic focusers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

9. Nanodiffraction and four-dimensional imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

10. Conclusions: further developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

1. Introduction

It has long been known that the strong electromagnetic

lenses used in electron microscopy may be applied to form

electron probes of sub-nanometer diameter by demagnifica-

tion of a small bright electron source, for electrons in the

energy range of a few hundred thousand eV. When focused

on a thin specimen, such probes can produce diffraction

patterns from regions less than 1 nm in diameter. Thus,

electron nanodiffraction (END) is possible. With a field-

emission gun (FEG) as a source, the diffraction patterns are

readily visible on a fluorescent screen and may be observed,

and recorded in a fraction of 1 s, by use of a low-light-level

television camera or a CCD camera. With a TV camera and

a video-cassette recorder (VCR), series of patterns can be

recorded at a rate of 30/s, either with a stationary beam to

study time-dependent phenomena or else during a linear or

two-dimensional scan of the incident beam over the

specimen for a detailed study of the variations of structure

over any small region.

At the present time, when there is an explosive growth

of the many aspects of nanotechnology and nanoscience,

it would seem obvious that END should be recognized as

an important tool for the study of the structures of

nanometer-size regions of materials and of the com-

ponents of nano-systems. However, the capabilities of

END have not been widely exploited. There are several

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doi:10.1016/j.micron.2003.12.002

Micron 35 (2004) 345–360

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* Tel.: þ1-480-9656459; fax: þ1-480-9657954.

E-mail address: cowleyj@asu.edu (J.M. Cowley).

reasons why very few groups have explored the

possibilities of the technique.

Among electron microscopists it may be thought that,

with the current possibilities for obtaining direct images of

structures with a resolution approaching 0.1 nm, the

complication of having to interpret diffraction data may be

avoided. Also, in most laboratories, the special modifi-

cations of the equipment required for an optimum

production and recording of END patterns are not available.

It is the object of this review to provide examples of the

applications of END to systems for which it is very difficult

or impossible to gain equivalent information by direct

HREM imaging or other available techniques. Although the

examples are necessarily limited to those from a small

number of laboratories, it is thought that the variety of

applications to systems of industrial, technical and biologi-

cal significance will be sufficient to make the case for

extension to a wider range of applications.

In principle, it is possible to obtain diffraction patterns

from regions of diameter as small as the resolution limit for

dark-field imaging in a STEM instrument, namely less than

0.2 nm, since that resolution limit is an indication of the

diameter of the incident beam probe that can be formed.

There are many intriguing possibilities, yet to be explored,

for exploiting END from regions of this minimum size.

There is an obvious association of END with STEM

imaging. In normal STEM imaging, only one signal is

recorded for each incident beam position. Part of the

transmitted beam intensity is recorded for bright-field

imaging or integration over part, or all, of the scattered

radiation is recorded for dark-field imaging. But enormously

more information is available if the distribution of scattered

intensity is recorded from each image pixel. Some mention

of the possibilities for making use of this information will be

made at the end of this article, but initially we deal with the

much simpler cases for which beam probes up to 1 nm in

diameter can be applied to produce diffraction patterns

which may be interpreted following the well-known basic

methods for diffraction analysis.

As suggested in Fig. 1, the incident beam focused on the

specimen is necessarily a convergent beam so that

the incident beam spot in the diffraction pattern formed

on the observation screen is a circular disk. For a perfect,

very thin crystal, each diffraction spot is then a circular disk

of the same size, as seen, for example, in Figs. 3 and 5.

Provided that the crystal periodicities are smaller than the

incident beam diameter, the diffraction spot diameters are

smaller than their separations and there is no overlapping of

the spots. Then it is readily shown that the spot intensities

are independent of the coherence of the incident beam and

may be interpreted just as for a parallel-beam diffraction

patterns such as given by the selected-area electron

diffraction (SAED) method.

If the objective aperture is made so large that the

diffraction spots overlap, interference fringes appear in the

area of overlap. Also, if there is any deviation from perfect

periodicity of the specimen, such as would give rise to

streaking or diffuse scattering in a parallel-beam pattern,

this may give rise to interference effects in which coherent

electron waves incident from different directions can

interfere, resulting in perturbations of the intensity distri-

bution. These effects must be recognized. But with this

proviso, the interpretation of the diffraction patterns can be

straightforward.

For many applications of END, the specimen regions

examined are necessarily very thin, as in studies of

nanoparticles and nanotubes. Then the diffraction intensities

may be interpreted in terms of the simple kinematical theory

of diffraction. When the samples are thicker, as in the case

of studies of defects in crystalline films, the dynamical

diffraction effects become significant. Then one of the

commonly available programs for the computing of many-

beam dynamical scattering may be used for their

interpretations.

Fig. 1. Diagram of the main components of a scanning transmission electron microscope. Nanodiffraction patterns are formed on the screen and are viewed

with a TV and/or CCD camera. Additional (condenser) lenses may be placed before the objective lens to control the incident beam size and intensity. Post-

specimen lenses may be included to vary the diffraction pattern magnification.

J.M. Cowley / Micron 35 (2004) 345–360346

In a dedicated STEM instrument, the source of electrons

is a cold FEG for which the effective source size is 4–5 nm.

The strong objective lens, with a focal length of about

1 mm, produces an electron probe at the specimen level for

which the dimensions are limited by the objective aperture

size and the lens aberrations. The beam at the specimen may

be assumed to be completely coherent so that waves

incident at any angle may interfere if scattered into the same

direction. A condenser lens is inserted to allow convenient

variation of beam size and intensity. The real or virtual

objective aperture limits the beam convergence angle. The

diffraction pattern is observed on a fluorescent screen. One

or more weak post-specimen lenses may be inserted to

govern the pattern dimensions. The fluorescent screen is

viewed with a TV or CCD camera through a suitable light-

optical system, as described in previous publications

(Cowley and Spence, 2000; Cowley, 2003). Electrons

from any part of the diffraction pattern may be transmitted

through an aperture in the screen to enter an electron energy

loss spectroscopy (EELS) analyzer. Deflection coils are

included to scan the beam over the specimen and to deflect

the diffraction pattern over the aperture.

Equivalent electron-optical systems are incorporated in

modern TEM/STEM instruments, and a number of results

with such instruments have been reported (Matsushita et al.,

1996; Hirotsu et al., 1998). Such instruments are usually

designed to optimize the high-resolution imaging functions

in the TEM and/or STEM modes and the requirements for

these purposes may limit, to some extent, the adaption for

optimum END operation, but suitable nanometer-diameter

beams of reasonably high intensity may be produced when

the instrument is operated in the ‘analytical mode’. In this

mode, the lens system is adjusted to give the maximum

possible intensity within a beam focused to a diameter of the

order of 1 nm, as is required for the best spatial resolution of

the microanalysis techniques of EELS or energy-dispersive

spectroscopy (EDS). Then the END patterns are essentially

the same as in STEM instruments.

END differs from convergent beam electron diffraction

(CBED) in that much smaller angles of convergence are

used so that the diffraction spots, for small unit cell

materials, are small with respect to the spot separations, and

the emphasis is on obtaining diffraction patterns from very

small specimen areas rather than observing the variation of

diffraction intensities with incident beam direction, inter-

preted in terms of the dynamical scattering theory.

In recent years, the provision of FEGs for TEM/STEM

instruments has allowed considerable advances in the

capabilities of the CBED technique. From CBED patterns,

obtained from regions of perfect crystal of diameter as small

as 10 nm, it is possible to make highly accurate determi-

nations of crystal structures, including electron distributions

in inter-atomic bonds for relatively simple structures. This

technique and its applications are well described in the book

of Spence and Zuo (1992) and the special issue of

Microscopy and Microanalysis (Spence, 2003).

2. Experimental procedures

The most convenient methods for the alignment and

adjustment of a STEM imaging system differ greatly from

those used for TEM. Low-magnification ‘shadow images’

(point-projection images) of the specimen are seen on the

fluorescent screen when the objective lens is greatly under-

or over-focused. The magnification of these images

increases to infinity and reverses as the in-focus setting is

approached and passed. Alignment of the electron-optical

system with the objective lens is ensured when the center for

magnification growth stays constant.

Close to focus, the shadow image is greatly

distorted by the aberrations of the objective lens.

Because of the spherical aberration of the lens, the

magnification may be large and positive for the over-

focused electrons that make large angles with the axis and

large and negative for paraxial electrons. For electrons

coming at some particular intermediate angle to the axis,

the cross-over is at the specimen level and the

magnification is effectively infinite. Then a ring of

infinite-magnification is visible in the shadow image. As

Fig. 2. Shadow images formed on the viewing screen. (a) and (b) are images of the edge of a crystal, far under-focus and almost in-focus, showing the infinite-

magnification circle. (c) An under-focus image of a thin crystal of beryl showing the Ronchi fringes.

J.M. Cowley / Micron 35 (2004) 345–360 347

shown in Fig. 2(a) and (b), this ring is clear for the special

case of a sharp edge in the specimen, when the effect is

similar to that for the ‘knife-edge’ test used in light optics

(Cowley, 1979), or when the specimen is periodic and the

characteristic distortions of the fringes due to the

periodicity, the Ronchi fringes, appear (Fig. 2(c)) as in

the method due to Ronchi for testing the aberrations of

large telescope mirrors (Ronchi, 1964; Cowley and Disko,

1980; Browning et al., 2001). Astigmatism of the lens

may be corrected by observing the symmetry of the

infinite-magnification loop or of the Ronchi fringes. The

dimensions of the infinite-magnification loop serve as an

aid to setting the defocus value. If the objective aperture

is then inserted at the center of the infinite-magnification

loop, it is assured that the microscope is correctly aligned,

stigmated and focused for operation, and a diffraction

pattern of the illuminated region of the specimen appears

on the fluorescent screen.

With suitable detectors, a bright-field or dark-field

STEM image, or both, can appear on the display tube

screens when the incident beam is scanned over the

specimen. An electronic marker may be positioned over

any feature of the image. When the beam is stopped at the

position of the marker, the diffraction pattern of the

chosen part of the specimen is produced and may be

recorded photographically or digitally. With a TV/VCR

system recording 30 diffraction patterns per second, it is

possible, for example, to record the diffraction pattern for

each translation of the beam by as little as 0.1 nm during a

slow scan along any chosen line in the image, or for any

two-dimensional scan over the specimen. One advantage of

this mode of END recording is that it can be assured that

each pattern is recorded with a minimum of exposure of the

specimen area to the electron beam, so that the radiation

damage of the specimen may be minimized.

When the intense bright source of a cold FEG is focused

on a small area of the specimen, the intensity of irradiation,

and rate of the radiation damage is necessarily high. The

END patterns from most organic and biological specimens,

and for many inorganic materials, can be seen to disappear

within a fraction of a second, although some samples appear

to be surprisingly stable. Techniques can be used to ensure

that END patterns are recorded with the first electrons to

strike a chosen part of the specimen. For example, the

specimen area may be viewed using low-magnification

images for which the incident beam is scanned along well-

separated lines. The END patterns are recorded with a TV–

VCR system from the time that the beam is stopped. Then

several END patterns are recorded for even very sensitive

specimen areas before the pattern disappears. Alternatively,

after a STEM image is viewed to allow selection of an area

of interest and for focusing, the END patterns may be

recorded at TV rates as the beam is scanned across an

adjacent area which may be imaged later.

By use of such minimum-irradiation techniques, END

patterns have been recorded for a number of materials,

including clay minerals (Fig. 3) and some organic crystals,

for which the END pattern for a stationary beam disappears

within a fraction of 1 s.

The recording of the END patterns may be made in

conjunction with any of the modes of STEM imaging. The

STEM image contrast depends strongly on the configuration

of the detectors. In accordance with the Principle of

Reciprocity (Cowley, 1969), the use of a small axial

detector in STEM gives the same image contrast as for

normal bright-field TEM with parallel incident illumination.

For a detector of increased radius, within the incident beam

spot of the diffraction pattern, the contrast may be reduced,

but the resolution may be improved (Liu and Cowley, 1993)

with an annular detector collecting most of the electrons

scattered outside the incident beam spot, the efficient

annular dark-field (ADF) mode, giving Z-contrast, was

introduced by Crewe and associates (Crewe et al., 1968).

The high-angle annular dark-field (HAADF) mode, in

which only those electrons scattered beyond the boundaries

of the usual spot diffraction pattern are detected, is now

popular for high-resolution studies of crystals and their

interfaces (Pennycook et al., 1996). Techniques making use

of a thin annular detector provide possibilities for the

detection of particular specimen components and offer

possibilities for bright-field imaging with improved resol-

ution (Cowley et al., 1995). For any of these modes, except

possibly the ADF mode, the collection of the image signal

does not interfere with the recording of the END patterns.

Fig. 3. Electron nanodiffraction (END) patterns from small kaolinite crystals, (a) parallel with the silicate layers, spacing 0.72 nm, (b) perpendicular to the

layers, (c) as for (b) but radiation-damaged after less than 1 s exposure to the electron beam.

J.M. Cowley / Micron 35 (2004) 345–360348

The SAED mode commonly used with TEM imaging has

the disadvantage relative to END that the minimum

diameter of the specimen region giving the diffraction

pattern is usually greater than 100 nm, rather than 1 nm.

However, for some purposes it has advantages in that, for

perfect crystals, the diffraction spots are sharp, the

resolution in reciprocal space is high and the dimensions

of the patterns may be measured with relatively high

accuracy.

There are several means by which the END method may

be modified to overcome some of the disadvantages relative

to SAED. The area giving rise to the diffraction pattern may

be increased by applying a small, fast scan of the beam

during the recording of the pattern. An under-focusing of the

objective lens has the effect of increasing the area of

the specimen illuminated and may also decrease the sizes of

the diffraction spots for sufficiently small crystals (Cowley

et al., 2000). In a recently devised method to obtain a

parallel-beam diffraction pattern from a small specimen

region (Gao et al., 2003), the condenser lens forms a cross-

over just in front of the objective lens so that the objective

lens focuses this cross-over on the fluorescent screen. The

diameter of the region of the specimen with near-parallel

illumination depends on the diameter of an aperture placed

before the cross-over, but may be as small as a few tens of

nanometer and the diffraction pattern spots may be

correspondingly sharp.

For many purposes, and especially in the exploration of

the structure and composition of nanoparticles in composite

assemblies formed by novel preparatory methods, the

combination of END with the microanalysis methods of

EELS and X-ray EDS can be extremely powerful. In a

dedicated STEM instrument, the electron-optical require-

ments for all of these techniques are similar. The detection

of the analytical signals need not interfere with the

observation of the END patterns. The requirement for

high intensity in a nanometer-size beam is the same in each

case, and switching from one mode to another can be made

readily. In addition to the compositional information from

the analytical techniques, the information on valence states

of the elements present, obtained from the fine structure of

the EELS edges, can be valuable in characterizing unknown

phases.

3. Nanoparticles

One of the first, and most industrially important,

applications of END was in the study of the structures of

the very small metallic particles in supported metal catalysts

and the relationship of those particles to the supporting

materials. It has been found that, even with atomic

resolution in transmission electron microscopy, it is

frequently difficult to interpret the images of the nanocrys-

tals involved, seen in random orientations (Tsen et al.,

2003). The END patterns usually give a more direct

indication of the particle structure and orientation.

For the frequently studied case of platinum particles in

near-amorphous alumina, END showed the particles to be

single-crystals and the sizes and shapes of the particles

could be found readily from dark-field STEM imaging

(Pan et al., 1987). One unexpected result of these studies

was that, in some cases, the particles were shown to include

the oxides of platinum. The alumina support, normally

assumed to be ‘amorphous’, was found to be microcrystal-

line. The degree of crystallinity and the structure of the

alumina were found to differ widely for catalysts samples

obtained from different commercial sources.

One interesting case was that of ruthenium–gold catalyst

particles on a magnesium oxide support (Cowley and Plano,

1987). The question posed in this case was why the addition

of the seemingly inert gold particles should increase the

catalytic activity of the ruthenium. No direct answer to this

question was found, but the unexpected result was that it

was discovered that, below a certain size range, the

ruthenium nanocrystals tended to have a body-centered

cubic structure, rather than the hexagonal structure of the

bulk material. Such variants of metal particle structures

have been reported in a number of cases (Uyeda, 1991).

Another complication in the structures of small crystal-

lites, particularly of the noble metals, is the occurrence of

twinning, and especially multiple twinning to give decahe-

dral or icosahedral particles. This has been studied

extensively using high-resolution TEM (Marks, 1994), but

END is required to determine the extent to which such

multiple twinning occurs in very small particles, a few

nanometer in diameter. In one case (Cowley and Roy, 1982)

it was shown that while gold particles in the 5 nm size range

were commonly multiply twinned, the degree of twinning

decreased with crystal size and few twins occurred for

particles in the 2–3 nm size range.

Currently, many experiments are being made towards the

preparation of nanocrystalline phases, in order to explore

the possibilities raised by observations that nanocrystalline

materials may have interesting physical properties and

chemical behavior. For example, a study has been made of

the formation of diamond-like material deposited on

platinum wires by a CVD process (Mani et al., 2002).

Spectroscopic analysis of the product suggested the

formation of graphite, diamond and amorphous carbon.

END of the product confirmed this analysis and gave

interesting further details.

The diamond-like phase present appeared in several

forms. Some appeared to be cubic with a lattice constant of

0.36 nm, as in macroscopic diamond, but the characteristic

absences of the diamond structure diffraction patterns did

not appear. The forbidden 200 and 222 reflections were

present and quite strong, even for nanocrystals for which the

production of these reflections by multiple reflection or

dynamical diffraction effects should not be appreciable,

suggesting that this was the so-called n-diamond with

J.M. Cowley / Micron 35 (2004) 345–360 349

a simple face-centered cubic structure (Mani et al., 2002).

This would imply a completely new form of bonding for

carbon atoms. The mystery of this observation remains

unresolved. Other phases present as nanocrystals, as well as

larger crystals, included the hexagonal diamond structure,

Lonsdaleite (Bundy and Kasper, 1967) and a phase that

appeared to be cubic with a lattice constant of about

0.42 nm; the so-called i-carbon, of little-known structure

(Matyushenko et al., 1981).

Nanoparticles on the flat or convex extended surface of a

large particle may be imaged by use of the scanning

reflection electron microscopy (SREM) technique (Cowley,

2002a). Their END patterns may then be recorded, with the

limitation that half the pattern may be obscured by the

shadow of the surface. As an example, small crystallites of

Pd imaged on the surface of a large MgO crystal were seen

to change their shape under electron irradiation. END of the

surface regions of the crystallites revealed that the Pd was

oxidizing to form the oxide, PdO (Ou and Cowley, 1988).

Similarly, crystals of Ag evaporated on a crystal of MgO

were seen to change and small liquid-like, but crystalline,

regions were seen to form at the junctions between the Ag

and MgO. END showed these regions to be composed of the

oxide, Ag2O (Lodge and Cowley, 1984).

4. Nanotubes, nanoshells and nanobelts

At the time of his discovery of carbon nanotubes, Iijima

(1991) showed that it was possible to observe SAED

patterns from individual tubes and these patterns were

important for determining the chirality of the tubes. Later,

diffraction patterns were also obtained from single-walled

carbon nanotubes (Iijima and Ichihashi, 1993), although the

patterns were necessarily very weak because the scattering

came from relatively few carbon atoms and the tube

occupied only a very small fraction of the selected-area.

Such SAED patterns have contributed to the knowledge of

the structure of nanotubes. However, useful diffraction

information can be obtained only if the tube is of uniform

structure and is perfectly straight within the selected region

examined, usually having a diameter as great as 100 nm.

For a detailed study of the structure of a nanotube,

especially when the tube is bent, imperfect or faulted,

nanodiffraction has many advantages. The incident beam

diameter may be much less than the tube diameter, so that

diffraction patterns may be obtained separately from the

walls and the interior of the tube, as in Fig. 4. The detailed

structure at bends or faults in the tubes may be studied. The

structures of nanoparticles included within a tube or

attached to its walls may be determined. The diffracted

beam intensities are sufficient for convenient recording,

even for single-walled tubes.

An extended review of the application of nanodiffraction

to the study of carbon, and other, nanotubes and related

structures has been given recently (Cowley, 2003). Here we

mention only a few examples to illustrate the capabilities of

the END method.

It is well-known that single-walled nanotubes (SWnT)

are characterized by a diameter and a chirality. Their

Fig. 4. END patterns taken from series of patterns recorded as the beam traversed multi-walled carbon nanotubes. Patterns are from one side, the middle and the

other side of the tube. Patterns (a)–(c) are from a tube of circular cross-section. Patterns (d)–(f) are from a tube of pentagonal cross-section and show that one

side is a flat graphite crystal and the other side is strongly bent. The strong rows of spots from left to right in (a), (c), (d) and (f) are the ð0; 0; 2lÞ reflections from

the graphitic planes of the walls.

J.M. Cowley / Micron 35 (2004) 345–360350

helix angle may vary from 08, corresponding to the arm-

chair configuration, to 308, corresponding to the zigzag

configuration. The determination of the helix angle is of

importance especially because it determines the electrical

properties of the tube. It has been pointed out (Qin, 2003;

Gao et al., 2003) that for SAED with parallel illumination,

for the diffraction pattern from a tube of cylindrical

symmetry, the diffraction pattern intensities are properly

described in terms of Bessel functions. A simplistic

interpretation made on the assumption of diffraction from

two planar sheets of atoms, the top and bottom walls of the

tube, may give appreciable errors. For END, however, with

an incident beam of smaller diameter than the tube, the

assumption that the pattern is given by just the top and

bottom layers of the tube, separated in orientation by twice

the helix angle, is justifiable. Therefore, the helix angle may

be measured as half the angular separation of equivalent

diffraction spots provided that the incident beam is

perpendicular to the tube axis.

An atlas may readily be made to show the appearance of

the END patterns for the various helix angles and tilts of the

tube away from the orientation perpendicular to the beam.

This provides a means for the rapid identification of helix

angles from the END patterns, thus allowing the statistics of

tube structures to be accumulated in a reasonable time.

It has been suggested that for the commonly used

methods of SWnT production, the distribution of helix

angles is random. From the statistics of helix angles

measured from the rapid recording of END patterns,

however, it has been shown that for small regions of a

sample, of the order of 1 mm in diameter, there is a strong

tendency for all the tubes in each region to show much

the same helix angle (Kiang and Cowley, 2004). But the

preferred helix angle varies greatly from one region to

the next.

For multi-walled nanotubes (MWnT) and also from the

near-spherical nanoshells, which often appear in MWnT

preparations, nanodiffraction patterns often show that

several helix angles are present. It has been shown, in

fact, that when there are many layers of graphitic carbon in

the MWnT or the nanoshell, there is a tendency for the helix

angle of the layers to change after there have been about

four layers having the same helix angle (Liu and Cowley,

1994a).

Another feature that is found in some preparations of

MWnT is that the tubes are not made of cylinders of circular

cross-section, as is usually assumed, but may have a cross-

section which is polygonal, and most commonly, pentago-

nal. The first evidence for this configuration came from

high-resolution TEM images which showed a different

spacing of the layers on the two sides of the tube image

(Zhang et al., 1993) END patterns such as those of Fig. 4,

obtained as the incident beam was scanned across a tube,

showed clearly that in such cases the one side of the tube

gave the clear diffraction pattern of a well-ordered flat

graphite crystallite, whereas the other side gave the fuzzy

pattern characteristic of strongly bent layers, with an

increased inter-layer spacing.(Liu and Cowley, 1994b).

Patterns from intermediate positions across the tube were

consistent with flat regions of crystal with sudden changes in

their orientation, such as would be consistent with a

polygonal tube cross-section.

Further evidence on the form of the cross-section of

MWnT came from observations of asymmetry in the

diffraction intensities. The projection of the potential of a

curved carbon layer is asymmetric, with the projected

potential rising sharply on the outside of the curve and

falling off slowly on the inside of the curve. When there

is an appreciable deviation from the single scattering,

kinematical, diffraction condition, there is an asymmetry

in the intensities of the diffraction spots on the two sides

of the origin spot. Measurement of this asymmetry then

gives a measure of the curvature of the carbon planes

(Cowley and Packard, 1996). In this way, it was shown

that the curvature of the planes of carbon atoms may

vary from zero up to a maximum, as would be

consistent. With a polygonal cross-section. Also it is

possible to describe in detail, the configurations of the

layers of carbon atoms at the ends of the tubes where

there are sudden changes in the layer directions to form

the closures of the tubes.

When carbon nanotubes are formed in carbon arcs in the

presence of metals, it is common to see crystals of the metal

or its carbide enclosed within the tubes. In some cases, the

enclosed crystals are large enough to allow their structures

and orientations to be determined by HRTEM imaging and

by SAED. In other cases, as with yttrium, the enclosed

material appears from HRTEM and SAED to be amorphous.

However, END shows that the material is a nanocrystalline

metal carbide and allows the orientations of the nanocrystals

relative to the inner walls of the MWnT to be determined

(Cowley and Liu, 1994).

In recent years, considerable attention has been given to

the formation of nanorods, nanowires or nanobelts, which

also show promise of important applications as structural

units or tools for devices built on a nano-scale. Oxides such

as ZnO form long, smooth nanobelts having widths of about

100 nm and near-perfect crystal structure with the hexago-

nal c-axis perpendicular to the axis of the belt (Pan et al.,

2001). Many other inorganic materials, including semicon-

ductors form similar structures. END has been applied, for

example, to the study of nanobelts of B and BN (Otten et al.,

2002).

For many of these preparations, the crystals are large

enough to give good SAED patterns from which the crystal

structure and the overall configurations can be deduced.

END becomes important mainly for investigating local

structural variations, as at crystal defects, bends, junctions

or minor attached crystallites.

J.M. Cowley / Micron 35 (2004) 345–360 351

5. Ferrihydrite and ferritin

Ferrihydrite is a naturally occurring mineral and is also of

current interest because it occurs in industrial wastes and is

a source of heavy-metal pollution of streams. Characteriz-

ation of this material, beyond its nominal composition of

Fe1.55O1.66(OH)1.33, has been made difficult because it

occurs naturally, or can be synthesized, only in a

nanocrystalline state. Two forms of the mineral have been

distinguished. One is named the 6-line form because the

X-ray diffraction pattern contains only six rather diffuse

lines. The other is the 2-line form, with an X-ray diffraction

pattern containing only two very broad lines. The difference

between the two forms has been regarded as probably due to

a difference in crystal size. A number of attempts have been

made to deduce the structure of the 6-line form on the basis

of the limited amount of data available, and a hexagonal

structure with a- and c-dimensions of 0.30 (or 0.56) and

0.94 nm has been proposed (Drits et al., 1993).

END from both forms of ferrihydrite gave clear single-

crystal patterns from individual nanocrystals. It was

immediately obvious that more than one phase was present

in each case. For the 6-line form, it was found that about

60% of the particles had a hexagonal structure, similar to,

but not exactly the same as, that proposed for ferrihydrite

from the X-ray data; but there were other phases present

including the known iron oxide phases of hematite,

maghemite, Fe2O3 (not readily distinguished from magne-

tite, Fe3O4, which has a similar structure) and a material

similar to wustite, FeO, with a face-centered-cubic structure

showing extensive faulting (Janney et al., 2001). The

hexagonal structure deduced from X-ray diffraction on the

basis of the assumption that only one phase was present,

inevitably led to only an approximate structure which could

be refined somewhat on the basis of the END data.

END of the 2-line material showed clearly that the phases

present were distinctly different from those in the 6-line. A

hexagonal phase was present but had a different structure

from that in the 6-line, and there was a greater proportion of

magnetite (or maghemite) (Janney et al., 2000).

The resolution of the problem of the structure of

ferrihydrite then raised the question of the nature of

the iron-containing cores of the molecules of ferritin

which provide the principle means for the transport and

storage of iron in the bodies of living organisms, from

bacteria to humans. Early electron microscopy suggested

that ferritin consists of a spherical protein shell containing a

core of an iron compound about 6 nm in diameter (Chasteen

and Harrison, 1999). The cores were said to be composed of

ferrihydrite and HRTEM studies revealed the 0.94 nm

periodicity of the hexagonal ‘ferrihydrite’ phase in some

cores (Massover and Cowley, 1973).

END of ferritin from horse spleen and from humans

shows that the cores are similar to the ferrihydrite mineral in

being poly-phasic. The various iron oxide and oxyhydroxide

phases present are much the same as in the 6-line

ferrihydrite although occurring in slightly different pro-

portions (Cowley et al., 2000). Table 1 summarizes the

results for the various ferrihydrite and ferritin samples.

Evidence has accumulated in recent years, from

Mossbauer and other magnetic studies that the ferritin

cores in the brains of human patients with neurodegenera-

tive diseases, such as progressive supranuclear palsy (PSP)

and Alzheimer’s disease, may have a different composition

from that of normal human ferritin (Dobson, 2001).

Evidence from HRTEM and diffraction patterns derived

from Fourier transform of the images suggested that some

magnetite-like phases were present (Quintana et al., 2000).

END patterns from the ferritin molecules from the diseased

brains have now given a more detailed account of the nature

of the ferritin cores involved.

For these ferritin cores from diseased brains, the most

common phase is that similar to wustite, face-centered cubic

with a ¼ 0:43 nm and strong faulting (which may reflect a

variable composition) and a magnetite-like phase is also

prominent. The hexagonal, ferrihydrite phase is present in

only small proportions (Table 1) (Quintana et al., 2004).

This different balance of phases in such ferritins can

presumably provide evidence of the differences in the local

chemistry in the diseased brains.

Some limited END studies of ferritin from plants

(phytoferritin) and from bacteria, suggest that the material

is less well crystallized than in the case of mammals, but

also tends to show a predominance of the wustite-like and

magnetite-like phases.

Table 1

Approximate percentages of phases present

Phases Double-chain hexagonal Double-hexagonal Magnetite-like Wustite-like Hematite

Samples

2-Line ferrihydrite 40 – 20 40 –

6-Line ferrihydrite 5 60 15 10 5

Horse, human ferritin – 60 10 10 15

PSP, AD ferritin 5 15 30 45 5

Horse, human ferritin: average for horse spleen ferritin and human liver ferritin. PSP, AD ferritin: average for ferritin from brains of humans with

progressive supranuclear palsy and Alzheimer’s disease.

J.M. Cowley / Micron 35 (2004) 345–360352

6. Amorphous and disordered systems: quasicrystals

In all so-called amorphous materials, there is some

degree of short-range ordering or even medium-range

ordering of the atoms as a result of the preferred packing

of atoms or of the strong tendency for preferred bonding

distances and angles. From X-ray diffraction or SAED, it is

possible to derive the pair-wise, nearest-neighbor corre-

lations of atom positions but not the many-atom correlations

over distances of 1 or 2 nm. The use of high-resolution

electron microscopy to determine the structures, with even

the best resolution available, presents severe difficulties

(Van Dyck et al., 2003). The question arises as to whether

diffraction using electron beams of diameter about 1 nm can

give any more complete information. A comparison of the

information gained from HRTEM and END for a system

with medium-range order is given by Hirotsu et al. (1998).

With a nano-beam, the diffraction patterns from a thin

film of amorphous material do not show the diffuse haloes of

the SAED patterns. For well-developed medium-range

order, the END patterns may include single-crystal spot

patterns, as in the case of Hirotsu et al. (1998). For less-well

developed order, they may show only patches of maxima

and minima of intensity which vary as the beam is moved

over distances much less than 1 nm as different configur-

ations of atoms are illuminated. The problem is how the

information contained in these intensity distributions may

be applied to provide descriptions of the local ordering in

the material. One approach would be to attempt to deduce

the actual arrangements of all the atoms present and then

make some statistical analysis of many-atom ordering

parameters, or else to deduce the presence of various

types of atomic clusters. But this process seems excessively

difficult and tedious and a more direct approach is needed.

Related problems arise in the interpretation of diffraction

results from disordered alloys and other compounds, where

sharp ‘fundamental’ reflections arise from the spatially

averaged, periodic lattice, but diffuse scattering is produced

from the local variations from the averaged structure. For

disordered binary alloys, such as those in the Cu–Au

system, the diffuse maxima in the X-ray diffraction patterns

or in SAED patterns have been interpreted in terms of

micro-domains of ordered structures (Chen et al., 1979). For

some oxides such as LiFeO2, TiOx, and related materials,

the configuration of diffuse loops and streaks in the SAED

patterns have been attributed to particular types of atomic

clusters (De Ridder et al., 1977). In each of these cases,

nanodiffraction patterns show additional intensity modu-

lations, which are strongly dependent on the beam position

and should, in principle, be able to provide more detailed

information on local ordering. However, the best approach

to the derivation of a satisfactory description of the state of

the medium-range ordering is still a matter for discussion.

One can envisage a process, whereby if the nanodiffrac-

tion patterns could be inverted by Fourier transform, the

atomic positions in the two-dimensional projection of

the structure of a thin near-amorphous film could be

determined. The difficulty is the well-known ‘phase-

problem’ of kinematical X-ray or electron diffraction: in

recording the diffraction intensities, the phases are lost. This

problem possibly is overcome by use of the methods, well-

known in X-ray crystallography, whereby a complete

structure may be solved if part of it is known. If an initial

nanodiffraction pattern is obtained from a known structure,

a second nanodiffraction pattern from a region overlapping

part of the known structure and a part of the unknown

structure may be solved to give the atom positions in the

unknown region. Continuation of this process for successive

movements of the nano-beam may then allow the atom

positions for the projection of a large area of an amorphous

thin film to be determined. The difficulty then is to deduce

the three-dimensional correlations of atom positions from

the two-dimensional projections. Another scheme would be

to perform statistical analyses of the Patterson functions

(autocorrelation functions) obtained by Fourier transform of

the END pattern intensities (Cowley, 1981a).

Such processes would be very tedious and exacting in

terms of both the data-collection and the data analysis. A

better approach is to devise a scheme whereby the desired

information is derived directly from the observations. One

such scheme is the variable coherence imaging method, or

‘fluctuation microscopy’, devised by Treacy and Gibson

(Gibson et al., 2000) in which the speckle in the dark-field

image is measured as a function of the cone angle for

‘hollow cone’ illumination in HRTEM. Variation of the

cone angle has the effect of varying the size of the region,

which is illuminated coherently by the incident beam. These

authors suggest that an alternative method, equivalent

according to the reciprocity relationship, is to use a STEM

instrument with a thin annular detector of variable radius.

These techniques give a measure of the average correlation

lengths for medium-range ordering in the atomic

configurations.

Some less-complicated, although presumably less-accu-

rate measures of medium-range ordering may be obtained

directly from observations of END patterns (Cowley,

2002b). The pattern of diffuse maxima and minima in the

END patterns varies as the beam is moved across the

specimen. The amount of beam movement for which a

particular diffraction maximum persists may be taken as a

measure of the distance over which a particular atomic

configuration extends, and hence the correlation length of

the atomic positions. Alternatively, if the nano-beam is

defocused so that it illuminates an increasingly large area of

the specimen, the variation of the size of the diffraction

maxima in the END pattern with defocus may be related

to the size of the regions with correlated structure.

It was the application of SAED and imaging in a

HRTEM instrument which first revealed the existence of the

quasicrystalline state in which there is orientational ordering

but no translational periodicity, with local five-fold

symmetries (Shechtman et al., 1984). During the period of

J.M. Cowley / Micron 35 (2004) 345–360 353

intense interest and extensive exploration of the quasicrys-

talline state, the question arose as to development of the

quasicrystalline ordering from an initial, amorphous state.

Did the local five-fold and other symmetries exist in the

liquid or amorphous states of the various alloy phases? END

patterns from thin films of Mn–Al alloys in the near-

amorphous, freshly sputtered state (Robertson et al., 1988)

suggested that they did. END patterns from small regions

can be expected to show such symmetries, of course, only if

the symmetry axis is parallel to the incident beam and if the

center of the beam is close to the symmetry axis. However,

the recording of END patterns such as those shown in Fig. 5

suggested that local clusters within the near-amorphous thin

films show the same symmetry elements as were observed in

the quasicrystalline ordered films.

7. Coherent nanodiffraction: crystal defects

In all of the above reports of applications of END, it has

been assumed that the patterns can be interpreted as if they

were SAED patterns except for having larger spot sizes.

However, in many cases, account must be taken of the fact

that the incident beam in a STEM instrument is almost

completely coherent. For the small effective source sizes,

4–5 nm, of the cold FEG, the coherence width of the beam

at the objective aperture level is much greater than the

aperture diameter, usually 10 mm. Hence, the convergent

beam radiation on the specimen is coherent, and electron

waves coming from different directions can interfere

coherently if scattered into the same direction.

It has been shown (Spence and Cowley, 1978) that for a

perfect crystal, the diffracted intensities are independent of

the coherence of the incident beam, provided that the cone-

angle of the convergent beam is so small that the

neighboring diffraction spots do not overlap. When the

spots overlap, the interference of waves coming from

different directions gives rise to interference fringes in the

area of overlap. The relative phases of the diffracted beams

determine the positions of the fringes. Observation of the

fringe positions thus provides the possibility for solving the

phase problem of kinematical crystallography, allowing a

unique determination of crystal structure. The application of

this method, called ptychography (Hoppe, 1982) is, in fact,

somewhat more complicated since the relative phases of the

diffracted beams depend on the chosen origin, i.e. on the

position of the center of the incident beam. The relative

phases must be inferred from the relative movements of the

fringes in the various areas of spot-overlap as the beam is

translated over the specimen (Spence and Cowley, 1978).

Fig. 5. END patterns from quasicrystalline sputtered Mn–Al thin films. Patterns (a) and (c) are from annealed films, showing 5-fold and 3-fold symmetries.

Patterns (b) and (d) are corresponding patterns from the near-amorphous, as-grown films, showing that the same symmetries are present in small local areas.

J.M. Cowley / Micron 35 (2004) 345–360354

More recent investigations of the possibilities have been

made (Plamann and Rodenburg, 1998).

For imperfect crystals having boundaries, disorder or

faults, the parallel-beam diffraction patterns show continu-

ous distributions of scattering in streaks or diffuse patches,

and in coherent convergent beam patterns, interference

effects can arise even when the diffraction spots do not

overlap for the perfect lattice. The first observation of

these effects was the appearance of a splitting of the

diffraction spots when the incident END beam illuminated

the edge of a small crystal such as a cubic crystal of MgO

smoke (Cowley, 1981b). Subsequent exploration of this

effect showed that the diffraction spots could be split into

two arcs or could appear as thin bright rings, depending on

the size and shape of the crystal edge (Pan et al., 1989).

In fact, it seemed possible that detailed observations on the

spot splitting for various positions of the beam around the

crystal edges could be used to deduce the complete three-

dimensional shape of the crystal.

Related spot-splitting effects can be observed for internal

discontinuities in the structure such as planar faults, when

the incident beam is parallel to the fault plane. At the out-of-

phase domain boundaries of ordered alloys, there is a shift in

the effective origin of the unit cell which results in a phase

change for the superlattice reflections but not for the

fundamental reflections. Hence when the incident END

beam illuminates a domain boundary, the superlattice

reflections are split, but the fundamental reflection spots

are not. The form of the domain boundary can be deduced

from the nature and direction of the splitting of the

superlattice reflection (Zhu and Cowley, 1982). Similar

splittings of particular groups of diffraction spots have been

observed for the case of stacking faults in face-centered

cubic metals (Zhu and Cowley, 1983). In this case, there is

no splitting of the spots for the fundamental reflections, for

which (for hexagonal indices) h þ k ¼ 3n; but the spots are

split for other reflections, and the nature of the splitting

depends on the nature of the fault.

The diffraction spots in the END pattern from a single-

crystal overlap when the incident beam diameter is smaller

than the periodicities of the projected crystal unit cell. The

diffraction pattern intensity distribution changes as the

center of the incident beam is moved around within the unit

cell. This is readily observed for thin crystals having large

periodicities (Cowley, 1981c). From the observation of such

effects, it is possible to deduce, for example, the centers for

local symmetries of the atomic arrangements within the unit

cell. In principle, this offers a means for the determination of

the structures of crystals having large unit cells. The

detailed intensity distributions of the individual patterns

may be calculated using the computer programs for the

many-beam dynamical diffraction theory, making use of

periodic-continuation approximation to take account of the

non-periodic nature of the incident beam amplitude

distribution (Cowley, 1995). There have been no appli-

cations of this approach to the study of the structures of

perfect crystals, but applications to the determination of the

form of crystal defects have been made.

It has been known for many years that some diamonds

contain planar defects, but attempts to determine the nature

of these defects using X-ray and electron diffraction and

HRTEM imaging were indecisive. Various models for the

defects were proposed, some based on the assumption that

an aggregation of nitrogen atoms was involved (Humble,

1982). To resolve this question, a series of END patterns

were recorded as the incident beam was translated in steps

of 0.02 nm along a line perpendicular to the trace of one of

these defects in a thin crystal. The modifications of the END

intensity distribution as the beam crossed the defect were

clearly seen (Cowley et al., 1984). Comparisons were made

with theoretical patterns calculated using many-beam

dynamical diffraction programs and a periodic-continuation

assumption for the various postulated models. Agreement

was found for the model proposed by Humble (1982),

involving a particular ordered configuration of nitrogen

atoms in the plane of the defect.

An application of END to a different type of crystal

defect is given by the study of the modulated structure of a

high-temperature superconductor in which the END pattern

intensities were seen to vary with the modulation (Zhu and

Cowley, 1994).

8. Nanodiffraction and holography: atomic focusers

The production of coherent, convergent electron beams,

focused to form sub-nanometer cross-overs, has relevance

for the attempts to improve the resolution capabilities of

electron microscopes by the application of the techniques of

holography. The most thoroughly studied forms of electron

holography for resolution enhancement are the ‘off-axis’

form and the in-line form involving through-focus series,

applied in HRTEM instruments (Volkl et al., 1998). It has

been pointed out, however, that in accord with the

reciprocity relationship, there is a STEM equivalent for

each TEM form (Cowley, 1992) and some of these involve

the same electron-optical configurations as for END.

In his original proposal for holography, Gabor (1949)

envisaged that the coherent convergent beam formed by a

strong electromagnetic lens is placed close to a thin

specimen and forms the greatly magnified shadow image

which is recorded as the hologram, containing the effects of

interference between the incident transmitted beam and the

waves scattered by the object. Reconstruction of the object

wave from the hologram intensity distribution is then made

by back-transform in a light-optical system (or, later, by

digital processing) to give the transmission function of the

object plus a conjugate image, which can be made to be far

out-of-focus. A straightforward back-transform would give

an image with a resolution corresponding to the width of the

cross-over formed by the lens. However, the width of the

incident beam for a coherent electron wave is determined by

J.M. Cowley / Micron 35 (2004) 345–360 355

the aberrations of the lens. If the back-transform from the

hologram includes a correction of the wave function for the

phase changes due to the lens aberrations, an image of

greatly improved resolution may be possible.

Tests have been made of this principle, applied to a

shadow image formed with a stationary beam in a STEM

instrument and digital reconstruction from the recorded

hologram (Lin and Cowley, 1986). However, the method

has severe limitations. Gabor applied the method to a black-

on-white object (transmission function equal to 0 or1) and

showed that it worked if the black parts are much smaller

than the transparent parts but, in general, the object must be

very thin and scatter weakly, so that a projection

approximation is valid and the phase changes in the object,

relative to vacuum, must be small. Also the perturbations of

the wave by the aberrations of the probe-forming lens must

be known with high accuracy. Also the field of view is

small.

The requirement for the correction of the lens aberrations

in the process of reconstruction from the hologram may be

avoided if a sufficiently small cross-over can be formed. It

has been suggested by Smirnov (1999) that a cross-over

much less than 0.1 nm in diameter may be formed by use of

an atomic focuser. The electrostatic field around a single

heavy atom or about a row of atoms passing through a thin

crystal and parallel to an incident electron beam can act as a

convex lens, with a focal length of the order of 2 nm and

giving a cross-over of diameter 0.05 nm or less. Various

means have been proposed whereby such atomic focusers,

or the periodic arrays of atomic focusers produced by the

rows of atoms in a thin crystal in axial orientation, can be

applied to give ultra-high resolution imaging in a TEM or

STEM instrument (Cowley et al., 1997). For a thin crystal of

gold in [100] orientation, computer simulations suggest that

if one row of gold atoms in the crystal is used to form a

cross-over as a source for Gabor-type in-line holography, it

should be possible to form images with a resolution of about

0.03 nm.

The limitation of the in-line type of holography to thin,

weakly scattering objects, and the difficulty that a conjugate

image is formed in the reconstruction process, may both be

overcome by combining the use of a small cross-over from

an atomic focuser with an off-axis holography scheme. By

placing an electrostatic biprism in the illuminating system

of a STEM instrument, two equivalent cross-overs are

formed in the object plane, as suggested in Fig. 7. One of

these passes through an atomic focuser, to form a fine cross-

over and can serve as a reference wave, passing through

vacuum, while the other passes through a specimen, and the

hologram is formed by the interference of the two waves on

a plane at infinity. Reconstruction from this hologram, as in

normal off-axis TEM holography, then gives a central

distribution and two side-bands. One of the side-bands gives

directly the complex amplitude distribution of the specimen

wave from which, in the projection approximation, the

phase function corresponds to the projected potential

distribution of the object. No correction for the lens

aberrations is needed and the resolution corresponds to the

size of the atomic focuser cross-over and so is about

0.05 nm (Cowley, 2003).

The observation of END patterns from the near-spherical

carbon nanoshells, often formed in conjunction with carbon

nanotubes, showed a splitting, or multiple splitting, of the

individual diffraction spots which at first seemed mysterious

but was explained in terms of an atomic focuser effect

(Cowley and Hudis, 2000). A graphite crystal, included in

one wall of the nanoshell, acted as an atomic focuser to form

a periodic array of cross-overs and this array could form

atomic focuser images of the periodicities within an almost

parallel graphite crystal in the opposite wall of the

nanoshell. One such image was formed within each

diffraction spot, as shown in Fig. 6. The imaging process

Fig. 6. END patterns from the graphite crystals in two walls of a carbon nanoshell, separated by about 100 nm in the beam direction. (a) shows a radial splitting

of the spots. The inner ring is of (1,0,0) spots. (b) and (c) taken with a larger objective aperture, reveal that in each diffraction spot there is an imaging of one

crystal by the atomic focuser action of the other crystal.

J.M. Cowley / Micron 35 (2004) 345–360356

applied also to non-periodic objects such as tungsten atoms

sitting on the nanoshell walls and the resolution was close to

the theoretical value given by the channeling of electrons

along the rows of atoms in a graphite crystal, namely

0.06 nm.

9. Nanodiffraction and four-dimensional imaging

As pointed out previously, in normal bright-field or dark-

field STEM imaging, a detector produces one signal from

some part of the diffraction pattern formed by the incident

beam and this signal modulates the intensity on the display

tube screen to produce the image. But, in principle, a much

greater amount of information is available since, for each

incident beam position, a two-dimensional END pattern is

produced on a viewing screen and all such END patterns

may be recorded and may be interpreted in terms of

structural variations in the specimen or else may be made

the basis for the derivation of images with ultra-high

resolution.

It was suggested by Rodenburg and Bates (1992) that the

data collected from the END patterns could be considered to

constitute a four-dimensional intensity function, with the

two dimensions of the image and the two-dimensional data

from the END patterns. They showed that, if a particular

two-dimensional section of this four-dimensional function

is taken, the result could give an image of the specimen

with twice the resolution of a normal STEM image.

Experimental tests made with visible light and with low-

resolution electron-optical imaging confirmed the correct-

ness of the principles of the scheme (Nellist et al., 1995).

A later formulation of the theory of this process

(Cowley, 2001) suggested that the possible improvement

of resolution is not limited to a factor of two, but, since the

phase modulations of the waves due to the lens aberrations

are effectively cancelled out, the resolution is limited only

by the size of the objective aperture and by the incoherent

imaging factors such as mechanical and electrical instabil-

ities. A one-dimensional test of the scheme, making use of

the essentially one-dimensional, non-periodic, projected

potential distribution of the wall of a carbon nanotube, gave

a reconstructed image with an apparent resolution of about

0.1 nm, using a STEM instrument having a normal

resolution of about 0.3 nm (Cowley and Winterton, 2001).

The END patterns, obtained with a large objective

lens aperture, appeared more like ronchigrams than just

a larger-beam END pattern, and were recorded at 0.1 nm

steps along a line perpendicular to the nanotube wall.

One difficulty with applying this scheme to the imaging

of more general, two-dimensional objects is that the amount

of intensity data to be collected and analyzed is enormous

for any but the smallest of specimen regions. The recording

time for the END patterns is necessarily long, which implies

that difficulties may arise from specimen drift and

irradiation effects. Also the difficulty arises, as in in-line

holography that the method is limited to very thin, weakly

scattering objects for which a linear-imaging approximation

applies. It has been pointed out that, in principle, these

limitations may be avoided if the collection of the four-

dimensional data is combined with an off-axis holography

scheme as in Fig. 7, and if the taking of the two-dimensional

section of the four-dimensional intensity function is done

during the data-collection process, rather than in the

manipulation of the four-dimensional data, by use of an

automated scanning scheme using variations of the biprism

voltages (Cowley, 2004). Then only one two-dimensional

scan is involved, the recording time is greatly reduced

and the ultra-high resolution image is produced rapidly. As

would be expected from the reciprocity relationship,

equivalent schemes, involving the collection of data in

modifications of the off-axis holography arrangement for

TEM instruments, also appear to be feasible.

10. Conclusions: further developments

Because so few people have access to instruments

optimized for the convenient observation and recording of

END patterns, the range of applications of the technique that

Fig. 7. Diagram of the action of a biprism, placed in the illumination system of a STEM instrument, to form two focused probes at the specimen level.

Interference of the waves from the two probes, one passing through vacuum and one through the specimen, forms a hologram on the viewing screen. There are

several modes of reconstruction from the hologram intensities that may give ultra-high resolution images.

J.M. Cowley / Micron 35 (2004) 345–360 357

can be reported is necessarily limited. However, it is hoped

that sufficient examples have been given to show the value

of the END data for a variety of problems in the materials

science of nano-scale systems. Potential applications to

show the structures of quantum dots and many components

of sub-miniature devices must abound. There are many

more applications to be made to poorly crystallized minerals

and to inorganic components of biological systems. With

the application of minimum-irradiation techniques and rapid

recording, some biological materials may also be investi-

gated. We hope to encourage the use of the facilities

available in modern TEM/STEM instruments, as well as the

few dedicated STEM instruments, for these purposes.

Another area where the few existing experimental results

suggest that a much broader range of applications may

usefully be considered is in the study of nanometer-size

projections or added crystallites on the faces of large

crystals or on any relatively smooth solid surface. Imaging

of such objects in the SREM mode gives better resolution

than the best SEM methods, and the structural information

given by the END patterns is not available from the

scanning probe microscopies.

Of the existing dedicated STEM instruments, few have

been optimized for the observation and recording of END

patterns, and of those few, some suffer from the ravages of

time and do not have even the limited stabilities of their

youth. This is unfortunate because a wealth of opportunities

awaits the systematic exploration of the capabilities of

coherent diffraction with small beam sizes. The few

experiments, which have been done, are sufficient to suggest

important possibilities for the detailed investigation of

crystal structures and crystals defects. The scattering angles

available for nanodiffraction are greater than those used for

HRTEM and the precision of the determination of structural

details may be correspondingly greater. Under dynamical

diffraction conditions, the diffraction intensities are sensitive

to the relative phases of the scattered waves, and the phase

problem of kinematical diffraction does not exist.

An important advance has come with the development of

the technique for obtaining diffraction patterns from very

small regions using parallel coherent beams of small

diameter (Zuo et al., 2003). These authors have combined

this microdiffraction method with the refinement of

structure using the iterative phase-refinement algorithms

technique of Feinup (1987) with an aperture image as a

support basis, and have derived the structure of a double-

walled carbon nanotube with high-resolution and high

contrast. This method may be applicable for greatly

improved structure determination of small periodic or

non-periodic objects. The success of this approach for

high-resolution imaging of arbitrary objects using X-rays

and electrons (Weierstall et al., 2002; He et al., 2003) also

suggests that its employment with coherent electron beams

should be productive. Modifications using coherent con-

vergent beams may also be feasible.

Acknowledgements

Most of the experimental work reported here was

performed with the instrumentation of the ASU Center for

High Resolution Electron Microscopy, and the author is

grateful to the Director and staff of the Center for their

support.

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