Application of electron optical techniques to the study of amorphous materials (submitted for the Special Issue in honour of David Cockayne)

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in honour of David Cockayne)

Archie Howie

To cite this version:

Archie Howie. Application of electron optical techniques to the study of amorphous materials (sub- mitted for the Special Issue in honour of David Cockayne). Philosophical Magazine, Taylor & Francis, 2010, pp.1. �10.1080/14786431003636105�. �hal-00595949�

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Application of electron optical techniques to the study of amorphous materials

(submitted for the Special Issue in honour of David Cockayne)

Journal: Philosophical Magazine & Philosophical Magazine Letters Manuscript ID: TPHM-09-Sep-0391.R1

Journal Selection: Philosophical Magazine Date Submitted by the

Author: 04-Dec-2009

Complete List of Authors: Howie, Archie; University of Cambridge, Physics

Keywords: amorphous solids, electron diffraction, electron microscopy Keywords (user supplied): coherence volume

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Application of electron optical techniques to the study of amorphous materials

(Submitted for the special issue in honour of David Cockayne)

A. HOWIE

Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, UK

Email address: ah30@cam.ac.uk

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Abstract.

Despite the sub-angstrom resolution capability of modern transmission electron

microscopes, the elucidation of the structure of amorphous materials by direct imaging methods is still challenged by problems of projection and data presentation identified over twenty years ago. The classical diffraction analysis, leading to the radial

distribution function not only retains its essential importance but has been extended to deal with small scattering volumes in inhomogeneous samples by the use of focused electron probe illumination but retaining orientation averaging. Fluctuation microscopy provides a significant advance in the detection and statistical representation of medium- scale spatial variations in amorphous structure. Issues of scattering coherence are crucial in both these two approaches. Further opportunities for use of focused probes include the study of angular correlations in nano-diffraction patterns as well as fine structure in energy loss spectra. Finally there may be a return to the direct imaging approach thanks to recent advances in electron tomography and in aberration-corrected depth sectioning.

Key words: amorphous solids, electron diffraction, electron microscopy, coherence volume.

1. Introduction

High resolution transmission electron microscopy (HRTEM) has made many decisive contributions to direct structure determination in materials science but, in the case of amorphous specimens, it has proved very difficult to rival let alone surpass the more modest, spatially averaged statistical test of short range order (SRO) offered by the radial distribution function (RDF) analysis deduced from classical diffraction data. Initial experience with HRTEM bright and dark-field images, summarized in §2 below, revealed two serious problems [1]. Firstly the images, being essentially 2-d projections of the 3-d structure, are usually dominated by the purely random statistical overlap of contributions from atoms too far separated in the beam direction to be structurally correlated. Secondly, in cases where significant structural effects were identified in HRTEM images, these were not usually readily appreciated by non-experts and it proved impossible to devise some simple statistical conclusion to supplement the classical RDF function. The use of hollow cone illumination or equivalently annular dark field (ADF) imaging in scanning transmission electron microscopy (STEM) was then proposed as a means of reducing the unwanted coherent interference effects between widely separated atoms [2] but not actively followed up at that time.

In this impasse, micro-diffraction and nowadays nano-diffraction provide attractive alternatives to direct imaging particularly in samples with obvious spatial variations and are discussed in §3. Detailed analysis of the spotty ring structure of these diffraction patterns has presented serious but interesting challenges. Much greater progress has however been made by azimuthal averaging of the patterns and extraction of local RDF functions making the usual assumption of isotropic structure [3].

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Almost two decades later, as discussed in §4, the hollow cone imaging ideas were substantially reworked and refined in the development of the fluctuation microscopy imaging method [4] through which spatial variations in amorphous structure are now routinely and successfully presented in the ideal simple angular scattering or radial separation plots [5]. Here the degree of spatial resolution and consequent inter-atomic scattering coherence have emerged as crucial parameters.

With continuing improvements in electron optics and in electron microscopy techniques it is often profitable to revisit previously intractable or difficult problems. We therefore conclude in §5 with a brief note of the possible impact on amorphous structure studies as a result of aberration correction (with improved depth resolution as well as lateral

resolution), of electron tomography (with the capability of full determination of 3-d structures).

2. Early work in HRTEM imaging

The bright spots visible in dark field images of amorphous materials [6,7] and later the prominent small patches of fringes found in bright field images taken with tilted illumination [8] were initially interpreted as arising from nm scale crystallites or other ordered regions. However image simulations soon demonstrated that the dark field structure could be an essentially random speckle phenomenon with bright spot sizes purely related to the spatial resolution of the microscope and with no systematic dependence on defocus for its overall appearance. Likewise, when random effects

dominate, the fringy appearance of bright field images is mainly determined by the image power spectrum and could be enhanced by a chromatic aberration filtering effect in tilted illumination [9]. By using axial illumination [10] and in some cases through focal series reconstruction [11], bright field images free of dangerous filtering effects were obtained where extended patches of fringes were generally much less noticeable. The image contrast mainly arises from alignments of atoms in the beam direction and in all but the thinnest samples such alignments involve atoms separated by over 3nm and thus reflect purely random statistics unless some unusual long range order is present. In such cases where the scattered waves have random phases, the only information provided by the high resolution bright field image is the power spectrum which is equally available from the diffraction pattern.

Image simulations carried out for a 1.5nm diameter crystallite surrounded by a region of random network showed that the crystal remained easily visible only when the total sample thickness was less than about 4nm [12]. For widely separated crystallites or other ordered regions comprising just a very small fraction of the sample volume, direct imaging would none the less often be a much more sensitive detector of their presence than classical diffraction broad beam data. These early micrographs gave a strong impression of the great variability of amorphous materials even when the averaged properties indicated by diffraction data seemed closely similar [1]. Amorphous Si or Ge films often seemed indistinguishable from random structures whereas silica was much more variable. In cases, where the images showed some noticeably larger patches of

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fringes in bright field or dark field spots larger than the instrumental resolution and exhibiting an optimum focus condition, more rigorous tests were devised and applied to assess their significance. These tests have included checks that the appearance of order disappeared and was not simply rearranged to new regions after phase randomization of the images whilst preserving the power spectrum [13] and autocorrelation measurements [14,15].

Although it was appreciated that stereo microscopy might provide a solution to the imaging overlap problem just described, it seemed in these early days virtually impossible to apply this successfully at the very high spatial resolution involved, particularly in the absence of obvious image markers enabling a particular region to be identified. Hollow cone illumination however allows images from different viewing directions to be incoherently and accurately superimposed as illustrated in fig. 1(a). In conventional TEM this can be achieved for a fairly small range of different viewing directions either with an annular illuminating aperture or by rocking the incident beam direction. By the reciprocity principle, the same result can be obtained more easily and over a larger angular range in STEM ADF imaging as indicated in fig. 1(b). The application of hollow cone dark field imaging to amorphous structures was thus investigated both experimentally and theoretically [2].

At each image point r = (x,y,z), where z denotes the plane of focus, the contribution to the coherent dark field intensity I from a particular incident plane wave component, defined by the vector K in fig. 1, is given by the expression

(

^,

)

jaj

(

j^,z zj

)

^exp

(

i ^. j

)

²

I r K = Σ ρ−ρ − − Kr (1)

Here the atomic image function for an atom j at position r_j = (ρρρρj, z_j) = (x_j,y_j,z_j), with z_j its distance from the focal plane, is given by

( ) ^{( )} ( ( ) ( ) )

q

ap

q j

j j

j

j z z i f K M K i i d

a γ θ σ

π

λ ^_^{ −}

∫

^{− −}



= 

−

−ρ q r r

ρ exp( ( )) exp .

, 4 ₂ (2)

where for simplicity the variation of the atomic scattering amplitude f and Debye Waller factor exp(-M) with position q in the objective aperture have been ignored. For later convenience in §3 and §4 below, we may note that when the radius Q of the objective aperture (or in STEM the probe forming aperture) is not too large we can ignore both the lens aberration term γ and the z component of q. Eqn. (2) then reduces to an Airy disc form

( ) ^{( )} ( )

j j j

j

j Q

Q Q J

K M K

f i

a

ρ ρ

ρ ρ ρ

ρ −

− −



 



= 

− ₂ ² 2 ¹

)) (

4 exp( π

π

λ (3)

For discussing speckle effects, it is convenient to split the coherent dark field intensity of eqn (1) into a sum of single atom and different atom contributions I₁ and I₂ respectively.

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( )

(

, ,

)

*exp

(

.(

)

exp

(

ρ cos^φ

)

,

, 2

2 1

j l j

l z l

j j l j j j

iK z

z iK a

a K

z I

a z

I

ρ ρ ρ

ρ

−

− Σ

=

∆ Σ

=

∆

≠

(4)

Referring to fig. 1(b) it can be seen that K has components Kz =2ksin²(α/2) and K_ρ = ksin(α). In general the speckle effect, arising from the superposition of atoms closely aligned along the optic axis, is dominated by the second term with say N(N-1) contributions compared with the N contributions from the first term. To include the spread of angular illumination (assumed incoherent) we have to integrate eqn. (4) over the angular ranges 0< φ <2π and α₁< α < α₂ corresponding to points in the condenser aperture or to points in the annular detector in the case of STEM. After the azimuthal integration overφ, the expression for I2 becomes

(

ρ α

)

α

α sin

sin 2 2 2 sin

sin 2 2 cos

1

0 2

2 ,

2 jl j Ajl kzjl Bjl kzjl J k jl

I 



 



 

 + 



 

 Σ 

= _≠ (5)

where Ajl =

(

ajal * +aj*al

)

Bjl =i

(

ajal* −aj*al

)

(6) ρ_jl = ρ_j −ρ_l z_lj =z_l −z_j (7) Integration over a sufficient range of α, will average out the oscillations of the Bessel function except for very small values of ρjl. Noting further that the Bjl terms will be close to zero when ρjl is small, the interference effect will also be restricted to atoms separated in the beam direction by less than zjl ≈ λ/(8sin²(αm/2)) where αm is an angle typical of the annular detector. Gibson and Howie [2] pointed out that the effect of these integrations is to introduce a cigar-shaped coherence volume within which the two atoms j and l must both lie if interference to occur. For the values of α in excess of 50mrad readily accessible in ADF STEM, we might expect this second effect to be useful in reducing the random speckle in typical dark field images of amorphous materials. Preliminary experiments suggested in fact that such reductions in speckle did occur leaving residual bright regions which could be more reliably linked to genuine regions of local order [2].

The problem of finding a convenient quantitative means of presenting any signs of local order discovered in this way was however not addressed though the possibility of mapping I² and higher moments of the dark field intensity was mentioned.

This apparently promising line of incoherent dark field imaging of amorphous materials was regrettably not pursued further at the time. Indeed it was not realized until some years later [16,17,18] that, in providing a basis for the general phenomenon of incoherent ADF STEM imaging, the simple effect of annular geometry described by these equations is a significant factor in addition to the arguments about thermal diffuse scattering that had previously been advanced [19]. As already noted above, for atoms with appreciable lateral separation ρjl, interference (i.e. transverse coherence) is suppressed by the annular detector geometry. For overlapping atoms such as those with separation distance zjl in the same column of a crystal, the role of phonons is more important in suppressing the longitudinal coherence [18].

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3. Nano-diffraction methods

To discuss the classical diffraction method it will be adequate here to start with the expression for the diffracted intensity from a monatomic sample neglecting constant pre- factors

( )

f

( ) (

K M K

)

jl

[

i

(

j l

) ]

I K = ² exp −2 ( ) Σ _, exp K.r −r (8)

In this approach, the overlap problem of high resolution amorphous imaging, is greatly reduced by making the assumption of statistical isotropy. We then find the Debye [20]

expression for the diffracted intensity I(K)

( )

= − + − Σ

_∫ ( )

dV

Kr r Kr g M f

M Nf

K I

jl jl jl

j

) ) sin(

2 exp(

) 2

exp( ²

2 (9)

Here g(rjl) is the pair distribution function measuring the probability of finding an atom j at distance r_jl from atom l. This equation already demonstrates some of the effects of projection for randomly oriented bonds since the first diffracted maximum occurs near Kr_jl = 5π /2 rather than at 2π where it would be for atom pairs aligned with K. Still more of the problems experienced in direct imaging are avoided in the classical diffraction approach by constructing the reduced intensity function Φ(K) = [I(K) – Nf²(K)exp(- 2M)]/ Nf²(K)exp(-2M) which measures the deviation from purely random behavior and can be directly related by a Fourier transform to the reduced density function G(r) = 4πr[g(r) – ρ] where ρ is the mean atomic density. A useful indication of the averaged significant structure then results in the form of simple plots of Φ(K) vs K or G(r) vs r.

This diffraction technique is long established in the scattering of X-rays and neutrons but, for the study of thin films, intergrannular layers and other nanovolumes, electrons are potentially unrivalled because of their high scattering cross section and our ability to focus them to small spots. There were some pioneering applications of electron diffraction to amorphous or highly disordered solids in the 1960s when removal of inelastically scattered electrons by energy filtering became more straightforward but the whole field has become much more active following the work of David Cockayne and his colleagues (for a detailed review see [3]). Their determination of the bond lengths in the C₇₀ molecule proved to be an early and striking success [21]. The consistency of their results with the full molecular structure was ensured by the ingenious procedure of relaxing the molecule through a simulated annealing algorithm [22] using –the measured G(r) as an interatomic pair potential. A subsequent and impressive comparison of their bond length data with various theories is given in [23].

These C70 results were obtained with broad beam illumination but spurred interest in nanodiffraction work making use of the massive developments in instrumentation that

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had taken place since the field was first pioneered [24, 25]. The extraction of the RDF function from typically 1-2 nm diameter local areas in for example intergrannular phase regions conveniently selected in TEM images has proved to be very fruitful. Individual nanodiffraction patterns display azimuthally structured or spotty diffraction rings so that azimuthal averaging has to be carried out before making the traditional analysis. These developments and the results obtained have been fully reviewed in [3, 26]. Here we concentrate on the effects on the simple plane wave illumination theory of using focused illumination and of possible departures from isotropy in such nano regions.

The effects on classical diffraction and RDF analysis of using an annular range of incident beam directions have been investigated experimentally and computationally for both incoherent illumination [27,28] and fully coherent illumination [29]. Completely incoherent illumination will not of course exhibit any spatial localization but the experimental spot sizes used have diameters typically about twice as great as could be produced with fully coherent illumination with the aperture used and so fall between these two extremes. The effect of an angular spread of incoherent illumination can be corrected by applying a deconvolution procedure to the measured diffracted intensity I(K). Although the need for a two-dimensional deconvolution was not clearly

emphasized, the experimental as well as computational results indicated that the main low-order peaks in the RDF function were not seriously affected [27,28]. In the case of coherent focused illumination, the effect was studied in amorphous Si both by simulation and by observation with different probe sizes [29]. The conclusion again was that the positions of the low order peaks in the RDF function are preserved for probe diameters greater than 1.2nm.

A different insight into these effects can be got by looking directly at the effect in eqn. (8) of including a range of incident plane waves as was done in §2. For the fully incoherent case we have to integrate the diffracted intensity over a range of incident q vectors within the illumination aperture shown in fig 1(b) (dividing by the aperture area for

normalization) to obtain the result

( ) ( ) ( ) { [ ( ) ] ( ) }

( ) ( ) ( )

jl jl jl

jl inc

jl inc l j j

l j inc

Q Q J Q

qdqd iq

K F

K F i

N M K

f I

ρ ρ π

ϕ φ

ρ ρ

ρ

1 2

, 2

2 cos

exp

. exp 2

exp

=

=

− Σ

+

−

=

∫

≠ K r r

K

(10)

The correction factor is just the Airy function of the probe forming aperture.

In the coherent illumination case the integration over q has to be done on the expression for the diffracted amplitude before calculating the intensity. The correction factor is then the product of two Airy functions.

( ) ( ) ( )

l l

j j jl

coh Q

Q J Q

Q J K

F ρ ρ

ρ ρ ρ

ρ ρ ρ

−

−

−

=2 ¹ − 2 ¹

ρ (11)

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To be detected in incoherent illumination (where no probe position is defined), it is then clear that the projected bond length ρjl must simply be less than the about 2a = 2π/Q, the diameter of the resolution cylinder indicated in fig. 2 (a). Longer bonds will contribute only if they are sufficiently aligned to the z axis. The corresponding peak in the I(K) plot will therefore move slightly outwards. For coherent illumination, both projected atom positions must lie within the resolution volume so many fewer atom pairs can contribute.

Summing up the results of many different probe positions effectively scrambles the phases of the incident plane waves and therefore reproduces the incoherent correction factor. Apart from restricting the volume sampled, coherent illumination will produce further filtering effects on bond lengths and orientations. Consider a bond of projected length ρ_jl < 2π/Q as indicated in fig. 2(b). The area available for its projected centre to occupy lies within an ellipse of major axis a = π/Q and minor axis b = π/Q – ρjl/2. The filtering factor is therefore roughly 1- Qρjl/2π.

For reliable application of the classical diffraction method it is obviously important to be confident about the validity of the assumption of structural isotropy. Significant failures, potentially signaled by an azimuthal φK dependence of the diffraction ring structure, could however be hard to detect in rather noisy nano diffraction patterns. Furthermore, any simple structural dependence only on θ, the angle with the optic axis would not result in any such obvious effect. In the case of thin film structures, a dependence of bond orientation relative to the film normal could easily be imagined and would not be obvious unless the film were tilted to large angles. Some idea about the likely importance of departures from isotropy can be obtained by reworking the Debye equation (9) for some specific angular bond angular distributions B(θ,φ). Instead of the isotropic function B₁ = 1/4π, we can consider for example a simple θ-dependent function B₂ = [(1-f) + 3f

cos²(θ)]/4π or a similar function relative to the x axis B3 = [(1-f) + 3f sin²(θ)cos²(φ)]/4π.

This last function could be relevant near a grain boundary or other interface normal to the x direction. Analytical expressions for these two examples can then be found to replace in eqn ( 9) the Debye factor D₁ = sinZ/ Z where Z = Kr_jl . We find

( ) ( ) [ ( ) ]

( ) [ ] [ ]

K

K Z

Z Z Z Z Z f

Z

Z Z Z f Z

Z f f

D

Z

Z Z Z f Z f Z f

D

φ

φ cos2

2

sin cos

3 sin 3 3 2

cos sin

3 sin 1 2

,

cos sin

3 1 sin

3

2

3 3 2 3

−

− −

− −



 

 +

=

+ −

−

=

(12)

In fig. 3, D₂(0.5) and D₂(-0.5) are compared with the Debye factor D₁. As expected, the diffraction ring moves outwards for f > 0 corresponding to a preponderance of bonds aligned with the z-axis. In fig. 4, D₃(0.5,0) and D₃(0.5,π/2), are compared with D₁. For f

> 0 the distribution B3 describes an enhanced bond concentration aligned with the x-axis.

The ring pattern exhibits as expected an azimuthal effect being moved inwards for K directions parallel to the x-axis (φK = 0) and outwards for directions normal to this. Apart from the shift in ring positions, it is noticeable in all these cases that there can be

considerable changes in peak heights as a result of departures from isotropy. An inward shift of the ring position seems to be correlated with an increase in height. For more

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realistic modeling of departures from isotropy, the parameter f would need to be a function of bond length r_jl. Eqns. (12) remain valid but are now functions of both r_jl and Kr_jl and not just Kr_jl.

4. Fluctuation microscopy

Almost two decades after the work described in §2, a much more fruitful way of using the hollow cone imaging approach for amorphous structure investigations emerged [16,30]. Fluctuation microscopy is still advancing [31,5] as so far the best means of investigating MRO. Rather than studying the dark field image intensity I(r) in full image detail, the fluctuations, i.e. spatial variations, of the intensity are measured via the second moment function V(K,Q) = {< I²(r) > / < I(r) >² } - 1. The observed dependence of V(K,Q) on the reciprocal space quantities K and Q (denoting the radii of the annulus radius and objective aperture respectively) can then be compared with computations for any assumed structure.

A crucial step in the development of fluctuation microscopy, running counter to most deeply embedded opinion in electron microscopy, was to appreciate the importance of operating at Q corresponding to a spatial resolution of 1.5nm or even worse. Whether this was realized in advance of the development of the method or was simply a consequence of the limitations of the hollow cone TEM instrumentation initially used is immaterial.

At such modest spatial resolution, we can ignore defocus and other aberration effects so that eqn. (2) becomes

(

j

)

j

^{( )} (

j

)

j i f K M K Q AQ

a ρ ρ  − ρ−ρ



 



= 

− ₂ exp( ( ) ²

4 π

π

λ (13)

where A(x) = 2 J1(x)/x is yet again the usual Airy function. The term ajal* + alaj* is then the main factor contributing to the coherent intensity in eqn (5) and requires that the two atoms concerned have projected positions in overlapping Airy discs. In the light of the reciprocity principle the situation is identical to that discussed in §3 for nanodiffraction work in the STEM. Fluctuation microscopy can indeed also be efficiently carried out in STEM mode [32,33].

As before, the final Bessel function factor in eqn. (5) gives the usual Debye factor when integrated over bond orientations. Since the annulus defines only a small range of K_ρ for each K_ρ value studied, there is no integration over α and the highly localized coherence volume introduced in §2 does not arise. The effective coherence volume is defined by the specimen thickness and the instrumental resolution as in fig. 2.

The importance of matching the instrumental resolution to the structure under

investigation by fluctuation microscopy has been very convincingly argued quite recently [34] in response to some observations purporting to show that the dependence of V(K,Q) on K conveys broadly just the same information as the diffracted intensity [35]. It

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appears however that these observations were carried out at too high spatial resolution (high Q) for the fluctuation microscopy technique to be effective.

Applications of fluctuation microscopy technique have yielded interesting results about the structure of evaporated amorphous tetrahedral semiconductors and the effect of annealing [36], light soaking [37] or low energy ion bombardment [38]. These initially surprising annealing studies have been to some extent confirmed by autocorrelation studies of HREM bright field images [15]. Fluctuation microscopy seems to have useful sensitivity to MRO lying in the 1-4nm region through higher order pair-pair correlation functions. Although the computations can be somewhat demanding, different atomic models can thus be tested against the intensity fluctuation data. The precise structural features, e.g. para-crystallites or voids, giving rise to MRO may often however be rather subtle and elusive [39]. Very recently, in Ag/In-incorporated Sb₂Te amorphous films, fluctuation microscopy has been successfully extended to detect sub-critical nuclei for crystallization and monitor their population changes following laser pulse annealing [40].

5. Conclusions and future prospects

The challenge of elucidating amorphous structures has certainly forced microscopists to consider more carefully issues of coherence in imaging and presentation of data.

Spatially localized nano-diffraction, retaining the classical analysis but with due attention to some of the coherence issues orientation averaging noted here, may offer useful

information about variations in SRO. The low-angle scattering region, which may be cut off by the need to use focused illumination in nano-diffraction, can nevertheless yield bright field images whose power spectra can also be used to construct the RDF function [41]. Nano diffraction may be less successful in dealing with MRO except perhaps when this is manifested in pair correlation effects giving rise for instance to the first sharp diffraction peak noticed in silica and some other amorphous materials [42]. For the more general investigation and characterization of MRO, fluctuation microscopy seems currently the best approach and has certainly enjoyed some successes in linking the variations detected in MRO to wider properties of the amorphous samples.

Following a preliminary attempt to find angular correlations in the short range order scattering from nano regions [43], angular correlation functions or “correllograms” were constructed from a large number of nano diffraction patterns [44]. Some rather weak evidence for a significant structural signal was found in amorphous indium oxide thin films. More recently it has been suggested that the fluctuation microscopy method may be usefully extended to examine the angular intensity variance function [45].

When several different elements are present, electron scattering may lack the ability of neutron scattering to use isotope effects but energy loss spectroscopy is a powerful alternative tool. Short range order in the vicinity of specific atomic species can be investigated through above-edge fine structure (for discussion see [5]). Spatial variations recently observed in the near edge structure obtained with a 50nm probe from a calcium-

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alumina-silicate glass indicate long-range fluctuations in composition as well as atomic and electronic structure [46].

The enduring mind set of electron microscopists, now reinforced with aberration- corrected equipment, will no doubt ensure that high resolution imaging of amorphous materials remains on the agenda despite the early somewhat disappointing history. The reduced depth of field available at higher imaging apertures has already provided some degree of depth discrimination by through-focal imaging in ADF STEM [47] which may be useful in reducing the overlap problem. At least amorphous materials will be free from the channeling effects which complicate these experiments in crystals. Combining the narrow coherence volume of ADF STEM with tomography is certainly something which should be tried with amorphous materials now that the tomography technique has been so enormously developed.

One can be confident that no single one of these approaches will provide all the answers needed. To make significant progress in this demanding field, electron microscopists will not only have to master several of them but also to arrange access to amorphous samples whose important structure-dependent properties are well characterized.

References

[1] A. Howie, J. Non-Cryst. Sol. 31, (1978) p. 41.

[2] J.M. Gibson and A. Howie, Chem. Scripta 14, (1979) p. 109.

[3] D.J.H. Cockayne, Ann. Rev. Mater. Res. 37, (2007) p. 159.

[4] M.M.J. Treacy and J.M. Gibson, Acta Cryst. A52, (1996) p. 212.

[5] M.M.J. Treacy et al, Rep. Progr. Phys. 68, (2005) p. 2899.

[6] M.L. Rudee, Phys. Stat. Sol. B46, (1971) p. K1.

[7] P. Chaudhari, J.F. Grazyk and S.R. Herd, Phys. Stat. Sol. B52, (1972) p. 801.

[8] M.L. Rudee and A. Howie, Phil. Mag. 25, (1973) p. 1001.

[9] W. Cochran, Phys. Rev. B8, (1973) p. 623.

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Figure captions

Figure 1. Scattering diagrams for hollow cone TEM imaging (a) and for nano-diffraction or ADF imaging in STEM. Because of the reciprocity principle, the two situations are theoretically equivalent but in practice the annular detector in (b) extends over a much higher range of angles α than the illuminating aperture in (a). The scattering vector K in (b) consequently has a significant axial component K_z as well as a radial component K_ρ.

Figure 2. Influence on atomic scattering coherence of the angular spread of imaging or illumination (the angle θ in figs. 1(a) and 1(b) respectively. The cylinder in (a) denotes the resolution volume with height equal to the specimen thickness and radius a = π/Q for an aperture of radius Q. Solid arrows denote bonds which scatter coherently for both coherent and incoherent illumination. Dashed arrows denote bonds which scatter coherently only for incoherent illumination. Coherent scattering from the remaining dotted arrow bonds is destroyed by the angular spread of illumination whether it is coherent or incoherent.

Figure 3. Dependence on Z = Kr_jl of D₁ the Debye function (denoted by ♦) and of the functions D2(f) for f = +0.5 (■) and for f = -0.5 (▲).

Figure 4. Dependence on Z = Krjl of D1 the Debye function (denoted by ♦) and of the functions D₃(f, φ_Κ) for f = +0.5, φ_Κ = 0 or π (■) and for f = +0.5, φ_Κ = π/2 or 3π/2 (▲).

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Howie Fig 1a

k

₀

k

_s

= k

₀

+q

= k

_in

+K+q α

θ

k

_in

K

q

Objective aperture radius Q

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Howie Fig 1b

k

₀

k

_in

= k

₀

+q α

θ

k

_s

4λ/πα λ/πα λ/πα λ/πα

_m²²²²

2 λ/πα λ/πα λ/πα λ/πα

_m

Coherence volume

K

q

Annular detector

Probe forming aperture radius Q

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Howie Fig 2

2b 2a 2a

a b

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Howie Fig 3

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

0 2 4 6 8 10

Kr

DD

Z

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Howie Fig. 4

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

0 2 4 6 8 10 12

Kr

DD

Z

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