Electron Microdiffraction

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS, VOL. 46

Electron Microdiffraction J. M. COWLEY Department of Physics Arizona Stare Universiiy Tempe, Arizona

B. Variants on CBED . . . . . . . . . . . . . C. Selected-Area Electron Diffraction ( D. Incident-Beam Scanning

B. Contamination A. Identification

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References . . . . . . . . . . . .

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I. INTRODUCTION

The term “microdiffraction” has been coined, with more respect for convenience than grammatical elegance, to suggest the diffraction of radiation from a very small volume of material. The smallness of the sample examined is defined in relationship to current practice. For X rays the dimensions of the single-crystal samples normally used are of the order of 100-500 pm,but special microdiffractiontechniques have allowed patterns 1

Copyright @ 1978 by Academic Press. Inc. All rights ofreproduclion in any form reserved. ISBN 0-12-014646-0

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J . M. COWLEY

to be obtained from regions a few tens of microns in diameter. The use of new high-intensity sources, including synchrotron radiation, may reduce this limit considerably. In electron diffraction the established practice is to obtain patterns from samples that are very thin (hundreds of A) in the incident beam direction but may be fractions of a millimeter in diameter in electron diffraction instruments or about 1 pm in diameter when the usual selected-area electron diffraction (SAED) mode is used in a 100 keV electron microscope. “Electron microdiffraction” thus refers now to the obtaining of diffraction patterns from regions much less than 1 pm in diameter and a few hundred angstroms in thickness or less. In principle, the smallest sample that can give a detectable diffraction pattern is a single atom. Already the scattering from single heavy atoms has been collected to give clear images in the scanning transmission electron microscope (Crewe and Wall, 1970)and has been made to interfere coherently with the transmitted beam to give phase contrast images in the conventional transmission electron microscope (e.g., Iijima, 1976, 1977). Given sufficient specimen stability the number of electrons that may be scattered from a single heavy atom in an experimentally convenient time (say, 1 minute) using the electron sources now available is about lo1’, which is sufficient to define the intensity distribution within a diffraction pattern with reasonable accuracy. The scattering of this number of electrons from each atom of an assembly of atoms should allow the relative positions and scattering powers of the atoms to be determined. Thus, in principle, the basis exists for the atom-by-atom determination of the structure of matter. In practice, of course, many difficulties of experimental technique, specimen stability, and interpretation of data prevent the attainment of this goal. However, the progress in recent years has been spectacular. Diffraction patterns have been recorded from specimen regions 20 A in diameter. The impact of this achievement on practical investigations in materials science has not yet been very great, partly because the techniques are in early stages of their development, but the potential of the method is obvious. A review of the history and current status of electron microdiffraction at this stage seems justified as a basis for discussions on the refinement of the experimental methods and the applications to the multitude of current problems involving structural variations over small distances in solids. The possibilities for electron microdiffraction arise because of the strong interaction of electrons with matter. The scattering cross sections of atoms for electrons are approximately lo6 times those for X rays of the same energy. The small thicknesses of samples used for electron diffraction have always been both a limiting factor and the basis for the special advantages of the method. Transmission electron diffraction developed slowly because

ELECTRON MICRODIFFRACTION

3

of the experimental difficulties of preparing samples in the necessary thickness range of 10-1000 A. The advent of the electron microscope provided the means whereby the thin sample areas could be recognized and the techniques for the systematic correlation of diffraction effects with crystal morphology could be developed. Reflection electron diffraction, both at low energies (LEED, 10-500 eV) and at high energies (RHEED, 10-100 keV) has been concerned with the scattering from the few top layers of atoms on crystal surfaces. Since with few exceptions the specimen areas used have been quite large (fractions of a square millimeter) we shall not be very concerned with these methods. Early in the development of electron diffraction instruments a single long-focus lens was added to provide a sharply focused pattern on the photographic plate (Fig. la) and this geometry is still used in many cases to observe fine detail in the diffraction pattern. To improve the resolution in the diffraction pattern, it is now common to produce smaller effective source sizes by using one or more strong lenses to demagnify the actual electron source.

(b) FIG. 1 . Electron optical arrangement for a diffraction instrument, [a) with the electron beam focused on the screen or plate and (b) with the electron beam focused o n the specimen.

For this configuration the beam diameter at the specimen level is defined by the limiting aperture and is usually several hundred microns. The very severe limitations on specimen thickness imply that only very rarely is it possible to obtain a nearly perfect single crystal covering a significant

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J. M. COWLEY

fraction of such a large area. Hence this lens configuration gives excellent averaging over orientations for polycrystalline patterns or mosaic or bent single-crystal films, but does not allow any well-defined correlations of diffraction intensities with the thickness, structure, or perfection of single crystals. The basic information on the scattering of electrons by crystals and the detailed comparison of experimental observation with theory depended on the development of methods to obtain patterns from much smaller regions. The first and most obvious means for decreasing the irradiated area of the specimen was to increase the power of the electron lens of Fig. l a to focus the beam on the specimen as in Fig. lb. The diffraction spots on the photographic plate were thereby spread out into shadow images of the limiting aperture, but a small amount of spreading is not important if we are interested only in the positions and intensities of the well-spaced pattern of spots given by single crystals. This idea was used, for example, in the electron diffraction instruments of von Ardenne et al. (1942), Hillier and Baker (1946), and Cowley and Rees (1953), in which reasonably sharp diffraction spots could be obtained from crystal regions about 1 pm in diameter. The idea of focusing the electron beam on the specimen had been introduced much earlier, of course, by Kossel and Mollenstedt (1939), who used much larger angles of convergence to obtain spectacular convergent beam electron diffraction (CBED)patterns from small regions of thin mica crystals. This work and its interpretation by MacGillavry (1940) were of great significance in revealing the strong dynamical diffraction effects that have been the dominant factor in determining the limitations and the unique capabilities of electron microdiffraction. The large-scale development of electron microscopy in the late 1940s and 1950s extended the range of electron diffraction work. The wealth of knowledge on the sizes, shapes, and imperfections of submicron crystalline regions provided a much better background for the interpretation of electron diffraction patterns. The development of the selected-area electron diffraction (SAED) techniques (Boersch, 1936) allowed the direct correlation of diffraction intensities with crystal morphology. The mutual advantages of electron diffraction and electron microscopy and their strong interdependence have made them inseparable to the extent that, except for the highly significant cases of exploratory research on diffraction phenomena, their combined use in a single instrument is standard practice for most materials science. The SAED techniques has suffered from the restriction that, in a 100 keV electron microscope designed primarily for high-resolution imaging, the area from which the diffraction pattern can be obtained is limited by the spherical aberration of the objective lens to be greater than about 1 ,urn.

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ELECTRON MICRODIFFRACTION

To overcome this limitation it was necessary to go to instruments having limited imaging capabilities. Goodman and Lehmpfuhl (1965) and their colleagues (Cockayne et al., 1967) developed the convergent beam diffraction technique with special instruments that allowed areas of 200 A diameter to be studied. Riecke (1962) developed instrumentation for “microbeam” microdiffraction. It is only with the recent production of high-resolution scanning transmission electron microscopy (STEM) instruments that the techniques have become available for the simultaneous observation of high-resolution images and diffraction patterns from extremely small regions. The diameter of the region giving the diffraction pattern may, in principle, be equal to the resolution limit of the microscope. In the following pages, after a brief outline of the relevant theoretical considerations, we shall describe in more detail the various techniques available for electron microdiffraction and then give some account of the nature of the information that can be obtained, with examples drawn from the applications already made.

11. THEORY OF IMAGING

AND

DIFFRACTION

A. Focusing and Imaging The aspects of electron optics relevant to our purposes may be discussed in terms of the familiar ideas developed for the physical optics of light (Cowley, 1975). We may confine ourselves to scalar wave theory, since polarization effects are negligible and the nonscalar effects of the magnetic fields of electron lenses on electron trajectories can be assumed, as a first approximation, to provide only a trivial image rotation. The monochromatic wave from a point source is described by the wave function in a plane a t a distance R from the source by $,(xy) = i(RA)-’ exp( - ikR) exp[ - i2n(x2

+ y2)/RA]

(1)

where we have used the small-angle approximation, which is valid for most cases of interest. For a more general source function $l(xy), the wave produced on this plane is $2(XY)

= $I(XY)

* $,(w)

(2)

where * indicates the convolution integral defined by f ( x ) * g(x) = J f ( X ) g ( x- X ) d x . For RI1 large compared with the maximum x2 y 2 values, this reduces to the Fourier transform relationship of Fraunhofer

+

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J. M. COWLEY

diffraction Y z ( u u )= Yl(uu) = Y$l(xy)

= JJ$l(xy)exp[2zi(ux

+ uy)]dxdy

where u = 2 sin 6,JA x y/RL

u = 2 sin 6JA z x/RI,

(3) and 8, and 8, are half the angles made with the z axis in the x-z and y-z planes. If such a wave falls on a lens, it is modified by multiplication by the transmission function of an ideal lens, which would exactly reproduce the object function, and by the function A ( u 4 expCiX(u4-J (4) where A(uu) is the aperture function, usually assumed to be unity within and zero outside an axially placed physical aperture, while ~ ( u u is) the phase change produced in the wave relative to the axial beam path. Usually ~ ( u uis) assumed to take the simplest possible form :

~ ( u u=) z A h 2 + +nC,A3u4

(5) where A is the deviation from exact focus and C, the spherical aberration constant. The wave function produced at a distance R,, near the focus of the lens, is then given by a second Fourier transform, Fraunhofer diffraction, operation as =

qw34A(uu)exp[ix(udl}

= $1

or

( R", -x-,

-y-

3

* 9[Aexp(i~)]

(7) W y ) = $I(% y) * CC(XY) + iS(XY)l where, in the final expression, we have referred the functions to the object dimensions by removing the magnification factors - R/R,, ignored terms of modules unity and trivial constant factors, and expressed the spread function in terms of its real and imaginary parts. The image intensity is then 4 X Y ) = I$l(XY)

* {CbY) + iS(XY)}12

(8)

For a point source represented by a &function, I(xy) = c2(xy) + s2(xy) = T(xy) (9) which defines the intensity spread function T(xy). If the object can be represented by an incoherently emitting region of intensity distribution

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ELECTRON MICRODIFFRACTION

Io(xy),the image is then given by summing the intensities from all points of the object and so is written

*W Y )

(10) Then T(xy) is the Fourier transform of the contrast transfer function of the lens for the incoherent imaging case, I(XY)

= IObY)

T(xy)= F{A(uu)exp[i~(uu)]* A ( - u , -u)exp[-i~(-u,

-u)]}

(11)

With this background we may describe the essential imaging steps in the various forms of electron microscopy. For CTEM the important, resolutiondetermining step is the action of the objective lens to form the first magnified image of the electron wave transmitted through the object. If i+b I(xy)is the wave at the exit surface of the specimen, the image intensity will be given by (8), magnified by the objective and subsequent lenses. In the practical case that the incident wave is not strictly monochromatic and the focal length of the objective lens varies slightly with time, the intensity distributions for the various wavelengths and focal lengths are added incoherently or, if the probabilities of occurrence for various values of A and f can be described by a distribution function D(A, f),

If the specimen is sufficiently thin in relation to the wavelength and the resolution, as discussed below, the wave function at the exit surface of the specimen may be described in terms of a transmission function q(xy),which multiplies the incident wave function :

(13) For many purposes it is a useful approximation to assume that $o(xy) is a plane wave of unit amplitude. The actual beam convergence in CTEM is usually of the order of l o p 3rad. However, for an increasing number of applications, larger convergence angles are used, as in the case of highresolution imaging with the specimen immersed in the objective lens field, in which case the forefield of the objective lens often acts as a short-focallength condenser to give angles of convergence approaching lop2 rad. The incident-beam convergence then has important effects on the image resolution and contrast (Wade and Frank, 1977; O’Keefe and Sanders, 1975; Anstis and O’Keefe, 1976; O’Keefe and Anstis, 1978). It is customary to consider the effects of beam convergence with the assumption that waves coming in different directions are incoherent so that the total image intensity is given by summing the image intensities given by plane waves for all angles of incidence. This would be appropriate i+bl(XY) = $ O ( X Y M X Y )

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J. M. COWLEY

if the specimen were illuminated by an ideally incoherent source without intermediate lenses or apertures. It is a good approximation for the conventional system in which the aperture of the condenser lens is illuminated by a hot-filament electron gun. In this case an effectively incoherent source of several microns diameter gives a coherence width of the radiation at the condenser aperture ( w x L/as, where asis the angle subtended by the source) of about 1 pm, which is very small compared with the usual dimensions of the condenser aperture, 50-200 pm. However if, as in some of the newer CTEM instruments, a field emission gun is used, the effectively incoherent source size may be less than 50& so that the condenser aperture may be illuminated coherently. Then the amplitudes of the waves incident on the specimen at various angles must be added. The incident wave is approximated by c(xy)+ is(xy), as in (7), where the form of the function is determined by the aperture and aberrations of the condenser lens system. For high-resolution STEM instruments, the high brightness of a field emission gun is necessary in order that a sufficiently high current of electrons should be concentrated into the small probe, a few angstroms in diameter, which is scanned across the specimen. It is a good approximation to assume coherent illumination of the specimen by a wave described by (7) with a &function source. The functions c(xy) and/or s(xy) thus have sharp peaks of form depending on the objective lens aberrations and defocus, with radially diminishing oscillations (Cowley, 1976b). For sufficiently thin specimens, the exit wave is given as in (13) by a transmission function $l(XY) = d

x-

x,Y - Y ) [ C ( X Y ) + i S ( X Y ) l

(14)

where X and Y are the coordinates of the center of the incident-beam probe on the specimen and we have assumed, for convenience, that the specimen rather than the incident beam is moved. On the distant plane of the detector, Fig. 2, the amplitude distribution will be given by the Fourier transform

FIG.2. Scheme for scanning transmission electron microscopy.

ELECTRON MICRODIFFRACTION

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of (14) as

yx,Y ( U ~ )= Qxr(uu) * A(uv)exp[i~(u~)]

(15)

where Q(u, u) = P q ( x y ) . The signal detected and used to form the magnified image is then given multiplied by a detector by integrating the intensity Ixy(uu) = ~Yxy(uu)~2, sensitivity function W(uu):

ZOb(XY)= J I x Y ( u ~ ) W (du u ~du)

(16)

As in the case of CTEM, the effects of finite source size and variations of I or f are included by the incoherent integration of intensities over the corresponding variables. The diffraction pattern (15) for a thin object will consist of a strong central spot surrounded by a weaker, wide-angle (to 10- rad) distribution of scattered electrons. The variety of image signals that may be produced by detection of all or part of the central spot (bright-field images) or all or part of the surrounding scattered radiation (dark-field images) has been described, for example, by Cowley (1976b). An aspect of electron microscopy that is of importance in the present context is the relationship of the imaging process to the formation of the Fraunhofer diffraction pattern. In the case of CTEM we have seen that the angular distribution of the scattered waves from a specimen is described in terms of the Fourier transform (3). The Fraunhofer diffraction pattern is produced where each of the component plane waves leaving the specimen is condensed to one point of a two-dimensional spatial distribution. This happens on the back focal plane of the lens, where the intensity distribution is I(uu) = p J ' , ( U U ) I *

(17)

with u = x f l , v = y f l in the small-angle approximation. If the lenses following the objective are used to magnify this back-focal plane distribution rather than the image plane distribution, the Fraunhofer diffraction pattern of the object, rather than the magnified image, will appear on the final viewing screen or photographic plate (see Fig. 3). The arrangement of the STEM instrument (Fig. 2) is essentially that of Fig. lb. A diffraction pattern is produced on the plane of the detector. The alternative configuration, Fig. la, can be provided by changing the lens excitations to produce a focused diffraction pattern but in this configuration a high-resolution image cannot be formed. It should be emphasized that for the convergent beam mode of Fig. lb, which is compatible with highresolution imaging, the intensity distribution of the diffraction pattern

10

J . M. COWLEY Specimen Oblect I

lens

Y ~

Back l o c a l plane Selected-area aperture Intermediate

lens

Prolector lens

Final

FIG.3. Ray paths for a transmission electron microscope, following the specimen, as used to obtain a high magnificationimage (left)and to obtain a selected-areadiffraction pattern fright).

given from Eq. (15) will be strongly dependent on the aperture and aberrations of the objective lens, including defocus.

B. Diffraction of Plane Waves For a very thin sample, the effect on the incident wave may be represented by multiplication by a transmission function q(xy) that describes the changes in phase and amplitude. The predominant effect is the change of phase due to the potential distribution &r) of the scattering matter so that, in the so-called phase-object approximation with a plane incident wave, the transmitted wave is given by the transmission function dXY)

= exPC -

W(xy)l

(18) where &y) = !4(r) dz, with the beam direction taken to be the z axis, and c = n/AE, where E is the accelerating voltage. Absorption effects may be included by assuming 4(xy) to be complex. The one-to-one correspondence between amplitudes at points of the incident and transmitted waves, implied by use of the transmission function, assumes no lateral spreading of the perturbations of the wave, i.e., no appreciable effect of the Fresnel diffraction smearing represented by Eq. (1). Rough estimates suggest that the spreading of wave perturbations during

ELECTRON MICRODIFFRACTION

11

transmission through a thickness T is of the order of (TA)’j2, so that for 100 keV electrons the spreading is 1 A for T % 30 8, and 3 A for T % 250 A. Since the atom sizes and separations in projection are of the order of 1 A, the thickness limit for this approximation is normally taken as 20-30 A. For even single heavy atoms the phase change represented by (18) may be quite large, exceeding n. However, for very light atoms the phase change is relatively small and one may approximate q(xy) = 1 - ia4(xy)

so that the diffraction pattern given by Fourier transform is Y(w)z Q(uu)z ~ ( u v-) ~ o @ ( u v )

(19)

This weak-scattering or “kinematical” approximation may be improved by adding the Fresnel diffraction effects. For each scattering element the transmitted wave is convoluted by the Fresnel propagator, Eq. (1) with R = z - z,,, and the out-going wave is found by integrating over z t o give a diffraction amplitude for a plate of uniform thickness T, Y(uu) = d(uu) - ia

s

@(uu, z)exp[ -2i~iz[(uu)] dz

(20)

where [(uu) is the distance of the Ewald sphere from the reciprocal lattice plane perpendicular to the incident beam. In the case of a three-dimensionally periodic potential distribution, such as a thin crystal, the diffracted amplitude is expressed in terms of the structure amplitudes y,,k[, the Fourier coefficients of the Fourier series describing the potential, as yhkl

rx - io@hkl[(sin

n[hklT)/nchkl]

(21)

where [hk[ is the hkl excitation error, or the distance of the hkl reciprocal lattice point from the Ewald sphere (Cowley, 1975). Since this kinematical approximation assumes that the scattered amplitudes are small compared with the incident beam amplitude, it is valid only for light atoms and for very thin samples. It fails for even thinner samples in the case of crystals viewed along the directions of principal axes, for which there is a progressive phase change due to rows of atoms aligned in the beam direction. Because for very thin crystals the scattering is approximated by that of a two-dimensional phase grating, the diffraction patterns can contain a very large number of diffraction spots, regularly arranged in a close representation of a planar section of the reciprocal lattice. The small curvature of the Ewald sphere of reflection in reciprocal space corresponds to the small spreading of the electron wave in real space due to Fresnel diffraction.

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J. M. COWLEY

For thicker crystals the factor in (21) involving the excitation error may be small for all but a small number of reciprocal lattice points so that only a few diffracted beams may be produced and for particular orientations a single diffracted beam may dominate the diffraction pattern. However, except as a rough first approximation, it is rarely possible to make the assumption, common in X-ray diffraction, that only one diffracted beam appears at a time (the “two-beam” case, when the direct transmitted beam is countec?). C . Dynamical Scattering

The single-scattering kinematical approximation may fail for a sample one atom thick. For very thin samples the phase object approximation (18) without the weak scattering approximation (19) can often serve because it includes all multiple-scattering processes. This is evident from the powerseries expansion of the exponential in (18), which on Fourier transforming gives Q(uu) = 6(uu) - iaO(uu) - $ 0 2 0 ( u u ) * O(uu)

+ &a3@ * (D * O + . .

*

where successive terms represent single, double, triple, . . . scattering. In general the Fresnel diffraction effects must also be included to give a full “dynamical” theory of scattering. This has been done in various ways (see Cowley, 1975). A simple, straightforward approach due to Cowley and Moodie (1957a) considers the progressive phase change of the electron wave by successive thin slices of crystal perpendicular to the incident beam, with Fresnel diffraction of the wave between slices. In the limit that the thickness of the individual slices tends to zero this gives the same results as the classical formulation of the problem by Bethe (1928), who solved the wave equation for electrons in a periodic potential field. On the basis of the Cowley-Moodie approach a computing method has been developed (Goodman and Moodie, 1974; Cowley 1975) in which the progressive modifications of wave amplitudes by slices of crystal of finite thickness are calculated. This method has been used for calculating diffraction patterns and images for crystals giving thousands of diffracted beams simultaneously (O’Keefe, 1975) and for crystals containing defects or disorder giving rise to continuous distributions of diffuse scattering (Fields and Cowley, 1977; Spence, 1977). Computer programs based on the matrix formulation of the Bethe theory (Hirsch et nl., 1965) have also been applied for many-beam diffraction problems, although usually for smaller numbers of diffracted beams, and the differential equation formulation of Howie and Whelan (1961) has proved valuable for computing images of dislocations and other extended defects in two-beam and several-beam approximations.

ELECTRON MICRODIFFRACTION

13

D. Diffraction in Convergent Beams The approximation of assuming the incident radiation t o be a plane wave of unit amplitude is sufficient for many purposes in that for parallelbeam, focused-electron diffraction patterns and for many applications of CTEM the range of angles of incidence present is too small to give any appreciable variation of difrraction intensities. However, for convergentbeam electron diffraction (CBED), STEM, and some high-resolution CTEM applications, the variation of diffraction conditions within the range of incidence beam directions is important. It has been customary in the past to calculate CBED patterns or CTEM and STEM images on the assumption of incoherence of the incident beam components: the intensities are calculated for each incident beam direction and are added, as mentioned in Section I1,A. The calculation of CBED patterns (Goodman and Lehmpfuhl, 1967) or of CTEM images (O’Keefe and Sanders, 1975) on this basis involves a large number of n-beam dynamical calculations in general, although for CTEM of very thin specimens an analytical modification of the contrast transfer function of the objective lens may provide a useful approximation, eliminating the need for multiple calculations (Wade and Frank, 1977; Anstis and O’Keefe, 1976; OKeefe and Anstis, 1978). The same approach of making a separate n-beam dynamical calculation for each angle of incidence may be used for the case of coherent illumination, as when a field emission gun is used, except that then it is necessary to add the amplitudes, rather than the intensities of the patterns or images given for different incident beam directions. It appears likely that, because interference effects will cause a more rapid variation of amplitudes than intensities with angle of incidence, a finer sampling of incident beam directions will be necessary in the coherent case, with a corresponding increase in the amount of computation required. An alternative method of calculation is possible for the STEM and CBED cases in which the diameter of the incident beam on the specimen is small. The incident-wave amplitude described by (7) is a localized, nonperiodic function. The transmission of this wave through a crystal may be calculated by use of the method of periodic continuation. The single incident beam is replaced by a two-dimensionally periodic array of nonoverlapping beams spaced at multiples of the crystal periodicity. A single calculation, with a very large number of beams, is then made for a superlattice having the periodicity of the beams. The diffraction pattern will then correspond to that for a single incident beam, sampled at a finely spaced array of points. This method has been used by Spence (1977) to calculate CBED patterns from small perturbed regions of crystal.

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J. M. COWLEY

It has been suggested (Cowley and Nielsen, 1975; Cowley and Jap, 1976) that in the case of coherent illumination, the interference effects near the crossover formed by a lens may lead to some surprising diffraction phenomena. The flow of energy in the beam is actually parallel to the axis near the focal point (Fig. 4a) and cannot be represented by the classical geometric optics picture of intersecting ray paths (Fig. 4b). Arguments or ideas based on the picture of Fig. 4b may be misleading. It has been shown by the calculations of Spence and Cowley (1978), for example, that the details of intensity distributions in a CBED pattern are strongly dependent on the focus of the objective lens although, for a perfect crystal pattern, this dependence appears only in the regions where the extended spots overlap.

-

fa)

A-

/

\ (b)

FIG.4. Ray paths (lines indicating the flow of energy) at the focus of a lens for (a) coherent wave optics and (b) classical geometric optics.

111. DIFFRACTION TECHNIQUES A . Convergent-Beam Electron Diffraction (CBED)

From the early work of Kossel and Mollenstedt (1939) it was obvious that CBED patterns from crystals gave more information than was contained in focused patterns for particular incident-beam directions. The variations of diffracted-beam intensities with the directions of incidence were clearly displayed in the complex distributions of intensity within the large circular disks formed by the diffraction spots from single crystals. The Kikuchi line patterns formed in the diffuse scattering outside the diffraction spots showed

ELECTRON MICRODIFFRACTION

15

intensity distributions that were obviously related but essentially different. A thorough understanding of the patterns came slowly. The first stage in the interpretation of the patterns in terms of Bethe’s dynamical theory came from MaGillavry (1940), who showed that with the assumption of two-beam diffraction conditions the intensity of a reflection as a function of excitation error [,,and the thickness T of a planeparallel crystal slab could be written

when w = &(h, th= 1/(2r~@~), and Qk is the h Fourier coefficient of the potential distribution of the crystal. For the special case of reflection at the exact Bragg angle, Ch = 0, this gives the well-known “pendellosung solution” with 1, varying sinusoidally with thickness, with periodicity equal to C h , the “extinction distance”. The intensity of the transmitted beam, I , , which from the conservation of energy must be equal to 1 - 1, in the absence of absorption, likewise varies sinusoidally. As a consequence, both bright- and dark-field images of crystals show “thickness” fringes if the thickness in the beam direction varies. For a crystal of constant thickness, the variation of intensity with angle of incidence, and so with [,,,given by (22) may be compared with the kinematical result, the square of (21). The kinematical result is illustrated in Fig. 5. Different angles of incidence of the incoming beam correspond to

FIG.5. Diagram illustrating the formation of a convergent-beam diffraction pattern in the kinematical approximation;in reciprocal space and (below)in the real space of the diffraction pattern.

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J. M. COWLEY

different positions of the Ewald sphere relative to the row of reciprocal lattice points and to different positions across the diameter of each of the circular diffraction spots. The intensity distributions across the spots therefore reflect the distributions of scattering power along the shapetransform extensions of the reciprocal lattice points, each having the form sin2(nchT)/(dh)*. For c h latge (w >> l), the two-beam approximation formula (22) approaches this form and the crystal thickness can be deduced from the periodicities of the fringes. For (h small, the intensity is dominated by the contribution of l@hl. By analysis of the patterns obtained by Kossel and Mollenstedt from mica, MacGillavry was able to deduce I@,( values in excellent agreement with those calculated from the structure of mica as then understood from X-ray diffraction results. The measure of agreement achieved must be regarded as fortuitous since knowledge of the structure of mica has since been modified by further refinements and the validity of the two-beam approximation is questionable for the patterns used. CBED patterns were obtained later by Ackerman (1948) from a wide variety of materials using an incident-beam diameter estimated to be 1 pm or less and by many subsequent authors. The interpretation of the details of such patterns in terms of two- or several-beam dynamical diffraction theory was made by Ackerman (1948), Fues and Wagner (1951), Hoerni (1950), and others, but the technique remained of rather limited academic interest until revived with the application of more advanced techniques and more quantitative interpretation by Goodman and Lehmpfuhl (1965, 1967).These authors initially used an electron microscope with the specimen placed near the back focal plane of the objective lens so that the strong lens gave a focused probe diameter of a few hundred angstroms at the specimen. Similar small probes, with improved operational convenience, were subsequently provided by a specially designed diffraction instrument (Cockayne et al., 1967) in the Melbourne laboratory. With these instruments, patterns such as Figs. 6 and 16 were obtained from very small, perfect crystal regions under carefully controlled experimental conditions and with sufficient control of the relevant experimental parameters to allow detailed systematic comparisons of observed intensities with those computed using accurate multibeam dynamical diffraction theory. The valuable series of results in the refinement of crystal potential distributions and the determination of crystal symmetries by this group will be summarized in Section V. In the last few years a further advance in CBED techniques has become possible with the introduction of field emission guns and STEM techniques. CBED patterns are readily obtained in both dedicated STEM instruments and scanning attachments for CTEM instruments, which increasingly use

ELECTRON MICRODIFFRACTION

17

FIG.6 . Convergent-beam electron diffraction pattern for a “systematic” row of reflections and some weak nonsystematic reflections. The center of the second-order spot is at the Bragg reflection condition.

field emission guns. A STEM instrument has electron optics ideally suited to CBED (Fig. 2).If instead of scanning the incident beam across the specimen the beam is held stationary, a CBED pattern of a fixed small area is formed on the detector plane. With such instruments diffraction patterns have already been recorded from areas as small as 20 A in diameter (e.g., Brown et al., 1976). In principle, the radius of the selected area from which the CBED pattern is obtained may approach the resolution limit of the microscope, which is currently about 3 A. For an ideal leiis having no aberrations and limited only by a circular aperture of radius u = uo, the beam intensity at the specimen is given by J f ( 2 7 ~ u ~ r ) / ( 2 nwhere r ) ~ , J , is the first-order Bessel function and r the radial coordinate. This has the well-known form of the Airy disk, a central maximum of radius r l = 0.61/u0 to the first zero of intensity. In terms of the angle M subtended by the aperture at the specimen, rl = 1.22,+. The central maximum is surrounded by concentric circles of intensity with intensity maxima decreasing approximately in proportion to r P 3 .Thus the incident intensity is by no means limited to a well-defined central spot. Integrated around the circumference of each concentric circle, the intensity falls off quite slowly, being proportional to r P 2 .Approximately 16% of the incident intensity lies outside the central maximum. This distribution and the even

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J. M. COWLEY

slower decrease with r of the wave amplitude of the focused spot become increasingly relevant as the size of the spot is decreased. A probe with central maximum 5 A in diameter passing through a thin crystal will give a CBED pattern reflecting the periodicity of the lattice even if the unit cell dimensions are much greater than 5 A. Each diffraction spot will take the form of a sharply defined disk and, although the disks may overlap, the fact that they are in a periodic array will be readily discernible. The diameter of the disk of intensity corresponding to each spot in the CBED pattern will be La,where L is the distance from specimen to detector screen or plate. Disks will just touch at the edges for planar spacings in a crystal specimen given by

d = 1/2sin9 = L/a

(23) If we assume incoherent STEM imaging, which is sometimes used as a first approximation for the dark-field STEM mode, the Rayleigh criterion gives a resolution for this case of

Ax = 1.221ia (24) which is slightly larger than the planar spacing (23), so that in order to resolve a spacing d in the STEM image, it is necessary to use an objective aperture that will make the CBED spots overlap by about 20%. These considerations must be modified by the presence of lens aberrations in practice. Spherical aberration will not affect the size of the diffraction spots, but it will modify the shape and size of the central maximum of amplitude or intensity at the specimen, it will modify the surrounding fringes, and it will modify the STEM resolution. It is well known that for bright-field phase contrast imaging of very thin specimens there is an optimum aperture size of approximately u,,, = 1.5Cs- 1/4jZ-3/4 (see Cowley, 1975) giving a least resolvable distance of approximately AX = 0.66C,"41314 (25) on the assumption that Ax = 1/umax. For dark-field imaging it may be assumed as a first approximation that, because the square rather than the first order of the spread function is relevant, the resolution is improved by a factor of about 2l'* (the figure for gaussian spread functions) and the constant in (25) has been given various values around 0.4. Thus with spherical aberration and coherent imaging a larger objective aperture may be used, giving larger spots and also better resolution (and it is usually assumed that the CBED spots will just touch for a d spacing just resolved). Hence it may be concluded that while STEM imaging can give information on spacings in the sample down to a certain dmi,, the CBED

ELECTRON MICRODIFFRACTION

19

pattern produced by the same incident beam contains information on spacings in the range of dmin and below as well as, in many cases, some information on spacings much larger than dmin. The usual reservations concerning the interpretation of diffraction patterns must, of course, be made. Under kinematical scattering conditions the “phase problem” applies if the CBED spots do not overlap. The relative phases of the reflections are lost when the intensity is recorded and only an autocorrelation function, or Patterson function, can be deduced, rather than the actual potential function, which would specify relative atom positions. For the more usual dynamical scattering conditions some information on relative phases of diffracted beams is present in the diffraction pattern but the problem of deducing these phases is complicated and in general is susceptible only to trial-and-error methods of solution. If the CBED spots do overlap and the incident beam has sufficient coherence, interference effects in the regions of overlap will, in principle, give information on the relative phases of the reflections (Nathan, 1976; Cowley and Jap, 1976).

B. Variants on CBED 1. Defocused CBED; Shadow Imaging In the lens in a CBED instrument is defocusedso that the small crossover is formed either before or after the specimen, the central spot of the CBED pattern will become a bright-field shadow image of the specimen. If the illuminated portion of the specimen contains a single crystal, each diffraction spot of the CBED pattern will become a dark-field shadow image showing the variation of diffraction intensity within the illuminated region (Fig. 7). For both the central beam and diffracted beams, the range of angles of incidence across the illuminated area will be the same as for a beam focused on the specimen. For a perfect, unbent, thin shgle-crystal plate of uniform thickness illuminated from an incoherent source, the only source of contrast in the spots would be the changes of diffraction intensity with incident-beam direction and the out-of-focus patterns would be exactly the same as the in-focus pattern. For other types of specimen, the contrast variations will

FIG.7. Diagram suggesting the formation of bright- and dark-field shadow images in a defocused convergent-beam diffraction pattern with the beam crossover before the specimen.

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J. M. COWLEY

depend on both the variation of diffraction angle and the variation of orientation and dimension of the lattice planes across the specimen area. In general this will give complicated images that may be difficult to interpret, but there are cases in which the variation of the structure is relatively simple and useful information may be derived. For example, Fig. 8 is an out-of-focus CBED pattern of a thin crystal plate containing a dislocation. The variations of lattice plane orientation around the dislocation line are clearly indicated by the deflection of the dark diffraction contour in the bright-field image and the bright line in the dark-field image.

FIG.8. Out-of-focus convergent-beam diffraction pattern from a graphite crystal containing a dislocation showing the perturbation of the bright- and dark-field extinction contours due to the local distortion of the crystal.

The use of the bright-field shadow images as a rapid and convenient method for associating information on specimen morphology with CBED patterns has recently been advocated by Dowel1 (1976). The resolution obtainable in such images is, in the incoherent approximation, comparable with that of STEM using the same lenses. It was shown by Cowley and Moodie (1957b) that in an ideal case the resolution could actually be better by a factor 2lI2 than for CTEM using the same lenses, but this does not represent an experimentally feasible imaging mode. 2. Wide-Angle CBED If the angle of convergence of the incident beam is greatly increased by using a very large limiting aperture or none, the CBED pattern from a

ELECTRON MICRODIFFRACTION

21

single crystal changes its nature. The individual round diffraction spots become large, overlap, and merge to give a more-or-less uniform background on which are superimposed patterns of black and white lines, somewhat similar to the Kossel lines of X-ray diffraction or the Kikuchi lines formed from electrons diffusely scattered in thick crystals. Figure 9 shows such a pattern obtained from a thin crystal of silicon.

FIG. 9. Wide-angle convergent-beam diffraction pattern from a thin crystal of silicon.

The geometry of these patterns may be understood in terms of the geometry of the crystal lattice. Parallel sets of lines correspond to reflections from various orders of reflection from a given set of planes. The prominent line pairs come from low-index hkl reflections plus the corresponding &2 reflections. Patterns of this sort have been used by Goodman and Lehmpfuhl (1968), Cockayne et al. (1967), and others to give rapid and convenient determinations of crystal orientations, using the symmetry of the patterns to identify the principal axiaI directions in the crystal. The intensity distributions in these patterns are necessarily complicated and highly dependent on crystal thickness, orientation, and perfection as well as on the focus of

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J. M. COWLEY

the incident beam, but interesting observations on this subject have been given by Fujimoto and Lehmpfuhl (1974). The specimen area giving the wide-angle CBED patterns is necessarily greater than that for the CBED patterns discussed earlier, because the contribution to the electron beam diameter at focus, due to third-order spherical aberration alone, varies with a3. However, beam diameters of a few hundred angstroms appear to be feasible. Smith and Cowley (1971) showed that for bent crystals the contrast shows striking variations with defocus. Pairs of strong black and white lines occur with a separation that is roughly proportional to defocus and to the curvature of the crystal in the direction at right angles to the lines. It was suggested that the separations of these line pairs could be used as a measure of crystal curvature. Also, since the defocused patterns took on some of the aspects of shadow images of extended regions of the crystal, perturbations of the lines by crystal defects, similar to those of Fig. 8, were observed and could conceivably be used to study local lattice strains. A further application of this type of wide angle pattern was suggested by Smith and Cowley (1975). The separations of line pairs in a pattern from a simple crystal of known structure such as silicon provide a convenient calibration of angles of electron beams leaving the specimen. This was useful, for example, in measuring detector angles in a STEM instrument for which the poorly defined power of electron lenses following the specimen made direct geometric measurements unreliable. 3. Grigson Scanning CBED In the STEM instruments designed by Crewe and co-workers (Crewe and Wall, 1970) and the commercial and other instruments based on this design, no provision is made for the direct observation or recording of the diffraction pattern by use of two-dimensional detectors such as fluorescent screens or photographic plates. Instead, recourse is made to the method developed by Grigson (1965) for use in conventional electron diffraction (HEED) instruments. The diffraction pattern is scanned over a single detector of small aperture using deflection coils after the specimen, and the signal detected is used to modulate either the intensity or the y deflection of a cathode ray tube. (Figure 15 was obtained in this way.) The advantage of this method is that a quantitative measure of the diffraction pattern intensity, in the form of an electronic signal, is provided in a form suitable for recording or display in a variety of ways. The main disadvantage is that it is very inefficient in its use of the electrons scattered by the specimen. Only a very small fraction of the scattered intensity is recorded at any one time so that, especially with the small detector sizes

ELECTRON MICRODIFFRACTION

23

needed to give reasonably high resolution in the diffraction pattern, the time to record the diffraction pattern is long and the radiation damage and contamination of the small specimen area illuminated may be severe. Often in practice the diffraction patterns given when the STEM instrument is operated in a high-resolution imaging mode is too weak for convenient observation by the Grigson technique. Instead a more intense larger diameter beam (20-50A diameter) is formed by use of a weaker, preobjective lens to give a sharper diffraction pattern of higher intensity. The difficulty is then that the correspondence between the diffraction pattern and the image is less direct and more uncertain. A scheme involving a twodimensional detector system, designed to overcome these difficulties, has been proposed by Cowley (1978). C . Selected-Area Electron Difraction (SAED) 1. SAED in C T E M

The use of selected-area electron diffraction with CTEM is sufficiently well established and well known to require only a brief summary. The principle is illustrated in Fig. 3. In the imaging mode (Fig. 3a) an aperture placed in the image plane of the objective lens will select the part of the image coming from a very small area of the specimen. If the focal length of the intermediate lens is changed so that the diffraction pattern formed in the back-focal plane of the objective, rather than the image, is magnified on to the final viewing screen (as in Fig. 3b), then only those electrons diffracted by the selected area of the specimen will contribute to the diffraction pattern recorded. For an aberration-free objective lens, the size of the selected area would depend only on the diameter of the selected-area aperture and the magnification of the objective lens and so could be made very small. However, the minimum size is severely restricted in practice by the spherical aberration of the lens. An analysis of the situation has been given, for example, by Hirsch et a/. (1965)and by Bowen and Hall (1975)and is illustrated in Fig. 10. For an object point on the axis, an electron beam diffracted at angle c1 ( x A/d, where d

FIG. 10. Diagram illustrating the limitation of specimen area from which a selected-area electron diffraction pattern can be obtained, due to spherical aberration.

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J. M. COWLEY

is the lattice plane spacing) will be displaced from the corresponding image point by an amount MC,a3, where M is the magnification. Hence for a selected-area aperture of radius Ro the maximum diffraction angle included in the diffraction pattern will be amax= (MC,/R,)1i3.Alternatively, one can see that an electron beam scattered at an angle LY from a point that is off axis by a distance ro will pass through the on-axis point of the image, the center of the selected-area aperture, if roM = MC,a3. Hence for a diffracted beam corresponding to a 0.5 I$ lattice spacing, for example, the uncertainty in the position of the diffracting region is of the order of CSct3= 8C,A3, with 3, in angstroms, and the useful minimum size of the selected-area aperture corresponds to a region of diameter about 8000 I$ of the specimen for 100 keV electrons with C, = 2 mm. These arguments give only a rough indication of the limitations of this method. It is usually assumed that the minimum useful selected area is about 1 pm for 100 keV electrons. However, it may be noted that the minimum dimensions of the selected area are strongly dependent on the electron wavelength. For electron microscopes operating at 0.5 or 1 MeV selected-area diffraction has been obtained from areas 500 A or less in diameter (Popov et al., 1960; Dupouy, 1976).In principle the diffraction pattern from a much smaller area of the specimen could be synthesized by choosing the appropriate parts of the intensity distributions of the diffraction patterns recorded for different amounts of defocus of the objective lens, but this procedure would be inconvenient and rarely feasible. 2. Microbeam Selected-Area Patterns The limitation of SAED due to spherical aberration can be avoided if the selected area of the specimen is chosen not by an aperture in the image plane of the objective lens but by restricting the incident beam so that it illuminates only a very small region of the specimen. This is the method used to obtain CBED patterns from small areas as described above, but a number of microbeam methods have been devised with the aim of getting as close as possible to a “parallel-beam” sharp diffraction pattern from a small area. Riecke (1962) produced a fine incident beam of small divergence by use of a triple condenser lens system. Two strong lenses were used to obtain a very small reduced image of the electron source and a third, long-focal-lengthlens served to image this on the specimen plane, giving an illuminated area approximately l00OA in diameter. Later (Riecke, 1962) he used a strong short-focal-length final condenser lens, actually the strong forefield of the objective lens, to produce incident beam spots in the specimen only a few hundred angstroms in diameter. The principle of this method is illustrated in Fig. 11.

ELECTRON MICRODIFFRACTION

25

FIG. 1 I . Use of a small crossover to produce a nominally parallel beam to give a microdiffraction pattern.

At one extreme, the beam defined by a small aperture could be focused on the specimen. This is CBED. At the other extreme, illustrated in Fig. 11, the electrons from a fine crossover could be focused by a short-focal-length lens to give a nominally parallel beam over a small area of the specimen. The appearance of parallel illumination is, however, an illusion derived from the geometric-optics diagram. The convergence angle of the beam on the specimen will be given by the ratio of the diameter of the crossover to the focal length of the lens. The same basic limitation applies to both extreme cases and all intermediate degrees of focusing. The diameter of the specimen area illuminated is inversely proportional to the angle of convergence. The "paraIle1-beam" case differs from the usual CBED case only in that the diffraction spots, being images of a crossover, are diffuse maxima rather than sharply defined images of an aperture. Correspondingly the area of the specimen illuminated may be more sharply defined, e.g., with roughly a Gaussian shape, rather than the slowly decreasing [ J l ( x ) / x ] * form for the usual CBED case. Thus the microbeam technique, used with a CTEM system to magnify the diffraction pattern and relate it to the image, can give results comparable to the CBED technique used in conjunction with STEM. For convenient use, however, it requires a specially designed condenser lens system not usually found in commercial electron microscopes.

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J. M. COWLEY

D . Incident-Beam Scanning

In the diffraction modes we have so far considered, a fixed beam is incident on the specimen and the intensity of diffracted electrons is measured as a function of the angle of scattering. The same intensities will be observed if, with essentially the same geometry, the scattered-beam direction is fixed and the intensity is recorded as a function of the angle of incidence, i.e., with a fixed detector and a scanning system used to vary the incident beam direction systematically (Fig. 12). Spec.

FIG. 12. Incident-beam scanning system to provide microdiffraction in a CTEM instrument.

This follows from application of the principle of reciprocity (Cowley, 1969a),which may be stated as follows: The amplitude (or intensity) of radiation at a point B due to a point source at A will be the same as the amplitude (or intensity) at the point A due to an equivalent point source at B. Applying this principle to all points of a finite incoherent source and a finite incoherent detector, we see that the intensity recorded with the detector in Fig. 12 will equal the intensity of the diffraction pattern produced by an incident beam from an incoherent source subtending the same angle p at the specimen. It must be remembered, ofcourse, that the illumination in a CTEM instrument is not necessarily equivalent to illumination from a finite incoherent source subtending an angle equal to the angle of convergence of the incident beam. Especially if a field emission gun is used, the convergent incident beam may be almost completely coherent and the simple reciprocity relationship of Fig. 12 no longer applies. In both STEM and CTEM instruments, deflector coil systems are usually present before the specimen and have been used to produce diffraction patterns by incident beam scanning. In STEM instruments, a weak, long-focus lens instead of the strong objective lens is sometimes used to give a focused, high-resolution electron diffraction pattern from a relatively large area of the specimen, as in Fig. la. Use of the scanning coils and a fixed detector then allows this diffraction pattern to be displayed and recorded, using the scheme of Fig. 12. The reciprocity relationship suggests that this arrangement is equivalent to the SAED method used in a CTEM instrument. The limitations are the same. We have seen that the effect of spherical aberration of the objective lens

ELECTRON MICRODIFFRACTION

27

on electron beams scattered through large angles puts a lower limit on the size of the selected area in the CTEM case. For the STEM configuration the effect of lens aberrations on beams incident at high angles will impose the same restrictions on the area from which the diffraction pattern is obtained. However, for the STEM case the effect of lens aberrations could be reduced by modulating the objective lens focal length in synchronism with the scan of the incident beam. The use of incident-beam scanning with CTEM does not suffer from this limitation of the SAED method. The diffracted beam that is detected can be the axial beam, which is magnified to give an image of optimum resolution by the microscope lenses. The selected area of the specimen can be chosen from the high-magnification image seen on the final viewing screen. A smalldetector aperture is used to select the specimen region and the intensity passing through the aperture is recorded as the direction of the incident beam on the specimen is varied. This method was suggested by van Oostrum et af.(1973) and has been used by Geiss (1976) to obtain diffraction patterns from regions as small as about 30A in diameter. It has the advantage over the microprobe methods of microdiffraction in CTEM that it does not require a nonstandard illuminating system. Also the heavy contamination and irradiation damage usually associated with a microbeam system are avoided because the normal broad-spot, near-parallel CTEM illumination can be used. Apart from the instrumental limitations of microscope instabilities and low signal strength, the method is limited only by the same fundamental factors as apply to the microbeam CTEM or STEM cases. In the absence of lens aberrations, the resolution obtainable in the diffraction pattern is inversely proportional to the diameter of the selected area of the specimen. The effect of spherical aberration of the objective lens is clearly seen in this case because it will limit the resolution of the final image and hence restrict the accuracy with which the selected area of the specimen can be defined.

E . Rejlection Microdiffraction The methods described above for electron microdiffraction from thin specimens by transmission may, in principle, be applied equally well to the glancing-angle reflection electron diffraction from flat surfaces of bulk specimens (reflection high-energy electron diffraction, RHEED) or to the near-normal incidence diffraction of low-energy electrons (LEED). While practical difficulties of electron optics have prevented any success with low-energy electrons, some limited results have been obtained for electron energies in the range 5-100 keV (Cowley et al., 1975; Nielsen and Cowley, 1976).

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For 100 keV electrons the angles made by incident and diffracted electron beams with a flat crystal surface are usually around rad. An incident beam of circular cross section will therefore intersect the surface in a highly elongated ellipse, with major axis 100 times the minor axis. Hence the selected area giving the diffraction pattern will be 100 times as large as for the transmission case. In order to reduce this area one could consider reducing the electron energy so that the diffraction angles would be increased in proportion to A. However, for a probe-forming lens of the same focal length and C, value, the area of the minimum spot will be increased by a factor proportional to A3I2. Hence the minimum selected area possible will vary roughly with ,I1/’. Similar considerations apply to the imaging of flat surfaces (Nielsen and Cowley, 1976).The resolution of the image formed by a scanning-microscope system will be approximately given by the probe diameter. As the accelerating voltage is increased and the diffraction angles decrease, the images formed by detecting the diffracted beams will become increasingly foreshortened and difficult to interpret even though the resolution in the one direction, perpendicular to the beam direction, will improve. These conclusions apply, of course, only to surfaces that are very nearly planar. RHEED intensities from very small regions will be very strongly influenced by any waviness, steps, or projections on the surfaces. The diffraction patterns and images may be almost completely dominated by contributions from the tips of small surface irregularities. In order to overcome the effects of this extreme sensitivity to surface morphology, the larger diffraction angles for lower-energy electrons are advisable. The energy range of 5-10 keV seems to offer the best compromise, although so far the beam sizes used for diffraction and diffraction imaging in this range have not been less than a few hundred angstroms (Cowley et al., 1975). F. Optical Microdiflraction

The optical diffractometer has become a valuable accessory in electron microscope laboratories and has been widely used for the evaluation of imaging conditions by observation of the optical diffraction patterns obtained from selected regions of the photographic negatives of micrographs. The photographic plate is illuminated by parallel coherent light from a laser source and the diffraction pattern is observed and recorded at the back focal plane of a long-focus lens. For images of periodic objects, the optical diffraction pattern can be useful in revealing the presence and the nature of the periodicities in the image. Since it is possible to obtain the optical diffraction pattern from particular areas of the photographic images of any size, the use of the optical diffracto-

ELECTRON MICRODIFFRACTION

29

meter has been regarded as an alternative to the use of electron microdiffraction techniques in the microscope. It has the great advantage that once the micrograph has been taken, it may be examined in detail before the areas for microdiffraction are carefully selected and the diffraction patterns can be obtained without further radiation damage or contamination of the specimen. The principal defect is that, except in special cases, the optical diffraction pattern does not have the same intensity distribution as the electron diffraction pattern, although the periodicities revealed are usually similar. The optical diffraction pattern may give valuable information when used in conjunction with lattice fringe images to follow variations of periodicities in crystals corresponding to variations of composition or ordering, as in the cases of the spinodally decomposed Au-Ni alloy and the partially ordered Cu,Au alloy examined by Sinclair et a/. (1976).However, applications of this sort must be made with care. The fringes in the image, corresponding roughly in spacing, but not in position, with the planes of atoms in the crystal usually arise as a result of strong dynamical scattering. It is well known (Cowley, 1959; Hashimoto et al., 1961) that under simple two-beam diffraction conditions the fringe spacing may vary if there are changes in thickness or orientation of the crystal. The spacing may also vary in a bent crystal at places where other strong reflections are excited. The more general n-beam diffraction case is even more complicated. The intensities of the optical diffraction pattern will resemble those of the electron diffraction pattern only under a very restricted set of conditions. For example, if the specimen is very thin and scatters weakly, the image intensity obtained in bright field at the Scherzer optimum defocus (Cowley, 1975) will be given approximately by Ie(xY) = 1

+ 2 g 4 ( ~* ~S ()X Y )

(26)

where &xy) is the projection in the beam direction of the potential distribution of the object. If the amplitude of the light transmitted through the photographic plate is proportional to Ie(xy),the optical diffraction pattern intensity will be given by I@(uv)~(uv)sinx(uv)1’, where @(uv) = 9 4 ( ( x y ) .This will be proportional to the electron diffraction intensity I@(uv)I’, only to the extent that the contrast transfer function of the lens, sinX(uv),is of constant amplitude and sign. However, even for this most favorable case, with optimum defocus, the optical diffraction intensities will be greatly reduced for small and large angles of scattering. Should there be deviation from optimum focus, or any appreciable dynamical scattering, i.e., if second- or higher-order terms must be included in (26), or if the relationship of the light transmitted through the photographic plate to the electron image intensity should not be exactly linear, further perturbations of the image intensity will result. This may be illustrated by

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J. M . COWLEY

reference to an idealized object with projected potential distribution

$(x)

=A

+ 2B cos 27rx/a

(27)

which, under the above assumptions would give electron and optical diffracpatterns

I(u) = A’S(u)

+ B16(u - l/a) + B?S(u + l/a)

(28)

i.e., a central beam and one diffracted beam on each side with amplitude B , modified by the contrast transfer function in the optical case. Unless o$(x) << 1, second- and higher-order terms will appear in (26),and (28)will be replaced by

I(u) = A’S(u)

+ 1B,”S(u- n/a) n

(29)

with appreciable intensities for several values of n. Even for an ideal optical diffraction arrangement the B,” coefficients will be modified in the optical diffraction pattern by the contrast transfer function of the electron microscope lens, which is in general complex, to give Bt # B,. If there is not a strictly linear relationship between the amplitude of the transmitted light and the electron image intensity, further higher-order reflections will be generated in the optical diffraction pattern, which will then be written

I&) = A’S(u) +

1 C,2S(u- n/a) n

(30)

with Cn# B.’ # B,. Thus in each case there will be a central beam and more than one diffracted beam on each side and in general the optical diffracted beam intensities will not be the same as the electron diffraction intensities. For greater crystal thicknesses or for different values of the lens defocus, the relative amplitudes and phases of the central and diffracted beams will vary. In particular, the crystal thickness or the defocus can be adjusted so that the diffracted beams are out of phase with the central beam so that instead of being given by (26)the intensity for the object (27)will be

I ( x ) = A’

+ 48’ cos ~ T C X / U= A’ + 2B2 + 28’ cos ~ T C X / U

(31)

Then the optical diffraction pattern will contain no first-order diffraction spots nor any of the odd-order reflections of (30), whereas the electron diffraction pattern will contain strong first-order reflections and all odd- and even-order reflections of (29). Thus the optical diffraction pattern may differ very markedly from the electron diffraction pattern in the relative intensities of the spots and also in the extent of the pattern. However, if these limitations are fully appreciated and taken into consideration the optical diffraction patterns may provide a valuable means for

ELECTRON MICRODIFFRACTION

31

analyzing images even in cases where strong dynamical scattering is known to exist but the image is known to give an approximate nonlinear representation of the projected potential, as in the images of complex oxides obtained with atomic resolution (Cowley and Iijima, 1976). Arguments have been presented (Iijima and Cowley, 1978) to justify the interpretation of the local intensity variations of diffuse scattering in an optical diffraction pattern from an image of disordered material even though the overall intensity distribution of the pattern is very different from that of the electron diffraction pattern (see Fig. 13).

FIG.13. (a) High-resolution electron microscope image of a disordered Nb,W oxide, (b) electron diffraction pattern from the same crystal, (c) optical diffraction pattern from the electron micrograph negative of (a).

IV. OPERATIONAL FACTORS A . Radiation Damage

For many specimen materials the most important experimental limitation on the use of microdiffraction methods will undoubtedly be the radiation damage produced in the specimen by the incident electron beam. This

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J. M. COWLEY

limitation is particularly severe for organic and biological materials but will be important for most substances other than the electrical conductors. Some energy is transferred from the incident beam to collectiveexcitations of the specimen crystals to give either correlated vibrations of the atoms (phonons) or collective oscillations of the nearly free electrons (plasmons), which decay rapidly, generating phonons. In either case the net result is a heating of the specimen. Other processes of inelastic scattering of the incident electrons result in ionization, leading to the breaking or rearrangement of bonds in the case of molecular or homopolar crystals or the generation of localized defects in ionic compounds. In thin samples, molecular fragments or displaced atoms may be ejected or evaporated into the surrounding vacuum. In many cases, even for metals and semiconductors, the enhancement of diffusion processes under irradiation may result in phase changes, crystal growth, or annealing, or any process that brings the specimen more nearly into equilibrium with its environment. A further source of radiation damage is the “knock-on” collisions of electrons with atoms, in which atoms are given sufficient energy to displace them from their lattice sites. Since the incident electron is relatively light, it must have very high energy in order to do this (e.g., 400 keV to displace copper atoms). Hence this type of damage is important only in high-voltage electron microscopy where it can be the predominant damage effect for metals for which ionization effects are relatively small. The importance of radiation damage effects for various specimen materials has been extensively investigated and discussed in relationship to highresolution electron microscopy (see, e.g., Glaeser 1974; Misell, 1977). The situation for microdiffraction may be judged by comparing the amount of intensity information required to define the diffraction pattern with sufficient accuracy with that needed to give a satisfactory representation of the image of the same area. A commonly made, rather conservative assumption is that in order to adequately define the image intensity, it is necessary to detect lo4 electrons per picture element. A smaller number may suffice for dark-field images. In order to characterize the intensity distribution in an electron diffraction pattern with comparable accuracy, one would have to detect a similar number of electrons, one the average, at each of a large number of points, perhaps lo5 points if there is a large amount of fine detail in the pattern or as few as 10 points if one requires to know only the intensities of the stronger spots in the diffraction pattern of a single crystal of simple structure. We may assume, as a reasonable compromise that one must detect a minimum of lo6 scattered electrons in order to define the diffraction pattern intensity distribution. Then the minimum area from which a diffraction

ELECTRON MICRODIFFRACTION

33

pattern can be obtained will have a diameter roughly 10 times the resolution limit for microscopy set by the radiation damage effects. Rough figures for this minimum diameter would then be, for example, l W A for 1-valine, 400 for polyethylene, and 30 A for phthalocyanine (Glaeser, 1974). As an alternative indication, we may go directly to the observations on the decay ofcrystal diffraction patterns that were used as the primary evidence for radiation damage effects in many cases. The radiation dose required to substantially reduce the extent of the diffraction pattern of organic materials, has been found to vary from 0.25 electrons/A2 for hydrated biological specimens to 130 electrons/A2 for phthalocyanines (Misell, 1977). If we assume that lo6 electrons will be scattered from an incident beam of lo7 electrons, the rpinimum diameters of the area that can be selected for microdiffraction then vary from 1 pm for hydrated biological material to 300A for the phthalocyanines. The use of microdiffraction from areas of 20A or less in diameter will therefore be restricted to a limited range of stable materials. These will mostly be electrical conductors for which ionization effects represent only transient perturbations of the cloud of free electrons. For most specimens the heating by the incident beam will be a relatively minor contribution to the radiation damage effect. The increase in temperature of an electron microscope specimen may be as much as 100 or 200 degrees, although it is usually much less than this. In spite of the fact that higher current densities may be used for microdiffraction, the temperature rise may be no greater because the cooling of the specimen by radiation and conduction of heat becomes much more effective as the volume of the heated region is reduced.

B. Contamination In the relatively poor vacuum systems of most commercial electron microscopes (about torr) the accretion of a layer of contamination on specimen surfaces is a well-known phenomenon, which may be a serious limitation to high-resolution microscopy. As the size of the incident beam decreases the importance of this effect increases. With an incident beam less than 100A in diameter, a mound of contamination thousands of angstroms in height may grow on each side of a thin specimen in a few seconds, completely masking the diffraction from the specimen (Knox, 1976). The contamination is believed to result from polymerization, under electron irradiation, of organic material in the system, coming from the pump oils, gasket materials, grease on the surfaces of structural components of the specimen stage, or from the specimen itself. It has been found by Isaacson et a/. (1974) that contamination may occur even in the baked,

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ultrahigh-vacuum system of a STEM instrument, presumably as a result of the introduction of mobile organic components of the specimen. The accumulated evidence suggests strongly that the contamination deposit is built up mainly as a result of migration of the organic molecules across the surface of the specimen. As the molecules enter the intense electron beam they become ionized, polymerized, and partially graphitized. Their mean free path within the electron beam is of the order of hundreds or thousands of angstroms. In normal high-resolution CTEM the incident beam is usually a few microns in diameter. As can be seen by subsequent observation at low magnification, a ring of heavy contamination may be formed around the boundaries of the beam. The central part of the incident beam is thus well guarded from contamination and a contamination rate as low as 1 &ec or better is commonly achieved. If the incident beam is less than 1000 A in diameter the contamination will cover the entire beam area. The dimensions of the mounds of contamination formed may be determined by subsequently shadowing the specimen with a thin layer of evaporated metal (Fig. 14). The observations of Knox (1976) showed an almost linear initial increase in the volume of deposited material with time. The minimum diameter of the mound of contamination formed with an incident beam of 30 A diameter was about 150 A. For mounds 2 pm high formed in about 400 sec of irradiation, the diameter at the base was about 2000 A. Obviously the further development of electron microdiffraction is possible only if contamination can be eliminated or, at the least, reduced by a very large factor. It is important that the amount of mobile organic material should be reduced to a very low level by use of a clean ultrahigh-vacuum system, by use of low-temperature specimen environments, and by removing

FIG.14. Electron microscope image of contamination mounds formed by an electron beam of small diameter (less than 100 A) in increasing intervals of time, with metal shadowing to indicate the heights of the mounds (after Knox, 1976).

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as much organic material as possible from the specimen grids and the specimen itself. Beyond this, various means have been suggested for preventing the migration of the organic material across the specimen surface. The observation of the “guard-ring’’ effect of the wide incident beam in CTEM suggests that low-level irradiation of a large surrounding area of the specimen with an auxiliary electron beam would be effective. Since this could interfere with the observation of the diffraction pattern, the alternative of an intermittent irradiation of the wider specimen region is probably more convenient. Lehmpfuhl has found that if a large specimen area is irradiated briefly with a high-intensity electron beam, the center of the irradiated area will remain free of contamination for a period sufficient to allow series of measurements to be taken. This technique is obviously of limited value for radiation-sensitive specimens, although the level of background irradiation may be several orders of magnitude less than the irradiation of the specimen area being examined. For specimens to be observed in clean ultrahigh vacuum conditions the most important source of contamination is the specimen itself and several means of pretreatment of the specimens have been reported as effective. Irradiation of the specimen with infrared radiation (Isaacson et al., 1974) apparently helps to clean it. Irradiation with ultraviolet light may be effective in fixing the molecules on the surface (Engel et al., 1977) and cooling the specimen reduces the mobility of the molecules. For heat-resistant specimens, baking at moderate temperatures may be very effective.

C. Instrumentaf Stability It is well known that the performance of STEM instruments is very dependent on their mechanical and electrical stability. Any rapid or irregular fluctuation of the beam position relative to the specimen, due to movement of the filament, vibration of the column, or stray electric or magnetic fields, will degrade the resolution. In microdiffraction the size of the selected area and the precision with which its position can be correlated with an image depend even more strongly on these factors. In STEM a slow drift of the beam or specimen will result only in a slight distortion of the image. A vibration of the instrument or a ripple on the deflecting or focusing fields will likewise produce only an image distortion if it has a frequency that is a multiple of the frequency of one of the scans. However, for microdiffraction any drift or vibration will be significant if it produces an appreciable motion of the beam relative to the specimen within the time taken to record the pattern (see Brown et al., 1976). It follows that the requirements for instrumental stability and freedom from stray fields are extreme and that optimum performance will depend on the development of

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recording systems of high efficiency to allow patterns to be recorded in the shortest possible time. The effect of these instabilities on the diffraction pattern intensities will be an incoherent addition of intensities from different specimen areas. It will resemble the effect of using a larger incoherent eIectron source (Cowley, 1976b).In general the coherent diffraction effects that depend on the relative phases of the diffracted wave across the beam diameter will be reduced. Some of the information that is unique to electron microdiffraction will be lost. V. INTERPRETATION AND APPLICATION A. Identification The most immediate and obvious application of microdiffraction is for the identification of the crystalline phases present in very small volumes of sample materials observed by electron microscopy. With the usual limitations of selected-area diffraction in CTEM, it is not possible to obtain clear evidence on the nature of small particles, precipitates in a matrix, or the components of a multiphase mixture when the sizes involved are less than a fraction of a micron. The possibility of obtaining a diffraction pattern from an area 30 A or less in diameter, identifiable in a microscope image, represents an important advance in technique. Applications of microdiffraction for these purposes to date have been mostly exploratory, designed to test the technique rather than to apply it systematically to the solution of structural problems, but systematic applications are beginning to appear, Geiss (1976) used his scanning technique with CTEM to observe diffraction patterns from single grains in an aluminum sample and from portions of single asbestos fibers. Carpenter et al. (1977) using the STEM attachment of a CTEM microscope obtained microdiffraction patterns from separate phases forming a lamellar microstructure in a'Cu -Ti alloy, and with similar apparatus de Diego et al. (1976)were able to sort out the diffraction patterns due to small precipitates in a Ta-N alloy. The detection of local ordering to form an ordered superlattice within a microdomain of an alloy has been demonstrated by Dunlop and Porter (1976) for metal carbides using the STEM/CTEM combination and by Brown et al. (1976)for Cu-Pt alloys using a dedicated STEM instrument. Chevalier and Craven (1977) have made a systematic study of ordering in the Cu-Pt microdomains with a dedicated STEM instrument, making estimates of the degree of order within 40 A microdomains from the intensities of superlattice reflections (see Fig. 15).

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FIG.15. Microdiffraction patterns obtained with a STEM instrument from microdomains of a Cu-Pt alloy approximately 40 A in diameter, showing the (hhh) row of spots. In (a) the direction of ordering is perpendicular to the beam; in (b) the direction of ordering is parallel to the beam.

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The difficulties of this method are those of normal selected-area diffraction, compounded by the difficulties inherent in the use of small regions barely resolved in an electron microscope image. A single diffraction pattern obtained in an arbitrary orientation is rarely sufficient to allow identification of a phase, even in the favorable case that there is so much known of the system in advance that a single feature of the diffraction pattern, such as the presence of superlattice reflections, is sufficient to give an unequivocal indication on the information required. For known phases, tilting the sample to a particular orientation will usually give the desired result. For unknown phases, a series of diffraction patterns in well-defined orientations may be required. With the microscopes now available it is often a very difficult and time-consuming task to obtain diffraction patterns from exactly the same, barely resolved region with the sample tilted through a series of widely different angles. It is easier to obtain diffraction patterns at random from a large number of different selected areas in random orientation if it can be assumed on the basis of independent evidence that all the selected areas have equivalent structure. There is a danger in this that one can always be tempted to assume on the basis of insufficient evidence that the one selected area giving a clear interpretable pattern is typical of all the regions that appear superficially to be similar in a micrograph. B. Symmetry

1. General Electron microdiffractionpatterns contain a great amount of information concerning the structure of small regions of matter. As in the more familiar case of X-ray diffraction patterns, the systematic derivation of this information involves the interpretation of first the geometry and symmetry of the diffraction patterns and then the intensity distributions. The differences from X-ray diffraction become evident early in this process, since it has long been realized that the well-known rules for X-ray diffraction on the evidence for crystal symmetry in the diffraction patterns do not apply for electron diffraction because of the very strong dynamical diffraction effects. The detailed exploration and understanding of the relationship of electron diffraction intensities to crystal structure and symmetry has been strongly dependent on the development of electron microdiffraction techniques. The diffraction patterns from larger areas of crystalline specimens are usually complicated by the averaging over a wide and undefined range of crystal thickness, orientation, and perfection, and the intensities vary strongly with all these factors. Except for a few cases of unusually perfect and uniformly thin crystal samples, it is possible to obtain sufficient control over

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all relevant experimental variables only by choosing a very small area of perfect crystal for which the thickness is shown to be uniform. The relationship of diffraction intensities to crystal symmetry differs from that for X-ray diffraction in several respects. The coherent multiple scattering (dynamical diffraction) renders the relationship more complicated but at the same time provides the possibility for deriving a greater amount of information. In situations for which more than two beams have appreciable intensity, Friedel’s Law does not hold, i.e., 1, # 1, in the absence of a center of symmetry. It is relatively easy in electron diffraction to find orientations for which the intensities of reflections corresponding to reciprocal lattice points symmetrically placed with respect to the origin are clearly not equal even though in geometrically equivalent situations. Hence the ambiguity of space group determination and in the determination of the sense of a polar axis, which limits kinematical X-ray diffraction, does not apply. This is demonstrated clearly in the CBED pattern from CdS, Fig. 16 (Goodman and Lehmpfuhl, 1968). In kinematical X-ray diffraction, the so-called forbidden reflections, which are systematically absent as a result of the presence of screw axes or glide planes in the structure, are essential guides for space group determination. There are many observations of electron diffraction patterns from thick crystals or crystal surfaces in which the forbidden reflections not only occur but appear to be the strongest reflections present. They can result from coherent or incoherent multiple scattering of electrons. If the reflection corresponding to the reciprocal lattice vectors h and g occur, the reflection h g can be produced by double scattering. In the coherent diffraction case, the wave amplitude for a reflection I), can contain contributions arising from waves I)h and $ , - h for combinations involving all h and also for combinations of three, four, or more wave amplitudes. The resulting intensity will depend on the relative phases of all these contributions. Under certain conditions as in crystal orientations of high symmetry, these amplitude contributions can cancel out systematically, giving zero intensities (Gjq5nnes and Moodie, 1965; Moodie, 1972).

+

2. Two-Dimensional Symmetry The conditions under which systematic absences (forbidden reflections) occur in electron diffraction patterns can most readily be appreciated by recourse to the real-space argument of Cowley and Moodie (1959; also Cowley et al., 1961) based on the concept of an initially plane wave progressing through successive layers of a crystal with scattering through small angles only. First we consider the approximation that the three-dimensional

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FIG. 16. CBED pattern From a thin CdS crystal in the [2130] projection showing the absence of a center of symmetry, but the mirror symmetry about the [OOOl] axis (horizontal) is clear (Goodman and Lehmpfuhl, 1968).

distribution of atoms in the unit cells can be ignored and only the projection of the unit cell contents on some central plane is relevant. This is equivalent to the assumption that the only reflections occurring are those corresponding to a planar section of reciprocal space perpendicular to the incident beam. The symmetry of the wave function for a planar incident wave after passing through one slice, one unit cell thick, will then be the symmetry of the transmission function exp[ - io+(xy)], where + ( x y ) is the projection of the potential distribution in the unit cell. Propagation of this wave by Fresnel

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diffraction to the next slice will not affect the symmetry since it is represented by convolution with a propagation function having cylindrical symmetry. If the potential distributions of the first and second slices coincide exactly when projected in the beam direction, the wave leaving the second slice will have exactly the same symmetry as that leaving the first. For a perfect crystal of any thickness, therefore, the wave at the exit face will have the same symmetry as that of the transmission function of the unit cell, projected in the beam direction. The systematic absences in the diffraction pattern will then be exactly those in the corresponding reciprocal lattice plane for the phase object approximation. It should be noted that these absences are not necessarily those given by kinematical theory. The symmetry of exp[ - io+(xy)] is not that of +(xy) in general. This is illustrated in Fig. 17a, which represents the [1101 projection

I-4

Wave function

(b-1) (b-2) FIG.17. (a) [110] projection of the structure of silicon or germanium. (b) Diagrams illustrating the difference of symmetry of the projection with Fresnel diffraction for a twofold screw axis with (1) the atoms at the same z coordinates and (2) with atoms at different z coordinates.

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of silicon or germanium. Kinematically, for spherical atoms, both the 200 and 222 reflections are forbidden. The 200 is absent because the projections of interatomic vectors AB and BC on the [200] direction are equal. However the potential fields of A and B overlap more than those of B and C . The function &xy) is additive for the overlapping regions but exp[ - icrc$(xy)] is not. Hence the projections of exp[ - io4(xy)]for the AB and BC pairs are not equivalent and the 200 reflection is not forbidden dynamically. On the other hand, this effect of overlap has less effect on the projections for the 222 reflection, which remains very weak, and no effect on the 110, which remains forbidden. These arguments are consistent with the results of n-beam computations made by Dessaux et al. (1977) using the matrix method and by O’Keefe (private communication) using multislice calculations. With even a small tilt away from the exact axial orientation, the condition of exact superposition of the projections of slices will no longer be valid and the forbidden reflections may appear. This is clearly seen in the C3ED pattern (Fig. 18) obtained by Goodman and Lehmpfuhl (1968) from CdS. The odd-ordered 00.1 reflections have zero intensity at the centers of their disks, which correspond to the exact axial orientation. These reflections are kinematically forbidden by the twofold screw axis in the crystal (a glide-line

FIG. 18. CBED pattern from a CdS crystal near the [lOTO] zone axis. The odd-order 0001 reflections (center horizontal row) show black bands corresponding to the dynamically forbidden reflections.

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symmetry of the planar group pg). The disks also have zero intensities along the 00.1 line since tilts in this direction do not destroy the symmetry element of the wave function corresponding to the screw axis in the crystal. However, tilts in other directions do destroy this symmetry relationship and the reflections gain appreciable intensity. Gjplnnes and Moodie (1965) have demonstrated that a further condition may give dynamically forbidden reflections, namely, when the incident beam is at the Bragg angle for the kinematically forbidden reflection. This may be understood by considering that atoms separated by half the lattice plane spacing will be illuminated by waves that are n/2 out of phase. The modifications of the wave functions due to dynamical effects in the corresponding rows of atoms through the crystal will then be independent and equivalent, giving waves at the exit surface that will cancel out when combined with an additional 742 phase change in the Bragg reflection direction. 3. Three-Dimensional Symmetry

In the previous section we made the assumption that the symmetry of a wave passing through a slice of unit cell thickness will be determined by the projection of the unit cell in the beam direction. This is not strictly true. An atom near the top of the slice will produce a perturbation on the wave function at the bottom of the unit cell that is different from that due to an atom near the bottom, because the Fresnel diffraction spread of the wave from the atoms will be different. Hence, as suggested in Fig. 17b, the wave function symmetry will reflect the symmetry of a twofold screw axis only if the atoms related by this symmetry element lie in a plane perpendicular to the beam or if, as suggested by Gjplnnes and Moodie (1965),the “forbidden” reflection is at the Bragg angle. For a threefold screw axis, the symmetry of the wave function can never be exactly that of the unit cell projection. These modifications of wave function symmetry due to the three-dimensional nature of the distribution of atoms will usually be small, especially for thin unit cell slices (10 A or less for 100 keV electrons) and the resulting nonzero reflections will be very weak and rarely discernible in principal orientations. They correspond in reciprocal space to the involvement of reflections associated with reciprocal lattice points not in the plane perpendicular to the electron beam. Calculations of the magnitude of their contributions have been made by Lynch (1971) for the favorable case of gold crystals in [1111 orientation. By analogy with X-ray diffraction jargon, they are sometimes referred to as “upper layer-line” interactions. They can be made to contribute appreciably to the intensities of CBED patterns by a suitable choice of crystal orientation, usually well removed from a principal axis orientation (see Goodman and Secomb, 1977).

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These three-dimensional diffraction effects introduce symmetry relationships quite different from those encountered in X-ray diffraction. A systematic evaluation of these relationships was made by Gj#nnes and Moodie (1965) and subsequently in more detail by Buxton et ul. (1976). Goodman (1975) has shown that, on the basis of these symmetry relationships, it is possible to develop a practical method for three-dimensional space group analysis without any of the ambiguities of kinematical X-ray diffraction. The most difficult determination is probably that of “handedness,” or the absolute configuration for noncentrosymmetric space groups. Goodman and Secomb (1977; Goodman and Johnson, 1977) showed that, with an appropriate choice of orientations, giving enhanced sensitivity to the three-dimensional dynamical diffraction effects, determinations of this sort can be made with a few observations plus some relatively simple calculations. In this way they confirmed the right handedness of a-quartz.

4. Crystal Symmetry Under dynamical diffraction conditions it cannot be assumed that the diffraction pattern reflects the symmetry of the unit cell contents. The electron wave interacts with the whole crystal, and it is the symmetry of the crystal that is relevant. The previous paragraphs have referred to cases that are ideal both theoretically and experimentally, in which the symmetry of the crystal is determined predominantly by the symmetry of the unit cell contents, with perfect crystal structures and crystal boundaries that introduce no further symmetry restrictions. However, these are special cases. Other special cases have been observed for which the effects of crystal symmetries, which differ from the unit cell content symmetries, are pronounced. The diffraction pattern is given by the Fraunhofer diffraction from the wave function near the exit face of the crystal. This wave function results from transmission through the whole crystal and retains the influences of all portions of the crystal including the entrance and exit faces and any faults encountered in between. A striking illustration of this is given by the CBED patterns of graphite obtained by Johnson (1972), which show very pronounced threefold symmetry in place of the usual, kinematical sixfold symmetry obtained with the beam parallel to the c axis. The threefold symmetry is that of a crystal having one or more stacking faults in the basal plane. While a very high proportion of the unit cells in the crystal have the usually hexagonal graphite structure, the lateral displacement at a fault plane reduces the symmetry of the wave function. An equally striking although more subtle example is given by Fig. 19, which shows a CBED pattern obtained by Goodman (1974) from a thin MgO platelet with faces parallel to 100 type planes, tilted so that the beam is in the

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FIG. 19. CBED pattern from a MgO platelike crystal with (100) faces with the incident beam along the [ 1 1 11 direction.

[1111 direction. If the entrance and exit surfaces had been perpendicular to the incident beam, the pattern would have shown the hexagonal symmetry of the crystal structure viewed in this direction. However, for the tilted crystal with the entrance and exit faces making large angles with the plane normal to the beam, the symmetry of the crystal as seen by the incident beam

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cannot be hexagonal but it can, at most, have twofold symmetry.The diffraction pattern of Fig. 19 shows only the mirror-plane symmetry indicated. C. Intensities: Structure Analysis

Following the determination of unit cell dimensions and crystal symmetry, the next stage in the quest for structural information is the interpretation of the observed intensities in terms of crystal structure. While the term “crystal structure’’ can refer to the atomic arrangement, or more precisely, the potential distribution, throughout the whole crystal, including surfaces and defects, we use it here in the more common, restricted sense, referring to the potential distribution within one unit cell of the idealized periodic structure. This is the equivalent of the electron density distribution determined by the processes of X-ray diffraction crystal structure analysis. The determination of crystal structure in practice usually consists of the determination of the magnitudes and relative phases of the Fourier coefficients of the potential distribution +(r) (the structure amplitudes Qh). It is the occurrence of strong dynamical diffraction effects that complicates the interpretation of electron microdiffraction intensities, making the derivation of structure amplitudes more difficult but at the same time making the method intrinsically more powerful. The “phase problem” of kinematical X-ray diffraction does not apply. Relative phases as well as magnitudes of the structure amplitudes can be derived from coherent diffraction effects within the crystals. Also it appears possible that, in favorable cases, the observation of coherent interferences between overlapping diffracted beam spots in CBED patterns may provide direct evidence on relative phases even under kinematical scattering conditions (Cowley and Jap, 1976; Nathan, 1976). Historically, structure analyses of crystals based on electron diffraction patterns from small single-crystal regions have followed one or the other of two extreme courses (Cowley, 1967). On the one hand, rough indications of structure have been obtained using patterns from specimens of poorly defined morphology and degree of perfection on the basis that, if there is sufficient averaging of intensities over crystal thickness and orientation, an interpretation in terms of kinematical theory will give reasonable results provided that forbidden reflections or other gross effects of dynamical scattering are absent or may be ignored. The assumptions of this approach are most nearly justifiable for light-atom materials. Recent examples of such analyses include the studies on relatively large areas of organic crystals by Dorset (1976) and of small areas of Ko.z,WO,, +,) using selected-area techniques by Goodman (1976b). At the other extreme are the highly accurate studies carried out under conditions for which the experimental variables are well defined, a small

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region of perfect crystal is selected, and the dynamical diffraction effects are not only taken into consideration but are utilized as an essential basis for the accuracy of the structure factor determinations. These methods have been applied to a limited number of materials for which suitable crystals are readily available, for which the dynamical diffraction calculations are not too complicated, and for which the accurate determination of structure factors is significant in relationship to theoretical results on the ionization and bonding of atoms in crystals. Very few of these investigations have relied only on the measurement of relative intensities of different reflections. The recent work of Goodman (1976a) using CBED patterns from areas approximately 100 8, in diameter of graphite crystals approximately 120A thick greatly extended the earlier CBED work of Hoerni and Weigle (1949). As well as showing that graphite crystals contain twinned or faulted regions and also areas having an orthorhombic structure, Goodman was able to select regions of perfect hexagonal structure and determine their thickness with an accuracy of better than one unit cell. By comparison of spot intensities with the results of detailed n-beam calculations he was able to refine the structure amplitudes of some of the inner reflections and attained an accuracy of about 1%. The use of CBED patterns to give rocking curves that may be used for structure refinement is more thoroughly established. The variation of the intensities of reflections with angle of incidence of the electron beam, as mentioned in Section III,A, is strongly dependent on dynamical interactions and can be used for determination of structure amplitudes with high accuracy. The methods used and early results have been reviewed previously (Cowley, 1969b)and will not be described in detail here. Early work on MgO (Goodman and Lehmpfuhl, 1967)used patterns such as Fig 6, due as closely as possible to a set of reflections lying on a line through the origin. The dynamical interactions between diffracted beams were thus limited to this so-called systematic set so that the dynamical theory calculations involved relatively few structure amplitudes. Later work by McMahon (1969) showed that neglect of all reflections except the systematic set could lead to errors that vary strongly with orientation but may be as small as 1% for orientations carefully chosen. Taking such effects into account, accuracies of perhaps p/, were attained for some of the inner MgO reflections. Determinations approaching these in accuracy have also been made for germanium by Shishido and Tanaka (1976). It seems probable that precision determinations of structure amplitudes by microdiffraction methods will continue to provide valuable data on particular problems of fundamental interest but will not be a popular pastime because of the considerable amount of care and effort involved. However,

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as the techniques of microdiffraction and the programs for n-beam dynamical calculations become more widespread, there will be a considerable expansion in the variety and sophistication of structure analyses applied to an increasing number of problems in solid-state science. D. Disordered Systems

Microdiffraction has obvious potential as a means for investigation of disordered systems, although the instrumental performance is not yet at the level where this technique can be exploited fully. The disorder of the structure may take the form of a disordered occupancy by different types of atoms (or vacancies) on the atom sites in an extended crystal or it may take the form of a disordering of the atom sites as in amorphous or heavily distorted crystalline materials. In each case, the questions of current interest relate to the possible existence and nature of small, well-ordered microdomains having dimensions in the range 10-50 A. It is very difficult to make deductions regarding these microdomains from the evidence of the usual diffraction patterns, which represent averages over diffraction effects from a very large number of microdomains having a great variety of shapes, sizes, and orientations. The possibility of obtaining diffraction patterns from individual regions of 1020 A in diameter is obviously important. For disordered alloys, considerable evidence regarding the presence of microdomains of an ordered superlattice structure has been obtained by inference from diffraction patterns and by direct observation in highresolution electron microscopes, particularly by use of dark-field images obtained from the diffuse superlattice diffraction spots. Although there are serious difficulties in interpreting high-resolution dark-field images (Cowley, 1973), the evidence is clear in some cases for the existence of well-defined regions in which the ordered superlattice structure is well established in one of its possible orientations relative to the average disordered lattice. These are mostly cases of quenched alloys in which local ordering has progressed to the extent of forming relatively large microdomains, as in the case of the Cu-Pt alloys studied by Chevalier and Stobbs (1976), where dark-field images showed highly reflecting patches averaging about 40 A in diameter. Microdiffraction from these regions (Chevalier and Craven, 1977) showed them to give the superlattice reflections corresponding to one or other of the possible orientations of the ordered, low-temperature Cu-Pt structures (Fig. 14). For the more purely short-range ordered alloys existing above the critical temperature for ordering, it has been concluded from X-ray diffraction evidence that a description in terms of microdomains can be made if

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the average microdomain dimensions are 10-30 A (Moss, 1962; Gragg et al., 1971) and that local configurations of atoms are not necessarily those of the low-temperature ordered structures (Clapp, 1971). Microdiffraction from small regions will clearly add further light on these questions. For amorphous materials the standard diffraction evidence on structure comes from the very diffuse ring patterns, from which it is possible to deduce radial distribution functions giving correlations in the distances between atoms and their near neighbors. From this it is very difficult to distinguish among the extreme structural models, the microcrystalline model that regards the amorphous material to be made up of small ordered regions 10-30 A in diameter, and the random-network model in which the deviation from an ordered structure is progressive and continuous, being governed by energetically possible deviations in local interatomic bonding. Observations by high-resolution electron microscopy of sets of parallel fringes extending over distances of about 15 A in images of amorphous germanium (Rudee and Howie, 1972) appeared at first to favor the microcrystalline model, but it was later shown that the evidence is indecisive because of an imaging artifact (MacFarlane, 1975). Microdiffraction patterns from thin amorphous germanium films have shown a mottled appearance in place of the smooth continuous halos given by larger specimen areas (Brown et al., 1976; Geiss, 1976). This is to be expected because the diffracting regions contained relatively small numbers of atoms in partially ordered arrays and there was little of the averaging over the large number of local arrays that takes place in the usual diffraction experiment. Clearly, microdiffraction patterns taken from even smaller regions can give valuable information on local atomic arrangements, even though difficulties will arise from the fact that the diffraction patterns will refer to two-dimensional projections of regions of films that are usually thicker than the lateral dimensions of the selected areas.

E. Microdifraction in Relationship to Other Techniques It has been emphasized in this chapter that electron microdiffraction is strongly and inevitably related to high-resolution electron microscopy, either CTEM or STEM. The relationship extends to the common instrumentation, the essentially complementary nature of the observations, and the body of dynamical diffraction theory used for both. Microdiffraction is also an essential component of the relatively recent concept of combining an array of techniques under the heading of “analytical electron microscopy.” It is seen that it is of enormous advantage, particularly in materials science, to combine in one instrument, or a few compatible instruments, the capabilities for imaging a specimen in bright

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or dark field, for obtaining microdiffraction to determine the crystal structure, and for making a chemical analysis of small areas by means of either X-ray microprobe analysis or electron energy loss spectroscopy. In modern microscopes, and particularly in those equipped with highbrightness field emission guns, it is possible to detect the characteristic X-ray emission of the elements contained in specimen regions as small as a few hundred angstroms in diameter. The minimum detectable mass of an element may correspond to fewer than lo6 atoms (Joy and Maher, 1977). By observing the energy losses of transmitted electrons due to the excitation of the inner-shell electrons of the specimen atoms, it is likewise possible to identify the elements in regions a few hundred angstroms in diameter (Silcox, 1977). This method is particularly suited to the detection of relatively light atoms for which the X-ray methods are less sensitive. In this context microdiffraction has been used to date mostly in its simplest form as a method for phase identification, but more sophisticated applications will undoubtedly appear with time. It is possible to envisage complete chemical and crystallographic analyses being made on an unknown crystal having dimensions less than 100 A. In further developments of the combination of techniques it may be possible to go beyond the practice of obtaining separate although related pieces of information by sequential application of the various methods. It has been suggested (Cowley, 1976a; Cowley and Jap, 1976) that STEM imaging and microdiffraction may be combined to enhance the power of both. The greatest amount of information concerning the specimen obtainable with the STEM technique will be derived by recording the diffraction pattern in the detector plane for each image point in the STEM image. Combination of diffraction information from successive image points may assist in the derivation of structural information on a much finer scale than the image resolution. Pattern recognition techniques based on the diffraction intensities may allow the more efficient detection of particular atomic groupings. With these and the many other possibilities now appearing it is evident that after a relatively long history of sporadic use and limited accomplishments electron microdiffraction may well grow to a technique of major significance in the near future.

REFERENCES Ackerman, I. (1948). Ann. Phys. (Leipzig) [6] 2, 19 and 41. Anstis, G. R., and O’Keefe, M. A. (1976). 34th Ann. Electron Microsc. Soc. Am. Meet. p. 480. Bethe, H. A. (1928). Ann. Phys. (Leipzig) [4] 87, 55.

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