CHAPTER 1

PRINCIPLES OF IMAGE FORMATION

J. M. COWLEY

DEPARTMENT OF PHYSICS, ARIZONA STATE UNIVERSITY

TEMPE, ARIZONA 85281

1.1 INTRODUCTION

Whatever their backgrounds, electron microscopists tend to bring to

the topic of image interpretation a set of preconceptions based on their
particular education and experience. The difficulty in presenting a
summary of the principles of image interpretation for a volume such as
the present one arises from the wide variety of backgrounds of the
readers.

In the early days of electron microscopy the background of optical

microscopy was familiar to most microscopists and provided useful al-
though incomplete analogies. Electron microscopists of the newer gener-
ation often have little experience of optical microscopy and tend to
base their assumptions on working experience with a particular area of
electron microscopy and the convenient concepts of that field. Biolo-
gists, for the most part, can get along with the useful assumption that
the image intensity depends on the amount of scattering matter present,
with the darker regions in positive prints corresponding to thicker re-
gions or heavier atoms. The limitations due to radiation damage, which
prevent them from using very high resolution, shield them from too much
exposure to the complications of phase-contrast imaging. The essential-
ly non-crystalline nature of their specimens shields them from the com-
plications of diffraction and the dominance of the daunting dynamical
scattering effects.

Materials science microscopists, on the other hand, are brought up

in a world of dynamical diffraction contrast, where thickness fringes,
bend contours and strain contrast predominate. A standardized set of
simplifying assumptions provide a relatively simple and very effective
guide through the labyrinth of three-dimensional diffraction processes
which are dominant for the useful specimen thicknesses and the usual
ranges of resolution.

J. J. Hren et al. (eds.), Introduction to Analytical Electron Microscopy

© Springer Science+Business Media New York 1979
2 Chapter 1

Yet for all electron microscope users the challenge of the many po-
tential advantages of using the high resolution capabilities of modern
microscopes, or of applying some of the recently developed techniques of
STEM, microdiffraction and microanalysis, requires that a more fundamen-
tal understanding be sought of the basic principles of image formation.
In this review, therefore, we will summarize the essentials of electron
scattering and imaging, referring the reader, where necessary, to appro-
priate texts for the more detailed expositions and for a wider range of
illustrative examples. For the sake of those who prefer words to sym-
bols, each topic will be first discussed in a loose, verbal fashion.
The mathematical statements which follow in each case are included for
the sake of those who prefer more exact formulations and may safely be
ignored without loss of continuity or concept.

For the basic ideas of geometric and physical optics, standard text-
books of optics such as those by DITCHBURN (1963) and STONE (1963) may
be consulted while for those with more background in physics, LIPSON and
LIPSON (1969) is sui table. Introductory treatments of electron optics
are found in such books as BOWEN and HALL (1975) and HIRSCH et al.
(1965). The mathematical sections of this review follow the treatment
of COWLEY (1975) which should be consulted for more complete develop-
ments of most of the topics. More specific references to these and
other sources will be given in relation to particular subject areas,
when appropriate, but we do not attempt to provide an exhaustive biblio-
graphy.

Electron Scattering and Diffraction

The terms "scattering" and "diffraction" are often used interchange-

ably. However, a common convention is that when "scattering" is not
used in a general, all inclusive sense, it refers to the case of a beam
of incident radiation striking a small particle (an atom or cluster of
atoms acting as a unit) and giving rise to an angular distribution of
emergent radiation which depends on the nature of the individual parti-
cles and not on their relative positions. This can be called the inco-
herent scattering case. The intensity distribution is the sum of the
intensities given by the individual particles acting independently.

The term "diffraction" is used when interference effects between

waves scattered by many atoms give rise to modulations of the intensity
distribution which may be measured to give information on the relative
positions of atoms. Diffraction may be considered as "coherent scat-
tering" when wave amplitudes, rather than intensities from individual
atoms, are added. However, it must be emphasized that it is the corre-
lation of atom positions, not the coherence of the incident radiation,
which is different for diffraction and scattering. The interaction of
electrons with atoms is the basis of both and there is a common mathema-
tical formulation. It is only the set of approximations made as a con-
venience in practice which distinguishes the two aspects.

Electrons are scattered much more strongly by matter than x-rays or

visible light. A single atom can scatter enough electrons to allow it
to be detected in an electron microscope. Monomolecular layers can give
strong diffraction effects with electrons but thicknesses of a
Principles of Image Formation 3

micrometer or more are needed for comparable relative diffraction inten-

sities for x-rays.

For even very large assemblies of atoms, the amplitude of scattered

x-rays is very much less than the amplitude of the incident beam. This
provides the basis for the common, very useful approximation, called the
"kinematical" or "single-scattering" approximation, justifiable because
if the amplitude of single scattering is very small, the amplitude, and
even more so the intensity, of doubly scattered radiation will be negli-
gible. Then the amplitude of the scattered or diffracted x-rays is
given as a function of the scattering angle by a simple mathematical
operation, the Fourier transform, applied to the electron density distri-
bution in the sample. When this angular distribution is observed very
far from a scattering atom (or group of atoms scattering independently
because there is no significant correlation in their positions), it is
described in terms of an atomic scattering factor, characteristic of the
type of atom and listed conveniently in tables. When diffraction by an
assembly of atoms is considered, one merely adds together the scattered
amplitudes for all atoms with phase factors depending on their relative
positions. For crystals, as is well-known, the regularity of the atom
arrangement leads to a reinforcement of the scattered amplitude in a
regularly spaced set of strong diffracted beams. The process of working
back from the observed intensities of these beams, through the Fourier
transform relationship, to the relative atomic positions is the founda-
tion of x-ray crystal structure analysis.

This kinematical scattering approximation is so elegant and so rela-

tively simple, as compared with multiple-scattering theory, that it is
used for electrons whenever it seems even half-way reasonable (and often
when it does not). But for electrons the scattering by atoms is very
strong. There can be appreciable multiple scattering within a single
moderately heavy atom and for very heavy atoms double and triple scat-
tering can rarely be neglected.

An important factor in this respect is that because electrons have

short wavelengths (0.037 i for 100 keV electrons) relative to atomic di-
mensions the scattering angles are small (around 10 milliradians). When
electrons enter a crystal in the direction of one of its principal axes,
the electrons travel along rows of atoms and electrons scattered by one
atom can be rescattered by any of the subsequent atoms in the row. For
crystals then, multiple scattering is very important. The diffraction
effects are strongly affected so that for electron diffraction by crys-
tals one must usually look beyond the kinematical approximation and deal
with the complications of the dynamical (or coherent multiple scat-
tering) theory.

There are favorable circumstances, involving the averaging over

crystal orientations and thicknesses, especially for light-atom mater-
ials, which can make the kinematical theory a good working approximation
for crystal structure analysis from electron diffraction patterns (see
VAINSHTEIN, 1964). For electron microscopy a similar sort of averaging
can simplify things in the case of very thick specimens with very small
crystal sizes, but for the more usual thin specimens there is no such
convenient means of escape since the intensity of each point in the
4 Chapter 1

image of a crystal depends on the full dynamical diffraction effects in

the immediate neighborhood of the point.

It is only because the scattering of electrons by atoms is very

strong that fine detail, at an atomic level, can be seen in electron
microscopes, but because it is strong, severe complications can arise
for all but the ideal cases of very thin specimens containing only light
atoms.

The Physical Optics Analogy

It was BOERSCH (1946) who first stated clearly the alternative view
that, instead of considering electrons as particles being bounced off
atoms, one could consider an electron wave being transmitted through the
potential field of the charged particles in a sample and the main effect
of the potential field is to change the phase of the electron wave.
Atoms or assemblies of atoms then constitute phase objects for electron
waves, as thin pieces of glass or unstained biological sections do for
light waves. In the usual idealized picture, a plane wave enters the
sample. The wave at its exit surface has a distribution of relative
phase values depending on the variations of the potential field it has
traversed, plus relatively small changes of amplitude corresponding to
the loss of electrons by inelastic scattering processes. The angular
distribution of electron wave intensity (the number density of electrons
detected) at a large distance from the object is then given by squaring
the magnitude of the wave amplitude in the Fraunhofer diffraction pat-
tern, given by a Fourier transform of the exit complex wave amplitude
distribution. This diffraction pattern intensity distribution will be
the same as is given by the single-scattering kinematical theory only if
the phase changes of the wave in the specimen are small. Large phase
changes correspond to the presence of strong multiple scattering.

On this basis we can build up a consistent theory of electron dif-

fraction and of electron microscope image contrast using direct analo-
gies with the concepts of elementary physical optics, including Fraun-
hofer diffraction, Fresnel diffraction, phase contrast imaging and the
wave optical formulation of the theory of lens action, aberrations, im-
age contrast and resolution.

Diffraction Patterns

If a plane parallel beam of radiation strikes a specimen the angu-

lar distribution of emergent radiation, as seen from a distance large
compared with the specimen dimensions, is the Fraunhofer diffraction pat-
tern. It is convenient to consider the intensity distribution as a func-
tion not of the scattering angle $, but of the parameter u = 2A- 1
sin($/2).

For electrons the wavelength A is small, the angles $ are small and
in the diffraction pattern on any plane of observation the distances be-
tween features are closely proportional to differences of u values [Fig.
l.l(a)]. In the two dimensions of the plane of observation, the
Principles of Image Formation 5

coordinates x,y are proportional to parameters u,v derived from the com-
ponents of the diffraction angles ~x' ~y'

For single atoms, or for random arrays of many atoms, the diffrac-
tion pattern intensities are proportional to the square of the atomic
scattering factor, f(u), and fall off smoothly with scattering angle
[Fig. l.l(b)]. Any systematic correlation between atom positions is re-
flected in a modulation of scattered electron intensity distribution.
For most biological and nonbiological materials considered to be "amor-
phous," the only correlation of atom positions is that due to inter-
atomic bonds of closely prescribed lengths. This gives a modulation of
the diffraction intensities with a periodicity roughly proportional to
the reciprocal of the bond lengths, but since there is no preferred di-
rection for the bonds, these modulations are smeared out into diffuse
circular halos and the percentage modulation of the intensities is usual-
ly quite low [Fig. l.l(c)].

The extreme case of correlated atom positions is the strictly per-

iodic arrangement of atoms in a perfect crystal. To the periodicity, a,
of the repetition in the crystal there corresponds the set of regularly
spaced sharp diffraction spots with separations proportional to lla
[Fig. l.l(d)]. For a very thin crystal lying nearly perpendicular to
the incident beam and having periodicities a and b in two perpendicular
directions, the diffraction pattern will be-a regular cross-grating of
sharp spots with separations proportional to lla and lib in the two di-
rections [see Fig. 1.lO(b)].

The perfectly periodic two dimensional case is an idealization

which is seldom realistic. If the lateral dimensions of the crystal are
small, the diffraction spots will be smeared out by an amount inversely
proportional to the crystal dimensions. If the crystal is distorted or
bent, the spots will be spread into arcs or the intensities will be
changed. If there are crystal defects, if the atoms are disordered on
the lattice sites or if the atoms have thermal vibrations, there will be
diffuse scattering in the background between the spots. Usually, of
course, the crystal is three-dimensional and the dimension in the beam
direction can have a strong effect on the presence or absence of diffrac-
tion spots, can make the diffraction pattern intensities highly sensi-
tive to small crystal tilts and can give rise to a multitude of compli-
cated dynamical diffraction effects.

Fig. 1.1. (a) The formation of a (a)

Fraunhofer diffraction pattern and
the intensity distribution along
one diameter for (b) a single atom
or a random array of atoms; (c) an
amorphous material, and (d) a sin-
gle crystal.
6 Chapter 1

Mathematical Formulation

The kinematical approximation is equivalent to the approximation of

excluding all but the first term in the Born series. Starting from the
integral formulation of the Schrodinger equation for a plane incident
wave of wave vector !o the wave function for the scattered wave is

~(r)
-
= ~o(r) ~rexp{-ik(r-r')}
4nr (r-r') CI>(r' )~(r' )dr' (1.1)

If we assume that in the integral we can replace ~(r') by the incident

plane wave exp{-i!oo!'} and consider the solution at-a large distance r
= R, this can be written

~(r) = exp{-ik R} -0-

+ (~/R)exp{-ik R}og(u)
-0-
(1.2)

where g(u) = ICI>(r)exp{-i(!-!o)o!}dr

= ICI>(r)exp{2ni~o!}dr (1.3)

and I~I = 2X- 1 sinCl>/2

i.e. the scattering amplitude g(u) is given by the Fourier transform of
the specimen potential distribution. This applies for any distribution
of scattering matter, as well as for the case of a single atom when the
scattered amplitude is the atomic scattering amplitude feu).

For any assembly of atoms at positions -1.

r., one can write

CI>(r) (1.4)

where the * sign denotes a convolution, so that

or.,
~f
g(u) = L . • e 2niu
--1. (1.5)
1. 1.

and the intensity distribution is proportional to

I(u)
-
= Ig(u)1 2 = I.I.f.f.*
J J1. 1.
exp{2niu(r.-r.)}.
- -1. -J
(1.6)

For the special case of a periodic object

(1. 7)

where h represents the vector to a reciprocal lattice point denoted by

the set of indices h,k,l, and the diffraction pattern amplitudes are
given by the intersection of the Ewald sphere (u = 2n(k-k )) with the
distribution - - -0

(1.8)

which is the set of weighted reciprocal lattice points.

Principles of Image Formation 7

In the physical optics formulation, for an incident wave ~ (xy) and

an object transmission function q(xy), the wave at the exit fage of the
specimen ~e(XY) is ~o(XY)'q(xy) and the Fraunhofer diffraction pattern
is given by the two-dimensional Fourier transform

'(uv) =II~e (xy)exp{2ni(ux+vy)}dx dy (1.9)

For a thin obj ect, the transmission function represents the change of
phase of the electron wave on traversing the potential field of the spec-
imen;

q(xy) = exp{-iat(xy)}, (1.10)

where t(xy) =
I,(r)dz and a =
n/XE where E is the accelerating voltage.
The assumption that at(xy)«l then gives the equivalent of the single-
scattering approximation.

The wave at any finite distance R from the specimen may be calcu-
lated by using the Fresnel diffraction formula

~(xy) = ~e(XY) * p(xy) , (1.11)

where p(xy) is the propagation function, given in the usual small angle
approximation by

p(xy) ~ (i/RA)exp{-ik(x 2+y2)/2R}. (1.12)

For thicker specimens, when Eqn. 1.10 does not apply, one can cal-
culate diffraction effects by dividing the specimen into thin slices and
applying alternately the phase change for each slice, Eqn. 1.10, and the
propagation, Eqn. 1.12, to the next slice. This is the basis for the
dynamical diffraction formulation of Cowley and Moodie (see COWLEY,
1975). Alternatively one can go back to the Schrodinger equation in the
differential form and find solutions for the wave in a crystal subject
to suitable boundary conditions as in Bethe's original dynamical theory
(see HIRSCH et aI., 1965).

1.2 THE ABBE THEORY OF IMAGING

One of the most important properties of a lens is that it forms a

Fraunhofer diffraction pattern of an object at a finite distance. In
Fig. 1.2 we suggest (using for convenience a ray diagram to indicate a
wave-optical process) that an ideal lens brings parallel radiation to a
point focus at the back-focal plane. If a specimen is placed close to
the lens, radiation scattered through the same angle t by different
parts of the specimen will be brought to a focus at another point in the
back-focal plane, separated by a distance proportional to , (in the
small-angle approximation). The intensity distribution on the back-
focal plane is thus the Fraunhofer diffraction pattern, suitably scaled.

The more usual function of a lens is to form the image of a speci-

men. Then radiation scattered from each point of the object is brought
together at one point of the image, as in Fig. 1.3. At the same time,
8 Chapter 1

however, the Fraunhofer diffraction pattern is formed at the back-focal

plane, as in Fig. 1.2. The imaging process may then be described as the
formation of a diffraction pattern of the object in the back-focal plane
(a Fourier transform operation) plus the recombination of the diffracted
beams to form the image (a second Fourier transform operation).

_f---,\o __--R---~

Fig. 1.2. The use of a lens to pro- Fig. 1.3. A ray diagram used to
duce a focussed Fraunhofer diffrac- illustrate the wave-optical Abbe
tion pattern in the back-focal plane. theory of imaging.

For a perfect, ideal lens, the reconstruction of the object trans-

mitted wave in the image plane would be exact as all radiation leaving
the object is brought back with exactly the right phase relationships to
form the image. For a real lens some of the diffracted radiation is
stopped when it falls outside the lens aperture. The phase relation-
ships are upset by lens aberrations. These changes can be considered to
take place on the back-focal plane where the wave function is multiplied
by a "transfer function" which changes its amplitude and phase. Corres-
pondingly, on the image plane the complex amplitude distribution of the
wave function can be considered as smeared out or convoluted by a
smearing function which limits the resolution and affects the contrast.

It would be convenient if we could make an equivalent statement for

image intensities, i.e. that the effect of lens apertures and aberra-
tions is to smear out the intensity distribution by means of a smearing
function which can be related (by a Fourier transform operation) to a
Contrast Transfer Function characterizing the lens. This is the usual
practice for light optics but it does not apply in general for electron
microscopy. The difference is that the usual imaging with light is in-
coherent; each point of the object emits or scatters light independently
with no phase relationship to the light from neighboring points. In
electron microscopy the conditions usually approach those for coherent
imaging, with a definite phase relationship between the waves trans-
mitted through neighboring parts of the specimen. It is only under very
special circumstances (which fortunately occur quite often) that the
relatively simple ideas developed for light optics can be used as reason-
able approximations for the electron case.
Principles ofImage Formation 9

For the coherent, electron optical case, the transfer function

which modifies the amplitudes of the diffraction pattern in the back-
focal plane can be written

T(uv) = A(uv)exp{ix(uv)} (1.13)

where the aperture function A(uv) is zero outside the aperture and unity
within it. The phase factor X(uv) is usually written as including only
the effects of defocus af and the spherical aberration constant Cs since
it is assumed that astigmatism has been corrected and other aberrations
have negligible effect. Then

(1.14)

The corresponding smearing functions, given by Fourier transform of Eqn.

1.13, are in general complicated and have complicated effects on the im-
age intensities except in the special cases we will discuss later.

Incident Beam Convergence

So far we have considered only the ideal case that the specimen is
illuminated by a plane parallel electron beam. In practice the incident
beam has a small convergence and this may have important effects on the
diffraction pattern and on high resolution images.

Two extreme cases can be considered. First, it may be appropriate

to assume that the electrons come from a finite incoherent source, with
each point of the source emitting electrons independently [see Fig.
1.4(a)]. This is a useful assumption for the usual hot-filament elec-
tron gun.

Then for each point in the source the center point of the diffrac-
tion pattern, and the whole diffraction pattern intensity distribution,
will be shifted laterally (and also modified in the case of relatively
thick specimens). The main effect on the image will result from the dif-
ferent effects of the transfer function, Eqn. 1.13, when applied to the
differently shifted diffraction patterns. In the extreme case that the
incident beam convergence angle is much greater than the objective aper-
ture angle, we approach the incoherent imaging situation common in light
optics, and all interference effects, including the production of out-
of-focus phase contrast, disappear.

The other extreme case of convergent illumination is that occurring

when the incidence convergent wave is coherent, e.g. when it is formed
by focussing a point source on the specimen [see Fig. 1.4(b)]. This
situation is approached when a field emission gun is used. Then it is
the amplitude distribution in the diffraction pattern which is smeared
out and there will be complicated additional interference effects within
the pattern (SPENCE and COWLEY, 1978). Provided that the convergence
angle is not too great, the effects on the image will be small except in
the case of images taken far out of focus.
10 Chapter 1

(a)

(b)

Fig. 1.4. Illumination of the specimen in a microscope with a conver-

gent beam for the cases of (a) a finite incoherent source and (b) a
point source and a lens used to give coherent radiation with the same
convergence.

Chromatic Aberration

The focal lengths of electron lenses depend on the electron ener-

gies and also on the currents in the lens windings. A variation in the
accelerating voltage (M), or a variation in the lens current (~I),
during the recording time will have the effect of smearing the intensity
distribution in the image. The intensities for different electron ener-
gies are added incoherently. The loss of resolution is given by

~ = Cca(6E
E
+ 2~I)
I
(1.15)

where Cc is the chromatic aberration constant and a is the objective

aperture angle.

In addition a fhange of electron energy gives a change of electron

wavelength A ex: E- 1 2 and this in turn affects the phase factor, Eqn.
1.14.

There are important chromatic aberration effects associated with

losses of energy of electrons which suffer inelastic scattering. These
will be considered in the subsequent sections.
Principles of Image Formation 11

Mathematical Fonnulation

For an incident wave amplitude tlso(XY) and an object transmission

function q(xy) the diffraction pattern amplitude on the back-focal plane
of an ideal lens is
, (uv)
e
= Q(uv) * , 0 (uv) (1.16)

where capital letters are used for the Fourier transforms of the corres-
ponding real space functions. With the transfer function

T(uv) = A(uv) exp{ix(uv)} (1.17)

the image amplitude is given by a second Fourier transform as

tIs(xy) = tise (xy) * t(xy) (1.18)

where the spread function t(xy) is given by

t(xy) = c(xy) + is(xy)

=FA(uv)cos(X(uv)) + i FA(uv)sin(x(uv)) (1.19)

and the magnification factor (-RjR o) has been ignored. For an isolated
point source the intensity of the image is It(xy)1 2 which is the spread
function for incoherent imaging. The corresponding contrast transfer
functi0IJ..\. for incoherent imaging is then given by Fourier transform as
T(uv)* T (-u,v).

For a plane incident wave from a direction (Ul,vl) the diffraction

plane amplitude, Eqn. 1.16, becomes
(1.20)

and the image, Eqn. 1.18, becomes

For a coherent source the image intensity from a finite source is thus

and for an incoherent source it is

(1.21)

Here S(Ulvl) is the source function.

The effect of beam convergence can thus be expressed as in Eqn.

1.20 by a convolution in the diffraction plane in the case of a thin ob-
ject for which the effect on the incident wave can be represented by a
two-dimensional transmission function. For thicker objects the formula-
tion is much more complicated.
12 Chapter 1

The effect of chromatic aberration cannot be represented by a convo-

lution in the diffraction plane. It is given by summing the image inten-
sity distributions for all wavelengths and focal length variations.

1.3 INELASTIC SCATIERING

Other chapters in this volume (e.g. Chapter 10 by Silcox) will deal

with the inelastic scattering processes in detail. Here we briefly sum-
marize the effects on the image formation.

Electrons may be scattered by phonons, i.e. by the waves of thermal

vibration of atoms in the specimen. The energy losses suffered by the
incident electrons are so small (about 10- 2 eV) that no appreciable chro-
matic aberration effect is introduced. For an ideal lens of infinite
aperture, all thermally scattered electrons would be imaged along with
the elastically scattered electrons and the effect on the image would be
negligible. For a lens with a finite aperture the effect of thermal
scattering is not significant for amorphous specimens. For crystalline
specimens, however, the elastic scattering is often concentrated into a
few sharp diffraction spots while the thermal scattering is diffusely
spread over the background. The objective lens aperture is used to se-
lect the incident beam spot or one of the diffracted beam spots. This
may cut off much of the thermal diffuse scattering and the loss of these
electrons can be attributed to an absorption function, included in the
calculations of image contrast by adding an out-of-phase term or by
adding a small imaginary part to make the effective crystal potential
into a complex function.

The other important means by which incident electrons can loose

energy is by excitation of the bound or nearly-free electrons of the
sample into higher energy states. This may involve collective excita-
tions, as when a plasmon oscillation is generated, or else single-
electron excitations. The excitations most relevant for imaging involve
energy losses ranging from a few eV to about 30 eV and scattering angles
which are much smaller than the usual elastic scattering angles.

For light-atom materials the number of electrons inelastically scat-

tered in this way may exceed the number elastically scattered.

The electrons which have lost energy will be imaged by the micro-
scope lenses in exactly the same way as the elastically scattered elec-
trons except that, because of the change of energy, the image will be
defocussed. The consequent smearing of the image will usually be about
10 R for thin specimens and so will not be important for low resolution
or medium resolution electron microscopy. For very high resolution im-
aging the inelastically scattered electrons will produce a slowly-varying
background to the image detail (on a scale of 5 R or less) produced by
elastically scattered electrons.

Even if the electrons inelastically scattered from a thin specimen

are separated out by use of an energy filter and brought to the best fo-
cus, the image which they produce does not have good resolution. This
follows because the inelastic scattering process is not localized.
Principles ofImage Formation 13

The interaction with the incident electrons is through long-range

Coulomb forces so that the position of the scattering event cannot be
determined precisely and the image resolution cannot be better than
about 10 R.

For thicker specimens the inelastically scattered electrons can be

scattered again, elastically and give a good high resolution image if
properly focussed. The situation is almost the same as for an incident
beam, of slightly reduced energy and slightly greater beam divergence,
undergoing only elastic scattering.

1.4 STEM and CTEM

It is easy to yield to the temptation of thinking about scanning

transmission electron microscopy (STEM) as an incoherent imaging process
since one can visualize in terms of geometric optics and simple scat-
tering theory how a narrow beam is scanned across the specimen and some
fraction of the transmitted or scattered intensity is detected to form
the image signal. This viewpoint can lead to serious errors. The
imaging process depends just as much on coherent, interference phenomena
in STEM as in conventional transmission electron microscopy (CTEM) given
equivalent geometries. For some of the special geometries used for STEM
in practice it is possible to approach incoherent imaging conditions but
it is unwise to assume this to be the case without a thorough analysis.

The relationship between STEM and CTEM imaging is demonstrated read-

ily by use of the reciprocity principle (COWLEY, 1969). Since this prin-
ciple is often misapplied, leading to statements that "reciprocity
fails," it is necessary to state it carefully, as follows. The wave amp-
litude at a point B due to a point source at A in a system is the same
as the amplitude at A due to a point source at B. This holds provided
that transmission through the system involves only scalar fields (no mag-
netic fields for electrons) and only elastic scattering processes. It
holds to a very good approximation for electrons in electron microscope
instruments if the image rotations and associated distortions due to the
magnetic fields of the lenses can be ignored and if inelastic scattering
effects are negligible. It holds in relation to intensities, not ampli-
tudes, in the presence of inelastic scattering if the change of energy
of the electrons is not detectable under the experimental conditions
(DOYLE and TURNER, 1968). Within these restrictions we see that the im-
age intensity must be the same for the equivalent points in Figs. 1.S(a)
and (b) representing idealized STEM and CTEM systems. We have drawn
lines to indicate beam geometries in these figures only as a matter of
convenience: the wave-optical reciprocity theorem does not imply the geo-
metric-optics notion of the reversibility of ray paths.

The reciprocal relationship is extended to finite sources of detec-

tors by the assumption that a detector adds incoherently the intensities
at all points of the detector aperture and the finite source is ideally
incoherent so that all points of it emit independently to give inten-
sities which are added incoherently at all points of the detector. It
is clear that STEM with a finite detector aperture as in Fig. I.S(a) is
equivalent to CTEM with a finite incoherent source as in Fig. 1.4(a) but
is not equivalent to CTEM with a coherent source as in Fig. 1.4(b).
14 Chapter 1

STEM

BC:==-iK: A CYEM

Fig. 1. 5. The reciprocal relationship between the essential elements

for STEM and CTEM instruments.

As a consequence of the reciprocity relationship it can be confi-

dently predicted that for identical electron optical components and equi-
valent sources and detectors, the contrast of STEM images will be the
same as for CTEM. In fact the whole range of phase-contrast and
amplitude-contrast effects, Fraunhofer and Fresnel diffraction phenomena
and bright-field and dark-field imaging behavior, familiar in CTEM, have
been reproduced in STEM.

The main differences in practice between the two forms of micro-

scopy arise from the differences in the techniques used for the detec-
tion and recording of image intensities. In CTEM a static two-
dimensional image is formed and the intensity distribution is recorded
by the simultaneous integration of the dose at all points in a two-
dimensional detector, usually a photographic emulsion. The STEM image
information is detected for one image point at a time. The detector,
usually a phosphor-photomultiplier combination, provides an electrical
image signal in serial form for display on a cathode ray tube or for re-
cording in any analog or digital form. A very important practical limi-
tation for STEM is that, in order to get the good signal-to-noise ratio
needed for high quality imaging, the number of electrons scattered from
each picture element within a short period must be large (10 4 or more).
For high resolution imaging this number of electrons must be concen-
trated within a very small probe size. The requirement is therefore for
a very high intensity electron source, such as can only be approached by
use of a field-emission gun.

STEM also suffers from the practical inconvenience that the picture
size is relatively small. The usual field of view of 1000 x 1000 picture
elements or less is much smaller than is provided by a photographic
plate.
Principles of Image Formation 15

On the other hand the fact that the STEM image appears as a serial,
electrical signal allows enormous flexibility for on-line image evalua-
tion and image processing and for recording of the image in an analog or
digital form for subsequent processing.

Other practical aspects of STEM will be treated in detail in Chap-

ter 11 by Humphreys in this volume.

STEM Imaging Modes

Where applicable, it is often convenient to discuss STEM image con-

trast in terms of the equivalent CTEM imaging configuration. For many
of the more important STEM modes, however, there is no convenient CTEM
analogue. Then it is preferable to discuss image formation and calculate
intensities on the basis of the STEM optics without reference to CTEM.

Figure 1.6 illustrates the STEM system. The beam incident on the
specimen has a convergence defined by the objective aperture. For every
position of the incident beam a diffraction pattern is produced on the
detector plane. This is a convergent beam diffraction (CBED) pattern in
which the central beam and each crystal diffraction spot will form a cir-
cular disc, which will be of uniform intensity for a very thin crystal
but may contain complicated intensity modulations for thicker crystals
(see Fig. 1.7).

The intensity distribution in the CBED pattern contains a wealth of

information concerning the atomic arrangements within the small regions
illuminated by the beam. The interpretation and use of these "microdif-
fraction" patterns obtained when the incident beam is held stationary on
the specimen is the subj ect of Chapter 14 by Warren (see also COWLEY,
1978a). For the coherent convergent incident beam produced with a field
emission gun, CBED intensities may be strongly modified by interference
effects (see SPENCE and COWLEY, 1978).

If the detector aperture is small and is placed in the middle of

the central spot of the CBED pattern (Fig. 1.6) the imaging conditions
will be those for the usual CTEM bright-field imaging mode. As the de-
tector diameter is increased the signal strength is increased. The
bright-field phase-contrast term for a thin specimen passes through a
maximum value and then decreases to zero as the collector aperture size
approaches the objective aperture size. For strongly scattering, thin
objects the second order bright-field "amplitude contrast" term in-
creases continually with collector aperture size. The collector aper-
ture size and defocus giving the best contrast and resolution can be
determined for any type of specimen by use of appropriate calculations
(see COWLEY and AU, 1978).

The most common STEM mode is the dark-field mode introduced by

Crewe and colleagues (see CREWE and WALL, 1970) in which an annular de-
tector collects nearly all the electrons scattered outside of the cen-
tral beam. This mode of dark-field imaging is considerably more effi-
cient that the usual dark-field CTEM modes (i.e. there is a larger image
signal for a given incident beam intensity) and so is valuable for the
16 Chapter 1

study of radiation sensitive specimens. Images may be interpreted under

many circumstances as being given by incoherent imaging with an image
intensity proportional to the amount of scattering matter present. This
convenient approximation breaks down for crystalline specimens when the
amount of scattering depends stongly on crystal structure, orientation
and thickness. Also it fails near the high resolution limits when inter-
ference effects can modify the amount of scattered radi.ation which is
undetected because it falls within the central beam (COWLEY, 1976).

The annular-detector dark-field mode has been used with great ef-
fect for the detection of individual heavy atoms and their movements
(ISAACSON et al., 1976) and also for the quantitative measurement of the
mass of biological macromolecules (LAMVIK and LANGMORE, 1977).

SOURCE

(a)

SPEC.

CBEDP

(b)

Fig. 1.6. The formation of a con- Fig. 1.7. Convergent beam elec-
vergent beam diffraction pattern tron diffraction patterns from a
in the detector plane of a STEM in- small region (less than 50~ dia-
strument showing the position of meter) of a MgO crystal for ob-
the bright-field detector aperture. jective aperture sizes (a) 20~m
and (b) 100~m.
Principles of Image Formation 17

When the annular detector is used to collect all scattered radia-

tion, however, the information contained with the diffraction pattern is
lost. In principle, the use of a two-dimensional detector array to
record the complete CBED pattern for each image point could add consider-
ably to the information obtained concerning the specimen. Structural
details smaller than the resolution limit could be deduced or recognized
by means of structure analyses or pattern recognition techniques (COWLEY
and JAP, 1976). More immediate and practical means for using some of
this CBED information include the use of special detectors and masks to
separate out particular features of the pattern (diffraction spots, dif-
fuse background, Kikuchi lines) for form image signals. Equipment de-
signed to allow the exploitation of these possibilities is now in its
initial stages of operation (COWLEY and AU, 1978).

A further means for dark-field imaging in STEM is provided by the

convenient addition of an electron energy filter to separate the elas-
tically and inelastically scattered electrons or to pick those electrons
from any part of the CBED pattern which have lost any specified amount
of energy. The resolution of such images may be restricted by the
limited localization of the inelastic scattering process but the method
has important possibilities in providing means for locating particular
types of atom or particular types of interatomic bonds within specimens
(see Chapters 7, 8, and 9 on ELS). The versatility of the STEM detection
system allows the inelastic dark-field image signals to be combined with
the signals from other dark-field or bright-field detectors in order to
emphasize particular features of the specimen structure.

Mathematical Description

For a point source, the wave incident of the STEM specimen is given
by Fourier transform of the objective lens transfer function.

~
o (r) = F[A(u)exp{ix(u)}]
- - = c(r)+is(r)
--
0.22)

where rand u are two-dimensional vectors. For a thin specimen with

transmission function q(r-R), where R is the translation of the specimen
relative to the incident-beam (or vice versa), the wave amplitude on the
detector plane is

(1. 23)

and the intensity distribution is IR (~) = 1 IjIR (~) 1 2. The image signal
corresponding to the beam position ~ is then

where D(~) is the function representing the detector aperture.

In the limiting case of a very small axial detector, D(~) is re-

placed by o(u) and Eqn. 1.24 becomes

J(~) = Iq(~) * t(~) 12 (1.25)

18 Chapter 1

which is equivalent to Eqn. 1.lS which applies for plane-wave illumina-

tion in CTEM.

1.5 THIN , WEAKLY SCATIERING SPECIMENS

For a very thin object we may assume (neglecting inelastic scat-

tering) that the only effect on an incident plane wave is to change its
phase by an amount proportional to the projection of the potential dis-
tribution in the beam direction. Writing this in mathematical short-
hand, if the potential distribution in the specimen is ~(xyz), the phase
change of the electron wave is porportional to

~(xy) = f~(xyz)dz. (1.26)

If the value of the projected potential ~(xy) is sufficiently small it

can readily be shown that the image intensity can be written as

I(xy) =1 - 2cr~(xy) * sexy), (1.27)

where cr is the interaction constant. The * sign represents a convolu-

tion or smearing operation.

This means that image contrast, given by the deviation of the in-
tensity from unity, is described directly as the projected potential
smeared out by the spread function or smearing function sexy) which
determine the resolution. Provided that the spread function is a clean,
sharp peak, the image will show a well-shaped circular spot for each
maximum in the proj ected potential, i. e. for each atom or group of
atoms. The smearing function depends on the defocus of the objective
lens and on its aberrations. For the Scherzer optimum defocus (SCHERZER,
1949), which depends on the electron energy and the spherical aberration
constant, the smearing function is a sharp, narrow negative peak [Fig.
1.S(b)] so that the image shows a small dark spot for each atom and this
spot is the clearest, sharpest peak attainable with a given objective
lens. For 100 keV electrons and Cs =
2 mm, for example, the optimum de-
focus is about 950 g underfocus and the corresponding width of the
spread function limits the resolution to about 3.5 g.

The imaging conditions are usually discussed in terms of the modi-

fication of the wave amplitude on the back-focal plane of the objective
lens. The equivalent of Eqn. 1.27 is that the diffraction pattern ampli-
tude is multiplied by A(uv)sin(x(uv)) where A(uv) is the aperture func-
tion and X(uv) is the phase factor given in Eqn. 1.14. In order to give
a good representation of the object in the image, the function sinx
should be as close to unity over as large a region of the diffraction
pattern as possible so that the diffracted beams will have the correct
relative amplitudes when they recombine to form the image. For the op-
timum defocus it is seen in Fig. 1.S(a) that the sinx function is small
for low-angle scattering but is near unity for a wide range of scat-
tering angles (corresponding to spacings in the object in the range of
roughly 20 to 3.5 g) before it goes into wild oscillations at high
angles. This suggests that in the image slow variations of potential
Principles ofImage Formation 19

.1f ; -1()()Q\

0.1

5. --- 10.A

(b)
·0.2

Fig. 1.8. The real and imaginary parts of the transfer function (cosx
(u) and sinx(u)) and the spread functions c(r) and s(r) for defocus {~)
-lOooR and (b) -400R. 100keV, C = 2mm, aperture radius u = o. 265X 1
S
(arrows).

will not be seen, detail in the range of 3.5 to about 20 R will be well-
represented and finer detail will be represented in a very confused
manner.

This mode of imaging is thus of very little use for biologists who
wish to see detail on the scale of 10 R
or greater with good contrast.
Their needs are discussed in the following section.

It is the common practice to insert an objective aperture to elimin-

ate all the high angle region past the first broad maximum of the sinx
function so that all scattered waves contributing to the image will do
so in the correct phase. Then the image will be interpretable in. the
simple intuitive manner in that, to a good approximation, it can be as-
sumed that dark regions of the picture correspond to concentrations of
atoms, with a near-linear relationship between image intensity and the
value of the smeared projected potential.

The restrictions to very thin weakly scattering objects [a$(xy)«l]

is often not so severe as might be assumed from initial calculations,
especially for light-atom materials. It is always possible to subtract a
constant or slowly-varying contribution to the projected potential,
since a constant phase change has no effect on the imaging (except to
make a very small change in the amount of defocus).
20 Chapter 1

For an amorphous specimen (for example a layer of evaporated car-

bon, 100 i thick) the phase change due to the average potential, '0, may
be 1 or 2 radians but since there are many atoms in this thickness, over-

'0
lapping at random, the projected potential may not differ greatly from
and the relative phase changes may be only a fraction of a radian
[Fig. 1.9(a)] . . Then the weak phase object approximation, Eqn. 1.27,.
holds quite well and this simple approximation is more likely to fail
because of the neglect of three-dimensional scattering effects than be-
cause of large phase changes.

On the other hand a single very heavy atom [Fig. 1.9(b)] may have a
maximum of projected potential no greater than for the amorphous light-
atom film [Fig. 1.9(a)] but the deviation from the average projected po-
tential is much greater and the error in the use of the weak phase ob-
ject approximation may be large. Similarly large deviations from the
average occur in the projections of the potential for crystals when the
incident beam is parallel to the rows or planes of atoms. If the same
atoms as in the amorphous film [Fig. 1.9(a)] were rebuilt into a crys-
tal, the projected potential could be as in Fig. 1.9(c) and the weak-
scattering approximation could fail badly.

(a) (b)
Fig. 1. 9. Illustrating the sub-
traction of the average potential
value '0from the projected poten-
tial distribution for the cases of
(a) a thin amorphous film of light
atoms, (b) a single heavy atom and
(c) a thin crystal. (c)

Beam Convergence and Chromatic Aberration

For the case of weakly scattering objects the consideration of

these complicating factors is greatly simplified. It has been shown by
FRANK (1973) and by ANSTIS and O'KEEFE (1976) for the particular case of
crystals (including cases when the weak scattering approximation fails)
that the effects of beam divergence and chromatic aberration can be in-
cluded by applying an "envelope function" which multiplies the transfer
function and reduces the contribution of the outer, high angle, parts of
the diffraction pattern. For most high resolution imaging it is one or
other of these envelope functions, rather than the ideal transfer func-
tion which effectively limits the resolution.

For current 100 keV electron microscopes used with the incident
beam focussed on the specimen to give the high intensity needed for high-
magnification imaging, the incident beam convergence may limit the reso-
lution to 4 i or more. For the 1 MeV microscopes which have recently
Principles oflmage Formation 21

been used for high resolution imaging with apparent point-to-point reso-
lution approaching 2 i (HORIUCHI et al., 1976), the limiting factor
seems to be the chromatic aberration effect due to lens current and high
voltage instabilities.

The results of calculations by O'Keefe, illustrating the forms for

these envelope functions and the consequent limitations on resolution,
are shown in Fig. 1.11.

The use of an envelope function to take account of convergence is

simple and convenient but is strictly applicable only for small angles
of illumination in CTEM or small collector angles in STEM. Calculations
for STEM (COWLEY and AU, 1978) have shown that if the objective aperture
size is correctly chosen, an increase in collector aperture angle can
give a slight improvement in resolution as the signal strength increases
to a maximum. Then for relatively large collector angles, approaching
the objective aperture angle, there may be conditions of reversed con-
trast with an appreciably better resolution. The influence of CTEM illu-
mination angle or of STEM collector size is, however, a rather compli-
cated function of the defocus, the objective aperture size, the spheri-
cal aberration constrant and the strength of the scattering by the speci-
men.

With care the resolution limitations due to beam convergence, chro-

matic aberration and such other factors as mechanic'\l vibrations or
stray electrical or magnetic fields may be reduced. It is pOSSible, for
example, to achieve a very small beam convergence with adequate inten-
sity for high magnification images by use of a very bright source such
as a field emission gun.

Then the resolution limitations represented by the envelope func-

tions will be less important. If a sufficiently large or no objective
aperture is used, diffracted beams from the outer parts of the diffrac-
tion pattern, where the transfer function is oscillating rapidly, will
contribute to the image. This will produce detail in the image on a
very fine scale, but since the diffracted beams are recombined with wide-
ly different phase changes, this fine detail will not be directly inter-
pretable in terms of specimen structure. Obviously some information
about the specimen structure is contained in this detail but in order to
extract the information it will be necessary to resort to indirect image
processing methods or else to seek agreements between observed image in-
tensities and intensities calculated on the basis of postulated models.

Most of the image processing techniques which have been proposed

depend on the idea that the difficulties due to the ambiguities of phase
and loss of information around the zero points of the transfer function
can be overcome by the use of two or more images obtained with different
amounts of defocus (see SAXTON, 1978). While some progress has been
made along these lines, few clear indications of improved resolution
have been obtained. The alternative of calculating images from models
of the specimen structure is realistic only for crystals.
22 Chapter 1

Mathematical Fonnulation

For a weak phase object, we assume a~(xy)«l and Eqn. 1.10 becomes

q(xy) = exp{-ia~(xy)} = l-ia~(xy)

For parallel incident radiation, the wave function in the back-focal
plane of a lens modified by the transfer function is

~(uv) = [6(uv) - ia~(uv)]'A(uv)exp{iX(uv)} (1. 28)

and on the image plane the wave function is

l/l(xy) = [l-ia~(xy)] * [c(xy) + i sexy)] (1.29)

where c(xy) and sexy) are the Fourier transforms of A(uv)cosX(uv) and
A(uv)sinX(uv). Neglecting terms of second order in a~, the image in-
tensity becomes

I(xy) =1 + 2a~(xy) * sexy) (1. 30)

so that we may consider the image to be produced by a system with con-

trast transfer function Asinx.

For dark-field imaging in CTEM with a central beam stop which re-
moves only the forward scattered beam, the 6 function and ~(O,O), from
Eqn. 1.28, we are left with only second order terms and obtain

IDF(xy) = {a~'(xy) * c(xy)}2 + {a~'(xy) * s(xy)}2 (1.31)

where ~'(xy) = ~(xy) - ~ and the average potential ~ is equal to ~(O,O).

For the case of STEM with a small axial detector, Eqn. 1.25 becomes

J(~) = I (l-ia~(~) * {c(~) + i s(~)}12

= 1+2a~(~) * s(~) (1. 32)

which is identical with Eqn. 1.30.

If the detector angle is not very much smaller than the objective
aperture angle, the bright-field image is given by replacing the spread
function s(~) by tI(~) where

(1.33)

where d(R) is the Fourier transform of the collector aperture function

D(u) of Eqn. 1.24. The form of tI(R) and its dependence on defocus arid
collector aperture size has been exPlored by COWLEY and AU (1978). For
the CTEM case, Eqn. 1.21 reduces to the same form, representing the ef-
fect of beam convergence from a finite source. For a small collector
angle in STEM or a small finite convergence angle in CTEM, the function
d(~) will be a broad, slowly varying peak and d(~)*s(~) = 0 since fs(~)
Principles ofImage Formation 23

dR = o. Then

or the effective transfer function is

A(~) sin(x(u)) * [D(u)·A cos(X(u))] (1.34)

Convolution with this narrow function will have the effect of smearing
out the transfer function and reducing the oscillations by a greater
amount as the period of the oscillations becomes smaller, i.e. the ef-
fect will be to multiply transfer functions such as those of Fig. 1.8(a)
by a rapidly decreasing envelope function.

If it is assumed that a dark-field STEM image is obtained by use of

an annular detector which collects all the scattered radiation (neglec-
ting the loss of scattered radiation which falls within the central beam
disc) the expression of Eqn. 1.23 with Q(uv) =
i~(uv) gives

(1. 35)

which is equivalent to the incoherent imaging of a self-luminous object

having an intensity distribution a2~2(R). The result of Eqn. 1.35 is,
of course, different from Eqn. 1.31. The two dark-field imaging modes
are not equivalent.

The approximation of incoherent imaging fails significantly for im-

aging of detail of dimensions near the resolution limit, i.e. when the
oscillations of ~(u) are of size comparable to that of the hole in the
annular detector. -

1.6 TIllN, STRONGLY SCATTERING SPECIMENS

As suggested above, the simplifying approximation of Eqn. 1.27 can

fail even for a single heavy atom. For thin crystals, and for amorphous
materials containing other than light atoms, we must use a better approxi-
mation. For sufficiently thin specimens and for sufficiently high vol-
tages (e.g. thicknesses of less than 50 to 100 R for 100 keV or 200 to
500 R for 1 MeV, depending on the resolution being considered and the
required accuracy) we can use the approximation that the transmission
function of the specimen involves a phase change proportional to the pro-
jected potential. The phase changes, even referred to an average due to
the average projected potential, may be several radians. We must then
consider not only the first-order term in a~(xy) as in Eqn. 1.27 but
also higher-order terms. For the case of parallel incident radiation,
Eqn. 1.27 is replaced by

I(xy) = 1 + 2sina~(xy) * sexy) - 2(1-cosa~(xy)) * c(xy) (1.36)

or taking only second-order terms into account

I(xy) = 1+2a~(xy) * sexy) - a2~2(xy) * c(xy) (1. 37)

24 Chapter 1

This means that there are two components to the image intensity. The
first, as before, is given by the projected potential smeared out by the
smearing function sexy), which is equivalent to multiplying the diffrac-
tion pattern amplitudes by A(uv)sin(x(uv)). The new term is given by
the square of the projected potential (or, more accurately, by the
square of the positive or negative deviation from the average projected
potential) smeared out by the smearing function c(xy). Use of this
smearing function is equivalent to a multiplication of a component of
the diffraction pattern by the function A(uv)cos(X(uv)) illustrated in
Fig. 1.8(a).

It is seen from Fig. 1.8(b) that for the optimum defocus, while the
function s(r) has a sharp negative peak, c(r) is rather broad and fea-
tureless. This means that for high resolution detail the sina~ term in
Eqn. 1.36 will give good contrast but the cosa~ term will be smeared out
into the background.

The situation for medium or low resolution imaging is best seen

from the curves of Fig. 1.8(a) or (b). The part of the diffraction pat-
tern corresponding to sina~ is multiplied by sinx which is close to zero
for the small scattering angles which correspond to slow variations in
projected potential. The portion corresponding to (l-cosa~) is multi-
plied by cosX which is close to unity for small scattering angles and
hence this will dominate the image contrast.

This situation is familiar in biological electron microscopy. In

order to get good contrast in the useful range of resolution (details on
a scale greater than 10 to 20 lb it is advisable to use a heavy atom
stain which will give a strong (l-cosa~) signal which is then strongly
imaged because of the relatively large values of the cosX function. A
small objective aperture size is used in order to remove the high-angle
scattered radiation which adds only a confused background to the image
and reduces the image contrast. Detailed calculations on models of
stained biological objects by BRIDGES and COWLEY (to be published) have
confirmed this interpretation.

The case of thin crystals is distinctive in that it involves more

quantitative consideration of more specialized diffraction patterns and
so will be treated separately.

Mathematical Fonnulation

Provided that we may use the phase-object approximation, the trans-

mission function of the object is written

(1.38)

and the image intensity for a parallel beam incident is

(1.39)
Principles of Image Formation 25

Extending this to the case of a finite incoherent source in CTEM or

a finite collector aperture, with aperture function D(u), in STEM, we
find that, for bright-field imaging when we neglect the amount of "scat-
tered" radiation included in the collector aperture, the image intensity
can be written

(1.40)

where Do is the integral over D(~), tl(!) is given by Eqn. 1.34 and

(1.41)

The form of the functions tl (!) and t2 (!) under various conditions of
defocus and aperture size has been calculated by COWLEY and AU (1978).

For a very small collector aperture size, Eqn. 1.40 reduces to Eqn.
1.37. If the collector aperture is increased until it is equal to the
objective aperture size,

t 1 (!) = 0,
t2(!) = c 2 (!) + s2(!).

Then the bright-field signal has the form

I BF (!) ~ Doll-a2,2(!) * {c 2 (!) + s2(!)}1

= Doll-IDF (!) I· (1.42)

where I DF (!) is, in this case, the dark-field signal calculated on the
assumpt10n that all scattered radiation, including that contained in the
central beam spot, is detected to form the signal.

1.7 THIN, PERIODIC OBJECTS: CRYSTALS

For very thin specimens, there is no difference between the phase

contrast imaging of periodic and nonperiodic objects, provided that we
limit ourselves to the Scherzer optimum defocus and use an objective
aperture to remove the higher angle parts of the diffraction pattern be-
yond the flat part of the transfer function. The projected potential,
or the sine and cosine of the scaled projected potential, will be imaged
according to Eqn. 1.27 or 1.36. Thus for thin crystals viewed down one
of the unit cell axes, the image is periodic with the periodicity of the
unit cell projection and shows the distribution of the atoms within the
unit cell in projection, within the limitations of the point-to-point
resolution of the microscope. With present day electron microscopes it
is possible to make a direct structure analysis of a crystal by direct
visualization of atoms in this way for a wide range of materials for
which the heavier atoms are separated in projection by distances of the
order of 3 R
or more (COWLEY and IIJlMA, 1977). Since the imaging does
26 Chapter 1

not depend on the periodicity, crystal defects may be imaged with the
same clarity as the perfect crystal structure, provided that the defects
do not have a three-dimensional structure which gives an unduly compli-
cated two-dimensional projection [see Fig. 1.10]. This provides a
unique opportunity for the study of the configurations of atoms in indi-
vidual imperfections of the structure and is rapidly broadening our un-
derstanding of the nature of the defects in a number of types of mater-
ial [COWLEY, 1978(b)].

(a) (b)

Fig. 1.10 . The image, (a), and the diffraction pattern, (b), of a crys-
tal of Nb 22 0 S4 . The image shows varying contrast of some parts of the
unit cell due to atomic disorder. The image contrast changes with in-
creasing thickness. 100keV, C = 2.8Bl1D, unit cell dimensions (indi-
cated) 21.2~ by 15.6~ (courtesy sFumio Iijima).

The imaging of crystals, on the other hand, makes it possible for

the first time to make quantitative correlations of image intensities
with known specimen structures. Such a possibility can provide an enor-
mous expansion of the power and range of applications of electron micro-
scopy. To achieve this, however, it is necessary to refine both the ex-
perimental techniques and the interpretive methods.

Crystals must be aligned with an accuracy of a small fraction of a

degree in two directions by use of a tilting stage. The crystal thick-
ness must be determined with reasonable precision. The amount of de-
focus, the aberration constants and the aperture size of the objective
lens must be accurately known.

On the other hand, it is rarely sufficient to use a simple approxi-

mation such as Eqn. 1. 36 to calculate image intensities. This phase-
object approximation has been shown to give significant errors for
Principles ofImage Formation 27

crystal thicknesses of about 20 g for 100 keV electrons. Reliable cal-

culations for image interpretation must involve the use of three-
dimensional, many-beam dynamical diffraction theory using either the
matrix formulation of Bethe's original dynamical theory of electron dif-
fraction or, more usually, the multislice formulation of dynamical
theory due to Cowley and Moodie (see COWLEY, 1975). In the latter, the
crystal is subdividided into a number of very thin slices perpendicular
to the beam. Each slice acts as a thin phase object. Between slices
the electron wave propagates according to the usual laws of Fresnel dif-
fraction. Standard computer programs for these operations are now avail-
able.

The degree to which agreement can be obtained between observed im-

age intensities and intensities calculated by these methods, taking into
account practical experimental parameters such as incident beam conver-
gence and chromatic aberration, is demonstrated in Fig. 1.11. The com-
puter programs can be extended to deal with cases of defects in crystals
by use of the assumption of periodic continuation, Le. it is assumed
that the image of a single defect will be exactly the same as for a de-
fect in a periodic array of well separated defects which forms a super-
lattice having a large unit cell.

This method for the study of crystal structures and crystal defects
has recently been extended to a wide variety of inorganic compounds and
minerals. Improved resolution, approaching 2 g has been achieved by use
of the high resolution, high voltage microscopes now in operation (e.g.
HORIUCHI et a1., 1976; KOBAYASHI et a1., 1974).

Special Imaging Conditions

It is well known that images of crystals can show details on a much

finer scale than the point-to-point or Scherzer resolution limits.
Fringes with spacings well below 1 g have been observed. HASHIMOTO
et a1. (1977) have shown pictures of gold crystals having details on a
scale approaching 0.5 g within the intensity maxima at the positions of
the rows of gold atoms. Pictures of silicon by IZUI et al. (1977) show
clearly separated spots 1. 36 g apart at the projected positions of
silicon atoms.

These pictures are taken without the objective aperture limitation

considered above. They correspond to situations in which the diffrac-
tion pattern consists of only a few sharply defined diffraction spots.
The requirement for clear imaging of the crystal periodicities is then
that these fine diffraction spots should be recombined with maximum amp-
litude and well-defined relative phases. This is quite different from
the requirement for a flat transfer function needed for imaging of gener-
al non-periodic objects or periodic objects with large unit cells. In
order to maintain large amplitudes for the relatively high angle diffrac-
tion spots the effects of beam convergence and chromatic aberration must
be minimized, not only by careful control of the instrumental parameters
but also by choosing the values of the defocus which make the transfer
function less sensitive to these effects.
28 Chapter I

Fig. 1.11. Structure images of Nb 12 0 29 taken at (a) 100kV (courtesy of

S. Iijima) and (b) 1 MeV (courtesy of S. Horiuchi). The inserts at
lower left of each micrograph are calculated images for 3sR thick crys-
tal of Nb 120 29 . The aperture functions below show the resolution condi-
tions under which each calculation was carried out. At 100 kV the physi-
cal aperture (A) at u = 0.30s/R limits resolution of 3.2R; the aperture
function due to a defocus-depth halfwidth of 100R (B) limits resolution
to 2 . 4R; while the aperture function due to an incident beam convergence
of 1.4 milliradian (C) restricts it to 3.sR. The combined effect of
these functions (D) results in an image of 3.sR resolution. At 1 MV the
physical aperture (A) and convergence aperture function (C) limit resolu-
tions to 1.9R and 1.sR, respectively. For the calculated image to match
the experimental result required a defocus-depth halfwidth of 500 R re-
sulting in the B curve shown. The combined effect (D) is virtually iden-
tical to and yields an image of 2.sR resolution (courtesy of M.A.
O'Keefe).

It is possible to achieve these special imaging conditions only for

crystals of simple structure in particular orientations. As HASHIMOTO
et al. (1978) have pointed out, the image intensities may then be sensi-
tive indications of the details of potential distributions in the crys-
tals even though they are in no way to be regarded as providing direct
pictures of the structure. The special imaging conditions, however, be-
come rapidly more difficult to achieve as the size of the unit cell in-
creases and are not relevant for the imaging of defects of the crystal
structure.

On the other hand, an improved appreciation of the special condi-

tions for imaging of periodic structures has lead to the realization
that some compromises are possible between the extreme situations. For
crystals having defects which disrupt the periodicity to a limited ex-
tent (relatively small changes of lattice constant) one can image the
Principles of Image Formation 29

structure with its defects to see detail which is finer than for a non-
periodic object although not as fine as for a strictly periodic object.
This is demonstrated, for example, by the pictures of defects in silicon
due to SPENCE and KOLAR (1979).

Mathematical Fonnulation

If a crystal is divided into very thin slices perpendicular to the

incident beam, the transmission function of the nth slice is

- exp{-ia~ (xy)}
n
where

(1. 43)

Propagation through a distance ~ =

zn+l - zn to the n+1 th slice is given
by convolution with the propagation function (Eqn. (1.12)]. Then we
have the recursion relationship

(1.44)

or, in terms of Fourier transform,

which, for a periodic object can be written

~ +l(h,k) = Lh k~ (h1k{)PA(h1k1)Q (1. 45)

n ,n u n+l (h-h ,k-k )
and this is an operation readily programmed for a computer with

(1.46)

for a unit cell with dimension a,b.

The image intensity is then calculated from the exit wave function
or its Fourier transform ~N{~) as

(1.47)

The condition that the image wave amplitude should be identical with the
wave at the exit face of a crystal is that exp{iX(u)} = 1 for all dif-
= =
fracted beams u h/a, v k/b i.e. that -

nAfA ~ + ~ J + ~nC s A3 [~ + ~ J2 = Nn (1. 48)

The case for N odd is included because it represents the case of an iden-
tical wave function shifted by half the periodicity.
30 Chapter 1

It is not possible to satisfy this condition unless a 2 and b 2

are in the ratio of integers. For the special case that a = b, (Eqn.
1.48) is satisfied for af = =
na 2 /A and Cs 2ma 4 /A3 for integral n,m (see
KUWABARA, 1978). The extent to which these conditions may be relaxed is
a measure of the limitations on the degree of crystal perfection or on
the finite crystal dimensions for which the image may represent the
ideal in-focus image of the crystal (see COWLEY, 1978c).

1.8 TIDCKER CRYSTALS

When the crystal thickness exceeds 50-100 i for 100 keV electrons
or 200-500 i for 1 MeV electrons there is in general no relationship
visible between the image intensities and the projected atom configura-
tions, (although for some particular cases the same thin-crystal image
is repeated for greater thicknesses). It is suggested that the image
intensities will be increasingly sensitive to details of crystal struc-
ture, such as the bond lengths, ionization or bonding of atoms and ther-
mal vibration parameters but since the images are also more sensitive to
experimental parameters such as the crystal alignment and the lens aber-
rations, the refinement of crystal structures by use of high resolution
thick crystal images remains an interesting but unexploited possibility.

Most of the important work done on thicker crystals has been on

the study of the extended defects of crYstals having relatively simple
structures with medium resolution (5 i or more) and no resolution of
crystal structure periodicities. The extensive work done on the form
and behavior of dislocations and stacking faults, in metals, semiconduc-
tors and an increasingly wide range of inorganic materials, together
with the theoretical basis for image interpretation in these cases are
very well described in such books as HIRSCH et al. (1965) and BOWEN and
HALL (1975). Because dynamical diffraction effects are of overwhelming
importance for these studies it is essential to simplify the diffraction
conditions as much as possible to make it relatively easy to calculate
and to appreciate the nature of the diffraction contrast. For thin crys-
tals with large unit cells, viewed in principal orientations it may be
necessary to take hundreds or even thousands of interacting diffracted
beams into account [see Fig. 1.10(a»). For the thicker crystals of rela-
tively simple structure it is often possible to choose orientations for
which the two-beam approximation is reasonably good; namely, when the
incident beam gives rise to only one diffracted beam of appreciable amp-
litude and these two beams interact coherently. The image intensity can
vary with crystal thickness because interference between the two beams
gives both of them a sinusoidal variation with thickness. Crystal de-
fects show up with strong contrast because changes in the relative phase
of the two beams result in different interference effects and so dif-
ferent intensities. The phase changes can be sudden as when the crystal
suffers a shear displacement at a stacking fault; or the phase change
can be more gradual as when the lattice is strained around a dislocation
line or other defect and the deviation from the Bragg angle varies as
the incident and diffracted beams travel through the strain field.
Principles of Image Formation 31

The standard dynamical diffraction theory, as originated by Bethe

and developed by many other authors (see HIRSCH et a1., 1965) is the
theory of the interaction of electron waves with a perfectly periodic
potential distribution bounded by plane faces. To adapt this to the
study of crystal defects it is usual to make a simplifying "column
~pproximation.1I For very thin crystals we have made the assumption that
the electron wave at a point on the exit faces of the crystal is in-
fluenced only by the potential along a line through the crystal to that
point in the beam direction. For thicker crystals we may assume that
the electron wave at a point on the exit face of the crystal is affected
by the diffraction, not along a line, but within a thin column extending
through the crystal in the beam direction.

The width of this column may be estimated in various ways which all
agree that it may be surprisingly narrow. For 100 keY electrons, for
example, and crystals several hundred ~ thick, the column width may be
taken as small as 5-10 ~ with errors which are not serious for most pur-
poses. For a microscope resolution limit of 10 ~ or more the column ap-
proximation serves very well.

To calculate the image of a defect it is necessary to calculate the

amplitudes of the incident and diffracted beams for each column of crys-
tal passing near the defect. The amplitudes for a column containing a
particular sequence of lattice strains and disruptions can be calculated
on the assumption that all surrounding columns are identical, i.e. that
the crystal is perfectly periodic in directions perpendicular to the
beam direction. The calculations are usually made by the difference
equation method of Howie and Whelan (see HIRSCH et al., 1965) in which
the progressive changes of the wave amplitudes are followed as the waves
progress through the crystal. Convenient computer techniques developed
by Head and colleagues (HEAD et al., 1973) have provided systematic tech-
niques for the identificatio~defects of many types. In many cases,
especially for thick crystals, it is necessary to introduce the effects
of inelastic scattering (mostly thermal diffuse scattering) on the elas-
tically scattered waves. This is done usually by the simple expedient
of adding a small imaginary part to make the structure amplitudes of the
crystal complex. It leads to a variety of easily observable effects.

Recent refinements of these dynamical diffraction studies of crys-

tal defects include the "weak beam method" which gives much finer de-
tails of defect structure at the expense of very low image intensities
(COCKAYNE, RAY and WHELAN, 1969). This method relies on the fact that
in a situation where both strong and weak diffracted beams, or only weak
diffracted beams, are present, the weak beam intensities vary much more
rapidly with crystal thickness or with change of incident beam orienta-
tion than do the strong beams. Thus the details of dislocation struc-
ture in an image formed by allowing only a weak beam through the objec-
tive aperture may be on the scale of 10-15 ~ whereas for dark-field im-
ages formed with strong reflections the intensity variations may be
stronger but show no detail finer than about 50 ~.

For the interpretation of detail on a very fine scale, the column

approximation may not be sufficient. A review of the more exact
32 Chapter 1

treatments which avoid this approximation has been given recently by

ANSTIS and COCKAYNE (1979).

Because the image intensity modulations for crystal defects are

strongly dependent on the angle of incidence of the electron beam in re-
lation to the Bragg angle for the operative reflections, the visibility
of defects may be reduced and the characteristic features of their im-
ages may be lost if the range of angles of incidence is too large in
CTEM or if the collector aperture size is too large in STEM. For CTEM
this usually does not constitute a serious restriction but for STEM, es-
pecially if a high-brightness source is not used, it may produce an un-
desirable low intensity of the useful images. One means for overcoming
this limitation of the STEM method has been suggested by COWLEY (1977).
If a slit detector is used in place of a circular detector aperture a
relatively large signal can be obtained for incident beam directions
having only a narrow range of angles of incidence on the operative re-
flecting planes. This is one of the situations for which the flexi-
bility in collector aperture configuration, inherent in the STEM mode,
offers considerable advantage.

Lattice Fringes

Within the limits set by incident beam divergence and the mechani-
cal and electrical stabilities of the electron microscope, it is possi-
ble to produce interference fringes in the image with the periodicity of
the diffracting lattice planes for any crystal thickness provided that
the objective aperture allows two or more diffracted beams to contribute
to the image. Under the usual operating conditions, with no careful con-
trol of the experimental parameters of the electron microscope or of the
specimen material, the information content of such images is very
limited. The position of the-dark or light fringes relative to the atom-
ic planes is indeterminate since this is strongly dependent on the crys-
tal orientation and thickness , the objective lens defocus and the cen-
tering of the incident or diffracted beams with respect to the objective
lens axis. The spacing of the fringes is usually close to that of the
relevant lattice planes but may vary appreciably if the crystal varies
in thickness or is bent or, in particular, if other strong reflections
are excited locally.

If care is taken to avoid complications from all of these factors,

however, some very useful data may be obtained by observations of lat-
tice fringes. Variations of lattice plane spacings corresponding to var-
iations in the composition or degree of ordering in alloys have been ob-
served (WO, SINCLAIR and THOMAS, 1978). Also the presence of defects
may be detected even though the perturbations of the fringe spacings or
contrast can usually give no direct evidence on the defect structure.

Lattice fringes may, of course, be observed in STEM, as in CTEM.

The accessibility of the convergent beam diffraction pattern in a STEM
instrument allows a rather clearer picture to be obtained of the condi-
tions under which lattice fringes are formed (SPENSE and COWLEY, 1978).
If the disc-shaped diffraction spots corresponding to the individual
reflections do not overlap, no interference is possible between incident
Principles oflmage Formation 33

beam directions with sufficient angular separation to produce interfer-

ence effects with the lattice plane periodicity. The region where dif-
fraction spot discs overlap is the region where such interference ef-
fects can take place, so that if the detector aperture includes the re-
gion of overlap, the image can show the lattice plane spacing.

It is not difficult to specify the detector aperture size and shape

which will give maximum lattice fringe visibility and image intensity
for any particular conditions of beam incidence, objective aperture size
and lens aberration.

Mathematical Considerations

In order to calculate the wave function at the exit face of a crys-

tal, using the column approximation, when there is a distortion of the
crystal which is a continuous function of distance in the beam direc-
tion, the multi-slice formulation of Cowley and Moodie, described above,
may be used. The solution of the wave equation, following the Bethe
formulation, in each section of the crystal which can be considered as
periodic is rarely feasible. Most commonly the difference equation form
due to Howie and Whelan is used. The changes in incident and diffracted
wave complex amplitudes due to diffraction from other waves in the crys-
tal, absorption effects and excitation errors, can be written

(1.49)

where ~ is the column vector whose elements ~h are the amplitudes of the
t!iffracted waves, § is a diagonal matrix whose elements are ~h =
d(hoR(z))/dz and R(z) is the vector giving the displacements of the lat-
tice points. The matrix ~ has diagonal and off-diagonal elements:

= th + iO/p~/4n

= a(~h -g + i~h' -g )/4n (1.50)

where th is the excitation error for the h reflection and ~h is the im-
aginary part added to the structure amplitude to represent the effect of
absorption.

For the two-beam case this simplifies to a simple pair of coupled

equations which may be integrated through successive slices of crystal.

For other than very thin crystals it is not possible to represent

the effect of the crystal on the incident wave function by multiplica-
tion by a scalar transmission function, as was assumed above. Instead
we may consider the action of a crystal to be represented by the action
of a scattering matrix on an incident wave vector, ~o(~), representing
the Fourier coefficients of the incident wave. Thus

~
-
= =-0
S~ (l.51)

and the matrix ~ =

exp{izM(h)/2k} where tl(h) is similar to the matrix A
of Eqn. l.49. This equation can also be-iterated through successive
34 Chapter 1

slices. For n similar slices !n = ~ lJ'o(h). This formulation follows

the concepts developed by STURKEY (19b2) and others and forms the basis
for a number of sophisticated and powerful treatments of diffraction and
imaging problems.

1.9 VERY TIHCK SPECIMENS

As the thickness of a specimen increases, the distribution of in-

tensity in the diffraction pattern is dominated increasingly by multiple
scattering effects. Diffracted beams traversing further regions of the
specimen may be diffracted again and again, both elastically and inelas-
tically. For crystals the spot patterns given by thin crystals are grad-
ually submerged under the diffuse background scattering. Diffraction of
the diffusely scattered electrons by the crystal lattice gives compli-
cated Kikuchi line configurations which in turn gradually lose contrast
and are lost in an overall broad background of scattering. For non-
crystalline specimens the initial rather featureless scattering distribu-
tion is successively broadened by multiple elastic scattering. Also be-
cause the electrons lose energy through successive inelastic scat-
terings, the distribution of electron energies becomes broader and
broader and the number of electrons which have not lost any energy be-
comes very small.

For CTEM the image resolution becomes poorer as a result of the in-
creasing angular spread of the electrons because the position at which
scattering occurs becomes less and less well-defined as the electron
beam spreads in the specimen. Also as the energy spread of the trans-
mitted electrons increases, the resolution suffers as a result of the
chromatic aberration of the lenses. Usually, in order to improve the
contrast, the objective aperture size is made small but this has the ef-
fect of reducing the image intensity rather drastically because, as the
angular range of the scattering is increased by multiple elastic scat-
tering, the fraction of the transmitted radiation remaining near the
central spot is rapidly reduced (Fig. 1.12).

For STEM there is the same loss of resolution due to the angular
spreading of the beam in the specimen. However, loss of energy of the
electrons by mUltiple inelastic scattering in the specimen does not af-
fect the resolution, since there are no imaging lenses after the speci-
men. Hence, the effect of specimen thickness on the resolution will be
less severe for STEM than for CTEM. For 100 keV electrons and specimen
thicknesses, of the order of a few micrometers, estimates suggest that
STEM will have an advantage over CTEM by a factor of about 3 (SELLAR and
COWLEY, 1973) although for special instrumental configurations this fac-
tor has been estimated to be as high as 10 (GROVES, 1975). With in-
creasing accelerating voltage this factor will decrease, being about 2
for 1 MeV electrons.

It has been shown that for STEM the best contrast is obtained for
very thick specimens if a very large detector aperture is used (or the
order of 10- 1 radians or more) (see Fig. 1.12). This has the added ad-
vantage that the image signal intensity can be as high as half the inci-
Principles of Image Formation 35

T = 5
T=
I I
...... ::-..
......

--
"-

r 'V 2ro r

(a) (b)

Fig. 1.12. Diagrams suggesting the change in the angular distribution

o~ scattered electrons and the reduction of the central peak of unscat-
tered electrons, for amorphous specimens of thickness T equal to (a) the
mean free path for elastic scattering and (b) five times this thickness.
The optimum objective aperture size, giving the greatest transfer of in-
tensity from inside to outside the aperture with a small change of thick-
ness, is indicated in each case relative to the mean scattering angle
for single scattering.

dent beam intensity so that there is no problem of decreasing image

intensity (SMITH and COWLEY, 1975). Also for STEM it is possible to get
dark field image contrast by use of an energy filter to separate elec-
trons which have lost less energy than the average from those which have
lost more. The desirable energy cut-off for maximum contrast obtained
in this way may be as high as several hundred volts energy loss for
thick specimens (PEARCE-PERCY and COWLEY, 1976). Again it appears that
STEM has a potential advantage in that the flexibility inherent in the
STEM detection system and the associated signal processing possibili-
ties, allows the optimum imaging conditions to be achieved in a relative-
ly straightforward manner.

In calculating image contrast for very thick specimens it is usual

to assume that the multiple scattering will mix up the relative phases
of diffracted beams to the extent that interference effects will be
washed out and a simple incoherent imaging theory can be used. The in-
tensities, rather than the amplitudes of multiply scattered waves are
added together. This assumption is so very convenient and the alterna-
tive of a proper coherent scattering theory is so forbidding, that it is
easy to ignore the fact that coherent scattering effects may be signifi-
cant for any quantitative image evaluation for even quite large thick-
nesses (SELLAR, 1977).
36 Chapter 1

Mathematical Descriptions

In the incoherent scattering approximation it is assumed that the

intensity distribution from the first slice of crystal is spread further
by scattering in a second slice and so on. For a single slice of thick-
ness 6z the intensity distribution may be written

where f2(u) is the square of the scattering amplitude, and the absorp-
tion coefficient ~ is given by ff2(u)du. Fourier transforming this in
terms of some arbitrary variable w,

The effect of transmission through n layers to give a total thickness T

= n6z is to convolute I1(u) by itself n times or correspondingly to
raise G1(w) to the nth power

G1n(w) = exp{-~T}[l + 6zP(w)]n

7 exp {-~T + TP(w)}, (I.52)

so that the angular distribution of scattered intensity becomes

(I.53)

The optimum detector aperture size for a STEM instrument with a

very thick specimen is found by finding the value of I~I for which the
differential of the intensity IT(u) with respect to the thickness T
changes sign.

The distribution of the number of electrons with energy loss is

found in the same way in terms of the mean free path for inelastic scat-
tering and the optimum energy cut-off for an energy filter is found by
finding the energy loss value for which the differential of this number
with respect to T changes sign.

1.10 CONCLUSIONS

One conclusion to be drawn from our discussions of the high resolu-

tion imaging of thin specimens is that no simple definition of resolu-
tion is possible unless the concept of resolution is severely re-
stricted. One can ask, for example, what is the smallest distance be-
tween two distinct maxima or minima of intensity in an image. This pro-
vides an operational definition of resolution of one kind, useful as a
convenient criterion to be used by instrument designers but it makes no
reference to the main function of an electron microscope which is to pro-
vide information regarding the structure of the specimen. In practice
this type of definition requires qualification in that, as is well-
known, point-to-point imaging is different from lattice fringe imaging.
In the latter, strong diffracted beams occur at large diffraction angles
Principles ofImage Formation 37

so that some contrast may be seen provided that the transfer function of
the lens is not zero. For non-periodic objects the diffracted wave amp-
litudes falloff rather uniformly so that outer non-zero parts of the
transfer function will multiply weak scattering amplitudes and the re-
sultant contributions of the outer parts of the diffraction pattern to
the image intensity will be so small that the corresponding fine detail
of the image will be of too little contrast to be detected. Thus, in
order to provide a reliable resolution criterion of this sort it is nec-
essary to specify the nature of the ordering and the degree of crystal-
linity in the specimen, but no independent evidence on these questions,
other than from electron microscopy, is available.

It is, of course, possible to consider only the extreme case of

near perfect periodicity and take the minimum observable lattice fringe
spacing as a measure of the "resolution limit." This is, in fact, a
good test of some instrumental parameters such as the mechanical stabili-
ty of the column, the specimen drift, interference from stray electrical
or magnetic fields and incident beam convergence. It is not a sensitive
measure of chromatic aberration and is insensitive to spherical aberra-
tion.

The use of a test object of completely random structure would pro-

vide a different basis for testing. But most "amorphous" materials, in-
cluding amorphous carbon films, are known to contain small regions which
are relatively well ordered to the extent that the diffraction patterns
given by individual picture elements (of diameter comparable with the
resolution limit) may contain quite strong maxima at high angles. There-
fore, the assumption of a scattered amplitude falling off fairly uniform-
ly with scattering angles is invalid.

From a different point of view we may choose to measure resolution

in terms of the ability of the electron microscope to provide a recogniz-
able image of the known structure of a test object. For very thin,
weakly-scattering objects the usual Scherzer criterion applies. The
resolution is assumed to be given by the reciprocal of the value of u =
2A- 1sin(,/2) at the outside limit of the flat part of transfer function,
sin(X(u)), for the optimum defocus. The objective aperture is chosen to
eliminate all the radiation scattered at higher angles, for which the
transfer function is oscillatory. As we have seen, this criterion for
measuring resolution cannot be used for strongly scattering and thicker
samples. Nor can it be used in general for dark-field imaging since for
detail near the resolution limit the intensity distribution of dark
field images often has no direct relationship to the atomic arrangement
in the specimen.

The situation is rather more favorable when the images of weakly

scattering objects can be reasonably well interpreted by incoherent im-
aging theory as in the case of STEM with an annular dark-field detector,
STEM bright field imaging with a large detector aperture (see Eqn. 1.42)
or bright field CTEM with a large angle of illumination (NAGATA et al.,
1976). It is a well-known result of light optics (BORN and WOLF, 1964)
that for incoherent illumination the resolution can be better hhan for
coherent illumination by a factor of about 1.5 (actually 21 2 for a
Gaussian spread function as is evident from Eqn. 1.35).
38 Chapter 1

Probably the best way to characterize the performance of an elec-

tron microscope is to determine the transfer function for the objective
lens for a thin phase object. This is consistent, although not identi-
cal, with the current practice in light optics. If the transfer func-
tion can be determined experimentally the resolution and contrast of im-
ages produced by the instrument for any specimen can be evaluated ac-
cording to any of the criteria which seem useful or by detailed calcula-
tion.

The most convenient method for achieving this is by use of an op-

tical diffractometer which provides the Fourier transform (or, more ac-
curately, the squared amplitude of the Fourier transform) of the image
intensity distribution. Provided that the specimen is a thin, weakly-
scattering object, the optical diffraction pattern intensity will give,
to a good approximation, the square of the transfer function, sin(x(u)),
multiplied by the intensity distribution in the diffraction pattern of
the object. Usually the specimen used for this purpose is a thin "amor-
phous" carbon film. Caution is necessary in practice to ensure that
none of the restrictions we have mentioned on the use of this method are
violated.

It has been found in practice that the transfer function measured

in this way may depend very strongly on a number of experimental factors
and so will show variations even for the same lens used on the same spec-
imen at the same voltage and defocus. One important requirement for the
development of more quantitative high resolution electron microscopy is
therefore a means for obtaining a rapid measurement of the transfer func-
tion, preferably as a continuous "on-line" monitoring of the microscope
performance. Some limited success in this respect has been achieved for
example by BONHOMME and BEORCHIA (1978) but further work in this direc-
tion is desirable.

The quest continues for even better and better resolution. One ap-
proach is to find ways of interpreting the image detail coming from the
outer parts of the diffraction pattern, beyond the Scherzer limit, by
use of image processing techniques. The other approach is to extend the
Scherzer limit. This involves the attainment of smaller spherical
aberration or smaller wavelengths without the sacrifice of other impor-
tant factors, since the directly interpretable resolution for the
Scherzer optimum defocus is given by
a ~ 0.6 C 1/4 AS/4 (1. 54)
S

The factor 0.6 is approximate and may be replaced by various factors de-
pending on the assumptions made. The improvement of Cs is difficult and
has a limited effect because of the 1/4 power. The decrease of the wave-
length by increasing of the accelerating voltage therefore appears to be
the most promising approach provided that the resolution is not limited
by the stability of the accelerating voltage or lens currents. These
latter factors do seem to be effective in limiting the resolution of the
current high voltage, high resolution instruments to about 2 R whereas
the theoretical resolution given by Eqn. 1.54 is more like 1.5 R or
less.
Principles of Image Formation 39

If the resolution is to be improved by image processing techniques

it is desirable, of course, to start from a situation where the directly
interpretable resolution is as good as possible. The extent to which
image processing methods may succeed depends then on the extent to which
the fineness of the image detail exceeds the limit of directly interpret-
able detail. This in turn depends on the extent to which the image de-
tail is limited by chromatic aberration, beam convergence, and similar
effects, in relationship to the limitations on interpretability set by
spherical aberration.

The next stage in the improvement of electron microscope resolution

will clearly be a very significant one from the point of view of applica-
tions in materials science. For most inorganic materials the interatom-
ic distances, seen in projection along favorable crystallographic direc-
tions, are mostly in the range 1.5 to 2.0 R. Improved resolution means
more contrast for the imaging of atoms. Thus the next factor of 2 in
resolution from the present commonly attainable limit will enormously
enhance the power of the electron microscope to give detailed informa-
tion on crystal structures and the all-important perturbations of crys-
tal structures due to the various types of defects.

An equally important direction of development is the addition of

microanalytical methods to the imaging capabilities. The addition of
information on crystal structure from microdiffraction patterns and on
chemical composition from the microanalysis techniques, based on the de-
tection of characteristic x-rays or characteristic electron energy
losses, forms the subject matter of most of the other contributions to
this volume. The recent commercial production of instruments to combine
all of these capabilities, with a limited sacrifice of performance in
anyone respect, provides the basis for a major reorganization of re-
search, especially in the materials sciences, aimed at a much more com-
plete understanding of physical and chemical properties in terms of the
atom configurations and atom interactions.

CLASSICAL and GENERAL REFERENCES

Anstis, G. R. and Cockayne, D. J. H. (1979), Acta Cryst., in press.

An excellent analysis of the various theoretical approaches to the
description of dynamical diffraction by crystals with defects.

Boersch, H. (1947), Z. Naturforsch 2a, 615.

One of the early works of this often neglected scientist who con-
tributed many important ideas to the subject of electron micro-
scopy, including much discussion of the possibility of observing
individual atoms.

Born, M. and Wolf, E. (1964), "Principles of Optics", Pergamon Press,

London.
The standard reference work for many years on the contemporary ap-
proach to optics.
40 Chapter 1

Bowen, D. K. and Hall, C. R. (1975), "Microscopy of Materials", John

Wiley and Sons, New York.
A more modern treatment than Hirsch et al. (1965) but on a more in-
troductory level, keeping as closely-as-possible to a nonmathemat-
ical description style: a little shakey on some points of funda-
mental physics.

Cowley, J. M. (1975), "Diffraction Physics", North Holland Publishing

Company.
A rather formidable book for the non-theorist which attempts to cor-
relate x-ray and electron diffraction with electron microscopy by
use of a common theoretical basis. Needs to be read in conjunction
wi th more detailed accounts, or prior knowledge, of the experi-
mental situations.

Doyle, P. A. and Turner, P. S. (1968), Acta Cryst. A24, 390.

The earliest and, for many purposes, definitive discussion of the
application of the reciprocity principle in electron diffraction.

Head, A. K., Humble, P., Clarebrough, L. M., Morton, A. J., and Forward,
C. J. (1973), "Computer Electron Micrographs and Defect Identifica-
tion", North Holland, Amsterdam.
A clear, systematic account of the methods and typical results for
calculating images of dislocations, stacking faults, etc., with
computer programs.

Hirsch, P. B., Howie, A., Nicholson, R. B., Pashley, D. W., and Whelan,
N. J. (1965), "Electron Microscopy of Thin Crystals", Butter-
worth's, London.
This is the "Yellow Bible" of materials science electron micro-
scopists. Produced as the result of a summer school, it c~ntains
an excellent account of the explanation of the contrast effects for
crystal defects in crystals. A second edition published in 1977
has a chapter added to summarize the progress since 1965.

Lipson, S. G. and Lipson, H. (1969), "Optical Physics", Cambridge Uni-

versity Press, Cambridge.
A book biased by the fact that both authors are physicists and one
is an outstanding crystallographer who contributed greatly to the
use of optical diffraction analogues for x-ray diffraction pro-
cesses.

Saxton, W. o. (1978), "Computer Techniques for Image Processing in Elec-

tron Microscopy", Academic Press, New York.
A definitive but difficult book on the basic concepts and methods
of image processing and a detailed description of the contributions
of the author and his colleagues to this subject, with computer
programs for the main operations.

Scherzer, o. (1949), J. Appl. Phys. 20, 20.

The clear, original statement on how to produce the optimum phase
contrast imaging for weakly-scattering objects.
Principles of Image Formation 41

Vainshtein, B. K. (1964). "Structure Analysis by Electron Diffraction",

Pergamon Press, Oxford.
An account of the work of the Soviet Union school which developed
and applied electron diffraction methods of crystal structure ana-
lysis: contains excellent accounts of the geometry of electron
diffraction patterns and the simpler approximations for electron
diffraction intensities.

OTHER REFERENCES

Anstis, G. R. and O'Keefe, M. A. (1976), in Proc. 34th Annual Meeting

Electron Micros. Soc. Amer., Ed. G. W. Bailey, Claitor's Publ.
Div., Baton Rouge, p. 480 and in press.

Bonhomme, P. and Beorchia, A. (1978), in Electron Microscope 1978, Ed.

J. M. Sturgess, Microscopical Society of Canada, Vol. 1, p. 86.

Cockayne, D. J. H., Ray, 1. L. F., and Whelan, M. J. (1969), Phil. Mag.

20, 1265.

Cowley, J. M. (1969), Appl. Phys. Letters 15, 58.

Cowley, J. M. (1976), Ultramicroscopy~, 3.

Cowley, J. M. (1977), in High Voltage Electron Microscopy 1977, Eds. T.

Imura and H. Hashimoto, Japanese Soc. Electron Micros., Tokyo, p.
9.

Cowley, J. M. (1978a), Advances in Electronics and Electron Physics, Ed.

L. Marton, Academic Press, New York, 46, 1-53.

Cowley, J. M. (1978b), Annual Reviews of Physical Chemistry, Ed. B. S.

Rabinovich, Annual Review, Inc., Palo Alto, 29, 251-283.

Cowley, J. M. (1978c), in Electron Microscopy 1978, Vol. III, Ed. J. M.

Sturgess, Microscopical Society of Canada, Toronto, p. 207.

Cowley, J. M. and Au, A. Y. (1978), in Scanning Electron Microscopy/

1978, Vol. 1, Om Johari, Ed., SEM Inc., AMY O'Hare, Illinois, p.
53.

Cowley, J. M. and Iijima, S. (1977), Physics Today 30, No.3, 32.

Cowley, J. M. and Jap, B. K. (1976), in Scanning Electron Microscopy/

1976, Vol. 1, Om Johari, Ed., IITRI, Chicago, p. 377.

Crewe, A. V. and Wall, J. (1970), J. Mol. BioI. 48, 375.

Ditchborn, R. W. (1963), Light (2nd Edition), Blackie & Sons, London.

Frank, J. (1973), Optik, 38, 519.

42 Chapter 1

Groves, T. (1975), Ultramicroscopy!, 15.

Hashimoto, H., Endoh, H., Tanji, T., Ono, A., Watanabe, E. (1977), J.
Phys. Soc. Japan 42, 1073.

Hashimoto, H., Kumao, A., and Endoh, H. (1978), in Electron Microscopy

1978, Vol. III, Ed. J. M. Sturgess, Microscopical Society of
Canada, Toronto, p. 244.

Horiuchi, S., Matsui, Y., Bando, Y. (1976), Jpn. J. Appl. Phys. 15,
2483.

Isaacson, M. S., Langmore, J., Parker, W. W., Kopf, D., and Utlaut, M.
(1977), Ultramicroscopy!, 359.

Izui, K., Furono, S., Otsu, H. (1977), J. Electron Microsc. 26, 129.

Kobayashi, K., Suito, E., Uyeda, N., Watanabe, M., Yanaka, T., Etoh, T.,
Watanabe, H., Moriguchi, M. (1974), in Electron Microscopy 1974,
Eds. J. V. Sanders, D. J. Goodchild, Aust. Acad. Sci., Canberra,
Vol. 1, p. 30.

Kuwabara, S. (1978), J. Electron Microsc. 27, 161.

Lamvik, M. K. and Langmore, J. P. (1977), in Scanning Electron Micro-

scopy/1977, Ed. Om Johari, lIT Res. Inst., Chicago, Vol. 1, 401.

Nagata, F., Matsuda, T., Komoda, T., and Hama, K. (1976), J. Electron
Microsc. 25,237.

Pearce-Percy, H. T. and Cowley, J. M. (1976), Optik 44, 273.

Sellar, J. R. (1977), in High Voltage Electron Microscopy, 1977, Eds. T.

Imura and H. Hashimoto, Japanese Soc. Electron Microsc., Tokyo,
p. 199.

Sellar, J. R. and Cowley, J. M. (1973), in Scanning Electron Micro-

scopy/1973, Ed. Om Johari, lIT Res. Inst., Chicago, p. 143.

Smith, D. J. and Cowley, J. M. (1975), Ultramicroscopy!, 127.

Spence, J. C. H; and Cowley, J. M. (1978), Optik 50, 129.

Spence, J. C. H. and Kolar, H. (1979), in press.

Stone, J. M. (1963), Radiation and Optics, McGraw-Hill Book Co., New

York.

Wu, C. K., Sinclair, R., and Thomas, G. (1978), Metal Trans. 9A, 381.