Instructions and comments on what to read in Williams books I – III
OVERVIEW Ch 1- 3
When you read through chapters 1-3 in Williams book, pay special attention to:
Abbreviation and acronym list before the Contents part – very useful!
Microscopist love acronyms!
Section 1.4, including the formulas
The concepts “mass thickness contrast”, “mean free path”
Section 3.7-3.9, atomic scattering factors, f, and how these are
assembled to structure factors, F. Equivalent to x-ray formalism.
Note that the more general von Laue equations for diffraction give the
same result as the famous Bragg’s law (3.22).
Do not get hung up on all equations! Many of them apply more to the diffraction
part than the HREM part, which is where we put the emphasis in this course.
You can safely leave many of them for the moment, until you really need them.
The concept of phasor diagrams (in 2.10) is overworked for our purposes – skip.
OVERVIEW Ch 5 - 8
During this step we are going to focus on the design of a modern High
Resolution Transmission Electron Microscope (HRTEM), in other words on the
hard ware. It is mainly presented in the Carter/Williams course book in the
following chapters:
Chapter 5: Electron sources
Chapter 6: Lenses, apertures and resolution
Chapter 7: How to "see" electrons
Chapter 8: Pumps and holders.
We are going to read the chapters with the main attention on the HRTEM
aspects, and the following parts of the chapters would be appropriate:
Chapter 5: 1, 2, 3, 4.
The different ways of emission electrons.
Two types of gun designs.
Chapter 6: 1, 3, 4, 7.
Different types of electron lenses.
Apertures in the TEM.
Depth of focus and fields.
Chapter 7: 7.2, 7.3.C, 7.5.
The viewing screen of the TEM.
TV cameras and Slow-scan CCD.
The dirty photographic emulsion.
Chapter 8: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
The different vacuums and its pumps.
What kind of specimen holder we could use.
Overview Diffraction Ch 9 - 21
Since this a HREM course, we will only pick the most necessary diffraction
parts. If you need it, brush up your Fourier transform knowledge by reading the
sheets on different FT.
Try to get through chapters 9-21 (sounds like a lot, but it isn't) in Williams book,
paying special attention to:
Ch. 9.1 A,B (Book I)
Figure 9.1 B is also known as Köhler illumination. In
figure 9.2 you see the effect of inserting an aperture; the convergence angle 
gets smaller. From the assignments of week 1 you remember that this
increases the depth of focus. It has also a crucial effect on the HREM resolution,
which ideally works with  =0 (parallel beam). We will look at this in the contrast
transfer function later. Standard value when recording real HREM images is
=0.5 mrad (small!).
Ch 9.3 A
Ch 9.6 B This is something you need for routine verification of what your
sample is, and what structure it has. Note that the Camera constant, L,
includes the wavelength, so if you change the accelerating voltage, it is
no longer valid.
Formula 9.2 is often written as d = L / d*, where d is the distance in
real space (in the crystal), in the order of 1 Å, d* is a distance in
reciprocal space, measured with a ruler in mm on the negative from the
microscope,  is calculated from the HT in picometers, and L is the
camera length if no lenses were present, and is about 1 m. Alas,
convert everything to SI units, until you get a feeling for what might be
a normal value.
BOOK II, Diffraction
Skip Ch 11-15. Read them when you need them for a specific case.
Ch 16 Read. Good to have seen where the extinction conditions in the
International Tables comes from. Close packings are very relevant for metal
particles. F=0 means that there will be no intensity (forbidden spot)
Skip Ch 17 until you need it. Fig. 17.9 possibly of interest (compare Fourier
transforms)
Ch 18 Skip, but: make sure you can construct a list like Table 18.1 of allowed
reflections using your knowledge from Ch 16. This should account for all the
rings you find in powder ring patterns from e.g. cluster samples.
Ch 19-21: Skip. Generally, if your sample is thick enough to give you Kikuchi
lines or details inside the CBED discs – forget about HREM!
PRACTICAL EXERCISE:
Those who have easy access to a TEM: (others look at fig 11.14)
1. Obtain a selected area diffractogram according to the procedure in 11.9.
2. Use the Diffraction Focus (intermediate lenses) to expand the spot to a
disc. Note a few things:
a) All spots expands to images of the illuminated area (limited by the SAD
aperture), thus, each spot contains information from the whole object
b) The central beam is very close to an absorption contrast image (strong
scattering=dark).
c) If you have a pattern with strong spots, the contrast missing in the
central beam can be found in the diffracted beam (complementary
images, “dark field images”). In a HREM image (phase contrast) all
these images are recombined into one
d) The imaged area further away from the central beam is slightly shifted
(due to spherical abberation). This means that you might get parts of
diffraction patterns from crystals just outside the selected area aperture
e) A simple but useful trick: When you have several small crystals inside
the SAD aperture (eventhough using the smallest one), and thus
several overlapping patterns, of which one is promising for HREM,
defocus the DP slightly, so you can just about see the image inside the
central disc, but still also a diffuse pattern at the same time. If you
move the specimen, the good crystal is the one which just leaves the
image when the good diffraction pattern goes out. Now you know which
particle to zoom in on.
HREM Imaging, Book III
You should focus your reading on the following sections:
Chapter 22: 1, 2, 3, 4, 5, 5A (skip 22.3C and 22.3E)
What is contrast?
Principles of image contrast.
Mass-thickness contrast.
TEM diffraction contrast (Two-beam conditions)
Chapter 27: 1, 2, 3, 4, 5, 7(not A and B)
Introduction.
The origin of lattice fringes.
Some practical aspects of lattice fringes.
On-axis lattice-fringe imaging.
Moiré-patterns.
Fresnel contrast
Chapter 28: 1, 2, 3, 4, 5, 6, 7
The role of an optical system
The radio analogy
The specimen
Applying the WPOA to the TEM
The transfer function
More on c (u), sinc (u), and cosc (u)
Scherzer defocus
This will probably give you something think about. But remember, this is the
basis for all HRTEM.
# During this step we are entering the chapters that are specific to highresolution transmission electron microscopy (HRTEM). The sections to read
are given in the outline.
Chapter 22
Chapter 22 introduces you to "Imaging in the TEM", and the difference
between amplitude and phase contrast is mentioned. But then this chapter
only deals with amplitude contrast and more specifically mass-thickness
contrast and diffraction contrast. There is no discussion about phase contrast
in this chapter. And although it is said that both amplitude and phase contrast
are present in a TEM image, phase contrast is dominating the traditional
HRTEM images. So…
Read chapter 22 (the sections mentioned in the outline) and be aware that
these contrast mechanisms are present in any image you will record. To what
extent is mainly dependent on the microscope setting and the specimen
(orientation, crystallinity…).
Some general comments to chapter 22:
In section 22.1 it is said that from digitally recorded images you can
enhance contrast electronically. And this is correct. BUT this can also be done
on an analogous TV-signal. You can of course also do similar tricks in the
photo lab with images printed from negatives, and do not forget that negatives
can be scanned. A scanned negative will give you a digitised image and then
you have access to all the fancy software again.
In the same section, the relation between intensity and contrast is
discussed. Remember that there is no such thing as "dark contrast".
Spend some time thinking about the fact that selecting one (or more)
beams by let's say an aperture will increase the contrast in the resulting
image.
Take a look at fig 22.2. Will setting A in ever give a HRTEM image? The
answer will hopefully be clear at the end of the week.
Please note that the arguments in the section about mass-thickness
contrast assume no diffraction contrast (e.g. an amorphous specimen). In
most practical cases (e.g. small crystalline particles) you will have both kinds
of contrast.
Think about the expressions "mass-thickness" contrast and "absorption"
contrast (section 22.3A).
Think about the contrast mechanism for mass-thickness contrast (e.g. fig
22.4). Larger aperture will give more scattered electron thus less massthickness contrast.
Ch 22.4
The combination of STEM and HAADF is very powerful these days when
analytical (S)TEM:s have a probe size with the size of an atom. The resolution
in HAADF STEM is actually better (0.14 nm) than the conventional HREM
point resolution (0.16 nm) on the Jeol 3000F microscope.
MASS THICKNESS (incoherent elastic scattering)
DIFFRACTION CONTRAST (coherent elastic scattering)
What is coherent/incoherent? If you do not remember check section 2.2.
In the first section you are told to "underfocus C2 to spread the beam". C2
is the second condenser lens known as brightness on JEOL machines and
intensity on Philips machines. This way to operate the microscope is probably
true for most cases, but the optimum way to focus C2 is actually dependent
on the lens design. So get to know your own machine and then figure out how
to spread the beam.
Chapter 27 deals with another contrast mechanism, phase-contrast. This is
the mechanism that mainly contributes to HRTEM images. It explains why we
can see lattice fringes and images with patterns that in some cases can be
directly related to the structure of a material.
Some general comments to chapter 27:
In section 27.2 some equations are introduced referring to chapter 13. This
is a chapter we have not covered earlier. Try just to roughly follow the
arguments so that you accept the statement in the grey block at the end of
this section.
Two parameters that appear in the equations are:
s= excitation error; seff=effective excitation error
These two parameters are defined in section 12.6 if you are interested. This
is one of the parts that have not been covered earlier but you were told to go
back when you need this information for a specific case.
Moiré patterns are nice to know about. They will appear when two
crystalline objects are imaged superimposed. You can create your own moirépatterns by drawing lines with varying spacings on transparencies and then
putting them on top of each other on a projector.
Be aware of Fresnel fringes. They appear whenever an interface between
two objects that changes the phase of the electron beam in different ways is
imaged together. They are focus-dependent!
Chapter 28 is THE chapter dealing with HRTEM. Now you should be aware
of the different contrast mechanisms. Next step is to carefully read the
CHAPTER PREVIEW. This section summarises what we are after in HRTEM.
What I would like to achieve here is for you to get a feeling for what information
we can extract from the image and how it is done. Try to read through the
chapter and get a general feeling for to process. Do not get stuck by trying to
check all equations. You are of course free to do that if you like. You will get a
simulation program for this rather complicated, but very important
concept of the CTF, Contrast Transfer Function. It is so complicated that
Olle used Excel as a simulation software. Others are available on the net.
Some general comments to chapter 28:
There are some things we need to agree on. One is that each point in the
specimen is somehow transformed into an extended region in the image,
which is the basis for equation 28.1.
The extended region in the image corresponding to a point in the specimen,
g (r), is given by the point, f (r), convoluted with the point spread function,
h(r-r'). Remember convolution? Otherwise go back to the notes about Fouriertransforms.
Read The radio analogy! It is simple and relates to something in every-day
life, yet it explains some central concepts in image formation.
So now we need to agree again. In the diffraction plane (the back-focal
plane) we will find the diffraction pattern. It contains all information about the
specimen, the higher the frequencies the further away from the central beam.
Please compare with the notes on Fourier transforms again.
The information in the back-focal plane then has to be transformed back
into the image. This is done by the objective lens, or as it might be put, "the
microscope does the maths for you". This sounds good but the problem is that
the objective lens not only "does the maths", it does some other things as
well. It introduces artefacts since the electromagnetic lens is far from perfect.
The function that describes how the information is transferred into the image
is denoted H(u) (where u is a reciprocal space vector) by Carter and Williams.
This is the contrast transfer function.
The contrast transfer function consists of the aperture function, the
envelope function and the aberration function.
Try to understand the meaning of the weak-phase object approximation
(WPOA). This is something to bear in mind. Transfer function theory is only
valid for weak phase objects. But still in practise it is very useful for most highresolution applications.
When applying the WPOA to the TEM, Carter and Williams switches
terminology. They define a new quantity T(u) that is very similar to H(u). The
difference is explained in 28.4. Note the note on terminology!
The transfer function describes how each spatial frequency (each
reciprocal lattice vector) in the exit wave function is transferred into the image
(intensity and phase). It actually shows how the objective lens acts as a filter
(compare with The radio analogy, 28.2).
Try to follow how Carter and Williams reach equation 28.34. This equation
is what microscopists usually mean when they talk about the contrast transfer
function. The parameters are:
Δf: focus, a negative sign usually means underfocus. Usually
around 0 to -100 nm.
λ: the wavelength
u: the reciprocal space vector
Cs: the spherical aberration constant. Measure of lens-quality.
Usually around 0.5 to 1 mm for HRTEM.
In 1949 Otto Scherzer deduced an expression for the optimum focus
setting that would give an extended region of transfer without zeros and
without too much loss of intensity. This setting is usually called Scherzer
defocus (eq. 28.35). Using this setting most of the information transferred to
the image will have reasonable intensity and have the "right" phase. This
should result in dark spots for areas of the specimen with high potential i.e.
where the atoms are. In such cases intuitive interpretation is easier.
If you read different textbooks you will see different expressions for
Scherzer defocus such as:
Δf= -(CsΔ)½
Scherzer focus in the provided Excel spreadsheet
Δf= -1.2(Csλ)½ Scherzer focus in the book
Δf= -(1.5)½(Csλ)½ Optimum focus in the provided Excel spreadsheet
- The last expression is sometimes called optimum focus and the first one is
as far as I know what Scherzer actually wrote in his paper. They are all valid.
It all depends on how much intensity transfer you are willing to sacrifice in
order to extend the resolution out in reciprocal space. You will get a chance to
play around with the transfer function during the exercise.
The transfer function we have deduced so far is that for a perfectly stable
lens acting on a parallel beam. This is hardly the case in reality. During the
next step, more of reality will be added to the transfer function starting on
section 28.8.
Ch 28.8
If we just regard the phase shift on the spatial frequencies in
reciprocal space by the CTF of the imaginary microscope, it would
continue infinitely for higher and higher frequencies, but oscillate
between positive and negative phase contrast faster and faster. In
reality, instrumental instabilities and chromatic abberration will
damp out higher frequencies, by forming an envelope on the CTF
(multiplied). The chromatic envelope (Williams: Ec; Olle: B(delta)
“Temporal coherence”) is due to instabilities in the lens currents
and high tension and the chromatic abberration in the lenses (each
“colour” of electrons have their own image plane).
The spatial coherence envelope (Williams: Ea; Olle: B(alpha)) is due to the
coherence of the electron source and is very much affected by the convergence
semi-angle, alpha.
Read 28.8
The chromatic envelope (Williams: Ec; Excel-sheet: B(delta))
The spatial coherence envelope (Williams: Ea; Excel-sheet: B(alpha))
These are the "classical"envelope functions. However, new electron sources like
the FEG and Cs correctors are changing this, and we may have to introduce
new ones in a near future.
Read 29.9
Minimum contrast! Passbands rarely used for straight-forward, interpretable
HREM
Go back and read 6.5
The abberrations: spherical, chromatic and astigmatism
Figure 30.5
How to recognise astigmatism in the Fourier transform
Ch 28.11 The Rose Cs corrector
The Rose Cs corrector was first built into a 200 kV FEGTEM in University labs
in Germany (Heider, M. et al, Nature 392 (1998) 768) and in a STEM in
USA. It was predicted by Scherzer 50 years ago (Scherzer, O., Optik 2,
(1947) 114-132). Now commercially available. This is a big step forward for
HREM!!!
Read 28.13
FEGTEMs and the resolution limit. Formulas 28.49 and 28 50 defines the
chromatic envelope. The delta E in 28.50 is about 1.5 eV for a LaB6 filament
and 0.3 eV for a FEG.
Ch 29.9
Note that the ONLY focus you can easily find with certainty and accuracy for an
amorphous thin film, is the minimum contrast focus, which is not the same as
zero defocus! It is 0.44 x Scherzer defocus, i.e. 17.8 nm for the 4000EX.
Remember this when you count clicks on the focus knob to find optimum
defocus in practise.
Below you will find a calculated CTF for the 4000EX microscope. The thickest
line is the damped transfer function. Rs is indicating the Scherzer resolution, or
point resolution. Note that the curve must be drawn for the optimum defocus.
Practically, you can find this by recording a HREM image of an amorphous
object, Fourier transform it, and at the frequency corresponding to the point
resolution, you will find a dark ring, since the contrast transfer is here zero for
the first time (starting from the center). Out to here, we have correct phase of the
beams contributing to the image.
The information limit (IL) is the highest frequency that can be transmitted by the
microscope, ignoring the phase of the electrons, an thus not getting a directly
interpretable image (atoms black). Especially for FEGs, the information goes to
much higher resolution than the Scherzer resolution, but it needs complicated
mathematics to make use of it.