Hello everyone.
Welcome to this material characterization
course.
From today's class onwards we will discuss
about transmission electron microscopy and
we have found so far the characterization
in terms of X-ray diffraction and also you
used electron microscopy that is scanning
electron microscopy and so on and then we
have enough background to take up this another
advanced variant of microscopy in terms of
electron optics and also in terms of diffraction
physics.
So with all that background we will be able
to appreciate this microscopy technique also
without any problem I believe and nevertheless
I will reemphasize some of the concepts then
and there whether it is diffraction physics
or an electron optical system wherever we
need to emphasize with respect to a transmission
electron microscopy.
So I will just begin the introduction of the
transmission electron microscopy course.
What I will do is I will first introduce a
very general introduction I give about this
technique what is that people expect or what
people want to do with this microscope what
are the information they get.
And then I will touch upon or I would say
that we will refresh whatever the basic diffraction
and then basic physics behind or optics electron
optics once again we will touch upon and then
I will take you to the instrumentation details
how what are the various parts and the functions
especially and then how they facilitate the
imaging system and so on.
Then in the later what I will do is I will
take to the diffraction in tem in much more
detail what are all the possible experiments
one can perform in TEM which exploits a diffraction
phenomena and what kind of information you
get and then I will focus on imaging part
and in imaging we will also discuss about
what kind of contrast mechanisms very briefly
because it is a part of a characterization
course which I would like to finish in 10
to 12 lectures the bay the fundamentals of
all this transmission electron microscope.
So if you look at this in the fundamental
of electron optics I will I would like to
look at this expression is much more carefully.
This kind of expression can be derived from
the de Broglie hypothesis and he showed that
in an energy of the electron can be related
to λ or you can say that you know if the
acceleration voltage increases you can bring
down the λ to the very small value.
Of course this where this expression is value
I mean valued or you can use this if you ignore
the relativistic effect and this is the basic
relation but if you if you do not ignore this
relativistic effect then you need to go to
this kind of an expression because at the
very high voltage the electron travels about
1.5 times the speed of light, so we cannot
ignore this relativistic effect and if you
ignore this relativistic effect then you can
use this relation and then you can approximately
derive λ is equal to 1.22 divided by E to
the power half where E is the electron volts.
Please remember electron volt is the energy
of the electron in acceleration and λ is
in nanometer.
For a 100 KeV electron we find that λ is
approximately equal to 4 pico meter which
is much smaller than the diameter of an atom
and we represent the acceleration accelerating
voltage of the microscope and eV represent
the energy of the electrons scope.
And then I would like to give you some very
brief you know introduction about this images
are the results which you can obtain from
an electron microscope.
Normally what we are interested is in a microstructure
micro structure containing features interms
of you know if it is a metallurgical sample
people are interested in defects and if it
is you know physics and then you are interested
in defect density and atomic positions and
so on.
And so but what you have to keep in mind in
totality I mean the TEM image gives a an average
image interms of the depth.
It does not have the depth sensitivity so
what you are seeing in the slide is the image
of an edge location in a gallium arsenide
layer band of dislocation threads through
the thin specimen from the top to bottom but
remain in focus through the foil thickness,
so what I try to say is from an TEM image
you will not be able to see these features
are in the top surface or the middle of the
surface or of the bottom of the surface.
So it gives what you are seeing is in a actually
a projection which we will see what I mean
by projection and what you are seeing is in
a an average you are we are not able to distinguish
whether these features are lying in the top
or whether this features are lying in the
middle of the foil or bottom of the foil.
So the TEM micrograph will not have that depth
sensitivity that is one information.
And what are the information you will get
from this TEM results in terms of crystallography
you get the basic idea of the crystal structure,
lattice repeat distance, specimen shape, crystallographic
symmetry, analysis of miniscule crystals you
can derive fine group and space group and
so on, so this is the typical electron diffraction
pattern one you can one can get from a TEM
and this is a TEM diffraction pattern from
a thin foil of aluminum lithium copper.
And these are the wide range of information
one can derive from the transmission electron
diffraction pattern.
So another very important information from
this you should know, as I mentioned that
what you are seeing that there is a photograph
where you see that two animals are standing
behind one another but it is appearing as
if the head of the animal is appearing in
the same both sides but which is not the true.
So the depth sensitivity is not available
in the TEM micrograph.
This is a point I would like to emphasis here
you have to be very, you have to keep that
in my information in mind.
All TEM information that we get images, diffraction
patterns, spectra is averaged through the
thickness of the specimen.
The most important information on have to
keep in mind whether it is an image whether
it is a diffraction pattern or a spectra is
averaged through the thickness of the specimen.
What is the thickness the thickness of the
specimen?
How it varies?
We will see when we prepare the sample preparation
and a typical thickness we will arrive at
and how that is affecting the imaging condition
that we will see it in the in the coming classes.
A single TEM image has no depth sensitivity
so this aspect has been illustrated in the
previous couple of slides.
And another important thing is you may get
a very different kind of features like this.
You may consider that one may think that this
is kind of a second phase particle and this
is another coarsening effect of the what precipitates
something like that but you have to be very
careful before we use this TEM results.
This could be simply not at all related to
the material at all it may be due to the some
of the you know irradiation damaged by the
electromagnetic radiation itself.
So unless you have a combination of all the
you know the typical requirement to interpret
the TEM results I will just mention it when
we come to the interpretation in an actual
TM results you have to be very careful you
cannot just unless you are well experienced
in this field it is very difficult to interpret
these results and then looking at this kind
of image you can be very easily misled because
you do not have enough supporting evidence
to prove that these are all second phase particle
or something like that.
So one has to be very careful about presenting
just a one bright field image or a dark field
image something like that and then talking
about a second phase particle and so on which
will be highly you know misleading or may
be completely wrong also.
So you need to have appropriate results in
combinations of a crystallography data through
diffraction and we have to prove the second
phase particle through the simple dark field
imaging and then you have to correlate with
that a bright-field amazing and so on.
So the what I am talking about all this are
different techniques which I will be explaining
it in due course of time.
So what I the information I want to derive
from this slide is you have to be very careful
about talking about the features in the in
a TEM micrograph just putting at the one slide.
You have to have additional information to
talk about the features of what you are seeing
in the micrograph.
Now the depth of focus in electron microscope
is very high which we have already seen it
I will play a small animation what you have
to look at it is this is the simple lens and
then you see that in α and β are the angle
correspond to the object side and an image
side and this crossover is projected here
and you can simply simple geometry you can
derive that αimage is equal to tan αimage
if you incorporate I mean if you consider
this triangle and this distance is Dobj and
this distance is at a dobj.
And if you see this is in a β objective and
for example then if you consider this geometry
triangle then you can derive an expression
like this for α image.
Similarly you can do the β this is distance
Dim and this is an αimage angle and this
is the dim then you can write considering
this triangle βobjective which is approximately
equal to tan βobjective is equal to dobj
by 2 that is this distance half distance divided
by Dobj by 2 this is this distance this half
distance you can write αimage by α and also
α sorry αimage as well as a βobjective
angles.
So what we do with this we can use this relation
to calculate the angular magnification in
the microscope that is MA is equal to αim
by βob this is angular magnification and
then we can also calculate the transverse
magnification which is MT is equal to Dimage
by dobjective and which is nothing but MT
is equal to 1 by MA that is inverse of angular
magnification is your transverse magnification
this is not small relationship so you should
appreciate this you have enough background
to appreciate this now.
And we also know that the depth of focus Dim
is equal to dobjective by βobjective times
MT square and the depth of field is Dobj is
equal to dobjective by βobjective.
So this is see the relation between the depth
of field and depth of focus with respect to
that ray diagram what we have seen.
So you have this background to appreciate
this we have already discussed enough what
is depth of focus and depth of field it is
just to give you a recap.
And now I will just show you a schematic where
you see that in the electron optical systems
they are mostly characterized by small aperture
angles leads to a decisive advantage where
the image focus is concerned.
So you can see that the rays which are converging
here and then converging there that is above
and below the objective you have a finite
distance where the image can be sharply focused
that is in general this schematics in general
displays the very small aperture angle effect
and then the concerned depth of focus.
Where α is the aperture angle and Df that
is this Df the most effective electron beams
spot size.
For a collection semi angle of 10 milli radians
and Dob of 2 A°, equation 2 tells us the
depth of field will be 20 nanometers.
So the Dob, dobjective of 2 A° if you substitute
that into the equation 2 what we have seen
you will get about 20 nanometers.
This means a specimen of this thickness can
all be in focus at the same time.
If you want to see a detail at the 2 angstrom
level we need to use a magnification of about
50,000 X.
Equation 1 tells us that under these conditions
the depth of focus will be about five kilometers.
If we only need to see two nanometers we can
use a magnification of 50,000 X and still
the depth of focus is 5 meters.
So all this ray diagram and the small-small
mathematical expressions illustrates a point
that the depth of focus in an electron microscope
is very high.
And then you will see that the this aspect
has been exploited in the at in a transmission
electron microscope hardware itself where,
you though you will see that image formation
is occurring in the fluorescent screen and
the on the table but your recording system
will be much below where you may have a plate
camera or a film camera or it is a CCD which
is much below but still whatever you are focusing
that image on a fluorescent screen will be
nicely recorded in a CCT camera of the same
focus.
So that is one evidence that the electron
microscopes have a significant depth.
So few more remarks on the depth of focus
the depth of focus is related to the depth
of field through the magnification M, where
D is equal to dM2 divided by α.
Compared to the object plane the extra factor
of M2 for the depth of focus arises because,
the image is larger by a factor M so the ray
intersections define in the image plane move
M times more rapidly than those on the object
plane.
Ray's of different angles that converge at
the same point on the image have mutual angles
M times smaller than what they left the object
plane.
So now we will try to demonstrate whatever
we have just read through it through a schematic.
You just observe that.
This is a glass lens and then this is an object
for example a solid line D1 and then see that
there is a small correction it has to be I
know that these two lines supposed to intersect
and then diverge and it has drawn as a parallel
line it is not true.
It has to be an intersection but I will you
assume that it is intersecting and then diverging
like this and then diverging in this direction
like this.
So you see that suppose if you assume that
the solid red arrow is an original object
and it is being imaged and what you are seeing
here is suppose if you see that you know the
distance d2 is the it is a limit of the blurring
image because the you know which when you
use this when you move this d1 slightly to
the distance d1 with the dotted line then
these two rays will trace like this you can
follow this.
Suppose if you assume that this is my solid
line original object if I move that into slightly
a position d1 then the divergence happens
and then the green ray will trace like this
because of that the distance d2 the blurring
will occur and your d1 is because of the mis-positioning
of the yours the image plane where you have
one here and one here intersection so it is
a mis- position of the image plane so that
causes a d2.
So what we have just seen as a if you go back
you can you can just verify this that is the
depth of focus is related to the depth of
field though it is a M2 times the I mean depth
of field that is that we can prove here.
Suppose if you have this distance is approximately
you know I mean the object here is magnified
here approximately 2.5 times.
This is to the scale and you can see that
if you square up this that is you can see
that 6.3 D1 times the D2 that means the depth
of focus is equal to six point three times
the distance of depth of field depth of focus
is M2 times the D1 so that you can geometrically
prove this illustration clearly shows that
the geometrical demonstration for the factor
M2 it is not M2 it is M2.
So that clearly shows about a depth of focus.
Now we will quickly review the resolution
of the electron lens.
We have already seen this in a fundamental
of electron optical system and just to recap
you would like to go through this.
Resolution is defined as minimum solvable
distance and then if you consider the theoretical
resolution if there is no abrasion at all
the resolution of any lens i.e, glass or electromagnetic
is customarily defined in terms of Rayleigh
criterion which is also a practical definition.
The criterion gives us a merit in terms of
the eye’s ability to distinguish images
of two self-luminous incoherent point sources.
A single point source will not be imaged as
a point even if no abrasions or astigmatism
are present.
I will play this animation for this two self-luminous
point sources which are trying to converge
and then these two point sources will be recognized
as a independent source only with the distance
of 0.61 times the lambda that we have already
seen it.
So you just recall that the earlier discussion
the finite size of the lens results in diffraction
of the rays at the outermost collection angle
of the lens usually by limiting aperture.
This diffraction results in a point being
imaged as a disc called the Airy disc.
So remember this is an Airy disc which we
have already seen it.
So I will not discuss that further which has
a cross section intensity profile.
And Rayleigh stated that if the maximum from
one source lies over the first minimum from
the other source then the overall intensity
profile exhibits a dip in a middle at about
80% of Imax so this also we have seen.
So the minimum of the next source will be
matching with the maximum of the first source.
So and this dip will occur at about 80% of
the Imax.
So this also we have seen previously.
The eye can discern this dip as two overlapping
images, thus indicating the presence of two
separable sorry, two separate objects.
Under these circumstances the distance apart
of two incoherent point sources is defined
as theoretical resolution of the lens and
rth and it is given by the radius of the Airy
disk or that is theoretical resolution is
equal to 0.61times λ by β.
And we have also seen about the spherical
aberration where Cs is the constant for a
particular lens called spherical aberration
constant and b is a semi angle of collection
in the objective lens the resolution of the
object is given by some combination of the
Rayleigh criterion and the aberration error.
So we will now look at the some treatment
by Hawkes gives the particularly a clear description
of how this combination leads to a value for
a resolution in a microscope.
So suppose if you include the spherical aberration
coefficient how it is going to be.
So this is a Hawkes’s treatment.
Suppose if you assume that we are taking a
spherical aberration into Rayleigh criterion
and take the combination of Rayleigh and spherical
aberration disks in the quadrature rth is
equal to r square th + r square spherical
aberration because of where r due to spherical
aberration is called (rsph2) to the power
1/2 we can now thus find how r varies with
βusing this relation or as a function of
β is equal to 0.61 times (λ by β)2 + Csβ3,
I mean to the power 2 whole to the power 1
by half but being the square root of all this
expression.
Since the two terms vary differently with
the aperture collection semi angle β a compromise
value exists when.
dr(β)/dβ=0 is if you can differentiate that
expression you will get this kind of value.
From this equation the optimum value of β
can be obtained like this βopt is equal to
0.77 times λ to the power 1 by 4 by Cs to
the power 1 by 4 so this is a called a spherical
aberration limited resolution.
And for 100 KeV electrons a λ is 0.0037 nanometers
for an instrument with a Cs is equal to 3
mm gives a βopt value of 4.5 milli radian.
So you have rmin is equal to 0.91 times (Cs
λ3) to the power 1 by 4 this expression that
gives the practical resolution of the microscope,
typically the value for the rmin is 0.25 to
0.3 nanometers but for high-resolution instruments
have the rmin which is approximately equal
to 0.15 nanometers.
So what we are now trying to say is we have
already stated that spherical aberration is
very important aberration which is very difficult
to eliminate from the lens.
So if you keep that spherical aberration into
a system and then how the resolution is getting
modified that is the bottom line and these
are all the small I mean steps or derivation
which demonstrates to you what is the how
the resolution expression get modified as
well as the how the β angle getting optimized.
So it is that is a basic information nothing
to get confused here.
So now we will again go back to some of the
basics.
Look at this animation what you are seeing
is suppose if you assume that this is a thin
specimen which being irradiated by the electron
beam and then your transmitted beam will have
or if the image of the specimen will have
an oscillation in the intensity that is scattered
electrons with varying intensity you will
see and also you have the incident beam you
have a diffraction pattern as well as the
forward scattered beam of electrons.
So these two you are going to get in the transmission
electron microscopy where you have a thin
specimen is placed.
So what you are seeing here is scattering
within the specimen changes both the spatial
and angular distribution of emerging electrons.
So that is the idea you have to appreciate
this the scattering within this thin specimen
changes both spatial angular distribution
of emerging electrons.
So that is that and you can see other schematic.
So the schematic is self-explanatory so you
have a background to understand this.
So a coherent incident beam is falling on
the thin specimen then you have backscattered
electron secondary electron and coherent elastic
scattering scattered electrons and then you
have direct beam, incoherent in a in elastic
scattered electrons and so on.
So if it is a bulk specimen there is nothing
like you see the forward scattered, I mean
signals forward scattered signals only you
get the backward scattered signals.
Only a thin specimens permits electrons to
be scattered in both the forward and backward
directions while the bulk specimen only back
scatters the incident beam electrons.
So very fundamental idea you know it but you
have to why we are saying this because in
transmission electron microscopy we use only
the forward scattered electrons we do not
look at the backward scattered electrons.
We will quickly rush through this basic idea
again once again.
Elastic scattering is usually coherent if
the specimen is thin and crystalline.
Elastic scattering usually occurs at relatively
low angles one to ten degrees that is in the
forward direction at higher angles for example
greater than 10 degrees elastic scattering
becomes more incoherent, inelastic scattering
is almost always incoherent and relatively
low angle that is less than one degree.
As the specimen gets thicker less electrons
are forwarded forward scattered and the more
are backward scattered until the primary incoherent
back scattering is detectable in bulk non-transparent
participants.
So this point you have to remember.
Forward scattering causes most of the signals
used in the TEM.
So the convenient definition of small angle
is about 10 milli radians in TEM.
We can control the angle of incidence of electrons
on the specimen and we will define the semi
angle of incidence as α.
In the TEM we use apertures and detectors
to collect the collect a certain fraction
of scattered electrons and we will define
any semi angle of collections as β.
We will define all this scattering semi angles
controlled by the specimen as θ and this
may be a specific angle such as twice the
Bragg’s angle where θ is equal to 2θB
or a general scattering semi angle θ.
So again you can look at the schematic how
the electron beam comes and this is an α
whatever we have just stated in the previous
slide you can just look at them as a schematic.
This is an α beam converging semi angle and
this is a specimen and then you have general
scattering angle θ and the collection semi
angle is β and this is your aperture and
this is an optic axis.
So I can play it again.
So that takes care of all the definitions
in a TEM and these are all the typical diffraction
pattern one get in a TEM and you should know
as a beginner what is the difference between
all four of them.
What you are seeing is a diffused ring which
typically comes from an amorphous material
and this is a single crystal electron diffraction
pattern and this is a poly crystalline single
I mean some poly crystalline electron diffraction
pattern as a ring sharp rings and this is
convergent beam electron diffraction.
So we will explain I will explain all these
things when we discuss the diffraction in
a TEM and what is the reason you see this
kind of a pattern that also will be discussed
in detail.
And I am just trying to give you an introduction
introductory feel that what kind of diffraction
you will be able to get from these transmission
electron microscopy.
So you have these four typical types of electron
diffraction is possible and then they are
very powerful I mean information it gives
you can derive little more significant micro
structural aspects from this diffraction pattern.
So we will go through them when we come to
that section.
And little more fundamentals again.
Let us again a recap the atomic scattering
factor f(θ) which is elastic.
f(θ) is a measure of amplitude of an electron
wave scattered from an isolated atom is proportional
to the scattered intensity.
f(θ) depend sin λ θ and Z.
It decreases as θ increases and it decreases
as λ decreases and it increases with Z for
any value of θ.
So we have discussed this aspects while discussing
the X-ray diffraction.
So you have enough background for this.
So I will skip this.
Which all inelastic process are occur in the
TEM?
Process that generate X-rays process that
generate other electrons something like secondary
electrons, processes that result from collective
interaction with many atoms.
There is a type of here atoms.
So these are all the general inelastic process
in a TEM.
And now you recall this animation I will introduce
an instrument through this animation.
What you are seeing is an electron source
first where you have the high voltage applied
and then you have anode, there is an aperture,
it is a condenser lens and then you have a
specimen, you have an objective lens and follows
by an aperture, intermediate lenses, projector
lenses and then final screen and what you
have seen is how the electron beam comes through
various apertures and lenses and falls on
the specimen and it produces some signals,
secondary signals.
And then it further transmits through some
of the electromagnetic lenses and apertures
and it falls on the.
So you have the you are now familiar with
this kind of an electromagnetic lines we have
already seen the functions of this and how
they are exploited here.
And also we have seen that you can look at
the corresponding light optical system where,
you have the condenser of condenser lens and
you have the specimen you have objective lens
and you have a projector lens and then screen.
So you have one is to one comparison with
the light optical system.
So both have almost similar I would say the
ray diagram except that they are all electromagnetic
lenses here it is.
So the convergence angles α are so small
that the ray diagrams are drawn with highly
exaggerated angles and while the beam in the
figure is not exactly the parallel to the
optic axis α under this condition is less
than I mean less than 10 to 4 radians that
is 0.0057 degree which is effectively a parallel
beam.
And then we will look at the electron sources
we will start with the electron sources.
This also we have seen it in the introduction
just for the sake of completion I will just
go through this.
TEM will use a thermionic source or a field
emission source and the two cannot be interchanged.
Field emission source gives monochromatic
electrons, the thermionic source are less
monochromatic in nature.
And this is the typical I mean electron source
or gun design.
You have this electron gun this is a Wehnelt
cylinder this is an actual photograph and
this is an optic axis and you have the filament
here and then you have the anode so corresponding
anode is shown here and you have the gun crossover
and applied voltages there and we have looked
at the function of this lens I mean the electron
source and it is designed earlier also.
A high voltage is placed between the filament
and the anode modified by the potential on
the Wehnelt cylinder which acts to focus the
electrons into a crossover with a diameter
d0 and the divergent divergence angle α0.
And then if you look at the thermionic sources,
for example it is a tungsten hairpin the tip
of tungsten hairpin filament and the distribution
of electrons when the filament is under saturated
and misaligned and then saturated aligned.
So you have this is an under-saturated and
misaligned beam will look like on the screen
and you have the under saturated aligned beam
will look like this and this is the saturated
beam.
So we will look at this when we you will evidence
this action while we operate the microscope.
And this is another thermionic source LaB6
crystal and the electron distribution when
the source is under saturated and aligned,
c is a saturated beam which will appear like
this.
This is a field emission gun tip.
Electron pass from the field emission source
showing a how a fine crossover is formed by
two anodes acting as an electromagnetic lens.
Anode one provides their extraction voltage
to pull the electrons out of the tip, anode
two accelerates the electrons to 100KV or
more or whichever is designed.
So again we are looking at a second time we
have discussed this.
So we use apertures in the lens lenses to
control the beam current and the convergence
of the beam hitting the specimen.
All lenses are imperfect, insofar as they
cannot get all the radiation emitted by an
object and so we can never create a perfect
image.
The image formed after each lens is rotated
by 180o with respect to the object.
We will see how this aspect is taken care
in the modern microscope when we discuss the
image and so on image formation in the TEM.
And a typical electromagnetic lens is shown
in the schematic.
You can see that these are all the copper
coils which is this is a, cross-section of
a electromagnetic lines which I have shown
in the fundamentals of electron optical system
as well.
So you have the soft iron pole pieces and
this is the this is a bore and this is the
gap and then you have the water inlet and
outlet for the cooling and this is an optic
axis, electron optic system.
The pole pieces surround the coils and when
viewed in a cross-section the bore and the
gap between the whole pieces are visible.
The magnetic field is weakest on the axis
and increases in strength towards the side
of the pole piece so the electrons are more
strongly deflected as they travel off axis.
So you can see that is why the schematic is
shown in this manner is because of this effect.
The bore to gap ratio is another important
characteristic of such lenses controlling
the focusing action of the lens.
When we pass a current through the coil a
magnetic field is created in the bore this
field is inhomogeneous along the length of
the lens but axially symmetric.
The strength of the field in a magnetic lens
control the ray path.
So though we have we are going through this
we have already seen the basic function of
an electromagnetic lines.
Just for the sake of completion and the recollection
I am doing this.
So we will continue to look at the some of
the instrumental details and then we will
go to the diffraction in TEM in much more
detailed manner.
So we will continue our lecture in the next
class.
Thank you.
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