Hello, welcome back to this Material Characterization
Course. In the last few classes we have just
reviewed all this Optical Microscopy variants
and it is working principles and live demonstrations
so on. Now, we will move on to the next domain
of Electron Microscopes. Like, I did it in
optical microscopy first let us review some
of the fundamentals of Electron Optics which
will be useful to understand the electron
optical system as well as electron lenses
design and its operation methods.
So far we have just looked at the light optical
rules and then we will see how these light
optical rules will be applicable to the electron
optical system. In this few lectures of Fundamentals
of Electron Optics, we will try to build a
background to appreciate the electron lenses
and their application to electron optical
system, and then we will also review the aberrations
which are encountered in this electron lenses
and then how to correct them in order to obtain
a better resolution of the microscope. So,
with these intentions in mind let us begin
our Fundamentals of Electron Optics lecture
with few remarks.
In fact, the paths of electrons in an electric
or magnetic field are identical to the ray
paths which is associated with light, where
glass lenses are the refractive medium. In
fact, this approach was first made by some
of the German scientists who applied this
analogy of the light optical system to the
dynamics of electron in the electron optical
system. In the case of an electric or magnetic
field, however, the refractive index is at
any point depends on the corresponding field
strengths. We will see how this is valid for
the actual electron optical system.
We will first discuss Electrostatic Lenses,
because the electrostatic lenses where the
first used in the electron microscope and
then their design and behavior were studied
then only this was adapted to electromagnetic
lenses. So let us review some of the primary
features or the theoretical concepts underlying
this electrostatic lenses. An electron beam
passing from a region of low potential V 1
to higher potential V 2 is on acceleration
observed to undergo refraction as defined
by Snell's law. Sin r by sin i equal to square
root of V 1 by V 2. We know that the Snell's
law which we have reviewed in the fundamentals
of optical microscopic system, so similar
thing is obeyed by this electron optical system
as well. This equation clearly mentions that;
this clearly demonstrate that your electron
beam also undergo a refraction according to
Snell's law.
Look at this schematic where we are demonstrating
the refraction and reflection of electron
beam on encountering the region of potential
difference. You see these two diagrams, first
we will describe this. First one, look at
this electron beam is encountering the potential
difference by this electrostatic lens where
V 1 is less than V 2 and then it undergoes
refraction, so where i is the angle of incidence,
r is the angle of refraction. On the other
hand, if you see that this is a electron beam
encountering the two electrostatic lenses
where the potential is reversed, where V 1
is greater than V 2 then your electron beam
undergo a reflection like this and then you
have the refraction also taking place in this
manner. We will see under what condition these
two are happening.
The electron beam on passing through a region
of potential difference with V 1 is greater
than V 2 experiences a retardation making
angle of refraction greater than angle of
incidence.
This is what we have just seen. So, where
i is very large then these two conditions
are valid. So for the refraction sin r by
sin i equal to square root of V 1 by V 2,
where i is smaller than sin inverse times,
square root of V 1 by V 2. For the reflection
where r prime is equal to i, where i is greater
than sign inverse times, square root of V
1 by V 2, where r is the angle of reflection
from the plane of potential zone. We will
go back and then see. So the plane of potential
zone which we referring somewhere here, and
then you see that i is equal to r i when the
reflection is considered. So, with this we
simply see that the electron beam exactly
follows the rules of a light optical system
and we will see what the additional points
are, we need to consider.
This schematic clearly shows that the cylindrical
electrostatic lens action, what you see is?
You see this electron beam coming and then
the diverged beam is going through this the
electrostatic field and then it is getting
converged. So, the converging action of this
electrostatic lenses very clearly demonstrated
in this schematic. So, an electrostatic lens
for V 1 is less than V 2 is thus observed
to act in an identical fashion to glass lenses
with respect to the focusing action on a divergent
electron beam. So this is what is clearly
demonstrated in this schematic.
Now, as I just mentioned before, the electrostatic
lenses where the one first developed for the
electron microscope and you can see in this
schematic that it is exactly analogous to
a glass lens system. So you see where a light
is coming and falling on this glass and then
it is converged in the right hand side, and
here you have this electrostatic lenses here
again the converging action is demonstrated.
In fact, the focal length, the front and back
focal length of this two lenses I mean in
this each system are equal. Hence, we will
see that that lens equation is exactly valid
in this electron optical system as well.
What I am going to show in this schematic
is, you see these are all some of the electrostatic
lens design for the Cathode-lens microscope,
and what you are seeing is a uni-potential
electrostatic lenses for a fixed focal length.
In this schematic it is clearly shown this
is for a fixed focal length. I can play this
schematic for you just to have a better capture
of the concept you see that electrostatic
lens and then the electron beam is forming
entering into this electrostatic field, and
you see that f focal length is fixed in this
situation.
In the second case, it is a variable focal
length where you have the combination of electrostatic
lenses for different field strength you can
also vary this focal length f 1 and f 2. You
can see that the first one coming through
f 1 point is lying are meeting at A 1 and
B 1 in the image plane and then the beam passing
through f 2 is falling on the image plane
in the point A 2 and B 2. So, you have the
variable focal length electron optical system
is demonstrated and what you see in the right
hand side is a simple right optical analog.
I just want to make sure that the electron
optical system is exactly what we have in
a light optical analog. You should not get
confused just because we are replacing this
light, I mean light optical system where we
use a glass lens as the refractive medium
instead of this refractive medium in an electron
optical system you have electrostatic lenses.
I hope this schematic gives you a nice comparison
between these light optical system as well
as the electron optical system, where the
electrostatic lenses are used or the cathode-lens
designs are adopted.
The electrostatic lenses we just discussed
about where, the electrostatic unipotential
electrons lenses the most useful for the incorporation
into a general electron optical system since
it is essentially analogous in function to
a single converging glass lens in a light-optical
system. This is what just we have seen. What
is unipotential lens? In unipotential lens,
the image and the object regions of the lens
are at the same potential with the consequence
that the refractive index is constant. So,
as I just mentioned that the front and back
focal length or I would say that the focal
length in the front and back focal plane are
same.
So, the focal length f is related to the object
image geometry in the form 1 by f is equal
to 1 by p plus 1 by q. The refractive power
of the unipotential lenses expressed by approximately
1 by f equals 3 by 16 times the integral from
z 0 to z 1 times V c by v 0 whole square dz,
which is function of the field strength.
I think with this few introductions to the
electrostatic lenses, we will now look at
how the electromagnetic lenses are being developed
into the modern electron microscopes. Since,
electrostatic lenses are analogous to the
optical system the same electrostatic lenses
also or I would say the electrostatic lens
design is adapted to electromagnetic lens.
Let us see how it goes.
The electromagnetic lenses are analogous to
the unipotential electrostatic lenses, which
are fundamentally analogous to a glass converging
lens in a light optical system. So, what that
we have to now understand is what this additional
magnetic field does to the electron path or
beam of electrons? Let us see, the action
of magnetic field on electrons is that any
deflection the electron experiences is proportional
to it is charge and mass. The magnetic field
exerts a force on a moving electron in a direction
normal to both the field and the propagation
direction of the electron. So what you have
to understand here is, the magnetic field
is going to produce an additional force in
a direction normal to both the propagation
and field direction of the electron. So, it
is perpendicular to both.
This is demonstrated in this a schematic,
you see this is the typical cylindrical type
electromagnetic lens action, it is a cross
section where you have all the circular slots
where a soft iron coil is being bound like
this, and this is the electron beam getting
into this a core of the lens and then you
see the field which is being generated, and
then you see all the electron beam is converging.
The magnetic field produces a force normal
to this field direction as well as the propagation
of the electron, so that means perpendicular
to this direction. So that produces a field
like this and which will have a kind of a
cylindrical shape with the radius r. We will
see how this is perceived.
Thus a magnetic field acting in a direction
parallel to an electron beam will not affect
it, while a field normal to the beam will
cause it to describe a circle with the radius
given by r 0 is equal to 1 by B square root
of 2 m V 0 divided by e. Where, r 0 is in
centimeters for V 0, the acceleration potential
in volts, and B is the magnetic field strength
in gauss. In effect, the electron in a uniform
magnetic field will describe a helical path,
please make a note of this. In a uniform magnetic
field describe a helical path, with a radial
extent limited by r 0. So what you have to
remember is this, this is r where you have
the circular beam are field is represented
around this region.
Now we will see how the other parameters are
getting affected? The refractive power of
the electromagnetic lens is given by 1 by
f is equal to 0.022 by V 0 times the integral
of from z 0 to z i H square dz. Where, V 0
is the potential through which the electrons
converging on the lens have been accelerated
and H is the magnetic field strength on the
z axis in gauss.
The field strength is related to the physical
design of the lens coil by 4 pi N I by 10
is equal to integral of z 0 equal to minus
infinity to z i equal to infinity H dz. From
which we can observe that the lens power is
proportional not only to the number of turns
N of the conductor, and the current flow I
but also to the extent of the field region.
Now, it is very clear from this expression
you can understand this, I will go back to
this you can understand the typical electromagnetic
lens and the number of coils which is being
used to produce this magnetic field in this
kind of a slotting system is going to be also
a function of your the magnetic field strength.
From hence forth in electron microscope you
are going to use only these kind of lenses,
electromagnetic lenses instead of what we
have seen already the optical analog.
Now, I will just play some of the schematic
where we will demonstrate the electromagnetic
system. I want you to go through this carefully
and then see what you observe then I will
explain one by one. You see that this is object
OA. I hope what all of you would have seen
this schematic once, I will replay this you
observe it again. What I am going to describe
from this slide is, here the primary difference
between the glass lens optical system or electrostatics
system to the electromagnetic system.
In a light optical system, you see that your
image inversion takes place, here also you
can see that OA the object is inverted and
it is not just inverted, inversion takes place
at 180 plus or minus phi 1 you have the additional
rotation takes place here, and if you have
the double lenses then it is further rotated
back to A B, but then you see that in the
additional rotation is added that is phi 1
plus or minus phi 2. So this is the primary
difference between the light optical systems
or electrostatic system with electromagnetic
system, you have image rotation takes place.
We will see the consequence and importance
of this image rotation when we deal with transmission
electron microscopy which I will deal with
later. So, carefully if you see the next schematic
the animation clearly showed that, you see
that the first lens has same strength as the
previous one so it has undergone inversion
plus rotation. But the second lens there is
a difference I hope you will be able to appreciate
this, you see that the number of lines has
come down that indicates the field strength
has come down. So you see the similar reaction
takes place here that means this rotation
also will come down. If you look at the third
schematic you see that inversion plus rotation
takes place and I have the second lens completing
the field is absent and you see that there
is no additional rotation that is the phi
2 is 0 the phi 1 which is generated by the
first lens remains in the image plane.
So, this particular schematic and with the
animation a clearly demonstrates the primary
difference between electron optical system
or electromagnetic lens system with the light
optical system. This is the only difference
you can if all you want to make a between
these two systems otherwise rest all the same.
Now, we will also look at the another schematic
where you see the clear animation shows that
electron optical system, where you have the
electron source usually it is a filament,
and then you have the condenser lens, and
then you have a specimen, and you have objective
lens, and then some of the additional intermediate
lenses, and then projector lenses and finally
the image. You see that 
similar analogue of optical system is also
shown you can see that animation very nicely
shown. Except the lens electromagnetic lens
action or you can see all this corresponding
components of the electron sorry, optical
system corresponding to the light optical
system. You can see the condenser lens which
here it is used to regulate the light and
here also it is being used to regulate the
electron beam and convert them onto specimen
that is a primary action. Here also the objective
lens will focus the light to the image plane,
the same action is done here the objective
lens, and then these two additional apertures
also help.
We will look at the details when we deal with
this especially the transmission electron
microscope. For the introduction, I just want
you to have a feel of these two systems in
comparison so that you do not have to feel
anything confusing they are all the same whatever
we have just looked at in the light optical
system as far as the instrument details are
concerned or the ray diagram is concerned.
First we will look at the electron gun. You
see that this is a typical schematic of electron
gun design, you have the filament, and then
you have the cylinder is called Wehnelt cylinder.
The grid cap is, I mean the filament itself
a cathode and then you have the anode. Then
you see that field strength is a kind of a
convergent, this is done by a negative bias
given to this, between a filament and this
anode which will not only accelerate the beam
and also concentrate the beam to this region.
We will see the importance of this in due
course. I just want to introduce this in the
beginning like this.
The filament is usually operated about 100
to 1000 volts less negative than the grid
cap, with the anode and the ground potential.
This is the bias, which I talked about. So,
filament is operated at 100 to 1000 volts
less negative than the grid cap. This arrangement
improves the stability of the emission stream
and because of the bias aids in concentration
of the electron beam.
If you look at the function of the condenser
lenses, it serves to regulate the intensity
of the electron beam in an optical system.
Also converge the beam onto the specimen object
of particular interest. The effective focal
length is determined by the expression of
the form.
F c equal to zeta c, c stand for condenser
and then V 0 is a potential, divided by N
c square and I c square. All c stands for
the a condenser, this is a focal length of
the condenser lens where zeta z the condenser
lens form factor it is a geometric parameter
and N c equal to number of turns of conductor
in the condenser coil system. Now, you will
understand what I mean by the condenser coil,
you have seen that the cross section of the
electron optical electromagnetic lenses so
you will be able to relate it very quickly.
So, V 0 is the acceleration potential of electron
beam in volts, I c is a condenser current
in amperes. So, it is clearly understand by
this expression this focal length of this
electromagnetic lens is related to these many
parameters.
Then if you look at the function of objective
lens, in an electron optical system especially
in a Transmission mode performs the same function
associate with glass objective lens in a light-optical
system. Focusing the electron beam to a final
area of resolution. Objective lens is a very
different from the other lenses primarily
in terms of the more constricted field parameters
necessitated by a shorter focal length through
the concentration of magnetic field strength
on the axis of the system. So, the objective
lens has a slightly different role in order
to bring the shorter focal length.
Obviously, the design will be slightly different;
you can see that it is slightly bigger. Even
if you go back to the schematic diagram we
have shown always the objective lenses shown
much bigger than the condenser and other intermediate
lenses because of the special action of this
objective lens. So, we will see that the focal
length is defined in an equation of the same
form f objective is equal to zeta objective
V 0 divided by NI whole square. Where, zeta
objective is objective lens form factor, N
is number of turns in lens coil, V 0 is acceleration
activating potential, I is objective lens
current.
So, you can see that nicely a drawn the schematic.
You can see that there is an additional a
hardware which is used called Pole Piece.
This is used to focus all this electron beam
in the column and this pole pieces completely
magnetized during the operation and you see
that the electron field or the electromagnetic
field strength is a focused using this two
pole pieces like. These pole pieces are used
in all the lenses whether it is condenser
as well as objective and other lenses.
Now, we will just see what are the types of
electron guns it is just an introduction,
we will see the details of a functions much
more all the details we will see when we actually
look at the system, but I just want to introduce
this types of electron guns. So to provide
a stable beam of electrons of adjustable energy
to have thermionic emission they are also
called emitters; example Tungsten and Lanthanum
Hexaboride. It is being also called a LaB6
or Lanthanum Hexaboride. These two are thermionic
emitters. Then you have another type called
Field emission guns, which has got three variants;
Cold field emission tip, Thermal field emission
tip, Schottky field emission tip.
So, what are the general characteristics of
electron gun? The important parameters for
any electron gun are the amount of current
it produces and the stability of that current.
The current emitted from the filament is called
"emitted current" ie. The portion of electron
current that leaves that gun through the hole
in anode is called a "beam current" ib. At
each lens and the aperture along the column
the beam current becomes smaller and it is
several orders of magnitudes smaller when
it is measured at the specimen as the "probe
current" ip.
So, how this gun performance is estimated?
Electron emission current, brightness, lifetime,
source size, energy spread and stability.
You will appreciate all this parameters when
we actually look at the operation of the electron
microscope and the some of the application
we will take up and then will explain the
each parameter how it affects the resolution
and the brightness and so on. Another important
parameter is, brightness is the most important
of all this because image quality at high
magnification is almost entirely dependent
on this parameter.
We have a definition for this brightness.
Electron optic brightness beta involves not
only the beam current, but also the cross-sectional
area of the beam d and the angular spread
alpha of the electrons at various points in
the column. Brightness is defined as the beam
current per unit area per solid angle, which
is represented by this equation beta equal
to current divided by area solid angle which
is nothing but ip divided by pi dp square
by n times pi alpha p square which can be
written like 4 i p divided by pi square d
p square and alpha p square. Where, the p
stands for probe current. We will see the
importance of all this parameters as and when
we relate to the microscopic operation as
well as the image quality and aberrations
and so on. So these are all very important
parameters to remember.
This is another schematic. This is from another
text book we have taken you can see the similar
filament and gun design, and we have already
seen the action of the gun and 
so on.
A high voltage is placed between the filament
and anode, modified by the potential on the
Wehnelt which acts to focus the electrons
into the crossover with diameter d 0 and divergence
angle alpha 0. So these two, just I want to
show d 0and the alpha. These two are controlled
by this lens design in order to focus the
electron beam.
This is the image of the tungsten hairpin,
the tip of tungsten hairpin filament and the
distribution of electron is when the filament
is under saturated and misaligned. Under saturated
and aligned and saturated. So, this is one
of the thermionic source, and these images
or at different different conditions and this
is under saturated and misaligned and you
have under saturated and aligned and you have
completely saturated, so you will understand
all this when we go to the operation of the
microscope especially in a transmission mode.
This is just for an introduction.
Another thermionic emission filament is, Lanthanum
Hexaboride crystal and the electron distribution
when the sources under saturated and aligned
and the one, is saturated. This is just for
your introduction of the electron gun source.
The next superior electron gun source is as
I mentioned it is a field-emission source.
So, where you can see that the field-emission
tip and you have this subsequent anode design.
Electron path from the field-emission source
showing how a fine crossover is formed by
two anodes acting as an electromagnetic lenses,
so you see that this is very fine and you
can also see this a photo graph, how sharp
the tip is so that is why you are able to
produce a very, very fine crossover of the
electron beam. The action of the anode one
is to provide the extraction voltage to pull
the electrons out of the tip. Anode two accelerates
the electrons to 100 kV or more. So, we will
look at the parameters are much more details
about this field emission gun as we go along.
These are some of the gun characteristics
you can look at it. Please remember the microscope
performance is a related to this electron
gun source and we will also see how it is,
but for the introduction I just want you to
have some basic knowledge about this electron
gun characteristic. You see the source your
tungsten hairpin, LaB6, Field emission, Cold,
Thermal, Schottky and in terms of brightness,
as I mentioned it is one of the primary requirement
of the electron gun and it is, lifetime, source
size, energy spread, and then beam current
stability. You can see that the field emission
gun have superiority over this thermionic
emitters in terms of brightness as well as
life time also in the probe size.
This is very important you see that thermionic
sources you can go up to 30 to 100 microns,
LaB6 can go up to 5 to 50 microns and here
we are talking about less than 5 nanometers.
You will all appreciate the importance of
the probe diameter when we discuss the operation
as well as image forming capability of a different
microscopes we will discuss and this is how
the field emission than a superior, because
it is able to form a very fine crossover or
less than 5 nanometers. Then also you see
that energy spread is also very a small compared
to the thermionic sources. You see the stability
is also much higher. So, with this I would
like to conclude this lecture and when we
come to the next lecture, we will discuss
another important aspect of this Electron
Lenses or Electromagnetic Lenses namely The
Aberrations. The aberrations and it is Effect
on Resolution or Limiting Resolution. These
aspects we will see in the next class.
Thank you.
