Hello welcome back to this material characterization
course in the last few classes we have just
reviewed all these optical microscopy variants
and it is working principles and life demonstration
so on and now we will move on to the next
domain of electron microscopes like I did
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 this 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 abrasions
which are encountered in this electron lenses
and then how to correct them in order to obtain
a better resolution of the microscopes. So
with this intention 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 rave
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. So 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 electro static lenses
because the electrostatic lenses were the
first used in the electron microscope and
then their design and behavior were studied
then only this was adopted to electromagnetic
lenses so let us review some of the primary
features are the theoretical concepts underlying
this electrostatic lenses.
So an electron beam passing from a region
of low potential V 1 to higher potential V
2 is on acceleration observed to under gore
fraction as defined by Snell's law sine R
by sine I equal to square root of V 1 divided
by V 2 so 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 so
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
their fraction 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
the electron beam is encountering the potential
difference by this electrostatic lenses where
V 1is less than V 2 and then it undergoes
refraction.
So where I is the angle of incidence are is
the angle of refraction on the other hand
if you see that this is electron beam encountering
the two electro static lenses where the potential
is reversed where v1 is greater than v2 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 this two are happening.
The electron beam on passing through a region
of potential difference with v1 is greater
than v2 experiences a retardation making an
angle of refraction greater than angle of
incidence so this is what we have just seen
so where I is very large then these two conditions
are valid so for the refraction sign R by
sign I equal to square root of B 1 divided
by V 2 where I is smaller than sign inverse
times square root of V 1 divided 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 divided 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 is somewhere here and then you
see that I is equal to R a when the reflection
is considered. So with this B we simply see
that the electron beam exactly follows the
rules of a light optical system and we will
see what are the additional points we need
to consider.
And this schematic clearly shows that they
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 v1 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 were 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 the 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 these two lenses. I mean in
this each systems are equal and 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 the what you are seeing is a unique potential
across static 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
then and you see that F focal length is fixed
in this situation.
And in the second case it is a variable focal
length where you have the a combination of
electro static lenses for a different field
strength you can also vary this focal length
F 1and F 2 you can see that the first one
coming through this 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 at the point a 2 and B
2 so you have the variable focal length electro
optical system is demonstrated and what you
see in the right hand side is a simple light
table taken on a log.
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 lens
so I hope this schematic gives you a nice
comparison between this light optical system
as well as the electron optical system where
the electrostatic lenses are used or the cathode
lens designs are adopted.
The electro static lenses we just discussed
about where the electrostatic unimposing potential
electron 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 unique potential lens in any potential
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 the back
focal plane a focal length are 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 divided by F is equal to 1 divided
by P plus 1 divided by Q so the refractive
power of the unipotential lenses expressed
by approximately 1 divided by F equals 3 divided
by 16 times the integral from Z naught to
Z 1 times VC by V naught whole square DZ so
which is a function of the field strength
so I think with this few introduction to the
electro static 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 are 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 unique potential electro static lenses
which are fundamentally analogous to glass
converging lens in a light optical system
so what that we have to now understand is
what this the additional magnetic field does
to the electron path or beam of electrons
so 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.
So this is demonstrated in this schematic
you see this is a typical cylindrical type
electromagnetic Lenz action it is a cross
section where you have all the circular slots
where a soft iron coil is being formed 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
so 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 perceive.
Thus a magnetic field acting in a direction
parallel to an electron beam will not affect
it why a field normal to the beam will cause
it to describe a circle with the radius given
by R naught is equal to 1 divided by B square
root of 2mv naught divided by E where R naught
is in centimeters for V not the acceleration
potential inwards 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 or not. So what
you have to remember is this is R where you
have the circular beam our field is represented
around this region.
So now we will see how the other parameters
are getting affected the refractive power
of the electromagnetic lens is given by 1
divided by F equal to 0.022 divided by V naught
times the integral from zero Z naught to Z
I H square DZ. Where V naught is the potential
through which the electrons converging on
the lengths have been accelerated and H is
the magnetic field strength on the z axis
in gas so the field strength is related to
the physical design of the lens coil by 4πNI
divided by 10 which is equal to integral of
Z naught equal to minus infinity to ZI put
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.
So now it is very clear from this expression
you can understand this I go back to this
you can understand the typical electromagnetic
lines 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 u are the magnetic field strength
so you are from henceforth in an electron
microscope you are going to use only these
kind of lenses electromagnetic lenses instead
of what we have seen already the optical and
lock.
So 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 a
object Oh a okay so I hope what all of you
would have seen this schematic once I will
replay this you observe it again 
okay what I'm going to describe from this
slide is the primary difference between the
glass lens optical system or electrostatics
system to the electromagnetic system.
In NL in a light optical system you see that
your image inversion takes place here also
you can see that way the object is inverted
and it is not just inverted even what inversion
takes place at 180 plus or minus π 1 you
have the additional rotation takes place here
and if you have their double lenses then it
is further rotated back to a B but then you
see that in the additional rotation is added
that is π1 plus or minus π2.
So this is the primary difference between
the light optical system 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 electron I mean 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 cleanses has same strength
as the previous one so it has undergone a
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 have 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 so if you look at the
third schematic you see that inversion plus
rotation takes place and I have the second
lens the completely the field is absent and
you see that there is no additional rotation
that is the π2 is 0 the π 1 which is generated
by the first lens remains the in the image
plane. You see that this is a object OA okay,
so I hope what all of you would have seen
this schematic once I will replay this you
observe it again.
Okay so what I am going to describe from this
slide is 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 way the object is inverted and it
is not just inverted even what inversion takes
place at 180 plus or minus you have the additional
rotation takes place here and if you have
their double lenses then it is further rotated
back to AB but then you see that in the additional
rotation is added that is plus 2 so this is
the primary difference between the light optical
system 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 electron
I mean 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 cleanses
has same strength as the previous one so it
has undergone a inversion plus rotation but
the second lens there is difference.
I hope you will be able to appreciate this
you see that the number of lines have come
down that indicates the field strength has
come down so you see the similar reaction
takes place here that means this rotation
also welcome down, so if you look at the third
schematic you see that inversion plus rotation
takes place and I have the second lens the
completely the field is absent and you see
that there is no additional rotation that
is the ϕ2 is 0 the ϕ 1 which is generated
by the first lens remains the in the image
plane.
So this particular schematic and with the
animation clearly demonstrates the primary
difference between electron optical system
our electromagnetic lens system with the light
optical system this is the only difference
you can fat all if you want to make between
these two systems otherwise rest all this
same.
Now we will also look at another schematic
where you see the clear animation shows that
electron optical system where you have the
electron source usually it’s 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 a similar analog of optical
system is also shown.
You can see that animation very nicely shown
so that except the lengths 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 that 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
the primary action and 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 helps.
We will look at the details when we look at
when we deal with this especially the transmission
electron microscope and for the for the introduction
I just want you to have a feel of these two
system in comparison so that you don't have
to feel anything confusing they are all the
same whatever we have just looked at in the
light optical system as far as we 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 cylinders called a Burnett cylinder
the grid gap is I mean the filament itself
a cathode and then you have the anode then
you see that the field strength is a kind
of a conversion this is done by a negative
bias given to this between 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.
So the filaments usually operated about 100
to 1000 waltz less negative than the grid
cap with the anode and the ground potential,
so this is the bias which I talked about so
filament is operated at one hundred two thousand
waltz 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.
And 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 converts 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 stand for condenser and
then V is a potential divided by N2 and I2
all C stands for the a condenser this is a
focal length of the condenser lens where Zeta
C the condenser lens form factor is the geometric
parameter and NC equal to number of terms
of conductor in the condenser coil system
now you will understand what I mean by the
condenser coil you have seen that cross-section
of the electron optical electromagnetic lenses
so you will be able to relate it very quickly.
So V naught is the acceleration potential
of electron beaming waltz IC is a condenser
current in and years so it is clearly understand
by this expression this focal length of this
electro-magnetic lens is related to these
many parameters.
And then if you look at the function of objective
lens in an electron optical system especially
in a transmission mode performs the same function
associated with the class objective lens in
your light optical system focusing the electron
beam to your final area of a solution this
objective lens is very different from the
other lenses primarily in terms of the more
constricted field parameters necessity 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 so 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 be the condenser and other intermediate
lenses because of this special action of this
objective lens. So we will see that the focal
length is defined in an equation of the same
form F objectives equal to Zeta objective
V0 divided ny n whole square.
So where Zeta objective is objective lens
form factor n is number of turns in lens coil
V0 is acceleration accelerating potential
I is objective lens current so you can see
that nicely drawn the schematic you can see
that there is an additional hardware which
is used called pole piece. This is an this
is used to focus all this electron beam in
the column and this whole piece is completely
magnetized during the operation and you see
that the electron field our electromagnetic
field current speed strength is focused using
this two pole pieces. 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's just an introduction we
will see the details of 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
you have turbine an emissions they are also
called emitters.
Example tungsten and lanthanum hex boric lab
6 it is being also called a lab 6 or lanthanum
hexode or it and these two are a thermionic
emitters and then you have another type called
field emission guns which has got three variants
cold field emission tip thermal field emission
tip hot key 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 the that
current. The current emitted from the filament
is called emitted current ie the portion of
electron current that leaves the gun through
the hole in anode is called a beam ib current.
At each lens and the aperture along the column
the beam current becomes smaller and it is
several orders of magnitude smaller when it
is measured at the specimen as the probe current
iP.
How this gun performance is estimated so electron
emission current brightness lifetime source
size energy spread and stability. You will
appreciate all these parameters when we actually
look at the operation of the electron microscope
and the some of the application we take-up
and then we'll 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.
So 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 β equal to
current divided by area solid angle which
is nothing but ip divided by ipd2 by n times
iαb2 which is can be written like 4 ip divided
by ipd2.
T-square and alpha v square so beware the
P stands for probe current we will see the
importance of all these parameters as and
when we relate the we relate to the microscopic
operations well as the image quality and aberrations
and so on so these are all very important
parameters to remember.
This is another Ruska Mastic which is just
part of our bread this is from the another
textbook we have taken you can see the similar
filament and done design and we have already
seen the action of the gun and so on.
So a high voltage is placed between the filament
and a node modified by the potential on the
Burnet which acts to focus the electrons into
the crossover with the diameter d0 and the
divers angle ϕ0, so these two just I want
to show d0 and the ϕ.
So these two are controlled by this lens design
in order to focus the electron beam and this
is the image of the tungsten harping the tip
of attraction hairpin filament and the distribution
of electrons 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 are two 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 the another thermionic emission
filamentous lanthanum hex a boric crystal
and the electron distribution when the sources
under saturated and aligned and do is saturated
this is for you were just an introduction
of the electron gun source.
The next superior electron gun sources as
I mentioned it is a field emission source
so where you can see that the field emission
tip and you have this subsequent down. I know
design and the electron path from the field
emission source showing you 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 fine crossover of the electron
beam and the action of the anode oneness to
provide the extraction of extraction voltage
to pull the electrons out of the tip and no
two axial rates the electrons to 100 kVare
mode.
So we will look at the parameters are much
more details about this field emission gun
as we go along and these are some of the gun
characteristics you can look at it please
remember the microscope performance is related
to this electron gun source and we will also
see how it is but for the introduction I just
want you to have a some basic knowledge about
this electron gun characteristics you see
the source your function harp in lapse-filled
emission cold thermal and short key and in
terms of brightness as I mentioned it is one
of the primary requirement of the electron
gun.
And its lifetime so size energy spread and
then bream beam current stability you can
see that the field emission sorry the field
emission guns have superiority over this thermionic
emission emitters in terms of brightness as
well as lifetime also in the probe size this
is very important you see that thermionic
sources you can go up to 3200 microns lap
six can go up to five250 microns and here
we are talking about less than five nanometers.
You will all appreciate the importance of
the probe diameter when we discuss the operation
as well as image forming capability of different
microscopes we will discuss and this is how
the field emission than a superior because
it is able to form a very fine cross over
less than five nanometers and then also you
see that energy spread is also very small
compared to the thermionic sources also you
see the stability is also much higher.
So with this I would like to conclude this
lecture and in when we come to the next lecture
we will discuss another important aspect of
this electron lenses are electromagnetic glimpses
namely the aberrations the aberrations and
its effect on resolution or limiting resolution
these aspects we will see in the next class.
Thank you.
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