Hello everyone welcome back to this material
characterization course.
In the last class we just reviewed the electron
optical systems and its governing principles
and electron lens design and its analogy with
the light optical system and I mentioned that
we will discuss the abrasions in this class
and as I mentioned in the fundamentals of
the optical system we have gone through all
the types of aberrations which the glass lens
will exhibit similar type of aberrations will
also the electron optical system will encounter
and then since we have already seen them in
much more detailed manner about what is the
each aberration and its definition.
I will just mention how this is taken care
in this electron optical system and then we
will take up some few examples and some of
the numerical significance of a spherical
and chromatic aberrations.
As we all know the spherical aberration is
very important and inherent to these lenses
in light optical glass lenses as well as in
the electromagnetic lenses and we also appreciate
that this is one particular abrasion which
directly influence the resolution of the microscope
and we will see them in little more detail.
So now we'll go to this.
Lens aberration & optical resolution with
regard to this electron optical system you
look at this schematic what you are just seeing
is a lens plane and where you have the range
of α that is aperture angle and then you
see that look at this ray tracing path and
then each ray is focusing at different - different
direction and basically and this is a region
we say that the disk of least confusion, and
then and if you look at this all this pair
of rays intersecting the image plane at the
different point and then eventually you see
this aberration disc in the image plane.
So the point you have to remember here is
it is a general schematic which is shown here
and whatever the aberration we talked about
whether it would be a simple astigmatism to
chromatic to spherical aberration all what
all these aberrations they do to this light
ray or electron beam they have they are directing
this electron beam into a different focal
point whether it is on axis or off axis that
we have seen.
So if you just think of all the aberrations
which impair the resolution of the optical
system or electromagnetic lens system it is
the total combination of all this aberrations
put together.
So you can consider this schema take a general
schematic where you see this the distance
δo is the defocusing condition and this is
also considered as the disk of least confusion
and then you see the aberration disk which
we have already seen in the beginning and
then you see that δ optimum with associated
with the electron lengths in general.
So the resolution becomes the total aberration
depending dependent to a large degree on the
half angle subtended from the image by the
lens aperture as designated α, this we have
already seen so I want to emphasis again please
have make sure that you understand this.
So you have the it is not a completely focused
condition this could be because of any aberration
but this is the smallest distance that is
δo is the least confusion and then and if
you look at this δ optimum which is much
larger in the image plane.
So you may see that you are at the defocusing
condition your image resolution is better
is that so it may be the case we will see
in the coming slide.
So let me read out some few introductory remarks
for these aberrations of the electron optical
or electromagnetic lens systems.
The ultimate resolution of the object signal
is influenced by mechanical flaws in the lens
design which produce imperfect lens field
pole piece fabrication and also by mutual
repulsion of electrons are a constricted point
that is a focal points lens aperture etc along
the optical axis particularly the focal points.
The variation in electron energy at various
points in the beam gives rise to image distortion
and contribute generally to loss of contrast
and sharpness.
So this is the fundamental point which you
have to keep in mind whatever happens in the
electromagnetic lens system it is the variation
in the electron energy at various points in
a beam you raise to image distortion and causes
the loss of contrast and sharpness.
The lens operations primarily responsible
for deviations in electron ray intersections
and concomitant loss image clarity may be
classified as geometrical aberrations, chromatic
aberrations and field effect aberrations including
a peak space charge of the electrons.
We will see one by one and the space charge
of electrons will be we did not discuss in
the light optical system we will see in this
system how it is affecting the resolution.
So it is just a recap of what we have seen,
the type of abrasions the this the schematic
I have just put it everything in one image
because we have already seen them in much
more detailed when we discussed in the light
optical systems so the first schematic shows
the coma effect and second schematic describes
the curvature of field and thirds schematic
shows the astigmatism and the fourth one is
lens distortion and the fifth one is spherical
aberration.
I will not describe them in detail because
you have already seen it.
If you have a doubt you can go back to that
lecture and then look at all this individual
defects and then make yourself clear about
this.
And it is the same thing I will only discuss
about how these defects are taken care in
this electron optical system.
In terms of coma it can be eliminated almost
entirely in electromagnetic lens by the establishment
of field conditions giving rise to unity magnification
and in terms of curvature field it is reduced
by properly shaping the electromagnetic lens
field.
The astigmatism on the other hand is correctable
by inserting stigmators in the appropriate
lens system to compensate the non-circularity
of the image beam profile on the image plane.
So what is stigmator?
The stigmators containing symmetrical arrangement
of tiny ferro-magnets are suitable permanently
magnetized components acts to circularize
the image and the lens distortion, the correction
of coma in an electromagnetic lens and currently
eliminates the lens distortions as well.
So what you should appreciate here is in electro
optical systems the most of your aberrations
is controlled by the field strength and the
field distribution.
In an optical system we just all the aberrations
were compensated with the an additional glass
lens here since all the focal length everything
is controlled by the field strength and your
aberrations also controlled by the appropriate
field strength and it is distribution in the
appropriate lens system.
So we will see the other aberrations.
Spherical aberration the correction of the
spherical aberration rest in the design of
lenses with special field distributions for
allowing smaller aperture angles to be attained
with the simultaneous detection in CS possibly
by a design aperture aimed at producing less
symmetrical lens field.
So as I mentioned this particular aberration
is very important and how much we can reduce
this will finally determine the resolution
of the optical system, and then we will see
them and its numerical significance in a few
minutes.
And chromatic aberration which is caused by
the fluctuations occurring in the lens coils
becomes simply a problem of electro electronic
regulation as do fluctuations in the cathode
and anode potentials.
To this extent this defect is correctable
however the energy losses resulting from the
inelastic scattering in the object cannot
be dealt with to the same extent and it is
overcome by operation of the system at higher
voltages.
The another important aspect of this electron
optical system is a space charge effect.
What is this space charge effect?
At the focal points along the electron optic
axis, the concentration of electrons into
small volumes produces a strong mutual repulsive
action and a concomitant tendency of the beam
to spread from the point of constriction,
that is from the point of focus this produces
an effective reduction in the associated accelerating
potential of the electrons and they lose velocity
and this problem is somewhat less at very
high voltage and where lower beam currents
are employed with an associated low beam intensity.
So this particular effect is specially belonged
to this electron optical system and you have
to remember the aberrations which we talked
about and its effect on resolution we simply
assume that or we simply do not consider the
specimen condition, for example whatever the
aberration we talked about we assume that
the specimen is pure and it does not have
any contaminating I mean constituents in it
or it does not react with the beam and then
produce its whole new product that will impair
the resolution.
So all this the treatment which we are talking
about are the compensating effect we talked
about assuming that the specimen is in the
ideal condition, okay.
So in the mathematical treatment which you
are going to look at is also in the similar
manner that we are not taking the specimen
effect that means we assume that specimen
is ideally prepared and it does not have any
contamination or any other reacting constituents
with the electron beam.
So now we will just take up this to spherical
I mean two aberrations first we talked about
spherical aberrations.
So what I am trying to write here is the image-forming
lens or the critical beam forming lens in
an electron microscope or microprobe system
in the objective lens, we always talk about
objective lens whether it could be a any image
forming lens or it could be an electron forming
then I mean you know electron microscope critical
beam forming lens or it could be electron
microprobe analyzer we always concerned about
the aberrations of objective lens.
We can describe the disk of confusion caused
by the spherical aberration as that is δSP,
δ0 is general notation for disk of least
confusion here, δSP is it is exclusively
caused by the spherical aberration can be
represented as two times CS α cube where
CS, the spherical aberration coefficient which
is also given by γo, Cs equal to γo times
Vo divided by (N.I) whole square bracket square.
So this expression you are familiar with already
this is potential, this is number of coil,
this is current which is which is which is
observed to be proportional to the square
of the focal length, so this can be written
as Cs equals to γo focal length of 
the objective lens and it V potential divided
by N.I whole square, where γ is a constant
ranging from hundred for strong lens and 150
for weak lens.
So, γ is a constant ranging from 100 for
a strong lens 150 for weak lens.
So similarly we will see this chromatic aberration.
So as we discussed earlier it depends on electron
energy loss and lens current fluctuation ΔI
and is express as δ chromatic disc of confusion
created by chromatic aberration can be related
to 2 times chromatic Cc α, δCr equal to
two times Cc that is chromatic aberration
coefficient α times ΔV by Vo whole square
plus Δ I by I whole square to power of half,
that is square root of the whole expression.
We write where Cc chromatic aberration coefficient
which is also given by Cc equals γo prime
focal length of objective where γo prime
coefficient is a constant varying from 0.5
to 1.0 for strong or weak lens action separately,
so you have this chromatic aberration constant
is equal to γo prime times focal length of
objective and γo prime is the constant varying
from 0.5 to 1.0 for a strong or a weak lens
action respectively.
So now what we will do is we will see that
how all this aberrations.
We will now try to write some expressions
for object resolution and then image quality
involving all this aberrations which we talked
about.
So we see that so what I have written is as
a consequence of the uncertainty principle,
the exact image displacement of electrons
diffracted from the object area will be subjected
to uncertainty of discrete line displacement
in the object of the order Δ x which is equal
to L.
So this least confusion line to line extend
the written as lambda divided by 2 sine α,
we are familiar with this expression and it
can be assumed like this.
So I read out again because of the uncertainty
principle the electrons diffracted from the
object area will be subject to uncertainty
of discrete line displacements in the object.
So if α is 0.01 to 0.001 radian then we can
write δLL is 0.5 λ divided by α.
For example you can write in a typical electron
microscope α is point zero zero three at
hundred KV your δLL could be roughly about
7Ao.
This will give you an idea a typical case
where you see that to δLL is how to appreciate
this.
So now we will include the lens aberrations
and see how this expression is modified.
Where 
so what I have done is where the lens aberrations
included in the real electron optical system
the ultimate resolution is given by considering
in addition to the diffraction uncertainty,
chromatic and spherical aberrations, the combination
of the error disk radii that is δ optimum
in the image plane is found from δ optimum
equals square root of δLL square plus δSP
by two whole square plus δCr divided by two
whole square.
If we consider 
limit of two points, we always talk about
point resolution as well as line solution.
You can consider these two if, you consider
two points in the image plane 
the optimum is given by and the resolution
limit of these two points is δPP that is
point to point disc of confusion, this also
you are familiar with already seen this where
again include spherical and chromatic aberration.
So if you include this the spherical and chromatic
aberration expression into this, the point-to-point
disk of least confusion, then you obtain δ
optimum point equal to square root of δPP
square plus δSP by two square plus δCr by
two square.
So these basic expressions are further modified
by several researchers and then we can write
one more general expression for disk of least
confusion, I mean we will talk about all this
much more detail out this is really going
to affect the practical resolution when we
look at the actual microscopic operation but
you should appreciate that the importance
of this two spherical and chromatic aberrations
how it really influence the resolution limit
of the optical system.
So what before we just look at this expression
if you recall this the ray diagram which I
showed in the beginning of this class where
you see that the δo was defined as disk of
least confusion in a defocused plain and then
if you look at the image plane where δ optimal
read on the image plane which is much larger
than the V0, δ optimum was much larger than
in the array discreet described which is larger
than the V0.
So that clearly implies that if you reduce
the field strength then you will automatically
get the better resolution, so to emphasis
this point these you get an expression for
δ0 itself that is what we have trying to
show here the minimum constriction of the
beam described as the disk of least confusion
on the defocus plain on the optical axis the
diameter of the disk of least confusion is
given by δ0 equals square root four times
δPP square plus Cs.α cube by two square
plus Cc.α. ΔV by V0 square.
Where δPP is equal to 0.61 times λ/α and
where Cs and Cc are the spherical and chromatic
aberration coefficient respectively, ΔV is
voltage change for a acceleration potential
V0 and α is an objective aperture angle.
So in this class I hope you have at least
have some basic idea about how this aberrations
in an electron optical system is considered
and its influence on the resolution of the
image and the microscope.
So now we will now go on to the actual electron
optical system especially we will start with
a scanning electron microscopy and its working
principle and its application from the next
class.
Thank you.
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