Dr. Dipankar N.

Basu
Department of Mechanical Engineering
Indian Institute of Technology, Guwahati

Module – 03
Nuclear Fission
Lecture – 07
Fission & role of neutron energy

Hello friends, welcome back to our MOOC’s course on the fundamentals of nuclear power
generation and today we are going to start a very interesting portion and a very important
module of our course which is the topic of nuclear fission. We have already studied the
fundamentals of nuclear power generation in the previous 2 modules where you got
introduced to the concepts of binding energy and we have discussed about actually how
mass can get converted to energy to give a huge amount of energy production because of
a very small amount of mass defect.

We have also discussed about the topic of radioactivity in our second module and there
we have seen that this is one particular figure which was also presented a during the second
module that is whenever there is an interaction between a particle and a nucleus.

(Refer Slide Time: 01:30)

We can have different kinds of interaction while we can initiate any such interaction by
hitting the nucleus with several possible particles of sufficiently large energy, but the
neutron is generally the most preferred choice and therefore we are discussing all these
interactions in terms of neutrons only, but all these discussions are equally true if we can
use some other particle to produce similar kind of effects, but generally it is universal to
use neutron as the initiating a particle.

And therefore, we are keeping our discussion restricted to neutrons only. Now when we
have already learned that whenever there is a neutron nucleus interaction, we can have
three types of possibilities one is scattering where the neutron and nucleus remains
separately, but because of their collision there is will be transfer of momentum and kinetic
energy from the neutron to the nucleus and we can have an elastic and inelastic version of
the same in case of elastic collision both kinetic energy and momentum are conserved,
whereas, in case of inelastic collision the momentum is conserved. But kinetic energy is
not conserved for the system because a good amount of energy may get released in the
form of photons inelastic collision is particularly relevant for neutrons having extremely
high energy level. The second kind of interaction is the absorption which is actually of our
interest in the second module.

In this case the neutron goes inside the nucleus and then either it may remain inside the
nucleus because of a radiative capture sometimes also called non-fission capture and to
and it produces just another isotope of the same parent nucleus, but the other possible
option or other possible following step subsequent state for the absorption is the fission
where after absorbing the neutron, the parent nucleus or the newly formed nucleus is
generally very excited to having high energy content and accordingly it can get splitted
into two components.

So, which is what we are going to discuss here and there is also third kind of interaction
possible which is transfer where neutron gets absorbed to the particle, but also leads to the
production of several other kinds of particles like alpha particles, protons or maybe
neutrons as well and the possibilities or probabilities of having any such kind of
interaction, having any particular interaction when one particular neutron strikes a nucleus
is given in terms of the cross sections and corresponding to each possible kinds of
interactions we can have one cross section definition, summation of all of them gives the
total cross section.
So, as we have already studied, see if our interest is to know what is the probability of
fission occurring when a nucleus of known energy strikes a known nucleus when a neutron
of known energy strikes a known nucleus that we can calculate considering sigma f
divided by a sigma total that is a fission cross section divided by the total cross section of
the same nucleus similarly we can also calculate the probability of any such other kind of
interactions and here as we are focusing primarily on this fission reaction.

So, this is one example here one neutron strikes a nucleus like in this example it’s a
Uranium 235, the neutron is consumed or absorbed inside the nucleus forming Uranium
236 which is generally an unstable nucleus and also having a very high amount of energy
content. So, it gets can get splitted into 2 daughter isotopes which is also associated
generally with release of several number of neutrons like 3 in this particular situation. And
also, huge amount of energy because the mass of all these products like in this case, we
have products in the form of krypton 92, barium 141 and also 3 neutrons. The combined
mass of all these 5 components together is generally found to be less than that of Uranium
236 which is the parent in the second step of reaction. So, corresponding mass defect gets
converted to energy.

(Refer Slide Time: 05:30)

Let us first start with a few examples of fission like first one corresponds to plutonium 239
which absorbs a neutron produce plutonium 240 and extremely unstable nucleus which
gets through a fission reaction to produce zirconium 103 and xenon 134 plus 3 neutrons,
but that is not the only possible way plutonium 240 can decay, because it happens only 73
percent of the total possible situations, but for remaining 270 sorry remaining 27 percent
cases plutonium 240 remains as it is rather it ejects or it loses its energy in the form of
gamma emission and comes to the ground state.

So, whenever the neutron strikes an Plutonium 239 and gets absorbed; firstly, here we are
talking solely about the absorption reaction that is the scattering and transfer reactions are
eliminated. So, whenever we are writing an equation as some nucleus plus a neutron you
are talking only about the absorption reaction; that means, neutron is going inside to form
another isotope of the same parent in this case plutonium 240 which is one neutron more
compared to plutonium 239. Now plutonium 240 after that is getting formed each can have
2 kinds of hits one is the fission which is the first one and in this particular example it
happens 73 percent of total about 73 percent of the total possible cases, but for the
remaining cases, it will only go through a gamma emission.

That means, it is actually a non-fission capture the other part of that absorption reaction
fission is absorbed by the nucleus, but there is sorry neutron is absorbed by the nucleus,
but there is no fission it just produces another isotope of the same and whatever amount
of energy, extra energy that has come with the neutron there gets ejected in the form of
gamma emission. Now how can we calculate these percentages we already know the
corresponding cross sections and that idea can be used like for plutonium 239.

Correspondingly with thermal neutrons, that is neutrons which are having energy of
approximately 0.025 electron volt. Its fission cross section is 750 barns and the capture
cross section that is non-fission capture cross section is approximately 270 barns. So, if
we try to calculate the possibility of a fission reaction here, then it is sigma f or let me
write properly it is a sigma f divided by total absorption cross section that is sigma f
divided by sigma c and this will come approximate to be equal to this 73 percent.

So, just by a knowing the values of this fission and absorption non-fission capture cross
sections we can get the idea about what are the possible percentages of having a fission
reaction and having a non-fission capture reaction after neutron has got absorbed. Another
example with Uranium 235 here, here absorption and neutron first produces U 236 and it
can either go through a non-fission capture like in the second case which happens about
15 percent of the total times to remain as Uranium 236 other is the fission reaction like
here it is producing krypton 94 and barium 139.

So, we know for thermal neutrons or cross sections for fission for Uranium 235, is 585
and for capture, it is 99 barns. Correspondingly, we get this, percentages a third set of
examples. Now with Uranium 233, another very popular isotope used in nuclear reactors
it after capturing the neutron produces Uranium 234 and almost all Uranium 234 isotopes
can go through fission reaction is a very high percentage of 94 percent because
corresponding fission cross section is extremely high compared to the capture cross section
only of about 6 percent cases we can see that Uranium 234 can eject gamma rays and come
back to a ground state.

So, there are possibilities of having both fission reaction and non-fission capture after the
neutron gets absorbed and we can get the ideas just by knowing the values of this cross
sections. Here I would like to introduce 2 terms one is fission fragments other is fission
products fission fragments refers to the immediate products of the fission like say if you
focus on the first reaction involving plutonium here these Zr 103 and Xe 134. These are
the fission fragments because these 2 are produced immediately from the fission reaction.
Similarly, for the last example involving Uranium 233, this Sr 94 and xenon 137, these
are the fission fragments, but there are several other kinds of products like we can see for
all these cases, we have neutrons getting emitted and also all these products of fission like
in the last example this strontium 194 and xenon 137 both of them generally are strongly
radioactive. So, after the fission reaction they keep on decaying a following their own
decay constant thereby producing several new isotopes and also releasing several other
particles like alpha particles or more neutrons or maybe electrons etcetera combining all
those things what we get that is called the fission products.

So, fission products definitely involve the fission fragments plus all the products of the
decay of the fission fragments plus neutrons and any other particle that may get emitted
during this fission reaction if we look further into exactly how a fission reaction happens,
it is a step by step of figures shown here.

(Refer Slide Time: 11:27)

In the first case, one nucleus is being struck by your neutron and that is getting absorbed
the neutron is getting absorbed in the second case which is shown by this b here in this b
now because the neutron is having a good amount of energy it is able to excite the newly
formed nucleus to a higher energy level and because of that excitation this nuclear start to
deform you can see; it is taking a dumbbell kind of shape in the third case the deformation
is even more drastic. It is just about to get split because you can think or just check out the
focus or the thickness here it is about to split it is almost similar to how our human body
cells or any live cell gets the divided into 2 components.

Just quite similarly it is this dumbbell shape produced during step b that is now even more
stronger and the two components are almost separated from each other which happens in
the step d. In the step d they will get separated into 2 components this is our component
number one, this is our component number 2 which we have already termed as fission
fragments plus we can also see there are release of neutrons there are 3 neutrons shown in
this figure and also in the previous slide we have seen the examples in all those 3 examples
3 neutrons are emitted.
But actually 3 is not a constant number, the emissions that are happening, number of
neutrons that gets emitted that may vary from anything from 1 to 7 also I would like to
just briefly go back to the previous slide where we have seen these examples, I must
mention here these are just one possible way the fission can happen. Let us take the second
Uranium 235, once Uranium 235 absorbs a neutron to produce Uranium 236 then it is not
possible to predict the next step; that means, out of this 2-step reaction the first step is
always the same Uranium 235 absorbing neutron to produce U 236.

But the second step there are infinite possibilities, it is only one possible options to have
krypton 94 and barium 139, but it can produce any number or any types of fission
fragments and also any number of neutrons. The number of neutrons can vary from 1 to 7
with approximately the average is being 2.5. So, it is not possible to say exactly which
fission reaction or exactly what we are going to get as efficient fragments. But it is for sure
that the newly formed the intermediate nucleus is going to get deformed into 2 components
and also will be releasing several neutrons number of neutrons can vary from 1 to 7. Now
just take a look at the extreme left of this figure what are the time scales that is involved.
The step from a to b to c to d, this total thing is happening within a time span of just 2 into
10 to the minus 14 seconds that is this is an extremely small time scale that we are talking
about within which this entire phenomenon of one parent nucleus being struck by neutron
and getting splitted into 2 components that happens and next the step e that is not precisely
the fission reaction rather that is related to the later part or the decay of the fission
fragments in will the fission fragments are radioactive. So, they go through their own
decay which is shown here you can see both the fission fragments are getting decayed.

And the nature of the decay has no relation with the fission reaction the nature of the decay
depends solely upon this fission fragments. And this fission fragments decay keeps on
continuing till both of them reaches some kind of stable nucleus. So, the decay process of
this fission fragments that depends on the half-life of the fission fragments themselves and
accordingly that can take a time span of seconds to a few years or maybe a few hundred
years also, but invariably that we finish when both of them lead to some kind of stable
isotopes and the fission product the term that we have introduced earlier that is primarily
added to that decaying that we are getting from the decay or radioactive disintegration of
all these fission fragments. So, primarily the decay of the fission fragments involves
gamma emission and beta emission.
(Refer Slide Time: 16:23)

So that their energy is lost in the form of gamma emission and also there is transmutation
because of the beta decay we shall be seeing that very shortly. Now let us take this example
of Uranium 235 absorbing one neutron to produce Uranium 236, these are the rest masses
for all 3 of them Uranium 235 he is having 235.043923 and similarly value is given for
Uranium 236. If we calculate the corresponding mass defect the rest mass of urine 236 is
found to be lower than the combined rest mass of Uranium 235 plus neutron and that is
having a value of mass different value of about 0.007 MeV which corresponds to 6.541
MeV.

This amount of energy is sufficient to induce the fission reaction that we have just seen.
And whatever we are talking about that is because of thermal neutrons that is the neutron
which is having very, very small amount of energy. So, we have not at all considered the
energy level of the neutron if we are making this reaction happens with an high energy
neutron then of course, that add energy of the neutron also kinetic energy of magnetron
Also will get added with this value, but unfortunately Uranium 235 remains the only
natural isotope which represents this kind of behaviour; that means, while there are infinite
number of natural isotopes presents in the universe Uranium 235 is the only one that can
experience fission reaction when it is being struck by a thermal neutron and if you
remember in our module 1, I have may have given some percentages of all these isotopes
presents in the nature Uranium, while Uranium is present in a good quantity.

But Uranium 235 is only about 0.7 percent of a natural Uranium core or ore as therefore,
it is present in a very small quantity and that is the only isotope that can go through a
fission reaction, but there are several artificial isotopes that can be produced and out of
this Uranium 233 and Pu 239 are 2 principal artificial isotopes that can also undergo fission
when being struck by low energy neutrons. So, the fission reactions are any thermal
reactor, what do you mean by thermal reactor I hope you remember thermal reactor is
something which works on thermal neutrons that is its working principle is based upon the
thermal neutrons are not fast neutrons.

So, for thermal reactors we can you primarily use these 3 isotopes as the main fuel to have
the fission reaction, one is Uranium 235 which is available in nature other 2 are Uranium
233 and Plutonium 239 both of them needs to be produced artificially, there are several
other isotopes which may undergo fission reaction, but not when is struck by thermal
isotope rather it requires much higher energy level to undergo fission reaction and that
energy can be supplied only through the kinetic energy of neutron say for example,
Uranium 238 can undergo fission only when it is being struck by a neutron of
approximately 1 MeV of energy. So, there are several, we can clearly see we can have at
least 2 types of isotopes in nature one like Uranium 235 which can undergo fission just
through the thermal neutrons.

Whereas there are examples in Uranium 238 which requires high energy neutrons
something at the level of fast neutrons to initiate the fission. Accordingly, all material
available or in related to the nuclear power generation can be divided into 3 categories
first is the fissionable material fissionable refers to isotopes which are capable of
undergoing fission after capturing either fast neutron or thermal neutron or maybe both
isotopes which are capable of undergoing fission reaction when that is being struck by a
neutron of any energy level either fast or thermal that we call fissionable material. So, this
example that we have just discussed Uranium 235, 233 plutonium, 239 Uranium, 238 all
of them are fissionable materials; however, the issue is that Uranium 235 can undergo
fission only by a thermal neutron, but Uranium 238 can’t.

So, while both of them are fissionable material, but only one is a fissile material which
refers to an isotope which is capable of undergoing fission only after capturing a thermal
neutron and also that is capable of sustaining a chain reaction and that is Uranium 233,
Uranium 235, plutonium 239, plutonium 241, thorium 233, all are fissile materials.
Uranium 233 is only the natural one, but others are all artificial. So, when we strike any
of these isotopes with the thermal neutron.

It can lead to a fission reaction and also we shall be seeing later on what we chain reaction
means, but for the moment you just keep in mind that a fissile material or to identify
something as a fissile material you need to have 2 conditions satisfied, number one the
isotopes should initiate fission reaction when being struck by a thermal neutron that is it
does not depend upon the energy of the neutron to initiate a fission reaction and secondly,
after the fission reaction happens it is capable of sustaining a chain reaction . Like the
example of Uranium 230 that was given earlier it is a not a fissile material from both counts
it continued a fission reaction with thermal neutron rather it requires only fast neutron to
initiate fission.

And secondly, it cannot sustain a chain reaction also and the reasons we shall be discussing
later on. But there is a third category of particles who are elements which are fertile
elements, fertile elements are not fissionable by thermal neutrons that is when that being
struck by thermal neutrons, they cannot undergo fission, but after capturing a thermal
neutron they can get converted to a fissile neutron or fissile a fissile isotopes rather that is
fertile and fissile are 2 kinds of materials fertile refers to after capturing a thermal neutron
it initiates fission and sustains that through a chain reaction, fertile on contrary it is not
able to initiate a fission reaction after capturing a thermal neutron, but it can get transmuted
to a thermal neutron.
Like this example of thorium 232 when it captures a thorium 232, it gets converted
rhodium 233 which can undergo 2 steps of beta decay to produce Uranium 233 which is a
fissile material. Therefore thorium 232 is called a fertile material, it cannot as undergo
fission by on its own, it is actually a fertile material because it cannot undergo fission by
on its own.

But by capturing thermal neutron it can lead to the formation of Uranium 233 which is the
fissile material and that is why you are calling it fertile. Another example of Uranium 238
which we have identified as a fissionable material, but not a fissile material it can also
capture a neutron thermal neutron that is to produce Uranium 239 which again goes
through 2 steps of beta decay to produce plutonium 239 which is a very strongly fissile
material. Actually, the all the fissile material that we are discussing here like Uranium 233,
235 plutonium 239 this is the one probably the strongest one.

And generally, not used for commercial power generation rather kept separately as war
grade materials that is for producing weapons nuclear weapons that is. So, i repeat we can
have 3 types of isotopes from nuclear reactions point of view or fission reaction points
point of view, one is fissionable; fissionable isotope can undergo fission by capturing
either thermal neutron or fast neutrons, second is fissile which can undergo fission only
by capturing a thermal neutron and hence and also capable of sustaining the chain reaction.

So, all fissile materials are fissionable, but all fissionable materials may not be fissile
because there may be examples which cannot initiate fission a thermal neutron rather
require fast neutrons something like Uranium 238. And fertile material is the third category
the isotopes which cannot undergo fission by capturing thermal neutrons but can get
transmuted to some kind of fissile neutrons fissile materials. There can be examples of
fissionable material, which are not fissile and not fertile as well like the example of
Uranium 234, Uranium 234 after capturing one neutron goes to Uranium 235, which
actually cannot undergo fission because of reasons we shall be discussing later on.

But that is one example of something, or you can it is actually Uranium 235 is a slightly
debated. You better can take the it is better to stick to Uranium 234. Now later we shall be
coming back to this later on it is a fissionable material, but it is not fissile neither fertile.
Now come to fission fragments.
(Refer Slide Time: 26:03)

As I have mentioned because of fission reaction, we have 2 fission fragments that gets
produced, but there can be from the same combination of neutron and nucleus we can have
different kinds of fission fragments produced and actually it is more like a probabilistic
phenomenon that which by which we can predict what can be the types of fission
fragments that gets produced that is while we cannot predict exactly what fission
fragments we are going to get but we can predict what is the most likely fission fragments.
Here the concentration of fission fragments as shown during the fission of on different 3
different isotopes.

Let us follow the red lines which corresponds to Uranium 235 you can see the 2 fission
fragments that we get they are never having equal mass number, rather it is generally
always true for any kind of fission reaction both the fission fragments have significantly
different mass number, like say for Uranium 235, once it captures you a neutron it converts
to Uranium 236. So, we can expect that there will be 2 fission fragments having a mass
number of 118, but that is not true while the average mass number for both the fragments
remains to be 118, but invariably we will find one fission fragments have a mass number
in the range of eighty to 110 while the other is having in the range of 120 to 150 and the
most probable one being somewhere here which is approximately 95 and other is
somewhere here which is approximately 137.
Similarly, for other isotopes, we can get other numbers while the range may be more or
less is the same, one between 80 to 110, other between 120 to 150, 60 or 150 slightly
above 150, but the most likely fission fragments for plutonium 239 will be having mass
number of 103 and 134. Now similar, if we focus on the atomic number, then it is likely
that the most probably fission fragments one will be having atomic number in the range
of 36 to 44 and other in the range of 52 to 60 and these figures shown here includes a very
large database of nuclear reactions involving all these 3 fuels and so we can almost take
these numbers for granted like for Uranium the most likely fission fragments will be
having mass number of 95 and 137 and for plutonium 239 in the range of 103 and 134.
There can be very some very, very rare cases a 0.2 to 0.4 percent where we can have a
ternary fission that mean along with the these 2; there can be a third fission fragments
which is invariably an alpha particle or may be a tritium.

Now let us compare the 3 lines that are shown here the green line corresponds to Uranium
233, the red corresponds to Uranium 235 and the blue line corresponds to plutonium 239,
if you observe carefully the gap between the most probable mass number for Uranium 233
is something between here to here, it is a large gap, but if we compare that with plutonium
the gap is something only a range of like this or if I write properly you will get it is
something like this ,the most likely fission fragments for plutonium 239, one is somewhere
here other is likely to be somewhere here and definitely they are much closer compared to
what we get for Uranium 233, if you follow Uranium 235 one is somewhere here other is
somewhere here. So, the distance between the most likely mass number for Uranium 239,
235 fission is larger than plutonium 239, but definitely smaller than Uranium 233.

So, what can we conclude from here, we can clearly see as the mass number of the fissile
material or fissile nucleus is increasing then the difference between most likely mass
number for the fission fragments that keeps on reducing and that is an important
observation I repeat as the mass number of the fissile nucleus keeps on increasing
generally its energy level also keeps on increasing, accordingly the distance between the
most likely the mass numbers of the most likely fission fragments that also keeps on
decreasing. In an extreme case of Fm 100, Fm 259 which is an artificial isotope we
generally get only a single peak that is it is likely to produce 2 fission fragments having
equal mass number; that means, as the mass number or energy level for the fissile nucleus
is increasing the both the fission fragments are there are larger chances of having 2 equal
or identical fission fragments.

(Refer Slide Time: 31:27)

Now, let us focus on the fission fragments it was mentioned that fission fragments can be
strongly radioactive in nature here.

One example is shown for Uranium 235 fission which is producing rubidium and cesium
or rubidium 93 and cesium 141, both of them are strongly radioactive. Some possible
decays are shown cesium can go through as you can count there are 4 steps of beta decay
it can go through from cesium to barium to La to Ce finally, leading to Pr and all these
steps of beta decay they are also accompanied by gamma emission in some very small
cases of 0.3 percent cesium 141 can first get converted to a cesium 140 and then finally,
going through 3 steps of decay to reach Ce 140, for rubidium again we can see there are 4
steps of beta decay to produce Nb 93 as the final product whereas, in certain situation it
can under also undergo a different route to lead to zirconium 92 here.

So, just forget the alternate or less likely or decaying option for the fission fragments if we
just consider the first line for both cases that is this line for cesium and this line for
rubidium then while because of the fission of Uranium 235 we are getting a rubidium 93
and cesium 141 as the fission fragments its fission products will include several other
possible into isotopes and it is most likely the final one.

Will be this one and this one and also there will be several electrons produced like in the
first case there are four beta decays and also same in the second case. So, there will be
eight electrons produced and also eight gamma photons plus 2 nitrogen, sorry, 2 neutrons
that has produced originally; that means, the fission fragments while it is restricted only
to 2 isotopes the decay of the fission fragments leads to a good basket of the fission
products. Another example of Uranium fission which is leading to the formation of
krypton 90 and barium 143 both of them are radioactive again you can see that is going
through 3 steps or rather 4 steps of beta decay in also it is producing electrons and new
anti-neutrinos and possibly also gamma emission. Similarly krypton is also undergoing 4
steps of beta decays to finally, producing Zr 90, that is while this particular fission reaction
is having krypton and barium as the fission products, but once the complete decay of all
the products are done then we are likely to get this one and this one and also several
electrons and neutrons and also large amount of energy as the fission products. Also, one
point I must add here because of the decay of all this fission fragments, we are also getting
some amount of energy; that means, the amount of energy that we get from the fission
reaction itself will also get a significant amount of addition from this decay of the fission
products and this total together can be a substantial amount. We shall be seeing the
numbers later on now it is it was seen that thermal neutrons can produce fission for certain
elements, but there are other elements which cannot have fission with thermal neutrons.
(Refer Slide Time: 35:14)

Rather they give or they can go through fission reaction only when they struck by a fast
neutrons, that is because the cross sections any cross section like the fission cross section
or capture cross section or maybe scattering cross section all of them depends strongly on
the kinetic energy of the neutron. Here the corresponding variation is shown for Uranium
235, you can clearly see as the neutron energy keeps on increasing energy level of the
neutron from extremely low level to mega electron volt level the corresponding cross
section values are also reducing, rather there are 3 clear zones we can identify one zone as
this one; one zone as this one and add another zone as this particular one.

And there are 3 distinct behaviors of the cross-section variation with the energy can be
seen. The first zone which is this particular one for corresponding to the slow neutrons
that is a corresponding slow neutron that is called the one by v region, you can see here
this cross section is reducing almost linearly with the kinetic energy and it has been
observed that this absorption cross section is nearly inversely proportional to the velocity
of the neutron and accordingly we can use this one by v relation to calculate the energy
level of any thermal neutron like say exam say here sigma a is inversely proportional to
the velocity. So, we can always say that is inversely proportional to the root of En.
σa(T1) = σa(T2) √(T1/T2)

En being the kinetic energy and hence as the kinetic energy of thermal neutron can be
directly related to the temperature of the surrounding, we can also write this to be
inversely proportional to the square root of temperature absolute temperature that is and
using this we can say suppose we are having we know the data of absorption cross section
for a certain temperature T 1, then the system changes its temperature goes to some
temperature T 2.

So, if you want to know the absorption cross section corresponding to the thermal neutrons
at this particular temperature, we can always calculate using this sigma a at T 2 will be
equal to sigma at T 1 into root over T 1 by T 2. So, these is a very straightforward
relationship and for all as all thermal reactors operate in this particular slow neutron zones.
So, we can easily use these numbers or these relations for calculation of the cross section,
absorption cross section. The second one is quite interesting which is called the resonance
capture zones you can see there are lots of peaks or undulations in the profile, this is
commonly observed between the energy level of 10 electron oh sorry one electron volt to
10 keV.

There are multiple peaks actually there are certain elements like indium which can have
just a single peak, but inevitably Uranium 235 or 238 or other fissile or fertile materials
can have large number of peaks there. So, at certain energy levels the cross section of the
neutron is quite high whereas certain other energy levels it can be quite low.

In fact, for elements like Uranium 238 the absorption cross section at this peak can be 100
times more than the base values and therefore, it can significantly affect the neutron
density inside the reactor. We in nuclear reactor most commonly we find the neutron
available as fast neutron level somewhere here and to convert that to thermal neutron we
have to go through some kind of process during which its kinetic energy will be reducing,
and it will be going in this direction.

So, while going through this direction it has to pass through this resonance capture region
and if when it is passing through a resonance capture region at that certain energy level
when this cross sections are high there is a every chance that the neutrons will get absorbed
by the fuel itself and this is a non-fission capture reaction sometimes also called the
resonance capture reaction and therefore, it is better to be avoided actually we cannot avoid
this, but it is always attempt is always made to pass through this region as quickly as
possible.
(Refer Slide Time: 39:42)

So, that the loss in the number of neutrons is not very significant and the third region is
the fast neutron region first neutron refers to in a neutron energy greater than one electron
volt during sorry 10 keV or 0.01 mega electron volt.

So, in this region it is generally not very important for power generation point of view, but
generally it is found to be the that the cross section is roughly proportional to twice of the
actual cross section area of the target nucleus, here we are talking about the actual cross
section or physical cross section not the cross section from nuclear point of view. So, the
variation of cross section with kinetic energy can have 3 clear zones. We have the one by
v relation applicable for the slow neutron or thermal neutron level, then we can have a
resonance capture zone where there are several peaks of cross section and then they are
the of there is the fast neutron region which is not very commonly used or considered for
analysing a thermal reactors.
(Refer Slide Time: 40:44)

These are the figures for 3 different fissile material that we are discussing you can see;
suppose, for thermal neutron is having an energy level of 0.025 electron volt. So, 0.025
electron volt is equal to 2.5 into 10 to the power minus 8 MeV.

So, from the figure it should be somewhere here and you can see here this red line
corresponds to the fission cross section and green line corresponds to the capture cross
section, you can see here the fission cross section for Uranium 235 is quite high a few
hundred generally about 600; the value was given in one of the earlier slide whereas, the
capture cross section is in the range of tens, no it is less than 100 at least similarly, if we
compare that with plutonium corresponding with the same level plutonium somewhere
here. Here also the fission cross is quite similar maybe slightly higher and the capture cross
section is also a bit high whereas, in case of Uranium 233, you can clearly see there is a
while in case of plutonium that gap between these two is not very large of course, it is a
logarithmic scale you have to be careful the vertical scale here is logarithmic.

So, small gap still can correspond to a large difference in the actual values, but Uranium
233 is having a significant difference between the capture and fission cross section and if
we think about the thermal neutron part somewhere here then of course, the capture cross
section is extremely small compared to the fission cross section.
(Refer Slide Time: 42:26)

Now if we compare between one fissile and one fertile nucleus. Uranium 235 and Uranium
238, very interesting for Uranium 238 is that the capture cross section is much larger
compared to the fission cross section almost throughout apart from very high energy level
that is when we are talking about energy level of one MeV or more that is fast or ultra-fast
neutrons, but below that the capture cross section is significantly higher than the fission
cross section like if we think about say the thermal neutrons which will be somewhere
here or maybe slightly more towards right here the capture sorry the fission cross section
is of the order of 10 to the minus 5 only. So, that is you can take it virtually to be 0 whereas,
the capture cross section is greater than 1.

So, capture is the most likely to phenomena which is, but in case of even 235 as we have
already seen the fission lines is above sufficient cross section is always greater almost at
any energy level fission cross section is greater than the capture cross section and also just
compare the resonance peaks that we are getting between the 2 materials like if you is in
the resonance zone say if we take this as the base level which is approximately 10 barns
then the largest peak that we are getting these are more or less at this level.

So, about 10 to twenty times larger peaks can be obtained now you compare that in case
of fission now sorry in case of Uranium 238 here if we take the base level to be something
here that is of the order of 0.01 the peaks can be much much larger a few hundreds. So,
there can be 10 to the power 3 times increase in the fission cross section at those peaks
and hence Uranium 238 exhibits a very strong resonance absorption behaviour and in any
thermal reactor.

(Refer Slide Time: 44:44)

If we are using natural Uranium as fuel which has more than 99 percent of Uranium 238,
there is every chance that a good number of neutrons may get absorbed by the Uranium
238 while passing through this resonance capture zone. Some numbers just to focus on the
previous points that I have mentioning just focus on the Uranium 235 and 238 here even
for a thermal neutron Uranium 235 is having a fission cross section of 583 and capture
cross section of about 100.

Whereas, the fission cross section is negligible for Uranium 238 and it is a very capture
cross section is also quite small. There are certain materials like say xenon is extremely
large capture cross section. So, that can actually act as a poison in the reactor which will
eat up all the neutrons available there by completely stopping any possible fission reaction
another thing you see for all these materials there is no fission cross section applicable or
that is their fission cross there will be no possibility of fission reaction for any of them.
So, they are not even fissile materials as well.
Now, if you come to the fast neutrons, for Uranium 235 the fission cross section is just
one, it was 583 here, it is just one here capture cross section has also see a found similar
amount or similar level of decay, but Uranium 238, actually has an increase in its fission
cross section that is still quite small, but significantly larger compared to its value
corresponding to the thermal neutrons. So, Uranium 238 can undergo some amount of
fission through fast neutrons, but the corresponding cross section value being quite low
that probability is also quite small. So, now, we know that it is important to convert a fast
neutron to the thermal neutron level as the fissile materials particularly like Uranium 233
or 235 or plutonium 239 has much higher values of fission cross section or absorption
cross section corresponding to the thermal neutron, corresponding to the fast neutrons their
cross section values are extremely small as we have just seen in the previous slide.

For the example of Uranium 235, we are talking about a difference of 583 and one.

(Refer Slide Time: 46:50)

So, we must reduce the neutron energy to the thermal energy level and then only we can
have a significant amount of fission inside a reactor and corresponding process is called
moderation. Moderation is a process of reducing the high kinetic energy of free neutrons
and get it converted to the thermal neutron this is one schematic representation say
generally, neutrons the chain this we shall be coming back during chain reaction of also,
but as we have seen whenever a fissile nucleus undergoes a fission there are neutrons that
is giving produce and those neutrons are generally fast neutrons because they will carry a
significant amount of energy released during this reaction. So, like in this case these are
the 3 neutrons that just got produced because of the previous fission as their fast neutrons
so they must pass through something called a moderator. A moderator works on the
principle of elastic scattering that we have discussed earlier during elastic scattering the
neutron collides with the nucleus and exchanges kinetic energy and momentum thereby it
loses his own energy and transfers a bit or a good part to the corresponding nucleus. So,
while passing through the nucleus the sorry passing through the moderator the neutron can
undergo repeated collisions, repeated elastic scattering and finally, coming out as a slow
neutron here the slow neutron of course, will offer much larger fission cross section.

So, that will lead to another fission reaction. This way a moderator helps in fission reaction
because suppose if this fast neutron is allowed to strike another nucleus corresponding
cross section being very low there is a very less chance of having a fission reaction, but as
we are using the moderator the slow neutrons will offer continued rate of fission another
thing when these fast neutrons initially strike the moderator nucleus their extremely high
energy level may lead to inelastic collision passing.

Some amount of excitation energy the corresponding nucleus, but after 2 or 3 such
collisions the most of the next collision steps are elastic in nature. So, collision for a good
performance from the moderator it must have a high value of scattering cross section and
also it is important that it has a low absorption cross section because if the moderator
which here itself is absorbing neutron then there will be no neutron left to induce the next
step of fission therefore, it is desirable that a moderator material should have very high
signal scattering cross section and extremely low preferably zero absorption cross section.

Some common choices are regular water, heavy water - heavy water refers to the water
where the hydrogen is actually a deuterium. So, heavy water and regular water both are
very good moderator, graphite can also be used beryllium also has found some applications
as moderator whereas, certain hydrocarbons has also been proposed recently the reason of
choosing these particles or choosing these materials we shall be seeing very, very shortly.
Now another term I would like to clarify here as we use the term heavy water to signify
water molecules which contains deuterium, quite occasionally that one light water is used
to indicate normal water, but we shall be using term regular water because light is not a
proper terminology.

(Refer Slide Time: 50:34)

So, we comes to the theory of elastic scattering here we are taking one nucleus of mass
capital M which is initially at velocity at 0 velocity the black one here is a nucleus and the
red one is a neutron having a mass of small m and coming with a velocity v it is
approaching the nucleus with the velocity V, once the collision happens, then it transfers
a part of its kinetic energy to the nucleus. So, both of them can get scattered in 2 different
directions here theta is the angle by which the neutron gets scattered from its initial line of
motion and similarly v prime is also getting scattered by some other angle.

So, as this being realistic scattering both energy and kinetic energy and momentum will
be conserved, conservation of energy can accordingly written as, before collision here one
refers to the state before or state before collision, two after collision, before collision the
nucleus is stationary. So, only kinetic energy is associated with the neutron.

So, half m v square represents the kinetic energy of the neutron after collision both are
moving. So, this is the kinetic energy of the this for the neutron this for the nucleus here
small v prime and capital V prime represents the velocity magnitudes of neutron and
nucleus respectively after the collision process and let us say E L is the energy magnitude
of this particular energy. Next is the conservation of momentum again initially the nucleus
is stationary.
((1/2) mv2)1 = ((1/2) mv’2 + (1/2) MV’2)2 = EL
(mv)1 = (mv’ + MV’)2
(m +M) vCM = mv
vCM = (m / (m+ M)) v

So, entire momentum is coming from the neutron and after that both of them are moving
this is a vector addition because momentum is a vector quantity. Next we define a term
called center of mass we can easily analyze this from this conservation of energy and
conservation of momentum point of view, but it is and by taking our laboratory or
wherever this experiment is going on as the frame of reference that is why it is often called
the lab scale and that is why this l subscript we are using energy at the lab scale, but
whenever you are talking about such kind of 2 particle collision it is better to go for a
center of mass approach center of mass is defined as an hypothetical body of mass which
is having the same amount of mass as the initial system, that is it is this mass is the
combined mass of the neutron and nucleus and also it is carrying the same amount of
momentum as the entire system.

So, the total mass of the body is small m plus capital M and if v cm represents velocity
then as I have mentioned the total mass will be equal to the total mass of the system and
total momentum will be equal to the total momentum of the system, now before collision
total momentum is small m into v that is the momentum carried by the neutron and as that
remains constant. So, this will be the total momentum of these center of mass, here this
blue ball shows a center of mass which is moving with a velocity v CM now. So, v CM
can be represented in terms of the velocity of the neutron center of mass; mass approach
has certain advantages and certain characteristics here total if we can or the center of taking
centre of mass as the frame of reference we can gain certain advantages once we know the
velocity and mass of the center of mass then we can calculate the velocity and momentum
for both neutron and nucleus with respect to that and that is by taking the centre of mass
as a new frame of reference and the advantage that we are going to get the total momentum
of the center of mass system will remain zero both before and after collision magnitude of
the center of mass velocities will not change because of the collision, but there will be
only change in the direction.
That is the direction of velocity vectors will change, but no change in magnitude and third
is total energy of the center of mass system will be less than the lab scale system due to
the motion of the center of mass itself. So, we shall be next discussing about the theory of
elastic collision by converting this lab scale system to a center of mass system.

But as we are running short of time today we shall be starting that in the next class onwards
where or next lecture where we shall be deriving the mathematical expression starting
from the kinetic energy conservation and momentum energy conservation, according to
the center of mass frame of reference we shall be deriving the expressions for velocities
of the neutron and nucleus after collision and we shall be seeing how much reduction in
the kinetic energy of the neutron that is possible and accordingly we shall be we shall be
setting up some criterion of choosing the moderator material.

So, thanks for your attention today hope to see you in the next lecture.