So, we discussed the basic reactions in neutron
induced fission of uranium. Natural uranium
has 2 isotopes; uranium 235 and 238; little
bit of uranium 233 also. And, we saw how a
neutron when gets into uranium 235. It can
induce almost instantaneous fission in the
time scale of 10 to the power minus 15, 16
seconds. And, we call it fissile material
235 uranium is a fissile material. It can
fission on absorbing thermal neutron low-energy
neutron. The other component uranium 238 which
is much larger abundance in natural uranium
that is not fissile. If you put neutron into
it most likely low-energy neutrons will not
create a fission in it.
So, today we will be talking about general
features or basic basics of nuclear fission
reactors which are used for power generation
from nuclear fission. All over the world including
India large number of fission reactors are
operating; in which one produces power from
this nuclear fission and uses it for various
purposes. Before going into that description
of typical fission reactor let me little bit
discuss more about the reactions of neutron
with uranium. We have already seen the nuclear
reactions with uranium 235 and 238 cross-sections.
How much is the cross-section of this induced
fission? So, that is one thing we will be
doing.
So, neutron and then uranium 235 this goes
to uranium 236 in excited state and that fission
in 2 parts. So, the cross-section and then
plus energy and some neutrons and so on. So,
the cross-section for probability for this
reaction depends very strongly on energy of
this incoming neutron. At almost 0 energy,
when it comes you know that the excitation
energy is higher than the activation energy
of that fission barrier and it takes place.
And, as you increase the energy of this incident
neutron, the excitation energy is more than
the q value. And, one may expect that since
the energy is much higher than the activation
energy; the fission cross-section will increase
the fission, probability will increase but
that is not the case with uranium 235.
As you increase the kinetic energy of neutron,
the fission cross-section goes down; this
is a typical 1 by v phenomena. We call it
1 by v dependence of cross-section on speed
of this low-energy neutron qualitatively.
You can think of if speed is low, it gets
more time to interact with the nucleus and
more probably to get absorbed into it. If
it is a higher speed, then it will just cross
through and the absorption cross-section will
be low and so on whatever. So, let me show
you how this fission cross-section changes
with the incident neutron energy. So, that
will be on the power point slide.
So, this diagram is in front of you; it is
schematic diagrams drawn do not take it very
seriously; these are not experimental data
but this is the trend. So, this is uranium
235 and the vertical side here is the cross-section
sigma in barn. You know barn 1 barn is 10
to the power of minus 28 centimeter square.
This is reaction probability is or reaction
mechanism decides this and the this probability
is measured in terms of these cross-sections.
So, this is the cross-section. And, the horizontal
side is the energy kinetic energy of neutron
in electron volts; both these access are logarithmic.
So, you have 10 to the power minus 2 here,
10 to the power minus 1 here, then 10 to the
power 0 here, then 10 power 1; that is 10,
this is 100, this is all in electron volts.
So, this is 1 kilo electron volt, then 10
kilo electron volt, 100 kilo electron volt,
this is 1 mega electron volt and this is 10
mega electron volt and so on. So, this is
horizontal axis is in logarithmic scale and
that is in electron volts. The vertical scale
cross-section is also in logarithmic scale
this is 1. So, this is 1 barn, and this is
10 barn, and this is 100 barn, this is 1000
barn, this is 10000 barn and so on. This is
0.1, this is 10 to the power minus 2 and this
is 10 to the power minus 3 and so on. So,
at almost 0 energy 10 to the power minus 3
electron volts here, this origin in this drawing
is 10 to the power minus 3 electron volt.
Here the cross-section is about 1000 barn
any standard this is a very high cross-section.
And, as you increase the energy of the neutron
this cross-section decreases, you can see
this cross-section decreases and thermal neutrons
at room temperature the kinetic energy will
be something like 0.025 volts.
So, that is 2 into 10 to the power minus 2
that will be somewhere here. So, this is the
thermal energy range. So, here if you ask
what is the cross-section, that cross-section
turns out to be about something like 600 barn
or 580 barn; this is for thermal electron.
Thermal neutrons means where the kinetic energy
is of this neutron is about say 0.025 electron
volt corresponding to thermal motion at room
temperature. So this is something of the order
of 600 barn or so and then it decreases.
Now, from 1 e v to 100 d e v this range. This
range this will be around 1 electron volt
and this will be around, say this is here
it is this is 100 e v here. So, little more
say 1 k e v this is the range; in which you
see lots of line going up and line going down
these are in fact resonances. So, the cross
section suddenly increases and suddenly decreases.
So, this the region where you have resonances
overall the cross section is decreasing and
but there are some fluctuations here. And,
then again here you do not distinguish those
though separate resonance peaks perhaps they
are all merged together and you get a almost
smooth variation.
Here is 10 to the power 6 that is 1mega electron
volt. This is also important number 1 mega
electron volts, 2 mega electron volts because
the neutrons that are produced in a fission
reaction the prompt neutrons; they are of
this energy something like 2 mega electron
volt energy. Similarly, the beta decay process
also gives neutrons delayed neutron; they
are delayed neutron, they are the energies
also large in mega electron volts. So, you
have energies like this and at these neutron
energies the cross section for fission in
uranium 235 is somewhere here which will be
around 1 barn.
So, from thermal energies to mega electron
volt energies the cross section decreases
3 orders of magnitude. Here at thermal energy
it was some somewhere around 600 barns are
like that whereas, in mega electron volt it
drops to 1 barn or so. So, a fraction of 600,
700 down as you compare these 2 energies that
is about uranium 235. Now, look at uranium
238. As you know in uranium 238 and the activation
energy is larger and the q value of that reaction
is smaller. The q value is only 4.8 mega electron
volt; whereas, the activation energy is something
like 6.6 m e v or so.
So, at thermal energies neutrons will not
induce fission in uranium 238 and that is
you can see the diagram here. This is uranium
238 and it is it is appreciable only after
this 100 k e v or so. Before this thermal
energy range or k e v range 10 kilo electron
volts and 100 e v; at all these places the
cross section for fission is almost 0 for
uranium 238. When it approaches that 1 m e
v point then the cross section starts building
up. And, the you know that is the barrier
penetration region. So, as the neutron energy
increases the barrier width to be penetrated
becomes smaller.
And, therefore the barrier penetration probability
becomes larger and hence the cross section
increases rapidly. So, this is that region
where cross section is increasing rapidly
from very low values to somewhat higher values.
And, then once this it reaches somewhere around
1.4 mega electron volts. So, that the fission
barrier that activation energy and the here
q value plus then neutron energy they become
equal or almost equal; after that you do not
have to penetrate the barrier. And, therefore
it just goes. So, that is this range; this
is 10 power 710 m e v, we do not interested
in this region.
So, 2 m e v, 1 m e v, 2 m e v the cross section
is here and that is less than 1, less than
1 barn, there is a less than 1 barn. The fission
cross section will be little less than 1 barn
if you have fast neutrons to induce this fission.
So, that is about the fission cross sections.
Now, apart from fission cross sections this
uranium 235 and uranium 238 they can also
absorb neutrons and still not fission. So,
that is the mechanism we had talked earlier.
It is radioactive capture most prominent non
fission mechanism is radioactive capture.
Now, you see this diagram. This diagram again
it is schematic is not experimental data as
such it is only shows the type of variation
it has. Here, what we are showing is neutron
radiative capture cross section. And, that
is this diagram is for uranium 238; on the
vertical side we have cross section sigma
again in barn logarithmic scale. So, 1 and
then here 10, then here 10 power 2 that is
100 barns, this is 1000 barns, this is 10000
barns and so on. So, this is the kind of scale
and on the horizontal side you have once again
neutron energies. And, this time it is not
logarithmic it is 10 electron volt here, then
20 electron volt, 30 electron volt, 40 electron
volt and so on say linear scale and this is
120 here. And, what are these cross section
for? These cross section is for a neutron
getting absorbed into uranium 238 but not
creating fission of course at these energies
10 electron volt or 50 electron volt or 100 electron volt.
It will anyway, it will not create fission
in uranium 238 because of the activation energy
because of the barrier height; but the neutron
can still get absorbed into it. And, not only
it can get absorbed into it; it has a large
resonances in this region. That means, the
uranium 238 energy levels are there. So that
these 10 electron volt neutrons or 40 electron
volt neutrons; they have a just sufficient
energy to excited this uranium 238 and that
particular level. And, hence the reaction
probability increases very rapidly the usual
resonance. So, you see that from thermal energies
or say 1 electron volt to say 120 electron
volt or 150 electron volt. You have very large
resonances look at the heights of the peaks
that we are showing here very large resonances.
The cross section is approaching 10000 barns
few 1000 barns from any standard, it is a
very very high cross section. And, this probability
of absorbing neutrons in this energy range
by uranium 238 is extremely high. So, these
resonances are to be kept in mind. In contrast
if you think of uranium 235 the low energy
neutrons can induce fission. We have seen
the fission cross section of uranium 235 lower
the energy larger the fission cross section.
But here also in competition to fission after
absorption of neutron in uranium 235 gamma
rays maybe emitted. And, once gamma rays are
emitted and energy of this uranium 236 that
is formed is lowered. So that the barrier
becomes important and the it has to penetrate
the barrier the this fission may not take
place. And, if further gamma ray energies take it
to the ground state of uranium 236 then it is permanently uranium 230s. Permanently means?
Whatever its own lifetime of alpha emission
so that is it. Now, the radiative capture
cross section at thermal energies by uranium
235 is about 97 barns. And, it gradually decreases
just like that fission cross section alright
you have seen the fission cross section that
decreases. Similarly, the radiative capture
cross section also decreases. Here it is starts
at about 1000 barns and they are the absorption
radiative capture absorption that starts at
say 100 barns 97 barns. And, then that also
decrease that also goes through the resonances
and so on. So, the magnitudes or the of these 2 events
that should be remembered; 97 barn is the
cross section for absorption of neutrons in
uranium 235 at thermal energies followed by
gamma ray emissions not followed by fission.
Whereas, the cross section for absorption
of neutrons in uranium 235 at thermal energies
followed by fission that cross section is
about 600 barns. And, both these cross section
decreases as you increase the neutron kinetic
energy. And, for uranium 238 the cross section
for fission starts around 1 m e v or so before
that the there is almost no fission. But the
absorption is there radiative capture absorption
is there, neutron getting absorbed in uranium
238 followed by gamma ray emission is there.
And, the cross sections for the that in the
range 1 electron volt to some 100 electron
volts. There are so many resonances and at
these resonances the cross section goes to
few 1000 of barns.
So, that is to be kept in mind and then there
are other processes also like elastic scattering
and inelastic scattering. And, neutron which
goes to a uranium 235 or uranium 238 they
it can just gets scattered elastically or
in elastically. For inelastic scattering there
will be some threshold because inelastic scattering
in some kinetic energy goes into internal
excitations. That means, it takes the nucleus
to its one of its excited state to some energy
goes into the nucleus. And, the total kinetic
energy available to these interacting particles
becomes slower. So, that excitation that that
gives you threshold, so any neutron of any
energy will not show up inelastic scattering.
For uranium 235 this threshold is 14 kilo
electron volt and for uranium 238 that is
44 kilo electron volts. So, for neutrons greater
than these threshold they can in elastically
scattered from these uranium nuclei.
And, elastic scattering can always take place.
There are cross sections for that normally
less than 10 barns also between 1 barn to
10 barn for this elastic and non inelastic
scattering. So, these are the process that
goes on. Now, a typical fission reactor. So,
that is now a general knowledge what a typical
fission reactor it is what are the components
and so on? But still let me tell those typical
words again look at the diagram.
So, this is again a schematic design of a
typical nuclear fission reactor. And, I am
only showing the core where this nuclear reactions
are taking place there is a lot outside this.
And, that react design is by any means it
is a very very complex thing but right at
the core weather nuclear reactions are taking
place what are the things? So, you have first
is you have a fuel rods. So, these are the
fuel rods I am showing these are white things
here these are the fuel rods; that white things
are I am showing. So, what are these fuel
rods? So, let me describe little bit on that.
Fuel is nuclear fuel that means the uranium
which will be most of the reactors involved
are based on this uranium.
So, that uranium which fission which gives
energy that is fuel but in what form it is
placed in the reactor? The uranium is first
mind from the rocks and then the uranium that
other rock part is to be separated and uranium
part is to be extracted from that. So, some
chemical bleaching or something is done to
get that uranium compounds out of that rock.
And, then these uranium compounds are further processed.
And, sometimes or most of the times it has
to be enriched because in natural uranium
as you know the uranium 235 is about 0.72
percent and uranium 238 is almost the rest
99.28, I can write if I neglect that third
digit or 275 like that. And, then the rest
is other things uranium 233 and so on. There
is a present story it will depend on time
although the time scales will be millions
of the years. But because they there these
are the alpha active things they will change,
so the concentration of uranium 235 and concentration
of uranium 238 that will change as a function
of time. But not with the time scale of few
months or few years or few decades in the
time scales will be in 10000 years or million
years or things like that but presently this
is the composition.
So, and as you know this is a fissile material, this is non fissile  slow neutrons will cause fission here and
only fast neutrons can do fission here is
lot of it. But and even if I go with fast
neutron fission the cross section of fission
is very low. That we have seen that uranium
238 line starts from here and then goes like
this and it is still less than 1 barn. Whereas,
the uranium 235 cross section at thermal energies
is very large about 600 barns and so on. So,
most of the fission that people depend upon
in a fission reactor is from this uranium.
But then with this small concentration the
design is more difficult, the size of the
reactor will become very large; although there
are some reactors operating with this natural
uranium.
So, normally what people do? People enrich
it and generally it is enriched to something
like 3 percent or 3 to 4 percent in uranium
235. And, how is that enriching done? Once
you have from the or you have separated out
that uranium compound which has uranium in
the form of uranium 235 and uranium 238, so
to enrich that part people make a gas out
of it something like some fluoride or something
and in the gaseous form. And, then it is taken
through some kind of a mesh or so.
So, that diffusion can take place. And, the
diffusion the rate of diffusion that depends
on the mass of that diffusing gas molecules.
And, so depending on we depend on the mass
difference this is u 235 and that is u 238.
So, that masses of those molecules will be
different and the diffusion rates will be
different. And, that is how some of this can
be separated out. So, some mechanism or you
can centrifuge these things. So, that because
of the mass difference in centrifuge things
will be separated. So, some uranium 238 is
to be extracted out is to be removed. If you
want to go from 0.7 percent to 3 percent or
4 percent how much uranium 238 you will have
to remove? That number comes out to be around
80 percent or so.
If you have initial concentration 0.7 percent
here and 99.3 percent here this is uranium
238 and this is uranium 235 that compound
form. So, if you remove 80 percent of it.
So, you start with say 100 units and 80 units
of uranium 238 you remove. Then, what you
will get here is if you remove 80 from here
it will be something like 19.3 percent here
or whatever is left is 19.3 units and here
it will be 0.7 units. So, if you remove 80
percent of uranium 238 species you get this
composition. And, how much is percentage of
this? This total is 20 and out of this 20.7 is here.
So, 0.7 in 20 units. So, 5 times of this 3.5
percent, to get this 3 percent or 4 percent
of enrichment you have to almost remove 80
percent it of that uranium 238 part. So, that
is how it is enriched. And, once again it
is then converted into solid oxide powders
and then pallets are made. Those pallets are
stacked one over other in some thin zirconium
alloy tube to make one long unit of that.
And, such tubes are then clustered together
several tubes and that makes one unit which
we call fuel rod. So, it has this enriched
uranium or natural uranium if you want to
work with that. And, that is shown here look
at your diagram once again the these white
things that with your red dots on your screen.
These I am schematically showing those fuel
rods which are constructed the way we have
said. And, then these rods are separated from
each other and the space in between is filled
with something called moderator. So, these
grey things here these are moderators. These
all schematic diagram do not think that if
you go to a reactor it will look like this
geometrically. So, these are moderators. Now, what is moderator?
So, once again let us talk about moderator.
So, what we had seen that the cross-section
for fission decreases 3orders of magnitude.
If the neutron energy is goes up from thermal
neutron to mega electron neutron. And, in
a fission reactor the neutrons which are used
to ignite these fissions they come from the
fission events themselves. The neutrons are
not supplied from outside for each fission
event. So, once fission starts we call it
chain reaction. Once fission starts each fission
produces few neutrons. And, those neutrons
are used to trigger further fissions all these
are known to almost everyone.
So, but then these neutrons which are produced
in fission events they are produced at much
higher energies; mega electron volt energy
some somewhere around 2 mega electron volt
energies. And, at these energies the fission
cross-section is much lower is about a 1 barn
or even less than that uranium 238 also can
fission. And, these energies but there also
the cross-section is much smaller is less
than 1 barn and uranium 235 also the cross-section
is much smaller. So, to utilize that large
cross-section at lower neutron energies is
what we do is we thermalize these neutrons.
These fast neutrons which are producing fission
events they are there kinetic energy is reduced.
And, how the kinetic energy is reduced? Kinetic
energy of the neutrons can be reduced by scattering
processes. So, neutron is the if you look
in terms of nuclei if you think then after
proton neutron is the lightest one. So, any
material you take carbon or beryllium or any
material you take neutron will be the lighter
one with one exception with protons. And,
there also the masses are almost equal almost
a very slight difference between proton. So,
you can treat them as equal mass. So, if neutron
collides with some other material and elastically
scatters then some kinetic energy will be
reduced. You can make a simple calculations
for head-on collisions.
If you have this small m is for neutron let
us say and it is going with some kinetic energy
some speed. And, then you have another nucleus
here with mass capital M at rest and this
neutron goes and hits this. Then, what happens?
If it is all elastic collisions no internal
excitations and so on, you can do a simple
calculation if this after the collision if
this goes with a velocity v 1 the and if this
goes with a velocity v 2. The separation will
increase. In fact since neutron is supposedly
lighter one and this other mass capital M
the other nucleus with which it is colliding
that mass is largest. So, neutron will in
fact come back to this v 1 will should turn
out to be negative literacy. So, the momentum
conservation will be m v is equal to m v 1
plus capital m v 2.
And, then if it is the elastic collision then
kinetic energy before the event is equal to
kinetic energy after the event. And, that
will leads to the equation that velocity of
separation is equal to velocity of approach.
So, velocity of approach here is v and velocity
of separation here is v 2 minus v 1. So, v
should be equal to v 2 minus v 1. So, this
is for elastic scattering. So, from here you
can get this v 1; we are interested in v one
that v 2 is find whatever material we had
kept for lowering down the energy of neutron
the moderator materials. So, that material
we do not we have not interested. What happens
to this velocity and how this gets it is distributed
into that material? But the neutrons which
is interacting with that v 1. So, eliminate
v 2 from here. So, m v is equal to small m
v 1 plus capital m v 2 and v 2 is v plus v
1. So, v plus v 1 and collect v 1 on one side
and other things on the other side. So, m
minus capital m into v, so I have taken this
term I have taken this term is equal to m
plus m times v 1. And, hence v 1 is equal
to small m minus capital m and divided by
small m plus capital m and into v.
So, as expected if capital m is larger than
small m this v 1 is negative. If a lighter
particle collides head-on with heavier particular
this lighter particular rebounds it comes
back. So, v 1 is negative and if they are
if with small m is larger than capital m then
of course this v 1 will be positive. So, kinetic
energy you can write kinetic energy off neutron
marked the total system kinetic energy of
neutron after the event final kinetic energy
will be half small m into v 1 square. And,
kinetic energy of the neutron before the event
is half small m into v square. So, this kinetic
energy final will be just m minus m by m plus
m square times kinetic energy is reduced.
Kinetic energy of the neutron only it is elastic
collision. So, kinetic energy of neutron and
that other particle with which it is colliding
that will remain same; that will that is not
going to change it is all elastic scattering,
but then the kinetic energy of the neutron
itself in which we are interested. So, that
kinetic energy is reduced by this factor m
minus m by m plus m square. Now, I am squaring
so that minus does not matter.
Now, you can see if you want quicker thermalisation
this kinetic energy to be reduced in few number
of collisions. What should be that case? This
factor should be as small as possible and
that will happen if this capital m is closer
to small m. If it is just equal then you have
see it become 0 in one collision head-on collision
remember. If you have 2 equal mass objects
colliding head on and one then the velocities
are interchanged. If they are equal mass particles
in elastic collision in 1 dimension is just
interchange but that is in 1 dimension. So,
here it will not be 1 dimension but is still
gives you an idea that lighter is the mass
of that moderator material which is used to
reduce kinetic energy of neutrons; that reduction
process will be more efficient.
Now, in lighter things what you can take lightest
one is proton hydrogen? So, you can put water
in it reactors are developed with water as
moderator. So, these fast neutrons if they
come to these water neutrons collide with
protons of hydrogen and then they thermalised
very fast. But then there are some problems
with water or with hydrogen with protons this
water if you think of this most efficient
way.
So, that the proton is there and this neutron
collides with proton. And, almost in one single
collision if it is head-on it gives of its
kinetic energy to that water molecule to this
p. But then you have this reaction that 1
h and plus neutron that goes to 2 h deuteron.
And, this reaction has sizable probability.
So, if you put water there and think that
this neutron will go hit and will transfer
kinetic energy it will. But in some of the
cases and the number of such cases may be
significant this neutron can just go and stick
to that proton and make deuteron.
So, it is all together lost from the chain.
So, that moderator material all this consideration
should be there this should not absorb the
neutron; at the same time it should reproduce
the kinetic energy. So, this is about the
moderation. And, why we need moderation outside
that fuel rod? This is because as you decrease
the energy of the neutron from mega electron
volt to thermal energies to perfectly utilize
that large cross-sections. It has to go through
that region of 1 electron volt to 100 electron
volt also. In that region if it is in the
same vicinity of uranium 238. That means,
the fuel rod the neutrons are slowed down
right in the fuel rods itself; there also
there will be scattering and the neutrons
will slow down.
But then as it passes through that region
say 1 to 100 e v during the slowdown from
mega electron volts to thermal energies. Thermal
energy is 0.025 electron volt or so 1 by 50
electron volts or even less. So, it has to
pass through this. And, you had already seen
that the radiative capture cross-section for
this neutron plus uranium 238 reaction is
very larger around 10000 barns or so. If these
slower neutrons in the energy range 1e v to
100 e v are in that fuel rod where uranium
238 is there even if it enrich it by uranium
235 by 3 percent, 7 percent is still uranium
238. So, all these neutrons are likely to
get absorbed in that uranium 238 and get captured
because it just gives off gamma rays and comes
to ground state. So, these neutrons are lost for further fissions.
So, that is why the design is such that the
fast neutrons once they are produced in that
fuel rod quickly they should come out of the
rod. And, then in that moderator area between
the rods you have all this moderator whatever
water or carbon graphite or beryllium or heavy
water or whatever that moderator material.
The choice of moderator material has to be
made it has to be a low z material, easily
available, cost-effective at the same time
it should be have lower cross-section for
any nuclear reaction that absorbs neutrons.
So, all those considerations are there but
whatever is the moderator material the scattering
of neutron takes place from these moderator
material. And, then the kinetic energy is
decreased to thermal energies before going
into a rod. And, no one will.
So, all this all random phenomena, so in that
fuel rod a fast neutron is created of course
there will be probability that it remains
in the rod but then it is a fast neutron it
is moving with high speeds. So, most likely
it will leave the rod early if the rod is
not too thick. So, it will leave the rod and
get into the moderator in the moderator with
some scattering events it with thermalize.
And, then it will still wander here and there.
And, then in this random wandering sometime
it will get into the fuel rod that thermalized
neutron.
So, the geometry is to be very nicely designed.
So, that after thermalisation it gets good
chance to get into the fuel at the same time
before thermalisation; when it is in the range
of few electron volt energies that time we
should not be able to get into the fuel rod.
So, that geometry has to be designed taking
into account all kinds of statistics and all
kinds of theories of probabilities and random
walks and this and that, so that most of the
time the fast neutron as it is created in
efficient event in the fuel rod comes out
in the moderator. And, then remains in the
moderator area till it is thermalized, till
it crosses that danger zone of 100 e v to
1 e v and comes below that.
And, then once it comes below that and once
it is kinetic energy is almost the thermal
kinetic energy. Then, you should get a sufficiently
good chance to get into one or the other fuel
rod. So, all these design has to be made.
So, that is why the moderator is needed. Now,
in the diagram once again look at your screen
those reactor designs. The other thing you
can see here are these control rods. We will
talk about that little later but why this
control rods are there? But these are rods
of materials like cadmium which absorbed neutron
with high probabilities. So, these rods are
there to absorb the neutrons. And, if you
find if the operators find that these nuclear
reaction is going faster then what it is designed
for these rods are inserted deeper into the
into this core. So, that they start absorbing some neutrons
and the rate of fission goes down that is
it. And, then you have this reflector outside
this is your reaction core in which you have
moderator and the fuel. And, then outside
here this is some kind of envelope which reflects
the neutrons; if neutrons try to get out of
this reactor core some of them will be reflected
back that is here. Then the heat that is generated
here if that heat is to be used to produce
electricity energy purposes, then that heat
is to be taken out some kind of coolant goes
into this reactor core gets heated because
of the heat generated in fission. And, that
heat is taken away from some other channel
and then used to drive turbine and this and
that. So, these are this is the typical design.
Now, let us look at what we call four factor
formula whole idea of this fission reactor
is that once it starts it is self sustained.
Once fission reaction starts that fission
reaction produces more neutrons and those
neutrons themselves produce more fissions
and more neutrons. And, it goes on as long
as you want. So, that is the thing it is known
as chain reaction. And, that is that is because
as you know this neutron plus 235 uranium
that goes to uranium 236. And, then finally
x plus y and then you have these neutrons.
So, with 1 neutron you get these neutrons
and on the average for uranium 235 this nu
is something like 2.5 or 2.45. So, on the
average 2.5 neutrons are produced and these
are of course fast neutrons they have to be
moderated and all that.
And, what controlled fission reaction will
be that out of this 2.5 one is used to trigger
the next fission. So, every 2.5 one should
be able to produce new fission. If it is more
than 1 then you are lost the it will just
explode. If it is less than1 then it will
die this will not sustain. So, these reaction
will stop it has to be just made at 1. So,
what are the processes where this 2.5 reduces
to 1 or whatever is the number? So, at certain
stage let us say at certain time you have n thermal neutrons  in the fuel. So, at a certain time say you
have these n thermal neutrons already thermalized
in the moderator. And, then these neutrons
have reached fuel rods to make fission.
So, this is one time instant or one time level.
Now, what happens to these n thermal neutrons?
The first is in the fuel these are all thermal
neutrons. So, they have high probability of
fission in uranium 238, 235 but then they
can also get radiative capture. They can undergo
this radiative capture reaction in uranium
238 as well as in uranium 235 both. In uranium
235 also if 600 barn is the cross-section
for fission 97 barn is the cross-section for
radiative capture. And, uranium 238 there
is no cross-section 0 cross-section almost
0 cross-section per fission. But then there
is a some cross-section for this radiative
capture we are talking of thermalized neutron.
So, that resonances are out of questions.
Those resonances are beyond 1 electron volt
but even this lower thing although it is not
going through those resonances some cross-section
is therefore, that non-fission kind of. So, you talk of effective  cross-section for fission; effective also
because you have both the species.
So, average on the average what is the fission
cross-section per nucleus that is there? And,
that will be say for this is let us say 570
or 574 barns for uranium 235 but then that
is only 0.7 or if it is enriched 3 percent.
So, say 0.7 percent and then you have that
is all 0 for that uranium that uranium 238.
Whereas sigma a if you see that will be 97
here, 0.7 here and then you will have some
cross-section there some 2, 3 barns whatever
some three barns or so. So, that is there
and then you have 99.78 by 100 and the fraction
sigma f by sigma f plus sigma a. This by this
fraction it is reduced because the fraction
which fissions that only give you nu times
n. So, from these n neutrons what I am looking
for from this capital n thermal neutrons;
when they fission how many new neutrons will
be created? So, not all these capital n will
go through fission this fraction because this
sigma a is for absorption without fission.
So, this fraction only this fraction this
is you can call it total cross-section. So,
out of total only this is going for fission
and rest is going for absorption. So, this
is for fission and this is for non fission
absorption; absorption without fission  that is sigma a. That fraction will be sigma a by sigma f plus sigma. 
So, do not multiply
by just 2.5 it is not n into that nu but it
is n into nu into this part. And, that is
written as eta.
Eta is nu times sigma f sigma a plus sigma
f and number of neutrons when this fission
take place will become n times eta. So, you had n thermal neutrons  and from this n thermal neutrons what you
get is  n times eta fast neutrons. So, you get this n times eta. And, what are the typical values
of eta? If it is natural uranium this eta
will be somewhere 1.3 or so for 3 percent
enriched this will rise and this will be something
like 1.8. So, that is the factor. So, from
2.5 by that remember uranium 238 is in large
quantity. So, it brings down this number drastically.
Now, these are fast neutrons. These fast neutrons
are supposed to come out of this fuel rods
as soon as possible. So, that they get into
the moderator but it is all random phenomena,
it is all statistical phenomena. So, before
coming to that moderator; some of these neutrons
can interact with the uranium 238 which is
there and produce what you call fast fission.
Fission of uranium 238 is possible at these
energies. These neutrons which are produced
fast neutrons they are at something like 2
mega electron volts or so. And, the fission
cross-section becomes considerably it is still
below 1 barn. But after 1m e v or so that
cross-section becomes significant. So, some
of these fast neutrons can induce uranium
238 fission before they come out of in fact
they do not come out of the fuel rod; they
get absorbed and they create fission. If they
create fission once again you will have enhancement
in neutron.
So, out of that capital n into eta most of
them come out in the moderator that is how
the geometry is designed. But some of them
which create fission in uranium 238 and do
not come out in moderator; they because of
that the number of neutron will be enhanced.
So, that is written by one more factor and
that is epsilon. So, epsilon. So, what that
epsilon? This is the enhancement factor due
to fission of uranium 238 by fast neutrons.
Now, you have n into eta into epsilon. So,
this many fast neutrons are there and they
come to the moderator. Then, in the moderator
they are suppose to thermalize but then some
of them even if they are in moderator they
are randomly moving here and there. And, you
have a fuel rod here; you have a fuel rod
here although the geometry is designed, so
that most of the time they will not see a
fuel rod before getting into thermal energies.
But it is all random phenomena apart of them,
a small fraction of them can always go to
that fuel rod surface. And, interact which
that uranium 235 while still in that dangerous
1 electron volt 100 electron volt range. So,
not all of them will be thermalized. So, that
the resonance that absorption resonance survival
from that. So, there will be a factor how
many of them survive? So, typically some 90
percent, 95 percent survive but still 5 percent,
10 percent neutrons do get trapped into that
resonances region. So, that is p. So, resonance
survival factor you can call it. Now, you
have n eta epsilon and p. So, these many have
survived that uranium 238 non-fission absorption
in 1 to 100 electron volt energy range.
And, they are thermalized but even if they
are all thermalized not all of them will be
able to get into the fuel rods. Because after
all there are some small cross-sections where
the moderator material or the other surrounding
material that is there can absorb this neutron.
So, the part of thermal neutrons which are
able to get into the fuel rods. So, that is
written as f. So, f is the fraction of thermalized
neutrons going in the fuel rods. So, that is here and then you get this n eta epsilon p f. So, these
where the thermal n thermal neutrons in the fuel rod and then here you get fast neutrons
this is in the fuel rod. Then, here it is
epsilon enhancement factor; fast neutrons
enhance the things then the part of it goes
into the moderator and survives that resonance
peak that is here. And, finally the fraction
f goes into the fuel rod, and so here once
again these are thermal neutrons in the fuel
rod same as this. So, this is 1generation.
In 1generation this n thermal neutrons in
the fuel rod has become n into eta into epsilon
into p into f thermal neutrons in the fuel
rods. So, in 1 generation capital n has become
this. So, it is changed by a factor k infinity we
I will I will tell next lecture what is this
k infinity? It is eta times epsilon times
p times f. So, in 1 generation the number
of thermal neutrons ready to fission in fuel
rod that changes by this factor. And, this
is known as 4 factor formulae for obvious
reasons. There are 4 factors; 1, 2, 3, 4.
So, this is known as 4 factor formulae.
