So, in the previous lecture I talked about
the reaction rates and the reaction rate is
R. If you write n x n y and sigma v average
value of that n x n y are the concentrations
number of nuclei of x of y per unit volume
and sigma is the cross section for that particular
reaction in those particular conditions, and
v is the relative speed between the two interacting
nuclei, and this we saw that the sigma v itself
is a function of v and since the velocities
are distributed according to Maxwell Boltzmann
distribution in the plasma.
So, you have to integrate over all varieties
of velocities involved and we saw that only
a small part contributes in this integration,
because when you do integration using this
Maxwell Boltzmann probability distribution
for relative velocities. This distribution
itself if you write in terms of energy, you
get two terms one dominated by this distribution
of velocities, another dominated by the barrier
penetration probability and then one rises
with temperature with a velocity, and one
falls with velocity, and the product is contributed
only by a small range in these relative velocities,
and then we can integrate and get this sigma
v. We also saw that as temperature increases
in the beginning this whole thing increases
and therefore, the reaction rate increases
but, if one goes to very high temperatures
which are not very relevant for fusion reactors
then but, in principle it is there that this
thing may come down. Now that reaction rate
can be calculated for any given concentration
of x plus y at a given temperature one can
look at different possible constituents which
can be used in a fusion reactor. And turns
out that for making a fusion reactor the most
attractive proposition is what you called
d t reaction deuteron and a triton this going
4 H e plus n d t.
So, this is D and this is D T reaction, This
has the best promise, if you look for low
z materials, the first one is 1 H proton,
it has its own difficulties will talk about
that later the most prominent difficulty with
that proton is 1 H plus 1 H. So, you have
2 protons here about 2 protons do not combined
to diproton you do not have Di-proton in nature
Di-proton is not a stable component. So, it
has to be 2 H 1. So, that means 1 proton has
to be converted into a neutron and hence this
e plus and nu and proton going to neutron
and positron and neutrino this is governed
by what we called weak interactions and that
interaction is really very weak.
So, in the nuclear time scales when the 2
protons are close to each other trying to
fuse in that short time itself, this week
interaction reaction has to take place this
beta decay has to take place and then the
total probabilities extremely small. So, this
is not at all a good choice then D is one
possibility and D T is another possibility.
So, on many different accounts this D T seems
to be much better making this helium and neutron
and it releases some 17.6 M e V of energy.
So, and this 17.6 M e V of energy it shared
by these two in the center of mass frame if
we look at this constituent particles here.
In the center of mass frame total momentum
is 0. So, here also the total momentum should
be 0.
And therefore, if this 2 H and 3 H, if they
come towards each other in the center of mass
frame and finally, make this helium and neutron
this helium and neutrons must go in opposite
directions. So, this is 4 H e going in this
way and neutron going this way and you can
make calculations the momentum should be 0
and therefore, the velocity should be in the
inverse mass ratio of.
So, the major part of the kinetic energy will
be taken away by this neutron and this smaller
part will be taken away by this alpha particle
here. So, that is how the energy will be distributed
now the main problem comes that to do all
these things the temperature needed will be
somewhere 10 to the power 8 Kelvin or. So,
at those temperatures when these particles
in the plasma you also have electrons remember.
So, you have electrons and the nuclei. So,
these particles move with velocities and then
they scatter from each other because of the
coulomb interactions general and the accelerations
produced in these charged particles could
be very large and specially for electrons,
because the mass of electron is small.
Once the acceleration is there these charged
particles emit radiation and that is a loss
of energy from that system the energy is radiated
in terms of whatever x rays or any wavelength
of gamma rays and the. So, this Loss in energy
by this accelerated electrons. I am saying
the positive charges also accelerated but,
since heavy the acceleration is small and
therefore, this radiation loss is also small.
So, most of the radiation loss is coming because
of the electrons. So, this is known as what
we call Bremsstrahlung radiation.
So, energy is coming out and one can make
calculations the electromagnetic waves those
equations are there radiation from accelerated
charge and one can work out how much will
be the total loss because of this. So, it
will be proportional to the concentration
of the positive ions, it will be proportional
to the concentration of the electrons, it
will be proportional to Z square, and then
it will be proportional to the relative speed.
So, k T half proportional to these, this is
loss in energy because of this Bremsstrahlung
per unit volume, and per unit time alright.
So, when you confine plasma of these positive
charges x and y which are going to fuse and
create this energy create energy means remember
is the rest mass energy which is going down.
So, energy is not created in that sense but,
yes usable energy that energy is being produced
and energy is being lost here. So, the first
condition should be that the rate at which
the energy is produced should be larger than
the rate at which this radiation is emitted
and this is the reaction rate. This is number
of fusion reactions taking place per unit
volume per unit time in each reaction. If
the energy emitted is capital Q.
Then, the energy that is being produced per
unit volume per unit time energy from fusion
per unit volume per unit time, that will be
that reaction rate n x n y sigma v and then
Q. This is the number of fusion reactions
taking place per unit volume per unit time
and from each reaction Q amount of energy
is coming out. So, this is the energy which
is generated in this fusion reaction in that
plasma per unit volume per unit time. So,
per unit time per unit volume and that should
be larger than this loss that is occurring
because of this Bremsstrahlung.
All these things are now we can we have the
equations one can calculate the energy which
is produced per unit volume per unit time,
the energy which is radiated. So, when all
these calculations are done for D reaction
deuteron reaction and deuteron triton reaction
D T reaction. Then the result I will be showing
on your screen.
So, here you see on the screen we have on
the x axis side horizontal side it is the
k T 1.5 k t here, this is as a function of
temperature we are plotting and this is a
log scale. So, this is 10 to the power 0,
this is 10 to the power 1, 10 k e v, this
is 100 k e V, this is 1000 k e V and. so on.
These are all in k e V’s on this side we
have two quantities plotted one is the rate
at which energy is being produced in fusion
and that is for D plus T. It is this black
curve here and for D plus D, it is this black
curve here.
So, at these energies 10 k e V around these
energies, we have D plus T is for a given
concentration, it is larger than D plus D
that is one thing and the loss bremss trahlung
loss that is plotted here in this dotted line
and a this access y axis is also in log. So,
it is energy per unit volume per unit time
again it is 10 to the power 2 here, and 10
to the power 3 here 4th is a log scale. And
on this log scale this is linear this Bremsstrahlung
loss is almost linear.
In this log scale and now you can see that
for low energies, the loss is larger, lower
temperatures, The loss is larger and the production
is small and this is the crossover time alright
this corresponds to something like k T is
equal to 4 k e V. So, k T is equal to 4 k
e V here this loss and the production will
match with each other, and for D-D crosses
at this point deuteron deuteron reaction it
crosses at this point and this is something
like 40 k e V. k T is equal to 40 kilo electron
volts here. k T is approximately 4 kilo electron
volts. So, that is how you have to choose
the operating temperature at least it has
to be larger than this, loss there are many
other things. So, it not only should be larger
but, should be much much larger than that.
So, that is the one consideration the temperature
has to be large enough. So, that the production
of energy exceeds far exceeds the loss because
of this Bremsstrahlung. And Z square is there.
So, D and D T Z is 1 z, 1 Z 2 both are 1 but,
if you go for higher heavier element fusion
than this, loss is also going to be very high.
Another justification why 1 should go for
the low Z materials coulomb barrier is low.
This loss is low and so on. Now, suppose the
plasma is confined at temperature high enough
so that, this loss due to this Bremsstrahlung
radiation is small as compared to the energy
that is being produced but, to take the plasma
to that temperature it has to be heated in
whatever fashion. So, energy has to be given
from outside how much is that energy and how
much is the fusion energy that we obtain from
that heating normally.
What happens at these high temperatures confining
it in a small volume for a long period? It
is very difficult some plasma particles will
leak through that confined volume one. And
if they leak through they will hit the walls
of the container and the container will be
heated and the plasma will be cooled. So,
generally this confinement to maintain that
high concentration generally it is done in
a pulsed way for presently it is of the order
of a one second or.
So, for a small time interval the high concentration
are maintained and after that the concentration
goes down and then next cycle goes in, and
for the next cycle again it has to be heated.
So, the heat that we are the energy that,
we are giving to heat up this plasma to those
high temperatures has to be compared with
the total energy output in that confinements
time. So, if that concentration is maintained
for say one second in one second how much
fusion energy we have gotten from system and
to heat up? How much energy we had given to
it? So that can be estimated.
So, suppose n represents the total nuclei
concentrations number of nuclei per unit volume,
total nuclei x and y both. So, if n represents
this and if it is hydrogen isotope deuterium
is this tritium. Then the electron you have
one electron. So, if n number of nuclei is
there you will also have n electrons per unit
volume, and if the temperature rise is capital
T, then the energy which is given  to take it to temperature
T will be three second n k T for the nuclei three-second, n k T for
the electron. So, that will be 3 n k T i you take the mixture deuterium and this tritium
50. So, the concentration of x is n by two
concentration of y is n by two.
So, how much fusion energy we produce in that
confinement’s time say tau. So, the energy
which is produced per unit volume per unit
time is n x, n y alright let me write n x,
n y and then sigma v and into Q this is the
energy produced per unit volume per unit time
multiplied by tau this is the confinement
time . So, this fusion reaction in one pulse
goes for this time tau. So, this is the total
energy that has been produced per unit volume
and here is the total energy, that we had
initially given to heat up the this thing
and this should be at least equal to this
should be larger the energy which we produce
from the fusion should be larger than whatever
energy input we have given to heat up. The
plasma we are neglecting the Bremsstrahlung
radiation, assuming that the temperature is
high enough. So that part can be neglected.
Now if you put n x equal to n by 2, n y is
equal to n by 2 this side will be n square
by 4 remember n this we are taking as the
total concentration of x and y. So, total
concentration is of nuclei is n and half for
deuteron and the D n T. So, this is n by 2
this is n by 2. So, n Square by 4 sigma v
Q tau. So, if we take the equality sign that
is the threshold minimum that has to be maintained
then you can write n tau you can take here
this side and let this n cancels with this
square here, and n tau you can then written
on this side everything else you can transfer
to other side. So, this should be equal to
12, 4 into 3, 12 k T here and then that divided
by sigma v and Q and tau is there we have
already written here.
So, it should be greater than this alright
it should be greater than equal to this is
the famous Lawson criteria. So, the concentration
and the confinement time that, this product
should exceed this quantity, one can calculate
it also depends on temperature and it has
a minimum as certain temperature for D T reaction,
for this quantity minimum at around say 20
K e v K T is equal to 20 K e v. So, this is
the temperature in fusion community temperature
is described in terms of K T and in the units
of kilo electron volts.
So, at this is minimum and it is around say
2 to the power 10 to the power 20. So, what
unit’s nuclei per meter cube and second.
So, the reactor design should be such that
the concentration into this confinement time
should exceed this number. So, that the energy
output that we get is more than what energy
we are giving. So, these are some of the design
considerations. Now the next topic is this
whole fusion idea started sometimes in 1950’s.
So, it is almost more than 60 years that people
had conceived people have thought of it people
has started working on it. Mid 50’s there
were already some experiments going on to
do. This fusion the 60 years down the line
we are still not having a nuclear fusion reactor.
So, where are the problems? So, the problems
are that one is this the main problem is confinement
main problem is for confinement this high
temperature plasma 10 to the power 8 Kelvin
or. So, at high temperature plasma has to
be confined in a small volume to get this
large concentration n and no material boundary
can be used to confine that gas because no
material boundary will sustain this. So, it
has to be done in some other way and the most
widely pursued root is magnetic confinement. What is magnetic confinement? Suppose you
have some container.
So, you have a container and you have some
gas molecules and which are going in random
directions with random speeds. So, the molecules
will go and strike here. So, if you take this
kind of says imagine this kind of cylindrical
thing then the particles which are going this
way. They will go this way of course, you
will have some boundary here also is not an
infinite thing, it is not a infinite tunnel
but, let us look at this the component of
velocity in this direction this component
will take the particle to hit this wall. So,
the particle will go somewhere and hit here
now if I apply a magnetic field in this direction?
What happens the velocity component in the
direction of magnetic field? Does not change
but, if there is a component of velocity perpendicular
to it goes around that magnetic field.
If that is the only component that goes in
circle otherwise, it goes in a helix this
is the simple F is equal to q v B q v cross
B. So, it bends if you have magnetic field
in this direction and velocity in this direction
then it bends, q v cross B. So, if it is positive
particle it will go like this. So, instead
of hitting this, if the velocity and the magnitude
of magnetic field they are proper then before
hitting it can make a turn and it can go around
this. So, this is the basic very fundamental
principle of magnetic confinement that by
magnetic field you can bend the charge particles
and if it bends then there is chance that
it will escape hitting the walls another interesting
thing in this is suppose you have a magnetic
field because this kind of field will talk
stop or is likely to stop or it has some potential
to stop this.
These particles to hit these walls but, in
this direction it is going and somewhere here,
there must be a wall if it is a container
there must be a wall. So, in this direction
also it has to be stopped. It is not allowed
to hit this it should not be allowed to hit
this wall also I give you another configuration,
which is known as magnetic mirror.
Suppose you have a magnetic field which is
non uniform and here the field is strong here
the field is weak. So, if you draw field lines
here the field lines are widely separated
here the field lines are more congested. So,
that means the fields must be field lines
must be go like this. So, you have larger
field here strong field strong B field and
here you have weak B field and suppose you
have a particular which is circling this magnetic
field and at the same time moving towards
this direction towards right.
So, a particle is there which is circling
it and then is going towards right. So, it
has a velocity suppose let me take this as
the x axis, this as the y axis and this particle
is going like this. Suppose the particle is
here, it is going into the board has as well
as it is drifting towards right. So, velocity
you can write as this x, y and z. So, minus
say v z k cap this is the velocity component
that is perpendicular to this x direction.
So, it is going like this. So, this z component
here and then there is a component in this
direction is going ahead also, this component
will be plus v x I cap and the magnetic field
has this field lines are bending the magnetic
field also has a component in this direction
and in this direction. These are the directions
of the magnetic field. So, it has a component
here. So, b is equal to this is some B x in
I cap direction this and then minus B y j
cap direction. So, if you calculate q v cross
B the force is q v cross B. So, q then v x
I cap minus v z k cap and cross product with
b x I cap and minus b y J cap. Now look at
the various terms, this I cap terms cross
product with I cap terms gives me 0 this I
cap cross product with J cap gives me k cap.
So, there is a force in k cap direction. So,
that component will change.
Now here k cross I is J cap. So, there is
a force in minus J cap direction. So, that
is a. So, there is a force this is the centripetal
forces circling. So, there should be a force
towards the center of the circle. So, that
minus J cap will be there and that last term
look at that last cross term here, so minus
and minus. So, it is these two minus makes
it plus but, then you have k cap cross J cap
which is minus I cap.
So, this will be something in J cap direction,
something in k cap direction and I cap direction
I am interested to write it explicitly it
is v z B y v z and minus I cap. So, v z here
I am talking of these two terms, this is going
to give me a product in I cap direction this
combinations. So, v z b y and then k cap cross
J cap is minus I cap. So, you have a force
in negative x direction, the particle is coming
in positive x direction and there is a force
which is in negative x direction so, it decelerates.
This is strong magnetic field stronger magnetic
field here is pushing the particle back that
is how it is acting like a mirror. So, this
third dimension also can be managed with magnetic
field so that it is confined in this side
also.
You can have similar structure magnetic field.
If the magnetic field is strong on this side
also it is some kind of a magnetic bottle.
So, particle is if proper field magnitudes
are there this particle which is coming towards
x axis will be pushed back and once it reaches
here and tries to go towards this stronger
field, once again it will decelerated and
will be reflected from this strong field region.
So, these are some of the simple designs only
to say to show that magnetic properly designed
magnetic fields can contain charged particles
in a volume of course, the things are very
complex because of this very high temperatures
random velocities, random directions speeds
large and so on.
So, the magnetic fields are to be designed
in a very careful manner that is what is being
done for the last 70 years 65, 70 years people
are designing magnetic field there are other
methods also for confinement like inertial
confinement, what we call inertial confinement.
So, that inertial confinement is another approach
which people are working very seriously and
this inertial confinement is to start with
solid pallet your deuterium and this tritium
at D and T that is in the form of a solid
pallet.
And that solid pallet is hit by from various
sides intense energetic particles all or photons.
So, let us started with lasers, so from all
sides put very high intense lasers on this
pallet, and then this pallet gets all that
energy, and it evaporates makes plasma there
that much of energy goes in. So, the temperature
rises and that is how the plasma is created
but, there is no confinement acts at it as
only because of the pressure, because of the
pressure of these beams that it is confined
but, once it is evaporated it does expand
the pallet does expand but, for whatever time
because of this beam from all sides compression
that shock waves that hit that shock wave
keep going in this material. So, for some
time it is confined that plasma is created
high temperatures are created and confinement
is created. The normally in this particular
design the confinement time is say nanoseconds
or. So, in magnetic confinement we are not talking
of seconds and minutes and. So, on but, here
it is still nanoseconds and the density of
course, are quite high because there it is
all gases here it is solid compressed by all
these things. So, the density is because of
that compression is very high and one still
reaches the same kinds of n into tau concentration
into confinement time which is comparable
to what one has in magnetic confinement. So,
this is another way on which people have worked
very seriously very intensively and again
once again there was some a period when this
particular inertial confinement results was
going low but, now again it is revived and
people are thinking this as another possible way.
Coming back to this magnetic confinement a
particular design is nineteen sixty eight
in soviet rules this design was constructed
and there we people found that, this concentration
and confinement that was very high as compared
to all other kinds of magnetic confinements
that was being tried. Since then almost the
entire activity has is going on which that
particular design which is known as tokomak design.
What is that tokomak design? As I told you here if you have a magnetic field in a particular
direction then the particles can go around
and in that transverse direction particular
fields can be applied to avoid collision with
the walls but, then on the longitudinal side
there is a problem alright that you have to
do some kind of reflection or something but,
if this cylindrical thing itself is bent and
joint to make some kind of ah motorcycle tube
structure, doughnut structure, Taurus structure.
So, the cylinder is bent. So, this is the
volume in which you create that plasma at
in this volume you try to confine it. So,
there are no ends has such. So, that confinement
in that direction is automatically taken care
of. So, you have cylindrical thing and that
whole thing is joint here like tube or Taurus
or doughnut of that type and the magnetic
field which are needed for confinement are
going like this. Just like here. If it is
a big cylinder and you are not worried about
the end we are only worried about this cylindrical
wall then you can create field like this and
this field will keep it moving.
So, similarly, here field of this type and
then the plasma will just go around this and
will be confined and how to create that field?
For that one can put large electromagnets
here you can put coils you can put coils around.
So, that those coils and pass current and
from there the magnetic field can be created
but, then in this design there is an inherent
problem and that is on the inner side on the
inner side the concentrations of this coils
will be larger on the outer side it will be
smaller because of the radius difference and.
So, the field will not be uniform inside this
volume the field will be stronger here and
the field will be weaker here alright in it
if a solenoid a cylindrical solenoid the field
is uniform everywhere but, if you have the
steroid described and then you put windings
there and from there pass a current and produce
a magnetic that magnetic field is strong the
on the inner side and weaker on the other
side and you have seen that stronger magnetic
fields tend to push the particles back.
And therefore, this magnetic its magnetic
field itself will help these particles to
go hit the outer walls on the outer side and
therefore, at such more complicated fields
are needed. So, what this is known as Toroidal
field of course, this is the basic that field very much there toroidal  on top of it another field called Poloidal
field  is added and that field is circling the Taurus like. So, this is the Poloidal field and this
type of field can be created by passing a
current along this line along the length of
this toroid along the length of this doughnut.
So, we have a several circles here. So, one
circle is for this is the magnetic field,
Toroidal magnetic field and then we have another
circle here which is say current. So, the
combination of these fields properly designed
this toroidal field and poloidal field is
able to contain the plasma for some reasonable
time. So, this is basic of Tokomak Design and this
design is being pursued in India we have in
Ahmadabad, Gandhinagar. We have this Indian
institute for plasma research where people
work on this design the name of that reactor
is adhithya where people have studied all
these things round the world people have worked
on this design. Once it came into light by
the soviet scientists in 1968 the current
status the current status is that several
countries United states, China, India, Japan,
South Korea and this Russia the seven European
union. These 7 partners have joined hands.
It is a big international collaboration almost
half the population of the world is represented
by these partners and they have now in collaboration
building up a machine known as I T E R.
So, International Thermonuclear experimental
reactor, so all these partners they are building
up different components and this whole thing
will be assembled here the status is that
it is being constructed in France, the construction
that groundwork there and the site started
2008 Likely to go for fusion in 2018. So,
the things will start producing fusion power
in 2018. So, this is the construction period
or the preparation period the partners are
doing their job in their places India is also
a partner in 2005 India joined this collaboration
and that i p r they are working on some kind
of this in the currently on the currently
operating machines joint this European Taurus J E T.
Joint European Taurus working for since, second
half of 80s 85 or. So, this is a machine which
has says maximum efficiency shown. So, 1997
or. So, it has produced a peak of something
like 16.1 megawatt and the energy produced
was around 75 70 percent of the input energy
given. So, that is the best. So, still the
energy being produced by these fusion reactors.
The most advanced types of fusion reactors
after 60 years of experience is still not
matching with the energy that is needed to
heat of the plasma energy that is being input
to the machine and this 70 percent 65 7 percent
it is taken as a big achievement but, then
all these reactors there are many more all
these reactors have given a very high level
of understanding of this behavior of plasma
and controlling the plasma in all those thing.
So, finally, the design for I T E R which
the scientists have made and which is approved
and on which things are going. This is the
design if everything works alright. It is
the factor that energy output divided by energy
input the factor is supposedly it will hit
the number 10.
So, this I T E R the goals is that 50 Megawatt
input and then 500 Megawatt output so, 10
times. So that is the kind of design have
been created to all simulation on all calculation,
on the all equations and this and that, it
is still experimental reactor International
Thermonuclear Experimental Reactor is still
not a commercial units to produce electricity.
If everything works out well then the next
phase which will starts something 2024 or.
So, known as demo, they are it is expected
that a prototype commercial fusion reactor
will be created and one can expect real output
somewhere in 2050 or. So that the current
scenario of nuclear fusion reactors but, people
are very enthusiastic because the they have
come a long way and now the kind of understanding
and the controls in the engineering that has
been developed through various reactors working
in different parts of the world.
This particular design has been made and it
is expected that they will succeed in doing
that and if it happens like that it will be
a big thing for our planet, because this will
be a source of energy where the fuel will
be abundant where the radioactivity problems
will be small and all kinds of ever increasing
energy demands can be met. So, that is how
it is it is going. So, with that I will close
this thermonuclear reactors terrestrial thermonuclear
reactors next lecture I will go how fusion
takes place in the sun no scientists working
there and still the sun is the or the starts
are the best nuclear fusion reactors where
hydrogen is being converted into helium and
fusion energy is being produced for billions
and billions of years. So, how that sun works,
how the star works little bit of nucleosynthesis
little bit of nuclear astrophysics that will
be our next chapter.
