Today, I would like to begin discussing a
very major application of plasma physics that
is thermonuclear fusion.
In this we will discuss basic fusion reactions,
primarily the deuterium tritium reaction that
is used to produce energy. And we will discuss
Lawson criterion for energy breakeven and
then we will discuss two approaches to fusion.
One is called magnetic fusion and second one
is called inertial confinement fusion also
known as laser driven fusion. And then we
will begin discussing some basics of toroidal
plasma confinement, which is in at the forefront
of fusion ratio is a device called tokomak.
Let me begin with the basic equation, deuterium
and tritium are considered to be the major
source of a major thermonuclear fuel. If they
can be fused the nuclei of deuterium and tritium
they could be fused then, they will produce
a helium nucleus called alpha particle with
atomic number 2 and atomic weight 4. Deuterium
has atomic number 1, atomic weight 2, tritium
has 1 and mass number is 3. Deuterium and
tritium are isotopes of hydrogen, if they
can fuse, the nucleus can fuse with each other.
Then, they can produce a helium nucleus plus
a neutron which has 0 charge, but mass number
is unity and in this process 17.6 M e V of
energy is released. The energy that goes to
alpha particle is 3.5 M e V and the energy
that goes to the neutron is more 14.1 M e
V.
Well, deuterium and tritium are isotopes of
hydrogen. Deuterium could be recovered from
sea water with at reasonable cost. And tritium
can be recovered from lithium, which is quite
in abundance in the soil and about 7 percent
of lithium is 3 Li 6. If you bombard this
with a neutron, then this will produce a alpha
particle 2 He 4 and tritium nucleus which
is 1 T 3, in the process 4.8 M e V energy
is released. And very significant amount of
sea water is deuterium, normally water is
H 2 O, but typically around 0.1 percent or
slightly less than that of sea water is due
to D 2 O. So, deuterium is available in sea
water and one can recover it at not too higher
cost.
The problem is that, if you want to undertake
this reaction, then the fusion has to, the
nuclei have to overcome a very strong repulsive
force between the two, because both are positively
charged. This has a proton in the nucleus,
this also has a proton, when they come very
close to each other, they repel via column
force. So, they have to be brought in within
the range of nuclear force, so that they could
fuse for there the temperature that are needed
are typically of the order of 10 to the power
8 degrees Kelvin. So that, the nuclei could
cross the column barrier and can fuse. A important
quantity of consideration is called fusion
cross action, which depend is strongly on
the energy.
So, if I plot here a graph fusion energy as
a function sorry; fusion cross section, this
is I am plotting in per meter cube; in meter
square. And I plot energy of deuterium when
it is bombarding or coming close to the tritium
and this I write in K e V, then suppose this
10 here, this is 100 here and this is 1000
here, this is logarithmic scale. And the fusion
cross section, if I write here 10 to minus
32 meter square and let me write 10 to the
power minus 30 here and let me write 10 to
minus 28 here.
Fusion cross section below 10 K e V energy
is small. So, it begins from here, it peaks
around 100 typically like this, the curve
was like this and then falls off something
like this. So, what is required is, that you
must have enough kinetic energy of tritium
nucleus may be 10 K e V or higher so that,
the fusion cross section is substantial. Even,
if the plasma temperature is 10 K e V in the
Maxwellian tail, if I plot the distribution
function, energy distribution function f of
electrons as a function of energy then this
goes like this. So, there are substantial
numbers of atoms or ions in the Maxwellian
tail.
So, even if the average energy at which f
peaks could be around 10 K e V, the number
of ions with 20 or 30 K e V energy are also
quite is very substantial and then they can
cause thermonuclear fusion. But 10 K e V certainly
is some sort of a guideline that one should
aim to achieve. So, people have focused their
attention in achieving plasma temperatures
of about 10 K e V and please remember 1 K
e V corresponds to about 10000 degrees Kelvin.
And hence 10 K e V temperature is around 10
to the power 8 degree Kelvin, 10 crore degree
Kelvin which is a huge temperature. No container
can sustain such high temperatures and hence
one should worry about the confinement of
such hot plasma, all state of matter is in
the plasma state at such high temperature.
So, primarily we are dealing with deuterium
tritium plasma undergoing thermonuclear reactions
of deuterium and tritium, and producing alpha
particles neutrons by other reactions. Well,
two things are important in here, number one
that one should be able to heat this mixture
to such a high temperature, this is first
requirement and second thing is that once
the mixture has reached that temperature of
10 K e V or so. And fusion reaction is started
taking place, then the alpha particles that
are produced in the reaction, they should
be trapped within the plasma and they should
be able to heat the plasma or sustain the
heating, sustain the temperature that is called
ignition.
So, two important considerations are that
one should achieve breakeven, means the nuclear
energy produced via D T reactions should exceed
the amount of energy that you have put in
to heat the plasma. And secondly, the alpha
particles that you produce should be able
to sustain the temperature.
Now, let me make some estimate of energy equilibration
or energy equality, that I want that the energy
output to be equal or greater than energy
in, if W in is the energy input to heat the
plasma and this is the energy output. Energy
input you can certainly think in terms of
particle density etcetera in temperature.
If your system has supposed electron density
as n, obviously deuterium density would be
half of it, because half the electrons are
released from by deuterium and half are released
by the tritium so, this will be n by 2. Tritium
density I can take to be n by 2 also and temperature
of suppose these 3 spaces they may differ,
that suppose typical temperature I choose
to be equal to T and the volume of the plasma
is suppose V.
So, the total number of particles in the system
are n V and the kinetic energy of every particle,
if I assume them to be a Maxwellian having;
Maxwellian distribution function is 3 by 2
into T, where Boltzmann constant is hidden
here, but this is a number of electrons. If
you count the total number of particles; electrons
plus deuterium ions plus tritium ions, then
this becomes 2 n so, this is the right equation.
So, total energy input that you have to provide
to initiate the process is so much, where
T has to exceed 10 K e V. Now, let us see
how much energy one can produce, if the confinement
time of the system is tau. Suppose, this mixture
is tends there for a duration of the order
of tau.
Let me estimate that, the number of fusion
reactions a particle will go, it is a deuterium
will undergo with tritium is defined as I
will call this nu f, which is equal to the
number of tritium ions per unit volume into
fusion cross section into velocity of deuterium,
thermal velocity of deuterium. Collision frequency
normally is a product of density collision
cross section into thermal velocity this is
the kind of thing. And in each reaction you
are producing energy say E f so, let me call
E f as the fusion energy per reaction which
is 17.6 M e V, fusion energy produced per
reaction, nu f is the fusion collisions each
deuterium ion is undergoing with the tritium
ion.
Now, to calculate the energy output W out,
this should be equal to total number of fusion
reactions. Now, in a unit volume the number
of tritium ions is n D, they go nu f each
of them undergoes nu f collision per second
fusion cross section collision per second
and in the process produces so much energy.
So, this is the amount of fusion energy produced
perturbed volume multiply by the volume, this
is the total energy output. If I put n D is
equal to n by 2, where n is the electron density.
And n T also as n by 2, then this can be written
as n square by 4, n by 2 from here, n by 2
from there, multiplied by sigma f into v D
into E f into V. And I want this to exceed
the energy, that you have put in which is
equal to n V into 3 times T.
Please remember, sigma f is a very strong
function of energy. And v D also depends on
temperature it goes as under root of temperature.
So, this equality if I put and please I made
a mistake here. This W out is the amount of
energy produced per second, if the confinement
time of the system is tau. Energy confinement
time I may call it tau E, then I must multiply
tau here also, this gives me one n will cancel
out, a product of density of electrons into
energy confinement time of the plasma, this
should satisfy certain inequality. And if
I take the equal sign here, then this is called
the Lawson criteria.
And when you put these numbers here, what
you find is, that density and energy confinement
in time product must exceed or should be of
the order of 10 to the power 14 per centimeter
cube into second at an electron temperature
of or plasma temperature of 10 K e V, at other
temperature this number will be different.
But primary goal of fusion research in last
30, 40 years has been to achieve plasma density
into confinement time product satisfying this
condition.
Well, as far as ignition is concerned, if
you want to have the burn continue by extracting
energy from the alpha particles, what you
require. That whatever energy is produced,
W out that you call sorry, let me reframe
this, what I want is that the energy produced
in alpha particles per second, this must be
equal or bigger than the energy lost via radiation
or other process is. If the plasma energy
that we call as energy in, which was 3 n T
into V, where V is the plasma volume, n is
the density of the electrons in the plasma
and T is the plasma temperature. This is the
energy that is at any instant of time contained
in the plasma particles.
If the plasma confinement time is tau E so
that, is that much time so much energy is
lost so, let me define a energy confinement
time as tau E, then this is the energy lost
per second, typically this is the estimate.
And energy produced per second will be how
much, I just estimated this quantity. So,
that quantity was W out is the total energy
produced in alpha particles plus neutrons
17.6 M e V W f. If I divide this by multiply
by this by factor of energy of a alpha particle
divided by rather let me put some symbol here,
E alpha upon energy released in a few reaction.
This is 17.6 M e V and alpha particle is 3
point M e V so, this quantity is like one
fifth so, this if this exceeds this number,
then we say that we have achieved ignition.
And you do not require any more heating mechanism,
but this factor energy released per nuclear
fusion reaction is E alpha for the alpha particle
and total energy released is E f which is
like 17.6 M e V. So, this ratio is one fifth
so what you require is this. But first generation
experiments on fusion focused on achieving
Lawson criteria this requires higher value
of n tau product.
Now, there are two schemes to achieve this
product, one is called magnetic fusion, where
you employ magnetic field to confine particles.
We have already seen that, if you have straight
lines of force like these magnetic lines of
force, then the charged particles may be electron,
if they move they are introduced in the system.
They will be confined across the line of force
though they can they are free to move along
the lines of force.
So, there is a confinement in two dimension
two transverse dimensions, but there is no
confinement if the lines of force are straight
forward in this direction along the length
of the lines of force. Then, we came across
the mirror confinement that even if the lines
of force are straight forward, but if you
can create a lines of force like this that
they converge like an a mirror machine, then
the electrons and ions which are gyrating
over the field lines here. When they come
close to the throat and if they have large
pitch angle, they can come back and they will
be confined in the system.
So, magnetic field can provide confinement
in three dimension, it is a different matter
that this is a not a stable confinement. Because
any small perturbation called interchanging
perturbation gives rise to disruption of the
plasma confinement or disturbance or destabilization
of plasma confinement. But certainly one possibilities
like this, that you can confine the plasma
by using magnetic field such that the lines
of force converge towards the end of the machine
and plasma is filled in here. Then, one can
think of a confinement by using close lines
of force and that is called toroidal confinement.
The toroidal confinement is like this. Suppose,
I take a current carrying wire of very large
length, then the lines of force along the
wire are circular, they close on themselves.
And if you can create a plasma somewhere here
in this region, suppose I create a plasma
in this region. So, the plasma if I can create
in this region, one would expect that the
particles in this region, because they are
moving along the field lines and they will
be gyrating about the field lines and may
be confined. But what we had seen that in
such a situation, because the lines of force
have a curvature R c was in the outward direction
and they also have a in homogeneity in the
magnetic field and that gives rise to drifts.
The electrons move in one direction and ions
in the other direction perpendicular to the
plane of this board by the v cross B force
and that produces a vertical electric field
perpendicular to the plane of the board. So,
what really happens is that, the electrons
undergo a force a drift which is given as
F cross B s, upon e B s square and this drift
is 0th order drift. And this force is primarily
the centrifugal force, which is m v parallel
square upon radius of curvature R c of the
line of force where the particle is located
and that is in the radial direction R c. So,
what really happens here is that, under this
force there is a negative sign here, that
if magnetic field is in this case is in the
z direction.
Suppose, I choose a magnetic field in the
z direction like this; sorry not magnetic
field, magnetic field is in the toroidal direction,
B s in this case is parallel to phi direction.
The current that produces this magnetic field
is in the z direction, in that case, this
is in the phi direction; this force is in
the radial direction so, R c radial cross
phi is in the z direction this is minus z
direction. So, electron drift is parallel
to minus z cap and ion drift is parallel to
z cap. So, a very simple configuration that
you have a current carrying conductor in the
z direction perpendicular to the plane of
this board, it will produce magnetic field
a circular line or close lines of force. But
this will produce a drift of electrons in
the minus z direction and ion drift in the
opposite direction.
So, there will be a charge separation created
between these two if you do not compensate
for it. And that charge separation will produce
an electric field E z and this electric field
will give rise to E cross v drift of electrons
ions both in the same direction and that turns
out to be in the radial outward direction
and plasma confinement is not possible. So
though, there is a good possibility that if
you can compensate for these drifts, you can
have particle confinement in the close lines
of force. However, this is a very serious
challenge to compensate for these drifts of
electrons and ions caused by curvature drift
and grad B drift. Grad B drift is in the same
direction as curvature drift is just adds
to it so, this is the serious matter.
What was really found in 1950s late 50s was
a theorist. That if you can have a current
in the plasma in the azimuthally direction
in the phi direction. The same direction in
which a DC magnetic field, azimuthally magnetic
field is produced by this current carrying
wire. Then there is a possibility to compensate
for to negate these drifts and that is a very
important contribution, we shall look into
this in detail either today or in our next
lecture. But that is a very important contribution
to plasma physics that just by passing a current
in the system in that azimuthally direction
just in the plasma have a current in the same
direction in which the magnetic field exist
in this direction, have a current in the plasma
then, you can have this.
This device is a; will produce at toroidal
plasma and a more effective or more economic
way of producing such a magnetic field is
not to have a single wire along z direction.
Rather a configuration that has come up is
like this that you have a torus, torus is
like this. So, close this on itself, close
this curve on itself like this and this one.
This is a torus like a bicycle tube which
if you straighten will be cylinder and if
you bend the cylinder, it will bend on itself
to form a torus. So, plasma is filled in this
here and by using magnetic field coils which
go like this schematically. You will produce
a magnetic field in this direction, this is
the direction of static magnetic field called
B phi. And you do not require a conductor
passing through the center of the torus, rather
these field coils are very close to the plasma
to this region, where you want to create a
plasma they can produce a magnetic more effectively.
So, and if you can treat this entire torus
as a secondary of a transformer, it should
be possible to induce a current also. So,
current is also you can create here in the
same direction and let me call as I phi, plasma
current. That can produce a suitable magnetic
field that can neutralize the drifts caused
by grad B drift and centrifugal force or curvature
drift. So, this is the kind of more economic
configuration for toroidal confinement and
this device is known as Tokomak 
and this is at the forefront of fusion research.
However, let me before I delve into Tokomak.
Let me also, talk to you about the other scheme
of fusion which is called inertial confinement
fusion. In this scheme, one employer one considers
a palate, a spherical palate of deuterium,
tritium. And you shine laser light from all
directions, usually 16 beams, laser beams
are employed they are launched from 16 different
directions on to a deuterium tritium palates,
whose radius could be of the order of a centimeter.
And it is a deuterium tritium mixture, where
if you calculate the atomic density of deuterium
tritium, then that will be typically of the
order of 10 to the power 22 atoms per centimeter
cube.
So, if you convert this deuterium tritium
when the laser arrives here, it converts this
deuterium tritium mixture. And this radius
of this from here to here is probably, I call
this as r 0 is around a centimeter, then a
hot plasma is created here. And as soon as
the plasma is created, this density is more
than the critical density for this laser.
Suppose, the laser frequency is omega, then
the plasma frequency that you produce is around
5 into 10 to the power 15 radian per second
and if it is a few times 10 to 22, then this
will be slightly more sorry this is omega
p. The frequency of the laser that you employ
here, either have 0.8 micron wave length,
they are called T I suffice laser or 1.06
micron wavelength called neodymium glass laser.
So, typically the wave length is around 1
micron. So, if I take 1 micron here, then
omega turns out to be typically 2 into 10
to the power 15 radian per second. Obviously,
this omega is less than omega p so, the laser
will not penetrate once the plasma is created
and once the plasma is created, it will deposits
energy only on the surface and surface will
expand. So, when the surface expands, you
have a lower density outer region called corona
and this is this happens within a fraction
of a peak of second. So, once the laser arrives
here, it quickly converts the plasma this
into a plasma and a hot outer region is; are
called corona is produced.
Well, the physics of this process we shall
discuss in detail later, but it would suffices
to say here, that the hot plasma which is
that the plasma which is create outside. It
will be part of it will be transparent to
the laser, because there will be some layer
here at which the laser frequency equals.
This is the layer some sort of a layer here,
at which omega p is equal to omega, inside
this omega p is bigger than omega the laser
frequency and hence that is a over dense plasma
as far as laser is concerned. So, laser will
penetrate up to the critical layer and it
will deposit a synergy is if get heated. What
you get a situation in which the outer region
is very hot and hence pressure is very large,
whereas, the inner region is relatively cool;
cold.
And there is a sharp pressure gradient created
near the critical layer and that will launch
a shock wave inward and compress the core.
So, suppose the core has been compressed to
a radius r 0 prime, this is the core with
in the process of compression will get heated
and where you expect thermonuclear fusion
reactions to take place. Suppose, you can
confine this plasma where the density as also
increase, when you compress the core the density
which was initially is so much will increase,
let me initial density let me denote by n
0, the new density I will call as n 0; as
n.
So, the modified density and the confinement
time of the compressed core, this should exceed
10 to the power 14 per centimeter cube into
second to achieve for the Lawson criteria.
This is what you require so, this is the density
of the compressed core and this is the confinement
time of the compressed core. Because compressed
core after this compressed, it will start
expanding because it has been heated. And
typical confinement time, you can estimate
by considering the process to be ambipolar
diffusion process or ambipolar expansion.
Let me explain what is ambipolar expansion,
suppose I have a core of deuterium tritium
heated, if I assume that the electrons and
ions are at comparable temperatures. So, if
T e is comparable to T i is still thermal
velocity of electrons is much bigger than
thermal velocity of ions due to their different
masses. So, what will happen, the plasma which
is filled in here have electrons moving with
large thermal velocity so, these electrons
will quickly get out they will come out here.
However, when they come out, there is a net
positive charge left behind and that push
these electrons back. So, the net positive
charge is left out in the inside and that
pushes these electrons back and does not allow
the electrons to leave, a distance more than
divide length. But this positive charge will
repel the ions themselves and consequently
ions will move out. So, once ion can go out
to a some larger distance suppose, ions can
come out to this region this distance, then
the electrons can go further out they can
move little more. And in this way the electrons
and ions as a assembly moves this charge separation
of the order of divide length which is quite
small. So, this is called ambipolar diffusion.
And ambipolar for ambipolar expansion, in
which the ions and electrons move together
they move together. And the velocity with
which this happens, it turns out to be of
the order of C s called sound velocity and
which is equal to electron temperature upon
ion mass. If the plasma is a mixture of deuterium
tritium, then this ion mass is some sort of
a geometric mean or some mean of tritium and
deuterium masses. So, this is the kind of
a speed one gets. And if, the initial radius
of the plasma is suppose r 0 dash, then the
time of confinement would be of the order
of, if the radius of compressed core divided
by C s.
Please note, that the original radius of the
palate that we consider was around a centimeter,
this compressed radius will be a only a few
mille meter. Sound velocity, if you calculate
at a temperature of about few K e V, then
this will be of the order of few times 10
to 7 centimeter or may be close to 7, 10 to
8 centimeter per second so, this is typically
of the order of a nanosecond. So, if it is
like a nanosecond or few nanoseconds, then
the density that you would require should
be certainly more than 10 to 23 or so to achieve
Lawson criteria, it means this compression
is mandatory.
But there is a beauty in there that, in the
Lawson criteria n tau product there is of
our interest, which I want to be more than
10 to the power 14 per centimeter cube into
second. This if the size decreases of the
palate, then this will increase as one upon
radius of the palate to the power 3 inversely
to it is varies inversely with the radius.
So, if the radius becomes half, the density
will become 8 times if it becomes one third
it becomes 20 times larger, whereas, the confinement
time scales as r 0 dash by C s. So, r 0 dash
means, when you take the product this scales
as 1 upon r dash square means, compression
is suddenly helpful to achieve this.
And one can achieve and people have achieved
this product, but obviously you require this
to be satisfied in conjunction with electron
temperature also to exceed 10 K e V. So, this
condition has to be satisfied in conjunction
with electron temperature of this order. Well,
this scheme we shall discuss separately in
one of the lectures of inertial confinement
fusion, it has made tremendous progress over
the years and I think I will return to this
in a separate lecture. Today, I would like
to take you back to toroidal confinement and
let me discuss some basic features of toroidal
confinement and then we shall return to more
details in our next lecture.
So, a primary thing in toroidal confinement
is 
to counter the grad B and curvature drifts
and for that people thought less a plasma
so, the tokomak if I have a schematic of a
tokomak like this. Where plasma is this is
the axis of the system, axis of symmetry perpendicular
to the plane of the board, which I call as
z axis. And let me these call this as my x
direction, this is my y direction and we use
a right handed coordinate system. So, what
you will have, that I am having a plasma which
has a toroidal magnetic field B phi and it
also has a toroidal current I phi. I phi is
not uniform, I do not know what kind of variation
I phi I should have to have a optimum confinement,
but what will I expect is that, this toroidal
this azimuthal current I phi will produce
magnetic field.
And the lines of force will be perpendicular
to this direction, if I draw a plane perpendicular
to phi direction, suppose I draw a plane like
this, then the lines of force will be circles
about this phi direction phi cap. So, I am
expecting just like in case of a wire which
was carrying current like this the lines of
force were circles. Here also I am expecting
the lines of force to be circles about these
bent lines of force, this should be circles
in a plane perpendicular to this, if I draw
a plane perpendicular to phi direction then
the lines of force will be like this. So,
this is my phi direction normal to the plane
of this board, then these will be the lines
of force, I will call this magnetic field
as B p poloidal magnetic field.
So, there are two kinds of magnetic fields
in the system, one is a magnetic field produced
by the external coils like these, they are
the coils that will produce a magnetic field
B phi and a current in the phi direction will
produce a magnetic field that I call B p.
I would like to discuss a constant of motion
and that will give you as a clue of plasma
confinement, just like in a mirror machine
I introduced a quantity called magnetic moment.
Here also I would like to introduce a similar
quantity.
Let me write down the magnetic field in the
plasma as p is equal to B phi, phi cap this
is the toroidal magnetic field produced by
external currents coil plus the magnetic field
produced by the plasma current, which I call
as B p also called poloidal magnetic field.
Now, as magnetic field can always be expressed
as curl of a vector potential, I can write
down B p also as curl of a vector potential
I will call as A p, the direction of A p is
in the direction of plasma current. If I am
taking the plasma current in the phi direction
then I expect this A p to be also in the phi
direction.
So, A p I will choose as A p and phi cap for
an axisymmetric system and this is the approximation,
we always make for tokomak. So, axisymmetric
system is the one axisymmetric for which all
the quantities like current magnetic field
as they do not depend on phi. Means, all points
inside the tokomak have the same likely or
having same value of magnetic field here,
here and here so, magnitude of this quantities
do not change with phi so, choose delta phi
equal to 0. Now, in this case I would like
to write down the azimuthal equation of motion,
azimuthal equation of motion. Which is 1 upon
R d dt of m R square phi dot is equal to the
force on the electron due to the magnetic
field which is v cross B force and I must
write down the phi component of this.
Now, this is the minus e is the charge of
the electron, if I write down the components
this will be v z and B r minus v r B z. B
r and B z are both the components of magnetic
field B p, because the toroidal magnetic field
is the phi direction and we are not interested
in that component. We are interested in the
radial and z components of b field so, which
are deducible from curl of A p.
So, let me write down the radial and z component
of magnetic field. So, B p we have; has curl
of A p and B p; radial would be how much,
this is simply called as B r radial component
of B p is B r and this will be equal to r
if I write down. I must write down a phi component
here, z here, minus z here and phi there.
So, this will turns out to minus delta delta
z of A p this in the phi direction. And B
z that I need will be equal to radial component
of this operating over the phi component turns
out to be equal to 1 upon r, delta delta r
of r A p, A p is a phi so, these are the two
components.
If, I substitute these two in the equation
of azimuthal motion, I get 1 upon r sometimes
people denote this a small r, sometime we
call as capital R, we are writing as capital
R is the simply the variable coordinate or
measure radius coordinate. So, this is d dt
of m R square phi dot is equal to minus e
v z B r. Now, v z I write down as say dz by
dt, this is my v z into B r, which is this
quantity so minus delta delta z of A p. Then,
a term minus v r, which I write down as dr
by dt, this is for v r multiplied by d z,
which is this term. So, it becomes 1 upon
r delta delta r of r A p this is what I get.
What I can do here? I can multiply this equation
by R and divide by R, I can write down please
this r and capital R no different.
So, let me use the same symbol here put a
capital R here rather than a small r. So,
if I do this and take small r common, I can
take this is equal to a minus time also common,
it becomes e upon capital R. And then this
can be rewritten as delta delta z of R A p
into dz by dt plus this term, which is delta
delta R of R A p into dR by dt. Here R, I
have taken inside the delta delta z operator,
because R is independent variable independent
of z so, I have taken this inside. You may
treat this as a full derivative of R A p with
time, because A p does not depend on time,
explicitly it depends only on R and phi r
and z and there is no phi dependence.
So, this entire quantity can be written as
e upon R into d dt of R A p and if you look
at the left hand side and right hand side,
1 upon R is common in both of them and consequently
you get a constant of motion and which I would
like to write down.
The constant of motion turns out to be m R
square phi dot minus e R A phi is a constant
of motion. This is very important constant
of motion and it has tremendous implication
for plasma confinement in a tokomak, just
like magnetic moment has implication for plasma
confinement in a mirror machine. And I think
implications of this we shall discuss in our
next lecture. We are coming close to end for
todays talk and in the next talk we will discuss
the implication of this on plasma confinement.
Thank you very much.
