SEMINAR
Nuclear Fusion and Tokamaks
Klemen Strni²a
Mentor: Jernej Fesel Kamenik
Fakulteta za matematiko in ziko, Univerza v Ljubljani
October 6, 2010
Abstract
In this seminar the basic principles of controlled nuclear fusion on earth will be presented along
with what seems to be the most promising solution, regarding current technological development,
to achieve it. I will discuss what nuclear fusion is and the choice of the optimal fusion reaction to
achieve fusion in controlled conditions. This is followed by the explanation of the role of plasma
and what seems to be the best way to contain and control it. The second part consists of the
description of how is this achieved in a special type of fusion reactor, the tokamak.
1
Contents
1 Introduction
3
2 Nuclear fusion
4
2.1
Fusion in the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Fusion on earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3
Thermonuclear fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.4
Break-even and ignition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.4.1
Fusion reaction rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.4.2
Thermonuclear power and losses . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.4.3
Ignition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.4.4
The Q factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3 Tokamaks
11
3.1
Magnetic connement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.2
The tokamak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.3
Quality of connement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.3.1
The β factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.3.2
Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.3.3
The q factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
Plasma heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.4.1
Ohmic heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.4.2
Neutral beam injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.4.3
Radiofrequency heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Tokamak reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.5.1
Fuel cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.5.2
Blanket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.5.3
Limiters and divertors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.5.4
ITER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.4
3.5
2
1 Introduction
The discovery of nuclear fusion is tightly coupled with the question of where does Sun get it's energy.
Physics in the 19th century had only two answers available, chemical and gravitational energy. With
the mass of the Sun measured, the known chemical fuels would provide sun's power output for only
a couple of thousand years. And gravitational energy, that is, work done by the gravitational force
contracting the sun, accounts for a couple of million years. None of those two explanations conform
with geological evidence about the age of Earth.
The realization that the energy radiated by the Sun is due to nuclear fusion followed three main steps
in the development of physics. The rst was Albert Einstein's famous deduction that mass can be
converted into energy with E = mc2 . The second step came with precision measurements of atomic
masses, which showed that the total mass of four hydrogen atoms is slightly larger than the mass of
one helium atom. That led physicists, around 1920, to propose that mass could be turned into energy
in the Sun if four hydrogen atoms combine to form a single helium atom.
When fusion was identied as the energy source of the Sun and the stars, it was natural to ask whether
the process of turning mass into energy could be performed on Earth. This presented enormous technical diculties since the necessary conditions were high density of reacting particles and temperatures
in the order of hundred million degree. In the Sun these conditions were created by the gravitational
force. The problem of how to contain fuel at such temperatures for the necessary amount of time
was attacked from two dierent angles. The rst idea was to conne it with magnetic elds and the
second one was to heat and compress it enough to burn before it can expand.
The considerable scientic and technical diculties encountered by fusion research programs have
caused them to stretch over the last ve decades. But the scientic feasibility of thermonuclear
fusion via the magnetic-connement route has been demonstrated by current experimental reactors
and ITER, the rst fusion reactor with a planned positive eciency, is being constructed. Progress
coupled with enormous available fuel reserves and almost zero pollution makes nuclear fusion one of
the most viable answers to growing energy problems.
3
2 Nuclear fusion
When the mass spectrometer was invented and the masses of individual atoms could be accurately
measured it became apparent that the mass of atoms containing more nucleons was less than the sum
of the masses of separate nucleons. The dierence in mass per nucleon is called the mass defect, and,
when multiplied by the velocity of light squared(according to E = mc2 ), it represents the amount of
energy associated with the nuclear force that holds the nucleus together.
Figure 1: binding energy per nucleon against the mass number for dierent elements
So if two hydrogen atoms could be combined to form an atom of helium the dierence in binding
energy would be released. Once this was known it was quickly recognized as the principle behind the
process fueling the Sun.
But even though the nal state, one helium atom, has lower energy than the two hydrogen atoms,
an energy barrier due to the mutual repulsion of their electrostatic charges must be overcome before
fusion can occur. The nuclei must be brought close enough together for the attractive nuclear force
to become stronger than the electrostatic repulsion.
Figure 2: the potential energy of the two nuclei against the distance between them
Quantum mechanics helps here a bit since the reacting nuclei can tunnel through the barrier and
their energies don't have to be strictly higher than the barrier for fusion to happen. With the WKB
approximation the tunneling rate T can be estimated to have the form of
2
T = e h̄
R x2 √
x1
2m(V (x)−E)dx
4
,
where m is the mass of the tunneling particle, E it's energy, x1 and x2 the limits of the barrier and
V (x) it's shape. If the energies of reacting particles are lower than the barrier fusion reactions will still
happen, but with the reaction rate slowed down by the exponential factor introduced by tunneling.
2.1 Fusion in the Sun
The energy release in the Sun(and other stars of equal or smaller size) is due to the reaction chain
known as the proton-proton chain. Because of the high temperatures all the atoms are fully ionized.
First, two protons combine to form deuterium(one of the protons turns into a neutron via the β +
decay process) and the resulting positron annihilates with an electron. The deuterium nucleus then
combines with another proton to form the light helium isotope 3 He.
p + p
e+ + e−
2
+ p
1D
2
1D
→
→
→
+
2γ
3
2 He
+
e+
+
γ
νe
0.42 MeV
1.02 MeV
5.49 MeV
From here there are four options, the most frequent being the process where two 3 He nuclei combine
to form 4 He, releasing two protons in the process. Overall, four protons are converted into one helium
nucleus.
3
2 He
+
3
2 He
→
4
2 He
+
2p
12.86
MeV
The net energy released is 26.7 MeV. The rst process is caused by the weak interaction and combined
with the need for the protons to tunnel through the Coulomb barrier makes this part of the chain
orders of magnitude slower than the rest, thus dening the reaction rate for the whole chain. The
average time required for two protons to form a nucleus of deuterium is 109 years.
This means that the sun has a very low energy release rate, about 276 W/m3 [8], which is about
a quarter of the energy release rate of the human body. This turns out to be rather fortunate. If
the fusion reactions took place as fast as we need them to proceed on earth, the sun would burn out
brighter, too fast for life to develop on earth.
2.2 Fusion on earth
When considering nuclear fusion on earth, exact imitation of the process in the Sun is out of the
question. The rst part, a weak interaction β + decay of a proton into a neutron, takes place too
slowly to be a source of energy on earth. But the following processes in the proton-proton chain
conserve the numbers of protons and neurons(are not weak interaction processes) and happen much
more quickly. So things look more promising if there is deuterium already present at the start.
The possible candidates have to be exothermic(obviously), have to involve low Z nuclei, since more
protons mean a higher Coulomb barrier, there have to be only two reactants because three body
collisions are orders of magnitude more improbable and have to conserve the numbers of protons and
neutrons. This is to avoid weak interaction processes which are at least three orders of magnitude
slower than processes facilitated by other fundamental interactions . The candidates that t these
criteria best are:
2
1D
2
1D
2
1D
2
1D
+
+
+
+
3
1T
2
1D
2
1D
3
2 He
→
→
→
→
4
2 He
3
2 He
3
1T
4
2 He
5
+ n
+ n
+ p
+ p
17.6 MeV
3.3 MeV
4.0 MeV
18.3 MeV
All of these reactions are used in nuclear fusion experiments. But with the practical aspect of using
nuclear fusion as an energy source in mind, the rst one is the most promising.
2
1D
+
3
1T
4
2 He
→
+ n
↓
↓
3.5 MeV 14.1 MeV
The estimation of how the energy released by this reaction is divided by the two products can be done
using classical physics. The reacting particles, as we will see shortly, typically have energies in the
order of 100 keV. And the energy released per reaction, calculated from the dierences in mass, is 17
MeV. So neither the reactants or the products are relativistic.
mHe = 4.01 u ,
mn = 1.01 u ,
EHe + En = 17.6 MeV .
The classical laws of conservation of mass and momentum give the energies of the reactants
EHe
mn
1
=
≈
En
mHe
4
⇒
EHe = 3.5 M eV ,
En = 14.1 M eV .
The reason why this reaction, usually referred to as the DT reaction, is preferred lies in the dependencies of cross-sections on temperature.
Figure 3: the cross section for the reaction DT, DD and DHe
It is seen from Figure 3 that these cross sections for the other reactions are considerably less than that
for the DT except at impractically high energies. The maximum cross section is at just over 100 keV.
Because of the tunneling eect mentioned above this is lower than the actual height of the Coulomb
barrier. For a DT reaction it is about 200 keV. That is why according to classical physics the Sun is
not nearly hot enough to burn hydrogen.
2.3 Thermonuclear fusion
So, to participate in fusion reactions, particles have to have energies of the order of 10 keV. This is
not even a minor problem to achieve with modern particle accelerators. It is relatively easy to study
fusion reactions by bombarding a solid target containing tritium with accelerated deuterium nuclei.
This is how the cross sections in Figure 3 were measured.
However, because the cross section is small, very few deuterium nuclei achieve fusion and the energy
spent accelerating the rest is lost. To achieve a positive energy balance the nuclei must fulll reaction
6
conditions for a sucient amount of time. Simply shooting a beam into a solid target or through
another beam fails to satisfy this because the nuclei lose energy too rapidly.
The most promising method of supplying the energy is to heat the deuterium-tritium fuel to a sufciently high temperature that the thermal velocities of the nuclei are high enough to produce the
required reactions. Fusion brought about in this way is called thermonuclear fusion.
Again it is important to mention that the required temperature is not as high as that corresponding
to the energy of maximum cross-section, roughly around 109 K. It is an order of magnitude smaller
because the required reactions occur in the high energy tail of the Maxwellian distribution of heated
particles. The necessary temperature is around 108 K, which corresponds to mean kinetic energy of
10 keV.
At such temperatures the deuterium-tritium mixture is fully ionized. This heated mixture of ions and
electrons is called plasma and the fourth state of matter, along solid, liquid and gas states.
With plasma temperatures mentioned above, there is no possibility of containing it in any conventional
container. There are two main options. One is that a magnetic eld can be used to guide the
charged particles and prevent them from hitting the surrounding solid walls. This is called magnetic
connement. The second option is to compress the fuel and heat it so quickly that fusion takes place
before the fuel can expand and touch the walls. This is called inertial connement. Tokamaks are
magnetic connement devices and as a consequence this text will focus on magnetic systems.
2.4 Break-even and ignition
One of the fundamental tasks is to determine the conditions required for a net energy output from
fusion. Energy is needed to heat the fuel up to the temperature required for fusion reactions, and the
hot plasma loses energy in various ways. Clearly there would be little interest in a fusion power plant
that produces less energy than it needs to operate.
The DT reaction produces an α particle and a neutron. The energy released by the fusion reaction is
shared between the α particle, with 20% of the total energy, and the neutron, with 80%.
The neutron has no electric charge, and so it is not aected by the magnetic eld. It escapes from the
plasma and slows down in the surrounding structure. The fusion energy will be converted into heat
and then into electricity. This is the output of the power plant.
The α particle has a positive charge and is trapped by the magnetic eld. The energy of the α particle
can be used to heat the plasma. Initially an external source of energy is needed to raise the plasma
temperature. As the temperature rises, the fusion reaction rate increases and the alpha particles
provide more and more of the required heating power. Eventually the heating from α particles is
sucient by itself and the fusion reaction becomes self-sustaining. This point is called ignition by
analogy with the burning of fossil fuels, where heat resulting from reactions triggers the next ones.
2.4.1
Fusion reaction rate
The rst step in determining the conditions for ignition is calculating the rate of fusion reactions. If
the two reactants have velocity distributions f (v) the total reaction rate per unit volume R is
Z Z
R=
σ(v 0 )v 0 f (v1 )f (v2 )d3 v1 d3 v2 ,
v 0 = v1 − v2 ,
where σ(v 0 ) is the cross section for the DT reaction. If we take the velocity distribution to be
Maxwellian and introduce new variables
7
fi (vi ) = ni
³ m ´ 32
mi v 2
i
i
e− 2T ,
2πT
V =
v1 + v2
,
2
µ=
m1 m2
,
m1 + m2
the reaction rate becomes
3
(m1 m2 ) 2
R = n1 n2
(2πT )2
The integral over V is
³
2πT
m1 +m2
Z Z
´ 32
³
e
−
m1 +m2
2T
¡
m −m2
1 +m2 )
V + 2(m1
¢2 ´
σ(v 0 )v 0 e−
µv 02
2T
d3 V d3 v 0 .
so that the fusion reaction rate is
³ µ ´ 32 Z
µv 02
R = 4πn1 n2
σ(v 0 )v 03 e− 2T dv 0 ,
2πT
Figure 4: the < σv > values for DT, DD and DHe reactions
and is usually written in the form of
R = n1 n2 < σv > ,
since < σv > is an experimentally measured quantity.
2.4.2
Thermonuclear power and losses
The thermonuclear power of a DT reaction per unit volume is the reaction rate per unit volume times
the energy released per reaction
PT = nd nt < σv > E ,
and since the ion density n is n = nd + nt
8
PT = nd (n − nd ) < σv > E ,
which is optimal for nt = nd = 12 n
PT =
1 2
n < σv > Eq.
4
Besides the thermonuclear power produced by the reaction there are also losses present, mostly due to
the acceleration of the charged particles. Because of their smaller mass the electrons undergo larger
acceleration than the ions, radiate much more strongly and are usually the only ones considered.
The electrons are accelerated in two ways. Firstly they are accelerated by collisions, the resulting
radiation being called bremsstrahlung. Secondly they are subject to the acceleration because charged
particles in magnetic eld move on circular trajectories in the plane perpendicular to the eld. This
is called cyclotron motion and the associated radiation is called cyclotron or synchrotron radiation.
The plasma energy losses are expressed with energy connement time τE . It is a measure of the rate
at which energy is lost from the plasma and is dened as the total amount of energy in the plasma
divided by the rate at which energy is lost.
τE =
W
.
PL
The average energy of plasma particles at a temperature T is 32 kT, comprised of 12 kT per degree of
freedom. Since there is an equal number of electrons and ions, the plasma energy per unit volume is
W = 3nkT .
The total energy in the plasma is therefore
W = 3nT kV ,
where the average is over the spatial dimensions.
In the context of magnetic connement, energy connement time is a measure of the quality of the
connement or better magnetic 'insulation'.
The plasma is heated by the α particles that are kept inside by the magnetic elds. They transfer
their kinetic energy, 3.5 MeV, to the plasma through collisions. Per unit volume
Pα =
1 2
n < σv > Eα ,
4
Pα =
1 2
n < σv >Eα V .
4
and for the total α particle heating
To get the total energy balance the α particle heating and external heating PH must equal the losses.
1
3nT kV
PH + n2 < σv >Eα V =
4
τE
9
2.4.3 Ignition
For a self-sustaining fusion reaction to happen, the α particle heating must at least balance out the
losses so there is no need for external heating.
µ
PH =
¶
3nT k 1 2
− n < σv >Eα V = 0 .
τE
4
This leads to the inequation(density and temperature are taken constant for simplicity)
nτE ≥
12kT
.
Eα < σv >
The right-hand side of this inequation is a function only of temperature and has a minimum around
kT = 30 keV
kT
= 5 × 1022 keV s m−3 ,
< σv >
Figure 5: the value of nτE required for ignition
so the required value of the ignition criterion would be
nτE ≥ 1.5 × 1020 s m−3 .
Usually τE is also a function of temperature and the optimum temperature comes somewhat lower than
30 keV. In the temperature range from 10 to 20 keV the DT reaction rate < σv > is proportional to T 2 .
Multiplying both sides of the equation by T makes the right-hand side independent of temperature,
while the left-hand side becomes the triple product
nT τE ≥ 3 × 1021 keV s m−3
The precise value in fact depends on the proles of plasma density and temperature and on other
issues, like plasma purity. A typical value taking these factors into account would be
10
nT τE ≥ 5 × 1021 keV s m−3
It is important to stress that the triple product is a valid concept only for T in the range from 10 to
20 keV where quadratic approximation of the reaction rate holds.
2.4.4
The Q factor
Another much used concept is the Q factor. It is a measure of the success in approaching reactor
conditions given by the ratio of the thermonuclear power PT produced and the external heating power
PH , that is
Q=
1 2
4n
< σv > EV
PH
At ignition, where PH is reduced to zero, Q approaches innity. Although the goal of a fusion reactor
is ignition it is possible to obtain a positive energy balance with a large Q. In this case the supplied
power PH is a cost on the system in that it involves recycling some of the reactor power with a
corresponding loss of eciency. A value of Q > 1 is not enough since there are losses when converting
the thermonuclear power(heat) into electricity and back again to heat the plasma.
3 Tokamaks
3.1
Magnetic connement
The most promising way of conning the hot plasma against it coming in contact with the walls is
with magnetic elds. The basic principle of magnetic connement is the so-called pinch eect. The
basic idea is shown schematically in Figure 6.
Figure 6: schematic showing the physical mechanism that causes an electric current to compress the
conductor through which it is owing
11
When an electric current ows through a conductor, in this case the plasma, it generates a magnetic
eld that encircles the direction of the current. If the current is suciently large, the magnetic force
will be strong enough to compress the plasma and pull it away from the walls.
It is possible to quickly estimate the force of the pinch eect with basic electrodynamics. Let's treat
the plasma is a cylindrical current carrier with radius R and uniform current density j in the axial
direction. The Ampere's Law and Lorentz force on a spatially distributed current
I
ZZ
Bdl = µ0
ZZZ
jdS ,
F=
j × BdV ,
give the expression for the force on the plasma per unit length, which has a inward radial direction.
F =
µ0 I 2
.
2πR
But it was soon observed that the plasma conned this way is very unstable. It wriggled and deformed
and quickly came into contact with the torus walls, as shown in Figure 7. Two examples of this are
the aptly named kink and sausage instabilities.
Figure 7: schematic representation of the kink and sausage instabilities in the plasma
The sausage instability can be qualitatively explained by taking into account the expression for the
force of the pinch eect. Because the force is stronger where the radius of the plasma is smaller, a
small perturbation in the radius will escalate because of the pinch eect.
Some, but not all, of these instabilities are tamed by adding an axial magnetic eld from additional
coils wound around the connement chamber. So to conne the plasma two strong magnetic elds
are necessary.
Figure 8: schematic of the magnetic mirror connement principle
12
In a straight tube the plasma will rapidly escape out of the open ends. One idea how to counter this
and still preserve the linear conguration was the magnetic-mirror machine.
A solenoid coil produces a steady-state axial magnetic eld that increases in strength at the ends.
These regions of higher eld, the magnetic mirrors, trap the bulk of the plasma in the central lower
eld region, though ions and electrons with a large parallel component of velocity can escape through
the mirrors. At higher plasma densities these losses are higher than the energy produced.
However, the problem of the open ends is solved by simply bending the tube into a torus. The plasma
is still contained in the same way(with similar magnetic elds) as in a linear tube. One component
of the magnetic eld goes along the major radius of the torus, around the main axis. This is called
the toroidal eld and performs the stabilizing function of the axial magnetic eld in the linear tube.
The second one goes along the minor radius of the torus, encircles the plasma inside. This is called
the poloidal eld and provides the pinch and keeps the plasma away from the walls.
Figure 9: schematic of the two toroidal magnetic connement principles, the stellarator and the
tokamak
Depending on how the poloidal eld is produced, there are two basic toroidal magnetic connement
principles. In the stellarator the poloidal eld is produced by external coils around the torus, just
like the toroidal eld. And in the tokamak the poloidal eld is produced by a current running in the
plasma in the toroidal direction(around the torus).
The main reason why stellarators have fallen behind tokamaks as an option for fusion energy production is it's complexity. The coils needed to produce the necessary magnetic eld have complex shapes,
which is a problem when they have to be made from superconducting materials. Also there is a lot
less room for heating and diagnostic equipment. But on the other hand, unlike tokamaks as will be
described in the next section, stellarators are not limited to pulsed operation.
3.2
The tokamak
The name is an acronym of the Russian words toroidal'naya kamera s magnitnymi katushkami, for
'toroidal chamber with magnetic coils'.
The basic features of a tokamak are shown in Figure 10. The toroidal eld Bφ is produced by coils
wound around the torus. The second magnetic eld, the poloidal eld Bp , is generated by an electric
current Iφ that ows in the plasma.
The two magnetic elds combine to create a composite magnetic eld that twists around the torus in
a helix. It is important to notice that because of the way the current is induced in the plasma, the
tokamak is inherently a pulsed machine. It is basically a big transformer, with a primary winding
outside, the central solenoid coil as the core and the plasma as the secondary winding. A magnetic
13
Figure 10: the coils producing the toroidal magnetic eld Bφ , the magnetic ux change through the
torus that induces the toroidal current Iφ and both elds in the torus of the tokamak
ux change through the center of the torus, that induces the toroidal current, is generated by a change
of current through the primary winding. Because this current can't be increased indenitely, there is
a limit to the time in which the toroidal current can be induced this way.
Figure 11: schematic view of a tokamak showing how the magnetic elds, due to the external coils
and the current owing in the plasma, combine to produce a helical magnetic eld
In present experiments the toroidal magnetic eld at the coils is limited to 10 T [4]. The basic shape of
the toroidal elds Bφ is obtained from Ampere's law, taking a line integral around a circular toroidal
circuit inside the toroidal eld coils, and neglecting the small poloidal plasma current
2πRBφ = µ0 Iφ ,
where R is the major radius coordinate and Iφ is the current in the toroidal eld coils. So the
dependence of the eld on the major radius is
Bφ ∝
1
,
R
the resulting eld at the center of the plasma would be around 5 T [4]. To achieve elds of this
14
magnitude superconducting coils are required. The poloidal magnetic eld is usually around ten
times smaller.
In total there are three types of coils. The central solenoid coil drives the toroidal current which
generates most of the poloidal eld. Then there are the toroidal eld coils and the correction coils
that change the eld to counter local deformations and instabilities.
3.3
Quality of connement
3.3.1 The β factor
The hot plasma exerts an outward pressure p = nkT . This outward pressure of the plasma has to be
balanced by an inward force and it is convenient to think of the magnetic eld exerting a pressure
equal to
2
Bφ
2µ0 .
The parameter
β=
2µ0 p
,
Bφ2
is dened as the ratio between the plasma pressure and the magnetic pressure of the poloidal magnetic
eld that compresses the plasma. It is the most straightforward gure of merit concerning the quality
of magnetic connement.
There have been many attempts to develop magnetic-connement congurations with β = 1, but the
most successful tokamaks require lower values of β for stability, typically only a few percent.
3.3.2
Instabilities
Some types of instabilities cause the sudden loss of the whole plasma, others persist and reduce the
energy connement time.
Even with the strong toroidal magnetic eld, tokamak plasma becomes unstable if either the plasma
current or the plasma density is increased above a critical value and the plasma extinguishes itself in
the timescale on the order of 10 ms [3]. These disruptions can be avoided to some extent by careful
selection of the operating conditions, such as plasma current and density.
Disruptions occur when the magnetic eld at the plasma edge is twisted too tightly. Increasing
the plasma current increases the poloidal magnetic eld, enhances the twist, and makes the plasma
unstable. Increasing the toroidal eld straightens out the twist and improves stability. The eect
of plasma density is more complicated. Increasing density causes the plasma edge to become cooler
because more energy is lost by radiation. Because hot plasma has a higher conductivity, the cooler
edge squeezes the current into the core, increasing the twist of the magnetic eld by reducing the
eective size of the plasma.
3.3.3
The q factor
One of the factors determining when a disruption occurs is the amount by which the magnetic eld
lines are twisted. The twist is measured by the parameter q , known as the safety factor, which is the
number of times that a magnetic eld line passes the long way around the torus before it returns to
its starting point in the poloidal plane
q=
∆ϕ
,
2π
15
where ∆ϕ is the change of the toroidal angle after which the magnetic led line closes. For a circular
cross-section torus it can be approximated with
q=
rBφ
,
R0 Bp
where r is the minor radius of the eld line and R0 the major radius of the torus. Large q indicates
a gentle twist and small q a tight twist. Usually the plasma becomes unstable when the parameter is
q < 3 at the plasma edge.
3.4
Plasma heating
In an ignited D-T plasma the energy losses are balanced by the plasma heating from the slowing down
of the α particles resulting from the fusion reactions. However, the fusion reaction rate is negligible
at low temperatures, and to reach the temperature required for ignition it is necessary to provide
additional external heating.
3.4.1
Ohmic heating
As well as generating the poloidal magnetic eld that provides the pinch eect, the toroidal plasma
current Iφ provides a way of heating the plasma. It heats it through electrical resistivity that is
consequence of electron-ion collisions.
This process performs two crucial functions in a tokamak, a very eective way to heat the plasma
and conne it, but it has one major shortcoming. The electrical resistance of the plasma falls with
temperature as
3
η ∝ T−2 ,
which comes from the temperature dependence of electron collision frequency in plasma. The most
obvious solution is to increase the plasma current, but increasing lowers the q value and causes
disruptions unless the toroidal eld is increased proportionally and there is an upper limit to the
strength of the eld because of the forces on the coils. The maximum plasma temperature attainable
with ohmic heating is up to 5 keV, or 5 × 107 K.
3.4.2
Neutral beam injection
One of the other two most promising heating methods is neutral beam injection. Beams of neutral
atoms(deuterium if we are dealing with a DT reaction) are injected into the chamber and heat the
plasma by collisions.
The beams used for injection heating have to be composed of neutral particles because ions would
be reected by the tokamak magnetic eld. Deuterium ions are rst accelerated to high energies by
passing through a series of high-voltage grids.
They are then neutralized by charge exchange in a deuterium gas target, and the unwanted residual
ions removed. The beams of neutral atoms are not deected by magnetic elds and can penetrate the
plasma, where they become reionized. After being ionized, the beams are trapped inside the magnetic
elds and heat the plasma.
The typical beam energy for present tokamak experiments is about 120 keV [3], but next generation
tokamaks and fusion power plants will require neutral beams much higher energy, up to 1 MeV [3].
16
Figure 12: Neutral beam injection for plasma heating
3.4.3
Radiofrequency heating
The other promising method that has been used since the start of tokamak research is heating the
plasma with electromagnetic waves. There are three resonance frequencies that are used for plasma
heating. And they all depend on the density of the toroidal magnetic eld Bφ . This means that they
heat the plasma locally because the toroidal eld has a dependance on the major radius.
The highest is the electron cyclotron resonance at around 28 GHz/T [3] which means from 100 GHz
in present tokamak experiments to 200 GHz in a fusion power plant. It corresponds to the frequency
of the electron cyclotron motion in the magnetic eld. The advantage of this method is that the
antennas that project the electromagnetic waves into the plasma can be farther away from the actual
plasma than with other methods and thus introduce less impurities into it. But the problem is that
sources for this frequency range still need additional development to be able to provide power output
in the megawatt range.
The second one is the ion cyclotron resonance at 7.5 MHz/T [3] for deuterium ions (it depemnds
on the charge-to-mass ratio). It corresponds to the frequency of the cyclotron motion of the ions
in the magnetic eld. The frequency range is from 30 to 60 MHz, in the range of commercial radio
transmitters which means that the technology is already developed. But the antennas have to be very
close to the plasma which introduces more impurities into it.
There is a third resonance frequency, midway between the ion and electron cyclotron frequencies,
known as the lower hybrid resonance. It falls in the range of 1 to 8 GHz. It has proved less eective
for plasma heating but is used to drive currents in the plasma.
3.5 Tokamak reactor
Up until now this text has been dealing with nuclear fusion and tokamak aspects that concern the
reactors in operation now. These are all experimental reactors with the purpose of studying plasma
properties, magnetic connement characteristics, operating regimes and so on. And the purpose of
these experiments is gathering knowledge and developing technologies needed to build a fusion reactor
that can function as a power plant, connected to an electrical grid producing electricity. The purpose
of this chapter is to describe how will electricity be produced and what additional characteristics have
to be implemented for an experimental reactor to be able to operate as a fusion power plant.
The basic principle of producing energy is very similar to that of a ssion or even thermal power plant,
the only dierence being how the heat is generated.
The reactor itself generates energy with fusion reactions. This is transfered out of the plasma by the
neutrons (carrying 80% of the reaction energy output) since they are not conned by the magnetic
17
Figure 13: schematic view of a tokamak operating as a fusion power plant
eld. They are absorbed in the material surrounding the plasma and heat it. The heat is then
transfered away with the coolant and electricity is generated with steam turbines.
For a thermonuclear fusion reactor to function as a power plant it has to have several key features
that are not necessary in experimental reactors while certain aspects receive additional requirements
which increase the technological complexity. The most important things are how the fuel cycle of the
power plant is managed, how is the heat and the products of the reactions(helium ash) diverted out
of the vacuum vessel and how will maintenance be performed.
3.5.1
Fuel cycle
The proposed fuel for a fusion power plant are deuterium and tritium. Deuterium is available in water
and the ratio with hydrogen atoms is around 1/7000 [1]. Taking into account the current energy
consumption and the amount of water in the oceans there are deuteruim resources for ≈ 1011 years
[1].
The situation with tritium is more complex. It has a half life of 12.3 years [8] and is virtually
nonexistent in nature. Therefore a power plant must operate in such a way that it generates tritium
in the course of it's operation, a process called tritium breeding. The most convenient way to produce
it is through two possible neutron-lithium reactions, since neutrons are already available
6
7
Li
Li
+
+
n
n
→
→
4
2 He
4
2 He
+
+
3
1T
3
1T
+
n
4.8 MeV
−2.5 MeV
Of these two reactions one is exothermic and one is endothermic but, since the kinetic energy of the
neutrons is 14 MeV, both are possible. Lithium is available as a mineral in the earth's crust and in
seawater. Reserves are estimated for ≈ 105 years [1] of present energy consumption.
This complicating factor in taking advantage of this is the fact that this process has to happen
somewhere where the neutrons are available, before they have deposited their energy. Which means
that it has to be inside the vacuum vessel next to the plasma.
18
3.5.2 Blanket
The structure surrounding the plasma, inside the vacuum vessel, is called the blanket. It's main functions are tritium production and extraction, transformation of neutron power into heat and collection
of the heat and shielding of the vacuum vessel and the magnetic eld coils outside.
Figure 14: Breeder blanket structure
Every DT reaction uses up one tritium atom and produces one neutron, and every tritium breeding
reaction uses one neutron. Therefore to have a closed cycle every neutron from the DT reaction has
to produce at least one atom of tritium. This could be achieved if pure lithium in liquid form would
be used as the breeder element in the blanket. The fast neutrons from the reactions would react with
7
Li and produce more neutrons, but this is not viable. Lithium is dangerous, it burns and ignites in
contact with air, reacts strongly with water and it is very hard to extract the produced tritium from
it. That's why lithium in the blanket will be in the form of lithium-lead, lithium-tin alloys or lithium
oxides(LiO2 , Li4 SiO4 ).
Due to the presence of other materials the probability of neutrons hitting lithium is strongly reduced.
Plus it is not possible to surround the plasma with a complete blanket of lithium, because the toroidal
geometry restricts the amount of blanket that can be used for breeding on the inboard side of the
torus. To still have enough neutrons they have to be produced extra for example with inclusion of
neutron multiplier elements like berylium that produces neutrons through
9
3.5.3
Be
+
n
4
2 He
→
+
4
2 He
+
2n
Limiters and divertors
The plasma in a tokamak is conned within closed magnetic ux surfaces and it ends where a ux
surface is interrupted by a solid surface. So there must be a well dened point of contact where the
plasma is limited.
One way to achieve this is with a limiter, named so because it limits the size of the plasma and
localizes the plasma-surface interaction. It also protects the wall from the plasma disruptions, and
other instabilities. For these reasons it is commonly made of materials such as carbon, molybdenum
or tungsten, capable of withstanding high heat loads, thermal shocks, produce little impurities and
have good heat transfer properties.
19
Figure 15: Limiter and divertor modes of operation
Introduction of impurities into the plasma is the main disadvantage of limiters and with this in mind an
alternative has been designed. The magnetic eld lines on the edge of the plasma can be deliberately
diverted into a separate space where the plasma can interact with the surface. The impurities produced
here can be severely inhibited from returning into the main chamber by the shape of the magnetic
eld at the passage. It can also remove the helium that is the product of the fusion reactions which is
crucial for the operation of a power plant and is a function that a limiter is not able to perform. The
disadvantage of the divertor is that it is a complicated part on the inside of the vacuum vessel and
needs extra coils to ne-tune the magnetic eld.
3.5.4 ITER
Since the start of tokamak research in the sixties there has been tremendous development concerning
plasma temperature, pressure and containment times. The value of the ignition criterion, the triple
product nT τE , has been increased by ve orders of magnitude.
Figure 16: Triple product values for dierent tokamaks from the earlier(T3) to most recent(JET)
Building on this advances is currently the biggest tokamak project, ITER, being built in the south
of France. It is still in principle an experimental reactor but with the goal of proving that a fusion
power plant can be constructed.
20
Fusion power
Burn time
Plasma current
Major radius
Plasma minor radius
Plasma volume
Toroidal eld at major radius
Toroidal eld at conductor
Heating/current drive power
500 MW
>400 s
15 MA
6.2 m
2.0 m
837 m3
5.3 T
12 T
73 MW (up to 110 MW)
Table 1: Principal parameters of ITER
The device should achieve extended burn with a ratio of fusion power to auxiliary heating power(factor
Q) of at least 10. Besides that objectives include development of technologies and processes needed
for a fusion power plant, including superconducting magnets, remote handling and advanced plasma
heating techniques. Also tritium breeding concepts have yet to be veried in operation.
Figure 17: Cross section of ITER vacuum vessel
It will have superconducting coils, the toroidal eld coils and central solenoid using niobium-tin cooled
to 4.5 K by supercritical helium and the poloidal eld coils using niobium-titanium. The divertor will
be made of tungsten with carbon plasma-facing components. The water cooling system is designed to
extract 750 MW of heat and the entire tokamak is enclosed in a cryostat. Plasma fueling is provided
by injection of deuterium and tritium gas and solid pellets.
The shielding blanket is divided into two parts. The front part has 1 cm thick beryllium armour, 1
cm thick copper to diuse heat, and around 5 cm of steel. The back part is a 35 cm thick shield made
of steel and water.
It is planned that once the top current is reached, with an inductively driven current(through the
central solenoid) of 15 MA, subsequent fueling and plasma heating will produce a run with a fusion
power of 500 MW with burn duration lasting up to 1 hour.
21
References
[1] John Wesson, Tokamaks. Clarendon Press - Oxford, 2004.
[2] C. M. Braams and P. E. Stott, Nuclear Fusion. Institute of Physics Publishing, 2002.
[3] G. McCracken and P. E. Stott, Fusion, Energy of The Universe. Elsevier Academic Press, 2005.
[4] John Wesson, The Science of JET. 2000.
[5] T. J. Dolan, Fusion Research. Pergamon Press, 2000.
[6] Paul M. Bellan, Fundamentals of Plasma Physics. Cambridge University Press, 2006.
[7] 2nd Karlsruhe International Summer School, Fusion Technologies.
http://iwrwww1.fzk.de/summerschool-fusion/download2008.html
[8] Wikipedia, Nuclear Fusion. http://en.wikipedia.org/wiki/Nuclear_fusion
22