SEMINAR

Nuclear Fusion and Tokamaks

Klemen Strni²aMentor: Jernej Fesel Kamenik

Fakulteta za matematiko in �ziko, Univerza v Ljubljani

October 6, 2010

AbstractIn 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 toachieve fusion in controlled conditions. This is followed by the explanation of the role of plasmaand what seems to be the best way to contain and control it. The second part consists of thedescription of how is this achieved in a special type of fusion reactor, the tokamak.

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Contents1 Introduction 3

2 Nuclear fusion 42.1 Fusion in the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Fusion on earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Thermonuclear fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Break-even and ignition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4.1 Fusion reaction rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.2 Thermonuclear power and losses . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4.3 Ignition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4.4 The Q factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Tokamaks 113.1 Magnetic con�nement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 The tokamak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Quality of con�nement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.1 The β factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3.2 Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3.3 The q factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Plasma heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4.1 Ohmic heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4.2 Neutral beam injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4.3 Radiofrequency heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.5 Tokamak reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5.1 Fuel cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5.2 Blanket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5.3 Limiters and divertors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5.4 ITER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

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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. Withthe mass of the Sun measured, the known chemical fuels would provide sun's power output for onlya couple of thousand years. And gravitational energy, that is, work done by the gravitational forcecontracting the sun, accounts for a couple of million years. None of those two explanations conformwith geological evidence about the age of Earth.The realization that the energy radiated by the Sun is due to nuclear fusion followed three main stepsin the development of physics. The �rst was Albert Einstein's famous deduction that mass can beconverted into energy with E = mc2. The second step came with precision measurements of atomicmasses, which showed that the total mass of four hydrogen atoms is slightly larger than the mass ofone helium atom. That led physicists, around 1920, to propose that mass could be turned into energyin the Sun if four hydrogen atoms combine to form a single helium atom.When fusion was identi�ed as the energy source of the Sun and the stars, it was natural to ask whetherthe process of turning mass into energy could be performed on Earth. This presented enormous techni-cal di�culties since the necessary conditions were high density of reacting particles and temperaturesin the order of hundred million degree. In the Sun these conditions were created by the gravitationalforce. The problem of how to contain fuel at such temperatures for the necessary amount of timewas attacked from two di�erent angles. The �rst idea was to con�ne it with magnetic �elds and thesecond one was to heat and compress it enough to burn before it can expand.The considerable scienti�c and technical di�culties encountered by fusion research programs havecaused them to stretch over the last �ve decades. But the scienti�c feasibility of thermonuclearfusion via the magnetic-con�nement route has been demonstrated by current experimental reactorsand ITER, the �rst fusion reactor with a planned positive e�ciency, is being constructed. Progresscoupled with enormous available fuel reserves and almost zero pollution makes nuclear fusion one ofthe most viable answers to growing energy problems.

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2 Nuclear fusion

When the mass spectrometer was invented and the masses of individual atoms could be accuratelymeasured it became apparent that the mass of atoms containing more nucleons was less than the sumof the masses of separate nucleons. The di�erence 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 ofenergy associated with the nuclear force that holds the nucleus together.

Figure 1: binding energy per nucleon against the mass number for di�erent elements

So if two hydrogen atoms could be combined to form an atom of helium the di�erence in bindingenergy would be released. Once this was known it was quickly recognized as the principle behind theprocess 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 beforefusion can occur. The nuclei must be brought close enough together for the attractive nuclear forceto 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 andtheir energies don't have to be strictly higher than the barrier for fusion to happen. With the WKBapproximation the tunneling rate T can be estimated to have the form of

T = e2h̄

∫ 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 andV (x) it's shape. If the energies of reacting particles are lower than the barrier fusion reactions will stillhappen, 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 chainknown 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 thencombines with another proton to form the light helium isotope 3He.

p + p → 21D + e+ + νe 0.42 MeV

e+ + e− → 2γ 1.02 MeV21D + p → 3

2He + γ 5.49 MeV

From here there are four options, the most frequent being the process where two 3He nuclei combineto form 4He, releasing two protons in the process. Overall, four protons are converted into one heliumnucleus.

32He + 3

2He → 42He + 2p 12.86 MeV

The net energy released is 26.7 MeV. The �rst process is caused by the weak interaction and combinedwith the need for the protons to tunnel through the Coulomb barrier makes this part of the chainorders of magnitude slower than the rest, thus de�ning the reaction rate for the whole chain. Theaverage 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 abouta quarter of the energy release rate of the human body. This turns out to be rather fortunate. Ifthe fusion reactions took place as fast as we need them to proceed on earth, the sun would burn outbrighter, 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 thequestion. The �rst part, a weak interaction β+ decay of a proton into a neutron, takes place tooslowly to be a source of energy on earth. But the following processes in the proton-proton chainconserve the numbers of protons and neurons(are not weak interaction processes) and happen muchmore 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 moreprotons mean a higher Coulomb barrier, there have to be only two reactants because three bodycollisions are orders of magnitude more improbable and have to conserve the numbers of protons andneutrons. This is to avoid weak interaction processes which are at least three orders of magnitudeslower than processes facilitated by other fundamental interactions . The candidates that �t thesecriteria best are:

21D + 3

1T → 42He + n 17.6 MeV

21D + 2

1D → 32He + n 3.3 MeV

21D + 2

1D → 31T + p 4.0 MeV

21D + 3

2He → 42He + p 18.3 MeV

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All of these reactions are used in nuclear fusion experiments. But with the practical aspect of usingnuclear fusion as an energy source in mind, the �rst one is the most promising.

21D + 3

1T → 42He + 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 doneusing classical physics. The reacting particles, as we will see shortly, typically have energies in theorder of 100 keV. And the energy released per reaction, calculated from the di�erences in mass, is 17MeV. 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

En=

mn

mHe≈ 1

4⇒ EHe = 3.5 MeV , En = 14.1 MeV .

The reason why this reaction, usually referred to as the DT reaction, is preferred lies in the depen-dencies 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 thatfor the DT except at impractically high energies. The maximum cross section is at just over 100 keV.Because of the tunneling e�ect mentioned above this is lower than the actual height of the Coulombbarrier. For a DT reaction it is about 200 keV. That is why according to classical physics the Sun isnot 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 isnot even a minor problem to achieve with modern particle accelerators. It is relatively easy to studyfusion 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 energyspent accelerating the rest is lost. To achieve a positive energy balance the nuclei must ful�ll reaction

6

conditions for a su�cient amount of time. Simply shooting a beam into a solid target or throughanother 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 suf-�ciently high temperature that the thermal velocities of the nuclei are high enough to produce therequired 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 correspondingto the energy of maximum cross-section, roughly around 109 K. It is an order of magnitude smallerbecause the required reactions occur in the high energy tail of the Maxwellian distribution of heatedparticles. The necessary temperature is around 108 K, which corresponds to mean kinetic energy of10 keV.At such temperatures the deuterium-tritium mixture is fully ionized. This heated mixture of ions andelectrons 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 conventionalcontainer. There are two main options. One is that a magnetic �eld can be used to guide thecharged particles and prevent them from hitting the surrounding solid walls. This is called magneticcon�nement. The second option is to compress the fuel and heat it so quickly that fusion takes placebefore the fuel can expand and touch the walls. This is called inertial con�nement. Tokamaks aremagnetic con�nement 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 fromfusion. Energy is needed to heat the fuel up to the temperature required for fusion reactions, and thehot plasma loses energy in various ways. Clearly there would be little interest in a fusion power plantthat 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 isshared 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 a�ected by the magnetic �eld. It escapes from theplasma and slows down in the surrounding structure. The fusion energy will be converted into heatand 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 particlecan be used to heat the plasma. Initially an external source of energy is needed to raise the plasmatemperature. As the temperature rises, the fusion reaction rate increases and the alpha particlesprovide more and more of the required heating power. Eventually the heating from α particles issu�cient by itself and the fusion reaction becomes self-sustaining. This point is called ignition byanalogy 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. Ifthe two reactants have velocity distributions f(v) the total reaction rate per unit volume R is

R =∫ ∫

σ(v′)v′f(v1)f(v2)d3v1d3v2 , v′ = v1 − v2 ,

where σ(v′) is the cross section for the DT reaction. If we take the velocity distribution to beMaxwellian and introduce new variables

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fi(vi) = ni

( mi

2πT

) 32

e−miv2

i2T , V =

v1 + v2

2, µ =

m1m2

m1 + m2,

the reaction rate becomes

R = n1n2(m1m2)

32

(2πT )2

∫ ∫e

(−m1+m2

2T

(V +

m1−m22(m1+m2)

)2)σ(v′)v′e−

µv′22T d3V d3v′ .

The integral over V is(

2πTm1+m2

) 32 so that the fusion reaction rate is

R = 4πn1n2

( µ

2πT

) 32

∫σ(v′)v′3e−

µv′22T dv′ ,

Figure 4: the < σv > values for DT, DD and DHe reactions

and is usually written in the form of

R = n1n2 < 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 timesthe energy released per reaction

PT = ndnt < 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 = 12n

PT =14n2 < σv > Eq.

Besides the thermonuclear power produced by the reaction there are also losses present, mostly due tothe acceleration of the charged particles. Because of their smaller mass the electrons undergo largeracceleration 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 resultingradiation being called bremsstrahlung. Secondly they are subject to the acceleration because chargedparticles in magnetic �eld move on circular trajectories in the plane perpendicular to the �eld. Thisis called cyclotron motion and the associated radiation is called cyclotron or synchrotron radiation.The plasma energy losses are expressed with energy con�nement time τE . It is a measure of the rateat which energy is lost from the plasma and is de�ned as the total amount of energy in the plasmadivided by the rate at which energy is lost.

τE =W

PL.

The average energy of plasma particles at a temperature T is 32kT, comprised of 1

2kT per degree offreedom. 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 = 3nTkV ,

where the average is over the spatial dimensions.In the context of magnetic con�nement, energy con�nement time is a measure of the quality of thecon�nement or better magnetic 'insulation'.The plasma is heated by the α particles that are kept inside by the magnetic �elds. They transfertheir kinetic energy, 3.5 MeV, to the plasma through collisions. Per unit volume

Pα =14n2 < σv > Eα ,

and for the total α particle heating

Pα =14n2 < σv >EαV .

To get the total energy balance the α particle heating and external heating PH must equal the losses.

PH +14n2 < σv >EαV =

3nTkV

τE

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2.4.3 Ignition

For a self-sustaining fusion reaction to happen, the α particle heating must at least balance out thelosses so there is no need for external heating.

PH =(

3nTk

τE− 1

4n2 < σv >Eα

)V = 0 .

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 aroundkT = 30 keV

kT

< σv >= 5× 1022 keV s m−3 ,

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 than30 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

nTτE ≥ 3× 1021 keV s m−3

The precise value in fact depends on the pro�les of plasma density and temperature and on otherissues, like plasma purity. A typical value taking these factors into account would be

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nTτE ≥ 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 to20 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 reactorconditions given by the ratio of the thermonuclear power PT produced and the external heating powerPH , that is

Q =14n2 < σv > EV

PH

At ignition, where PH is reduced to zero, Q approaches in�nity. Although the goal of a fusion reactoris ignition it is possible to obtain a positive energy balance with a large Q. In this case the suppliedpower PH is a cost on the system in that it involves recycling some of the reactor power with acorresponding loss of e�ciency. A value of Q > 1 is not enough since there are losses when convertingthe thermonuclear power(heat) into electricity and back again to heat the plasma.

3 Tokamaks

3.1 Magnetic con�nement

The most promising way of con�ning the hot plasma against it coming in contact with the walls iswith magnetic �elds. The basic principle of magnetic con�nement is the so-called pinch e�ect. Thebasic idea is shown schematically in Figure 6.

Figure 6: schematic showing the physical mechanism that causes an electric current to compress theconductor through which it is �owing

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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 su�ciently large, the magnetic forcewill 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 e�ect with basic electrodynamics. Let's treatthe plasma is a cylindrical current carrier with radius R and uniform current density j in the axialdirection. The Ampere's Law and Lorentz force on a spatially distributed current

∮Bdl = µ0

∫∫jdS , F =

∫∫∫j× BdV ,

give the expression for the force on the plasma per unit length, which has a inward radial direction.

F =µ0I

2

2πR.

But it was soon observed that the plasma con�ned this way is very unstable. It wriggled and deformedand quickly came into contact with the torus walls, as shown in Figure 7. Two examples of this arethe 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 theforce of the pinch e�ect. Because the force is stronger where the radius of the plasma is smaller, asmall perturbation in the radius will escalate because of the pinch e�ect.Some, but not all, of these instabilities are tamed by adding an axial magnetic �eld from additionalcoils wound around the con�nement chamber. So to con�ne the plasma two strong magnetic �eldsare necessary.

Figure 8: schematic of the magnetic mirror con�nement principle

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In a straight tube the plasma will rapidly escape out of the open ends. One idea how to counter thisand still preserve the linear con�guration 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 throughthe 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 plasmais still contained in the same way(with similar magnetic �elds) as in a linear tube. One componentof the magnetic �eld goes along the major radius of the torus, around the main axis. This is calledthe 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 calledthe poloidal �eld and provides the pinch and keeps the plasma away from the walls.

Figure 9: schematic of the two toroidal magnetic con�nement principles, the stellarator and thetokamak

Depending on how the poloidal �eld is produced, there are two basic toroidal magnetic con�nementprinciples. In the stellarator the poloidal �eld is produced by external coils around the torus, justlike the toroidal �eld. And in the tokamak the poloidal �eld is produced by a current running in theplasma in the toroidal direction(around the torus).The main reason why stellarators have fallen behind tokamaks as an option for fusion energy produc-tion 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 lotless room for heating and diagnostic equipment. But on the other hand, unlike tokamaks as will bedescribed 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 coilswound around the torus. The second magnetic �eld, the poloidal �eld Bp, is generated by an electriccurrent Iφ that �ows in the plasma.The two magnetic �elds combine to create a composite magnetic �eld that twists around the torus ina helix. It is important to notice that because of the way the current is induced in the plasma, thetokamak is inherently a pulsed machine. It is basically a big transformer, with a primary windingoutside, 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 thetorus 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 changeof current through the primary winding. Because this current can't be increased inde�nitely, there isa 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 coilsand 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 ofthe toroidal �elds Bφ is obtained from Ampere's law, taking a line integral around a circular toroidalcircuit inside the toroidal �eld coils, and neglecting the small poloidal plasma current

2πRBφ = µ0Iφ ,

where R is the major radius coordinate and Iφ is the current in the toroidal �eld coils. So thedependence of the �eld on the major radius is

Bφ ∝ 1R

,

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 tentimes smaller.In total there are three types of coils. The central solenoid coil drives the toroidal current whichgenerates most of the poloidal �eld. Then there are the toroidal �eld coils and the correction coilsthat change the �eld to counter local deformations and instabilities.

3.3 Quality of con�nement

3.3.1 The β factor

The hot plasma exerts an outward pressure p = nkT . This outward pressure of the plasma has to bebalanced by an inward force and it is convenient to think of the magnetic �eld exerting a pressureequal to B2

φ

2µ0. The parameter

β =2µ0p

B2φ

,

is de�ned 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 qualityof magnetic con�nement.There have been many attempts to develop magnetic-con�nement con�gurations with β = 1, but themost 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 theenergy con�nement time.Even with the strong toroidal magnetic �eld, tokamak plasma becomes unstable if either the plasmacurrent or the plasma density is increased above a critical value and the plasma extinguishes itself inthe timescale on the order of 10 ms [3]. These disruptions can be avoided to some extent by carefulselection of the operating conditions, such as plasma current and density.Disruptions occur when the magnetic �eld at the plasma edge is twisted too tightly. Increasingthe plasma current increases the poloidal magnetic �eld, enhances the twist, and makes the plasmaunstable. Increasing the toroidal �eld straightens out the twist and improves stability. The e�ectof plasma density is more complicated. Increasing density causes the plasma edge to become coolerbecause more energy is lost by radiation. Because hot plasma has a higher conductivity, the cooleredge squeezes the current into the core, increasing the twist of the magnetic �eld by reducing thee�ective 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 �eldlines are twisted. The twist is measured by the parameter q, known as the safety factor, which is thenumber of times that a magnetic �eld line passes the long way around the torus before it returns toits 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 circularcross-section torus it can be approximated with

q =rBφ

R0Bp,

where r is the minor radius of the �eld line and R0 the major radius of the torus. Large q indicatesa gentle twist and small q a tight twist. Usually the plasma becomes unstable when the parameter isq < 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 downof the α particles resulting from the fusion reactions. However, the fusion reaction rate is negligibleat low temperatures, and to reach the temperature required for ignition it is necessary to provideadditional external heating.

3.4.1 Ohmic heating

As well as generating the poloidal magnetic �eld that provides the pinch e�ect, the toroidal plasmacurrent Iφ provides a way of heating the plasma. It heats it through electrical resistivity that isconsequence of electron-ion collisions.This process performs two crucial functions in a tokamak, a very e�ective way to heat the plasmaand con�ne it, but it has one major shortcoming. The electrical resistance of the plasma falls withtemperature as

η ∝ T−32 ,

which comes from the temperature dependence of electron collision frequency in plasma. The mostobvious solution is to increase the plasma current, but increasing lowers the q value and causesdisruptions unless the toroidal �eld is increased proportionally and there is an upper limit to thestrength of the �eld because of the forces on the coils. The maximum plasma temperature attainablewith 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 neutralatoms(deuterium if we are dealing with a DT reaction) are injected into the chamber and heat theplasma by collisions.The beams used for injection heating have to be composed of neutral particles because ions wouldbe re�ected by the tokamak magnetic �eld. Deuterium ions are �rst accelerated to high energies bypassing through a series of high-voltage grids.They are then neutralized by charge exchange in a deuterium gas target, and the unwanted residualions removed. The beams of neutral atoms are not de�ected by magnetic �elds and can penetrate theplasma, 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 generationtokamaks and fusion power plants will require neutral beams much higher energy, up to 1 MeV [3].

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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 theplasma with electromagnetic waves. There are three resonance frequencies that are used for plasmaheating. And they all depend on the density of the toroidal magnetic �eld Bφ. This means that theyheat 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 GHzin present tokamak experiments to 200 GHz in a fusion power plant. It corresponds to the frequencyof the electron cyclotron motion in the magnetic �eld. The advantage of this method is that theantennas that project the electromagnetic waves into the plasma can be farther away from the actualplasma than with other methods and thus introduce less impurities into it. But the problem is thatsources for this frequency range still need additional development to be able to provide power outputin the megawatt range.The second one is the ion cyclotron resonance at 7.5 MHz/T [3] for deuterium ions (it depemndson the charge-to-mass ratio). It corresponds to the frequency of the cyclotron motion of the ionsin the magnetic �eld. The frequency range is from 30 to 60 MHz, in the range of commercial radiotransmitters which means that the technology is already developed. But the antennas have to be veryclose 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 e�ectivefor 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 thereactors in operation now. These are all experimental reactors with the purpose of studying plasmaproperties, magnetic con�nement characteristics, operating regimes and so on. And the purpose ofthese experiments is gathering knowledge and developing technologies needed to build a fusion reactorthat can function as a power plant, connected to an electrical grid producing electricity. The purposeof this chapter is to describe how will electricity be produced and what additional characteristics haveto 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 di�erence being how the heat is generated.The reactor itself generates energy with fusion reactions. This is transfered out of the plasma by theneutrons (carrying 80% of the reaction energy output) since they are not con�ned by the magnetic

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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 thentransfered 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 featuresthat are not necessary in experimental reactors while certain aspects receive additional requirementswhich increase the technological complexity. The most important things are how the fuel cycle of thepower plant is managed, how is the heat and the products of the reactions(helium ash) diverted outof 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 waterand the ratio with hydrogen atoms is around 1/7000 [1]. Taking into account the current energyconsumption 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 virtuallynonexistent in nature. Therefore a power plant must operate in such a way that it generates tritiumin the course of it's operation, a process called tritium breeding. The most convenient way to produceit is through two possible neutron-lithium reactions, since neutrons are already available

6Li + n → 42He + 3

1T 4.8 MeV7Li + n → 4

2He + 31T + n −2.5 MeV

Of these two reactions one is exothermic and one is endothermic but, since the kinetic energy of theneutrons is 14 MeV, both are possible. Lithium is available as a mineral in the earth's crust and inseawater. 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 happensomewhere where the neutrons are available, before they have deposited their energy. Which meansthat it has to be inside the vacuum vessel next to the plasma.

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3.5.2 Blanket

The structure surrounding the plasma, inside the vacuum vessel, is called the blanket. It's main func-tions are tritium production and extraction, transformation of neutron power into heat and collectionof 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 breedingreaction uses one neutron. Therefore to have a closed cycle every neutron from the DT reaction hasto produce at least one atom of tritium. This could be achieved if pure lithium in liquid form wouldbe used as the breeder element in the blanket. The fast neutrons from the reactions would react with7Li and produce more neutrons, but this is not viable. Lithium is dangerous, it burns and ignites incontact with air, reacts strongly with water and it is very hard to extract the produced tritium fromit. That's why lithium in the blanket will be in the form of lithium-lead, lithium-tin alloys or lithiumoxides(LiO2, Li4SiO4).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 toroidalgeometry restricts the amount of blanket that can be used for breeding on the inboard side of thetorus. To still have enough neutrons they have to be produced extra for example with inclusion ofneutron multiplier elements like berylium that produces neutrons through

9Be + n → 42He + 4

2He + 2n

3.5.3 Limiters and divertors

The plasma in a tokamak is con�ned within closed magnetic �ux surfaces and it ends where a �uxsurface is interrupted by a solid surface. So there must be a well de�ned point of contact where theplasma is limited.One way to achieve this is with a limiter, named so because it limits the size of the plasma andlocalizes the plasma-surface interaction. It also protects the wall from the plasma disruptions, andother instabilities. For these reasons it is commonly made of materials such as carbon, molybdenumor tungsten, capable of withstanding high heat loads, thermal shocks, produce little impurities andhave good heat transfer properties.

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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 analternative has been designed. The magnetic �eld lines on the edge of the plasma can be deliberatelydiverted into a separate space where the plasma can interact with the surface. The impurities producedhere 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 iscrucial for the operation of a power plant and is a function that a limiter is not able to perform. Thedisadvantage of the divertor is that it is a complicated part on the inside of the vacuum vessel andneeds 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 concerningplasma temperature, pressure and containment times. The value of the ignition criterion, the tripleproduct nTτE , has been increased by �ve orders of magnitude.

Figure 16: Triple product values for di�erent tokamaks from the earlier(T3) to most recent(JET)

Building on this advances is currently the biggest tokamak project, ITER, being built in the southof France. It is still in principle an experimental reactor but with the goal of proving that a fusionpower plant can be constructed.

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Fusion power 500 MWBurn time >400 s

Plasma current 15 MAMajor radius 6.2 m

Plasma minor radius 2.0 mPlasma volume 837 m3

Toroidal �eld at major radius 5.3 TToroidal �eld at conductor 12 THeating/current drive power 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(factorQ) of at least 10. Besides that objectives include development of technologies and processes neededfor a fusion power plant, including superconducting magnets, remote handling and advanced plasmaheating techniques. Also tritium breeding concepts have yet to be veri�ed 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 cooledto 4.5 K by supercritical helium and the poloidal �eld coils using niobium-titanium. The divertor willbe made of tungsten with carbon plasma-facing components. The water cooling system is designed toextract 750 MW of heat and the entire tokamak is enclosed in a cryostat. Plasma fueling is providedby 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, 1cm thick copper to di�use heat, and around 5 cm of steel. The back part is a 35 cm thick shield madeof steel and water.It is planned that once the top current is reached, with an inductively driven current(through thecentral solenoid) of 15 MA, subsequent fueling and plasma heating will produce a run with a fusionpower of 500 MW with burn duration lasting up to 1 hour.

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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

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