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GOVERNMENT OF INDIA ATOMIC ENERGY COMMISSION

TOWARDS FUSION POWER by G. Venkataraman

REACTOR RESEARCH CENTRE KALPAKKAM 603 102 TAMILNADU, INDIA 1975 R.R.0.-7

GOVERNMENT OF INDIA ATOMIC ENERGY COMMISSION c- l o oi 04

TOWARDS FUSION POWER

G. Venkat axaman

REACTOR RESEARCH CENTRE KA1PAKKAM 603 102 TAMILNADU, INDIA 1975 PREFACE

Earlier this year, the author gave a series of lectures on Nuclear Fusion at this Centre. The audience consisted predominantly of engineers involved in fast reactor technology, none of whom were previously exposed either to fusion or to plasma physics. Subsequent to the lectures, there were many demands for the notes, and in view of their popularity, it seemed worthwhile to make them available as a report so that they could reach a wider audience. In offering this report, the author v/ishes to emphasize that it lays no claim to being authoritative since his previous acquaintance with plasmas was restricted only to certain aspects of solid state plasmas„ Rather it is offered as a • broad general review for the non specialist who has a cultural interest in fusion. The level of discussion is maintained low enough that it can be followed by scientists and engineers in diverse disciplines. It is hoped that from a perusal of this report, the reader will obtain a perspective view of both the march towards fusion, and of future prospects.

The assistance of Shri N.S.Thampi, Shri P.Subba Rao and Shri K.Joseph in the preparation of this report is gratefully acknowledged. Special thanks are due to Shri Thomas Kutty for assistance with the figures.

Kalpakkam G. Venkat araman 1975 CONTENTS

Page No.

CONCERNING FUSION REACTIONS AMD FUSION POWER SYSTEMS IN GENERAL 1.1. Introduction 1 1.2. Fusion Reactions 3 1.5. Need for a Hot Plasma T> 1.4. Lawson's Criterion 15 1.5. Pulsed Versus Steady State Machines 21 1.6. Summary 24

2, PLASMA BEHAVIOUR AND DESCRIPTION 26 .1. Definition of Plasma 26 ,2. Plasma Production 32 .3. Approach to Description of Plasma Behaviour 32 ,4. Relativistic and Quantum Effects 34 Orbit Theory 36 >6 o Statistical Approach - An Outline 42 » ! a Vlasov Equation 45 2.8. Concerning Screening 46 2.9» Effects of Collision ~ Modifications to the Vlasov Equation 50 2.10. BBGEY Chain 54 2.11. Macroscopic Approach i>6 2.12. Waves and Oscillations in Plasmas - General 62 2.13. Normal Modes of a Cold Plasma 63 2.14. Effect of Temperature (Including Damping) 73 2.15. Effect of Surface " 76 2.16, Plasma Instability 77 2.17. Energy Losses from Plasma 81 2.18, Chapter Summary 86

3. TOWARDS MAGNETIC CONFINEMENT 89 3.1. Intro duet ion 89 3.2. General Remarks on Confinement 89 3.3. Pinch SysteiAs 93 3.4. Mirror Machine 101 3.5o Toroidal "Systems including the TOKOMAC and the STELLARATOR 105 ii

Page Ho, 3-6. Other Confinement; and TIT Mae nine Concepts 116 3-7. Quo Vadis? 125

4. LASER INDUCED FUSION 130 4.1. Production of Plasma by Laser 131 4.2. Implosion by Laser 139 4.3. [Typical Results for La3er Implosion 147 4»4. Problems of Laser Implosion 150 4.5, Prospects for High.-Powered Lasers and Future Expections 160 4.6. Some Other Ideas 163

5. HIBRID SYSTEMS 166

6. PROBLEMS OF FUSION TECHNOLOGY 180 6.1. Introduction 180 6.2. Description of a Typical MCTR 180 6.3. Plasma Heating 186 6.4. Vacuum Probleu 188 6.5. Materials of Construction 189 6.6. Radiation Damage Problems 192 6.7. Blanket Problems Including Tritium Breeding 199 6.8. Magnet Problems 207 6.9. Power Generation 215 6.10. Possible Hazards Associated with Fusion Reactors 219 6.11. Laser Driven Systems 221 6.12. Concluding Remarks 225

APPENDIX 229 1, CONCERNING FUSION REACTIONS AND FUSION POWER SYSTEMS IN GENERAL

1.1. Introduction

The term nuclear fusion applies to the combining of two nuclei to form a bigger nucleus, AS in the case of nuclear fission, fusion offers the possibility of power generation becavise often a fusion reaction is accompanied by the release of energy.

Fusion reactions are not new and have been known since the thirties, The realization that nuclear fusion could provide a massive source of energy appears to have come during the Second ¥orld War, unfortunately in a destructive context. It is interesting that in contrast to the history of fission, the destructive application of fusion has preceded the constructive. To amplify, the first controlled nuclear fission chain reaction was achieved in December 1942 while the first fission bomb was exploded in July 1945. In the case of fusion on the other hand, the first device was exploded in 1954 while the controlled release of energy is yet to be achieved J

Although the early interest in fusion was mainly from the weapons point, it was realised even then that fusion offered an alternate source for power generation. Accordingly, work on the controlled release of energy from fusion was contemplated or started in many laboratories by the early fifties but unfortunately, much if not all of this remained classified, largely on account of the international tension then prevailing - Indeed there was even no acknowledgement that such work was in progress. The first emphatic public prognostication that controlled fusion was possible, was made by Homi Bhabha in his Presidential Address to the First -2

International Conference on the Peaceful uses of Atomic Energy held in Geneva in 1955« Bhat>ha declared, "It is well known that atomic energy can "be obtained by a fusion process as in the hydrogen bomb and there is no basic scientific know-ledge in our possession today to show that is impossible for us to obtain this energy from the fusion process in a controlled manner.. The technical problems are formidable but one should remember that it is not yet fifteen years since atomic energy was released in a pile for the first time by Fermi. I venture to predict that a method will be found for liberating fusion energy in a controlled manner within the next two decades. When that happens, the energy problem of the world will truly have been solved for ever." This statement literally took the lid off, and shortly thereafter (controlled) fusion physios and technology became declassified subjects. The fi - 't step in this direction was taken by Academician Kurchatov who in 1956 delivered a lecture at Harwell on some controlled fusion oriented work done in the Soviet Union. The ball having been set rolling, the various laboratories soon began to vie with each other in describing their respective fusion programmes, following which inevitably came the journals, international conferences and the like devoted completely to fusion.

Nearly two decades have passed since Bhabha's forecast, but fusion power has not yet become a reality. This, however, has not been for want of trying* A considerable amount of effort has been put in all over the world, and the delay has been due to the appearance of unexpected problems. These took a long time to understand and solve, and after several years of inching forward, the climate today is one of optimism. -3-

In what followss we shall broadly survey these developments.- We start off with a brief description of fusion phenomena and "the conditions required to be attained for the controlled release of energy. It will turn out that the latter requires the production of a hot plasma, and for this reason we shall give some attention to the properties and the "behaviour of a plasma, Magnetic confinement „ the first major attack launched on taming fusion power will be reviewed next- This will be followed by a discussion of laser-induced fusionj, an alternate prospect that has created widespread excitement ^recently. Finally, looking into the future; we shall outline the nature of fusion technology and the problems likely to^be faced there- Interspersed will be a few related digressions.

1*2, Fusion Reactions

As already mentioned, nuclear fusion like nuclear fission, is often accompanied by the release of energy, and for the same reason. This may be better appreciated from Fig.1*1 which shows a plot of the binding energy (BE) per nucleon as a function of mass number.^(1") ' The higher the BE/' nucleon., the more stable will be the nucleus which explains why heavier nuclei tend to fission while lighter nuclei favour fusion ( - some "impetus" may be required though to initiate these processes). Both in fission and in fusion, there is a net decrease of mass which reappears as energy, fusion being the more "efficient" of the two processes. To see this,, assume that a nucleus of mass number 240 undergoes fission into two fragments of mass 120. From Fig.1.1, I I

30 60 90 120 150 180 210 240 270 Mass number A > Pig.1.1. Binding energy per nucleon as a function of mass number. (After Marmier and Sheldon, fef.1) -5-

(BE/A) after fission - (BE/A) hefore fission X? 8.5 MeY/nucleon - 7.5 MeV/nucleon « 1-0 MeV/nucleon .

The corresponding figura for 'the fusion reaction D + D H + H is =2 (7*075. -1.113) ^ 6 MeY/nucleon, which is much higher than the gain achieved in fission. Per nucleus therefore, fusion reactions liberate more energy than fission. In passing it is worth observing that nuclear fusion is similar to the fusion of molecules to form bigger molecules e.g. Oxygen + Hydrogen —> Water; only.the energy liberated in the former process is about 10 times larger.

Many examples of fusion are available but from the power production angle it is sufficient to confine attention to just a few. Digressing for a moment,, we remark that the Sun produces energy via fusion, and in fact the Sun may be loolced upon as a thermonuclear reactor employing gravitational confinement J The early work of Bethe and Weizacker on energy production in the Sun is quite well known; however, their ideas have been subsequently modified and presently it is believed that the following two cycles (for hydrogen to helium conversion) are the important ones:vw

(i) P + P -* D + (t • + V + 0.42 MeV P + D -> He3 + tf -: 5.3 MeV

p "5 4. He + He -> He^" + 2P + energy t (from another PP, PD set)

Net effect: 4P —> He 4+2/a + 2 V + 2 tf + energy -6-

+ 5+P~^D+. A +>> + 0,42 MeV P + • D -> He- IT •:• 5.5 MeV

He4 + He3 Be' + tf

Li7 + V + Y

Li7 -> 2He4

4 + / Uet effect: 4P He + 2/i +2y+23 + energy

5he first set is believed to have been responsible for energy production during the early stages of Sun's history and the second set is considered to he more important at the present stage of Stm's life on account of the higher central temperature and helium concentration- Reverting to a consideration of fusion power, the most important reactions from this angle are:

U + D r >- He3 (0.8(0,82 MeV) + n(2„45 MeV) (a)

1 $> £ (1#1 MeV) + H(3.02 MeV) (b)

3D + T He4 (3.5 MeV) + n(U.1 MeV) (c) 1) + He- He4 (3,6 MeV) + H(H»7 MeV) (d)

(1.1) 3}he principal feature of these reactions is that they are all based on deuterium of which there is practically limitless supply in the oceans (isotopic abundance of D is -7-

0„015$). Besides, the cost of recovery is also relatively small, being of the order of a few cents per gm. Based on availability, then, the D-D reaction would seem the obvious choice for founding a power programme. Unfortunately, there are some drawbacks. Firstly the energy per event is not as high as in the case of D-T, for example. Secondly, and this is the more important consideration^ the temperature and the confinement conditions are far more severe for D-D than for the D-T system. On the other hand, the D-T scheme while attractive from the energy release and ignition temperature angle,* suffers from the disadvantage that tritium (i) is not found in nature, and (ii) is radioactive. These are, however, surmountable problems, and the former can be tackled by resorting to breeding, concerning which more.will be said later. Although He^ is not available in Nature, the (D + Sr) reaction has been included in our list because it has been considered as a candidate in some prospective thermonuclear machines employing direct conversion to electricity (see last chapter),. Presumably the required He can be bred elsewhere.

1 «5- Need for a Hot Plasma

Per 3e„ the fusion reaction is not difficult to achieve, and in fact the (D-T) reaction is used quite routine­ ly in many laboratories to produce 14 MeV neutrons. This is done with an accelerator as schematically illustrated in Fig. 1.2. Deuterium ions produced in the ion source are accele­ rated down a column to about 150 keVj, and allowed to impinge on a metal target in which tritium is adsorbed. The incoming

* Why ignition is involved will become clear shortly. -8-

N SOURCE ^D BEAM JARGET

POWER SUPPLY

Pig.1.2. Schematic drawing of an electrostatic accelerator used for producing H MeV neutrons via D-T reaction. The target is usually a Zirconium foil with tritium absorbed. -9- deuterons must have sufficient Kinetic energy to overcome the Coulomh harrier of the tritium nuclei if fusion is to occur. ("5) A reference to Fig. 1.3 shows that 150 keV is an ideal energy since there is a resonance there. Although adequate as a neutron generator, the system of Pig.1.2. is totally unsuitable as a power source for the reason that

^D-T fusion^ 10"24**2) <£ ^-io* collision^10 cm *

In other wordss only ahout one out of every ~\0 incident deuterons has a chance of causing fusion. Therefore,

17 Me7 Energy gain - ****& ffffifl ^ A Energy input 104 x 150 KeV

-2 •-* 1Q , which is clearly unacceptahle.

The principal message of the above exercise is that accelerators are not useful as power generatorsr If gainful energy production is to occur, then-the system must be such as to sustain spontaneously a large number of nuclear collisions at high enough velocities. One way to achieve this would be to take a mixture of deuterium and tritium gases, and heat the mixture to a very high temperature v*> 10 degrees K, At this temperature the gas will be completely ionized. i,e. will consist of deuterons, tritons and electrons. Even though the average energy v/ill be only v^ 10 keVl, many nuclei will have energies v-^ 100 keV on account of the Maxwellian tail; -10-

10 1 • ' • „_. .. I m • ,'~-\ / SI —- k ^v

\ 1 -/ 6 -?=^:: -> - H z/ : ™ ^ -S( /

in '"

< c m * 2 ^ T1 0 u Ul V) i / / if/ 10" *

W/ H / / / / 3 tor =/ ..I fc -^ L "ii y/ ' /

! 6 16 10 /2 0 l40 GO BO 100 200 400 600 <000 OCUTEROM ENERGY (KEV) lig.1.3. Variation of the fusion croS3 sections as a function . of the deuteron energy for D-D, D-T and D-He3 i(lt reactions. Observe the resonance around 150 keV for the D-T reaction. (After Glasstone and LoVberg, ref.3). -11-

such nuclei can collide and fuse., liberating energy. Since the medium is hot? there will always "be nuclei available with requisite energies to'undergo fusion. It thus emerges that a fxxsion pov/er system must essentially consist of a very high temperature gas; the latter will be in a highly ionized state and thus constitute a plasma (to be formally defined shortly). •Since the fusion reaction in the hot plasma is driven by the kinetic energy associated with heat motion, one often refers to it as a thermonuclear (TN) reaction, The problem of generating power from fusion is then essentially one of (i) producing a hot plasma, and (ii) arranging conditions for the sustenance of TIT reactions long enough to extract energy. If the process is achieved with explosive violence, one has a bomb; with controlled release of energy, one has a thermo­ nuclear reactor (TNR),

It is worth adding here a historical note concerning the mild excitement created in the fifties by the experiments ( Z) of Alvarez and his colleages at Berkleyv/ wherein it was demonstrated that fusion reactions can be catalyzed by cc- mesons at low temperature J The idea is ingeneous. I^A. mesons are a species of particles rather like

electronsr, but with a mass ""> 212 times larger. The p- 's come in three varieties, positively charged, negatively charged and neutral. They result from the decay of another species of mesons v'z. the TV • mesons, which not only occur in cosmic rays but are also copiously produced in high energy accelerators. (In fact, there is even a meson factory which has recently been set up specially for meson physics experi­ ments). What . . the Berkley group showed was that when a target of liquid hydrogen is irradiated" with a M-~ beam, a -12- few events v/ere observed in which the muons catalyzed fusion. The procacs may "be understood as follows: To start with the if attaches itself to a deutron (or a proton) to form a neutral JJL- mesic atom i-e. a hydrogen-like atom with the jJL~ replacing e~. This muonic atom which is electrically neutral could easily get attached to another deuteron (say) to form a molecular ion D0 . However, in contrast to the usual ion, the one under consideration is bound by a Lt ~ • The mass of U.~ being greater than that of the electron, the binding energy is also larger. In this tight grasp> the vibrations of nuclei will quite frequently bring them into coincidence with a chance for fusion (of the D-D type). The meson will of course get released in the process and could trigger more reactions of a similar nature- The time a HI' would require to complete a fusion process is estimated as 10~ sec, On the other hand-, the'lifetime of —6 the |A_ is also small (v^10 sec) so that a large and steady stream of u. 's is required to keep the fusion process going, which makes the process economically unattractive.

Returning to the mainstream? we canr having "broadly recognized that the road to fusion power is via thermonuclear reactions in a plasma, now sharpen the focus "by raising a series of specific questions as follows:

i) How is the plasma to be produced, heated and confined? ii) How to keep energy losses to a minimum? iii) How to achieve more energy output than input, and finally, iv) How to harness the energy released, i.e. produoe electricity. -13-

1.4. lav/son's Criterion

The efforts made to find answers to the above questions will be reviewed in the next few chapters,. Prior to that, however it is appropriate to define a criterion of performance for any proposed TIT system, which will provide an assessment of the economic viability of the system concerned. Such a criterion' was first spelt out by Lawson, and will be presented here in a rudimentary form. The essential idea is to estimate to what temperature the system must be raised when the useful energy released at least matches the energy inputs To simplify matters,, two assumptions will be made (i) that the gas retains all the energy supplied to it by the input source, and (ii) the output device converts the energy released in fusion to a usable form (i.-e. electricity) with an efficiency of unity.

Consider then a thermonuclear mix heated instanta­ neously to a temperature T to create a plasma. The energy input necessary is given by

Ein ^ 3nkBT

where n is the total density of nuclei and is the sum of

individual densities n. and n2 of the two reacting components.

For a D-D system, n.. = n2 = n/2. The 3nte*J£ term represents the energy required to heat the gas. Electron binding

energies are neglectedi but the contribution from electrons is included. Sxippose now the TIT reaction occurs in the plasma and lasts for a time T ' (often called the confine­ ment time). The energy output will then be given by ' •14-

Eout ^ •' RBT where R denotes the fusion reaction rate/cnr of the mix, and E is the energy liberated per fusion event. Therefore, ignoring losses,

„. „ Eout REX Sai„ . a =5— . 35^

and must be equal to one for break even. A meaningful system must therefore at least satisfy the criterion G ^ 1 i«e.

L(T) 5 M > -_L~ . (1.2) n x 3nricBT

ITow the rate R at which a reaction proceeds in a uniform gas mixture may be written in terms of the cross section

a1&1)a2fcT2> III -J2| ^"(I1 -j2>dXld^' Assuming Maxwellian. distributions, and introducing the variables v = v* - vol V = (ra-v, + m0v0)/(m. + mQ) and subsequently integrating over V,

R » n^g (o-v>T (1.3)

where 00

V/hile the exact form of cT(E) is not known, one may assume that the collisions are dominated "by Coulomb interactions and hence v/rite after G-amov,

C(E) « AE~1 exp'(-BE"^),

(7)K,J which eventually gives

2 3 1/3

Figure 1*4 shows a plot of \0*vXi versus temperature for the fusion reactions mentioned earlier, ("5)w/ The superiority of the D-T reaction at lower temperatures is clearly evident.

Tv/o conclusions which emerge from above are: (i) The r,h-s« of Eq.'(1-2) depends mainly on temperature, and (ii) for a given, fuel mix* the product (nX ) alone is fixed and not n and X 'individually. Thus different systems are possible with widely varying values of n and "C but having the same (nX ) • value. The above analysis suggests that once the plasma is heated above a threshold or "ignition" temperature, gainful release of energy should be possible,, the gain increasing with temperature,, Unfortunately, this is not true on account- of various losses.

Energy can "be lost from a hot gas in two ways,

namelyp radiation and conduction, More will be said about these later but for the nonce we remark that with these included, I>(T) is not a monotorjieally increasing function of T but rather has a peak as sketched in Pig.1.5 . Prom . this it is clear that -16-

i 1 t 4 6 8 tO to 40 €0 90 tOO 800 400 600 •000 KINETIC TEMPERATURE (KEVI

Pig.1.4. Variation of <&v? with temperature. (After Olaestpae and :Lov&aa^ SfflJf.3). -17-

(i) for every fusion reaction there is a critical (nT) value,

there is a (ii) for eveiy (nU ) ^(nr)Gritical9 lower as well a higher limit of temperature for the fulfilment of the lawson Criterion.

•The aim must therefore he.to design a machine which has a (nT) value greater that the critical, and arrange for it to operate at a temperature which lies within the two hounds. At a given temperature, the power density would naturally increase with the particle density, the expected variation being as sketched in Pig.1.6^ \ For machines where the plasma is expected to he in the "gaseous" form, the particle densities anticipated are in the range 10 ^ - 10 / •z cm , correspondingly, ~C is expected to he of the order of —1 —3* 10 - 10 sec. (Later we shall discuss laser driven fusion where n will he 10^" times normal solid density! In this case, "C will shrink to a few nanoseconds.)

It is important at this stage to recognize the relationship of the "confinement time" occurring in Lawson's Criterion to the characteristic time t„ for Coulomhic collisions between the ions. It is found thatv '

1 5 2 t s u-> Ix. /

where 3?i is the ion temperature. If now we define Q = (X/b3), then it is clear that Q is a measure of the numher of collisions ions will make in one confinement time. ]?usion reactors are often divided into two classes depending upon whether Q ^ 1 or Q <^1. Systems in the former class would -18-

1-5x10

5x10 I-

Kf 10? IC? ICP TEMPERATURE IkeV)

Pig.1.5. Lawson function L (T) for the DT reaction. -19- correspond to low temperature reactors. In such machines, the ions would make several collisions in one confinement time, and the velocity distribution of the ions would closely approach an isotropic Maxwellian corresponding to 'J?.. Reactors with Q <1 1 would be "high temperature" systems, in which the velocity distribution would deviate substantial­ ly from Maxwellian on account of the relatively few collisions. One example of such a distorted distribution would be that associated with a beam of high energy particles. The concept of a temperature in this case clearly requires a stretching of the familiar principles of statistical physics, and is to some extent notional.

Y/hile on tie subject of plasma temperature it must be remarked that electrons and ions may have different temperatures T and T. respectively, with TQ being usually greater than T.. The existence of two constituents of a physical system at. different temperatures is not altogether unusual, and those familiar with magnetic resonance will recall the possibility of nuclei in a solid bei

The time T occurring in lav/son's criterion requires a further comment. If one looks more carefully into the energetics of a thermonuclear machine, then many characteristic times may be recognized such as the time for which the high temperature lasts,* ' and the times associated with the leakage of the plasma particles by various mechanisms. Prom an operational point of view, leakage processes will determine the plasma confinement, and can be clubbed together Into a single characteristic •time only in an approximation. This time in turn can be -20-

10" rf «f «T DENSITY (NO/CM3)

Pig.1.6. Fusion reactor power densities plotted as a function of fuel-iqn density of D-T mix', at a temperature of 100 keV. The situation ^. *;. corresponding to two typical reactors is arao *> shown along with, their temperatures ana confining fields. (After Post, ref.9). -21-

coinpounded in some sense with the time associated v/ith the high temperature state to yield a single characteristic time T". It is such a grand average we shall consider in Lav/son ' s criterion.

1*5. Pulsed Versus Steady State Machines

The fact that an initially prepared plasma has associated with it an average characteristic time Z (in the sense described above), would seem to imply that laboratory thermonuclear reactions must necessarily be pulsed, the cycle being - plasma creation: and heating, nuclear burning, and quenching. Indeed this is true for many systems as will be seen subsequently. For example, in reactors where fusion is achieved by shining an intense laser pulse on a DT pellet (Chapter 4), it is expected that the TN burn v/ould start at the centre of the pellet and propagate outwards, rather like a spreading fire. In this case, the reactor operation will be necessarily pulsed since the burn obviously cannot last after the fuel is consumed (in fact it is expected to be quenched even earlier). On the other hand, in 3ystems where the plasma is confined by a magnetic field, one could have both pulsed and steady-state operation. In a steady-state machine, the plasma would be ignited, and as the burn peters away (due to loss of temperature and/or fuel), fresh fuel and heat are added to keep the cycle going. The Sun is an example of such a steady state system v/ith this particular feature that the heat required for the burn propagation is supplied by the reaction itself. Such self absorption, hov/ever, is generally not possible in terrestrial plasmas (except of the super high density variety - cf. -22-

chapter 4) and the siistenance heat must be by external source,

The different possibilities have been analysed by (11) Millsv ' v/ho notes that a machine proposed as a reactor using plasma confinement may fall into four categories depending on the (n~) values, (See Pig. 1=7). For (nT ) less than the Lawson-Criterion, the machine operates uneconomically, producing less than the input power. If (n T) is greater than that specified by Lawson's criterion (12} but less than another specified by Spitzer, ' then a steady-state rea.ctor is possible but a continuous supply of energy from outside is necessary• T^e Spitzer condition is obtained by equating the portion of the released energy that is deposited in the plasma to that required for heating to plasma temperature, the cold fresh fuel injected to

* Even in the case of self absorption, for chain reaction and steady burning, it is nec;?s3ary that the heat deposited shoufl not leak away. In the next chapter we shall consider several leak mechanisms and it will be found that they involve in many cases the transfer first of the reaction energy to the electrons, If this process can be inhibited then correspondingly leakage could be minimised. Glasstone and Lovberg^ ' calculate the minimum size of a plasma having self sustaining thermonuclear reactions corresponding to different assumptions for energy loss. In every case the reaction heat is assumed to be wholly absorbed^ When the loss is primarily through black body radiation/ the minimum radius for a D-D system is is> 600 miles J If cyclotron radiation is the dominant leak mechanism, the minimum radius works out to ^ 95 metres- -23- (nr)

Pulsedt reactor s

Steady state rtactor 1 SPITZER'S possible with no continualr*

LAWSONS <*=» CONDITION no usefu1l reactor

FUEL OUTPUT HEAT

(b)

COLO ^OUTPUT (c) FUEL REACTOR o_ Pig.1.7. (a) Shows the different regions of (n,T ) classified as per Mills.C'O) (b) Shows the input/output of a reactor with (n,r ) between Lawson's and Spitzer's values. (c) Shows the situation for a reactor operated at the equilibrium condition. -24- replenish losses. This is to be contrasted with Lawson's condition which is obtained by equating the electrical energy recovered to the thermal energy necessary for ignition. If nT= (nT)s-oi^r,p.. then there is equilibrium and the reactor can be refuelled with cold fuel' ("see Pig.'1.7). In this case the heating of the injected fuel is'accomplished mainly by the energy deposited by the charged particles produced during fusion (i,e. the alphas in the.case of D2 reaction). A reactor operated under the Spitzer condition is .thus rather like an oil furnace which, after ignition,' is kept going by spraying fresh oil. For this case, the ion and the electron temperatures will remain constant with time. However, operation at this condition will be complicated since the equilibrium may not be stable, and feed back control may be necessary.* If nr exceeds the Spitzer value, then electron temperature runaway will occtir causing the electrons to stream .in. some particular direction and hit the container walls, thereby destroying confinement-after some time. In this situation only pulsed reactors, are possible.

1j 6. Summary

Ih this chapter we have briefly reviewed the basic ideas concerning fusion reactions and have" noted that the controlled release of fusion energy must necessarily be through a hot plasma. In this . context, w!as "introduced the Lav/son's Criterion, which plays a role somewhat analogous to

* Even for a steady-state reactor, the time ~C has a meaning. In this, case it refers to. the., time .agiven. "batch" of reacting nuclei are confined..'*'.".-.. . -25-

k ^ in renctor physics. The difference of course, must be recognized; the former refers to break even and the latter to sustaining a chain reaction. So far no system has been developed which fulfils the Lawson requirement. However, thanks to the effort put in during past two decades, it is hoped and believed that the summit will be climbed shortly!

REFERENCES

1. P.Marmier and E.Sheldon, Physics of Nuclei and Particles (Academic, New York (1969), Vol.1. 2. I.Kaplan, Nuclear Physios (Addison Wesley, New York, 1971). 3. S.Glasstone and R.H.Lovberg, Controlled Thermonuclear Reactions (Van Eostrand, New York, 1960). 4. L.Alvarez et al., Phys. Rev. 1^5, 1127 (1957). 5. J.D.Jackson, Phys. Rev. j.06, 33Q (1957).

M 1 6. J.D.lav/son, Proc. Phy3t Soc. (London) B 70, 6 (1957). ': 7. W.B.Thompson, Proc. Phys. Soc. (London) B?7Q-« 1 (.1957). 8. T.S.Green, Thermonuclear Power (George Newnes, London, 1963). 9. R.P.Post, Ann. Rev. Nucl. Sci. 20, 509 (1970). .10. L.A.Artisimovich, Controlled Thermonuclear Reaction (Gordon and. Breach, "New York, 1964). ! 11. R,G.Mills, in Proceedings of the Nuclear gusion Reactacs Conference 1969 (UKAEC, Culham Laboratory for BNES, 1970), p.322. 12. L.Spitze* et.al., USAEG Report NYO-6047, Washington, D.C., (1954). -26-

2. PLASMA BEHAVIOUR AND DESCRIPTION

la this chapter we shall describe a plasma and •briefly outline how its properties are analysed and studied Toy theoretical plasma physicists. Such a discussion will form a useful prelude to the next chapter wherein we shall review attempts made to achieve plasma confinement. It is worth, emphasizing that our treatment here will .he largely descriptive than deductive. This is quite adequate for our i purpose which is to give the reader the general flavour rather than the equipment to solve problems associated with the plasma.

2.1. Definition of Plasma Mil"11 I' •— ••!•••• •• li •" •"•' ' '»«• A plasma may be very simply defined as an ionized gaci. Sometimes,' plasma is also referred to as the fourth state of matter.- The reason for this can be appreciated by following through what happens when a solid is heated. The sequence is illustrated schematically in Fig.2.1 wherein the transformation of the solid to liquid and subsequently of the liquid into gas, is well known. If the gas is further heated, a stage will be reached when it begins to ionize, i.e., when the "heat of ionization" is added. At this- point, the medium will consist of ions,* electrons and unionized or. •'' neutral atoms /molecules. Such a system is usually described as a plasma. Many subcategories may be identified like weakly ionized plasmas, intermediate plasmas and strongly

* The ions may be molecular or atomic depending on the substance involved. Furthermore, there could be several chemical species, and the ions of same species could occur with, varying charges. ••}

i •h f\J PLASMA

Energy

Fig.2.1. Relationship of plasma to the other known states of macter Energy input normally takes one through the sequence : solid —± liquid plasma. " '" gas -28- ionized plasmas, and criteria can "be laid dovra to distinguish them. Nowadays, the word plasma is generally applied only to a fully ionized system.

It is interesting that though rare on earth, over 99$ of the universe is in fact composed of plasma. In Fig. 2.2 we illustrate the varying conditions of density and temperature under which the plasma state is realisahle. Noteworthy is the fact that the figure includes a mention of the so-called solid state plasma concerning which more will be said shortly.

An important feature of the plasma is that although composed of charged particles, it is macroscopically neutral i=e., if we consider any representative macroscopic volume, then there will be in that volume, as many positively charged as there negatively charged particles. More formally, if n , n. and n denote respectively the densities of electrons, positive ions and neutral molecules, then macroscopic charge neutrality implies n = n. = n (say). This neutrality is due to the very intense electrostatic forces that arise as soon as n tends to become different e from n. locally. In fact, so strong are these forces that if we try to create a charge imbalance by artificially separating the positive charges from the negative, the two will rush towards each other on release of the constraint. Some overshoot and oscillation may b» expected; charge density oscillations so resulting are referred to as plasma oscillations. We shall learn more about them later. The equilibrium state of.an ionized gas may be -29-

10

SUN

10

10 IONOSPHERE >- CD cc

LU /JTEONP)

YLROO M TEMPERATURE

OUTER SPACE J L L jo _u _18 26 10 10 10 10 # 10 DENSITY (cm)

Pig. 2.2. Different forms of plasma plotted on a temperature- density plane. CTR refers to controlled thermonuclear J?© £tCU027 •

T.K. -30-

characterised by n, T the temperature, and d = (n/n + n ) the degree of ionization. In true thermodynamic equilibrium,

c| , n arid 'J? are interrelated. For instance, if we have a gas of atoms which can exist only in two states- the ground state and the ionized state, then the various quantities are related byu '

3/2 Vi (27[ml£«E) - SjS* exp (-B^/k T) no B where P e , Pi. and P o are the statistical weights for electrons,- ions and neutral atoms, and E ^ is the energy separation between the two states. This is a simplified version of the well-known Sana Ionization formula. As is evident, at very high temperatures, the gas will be completely ionized. The DT plasma which we shall frequently consider here will be assumed to be in this state. In fact it will consist not of Dp, Tg and OT+ ions (*->e' 'tne isotopic counterparts of Hp) but of bare deuterium and tritium nuclei and electrons.

To recapitulate, a plasma may be defined as an assembly of charged particles that has overall electrical neutrality. The number of distinct charged species constituting the plasma determine its components. For example, a hydrogen plasma i.e. a plasma consisting of protons and electrons would be regarded as a two-component plasma. On occasions, one has models for a two-component system wherein attention is focussed only on one member. -31-

The other member is assigned a minor role, and the system is therefore regarded effectively as a one-component plasma. A well-known example is that of conduction electrons in metals. Consider, for instance, metallic sodium. This may he regarded as made up of an array of Na ions immersed in a sea of electrons. Often, for describing the electronic "behaviour, the crystal is visualized as a sea of electrons coexisting with a uniform background of positive charge. The latter is further regarded as rigid and is assigned merely the role of providing overall charge neutrality. Such a system is referred to as an electron ga3. and is an example of a two-component system modelled as a one-compo­ nent one. As with metals, one can think of plasmas iija semiconductors also. Here one could have electrons and/or holes (produced by impurities, for example), and these, together with their background compensating charges, can be regarded as constituting a plasma. Another example of a "condensed state plasma" would be liquid mercury. Here again, one has' a two-component plasma but if required, attention could be focussed only on one of the components at a time. If the electrons are chosen for attention, one has effectively an electron gas. On the other hand attention could also be directed to the positive ions (assigning to the electrons the role of instantaneously correcting charge imbalances), once again obtaining a one- component model.

Mention of mercury makes it pertinent to refer to the electrical conductivity of plasmas. Now a coaoUctor is usually looked upon as a material in which there are electrons free to move about. When subjected to an external -52- electric field, the electrons immediately respond 'in such a manner as to screen the field and nullify it in the interior. However, by making the conductor a part of aa electrical circuit, the field in the interior may "be sustained. Though exposed to a field, the electrons do not get accelerated indefinitely since they tend to lose energy to the lattice "by collisions. This is the origin of the veil-known ohraio heating, and may "be characterised in terms of a quantity known as conductivity. For a solid conductor, only the electrons contribute to conductivity but in a liquid metal like mercury, the ions too influence the electrical behaviour. A plasma "being akin to liquid mercury in some respects, is often treated as a conducting fluid. Of course ascribing fluid-like properties such as flow etc. 22 can "be"done only at high particle densities ( in 10 charged particles/cm ). The plasmaswe shall consider (in this and the following chapter) are more akin to a conducting gas since the densities are £L 10 /cm . 2,2. plasma Production

Plasmas can be produced in a variety of ways such as thermal, injecting an ionised beam into a gas, photoionization, a.c. and d.c. discharge. Not all the methods mentioned above are relevant to the production of thermonuclear plasmas.

2.5 . Approach to Description of Plasma Behaviour

We now come to the question of how to describe and study the dynamical behaviour of plasmas. As we shall -33-

shortly see, tb.ere is available from statistical mechanics, a method capable of providing a complete description o-f plasma behaviour. Unfortunately j, this is an utterly formal approach of little practical value. At the working level, simplifications are possible, and, depending on the circumstances, varioua view points are adopted. In one extreme, one analyses the behaviour in terms of the trajectories of the individual particles. It is assumed that the particles do not interact \tfith each other and that they are influenced only by electric and magnetic fields. In other words, one studies the plasma by following through the orbits executed by the constituent particles in the fields that may be present.* It must be intuitively obvious that this single-particle approach is meaningful only when the particle densities are very small. Though not rigorous, this approach enables the visualization of the gross aspects of plasma confinement as will be seen in the next chapter.

\tfhen the density of particles increases, jnter- particle interactions begin to assume importance, and at this stage the single-particle approach becomes inadequate. Resort'is therefore made to methods, akin to those used in kinetic theory of gases except that here the constituent particles are charged and not neutral. As the density and/ or interaction becomes even larger, the gas-like approach

* If necessary, particle interactions may be included to a limited extent by taking account of the fields produced by all the other particles on the representative particle whose orbit is under study. -34-

ceases to "be useful, and appeal must he made to fluid- mechanical techniques. The methods of study referred to above may "GO sehamatically summarized as follow:

Density: nearly aero > small >, very large

Techniques single-particle") __^^ ionized gas? ^conducting model/orbit V ~~~^ model J ^ fluid model theory ^ later v;e shall amplify on the above mentioned techniques.

2.4. Relativistic and Quantum Effects

A 'brief comment nest OH relativistic and quantum, effects. Recalling that relativistic effects become significant only when particle velocities approach that of light and that velocities depend on particle temperature, it is easily estimated that for electrons, such effects assume significance- only for temperatures ^,10 K while for protons the temperature must he > 10 ^ K. Relativistic effects are thus ignored except for very high temperature plasma wherein also the corrections are significant only for- the electrons.

Qua&tum effects become important only at high densities aaad/or "low" temperatures. These criteria can "be amplified as follows: Suppose all the electrons in the plasma are assumed to fill a Fermi sea. In other words, all. available energy states from zero upwards are filled, each -35-

© Core of White Dwarf O Ionosphere E3 Gas discharge

A

Relativistic Plasmas

JO) u V) Classical Quantum o Plasmas Plasmas

O

DENSITY (log scale) Domains where relativistic and quantum treatments must be applied in discussing plasmalfbehaviour. In the low temperature and low density region, a classical treatment is adequate. -36

eaergy state with two electrons cori'es ponding to the two possible spin ox*i«s stations. The maximum energy then deflates the Fermi eaergy Ep. If E_, <^ EgT, then a classical approach is adequate. Such a situation caa occur either on. account of very low density, or very high temperature (or of course both). For the reverse situation viz. E,p ^ knT' a I'U-sjatxim description is necessary. Fusioii plasmas audi as are encountered in magnetically confined systems, may he treated classically since "both the density and the temperature are favourable for this. On t&e other haftd, plasmas in metals must be handled via qua&tuai mechanicsj however oae may get away with a classical approach if one is considering a semiconductor at room temperature, Quaatum effects may also have to be allowed for is the siiper dense plasmas prodticed by laser implosion (Chapiter 4-).

2.5 „ PJlMtJChe crv

We begin now a brief survey of the different methods available for describing plasma behaviour, starting with orbit theory. The latter is the name given to an approach where the plasma behaviour is visualized in terms of the orbits of the individual constituent particles. Obviously, a description of the actual, complex trajectories likely to occur in real plasmas is not possible. However, the analysis of orbits in simple field geometries can be used as guideline fox* understanding plasma behaviour under certain circumstances. -57

In analysing trajectories, the "basic force equation to be used is

where the notation is self evident, ^y solving this equation, orbits for various field geometries may be traced. Usually gravitational field is ignored in the analysis of terrestial plasmas.

We shall consider a few simple examples of the application of (2.1). Suppose we have a particle which is subjected to a magnetic field H in a direction perpendicular to v as illustrated in Pig.2.4a. If there are no electric fields present, the magnetic field will bend the particle around in such a way that at every instant, v remains perpendicular to H. The net result is that the particle will precess around the field lines in a plane perpendicular to them. Prom (2.1), the gyration frequency for the electron, for example, is readily estimated as

c mc a result quite possibly familiar to the reader. A positive ion in a similar situation will also gyrate but in the opposite direction. The quantity ' CJ is referred to as the cyclotron frequency. If the initial velocity of the particle is not strictly perpendicular to H, then only the perpendicular component of v would be affected by the -38-

-*Y

1 K

GUIDING CENTRE

Fig.2.4• Behaviour of an electron in a uniform magnetic field. $U If the electron is initially moving in the .JJ plane before the application^ then the field vrillSorce it into a circular orbit as shown in (a).. . If the electron had a Z comparent of velocity, it will spiral aa shown in (b). T.K -39-

magnetic field, and the particle trajactory in this case will "be a helix (Pig.2.4b). For the field configuration shown in Pig.2.5, the electrostatic forces can either assist or resist the centripetal forces associated with cyclotron motion, the overall result "being particle drift in a loop or more precisely, in a cycloidal-type orbit.

For complex situations such as obtain in the interior of the plasma, trajectory visualization as above is clearly impossible, hut the situation is salvaged to some extent by two concepts to which reference may now be made. The first is the guiding-centre approximation which, from a physical point view, implies that the total motion of the particle can be regarded as a superposition of the cyclotron motion as executed in the absence of electris fields as well as non uniformities in magnetic fields, and the drift of the orbit centre caused "by the effects above excluded. The instantaneous position of the gyration centre is referred to as the guiding centre whence the name for the approximation. Prom a formal point of view, it is supposed that at every point, the particle velocity v is broken into two components as v= v. + v„, where v0 is the velocity of cyclotron motion in the absence of perturbing effects, and v., is the drift velocity of the orbit centre caused by the perturbations. Referring back to Pig.2.4, it will be observed that the guiding centre is clinging to the field line while the particle is tracing a helix. Movements of the guiding centre are also indicated in Pig.2.5. When thermal effects are present, the guiding centre concept may continue to be used; only, in this situation, the motion of the guiding centre J?ig.2.5. Behaviour of charged particles under crossed electric and magnetic fields.. The top part of the figure shows the force balance as a positive ion subjected to an electric field as indicated and magnetic field directed into the paper, e denote the electrostatic force, G the centripetal force and'r the resultant. * Both electrons and positive ions move on a cyeloidal path.

J.K. -41 r

will resemble that of a Brownian particle.

The other idea used in simplifying trajectory analysis viz. adiabatic invariant is based on the noticfflttaat during the complex motion of the particle, certain quanti- "Ciesare conserved or are approximately so. One example of such a constant of motion is the particle energy. The constant of motion more frequently considered is, however, the-magnetic moment, which in fact comes as a package deal with the guiding-centre approximation. Now ac:;charged particle (e.g. electron) executing circular motion may be regarded as a current loo.p, and may be assigned a m^paetic moment U* given by

huT * evr where

myc r = IT denotes the radius of gyration. In a situation'-where the magnetic field varies both in space and time, ^L m will also vary. However, if the field variations are "small'1 in some sense, then the analysis of particle motion can be simplified by assuming U-m is approximately constant. Such a concept is frequently used in conjunction with the guiding-centre approximation which then implies that the magnetic moment of •tot xorbits executed about the instantaneous position of the guiding centre remains constant as the centre itaelf wanders -42 T

arotand. One immediate consequence is that the flux enclosed © r H 1 ex L by the orbit remains a constant (since P m = £ —5— r. -*KS=- 5; f> whei-e (p is the flux), so that the magnetic flux ia often visualized, as being "frozen" into the particle path.

2.6. Statistical Approach - in Outline

She statistical approach Is the antipode of the orbit description considered so far. Even here various shades of treatment are available. The most formal of -these centres around the LiouvUle equation

• |£ + |Vf H] = 0. (2.2)

Here is the IT-particle distribution function and has the following meaning, let r. and p. denote respectively the position and momentum of the i th particle in the system. Then

f (£i»£2» •••&.» "'3:$'* 5i» J?2» •*•

&• ••• £***> d£i ••• dJ;i ••• d*ff d£i ••• *&

is the joint probability that at tine t, particle 1 ia * * • located at r., in the range dr. and vith its momentum p. in the range dp., , particle 2 is located at r_ in the range dr«

•with its momentum p0 in the range dp, and so on. Furthermore, H is the Hamiltonian of the system and Lf » H-J denotes the commutator of £^ ' with H. By solving the LionvIIOe equation -43-

under appropriate boundary conditions, f^ ^ may be obtained' and in this way one can have a very- detailed knowledge of the microscopic behaviour of the system. While elegant as a formal framework, the N-particle distribution function £^ ' is both complicated and cumbersome. In practice it is convenient (and indeed inevitable) that f^w' be reduced to distribution functions involving only small numbers of particles by averaging over the coordinates and momenta of the remaining large number of particles. The simplest of the distribution functions so derivable is the single particle function f'!' defined as

(^•JJ P-J »" ) =-J>\ J 1 &\^J P ° • o-^wl P-] • • • » • ° • »Pv( t ) X dr0 . o o dr„ dp9 ... dp.T. (2.3)

In contrast'to f^ ', fS ' now depends on the coordinates and momenta of a" single particle and is therefore more tractable.

In plasma physicss one often considers the related function f^ '(rp v;t) (hereafter simply abbreviated as f) rather than f (rB P»t) defined above.

We shall shortly see how the macroscopic "variables common- lyPemployed for describing the plasma can be defined in terms of,j, thereby obtaining microscopic insight into macroscopic quantities. But first a discussion on the equation analogous -44-

to fcne I/iouvi'lie^.equation satisfied by f*. This has the form

f f 44 * T0 V -n + ^ • V „ = "something". (2.4a)

Here p denotes the force acting on the particle, and for a charged body,

F = q(E + v X B), (2.4b) ignoring gravitational effects. The second term on the l.h.s. of (£.4) expresses the influence of diffusion phenomenon, while the third expresses the action of the forces to which the constituent particles are subjected. The "something" on $&e r.h;s. incorporates the effects of particle interactions, asidp depsxi&ing on the circumstances, various choices can be made for it leading to different approximations, if there are several species in the system, then an equation similar to (2.4) can be written down for each of them. The fields entering (2.4) (via 3?) refer not to the external fields but to the total i.e. external plus induced fields, and are governed by Maxwell's equations

$ Parenthetically it may be remarked that studying the dynamical behaviour of the plasma via Iiioville equation supplemented by an appropriate initial condition (i.e. "maximally non-committal with regard to all missing information") is equivalent to solving the Newton's equations for all the particles subject to the initial conditions obtainable in a statistical way. The latter, incident ally 9 is the approach adopted in computer simula­ tion of many-particle systems like liquids. -45-

tf. B = - 47t V-. P

7X E = - A H 7. H = 0 VX H = - J (E + 41TP)

D = € E VA.A, iA^- E + 4TP . (2.5)

The currents and charge densities entering Maxwell's equations themselves depend on f for

? (r, t) = qn (r, t) = q f(r, v; t)d£ , (2.6)

J(r, t^qfvfdv'. v ^ >- = qn(r, t) , (2.7) where

Equations (2.4) and (2.5) are thus coupled, and to find the distribution function, f, the entire system of equations must "be solved self consistently.

2.7. Tlasov Equation The simplest version of (2.4a) is the Ylasov equation -46-

often referred to as the collisionless Bo It zmann Equation ( eBE) for the reason that it is valid for a system in Which the particles do not suffer collisions. It might appear somewhat strange to regard a system of charged particles as being colli'efonless (i.e. as having no interactions) when in fact they do interact and that too via Coulomb forces which are well known to be long ranged. Actually some recognition to electrostatic forces has been given in the above equationf and It lies buried in the term P. The mathematical justification of this statement is beyond our present scope but it merits mention that when dealing with an assembly of charged particles, the interactions are often sort of split into long-range and' short-range parts, The description of the long-range part is given in terms of fields and is effectively incorporated into P in the present case. Parenthetically it may.be remarked that"the long-range part of the interaction between pfejrged particles basically . r leads to screening, and can^afc»n required, be doacribfd via a dielectric constant. There is still the seaidual so-called short-range part of the Couleab interaction', and in the VlaSQv equation this is ignored.

2.8. Concerning Screening

The question may be asked as to how the short-range part of the interaction may be desoribed just as the long-range part may be characterised in term* of macroscopic fields and the screening produoed by them. Like the concept of screening, -47-

this problem too has received attention' a long time ago, the pioneer in this field being Dehye who became interested in this problem in the course of his studies on electrolytes. The basic idea is simple.

Suppose we have a plasma and introduce into it a "test" particle of charge Qe at r = 0 say.. This test charge "will attract particles of the opposite charge towards itself, and repel like charges. In this way the electrostatic potential presented by the test charge will be screened or shielded. Further analysis' shows' that if a probe of unit positive charge is introduced into the medium to measure the potential presented by the test charge, the prlpbe will see the screened potential

4>tr) . *? **(-X) •r r \ *$< (2.10) instead, of the usual (1/r) potential. The quantity "Vp is variously referred to as the Debye radiua, shielding/screening radius £tc., and is given by

,\k_ T i V(,Hb> • * (2-11) "When quantum effects are important, there is an analogous screening radius

\ « -£E--= [WETP. (2.12) A quantum kTp j .—j V -48-

where lc_ is the so-called Thomas-Fermi wavevector. Figure 2.6 shows a comparison of the shielded and unshielded FOtC23tio.ls. The essential feature is that if screening is reckoned., the Coulomb interaction "between particles at "large" distances from each other can be ignored. It is customary to regard (in a hand-waving sort of way) ^\p as a cut-off distance beyond 'which Coulomb interactions may be ignor-ed. We shall return to this point later but for the nonce, \*e remark that for shielding to be effective; sufficient number of particles must be contained within a sphere of radius %ji or the Debye sphere as It is sometimes called. Squivalently, one demands that the average jjater-particle distance "be considerably less than the Debye length,." i.e.

1 ( 1 y5 « *J> m (2.13)

To summarize then, when dealing with many particle systems? it is convenient to split the Coulomb interaction slpheoiatically into long- and short-range parts. The former is described through appropriate fields, and provides a means of discussing'the screening action of the particles (through the equation D = £E for example). Once the screening action is so disposed off, the residual short- range interaction need be considered only if the distance r between the two concerned particles is less than ^-p . Also the law of interaction is taken as in (2.10). Figure 2.7 offers a pictorial paraphrase of the above statement . -49-

BARE COULOMB POT

4> SHIELDED POT

•T

Pig.2.6. Schematic drawing of the hare and the screened coulomb potentials.

COULOMB INT.

LONG-RANGE SHORT- RANGE PART * PART In terms of mac In terms of electric fields screened potential

Note Both descriptions listed above must be included' for complete accounting of electrostatic effects. 7ig.2.7. Compartmentalized description of coulomh interaction in terras of its short-range and long-range parts. -50-

2.9. Effects of Collision - Modifioatibns to the Vlaaov Equation

We now return to the Vlasov equation, and consider the improvements that can be effected by way of including the effects of collisions. Whereas the Vlasov equation shows how f changes due to various "slowly varying" long- range interactions, what we now require is how f would be affected by "rapidly varying" short-range interactions (i.e. collisions). Essentially this involves making a suitable specification for the "something" that appears on the r.h.s. of Eq.(2.4a).

j In the kinetic theory of gases there is,'* iwell established and time-honoured procedure for specifying this "something". One assumes that particles behave like billiard balls and that two-body collisions alone are important. In other words, one supposes that interactions can be viewed as binary hard-sphere collisions, and that between two collisions, the particles are not. subjected to any forces, pursuing as a result, straight trajectories.

Eased on such an assumption, it can be sh^m that

"something" = (f- fr -f£.) Iv-v. f c~(v- -vi') dil' dv,, ZroJ

(2

where

f - f(r, v, t) f1 - *<*fli» t)

f f'= f(r1t v\t) 1 - *<£' X-I> t) and dH. la a solid angle element about the direction JX. rr(v -v\,) <*#-' *s *&* creflBsection for a particle to acatter fr|ha a velocity v^ to a velocity jr.. with a spread dia' in the final orientation. With (2.14) in (2.4), one obtains the famous Boltemaim equation, the r.h.o. now denoting a forcing function which describes hew binary collisions cause fluctuation*. The reader familiar with reactor theory -would nb doubt have recognised th* similarity to the neutron traneport equation.

Vhen multiple collisions are important, other representations for "something" becomes neoessary. One simple trick is to write

"something* » (-&% ) collision and adopt a so-called relaxation model to describe ( 7>£/~dt) collision. The essence of such a model is that if the local equilibrium situation is desoribed by a function f (say), and further that if the distribution function f is disturbed from equilibritm (so that f 4 fQ), then collisions restore f to f exponentially with a relaxation time X\> -5Z-

( vstime Detween*colliaions), leading t©

^collision raT^' *«>

A teere sophisticated approach to multiple collisions is via the Fokker-Jlanck equation

^* colliaion «. *^

Here ol, ^3 denote Cartesian indices. the eaeaa square fluctuation in velocity. Ferisally, tltee© averages ara defined «a below:

a ^2„9li>5 wfcesre ^v. Av) denotes the pr©lability that a particle chcuftgee its velocity from v to v * £v in a time- &refcaBTra2. At du« .-fee multiple celliaiena. It i& aeaisasd that P does nt»t explicitly depend en time &nd that it oantrol@ .&!» -53-

evolution of f via the equation

f(r, vp t) = [f(r, v-Av, t-.^t) X P(T-iT, AT) d (Ay).- (2.18)

In fact, the Fokker-Planck equation is deduced from (2.18) "by a Taylor expansion.

From a physical poijjt of view, the Fokker-Planck equation has two terms of opposite signs which could result in a net change in f due to collision. The first which involves /(^irS is referred to as the dynamical friction term since m

To apply the Fokker-Planck equation to' an actual situations ^ir') and ^AT^AZ^^ must be evaluated, bearing in mind that collisions are Coulobmic. . Now the problem of Coulomb scattering (or Rutherford scattering as it is often termed), is well known. The differential scattering crosssection is given by.

%-Zie^* •sp-9v ^Uv+c^/2; <2-i9) -54-

where z1e and z2e are the cha/ges on the two colliding particles, y^ is their red'iced mass, g the magnitude of their relative velocity, and ?G is the scattering angle. As the reader may "be aware, the angle of scattering depends on the impact parameter p, the dependence being'proportional to cot ("P/2). Note that at % = 0, the differential crosssection becomes infinity which tells us that the probability of a very small angle of deflection is extremely Large. This singularity presents a difficulty which, however, can be circumvented when we recall our earlier remarks about screening, Debye sphere etc. According to those arguments, once screening is built in (i.e. once electric fields produced by the particles are incorporated into the formalism), explicit particle interactions need be considered only within the Debye sphere. Stated differently, collisions with large impact parameters need not be considered, and it io precisely for such a situation that the difficulty associated with the singularity arises. Thus any treatment that seeks to go beyond the "Vlasov equation need only consider those collisions far -which the impact parameter is less than J\j>". The long- range effects are built in separately thrqugh.the electric fields,.,and in this way .all the consequences of. electrostatic interaction are allowed for (see also Pig.2.7).

2-10- BBGKY Chain

Reference may next be made to the Born', Bogoiiubov, Green, Kirkwood aJid Yvon (BBGKY) hierarchy of equations. These spring from the fact that if one starts systematically from the Liouville equation and then de&oces the equation satisfied by -55-

Pifi»2.8. Schematic drawing of the orbit of an ion scattered by another via coulomb forces, p is the impact parameter and % uhe angle of scattering.

.dig.2.9- .iiispm-aiotf relations lor a plasma in the absence ol a magnetic field. The degeneracies are shown in {ar.inthesis. The dotted line is the corresponding curve for (pure) electromagnetic radiation. Observe ttiat the doubly degenerate bracii Vjecouies photon-like for large q. -56

f y one fxnas tnat the undefined quantity "something" on the r.Ls. of Eq..(2.4a) involves the two-particle function f* '

(r,9 v„? r09 v0? t). Thus f cannot be solved for unless fA ' is known. If an evolution equation is set up for fv , it turns out that it involves the three-particle function ("5)fw/ ;

the equation for £^' involves f^J g^ so. on<> j^e oiim^ down from the Liouville equation has thus produced equations which though simpler than the starting oneD are infinite in number and are all mutually coupled? This hierarchy is called the BBGKY chain, and the problem is to cut it of£ at seme stage and obtain a closed 3.oop. Such exercises are required o».ly If gr-eat rigour and sophistication is called for. We shall not consider these here but instead turn t© a macroscopic description of the plasma wherein also a similar chain problem is encountered,

2.11. Macroscopic Approach

The distribution function approach described abeve

givas a fairly useful and comprehensive9 microscopic description of the plasma. However, often the distribution" function cannot be solved for,, in which case the above mathematical apparatus loses its utility. One must then be satisfied with*a oearser description ©f the system. Such a methodology' is already (available to us from fluid mechanics wherein it may be recalled that one 'uses macroscopic variables and sets up equations governing these variables, obtaining thereby a basis for discussing hydrodynamics. Actually, the hydro dynamical equations were set up as early as last centtiry usiug quite independent (but phenomenological) .considerations „ Subsequent work has shown how these equations -57-

could be deduced as appruxiuidtioiis fj.'s*m on.u BoliiiJUiajCi:a equation. A similar trend has occured in plasma physics also, and we shall now "briefly consider the macrascopic equations governing a plasma, or the magnetohydrodynamic equations as they are ©ften'called.

Recall first that the state of a fluid in hydrodynamics is described by certain macroscopic quantities such as density9 pressure etc With the hind sight of statistical mechanics, these quantities may be expressed in terms &£ f as fellows; density- n(r, t) = |f(r, v, t)dv mean fluid -velocity- u(r» "O = = ?r W^ (r» vs> t)dT kinetic pressure p (r, t) tensor- °^^ v- s= m ' f(v~u) (v--u) f dv J ,^«. WW* Ol >M |A /5k \A<« tensor of thermal energy flux or Q , ir, t) heat flow- <*pr ^ = m f (v-u) (v-u) (v-u) f dv, (2.20)

are where, as before, c< 5 f2 ? V' ' Cartesian indices. Other higher order tensors may be defined but they are not particularly useful. The four macroscopic parameters defined above are the only on^s encountered in classical hydrodynamics„ Of these, the meaning of the first two is evident. The tensors p and Q however need an explanatory comment. The former has the dimension -58- ot: energy density, and is a measure ©f thermal motions in the fluid. If all the particles had the same velocity, then u_ will "be equal to v and p will vanish.. It is shown in "books on fluid nEchanics that the divergence of p "behaves as a force per unit volume, a fact which is responsible for p being called a pressure tensor. If p has the structure

P =

1 then the kinetic pressure is said to "be a scalar or isotropic. The tensdrial character is reflective ©f medium anisotropics. The tensor 0 has the dimensions of energy flux whence the name heat flow tensor. The pressure tensor is symmetric i.e. hun - ykcL » similarly the heat flow tensor is symmetric in the interchange of any two of its three indices. It is interesting to observe that the local temperature T(r, t) is related to p via

5 1 \-> U' r 2 - k T(r, t) = - l_y PDt0i( » *)• ( -21)

Setting aside the special complications arising in a plasma, let us explore a little further the fluid mechanical aspects. Now there are three well-known equations involving the above parameters which form the cornerstones of traditional hydrodynamics. There are respectively the equation of continuity (which essentially expresses the conservation of mass); the equation of momentum transport -59-

(arising from momentum conservation) and the heat transport' equation (expressing energy conservation). The momentum transport equation is the fountainhead of the well-known Uavier-Stokes equation, and likewise, the energy transport equation is the source of Fourier's equation for heat conduction. It turn out that these heuristic equations can ;'3Sf fact be deduced from the Boltzman equation by taking moments in velbcity space (i.e. by multiplying by powers of v and integrating over the volume).

Consider first the macroscopic equation obtained from the collisionless Boltzman equation by considering the zero order moment. We recall that the CBE is given by

•M7) « + v>v.V_~ v^r f + '£m. w.Vvf = 0.

Multiply throughout by (v)r with r = 0 and integrate. Using the standard trickB of vector integration it can be shown

* v. V f dv = JV_. n (r, t) u

p. Vf dv = 0. -60-

Ihus the zeroeth moment yields if *&• f-7-° <2-2')

which, is the familiar equation of continuity. Similarly, taking r = 1 leads to

K ' fi= *>!*/* P (2.24)

from which, "by'suitable manipulations, the well-lcnown Navier- Stokes equation of hydrodynamics can "be obtained.

The second moment (r = 2) yields a slightly more complicated equation which we shall not write down. It suffices to record that what is obtained is an equation for -^rp^jj , i.e. the time evolution of the pressure tensor, and that it involves ££C

Attention is drawn to the fact that the macroscopic equations are not closed. The evolution equation for n involves p; the corresponding equation for p involves Q and so on, and" one.is faced with a hierarchy as in-the BBGKY. chain. Once again truncation must be resorted to, and this -61-

is done by by invoking suitable physical assumptions. 3?his strategy ia well known in hydrodynamics, and analogous methods have been adopted in plasma physics also. For example, one could make the simplyfying assumption that the plasma is isotropic and therefore capable of being characterized at each point by a aaes density f> , scalar pressure p and a singla- temps rature. It can further be assumed that the system is adiabatic and that it behaved as a perfect conductor. Under these circumstances, the first three moment equations become closed and are;

-r-x- + ay..u + u. V„n '«— 0

<5> (& + u. V ).u - - Vp + i-(j x3B).

(^r + U.Vr) P • *9 i*7 •») - where ~l£ is the ratio of specific haats. The above equation must of course be supplemented by Maxwell's equations, and the whole set known as hydromagnetlc equations .is. widely used by plasma physicists. A more elaborated version at truncated and close? equations has been obtained by Chrv, CFoldbeorger and ( 2) Low. ' The derails of their model are Xow»ver too complicated for diacu»oJ,oa hare. -62-

2.12. Waves and dbcillations in Plasmas- General n _ : The next topic we wish to consider concerns waves and oscillations in plasmas, a subject of great practical importance especially in the context of plasma stability.

Waves and oscillations in fluids are well known; analogous phenomena are possible in plasmas too, in fact with greater variety since here the fluid particles are charged and can both generate as well as respond to ^fields.

Basically two approaches are possible in the study of the natural oscillations of a physical system. One may either set up the equations for describing the normal modes, or else imagine a hypothetical experiment in which the system is given an infinitesimal disturbance and the resulting oscillations are observed. In the latter approach, therefore, one is essentially looking at the forced oscillations of the system. However, the natural frequencies are discernible as resonances in the forced oscillations. Both techniques are employed in the study of plasma oscillations, although the latter is more favoured on *Jcount of its power, not withstanding the complication that the type of response observed is governed to some extant by the disturbing force applied.

The de|»cription o± the plasma itself may be done in a graded manner. One may start with a cold plasma, and analyze the normal modes by considering the density fluctuations self consistently in conjunction with Maxwell's equations. At higher temperatures, the thermal motions of the particles -63-

baoome important, and the material description oust be vi®, the macroscopic equations (if one is interested la laxge wavelength oscillations) or via the Boltaaann equation (if the focus is on small wavelength phenomena). In either case, the matter equations must of course be supplemented

by the field (i0e. Maxwell*s) equationa.

2.13- formal Modes of a Cold Plasma

Let us now direct specific attention to the normal modeideas of a cold, cone-componen t plasma. 3b«i constitutive equation Is

D - (B + 4WP) - € 8 (2.26)

where D is the dielectric displacement vector, P tfee polarisation and E denotes the electric field. The dielectric tensor € denotes comprehensively the material properties. Equation (2.26) must be next be supplemented with Maxwell's equations:

V- E ax - 4?t V.* <'a)

7. H at 0 (b) \*r- ww V X E rs -n (c)

X? X H K - J (B • *1P) .

Assuming wavelike solutions of the form s«p l(q.r - ««>*), (2;2-7o) and ~ *~ -64-

(2.27d) become

q x E = (tO/c)H (2.28a)

q x H = -(Cj/c)(E + 4TTP). (2.28b)

Taking the cross-product of (2.28a) with q and remembering the rule for triple cross products, we obtain

2 2 • (q.E) q - q^E = - co / -2 <£ ( U) ) E , or in matrix notation, [£€(*)- [4,- |]V]E -0 ,:^| ?2-29) where E is a three component column vector, ££ is the unit matrix of order 3 and

where ^6£ denotes the ct th Cartesian component of the unit vector °V z ^ / \°^ \ . For oscillatory solutions of (2.29) with non-vanishing fields, one must have COi £M - UrSKIH ' ** (2.30) This is the well-known Presnel's equation of optics, and applies to electromagnetic wave propagation in any medium characterized by a dielectric tensor ^(do). In particular, -65- thie is also applicable to plasmas with an appropriate interpretation of € ( Q).

Assume now that there is no external magnetic field present. We next introduce a characteristic parameter ^p defined by

2 2 oy m ^ne* P -m where n is the electronic density and m the mass. The quantity COp is the plasma frequency to which a brief reference has already been made. For the system under consideration,

Vhen introduced Into'-{-2.30), w* obtain three colutions via. r

fc- C<*>/ = JL—--_ doubly degenerate • .£*) to : (5.52) Figure 2.9* shows a sketch of the to •/• q curvas resulting fro* (2.32). Such plots are know as dispersion curves or dispersion relations. Associated with each (G), q) value, there are waves with phase factor ezp i (q.r - Ut). The waves involve both field and charge density oscillations in some fashion. The solution of (2.32a) corresponds to longitudinal waves i.e. waves for

* See page 55 -66- which E is parallel to q. Such waves have largely the character of charge density oscillations*, which, as noted earlier, arise on account of electrostatic forces, particularly their long-range part. One may intuitively expect that this type of cooperative phenomena may.not/be possible when the wavelength becomes small, and indeed detailed analysis shows such modes cease to exist when the wavevector becomes larger than the inverse Debye length. The broad features described above hold in a quantum picture also, the only difference^being that the energy associated with each normal mode is described in terms of quanta referred to as plasmons. The solutions of (2.32b) on the other hand lead to transverse waves i.e. with E \ q. While in free space waves with electric field normal to the direction of propagation would be characterized:.. as electromagnetic waves, in the plasma they are of a hybrid variety, having both plasmon like- and photon-like character. The former dominates for q -> 0, while ;bh.e latter becomes significant for q> (t^/c).

The presence of an external magnetic field can make considerable change to the normal-mode spectrum described above. Assume for simplicity the magnetic field H is uniform, and choose its direction as the Z - axis. For this case:

•KX. 0-H(j € COD) - - °\ & 3b O 0 o (2.33) -67-

Sometimes, ths coordinate frame (x, y, z) is replaced "by another viz, (1, r, z) where 1 and r denote respectively

In the (1, r, z) frame, 6

€ to>) -

e)

io^Oj -C0t) (2.55)

with 6J _ * (eH'/mc) denoting the cyclotron frequency. The collective mode spectrum is again given by Prosnel's equation (2.30) with € £u>\ as noted above. The solutions naturally will depend on the direction of. q with respadr to H . and are illustrated in Pigs.2.10 and 2.11 for q _l_ H„ and q | H . In the former case there are three solutions, one given by -68-

t 3

HI

c* /! ^ / 3

Dispersion relations similar to those in previous figure ^but for waves propagating parallel to the magnetic field. R and L denote right- and left- circular polarization. The lowermost R branch is the helicon branch. Despite its electromagnetic character, the phase velocity is orders of magnitude smaller than that of light -69-

do*- (2.36)

and two more by

As in (2.52) therefore, there are three solutions, two of them modified. For q |\ H , one obtains

<££ = <£.cu) co* (a)

1£. = €^-Cu>) '00 a;

0 * G^LoD) . (c) (2.38)

These equations lead in all to eight solutions, four with positive frequencies and four with negative frequencies. The two sets are identical and only the positive frequencies are plotted in Fig.2.11. The markings L and R refer to left- and right- circularly polarized waves which are derived from (2.38a) and (2.38b) respectively. Observe in passing that for the R branches, the polarization of the wavsi. is in the same sense as the cyclotron motion of the electrons.

The lower of the two R branches merits particular -70-

attention. This is a photon-like branch (for small q), but the phase velocity can be quite different from c. Shese waves have been christened by solid state physicists as helicons. They were however known much earlier to ionosphere physicists but as whistlers, the name given to a whistling type of sound heard frequently in radio communication. The origin of this is believed to "be due to lightning flashes. The latter produce electromagnetic waves of a wide rarag© of frequencies but these travel through the ionosphere along the •agnatic field lines of the easth with different phase velocities so that the pulse gets distorted leading eventually to a rising or falling note like a whistle.

The behaviour of helicons in solids is particuarly spectacular. For instance, if one propagates an electro­ magnetic wave of frequency 10 KHz into sodium held .at 4°K and in a field of '1CT gauss, then one finds that **». »©de propagating as a helicon has a phase velocity of ^> 10 cm/ seci<5>

There are additional features associated with plasma waves which appear in two component plasmas„ The Electrons and holes in bismuth offer an example of such" a plasma system. In this case, one can have long wavelength waves known as Alfven waves, after their discoverer who found them nearly two decades ago while studying suns pots. The mathematical analysis of waves in two component plasmas proceeds as before, with -71

e ^r " ajtu3dLWce^ t+>Cb>±Uici) (2o59) where ^u^ and ^b-^, denote the plasma frequencies of the electron and hole systems respectively, and 6Jc&r and k^^, the corresponding cyclotron frequencies.

The dispersion curves for q \j H are sketched in Fig.2.12. The waves for q->0 associated with the lowest branches are referred to as Alfven waves. In the usually advanced physical picture of these waves, one regards the transversely oscillating magnetic field line as akin to a vibrating piano string. Recalling that in the guided-centre approximation the charged particles gyrate clinging to the field lines, we can visualize the present situation as if there are "beads" attached to the string. There must of course be two species of beads corresponding to the two components. These produce an effective loading of the piano string and modify the wave propagation from that corresponding to the unloade^case (i.e. propagation of transverse electro­ magnetic waves in free space). Thus one has transverse field vibrations accompanied by transverse mass motions, and these hydromagnetic waves are the ones referred to as Alfven waves. As the wave frequency increases, the wave splits into two circularly polarized waves (cf. Pig.2.12) since a distinction in the sense of gyration now becomes discernible, -72-

WAVE VECTOR—> Dispersion relations for q. [\ HQ for a two-component pla3ma. The long-wavelength 1'owrfiequeticy modes are referred to as AlfVen waves. -73-

2.H« Effect of temperature (Including Damping)

While the cold-plasma approximation is adequate for solid state plasmas, it is hardly satisfactory for thermo­ nuclear plasmas. At elevated temperatures, the random thermal motion of the electrons makes possible energy exchange in various manner between the plasma and the electromagnetic field, a process which can lead both to wave damping as well as wave amplification, depending upon the circumstances. The latter process is important in the context of plasma instabilities. As previously remarked, wave propagation can be studied for both macroscopic as well as microscopic situation, using respectively the macroscopic equations (e.g. those of Chew-Goldberger-Low) and the Bcltzman equation, in conjunction with Maxwell's equation. An option is'also available regarding the technique of analysis, but it is customary to rely on the response function approach to do a dispersion analysis. In this, one supposes that the system is initially in equilibrium, and that it is given a small •displacement. The first order changes in -tiie various quantities are then calculated using appropriate equations, which may be suitably manipulated to focus on the response of any desired quantity. This could for example be the net field (external plus induced) resulting due to a perturbation, which can be expressed in terms of the equilibrium properties and the parameters describing the perturbation. The analysis is further divided into a study of the longitudinal and a study of the transverse component. Por the former; one " obtains typically, -74- where the scalar function a(q, w) is determined "by the initial conditions and b depends on the intrinsic properties of the plasma. Of interest are the singularities in"the response, and these evidently will arise due to poles of the numerator as well as zeroes of the denominator. The analysis of the latter leads to the natural frequencies of the system and their life -times. Analogous procedures are possible for the transverse component q x E.-

The response-function method of wave propagation analysis has the virtue of not only yielding the normal mode fire^Rsncies,, but also their possible damping.

Mechanisms for damping of plasma waves are many, important among these being landau damping and 'cyclotron damping. Consider a plasma oscillation wave.- The particles participating in the wave motion will of course have their own velocity distributions but those having a velocity component along q equal to the phase velocity of the wave, will be on "speaking terms" with the wave. The possibility exists of the particles either gaining energy from'the wave or losing energy to it. The former, referred to as landau damping, occurs, when the particle velocity is slightly smaller than the phase velocity. Figure 2.15 seeks to illustrate schematically the process. Shown here ;is the potential distribution set up by the plasma wave. ;The two naturally travel together. As for the particles in the medium, their response is governed by the potential wave, and the velocity component they have in the direction of wave propagation. A particle moving in the wave direction with a velocity slightly smaller than the wave will tend to -75-

==d

Figure to illustrate 'the deceleration of a charged particle with x'espectit'o to a wave associated with pl:>.Kt)ia oscillations. i"n* wavy lines above denote the pocential distribution associated with charge fluctuations. She waves travel to the right with a velocity equal to the length of the solid arrow. A typical particle initially at A movea' with velocity given by open arrow. At a later instant it will be at 0 with respect to the potential wave. In this way it will "oscillate" in the potential well as illustrated at the bottom and get trapped there. Both faster ana slower particles tend to eventually get trapped at the well minimum. In the former cuse- illU3!;rated here - the wave gains energy. In the latter case the wave is damped (Landau, damping). -76- lag behind, and thus "roll" into the "valley". In so doing it day overshoot, and oscillate "back and forth but it will eventually settle down at the "bottom thereafter travelling in phase with the potential wave. The net effect is 1hat the average velocity of the particle increases, the energy gain occuring at the expense of the plasma wave, leading to a damping of the latter; The converse effect where particles with velocities slightly greater than the phase velocity get trapped in the potential well is also known, and in this case thea?e is wave amplification.

Cyclotron damping is another powerful dissipative process which, like landau damping, results in the transference of wave energy into that of single particle motions, viz. that of cyclotron motions.

There are of course many other types of damping but the two mentioned are the dominant ones.

2,15. Effect of Surface

The foregoing considerations of wave propagation apply to an infinite plasma, a fact which though not mentioned, is implicit. On'the other hand, terrestrial plasmas are always bounded which' means that from a practical point of" view, the above normal-mode analysis must be repeated taking account of the presence of surfaces. Recalling that fluids can exhibit specific oscillatory behaviour associated with their surfaces, ±t may be anticipated that analogous additional oscillations ("both of the damping and exponentially growing variety), will occur in plasmas. These additional features must be taken in -77-

conjunction with those sketched earlier when considering the problem of plasma stability.

2.16. Plasma Instability

This iB a natural point for discussing plasma instability, a crucial problem in controlled thermonuclear research.

We have earlier noted that magnetic confinement offers a promising way of achieving controlled -thermonuclear reactions. Unfortunately, such a confined plasma can never be in equilibrium (where by equilibrium we mean a state in which all macroscopic quantities characterizing the plasma are constant in time). This can be immediately seen in an intuitive way since a confined TN plasma would be at a very much higher temperature than its surroundings (i.e. Utie lab). The natural tendenoy for the plasma therefore would be to cool down to the surrounding temperature, a process accomplished by expansion which in effect means loss of confinement. Instabilities aid the transition from the unstable to the stablft «qa±librium state, arid hence the interest in them.

To illustrate how stability may be analysed, consider an one-dimensional system with a maSs point moving in a potential as illustrated in Fig.2.14!. It is obvious that point C is not a position of equilibrium while A, B, and D are. However,. among the latter some distinctions can be made. While B is a stable equilibrium- A ia unstable; D comes in the intermediate category of metastable. The example discussed -78-

B

?ig,2. •.,!. 'JO-jemafcio drawing illustrating the concept of stability. Shown here is the potential distribution _f J. one-ai^ncional systera. Position B is ;>w>.:•.•': Lv aa equilibrium. A is one Xvo bui u"it.,Jj.-.r;iii».. '!) is a rac-tats table equilibria-;:. At C the »•...•'.<..?. ..-•:> ,-iov. in equilibrium.

+

, . (a) (b) Pig.2.15. Schematic drawings to illustrate the influence of sagnetic field lines on plasma stability. The shaded regions show the plasma. In (a), the plasma if it expands, enters a region of lower field and -cnerefore encounters less m-^jnotic pressure. The configuration with field lines rionvex is unstable. In (b) the situation is opposite and the conilguration is stable. -79- can be generalized to three dimensions, and one may from a study of the configuration energy, conclude whether a given configuration is stable or not. A more detailed appreciation of the stability of a particular configuration requires an examination of the dynamical response to a small perturbation, and it is only such an inspection that permits a distinction, between positions A, B and D in Pig.2.14, for example.

Both static and dynamic method of stability analysis are employed in the study of plasmas. One useful guideline to emerge from such a static analysis is illustrated in Pig.2.15 where a section of the plasma is sketched, as also the field lines in the neighbourhood. When the field is concavely oriented, the plasma is unstable since magnetic flux and therefore magnetic pressure decreases outwards. For the field convex geometry, on the other hand, the plasma will encounter higher pressure if it tries to expand into the field region, and the configuration is therefore a stable one.*

It should not be surprising that instabilities depend to a large extent on the natural frequencies of the system. They may be of• various types, ranging from gross or shape instabilities to microinstabilities. The former are often called magnetohydrodynamic instabilities while the latter include convective and turbulence-like phenomena. The great optimism ' that prevailed amongst fusion researchers during the early days was subsequently tempered largely by the discovery of instabilities, and Pig.2.16 illustrates the

* This criterion was originally proposed by Teller. -80-

OS g 10 c o w u Qi a. o 5 2

o

1950 1960 1970

Fxg,2.-i6. The increased preoccupation with plasma instabilities her fifties and early sixties is illustrated -81 «•

growing preoccupation with this phenomenon over the years. However, thanks to the intensive work of the last two decades, it may be aaid that "by and large enough understanding of instabilities has been acquired as to be able to weaken or altogether suppress them. The comer has "been turned it appears.

2.17. Energy Losses from Plasma Upto this point, we have got considered the possibility of energy escape from the plasma, losses can occur via a variety of processes, prominent among them being diffusion, radiation,, charge exchange and thermal conduction. Apart from diffusion, the other processes by and large do not involve losd of matter from plasma.

Diffusion is a phenomenon that is characteristic of a system in which the members frequently collide. As a result, the particle tra;j«c-fcories are not well defined but quite ragged and random. The driving force is concentration gradient, and in the case of a confined plasma (our main interest here), there will invariably be a density gradient from the interior to the outside, offering scope for diffusion. The particle current J due to diffusion is given by

J =• nVv\ s - BV n . (2.4<>9 w^ ^XA^ r »». In general, (TS the mean veloclliy would have contributions UVJA^J. UJICUI uuc uu kuxxuaxDu giauxcuv; xux IUX-LI? axuuauxuu \j±x-^-j the diffusive part^'of J is related to KJ n as above.

•\ -82-

Assuming for simplicity that velocity is fixed, the diffusion coefficient D is given by

D = (1/3)(1V)

•where I is the mean free path. In a plasma, the ions and electrons will have different velocities, with usually v "> v.. However, the mean free paths are usually comparable so that D > D., . No cognizance has been taken here of possible space charge effects. This is permissible for very low densities but at high densities when space charge effects becomes important, additional complications occur. Basically, if a perturbation of one species of charge develops, the electric field resulting from the charge imbalance i.e. the space-charge field, can become strong enough to drag the oppositely charged species along in an attempt to preserve charge neutrality. With respect to diffusion, this implies an inhibition.

In the presence of an external magnetic field, the diffusion phenomenon becomes further modified. Consider, for example, a uniform magnetic field to be pervading the plasma. If there were no collisions among the particles, then provided there are no electric fields, the guiding centre would remain fixed on a given field line, and diffusion across the field line cannot take place. Collisions can lead to such a transverse diffusion but a detailed mathematical analysis on the basis of classical models shows that diffusion is inhibited in a direction perpendicular to the field lines but is unaltered parallel to the field direction. In an obvious notation, D^ <" D|( f= D„ \). Space charge effects are -83-

expected to reduce Dx er»n further. Experimental work on diffusion of plasmas across field lines has revealed' that the rate is often faster than that predicted by above considerations. In other words, the inhibition is not as much as anticipated; This anomalous behaviour is attributed to plasma turbulence and was first analyzed by Bohm. Though once a great source of worry, it is generally believed at present that instabilities and losses due to Bohm diffusion can be curbed.

In contrast to diffusion;, radiation loss does not Involve loss of matter. Energy depletion occurs via emission of electromagnetic radiation, a process which is to be expected since the' plasma consists of charged, particles moving in electric and magnetic fields. The first and the most obvious source of radiation is of the black body type. Recalling that the intensity of such radiation goes as T , one would expect black body radiation to be a serious source of loss. Fortunately, a black-body by definition has to be opaque, and terrestrial plasmas are too thin for this. This type of energy loss may therefore be ignored. (See in this connection the footnote in chapter 1 concerning the size of self-sustaining TN systems).

A more important source of radiation loss is bremstrahlen (literally breaking radiation). Bremstrahlen occurs whenever a charged particle is decelerated by a collision process. The energy loss suffered is emitted as electromagnetic radiation, and can occur in the following types of collisions; -84-

el - el, ion - ion, el - ion, ion - neutral atom. el - neutral atom,

Considering electrons, the power P^, radiated when the deceleration is Drought about by a particle of charge Ze is proportional to (Ze) X //T. This implies that the presence of even trace impurities of moderate and high atomic numbers in a TN plasma can be disastrous. For example, it is estimated that a 1 atomic percent of oxygen impurity will increase the rate of energy loss via brerastrahlen by 77#! This loss will determine the ignition temperature which naturally will increase due to enhanced emission.

There is another way in which a plasma may radiate electromagnetic energy. Earlier we noted that in a magnetic field, charged particles can gyrate. The oentripetal acceleration of charged particles is accompanied by emission of electromagnetic radiation, referred to as cyclotron (or synchrotron) radiation. The intensity of such radiation. associated with electron gyrations will be significant for a TN plasma. Due to this radiation loss, the electron will slow down in its gyration, leading to a cooling. In contrast to bremstrahlen emission which is mostly in the UY and x-ray region, cyclotron radiation is mainly in the microwave and IR - region. As a result, cyclotron radiation has a greater chance of being self trapped than bremstrahlen, and is therefore not a source of concern except at high o temperatures when the T~ behaviour takes over making it more -85-?

important than bremstrahlen (which goes as i^

Charge exchange is yet another mechanism for energy loss from a plasma. The phenomenon may be understood by caisidering, for example, a "cold" neutral atom (hydrogen say) which strays into the plasma from the container wall. This neutral atom upon collision with an ion could conceivably transfer an electron, leading to a reaction of the following type:

, vneutral v+ *y „+ ..neutral (HWa + \ot ^ Hcold + Dhot

The plasma particle'having been rendered neutral is no longer subject to the effects of the confining field and nay therefore escape, carrying away incidentally, energy. In addition, the particle upon reaching the container wall, could liberate more'neutral atoms by collision, offering fresh scope for charge exchange process.

Apart from'the processes described above, energy can also be lost by simple thermal conduction across a temperature gradient. Such a gradient will necessarily exist in laboratory'thermonuclear plasmas. In the absence of a magnetic field, the charged particles will behave like neutral atoms to a first approximation, as far as thermal transport is concerned. With magnetic field present, anisotropy can result as in the -case of diffusion. While thermal transport along field lines remains unaffected, conductivity across field lines becomes diminished. Prom a 86-

practical point of view this implies that conduction losses v/ill not be a serious problem if the plasma is confined in a toroidal container (see next chapter) since in 3uch a system, the field lines will be generally parallel to the annulus in which direction the temperature gradient does not exist. The same would not be true of cylindrical containers where serious conduction losses could occur through the ends.

2.18. Chapter Summary

This chapter has been rather long but unavoidably so since we have attempted a survey of the whole of plasma physics. Starting with the definition of n Plasma, its characterization, the identification of + lous types, and their occurence, we moved on to th tods available for description of plasma behaViour. At the one extreme of low plasma density, an overall description is possible in terms of particle trajectories, in particular under various configurations of externally imposed electric, and magnetic fields. Trajectory description naturally becomes difficult when one recognizea the possibility of random thermal motions etc. but even here, the situation can be salvage-?. tftth approximations like the guiding-centre approximation. When the plasma density becomes significant, orbit approach ceases to be meaningful and a statistical approach becomes necessary. While a variety of descriptions for the matter behaviour are possible within the latter framework, the field description remains the same, and is done via Maxwell^ equations. The various combinations of Maxwell's equations with the matter equations lead to different models for the plasma. The chart -87-

in Pig.2.16 gives a schematic summary. The different approaches cited in the above figure can "be used, among other things for exploring instabilities„ The latter refers to any cooperative plasma motion that can regenerate itself, and can be either of the gross variety or be micro in character. The understanding of instabilities is facilitated by a study of waves and oscillations in plasmas, particularly waves of the growing type, naturally, wave motion study must take into account the effects of temperature, and surfaces. The latter is especially necessary when exploring gross instabilities, lastly, we have touched briefly upon the processes by which plasmas may radiate away energy. Such losses can prove to be a serious obstacle to the attainment and sustenance of high temperature conditions.

With this background, we are ready for a tour of the various proposed thermonuclear machines, for which the reader may turn over, to the next chapter.

REFERENCES

1. J.L.Delcroix, Plasma Physics (Wiley, New York, 1965) Vol.1. 2. G.Chew, M.L.Goldberger and F.E.Low, Proc. Roy. Soc. (London) A 256. 12 (1,956). 3. R.Bowex-s, C.Legendy and P.Rose. Phys. Rev. Letters 7, 339 (1961). -88-

FIELD LIOUVILLE EQN. PLASMA EON. FOR N.PARTICLE DIST. FN. NOT VERY USEFUL.

BBGKY CHAIN CHAIN LINKS f(l) TO HIGHER ORDER DIST. FNS. " CBE ' .EQN. FOR f(l). COLLI­ SIONS IGNORED U. MAXWELL'S NO SHORT-RANGE, INT.: - . EQNS; BOLTZMANN EQN. COLLISIONS INCLUDED. FOR MULTIPLE COLLL • SIONS, MUST USE RELAXATION .OR FOKKER PLANCK MODEL.' ,

MACRO EQNS. \ ANALOGUE OF CLASSI. CAL HYDRODYNAMIC EQNS. HIERARCHY OBL .. TAINED IF DERIVED FROM CBE. TRUNCA­ TION VIA PHYSICAL ASSUMPTIONS.

Fig.2.17. Summary of tne various descriptions ?ivpn «* TO - JS^f by C°^«S theCflSld°andgmISr?J-.PlaSraa equations. -89-

3. TOWARDS MAGNETIC CONFINEMENT

3.1. Introduction

The stage is now set for a consideration of the systems proposed for achieving the controlled release of fusion energy. Briefly, the latter involves

(i) heating a small quantity of fusion fuel to a temperature greater than the ignition temperature producing thereby a thermonuclear plasma, (ii) isolating, the plasma, and (iii) converting the energy liberated in fusion into a useful form i.e. electricity.

Although the technical problems associated with (i) and (iii) are"substantial, they are "solvable" within the framework of existing technology. Item (ii) is the real thorny problem namely, how to confine or hold together the hot plasma long enough to realize a useful output of energy. This chapter in a sense is a catalogue of some of the attempts made to overcome this difficulty.

3.2. General Remarks on Confinement

As already indicated, magnetic confinement has been envisaged right from the beginning as the only likely means of holding together a hot plasma for long periods. A variety of magnetic confinement schemes have been suggested? and in the early days, many of these were actually expected -90-

to lead to workable thermonuclear machines. However, it soon became evident that magnetic confinement was far more difficult than anticipated, and it turned out that many of the experiments originally designed to achieve fusion became converted into investigations in plasma physics. Several ways of achieving magnetic confinement are possible but no one particular approach seemed distinctly superior at that time, and-it was inevitable that many of them were actually tried.

A few general thoughts on magnetic confinement are desirable as prefacing remarks before discussing the various schemes tried. Grossly speaking, magnetic confinement can be visualized by analogy to the confining of a gas through external pressure-only in the present case the pressure is provided by the magnetic field. To sse this bettefr,consider the macroscopic equation (2.24). Assuming a steady state and for simplicity that the pressures are isotropic, we may write

n^e I E + (1/c)(v± x B) - VPi

-n e f E. + (1/.C)(V x B)~| - V P«>

where the subscripts i and e denote respectively the ions and electrons. Since n^ = n = n, we get

d/c)(3 x B) = yp -91-

where en(v. - v ) «= 3 and p = p + p. . If the field does not vc^ry with time, Maxwell*s equations give

Vx B - (1/c)47tj v/hence

(l/4Tt)(V3: B) x B =Vp. (3.1) V^- w*^

This is usually referred to as the condition for hydromagnetic equilibrium. Recalling from vector analysis that

( V x B) x B » -i V BB2 + (B. 7 )B,

and further that (B.V) B vanishes if B is constant and parallel everywhere (- we are assuming the presence of a •uniform field for simplicity), (3. 1) simplifies to

v(p;^} = ao that

„/„N . ] _ P(r) +-^-- 0- (3.2)

2 The quantity B /8 "ST has the dimension energy density or pressure and may "be interpreted as magnetic pressure. If B is the value of the external magnetic field confining the plasma, then applying (3.2) at the boundary and remembering p = 0 outside the plasma, we have -92-

Clearly B(r) <£ B i.e. the plasma tries to exclude the magnetic field and thus exhibits diamagnetiBm.* This may be visualized in terms of a surface current which produces a field counterbalancing the external one. If the system were a perfect diamagnet, the external field will be totally excluded. In practice, there is always some penetration. Note that p refers to-plasma pressure perpendicular to B. •

It is customary to introduce in the context of magnetic pressures, a parameter A defined by

? £ I 8K

For the case of the steady magnetic field under consideration!

p. i- # <,.

A thermonuclear machine is to some extent characterized by its |3 value since it is a measure of the ratio of fusion energy density (proportional to p), and- the magnetic energy density. Too low a plasma density implies too inefficient a use of the confining field xelative to the fusion power generated. -93-

An idea of the confining fields required may be 15 / 3 had from the above result. Talcing n^10 "Vcm and T **> 10 keV, we have on equating the kinetic and magnetic pressures,

nk 'J? which gives for the case considered,

A 5 BQ v^ 1CT" - 10 gauss.

3.3. Pinch Systems

The pinch effect was among the first ideas tried out for achieving controlled thermonuclear reactions, and refers to the "self focussing" or constriction of a stream of charged particles due to the magnetic field produced by its own current. The basic concept actually predates considerably the fusion programme, having been discovered in 1934.

Figure 3.1 illustrates schematically representative stages in the pinch sequence for both the linear and the toroidal pinch. The magnetic lines of force set up by the discharge current act as a hoop, and produce a constriction, pulling the plasma away from the container wall. A simple relation may he derived between the discharge current I (amps) and the field produced. "Using the Biot-Savart law, the azimuthal field B£. (gauss) is given by Fig.3.1-. Schematic illustration of the pinch effect in linear and toroidal systems, (a) depicts the situation at the begining of the discharge. 'She. plasma is diffusej as the discharge current increasess the associated field does likewise, pinching the plasma as shown in (t>). -95-

Be = l/5r where r (cm) is the radius of the discharge. By equating the magnetic and kinetic pressure,

or

I2 06 T.

To x'each a temperature of 10 K say, under typical fusion conditions (-cross section area of discharge v^ 1000 cm2 ; 1 ^ % N ^10 /cnr), a pinch" current of more than a million amp is required.

The simplest way to produce, a pinch discharge is to apply a high voltage between two electrodes at opposite ends of a straight tube containing gas at low pressure. Such a system is referred to as a linear1 pinch, and from the thermonuclear point, has the disadvantage that ions bombarding the electrodes can liberate impurity atoms leading to brehmstrahlen losses etc. An obvious way of solving the electrode problem is to use a toroidal tube to contain the gas. Excitation of the plasma can be done via induction, using the toroid itself as the secondary of a transformer. More will be said about i roidal systems later.

The foregoing discussion assumes implicitly that the discharge is steady and continuous or that the time -96-

variations in the discharge current are smaller than typical processes associated with" the plasma" suo'fe"as.di##asion„ sound transit time etc. Frequently^, pinch currents are set up by the rapid discharge of a condenser, and for this situation^ the dynamical aspects have to 'be carefully considered. An interesting outcome of the dynamic pinch (or shock pinch as it .is often called) is that when it reaches its smallest radius, all particles acquire a radial velocity approximately equal to the collapse velocity. If the contracted column can he kept long enough for thermalizatio% of the ions to occur, then the additional velocity acquired' ' due to pinch could he converted to plasma heating. The hope was that such heating would accompany confinement, and lead to thermonuclear burn.

Unfortunatelys> these hopes proved to be unrealistic, and almost from the' begining, the pi:£ph programme was beset with instability problems. Figure 3.2 shows some well-known examples of gross or macroscopic instabilities. These can be studied by normal-mode analysis, starting with the macroscopic equations and appropriate boundary conditions. If a cylindrical coordinate system is used;, the wave-like disturbances may be described in terms of a displacement amplitude given by

§ = |(r) exp (i(m9 + kz + wt)).

The m-values of the kink-and sausage-instabilities are shown in Pig.3.2. -97-

IA

B I T i y 9~)

r small \ r large Bo largeJ Be smalt

(a) Sausage instability (m=0)

Be large. PLASMA

Be sma IL/.

(b) Kink Instability (m=1)

(c) Flute, instability

5.2, Illustration of some of the gross instabilities that have plagued controlled fusion research. -98-

One method suggested for reducing the ansf;abilities is the introduction, of an axial magnetic field and a conducting vail (see Pig.3-3). The axial field lines provide necessary longitudinal "tension" to inhibit sausage and icink instabilities.* The presence of the conducting •wall causes a flux crowding if a'kink instability occurs, •which then produces a counter force attenuating the kink. The wall thus serves as an additional sentinal against kinks. Alternately, one may take the view that image currents induced in the conducting wall repel the pinch and in effect help straighten out possible unstable undulations. Whereas the axial magnetic field serves to stabilize the short "wavelength (i.e. sharfc) kinks, the conducting wall attenuates the long- wave length kinks.

Ins pit© of such innovations, linear pinch systems have generally failed to fulfil their early promise. They have, however, played a useful role, in highlighting the macroscopic instabilities. Experience gained in this area has proved useful elsewhere subsequently.

The theta-pinch (©-pinch) is yet another system employing the magnetic compression concept but in a different Xopn as compared to the linear pinch. In its simplest version, it consists of a wide single-turn solenoidal coil wrapped around an insulating tube (Fig.3.4). The gas is held inside the tube and ionized in a convenient-manner, following which, a heavy discharge is sent through the

* Recall from Chapter 2 that f-Leld lines behave like rubber ' bands. -99-

Pig.3.3. Stabilization of the linear pinch. The usual magnetic field lines -in a linear pinch are azimuthal. 1'he addition of an axial field (Br/) provides "backbone" and helps stabilize the. plasma against gross instabilities... The conducting wall also does likewise. -100-

Fig.3.4. Illustration of the circular pinch system. (a) shows single turn conducting wall wrapped around the insulating tuhe containing the plasma. (b) shows schematically the discharge circuit. (c) shows the constriction of the pla3ma. -101-

solenoid with the help of a condenser "bank. The rapidly increasing current in the conductor produces an oppositely directed current s'&eat in the plasma. As a result9 a magnetic field! is produced in the annulus between the current sheets which drives the plasma inwards. Besides producing confining action, the high compression causes plasma heating, favouring condictions for a thermonuclear burn. Noteworthy is the fact that as compared to the linear pinch, the position of the fields and current line are interchanged; the two configurations are thus orthogonal (Pig.3.5).

The 9-pinch too has revealed many instabilities, particularly of the flute variety (pig.3.2). At the present time, it may be said that the system has emerged from the woods as a strong candidate aaong those most likely to succeed.

3.4. Mirror MaQhiae

In. the linear-pinch* confinecsent is sought to be achieved by the magnetic field generated by the plasma. An alternate scheme may be envisaged wherein the field is produced externally„ The field geometry in this case belongs to a class of systems known as aagnetic bottles. Figure 3.6 shows a schematic drawing. Surrounding the plasma tube is a coil with windings suitably adjusted to give a strong concentration of field at the ends as represented. Such a system is often referred to as a airror machine. To understand the mirror action, consider a plasma particle moving in the system. Two conserved quantities (adiabatic -102-

9_ PINCH LINEAR PINCH (a)

A H2-

(b) .5-5. Oonrparis^tj of tne field and current directions in the linear and circular pinch. She two configurations r-.xv "orthogonal", i.e. the positions oi' the current and field lines are interchanged. -105-

(a)

m (b)

O •(c) UJ

POSITION

(d)

LgO.O. ; shov.'.-t th'? plasma tube and th« field coil. C £fc» ' ;.'.'vt! bh«s increased wiming daaaitj kt ends. Cb) sViCw-3 "..•".« i'ialci. lines; -the end sections of tlvi- coils or- aliov/n as'.solenoids for emphasis, (c) akcitciias the field intensity along- the. axis.. niiovi> i&o spiralling of vie particis into a t.iglvce.r helix sr(d its reflection at the end. -104-

invariants) identifiable are: (i) the total kinetic energy W = W|| + Wi (where the subscripts refer to components parallel and perpendicular to the field), and the magnetic moment ^- = (W_L./B) (see sec.2.5). If B& and B^ denote the fields at the centre and the end of the tube, then from the above constraints,

a +w w +w w n 2 = ,r 2-

w_La Ba Bb

Therefore

,, a w 1). II "a. < ITa

Since (B^/E^) > 1, (W.a - w *) is positive i.e. W b is smaller than ¥ a implying that there is longitudinal slowing down of the particle as it moves into a region of stronger field. Accompanying this of course is an increase of the transverse kinetic energy, implying a more rapid gyration of the particle as illustrated in Fig.3.6d. Deeper analysis shows that the particle can be totally reflected at the mirror. In particular, if 9 is the angle the initial velocity makes with the machine axis, then only if the condition '

Sin e > V^ -105

is satisfied, is the particle reflected at the mirror. In: other words, there exists, at either mirror, a loas cone with semi-angle 9C given by

% - sin"~1 7~v\. • and particles with velocity vectors in this cone will escape.

Like its cousin the linear pinch, the earlier versions of the mirror machines were also prone to ._ macroscopic instabilities. Prom Teller's criterion we know that if a magnetic well is created1'{- field increases in every direction from a certain point) then the plasma can be contained and that at least macroscopic instabilities can be overcome. Referring to the mirror machine, we observe that there is certainly a well along the mirror axis. However, in the radial direction, the field decreases towards the walls, offering scope for instabilities e.g. the flute instability. ' An elegant solution to this problem is to use the so-called Joffe bars (Pig.3.7a). These are conductors arranged so as to provide'in the X-Y plane, the field pattern shown in Fig.3.7b. Prom the field geometry it should be clear that the plasma finds itself in a near three dimensional well and is therefore less prone to instabilities.

3.5. Toroidal Systems, including the TOKOMAC and the , STELLARATOR

The confinement geometries considered thus far have been linear. Other geometries are possible, and of -106-

(a)

•(b) Fig.3.7. St __..abilizatio- n oi plasma in.mirror machines agcu.ua i: instabilities curough Jpfie hars. (a) showa'the additional conductors and the direction of ourrurti carried by them, (b) shows the field pattern in & sectional plane. Observe that the field increases everywhere away from the axis contributing to radiax stability. -107- theae, toroids have received considerable'attention. Two "basic variations of the toroid system are possible as illustrated in pigs.3.8 and 3.9". ' In one, the magnetic field lines are parallel to the mirror axis of the torus, •while in the other, the field lines encircle the plasma. Of the two, the system in Fig.3.8 has a natural tendency to instabilities since the field geometry is favourable for pushing the plasma to the walls. On the other hand, in the system shown in Pig.3.9, the plasma when it moves towards the wall, drags the field lines along, modifying the field profile as shown at the bottom of Pig.3.9. This is possible because the field is generated by the • plasma itself, and can therefore readjust to plasma / redistribution. By going off-centre, the field is equalized and further drift of the plasma is arrested.

Several experiments based on toroidal bottles were tried in the fifties, prominent among them being the Perhapsatron*, Zeta and Alpha. We describe here briefly the Zeta Experiment since it is representative of these attempts. Figure 3.10 shows a schematic drawing of the Zeta set up. The plasma tube forms the, secondary of a transformer system, and consists of an aluminium torus of 50 cm internal radius and an average major radius of 1.5 m. The gas in toirttf (deuterium) is preionized by r.f. discharge, and subseounfcly subjected to a unidirectional current pulse by rapidly

* Perhapsatron is the name given by Tuck (of Los Alamos) to his experiment in response to a skeptical colleague of his who upon hearing what Tuck was hoping to achieve, called the device "impossiblitron". (A.S.Bishop, Project Sherwood, Addision-Wesley, 1958, p.25). -103r-

MAJOR AXIS SOLENOSD^ MAGNETIC FIELD ALONG AXIS

© © FIELD

.PRESSURE 4 K .LESS PRESSURE ON OUTSIDE

.i?l£.3.8. Toroidal system with field lines along minor/axis. The field i3 produced by a ooil wrapped, around aa illustrated. Also shown are: The position- o'f'-.the- plasma in a cross sectional plane, and the radial behaviours of the magnetic field and magnetic pressure. -109-

AZIMUTHAL METAL WALLED FIELD TORUS

PLASMA./

FIELD t nu

FIELD

Fig.3.9 • Toroidal, system with azitnuthal field. Initially the plasma v/ill he in the centre of the tube, e.sd as with a single-turn coil, the field v/ill be stronger on the inside than the o\itside. i'he excess magnetic pressure on the inner surface then drives the plasma outwards till pressure is equalised. -110-

discharging the condenser in the primary circuit. Peak currents attained were in the range 1(r - 10* amps, and the duration of the pulse to 3-4 ms. A Bteady axial magnetic field of around 300 gauss was applied.

t In spite of the fanfare accompanying them, none of these experiments achieved controlled 3eN fusion, the principal problem plaguing then being plasma instability. We have already seen that one trick for overcoming gross instabilities is to exploit the magnetic well concept. Another is to use "magnetic shear" which is achieved by having a "nested" or "rafted" field configuration as illustrated in Pig. 3.11. Such a complex-.'. interwoven net- - work of field lines generally manage to exert enough pressure to keep the plasma particles away from the wall* The problem is to achieve this stabilizing rafting by suitably combining the effects of the fields produced by the pjLaama on the one hand, and the externally-produced fields on the other.

One highly promising system to evolve out of the original toroidal machine through incorporation of magnetic she ax is the TOKOMAC developed ¥irst in the USSR, and subsequently tried out in many other countries, figure 3.12 shows the tokomak configuration. In overall geometry, it resembles the Zeta but the difference lies in the strength of the externally applied field. The combination of the externally generated field and that produced by the induced current in the plasma provide the required shear. In Chapter 6, we shall describe some power station concepts built around this highly promising candidate, the Tokomac. 111-

GAS DISCHARGE IN TORUS

?iii,.''«10. Schematic drawing of the Seta and its discharge circuit. -112-

rig>3.11. Illustration of the rafted i'ieid coxicept. The field lines in different layers are slightly rotated with respect to each other leading to magnetic shear. (After Bishop, cited in footnote on p. 107) -113-

Earlier we saw that the simple toroidal system ! depicted in Fig.3-8 "had the disadvantage that there existed a magnetic field gradient perpendicular to the direction of the field. The consequences flowing from this situation are illustrated in Fig.5.13, using the principles discussed in the previous chapter. It may he observed that the electrons and the positive charges drift ^n opposite directions. In the toroidal system, such a drift will lead to space charge* effects and hence strong electric fields. The latter in conjunction with the axial magnetic fields, will then drive the particles towards the wail, destroying confinement. One main reason why. all this happens is that the magnetic lines of force close on themselves, producing thereby a concentra­ tion of the flux lines near the inner wall. It was noted by Spitzer that if the magnetic degeneracy (i.e. the closing of the field lines on themselves) is removed, then the possibility exists of avoiding space-charge and related effects. Essentially this requires a mechanism "by wnich the field lines get displaced a little when they complete a circuit. Fields possessing such a property are said to exhibit rotational transform. A conceptually simple (hut complicated in real life!) means of achieving this is to have the tube made in a figure-of-eight configuration (Fig. 3.14). Consider now two sections .of "th^s device as shown in Fig.3.15. The line of fbrce at A*..will, when followed through, appear at A~ at the opposite end, having moved from the inner to the outer wall. The net result is that possible space-charge effects get neutralized during Particle orbiting-. Such a device with rotational transform v/as called Stellarator by Spitzer. like, many other starxers, the Stellarator was conceived with great hopes, and several Fig.3.12. Schematic drawing of the Tokomac system. Externally it looks similar to the Zeta. At bottom left is shown the variation of the azimuthal and radial fields for the zeta configuration. The corresponding variation for the Tokomac is shown on".the right. (After Post, ref.4). -115-

ELECTRON PATH

POSITIVE iON PATH ^ja&r

g*'5.13. Ead.fi; of electrons and ponitive ions in the preasnco of a magnetic fiold (into the peper) and a field gradient at right angles to the direction oi tots field* Observe that the two. species drift in opposite directions. -116-

versions were built with the expectation that they would function as thermonuclear reactors. However, as with the other experiments, instabilities proved to be the main hurdle and much patient study was required to identify and understand them. Over the years these studies have led to several imprpvqments in the basic concept. Indeed even the geometry of the machine ha3 changed. In the present day systems the figure-of-eight arrangement is no longer used, rotational transform being achieved with an ordinary torus through the help of appropriate coils (see Pig.3.16). It may be noticed that at this point, externally, the Stellarator looks rather like a Tokomac. It is worth mentioning in passing that the Tokomac also possesses rotational transform which is why it is superior to the simple torus. The main difference between the Stellarator and the Tokomac is that in the for*err the shear ia produced entirely by a combination of external fields, whereas in the Tokomac, the field due to the induced current in the plasma also £s exploited,

3.6. Other Confinement sfcd TH Machine Concept a

We collect together here a few odds and ends. Some of them are of historical significance only. The ideas mentioned have no doubt contributed to progress in some way but So not represent the mainstream of current thinking.

1st r on

The Astron is a novel concept proposed by Christophilos, wherein relativistic gyrating electron* are -117-

wj^issffl S&ii

r—» A' 4. 4IP **jf_ • —-31j"^JJ^jj-pStf?ay;:jj| Sfeg^£||ipj *^m>

?ig.3.14. Figure of eight configurations of Steliarato'r. (After "Bishop, loo.cit). -116-

Pig%3.15t Croso-sectional view of the Stellarator illustrating rotational transform. A line of foroe passing through A1 will make subsequent transits through Ao, A,, A. and appear after one .round at B . The angls A. OB. equals «;G where 6 is the 1angle of inclination of the encls. -r.3-

AXIAL FIELD WINDING

HELICAL WINDING

STELLARATOR

j'ig.3.16. Sketch of a modern S'oell^rator, (After Post, rei',4). -120- used to generate the confining field. The principle may be understood by reference to Pig.3.17a which depicts schematically a cylindrical tube with an external coil. The magnetic field pattern is not unlike that in a mirror machine but the similarity ceases at this point for unlike in the mirror machine, one injects into the system axially, high energy (to few MeV) electrons. The latter gyrate in the field producing a large circulating layer of current - the E-layer. This circulating current layer then begins to set up its own magnetic field, modifying the original pattern as indicated in Pig.3.17c. At sufficiently high circulating currents, a magnetic well develops which can be utilized to confine a plasma. The plasma is in fact allowed to be formed after the generation of the E-layer by injecting D and 3D atoms/ions. The ions pick up energy by collision with the electrons, and get trapped in the field; under suitable conditions they undergo a thermonuclear burn.

Homopolar Generator

This is the name given to a madtine in which the plasma rotates as a whole. The origin goes back to a suggestion by Snyder in 1955 who noted that a rotating' plasma could perhaps withstand instabilities. A schematic drawing of a rotating plasma system (-called homopolar on account of the similarity to the flywheel of a conventional homopolar generator), is -shown -in Pig.3.18. The system consists of two concentric electrodes held between insulating walls. The whole apparatus is placed in a magnetic field, and a gas discharge is induced by passing a current between the electrodes. When the discharge is first -121-

E-LAYER MIRROR COt. l^>fo<3><3>^^

ELECTRON LECTRON PATH ACCELERATOR

(b)

(c)

••'iffrs."*>. 3 . IV (a) shows "the section o.f the Astron. ^r *^Pi°a! tVa £leld 1^?s ^fove the injection cf the fast electrons, (c) High enarfty electrons injected into Mi = ir.ach.ine form ^ circulating lyyer of current -.:bout the ceiur.-vi axis. This current produces a magnetic field which modiries -ae original field to give a configuration of ii^la lines cloaing on themselves." DT ions introduced in this magnetic well get heated up by collision with the electron- end produce X UE1 OC . -122-

passed, the gas breaks do-wn and plasma is pinched away from the insulating walls. The radial current, flowing across the magnetic field produces a torque on the plasma causing it to rotate as a whole "between the insulating plates. The centrifugal motion of the plasma sets up an azinufchal current and an accompanying axial magnetic field. The latter, combines with the steady axial field already present to produce a net bowing effect, the "tension1* of which helps to trap the plasma, and prevent it from coming into contact with the outer wall.

Cusp Geometry

The great difficulties experienced during the mid-fifties on account of instabilities resulted in many ingeneous ideas and suggestions for overcoming them. The cusped geometry is one such, the name, referring to the field configuration (see Fig.3ol9). Observe that the field lines produce a trough and therefore a tedency to stability.

PCX and the OGRA

DCX is the acronym for the direct current experiment originated at Oak Rige for studying the trap­ ping of ions in a mirror machine. A beam of molecular ions of several hundred keV is injected transversely into a steady state field. An arc is then struck between tvo carbon electrodes causing the dissociation of the molecular ions into atomic ones. The positions of the arc and of the entering molecular deuterium ions are ^125-

DRECTUN OF MAG. FIELD J

PLASMA

JULATIN8 CONDUCING* WALL WALL

i'iC.S.I'Ji oolv^io/bic graving of ths lionoijolir gcrr-r^'or, T'.io bfv; p-ri; cho'vs 'the nut-aw:.?/' Yiev f-ru:'. the. bo it,?.-'. hcl? fie :?::eV! l\ras. -124-

PLASMA

Fi£«3»l9. Cuap geometry. The top r.hows the perspective view vjid the bottom the sectional view. -125- choaen oo that the circular orbit of the dissociated atomic ion bean is concentric with the magnetic axis. It was originally hoped that fusion reaction would be achieved in the machine but this has' not come;;ab6utl

The OGEA (-derived by reversing the initial letters of the names CJolovin and Artisimovich) is a first generation Soviet' mirror experiment'similar to the DCX in that plasma buildup is through the injection of molecular ions. However, dissociation is sought to be achieved not with a carbon arc but collisions with molecules and ions.

3.7. Quo Vadia?

This chapter has attempted to capture the highlights of the work done during the last two decades concerning magnetic confinement. The brevity of the chapter hardly does justice to the enormous effort that has been put in, much less to the difficulties and frustrations experienced. We recognize this shortcoming and would be satisfied if the reader has at least obtained the broad flavour of. the programme', and the diversity'of approach in particular. IJooking back, the optimistic prognosis made in the mid-fifties looks most daring iateed. The discovery of instabilities of course promptly- introduced a sobering influence directing the attention of researchers to basic questions.* As Artisioovich has

* Recall also Pig. 2.16 -126- remarked, this-'lead to a hundred flowers blooming in the sense that a large, number of conceivable approaches to overcoming instabilities have been given a try. The position after two decades' of hard work is that:

(1) A good understanding exists of the basic equations of plasma physics, and their adequacy to describe realistic situations.

(2) The broad principles of magnetic confinement have been established.

(3) One knows how to tame gross instabilities by using magnetic wells, magnetic field shear stabilization etc.

(4) A fair amount of knowledge concerning microinstabi- lities has been accumulated but difficulties from this source have not yet been fully eliminated.

After evaluating the various contenders, Post feels that at the present time, there are basically three systems which aire close to creating and confining plasma at fusion conditions. These are the Tokomac T-3 at the Kurohatov Institute, Moscow, the Theta Pinch Scylla-IV at Los Alamos, and the Mirror Machine 2-X at Liverraore. The operating conditions of these systems are compared in Table 3.1, while the march towards full-fledged reactors is illustrated in the form of T-n-X plot in Pig,3.20.

Qualitatively, fusion research is in a very much healthier sta^e as compared to the doldurms of the late -121-

4 T (kftv)

MnrlUK REACTOR

N (crrf1)

PULSED REACTOR Fi«j.5.n0. Three-dimension n ~ t - C plot, illustrating ths aocompliaha«nt3 of the cur-rant hopefuls, • namely 't 'A2~* and ,3J-3 machines. 5-IV has to nov$ xpwaras the goal marked pulsed' reactors and 2-X towards the go-il mirwr reactor, Th© goal fox- xokoinaea is not ojiown in order to avoid artistic complications. ,1

Table 3.1

COMPARISON OP P3RF0RMANCE CHARACTERISTICS- OF SOME IMPORTANT TN SYSTEMS EMPLOYING MAGNETIC CONFINEMENT, (4)

T5 S IV * 2X

r3 13 16 a. (cm ) 3 x 10 3 x 10 4 x 10,1 3

T4. (keV) 0.4 '•6

V (eV> 800 300 . 250

sec -2 ,-6 r3; z ( ) 1.5 x 10 3 x 10 1 x 10

Plasma volume (cm ) 5 x 105 60 5 x 10-

B (kG) 35 120 12

5 11 10 10 n (cnT ) 4.5 x 10 9 x 10 4 x 10 -129-

fifties. A big gap still exists between present performance and that expected in viable systems but it would appear that it can be bridged if sufficient effort is made.

REFERENCES •

Detailed references are not given in this chapter since to do so would make the list very long. We therefore merely cite the following books and articles where more information may be obtained.

1. A.8.Bishop, Project Sherwood ( Addison-Vfesley, New York, 1958). 2. T.S.Green, Thermonucle&r Power (George Newnes, London, 196?)> — - .

3. S-Glasstone and R.H.LoVberg, Controlled Thermonuclear Reactions ( Van No3trand, Hew York, 1966). ' 4. R.F.Post in, Annual Review of Nuclear Science (Annual Reviews Inc., Palo Alto Calif. 1970;, Vol.20, p.509. 5. L.Artisimovich, Controlled Thermonuclear Reactions (Gordon and Breach, New York, 1964). -150-

4. LASER'INDUCED FUSION

During the last two or three years, tremendous exoitem.ent has gripped fusion researchers consequent to the realisation that lasers offered a-tot ally'"new approach to controlled thermonuclear reacti ons. The "basic idea is to heat to TN ignition, a solid pellet, of fusionahle material by shining upon it an intense pulse of radiati on from a high power laser. Confinement problems do not arise in this scheme because the TN burn is completed before the pellet disintegrates; in other words,, the hot plasma is held by J.ts own inertia whence the name inertial confinement for this scheme.

Laser produced plasmas are not new. Within two. years of construction of the first ever laser by Maiman (around 1961), pulsed ruby lasers were developed which could q deliver an intense pulse with a peak power of 10 v/atts at a wavelength of 0.69 |Am. Even prior to this, Teller and his colleagues foresaw that .the'.laser could, be harnessed for TN ignition, and initiated computer calculations. (The crucial concepts were declassified only recently,. and Teller has expressed the hope -that its impact would..be comparable with that of the talk given by Academician Kurchatov at Harwell in 1956. See Chapter 1), AS laser technology developed, 3mall scale plasmas were in fact produced by laser irradiation, and in 1968 Academician Basov and his colleagues at the Lebedev Institute near Moscow, succeeded in observing neutrons generated in a laser-produced plasma,. Within a year similar experiments were reported from France and the United States, 5 and total neutron shields in excess of 1(r from the D-D reactions have been.measured using targets, of deuterium with -151-

incident laser-beam, enei'gies of between 50 and 250 joules.* Interest has naturally quickened and plans have been initiated in several laboratories to produce lasers which would herald the era of laser-induced fusion, Even as laser builders, were preparing to come to grip with the problems of developing such lasers, scientists of the Lawrence Livermore- Laboratory announced that computer calculations carried out by them strongly suggested the possibility of using an implosion technique . . v:hereby the threshold power for laser ignition of TN reactions could be substantially lowered over previously estimated values. This relaxation of the laser power requirement has in effect brought laser-induced fusion almost "within grasp", making understandable the wave of enthusiasm currently sweeping the laser camp. We shall briefly review some of -these developments in this chapter, starting with basic ideas concerning plasma production by lasers and leading up to the implosion idea. "Discussion of power stations based on laser-induced fusion is reserved for the last chapter.

4.1• Production of Plasma by Laser

Let us start by considering a laser beam .whose output is focussed by a high speed lens. If I is the intensity at the focus, the electromagnetic density is v> ( I/.c ), and currently attainable values of this quantity are w<->'6.7 x 10 atmospheres (corresponding to intensity levels of 10 tf/cm ). If such intensity is focussed on a solid surface, it immediately

* It has been suggested (Physics Today,.August 1974) that these neutrons could have been produced by processes other than TF reaction (O e.g. by collision of lbns with deuterium in the walls. -132- results in the production of a plasma 3heath which envelopes the surface'; the subsequent behaviour of the system depends to large extent on the optical properties of the plasma sheath. In the case of a metallic surface, the formation of the plasma is understandable since the light can be heavily absorbed by the conduction electrons leading to emission of some free electrons from the surface which then get heated to a high temperature. For insulators the situation is not so simple as they are usually transparent to optical radiation implying in turn poor energy deposition and therefore poor heating. However,, these considerations become invalid at very high intensities on account of non-linear phenomena, and it turns out that as a result of the latter, some surface ionization always occurs. The exact mechanism by which the first free electrons are produced is not known' preoisely although several possibilities like multi-photon ionization (absorption of several photons at the same time to eject a photoelectron) have been suggested. These electrons are accelerated by the electric field of the laser light and liberate further electrons from the surface by collisions, and very quickly there is an avalanche. Once a plasma develops, the subsequent events can be described by visualizing a layered medium i.e. a solid surface with a plasma envelope, with the light predominantly absorbed in the plasma region.

Figure 4.1 shows th« time sequence (as computed) when a 50 Um hydrogen foil placed in vacuum is irradiated with 12 2(1} a moderately powered ruby laser (^10 w/cm ). Assuming a s.nall initial concentration of free electrons, a strong absorbing layer forms at the front within 0.1 ns. The electrons in this layer co.n pick up energy partly on account of "Inverse Brehmstrahlen". Absorption is also possible if the plasma -155-

4.1. Computer simulation results of plasma I'onr. vbic;; ..men a 50 p. hydrogen foil is irra.liat •,, with a laser "beam. Plotted here av.'5 tho Jo.oaiLy, velocity and temperature piofiies .-vp various instants. (.U'tar Muloer et'al., 1975). 134- frequency of the surface cloud is higher than the light frequency. This v/ill be the case once the electron density builds up. Due to this heating, the piaanabegins to expand into the vacuum causing a rarefaction near the surface. This enables the laser beam to penetrate into the target by- evaporating layer after layer, and as material is removed from the surface and ejected into the vacuum, the recoil generates a shock yave reminiscent of the thrust developed by an ion rocket. The ablation of the surface, propagation of the shock and the depth of penetration of the laser beam are all inter-related process, and Pig.4.1 summarizes the overall situation, points worthy of attention here are:

i. Most of the laser energy is absorbed in the electron cloud but only a small fraction Is used up in the .formation of the shock. The shock v/ave balances the momentum of the expanding plasma and since the density of the pla3ma is vastly different from that of the medium, only a small fraction of the laser energy is transferred to the shock (u» .£$#). The rest goes up to heat the electron cloud.

ii. The shock pressures for the geometry considered are ^.106 *tttoaph«r©e. •

lii. The compression is due entirely to the shock, and

iv. the density increases but not by orders of magnitude. -135-

' \fi\en tli6 incident intensity i.y raised, the 3.1.tuition bo.c-oroer; 'more c:ornpl.icate.'i ---.s the plasa?;. production j.s doterninsd fcy light absorption, as v/c-ll as heat conduction, t-v:. illus'tr-rfced in ?i r./, 4 . 2, Here-'.>•"? ?,e? tJiat at higher iiiai'floril; intena-.tieo, the heating, .sxtear.s to regions not yet reached by the lx.f;ix be^ra; thii.; i-; CV-J to th^r-nr..! diffua:! on. in brief, laser irrodiavion of a yolid taav^t pr-.-juuerj;:-: boih compression and intense heating, auggeating its possible application for triggering TN reactions. Assuming that the required tauperature can he reached, the time: for vrhlch the l:-r:-^r pulse must last is i^ (n"C)-r . . v./*1,,.. ^-», * „„.„ „„,• s yLav.'3on under compression Tn i:h? other hand, during; this irradiation time, the target mu.r>t hold together by inertia which means that the disassembly •) i'-ie which is ts\ (linear dimension/hypersonic velocity) (see ?i~. '..1 ) munt equalX • A crude estimate of. the quantities involved nay now bo obtained as in Chapter 1 from a consideration of the break-even condition. At there (see 13*1.(1.2)), 12^1

*p where we have taken n^- iig- n/2. Suppose now that the energy release occurs by the irradiation of a spherical target of r.-.dius R, The life tine ~Q of the pellet in its fully heated condition is controlled by the expansion v/ave proceeding into the plasma, given approximately by

r «.« R/V. (1.3) -136-- '-

J.- *\

X CD

1 U*M-*JMJ* -ff«fccaffj|a*s

K^«sitfc>a^j«fcR?stc-ris=!^*AMa snjsrejMttianMfi

/"*

2: «*» «£Z5w-

6 CD UJ

DISTANCE INTO FOIL Pig.4.2. Schematic drawing of the light penetration" ana electron temperature distribution in a plasma produced in a foil by laser irradiation. Both r.'Lo+o refer to the same time lapse alter start of las-^r irradiation. Only, the top figure corresponds to laser power ^ 1012 w .,n(i the lower one to •>% 10^5 w. Observe that in the latter case, the heat penetrates much further than the light. !Phis is due to thermal conduction by the electrons. -137-

This'is also the inertial confinement time or the hydrodynantic disassembly time as it is sometimes called.

For v it is reasonable to take the value (10/3 kBT/mT)~t the sovmd velocity in a fully ionised gas (ml.• = average mass of the ions).. Prom (4.1') and- (4.2) we get

». a^x QL *±£2T (4.5>

The corresponding inpvit energy is 11/2 2 5/2 3 4 •• 12 "l0 1 ^ V ^ G 5 &* ^^57T" -y "J? T (4.4)

For a Given density n and gain G, the minimum of the input ("11/? energy depends on the minimum of the expression A (WO > K

various fuel mixtures. Taking 1,^- «= 1° keV, ..and mT = 4-.2X10-24 gm.<2) ' '

R = 21 2 51 min 7(G/n)10 cmf Wmin = 6 (GVn )10 Joules.

22 ^ .7or n ^5 x 10 /cm , the break even (G = 1) values are R vo.14 cm W'w>2.4 MJ and "C\J* 1.4 ns. These estimates are quite crude but sufficiently good ballpark values, Figure 4-3 summarizes break even power estimates made by various authors, the wide spread arising from variations in •138-

ENERGY (J) ,10 10 «—.CHU *~ LINHART

8 IP kg TNT-*10 S_ JOHNSON & HALL _,WITKOWSKI — HORA 8..FIRSCH Vf ~}BASOV & KROKHIN

1g TNT-*10«

Pig.4.3. Breakf-even energy in Joules for laser fusion as estimatefl "by different workers. (After Mulser et al,, 1973). -139-

assumptions, A significant feature is that all these calculations consider fusion plasma at solid state densities. In 1972, a qualitative change in the picture was "brought ( 2 ) about by the announcement* of Nuckoll3 et alv ' of Lawrence Livermox'e Laboratory that using a properly shaped laser pu,lse, compressions upto 10 times normal solid densities are possihle, implying a corresponding reduction in the laser threshold energies. 4.2, Implosion by Laser To appreciate the essence of the idea of the Liver mo re group, let us. assume that a 1mm radius "JDT pellet ( \^1 mg) is irradiated with a 1 MJ laser. Assuming full absorption, this would raise the pellet temperature to '!0-'°K and initiate a thermonuclear burn. J?or the size under consideration, the confinement time is ^2 x 10 -10 GOO and the burn achieved during this time is insufficient for break oven. On the other hand, the burn time as estimated from the break even condition must be 1000 times longer i.e. . * 10 sec (see So.(4.1))- If however the same pellet is compressed to a radius of 0.1 mm (and thereby 1000 times in volume), the inertial confinement time would be reduced by a factor of ten while the burn time (determined by (n-C )-rawson) would be reduced by a factor of thousand. Substantial fuel burn up is thus possible, making it- conceivable to achieve more energy release than is consumed by the laBer. In. fact, according to the Livermore group,

*The announcement was made as a post-deadline paper to the Seventh International Conference on Quantum Electronics, held in Montreal 8-11 May 1972. A formal note was published shortly thereafter in Nature (reference-2), -140-

".,-..... any laser-fusion power-generation scheme "based en heating of a Til fuel at normal density is doomed to ( 1 ) failure . " v ' It would appear from above that compression is a must if laser induced fusion is to he achieved. However, the details of compression desirable and the manner in wh:c3. it is to be produced are quite involved questions. For example, v/e have sesn earlier that if the laser produces merely uniform heating and no compression then the break­ even energy is in the range of several MJ. Suppose nc;; that the laser acts as a hydrodynamic driver compressing the pellet surface uniformly. In this case one effectively has a single shock and a mathematical analysis has shovnv "v '' that the laser energy requirement is not reduced very much &p&x that of the uncompressed sphere. To reduce the laser threshold, a more complicated arrangement is required wherein the laser first acts mainly as a piston compressing the pellet and in such a manner that the central region of the - compressed pellet is heated to ignition temperature while the rest of the compressed pellet is as cold as possible. The latter is essential to minimize the over-all energy requirements. u?he selective heating of the central core above j.s achieved by the convergence at the centre of the shocks that produce the compression. Upon converging ore the centre, the kinetic energy of motion of the shocks is converted to internal energy. Once the central core is ignited, to fusion, a burning front develops propagating outwards into the previously colder regions of the pellfet. Provided sufficient energy released in fusion is deposited in these regions, the burn can be to some extent self -141- sustalning* and at this point laser power, may, no longer be required. In brief, what i3 required:is: a-la3er pulaei of suitable wavelength-time profile-'i'sot-thatr. the"pellet^ia- apherioally imploded in a oonvergenttfashion, resulting eventually in a compressed y«llet"compacted:.to about 10,000 time3 the normal density. At, this«Juncture, the.laser:in a final burst, delivers an intense beam:lasting Just,a few pico seconds, heating the supercempressed^mafcter:to- TH*ignition. The burn then,, propagates.outwards,.lasting until various.: quenching me chani sms t ake ove r ••• and•" suppress* the re action.

An appreciation of the laser energies-required in. this scheme can be had from Pigs.4.4 and 4.5. The.firBt of these shows a plot of the compr'-ssion versus the-energy required to achieve such a compression.. For. the desired compression of 10 , something like 2 x 10.' J/gra of;energy must be delivered, implying'a laser" power requirement-.of 10 J/fe.n of DT, assuming that only 5^ of;the laser-energy is used in compression. This ccmprcssed."matterrwill: release energy upon ignition, and' assuming 30#:.burn up} .the? energy release will be *\1C J/gn; suggesting;a gain of «-v100. However, this is not the true-gain since the laser; efficiency and the efficiency of energy recovery;-from the TIT reaction have not yet been allowed for. If" these:are-arbitrarily assumed to be 10$ each, then the apparent" gain- is completely wiped outt Fortunately, this is not the full story,, for, when the burn starts, the alphas produced in the DT reaction are all stopped in the pellet itself on- account of the. tremendous density. A substantial fraction of the neutrons are also stopped for the same reaBori, the net effect b*lr# tft*t the -142-

I I- —sq - "H"-" Tma 'i 10" c •.

' • •' '- — .**-> j- 'l

" * ? . b B "v;"": ----- .-- ~^-* 10 - •, • -.^^^^ >^ — -• J71 A 107 v^ — . - CD t LU VMINIMUM COMPRESSION ENERGY. UJ 1CT —

— / —

10 I. ... .,.1. I I', J., , ] . io° id 102 id3 xi io5 io6 10' COMPRESSION (Liquid density 1)

Pig.-l.'U Plot'of the energy required to corapress pellet as .a .function .of compression. G axv-i A indicates the laser energy if the laser merely provides a trigger and the burn is by proroga­ tion. Curve B applies if the trigger and'the burn is by laser. C indicates the energy released for a 30$ burn up of the pellet (1M dia). -143-

COMPRESS^ (LIQUID DENSITY* 1)

Plots c£ the energy gain versus eo/p.rression in laser-in.Juced fusion. Jesuits are shown for two laser?.;, one with r. cofcr.l aiirgy out cf 104 J and another a hundred times more powerful. In each case, tho dc.shed curve corresponds to a propagating burn and the full line to a uniform ignition, -144-

energy of the reaction products helps in the heating and thus in the propagation of the "burn (as in the case of ordinary- fire), which reduces considerably the laser power requirement ("by a factor of 100, it is estimated), bringing it within grasp of current technology.

Turning now to the details of the implosion approach, the concept of using implosions for high compression is in itself not new. It was exploited in the mid forties in nuclear weapons, and it is not surprising that workers at Livermore and Los Alamos turned to the implosion idea to achieve supercompression of fusionable matter. There is of course a qualitative difference "between the two problems in that in nuclear weapons, implosion is produced by chemical explosives fi magnifying the pressures from i^10 t o ^10 1 atmospheres whereas in controlled fusion oxse is seeking to produce ir-ulosion with laser generated ablation, attaining pressures 12 of the order of 10 atmospheres J

Figure 4.6 shows a schematic drawing of the pellet and its environments at a typical instant during the implosion. The pellet proper is surrounded by a low density hot plasma which is isotropically illuminated by focussed laser beams. Most of the light is absorbed in the envelope, maximum absorption taking place in a narrow region termed the critical surface. The formation of the tenuous envelope is in itself a complex process, and is probably accomplished within about 0.1 nsec after laser irradiation starts. One mechanism of light absorption (at the critical surface) is inverse Brehm- strahlen, already mentioned. Significant anomalous absorption due to non-classical processes is also possible,, dominant among which are those where the incident electromagnetic energy ESCAPING IONS. J.ASER RADIATION

OUTER CLOUD SHOCK. FRONT.

•fc. \J1

\_HOT ELECTRONS ABLATING SURFACE.

COMPRESSED SHELL CRITICAL SURFACE

I'ift - 4.6. Schematic view th-£- ! iit-i; •.).'.•'ii .Lnr-t :;'.- T •.i.'.u-ing the implosion showing the outer'envelope, Zhz c; -.•:.-.I i?n3orpt.J.on surface, the shifting layer and fcb? coI'IV-TA;'.?•:• •»..'. fron-:. -146-

excites one or the other collective modes of the plasma. Processes suggested include:

Phnton decaying into ^noxon . & ^ piaamon + Phonon Plasmon + Plasmon ->• Plasmon + Photon

^ Phonon + Photon

The last two are well known in non-linear optics as Stimulated Raman Scattering and Stimulated Brillouin Scattering respectively.

Light absorption dvc to one or other of these mechanisms results ultimately in the production of 'hot' electrons. These hot electrons are transported inwards from the critical surface by "hernial gradient and heat the pellet surface, ablating It. The ablated material explodes outwards and the escaping ions are heated by ion- electron collisions in the .-ix-sio-sphere. More important however are the reaction forces generated by the escaping ions which accelerate the pellet surface inwards causing a spherical shrinkage. At this point we must recognize that compression arises both on account of continuous physical inward movement of matter as v/ell as the shocks (generated by laser impact) v/hich travel inwards. Besides compression, there is of course also the heating. The problem, as already mentioned, is to control the various factors such that matter is first supercompressed before superheating to TN ignition. This sequence is important -147-

hecause premature heating will render compression difficult and very high densities may not "be attained. A formal mathematical analysis of the problem is exceedingly complicated, and resort is therefore made to computer simulation to obtain insight into the problem. Such exercises requires several hours on the most sophisticated computers for each case history. It has emerged that control of the sequence of events can be had by tailoring the laser pulse i.e. by requiring the laser, power to build up in a particular way. In essence one starts off with low laser power, the rising power producing increasing compression in such a manner that there is.no excessive heating until the full 10 fold compression is-achieved., when-' ignition is effected with a final burst of energy" .:f rot*, the laser,

Pigure 4 *7 illust.r:.-'•••? schematically the escalation of power during the implosir.-': It is worth not'ing that radiation pressure drives an ''electron piston" which in turn operates a "ion recoil piston1''; .being both convergent, the compression is effectively ic.yussed, implying a pressure magnification. •• ... •-,..,...

4.3. Typical Results for Laser'Implosion

To compensate for the pos'sible lack of clarity in our description of laser'implosion0 we shall now present from several sources, results of typical calculations which, hopefully, would offer a detailed idea both as to the sequence of events and the order of magnitude of the various quantities involved. Pigure 4,8 shows the first ever results reported by Nuckolls et al* ' Plotted here are the outcome of a computer experiment involving the implosion of a 0.4mm radius 2 2 lrf° W/CM2 f r^> tC^-tePw/CM

ICp-tFw/CM2 -tfWcMi

I tPw/CM1-

Kg.4.7. Schematic drawing to illustrate the concentration of laser power and thus the escalation of the compressive pressure. The top part shovs the focowsiag of the laser beam on the critical surface by tiie lens. (The en.rgy absorbed " at the critical surface is transported inwards by the hot electrons to achieve a further increase. ?ne shock v/ave generated at the ablating surface converges to produce, astroiioraicai pressures! .•i,:.-4.y. Compitt.,5.- simulation results lor laser implosion oo r, ••., ,131 radius DT peilox =y a 60 ivj 1-, ...-.• (Alter iiucjcolls est, al ref '•>) -150-

D!T pellet "by a 60 IcJ laser operating at I jwn wavelength. The next figure shows similar results reported by the same group ( 5 ) none tine latsr in Physios Today. ' In this case "che temperature and density profiles et representative instants are available, and the existence of the compressed shell can he seen "clearly. Also evident is the ratio of the ahloted to compressed matter. Further snapshots of the implosion are shown, in Fig,4.11 which is based on frames of a computer ( 3 ) generated, movie prepared "by the Liverrnore group. ' ?i,?ure X* ) v i\s I 2 & 4.13 show results obtained by Lrueckner and Jorna ' P.or. a .,?,se history whose features are shewn in Table 4.1. Observe i.n this co.se the steps in density associated with the various shock pulsos arising from the steps in laser power. Implosion of hollow pellets have also been staled (these have sonic advent *.~es which v;ill be 'pointed out (7 ^ shortly). Some results obtained by the Los Alamo3 groupv are shown in Pi^..* „14 wherein is depicted the life history -2 of a 7.5 ••'T I)'?, shell of initial inner radius 5.4 x 10 cm and thickness A R - R/56. The laser inijut energy is 5.3 kj, the 13 pulse duration ^30 ns and peak power ^3.8. x 10 -'7/. The differences in 111571 itudes in these various results arise mr.inly from differences in the detailed assumption. The broad features o.f implosion may however be same. '",.'} . Problems of Laser Implosion Two important problems likely to be faced in laser implosion may now be mentioned. The first concerns the dejroe to which spherical symmetry is required. Now in the compression of a sphere by 10 fold, the radius decreases by -151-

TIME (nsec) Fig.4.9- Further results on simulation of D'J? fusion by laser implosion. At bottom right, v/e see the laser pulse power rise in steps. At the top right are ilLusi'lrated the history of the ablating layer and the pressure build up. 2op Left shoves density and temperature profiles at two instants. (After Nuckolls et al. r.ef.5"). -152-

Fig. /(. 1 0_. laser-induced pellet compression. Plot ced here are trajectories of representative particles in the pellet as a function of 'time.'• Distances are in microns and time is in nanoseconds. Observe that for some particles the trajectory is outw?.rds. This corresponds to ablation. Other neve inward3 corresponding to compression. (After Kidder, ref.6). -153-

jtfk ; t«Ons t*21-3439 ns ; -^xs PmsO-21 ?m = 4-75 —%

t« 24 0319 ns t =24-0453 ns >m« 1224-94 Pm»2761-24

" H . % \ . \

t*240493ns t*240520 ns Pm =3099-04 s ^S>Pms1381-53

Jx1(fK <^ C • v2x10lKt\ .

DrrrW.iti._?;:?• baaed on typical frr \,.v'. o.i.' a c;.: v--i;r.~ i' gr;::er,»tow. i'il;:i o/. Dl1 fusion. Ji.-c las/-r : 5-'r k-j. 'J?he hat^ied portion sn^.-i^ pellet dashed lines ih:j electron i&otfierais and • lie line;:: Ion Teir-ps.:--..cures. In 1 , 2 and 3, width .'.3 - ILOO U. In 4, 5 auc- 6 i\ i -154-

gable 4-1

Initial diameter OJ 'DT sphere 0,5 nivn

Laser pulse 6,5 >: 1011w 0 - 5.47 nsec 12 6.3 x 10 V/ 5.47 -7.21 nsec linear rise 6,3 x 1 012 - 4 x 101S/ 7.21-7.42 nsec laser energy absorbed 60.1 !cj

Central density (avera/jed over 1 Q?» of radius) 15S0 gai/cm' fraction oi' sphere iraploded to a density :;;rrather thai; 100 1 2# £m/era I'ime for which central region is compressed to gi'eeither than 100 ^in/oLp." 1.8 x 10"11 sec.

Energy input in com­ pressed region at t ime of max. compres- siorj 2-.9 kJ Total fusion yield 510 \:J -155-

10 100 1000 R (MICRONS) Pig.4.12, Density pro.i'ilca at,various times during laser i.v.ploaioa corresponding to case history in Table' 'a.-'-!. Tima is in' units of 10~1° sec. The step xnpresents shoc-c due to laser pow-i-r pulse. Observe rapid density increase in irinal stages. ' -156-

0 10 20 30 W R (MICRONS)

Fig.4.13. Propagation of burn.front corresponding'to ca3e history of Fig.4.12. -157-

about a factor of twenty. If the pellet shape after implosion must be nearly spherical ( this must be so as otherwise convergence effects v/ill hot be present), then a high degree of spatial and temporal uniformity in the implosion is reouired. This can be achieved by irradiating the pellet as" uniformly and as synchronously as possible v/ith multiple beams (see Fig.4i15)» The asymmetry in the irradiation geometry, if any, is wiped out in the atmosphere (that rabidly develops around the irradiated pellet) by the multipi-; scattering of the electrons during the process of carrying energy from the critical to the ablating surfaoe.

The second problem arises from 'improper' heating of the outer layer electrons. Such improper heating can lead either to preheating or decoupling. The former refers to the heating up of the central core ahead of the propitious moment, and could arise if there are excessive number of high energy electrons in the electron distribution. These supertharmal' electrons may .penetrate deep into the core (unlike the bulk of the electrons which stop at the ablating surface), and cause premature heating thus, making compression more difficult. Such a deep penetration is possible on account of the relatively longer range of the superthermal electrons. Situations could also arise when, superthermal electrons generated in.one part.of the outer mantle shoot past the core to the opposite side to enter the (optical) absorption layer there. Here they will get heated to still higher en?rgies v/ith larger mean free paths, and the process could snowball resulting in the near decoupling of the pellet from the laser heated electrons. It is necessary therefore to tailor the laser pulse suitably to avoid both preheating -153-

x«sm T«t:*01:

.0° - I -•—•. ^

_ i l 1 *• ,n« i i i i. i ft i. t i_i . 111 I ,1, ^i i i ..1 1 1, tIM> »•JUL JL. . T« 29.972 irit) T« 30.093 (in) 7S\ lv/ < 1 1 '1 1 1 1I 1 l I I I t I I I I I

- ' • -•

KZZ-••— - l^J V — 1 1 l'l II .03 .10 .0 R (cm)

Pig.4.14. Time his!;ory of density and temperature oi' a 7*5 ugu m she.!i heated "by a 5.3 icJ 002 laser. The + curves refer tc ion temperatures and the dot-dash curves t eiecti-ou t emperatures. (After Clarice et al ref.7). -s*- Pig.4. J"3. Schematic illustration of tiv^ isotropic iiruminaTiroir-oi— peiiet by 'eonvergont beams, ill these beams are derived from the same laser.

\J1 -*» I

LASER - . • • (OSC) AMP AMP AMP .FUEL

y- ......

?i~. -!. 16. >oho..,riTio iiuiotration of t-i.: builfl. up of a. laser beam by a chain of optical amplifier. -160-

nnd decoupling. Pai'anthetically it may be mentioned that decoupling effects are lesser in hollow pellets.

4-5> Prospects for Hi^h Powered Lasers and Future Expectations

The computer experiments discussed above have shed some light on the type of laser that is needed to achieve fusion. In specifying a laser for this purpose, one must consider several facts. Firstly there is the question of absorption of laser light by the plasmar This depends among other things on the wavelength, and generally speaking one would"like a short wavelength laser. (Even better would be a laser whose wavelength can be changed v/ith time to optimise absorption! This of course is not easy but could conceivably be done v/ith a series of lasers operating in sequence). Next one must consider the pulse shape* This is all important, because the implosion sequence is very much dependent en the power-time profile". The duration of the pulse is yet another factor. Thanks to the implosion scheme wherein one may envisage the utilisation of the

The Nd-glass laser consists essentially of a glass Table 4.2

LASERS FOR FUSION (ADAPTED FROM REF. 5)

wave Energy Pulse Max. Output Laser length Efficiency storage width short pulse (microns) (j/litre) (ns) (J)

Ha-gUss 1.06 0.2 500 0.02 350 350 1 1 1 C02 10.6, 5 * *° ?

Iodiae -1.52 0.5 30 0.6 12

DeairatiU 0.3-0.5 5 100-1000 0.1-1.0 : 10* - 10* -162-

Liatxix containing .Nd"1 ions, pumped by xenon flash lamps. The laser radiation is in the infrared, having a wavelength of 1.06 Urn hut by harmonic generation, wavelengths of 0.53 U-m and 0.2G5u.m can also be'produced with fairly significant efficiencies. ;'Tne cOg laser provides 10.6 jj.m radiation associated wi-th transitions between the vibrational levels of the molecule in its ground electronic state. High efficiencies are attainable but an important drawback is the difficulty of harmonic generation. The iodine laser operates at 1 ,515Um, the lasing action being connected with transitions between electronic levels of neutral iodine atorr which is produced in an excited state by photo dissociation of CP-.I with xenon flash lamps. This system is being extensively studied in West Germany, the cheapness of the laser medium ( to be. contrast eel "with Nd -glass) being an important attraction.' Control .is however proving to be a problem, the laser being prone to parasitic oscillations on account of its high gain.

Considerable effort is* under way to develop lasers of the desirable typo. Notwithstanding these, the Nd-glass lsser system seems to have been chosen by most of the laser fusion laboratories as the prime vehicle for preliminary implosion experiments. The systems are all quite similar, and consists of .a laser oscillator («>»10 ^J) whose output is boosted by a cascade- of amplifiers (see Fig.4.16). There will be several parallel cascade chains and their outputs are eventually focussed onto the target from various directions. A giant $20 million facility of this type is under ccnstruction at livermore. Scheduled'for completion ^1977f it will be ^s\ 55 metres- long and will produce 10 kj of energy in about -163

100-500 psec with a peak'power of (s> 20-100 terawatts. Meanwhile some small scale implosion experiments have already been carried out. At a workshop held in Renessler Polytechnic, Basov and his colleagues *• ' reported of a experiment in which a sphere of dexiterated polyethylene was spherically imploded with a 9 "beam geometry. Density compressions to around ?0 gms/cm"' were claimed. More recently, a similar report has emanated from IQ-13 Fusion Inc. in Michigan U.S.A. ' In this case a Hd-glass laser beam was split into two and delivered on to a Dri target from either side. The incident energy was f 80-120 J and the target diameter o* 60-80 m. 5 About 3 x 10 neutrons were observed and compressions of the order of 50-100 were claimed. Dvidently there is much more ground l"o be covered and presumably one would have to await the completion of the giant Livermore facility before the thousand fold compressions mentioned earlier can be achieved.

'!,6. Some O'Jher Ideas

Before concluding, mention may perhaps be made of the suggestion of.Deu: et al., ' who speculate that a fast .chain reaction could be set up in a supercomprsssed DT plasma even at low temperatures I 'fhese authors call attention to an (11) old suggestion of G-ryzinskiv ' who noted that if DT plasma is compressed sufficiently as to make the Permi velocity ^velocity of ions resulting frpra the D£ reaction, then the ions so produced would lose energy not by collisions with electrons but by collisions with other ions. Following up this idea, Dar et al, report preliminary calculations to demonstrate that if the electron density is <^ 102 7 cm— 5 , then the electrons dominate the slov.dng down of fast ions shot thx-ough the medium -164-

whereas for densities > 10 cm"" the electrons cea3e to be effective. Under these conditions the DH? reaction once triggered., can be self-sustaining by virtue of the energy picked up by the other ions from the reaction products, and in this way one could conceive of a true chain nuclear reaction (as in fission) and not burning due to external heat, Obviously, if such a thing were possible, it would have important implications ;ror the laser threshold. At the present tine the idea seems to be in its infancy.

Suggestions have also been made that high current pulsed relativistic electron beams could be harnessed to produce compression and therraonulcear ignition. Such speculations are the result of tremendous advances made recently in accelerator technology. At the present time, for example, a 1 MJ pulse of width ^ 50 ns of relativistic electrons with energies a fev/ MeV is quite practicable, and the performance is expected to improve q\xite significantly (12 ) in the immediate future. Bobinv ' has given an interesting evaluation of the comparative features of laser-induced versus electron beam-induced fusion. To the present aivbhor, however, it would appear that laser-induced fusion is receiving greater backing as of now. .-16.5-

REFERENCBS

1. P.Mulser, H.Siegel and S.Witkowski, Phys. Letters 60, 189 (1973). ~~ 2. J.Nuckolls, L.Wood, A.Thiessen and G.Zimmerman, Nature, 239, 139 (1972). 3. J.L.Emmett, J.Nuolcolls and L.Wood, Scientific American 230, 24 (1974). 4. K. A.Brueckner and S.Jorna, Rev. Mod. Phys., 46, 325 (1974). """ 5. J.Nuckolls, J.Emmett and L.Wood, Phys. Today, 26(8), 46 (1973). —^ 6. RoE.Kidder, Nucl. Fusion, t£, 53 (1974). 7. J.S.Clarke, H.N.Pisher and R.J.Mason, Phys. Rev. Letters 30, 89 (1973). 8. N.G.Basov, O.N.Krokhin and G. V.Skilzkov, in Laser Interactions' edited "by H.J.Schwarz and H.Hora rpTe"mim, New York, 1972), Vol.2 p.389.

9. Physics Today, 27(8)f 17 (1974) 1 0. A.Dar, J.Grunzweig-Genosaar, A.Peres, M.Revzen and A.Ron, Phys. Rev. Lettera £2, 1299 (1974). 1 1. M.Gryzinski, Phys. Rev. 111. 900 (1958). 12. J.L.Bobln, Nucl. Fusion, J^, 553 (1974). 166

5. HYBRID SYSTEMS

In this chapter we shall consider Fusion-Fission hybrid systems which envisage the coupling of the 14 1'IeY neutrons (born in DT fusion) with "either fissile or fertile nuclei, to px*oduce an amplification of the fusion neutron source strength as also the energy. A well-known- example- of such a system in the nralii-negaton bomb which is essentially a hydrogen bomb coupled to a iPJ"" (i.e. depleted uranium) mantle- However, here our concern is with systems in which energy is released in a controlled'fashion. Such systems also were considered as early as the fifties when the hydrogen bomb was being developed, but much of the work done then was classified with the result that the hybrid power generator concept was rediscovered several times 1? "* ' Recently there has been a revival of interest in the concept particularly as calculations suggest that the Lawson condition can be significantly lowered on account of the utilization of the energy released in the fission compartment..

Both short-term and 1'on^-term use3 could be envisaged M ) for the hybrids,x ' In the former category, for example, one could think, of a system in which the technological problems of pure fusion reactors can be studied. Such a device would not be a net energy consumer and yet the plaorua could perhaps be contained with existing knowhowr on account of the lowering of the Lawson Criterion. A second possible objective would simply be to breed rapidly fissile nuclei for fission power reactors. Undoubtedly, this is also the objective of the breeder programmes that have been initiated all over the world. However, the doubling times of fast breeders are relatively large and could conceivably exceed the power doubling rates, in which case there -167- is the likelyhood of shortfalls in conventional nuclear programmes.* The rich neutron economy of the hybrid system offers promise of a smaller doubling time and thus a means of accelerating fission power programmes, should such an emergency occur- The protogonists for hybrids expect use for such systems even after pure fusion capability is achieved'. Among the uses contemplated are tritium breeding for non- tritium producing plants and as a burner of Plutonium and other actinide wastes of fission power economy. It is also possible that some of the hazardous fission product nuclei could be economically transmuted by neutron capture in hybrids to less hazardous species.

Much of the early work on hybrids was restricted to outlining the basic concept. The first detailed study of neutron economy in such a system appears to have been carried ( 2 ) out by Lontai of MIT. ' The configuration considered by him is schematically illustrated in Fig,5-1 and consists of a blanket surrounding the DT plasma. An enerpy transport current of 5 I-Itf/m'" of H MeV neutrons into the blanket was assumed and the neutron balance in that region was calculated using multi-group transport codes developed by fission reactor theoristsc The results were not encouraging from the point of relaxation of the Lawson criterion but the presence of fissile material in the blanket did indicate an enhancement of the power output suggesting that such blankets might be considered for power amplification when fusion reactors became feasible.

* Evidently, we are here referring to the pre-fusion era? -168-

WALL 1cm THICK

Pig.5.1. One of the hybrid blanket configurations studied by Lontai- The uranium in the attenuator is for providing power amplification through fast fission. -169-

( 3 ) Lidsky,v ; also of MIT, has pursued the idea further. He calls it the Symbiosis Scheme* and advocates it for the purpose of "utilizing the strength of fission and fusion power systems." More specifically, he considers a self-contained power generating unit comprising a power-consuming (!) TNR and a fuel-consuming Molten Salt Reactor (MSR), as schemati­ cally illustrated in Pig.5.2. The fusion reactor blanket

contains the salt mixture LiF + 3eFP + ThF., and neutron 232 233 absorption in Th . results in the production of U which is the fuel for the KSR. The latter is a thermal reactor 232 233 operating on the Th - U cycle, i.e. it also utilizes excess neutrons to convert the fertile thorium into fissile uranium. It is worth noting that the MSR does not function as a breeder but merely as a converter. In other words, its breeding ratio is less than unity; it is in fact 0,96, The combined system however acts as a breeder, and Lidsky argues that the appendage of a TNR to the fission reactor does not imply an economic penalty. On the other hand it has the advantage of a smaller doubling time, i.e. of the order of 10 years a3 compared to the 15-20 years anticipated for the Molten Salt Breeder Reactor.

Studies of hybrids have also been carried out by ( 4 ) ( 5 ) LeeK ' and Leonard et al. ' and these too showed significant reduction in Lawson Criterion while achieving energy multiplication from:fission. Table 5.1 gives a comprehensive summary of stich studies.^ '

* Dictionary meaning: Mutually beneficial partership between organisms of different kinds. -170-

(d)

U.233

(b)

CJRJ CoRi CjRi Ci«co-Ci

(c)

n,(R,U,-Wi.) NET OUT PUT Fig.5.2. Schematic drawing.of the fusion-fission hybrid considered by Lidsky. (a) shows the energy and fuel flow (b) depicts the breeding aspects and (c) the energetics. Legend; G - fuel nuclei produced per event. N - total number of fissile or fertile nuclei. R - fission or fusion rate. ' Subscript 1 - DT cycle; 2 - fission reaction.

C, nor of fissile nuclei produced per DT event. U., energy / fusion event • . Up ensrgy / fission event Wr power required to heat equilibrium plasma.

n1 efficiency for electricity prodn - fusion.

n2 efficiency for electricity prodn - fission. -171-

Tahle 5.1

COMPARATIVE FEATURE OF SOKES HYBRID CONCEPTS.^- ^

Pure Lontai Lindsky Lee "^J1^ Characteristic Fusion (Ref.2) (Ref, 3 ) (Ref.4) (fefTj?)

Tritium production (per source neutron) 1-4 .1,2 1.005 1.18 1.06 Fissile Production (Per source 2.49 2.7 neutron) 0-.1 0,4 Blanket fission {VPT source neutron) — 0.2 •*• 1.57 2,6 Blanket energy (Me 7) (per source neutron) 18,3 40 22.4 306. 500 Plasma density n confineaient time 10U , m 101* 2-7 x 10U 3 > 5 x 1013 ?raction of Lawson 1,0 0.5-0.8 0.25 0.16 l'i keV _ _ 20 6c 10

1-; MeV v/ai:. load C?.7/m2) 0-85-10 JsO 1.0 12.8 0-05 -172-

Our discussion of hybrids will not be complete wiuiout a reference to systems employing laser triggering. Based on the possibility of generating intense pressures via laser implosion, Y/interberg*" ' and Askar'yan et al. "^ ' independent­ ly suggested that supercriticality can be achieved with 233 23S microgram quantities of fissile material like U , U JJ and Pu 239 . The broad principle i3 similar to that employed in conventional nuclear devices except that implosion is now sought to be produced via a laser rather than through chemical explosives. The implosion pressures involved are, of course, vastly different as already seen in an earlier chapter. It is expected that under the action such intense pressure, the fissile material will be compressed to about 250 times its normal density reducing thereby the minimum mass and minimum siae required for critieality. For instance, Winterberg^ ; estimates that criticality radius under laser implosion will be CA 2.6 x 10 cm as compared to the 4 - 5 cm for a bare fissile sphere at normal densities.. The critical mass too comes down from the kilogram level to fractions of a gram I Winterberg further claims that if the fissile pellet is surrounded by a fusionable mantle (like DT say), then the —3 critical radiae^ comes down to is) 4.7 x 10 cm. This apparently io the result of the neutrons born in fission being partially reflected by the hydrogenous mantle. Besides providing a reflector, the DT blanket also contributes to the energy output on account of the triggering of the TIT reaction by the fission heat: The neutrons liberated by DT fusion lead to enhancement of the fission rate, in turn intensifying the fusion rate and so on, resulting in an overall cumulation of fission and fusion yields. Note that in contrast to systems- -173- considered earlier v/here fusion neutrons triggered fission, here the situation is reversed with fission being employed to trigger fusion as in a hydrogen bomb.

Vinterberg'8 note as also that of Askar'yan et al. essentially sought to focus attention on the concept, and such numbers as were reported were illustrative back-of .'-the envelope calculations- Understandably, several authors have tried to make more realistic estimates using, in particular, sophisticated multi-group transport codes developed for applications in reactor physics. For example, Krumbein^ ' has made calculations noting the fact that the assumption of uniformly high density throughout the compressed pdlet is probably not valid. Instead he assumes that there are three regions, the inner most one with a radius 0.1 R having _ c s matter at density 0..75 I (where ? n I "the nominal maximum density at the pressures involved), a central zone extending to a radius of 0,5 K having a density 0.07 0,1 R ), and an outer zone with a density of 0,025 9 • Some of his results are summarised in Table 5.2. 1 m Similar calculations have also been made by Seifritz and Ligoufw 9 ') who report details of the time behaviour, of the phenomenon. In a typical calculation, they consider a Pu pellet of 6;24 mm total diameter reflected by a 1.7 mm thick Li B shell, which is imploded with a laser pulse of energy 4-7 MJ, to a super critical state of k »« = 1.25. The Li in the mantle provides triiions for the DT reaction via the reaction

Li + n (from fission here) —^ T + c£ , -174-

On implosion, the radius of the core and the thickness of the —2 — "*i reflector shrink to 2.16 x 10 cm and 4 x 10 -* cm respective­ ly, corresponding to a density increase of <-" 250 for the core and uo 4150 for the reflector, The neutronic excursion in the fissile material releases in a 0.8 nsec hurst a fission energy of f» 7200 MJ (equivalent to 2,3 x 10 fission or 1C61 tons of TNT), sufficient to heat the reflector to TN ignition. On account of the enormous enhancement of the reflector density, all leakage neutrons from the fissile medium are absorbed in the reflector leading to the production of tritium. Both the tritium and the alphas remain in the reflector due to the reduction of the range at elevated densities, supporting the TN process, Typical results are shown in Fig.5.3 which displays the time behaviour of various quantities around the disassembly time. An interesting feature is that the kinetics are significantly different from that of a conventional nuclear device. This is largely on account of the strong time dependence of the excess reactivity kex(t) = keff.(t) - 1 owing to the rapid dep^tion of fuel atoms, which in turn is a consequence of the enhanced densities. Thus the "fast rising chain reaction 'outwits' early disassembly, resulting in a mien more narrow bur3t and higher burn up than in a fission bomb". HO) In a subsequent paper, Seifritzv ' has given attention to the approach to supercriticality of the fissile system and that of TN ignition. The problem is to avoid a stochastical behaviour during the start, of the fission chain reaction* In a fission bomb, this is secured by providing a Po - Be neutron source but this is clearly not possible in the laser-imploded system on account of. Bias' limitations. -175-

« 5- '2 \z *r - 11 16 8 10 £ •3 - «(t)y \\ MM 4 - \ \/ 09 Q 2 \ \ 1 0 i \ \ .el 60 6-5\>.\ \ 70 V -2 - \\ -4 - \\ - -6 "I •8

TIME (n sec)

'2

X u. z § »— Ul z 6-5 TIME (nsec)

"' -- - - -••• • •>:v:~io• n OJ a r-luloniu.. peiii't auicrounded by a Li D -..ancio .'J:iovm tisre are v-.r- u":i ;•.ni co around •"•i •• .r."xy ciiac , ..•'o.:1 ocrpaiison, a 24~;:ilo tor; ?c". ••;. :ic«&l riUolerii" device will have a l;ursc ci.Tie ( ^. --O'a energy relenss) 02? around 70 nanosecond. .Z'he bum 'up v/il;. be only around 1< ;.i!Ol'ons l.'i liiti pr•:'•-.• it exr^.p'ie it .ce^choc ^ r;0^. ( A;: her 3ei.c'rix 3 an; Li/ cv;, • r^i. 9 ) . -176-

However, one may contemplate partial preheating of the mantle to generate some 14 MeV neutrons (via DT reaction) which could act as the source neutron. Such preheating could conceivably be done by the initial portion of the laser pulse itself, or via with a pulse of fast deuterons or relativistic electrons striking the pellet simultaneously with the laser pulse. In 11 this way, a sufficient neutron source pulse ( i»10 n/sec) during the first neutron generation could be delivered when maximum compression is achieved.

Speculations have been made on the possible nature of a power station using the laser fission fusion (LFF) C concept. In essence, such a station will resembe that based on pure fusion i.e. the reactor will be a pulsed system with "micro explosions" occurring (possibly) several times every second. As remarked earlier, those explosions would be comparable to that generated by about a couple of tons of TM. The energy released will be carried by the fission debris and the neutrons (born in fission and fusion). While the kinetic energy of the neutrons could be absorbed in a blariket surrounding the combustion chamber (see next chapter), the energy of the fission fragments could be converted directly into electrical energy by making the fragments do magneto bydrodynamic work oh a magnetic field imposed from outside.

Generally speaking, theoretical calculations concerning imploded hybrids can be said to be at a lower level than corresponding effort in respect of pure fusion. Admittedly, sophisticated transport codes are being used but the kinetics is still being discussed on a compartmentalized -177-

basis. In other words,- these calculations do not consider the details of the production of the compressed state "by the laser pulse; rather they assume.that such a state has already been prepared and seek to follow up the subsequent events. This is clearly inadeqxiate and one must consider simultaneous­ ly how the laser compresses the mantle and the fissile system, and the time-dependent interactions between the two parts. Ifo doubt for "first-order" calculations such details may he ignored a3 triggering of fission requires merely a compression as compared to the triggering of fusion which needs both compression and heating, whence one may assume that fission chain reaction precedes thermonuclear ignition. However, both from the point of determining the statistics of initial neutron growth and from the point of determining the energy gain accurately, more detailed calculations are required. It is also necessary while making such calculations, to examine critically the assumptions made in ordinary reactor theory on account of the vastly different conditions of density in an imploded pellet as compared to those in a nuclear device, For example, at a certain stage in the growth of the neutron population, neutron-neutron interactions may become important and cannot be ignored as is usually done.v ' On the other hand, not withstanding the possible energy gain, IFF may not be attractive from a technological point of view on account- of the hazards associated with fission-product activity, in which case the incentive to pursue further the calculations will be lost, -Whether interest in such systems will survive, time alone can tell. 178-

Table 5.2 CRITICAL RADII OP FISSILE PELLETS (etf

Laser Pellet Normal Critical radi-as(cm) itsplosion Composition. density Laser implosion (Layered (Uniform density) density)

U23 3 5.65 0.023 235 U 8.28 0^033 Pu'23 9 4.93 0.020 0.25 239 Pu 4.27 0.0028 0.075 (D sb&IL » 1 .Oca) (L shell = 0.005cm) (D shad = 0.07 cm) 255 U +D 0.0026 (D shell-0.005cm)

REFERENCES

1. B.R.Leonard, Hucl. Tech. 20, 161 (137-25). 2. L.F.Lontai, Technical Report No.436, Massachussetes Institute of Technology (1965). 3. L.M.Lind3ky, in Proceeding^ of the Naclear Fusion Reactors Conference 1969 UTKABA. Culham Laboratory for BNES, 1970), p.41. 4. J.D.Lee, in Proc. 7th Conf. Inter Society Energy Conversion Engineerlnp, (American Chemical Society. 1972), p. 1294 -179-

5. B.R.Leonard, Jr., and W.C.Wollcenhauer, BNWL-SA-4390, Pacific Northwest Laboratories (1972). 6. F.Winterberg, Nature 241, 4*9 (1973). 7. G.A.Askar'yan, V.A.Nomiot and H-S.Raoinovich, Sh. ETP. Pis Red 17, 579 (1973) (Sov. Phys. JEIP - Letters 17.. 424 (T973)). 8. A.D.Krumbein, Trans, Am, Nucl. Soc. IS, 19 (1974). 9. W.Seifritz and J.Ligou, Trass. AM. Steel. Soc. 18„ 18 (1974). ~~ 10. W.Seifritz, Trans. Am. Nuol. Soc. (1974 Winter Meeting) 11. S.Ganesan, private communication. -180-

6. PROBLEMS 0? FUSION TECHNOIOGY

6.1. Introduction

Although the plasma is yet to, be tamed, optimists are already looking ahead to oomplete power stations "based on Tfi Reactors. Such stations must oomprise

(1) the basicreactor where fusion occurs, (ii) a blanket which receives the heat, provides cooling and a facility for breeding (of tritium), and (iii) a power generation system.

To sharpen the focus, reference designs have been formulated for example* for reactors based on Tokomac,v( 1) ' mirror machine, ' Astron^ ' and for laser driven systems.^- ' Such exercises have been very productive in bringing to light the problems likely to be encountered when fusion technology beoomes a reality.:

Attention will now be directed to some of the problems of fusion technology both in respect of magnetically confined reactors (MCTR) and inertially confined reactors (ICIR).

6.2. DegoTJLptlon of a Typical HCTR 1 • •• s Earlier we described some of the more promising magnetic confinement systems. The development of these as power sources requires a consideration of engineering feasibility. Questions arising in this context" are: -181-*-'*

(i) Capital cost, (ii) life and' reliability of major components in the system, and (iii) maintenance of the system.

Translated into practical terms,.these imply finding; solutions to various technological problems such as choosing proper materials of construction, designing a plasma container that will withstand both thermal and radiation blasts, designing large magaets to produce the required high intensity field3, designing heat extraction systems, tritium breeding and removal etp.

To obtain a feel for these problems, let -dk consider the schematic diagram of a fusion reactor given in Figure 6.1, Starting from the centre, we have the plasma, then the vacuum, the vacuum wall, the blanket (which serves the dual purpose of heat extraction and tritium breeding), the shield and finally the magnet coils. Figure 6.2 shows a cutaway view of a conceptual thermonuclear reactor based on Tokomak. ' The next figure shows a crowi section view of the Princeton Reference Design Model (PKDM). The magnitude of the reactor may be better appreciated by a reference to Table 6.1 which summaries' the principal specifications/ ^ The reactor itself is an eapty tube of ^j.5 a mean diaaeter, in the Bhape ef a torus of " diameter 17.7 m. The first wall, i.e. the vacuum wall, consists of 210 heavy-walled metal tubea, 6 COL in diameter, welded together. Helium flows through the tubes at rates high enough to remove the energy depositod in the wall. -182-

ELD PLASfcdA

MAGNET BLMK

Pig.6.1. Schematic diagram of a fusion reactor employing magnetic confinement. -183-

Taplc 6«1 (6)'. CHARACTERISTICS OP PRDM FUSION POWER REACTOR

Magnetic field at centre of plasma 64 kg Confinement time 1.1 sec Fuel burnup per pass 3»9# 21 Total reaction rate 1.33 x 10 /sec Daily fuel consumption

D2 0.390 kg T2 0.575 kg Fuel pellet 1 mm DT solid pellet Plasma volume 70Q m3 Vacuum wall Surface 800 m2 Power 1030 MW Coolant He (214° -• 620°C) 630 kg/sec Divert or Surface 2400 m2 Thermal power 90 MW Coolant He 57 leg/see Blartke.1? Daily consumption of Id6 1.13 kg Tritium productioni 0»585 kg/day Breeding ratio 1.016 Thtermal power 3710 m Tritium doubling time 2.5 years Daily ash production Hel&um 1.51 kg Hydrogen 0.01 kg Net power generation at 405$ efficiency 1840 MW itlHlISilllJlJl iljlil! iflkPPDaxpdfit: 9) 9

6 .

vo

P^ -183-

Table 6.1 CHARACTERISTICS OF PRDM FUSION (6)' POWER REACTOR v

Magnetic field at centre of plasma 64 kg Confinement time 1.1 sec Fuel turnup per pass 3.9# Total reaction rate 1.33 x lO^VseO 4 c Daily fuel consuaption,

*2 0.390 kg *2 0.575 kg Fuel pellet 1 mm DT solid pelle t Plasma volume 70Q m3 Vacuum wall Surface 800 m2 Power '1030 m Coolant He (214° - 620°C) 630 kg/sec Divertor Surface 2400 m2 Thermal power 90 m Coolant He 57 kg/see

Blanket /• Daily consumption Of Id: 1 ."13 kg Tritium production 0.585 kg/day Breeding ratio 1.016 Thfermal power 3710 m Tritium doubling time 2.5 years Daily ash production Hei&um 1.51 kg Hydrogen 0.01 kg Net power generation at 4O56 efficiency 1840 M • 10 i—

}i! ii 9 ji if llllllllilihlliiililll . M n *

o

•V "

H -185-

VflCUUW PUMPS

OH AND DIVtRTOR COILS ITYPtCALM) \

Pig.6.3. Cro3s-sectional view of the Princeton Reference Design Model (PRDM). (After ref.6). -186-

Solid pellets of D.T, 1 mm in diameter-(containing 3J* argon and a like amount of hydrogen) are fired into the reactor tube by the injectors (of which there are 4-0 distributed around the periphery) at velocities sufficient to penetrate deeply into the plasma; before vaporization and ionization occur. Within the confinement time of 1.1 sec, the resulting ions drift toward the reactor, wall and are skimmed off by-the.divertors assisted by the vacuum systems. The net output is ur\ 500O JW, the electrical rating being ^ 1840 MW.

6.3. Plasma Heating

Reference designs such as the o4e described above provide a basis for investigating the technological implications. Consider, for; example, the "plasma beating problem. Over the years much attention has! been devoted to this, and it is generally felt that ttfe problem is Solvable if inot already solved. Various methods for plasma heating have been proposed. One of He he simplest, conceptually, is ohmic heating., in this method, an ionized gas is first formed in the confinement chamber by fome suitable means, e.g. rf discharge. A.current is -then arranged to be set up in the plasma which.then produces the required heating via ohmic, processes.. There are various 3/ays of setting up the plasma current:., and: In toroidal systems one usually does this by explpiting. the transforBssr principle as mentioned in an earlier chapter. A. second method of plasma heating is. by compressing; the plasma wit& a rapidly varying magnetic field. If the compressi «* li not too rapid, it can be regarded as. adlabatic. On theiot&er I. -187- hand, it is also possible to envisage a sudden compression e.g. due to a shock wave, leading to non-adiabatic processes. Either way, the net result of compression is the increase in the internal energy of the plasma, and therefore ^he temperature.

Yet another method is netural "beam injection. In this, accelerated ion beams are first formed; the ions are jjhen readered neutral by passing them thfconghla neutralizer cell where they pick up the required number of electrons to become electrically neutral. The neutral beam thus formed, when injected into the plasma can penetrate it quite easily and transfer its energy to the plasma*by collisions.

Several other such methods are available, and after an evaluation of these, the designers of PRDM have settled on the so-called Transit Time Magnetic Pumping

(IIM!Pj. ' This involves the imposition upon the static Bz of a travelling wave of magnetic field

B = B0 oos(«*)t - kZ)

with the wave phase velocity (60/k) roughly equal to the mean ion velocity. Ions whose velocities are sufficiently close to the wave phase velocity are "picked up" and Imparted energy through the interaction between the ion magnetic moment and the travelling magnetic field gradient. The net effect is to cause a distortion to the Maxwellian, the distorted distribution'site eouently settling down to a new Maxwellian corresponding to a higher temperature. The eneigising coil is expected to be located between the plasma and the -188- vacuum wall, and must naturally be provided with cooling, in addition, of course, it must be able to Withstand radiation* damage. The TTMP system requires an, initially prepared plasma with a temperature of about 1 ke\v. The input power is expected to be several megawatts, applied at around 100 kHZ. Fairly good efficiencies are anticipated, with over 50j6 of the applied power absorbed in the plasma. One of the advantages claimed is that the plasma warming can be done in a slow and controlled manner, once ignited, the reaction products may be'expected to contribute to further heating, fresh fuel being added by injecting frozen pellets (nick named "snow ball in hell" method!) •1. Somewhat similar to spraying oil in a oil burning furnace to replenish losses. (See also Fig. 1.7^, 6.4. Vacuum Problem

Next consider the vacuum problem. The thermal output of ^ 5000 MW (of PRDM) is achieved by U 1.3 x10?1 fusion reactions per sec and this amounts to only 4# fuel burn up. It turns out therefore that close to 0.1 mole or ^6i 10 plasma particles (D+, !P+, He+) must be removed from the reactor every second. Roughly an^qual number of D and T particles must, of course, be reinjected every seoond to provide new fuel. To handle this amount of gas -Jfequires a throughput of ^ 3000 Torr litres/sec. In turn, to maintain a base pressure of 10~5 Torr, a pumping speed of w\ 109 litres/sec is required which is -very large indeed. It is worth noting that the problem of 22 sweeping away 6 x 10 particles per sec is not merely ... one of vacuum technology. These particles carry about -189-

150 MW (!) of power which complicates matters "by introducing a heat removal problem. Further these particles can cause damage when they collide with any surface. According to Post/ ' the vacuum problems though large "can be labeled solved".

An alternate scheme is also possible which is lees demanding in pumping speed (less by nearly a factor of 20^°'). In this one employs, a "divertor", and the ions constituting the ash are trapped by being allowed to strike a suitable surface in the divert or reducing thereby the pumping requirementBjsin.ee the ash does not have to be swept away by creating a vacuum. Instead, the ions are taken away by continuously moving the trapping surface, and liberating them elsewhere as illustrated schematically in Fig.6.4. The development of divertors for reactors poses no serious problem since experience with small scale divertors is already available from Stellarator experiments.

6.5. Materials of Construction

Turhing next to the systems external to the plasma, the most important design problem fadBS is allowing for 4 ation damage to the materials of construction. -~e bulk of the fusion energy, it will be recalled, appears in the form of 14- MeV neutrons. Prom the point of energy recovery and tritium breeding, these neutrons must be slowed down and absorbed in the blanket. At the same.time care must be taken-to see that excessive damage is not caused to the.structral parts exposed to -190-

INJECTOR REACTOR

ASH SEPARATION h— He Ash

MAKE UP T2*D2

wmwmmmmmm PLASMA

TO INJECTOR

Fig.6.4, Schematic drawing showing the divertor and i«H role in sweeping the "ash in a fusion reactor -191- the neutron flux. Fortunately, considerable experience regarding neutron-produced damage is available frcm fission- reactor technology, "based on -which some promising candidates are identifiable, and the,ae are -shovm in Table 6.2.(1°) Table 6.2 CANDIDATES FOR STRUCTURAL MATERIALS IN FUSION REACTORS^10)

Limitations St. steel Nb - 1# Zr Ho V Max. operating tempo (°C) 500 1000 1100 850 Max. temp, of compatibility with Li 500 1100 1100 ? Max. temp„ of compatibility with E 850 1100 '1100 ? Sputtering jpatio fo£ ' 20 keV denterons 0.001 0.01 ? Ductility 5% at 5g at 0£ at 20# at (estimated 450°C 425°C for 425 °C 550°C from exposure for fluence for for to fission fluence fluence fluence neutrons) 22 07 3 x 10" 3 x 10^ 3 x^lO22 r4x1020 Weldability Excellent Excellent Poor Good

After heat from induced activity 1000 h. after shut down 0.05 0;001 (W/Mtf(th) of reactor output) -192-

6.6. Radiation Damage .Problems

A "brief digression on radiation damage probleaa is pertinent here. The problem aa far as structural materials* are concerned, can b£ divided into two parte, surface damage and "bulk damage. Surface damage is particularly important in respect of the inner part of the' vacutui wall, which will he subjected to intense fluxes of high energy ions and neutrons. A variety of surface disruption* can occur, the two most important being sputtering and "blistering.

Sputtering occurs when metallic target is "bombarded by high energy ions, neutrons or neutral atoms. This may result in the displacement of atoms in the target from their normal lattice sites, leading, in some cases, to the ejection of the atom from the surface.

The word blistering refers to the appearance of blisters on the surface consequent to the bombardment with gaseous ions like helium. Figure 6.5 shows an example of blisters observed.on rhenium when bombardedwwith 20-JceT He+ beam.' ^ Upon reaching the target,-the projectiles penetrate a certain distance before coming to a stop, and near the end of their range they produce many vacancies (and naturally interstitials). Since the .solubility of hydrogen and helium in most metals is smallV»oat of the gas precipitates as bubbles in this terminal region which. subsequently grow by capturing the large.awfeer of vacancies in the neighbourhood. Eventually the bubbles develop e^fyigh pressure to deform the metal •surface causing a blister-like appearance (see also Pig. 6.6). * The problem of. radiation damage in the magnet material will be touched upon later• ; :\ -195-

" *V 111* vJ^ S&V ** f81

*W^,Jk^ MaWr*^ * ****** *'

Fig.6.5- Blisters on. rhenium produced by ion bombard­ ment . (After ref.1.1 ). -194-- I ill

DAMAGE U) ZONE

III 1

••••:.•••.•.• •*-. • • • ••. •••'••

BLISTERS

tc) «cu

Fig .6»6.. • Deve.lopnwnt • of /bli stera following bombardfltont VMifi heiiujalone. (a) ahowa; the initial Btagaait in (b) some of the Implanted helium atoms aid •; the. yacaneleB produood iqr them haw nucleated ihto.Baall Tjuhbleo. At high doses, the bubbles ooal-erude to form bliaters on aurfaoe afi ehown iij;(o). • -195-

( fb"\ Table 6.3 gives a typical estimate for wall erosion.v , ' It must he remembered that ajjart from erosion, blistering and sputtering can cause contamination of the plasma by metallic ions, leading to quenching of fusion. (Recall Sec.2.17).

In addition to damaging the surface, the neutron radiation in a fusion reactor can also cause considerable bulk damage to the structural material. While the.-total 14. 2 flux ( ys\ 10 n/cm sec) will be smaller than in fast breeders, the ratio of excess neutrons produced per megawatt in a D-T reactor to that in a fission reactor will be larger by a factor of two or three. More importantly, in a fusion reactor, most of these nentrbns will leayo the production zone and impinge on the (Ktrucfctiral material. Ihjtte, together with the higher energy of fusion neutrons, can lead to considerable bulk damage. One effect of neutron irradiation which may be immediately anticipated i/3 embrittlement or loss of ductility, a phenomenon already known from experience with fission reactors. In addition, neutron irradiation studies carried out in experimental fast breeders have shown that void formation occurs leading in turn to swelling. Notwithstanding these pointers, quantitative estimate of neutron damage in fusion reactors will require extensive studies, particularly in view of the fact that there is already evidence of a qualitative difference in the pattern of damage caused by fast neutrons in reactors and 14-MeV neutrons. For example, using fairly intense beams of 14-MeV neutrons (fluence i>, 10 ^ n/cta ). Merklev ' has shown that damage in this case ia ^reduced -196- ENfeRGY (MeV) K 10 5 2 1 05 0-2 CM 005 002 i i I i 'i i "-T—r i 20 interva l

X 10 XEBR n

• Flu x /Uni t letharg y _T WISCONSIN CTR ' 0 'T""fff"T" {3 12 3 U 5 6 7 LETHARGY

Fig,,6,7. Comparison of the neutron spectrum of a typical fast reactor and that expected.in a fusion reactor. For convenience, the x axis is given in terms of U the logarithmic energy decrement (U = In (E/E) where EQ is the energy of the neutrons) produced by fission or fusion as the ca3e may be. (After Kulcinski et_al ref.12). -197-

Table 6.3 EROSION RATES OP FIRST WALL IN WISCONSIN FUSION REACTOR DESIGN (U.WMAK - I). #4«N WALL MATERIAL IS 316 - S3 (20# COLD WORKED)*1 ./

.,. Mean Energy Flux Er'oSion Rates Ion \ (keV) (cm2/sec) (mm/yr)

Sputtering D+ 23 ,6.4 x 1013 0.0047 T+ 23 6.4 x 1015 0.0070 He 23 4.7 x 1012 0.0026 He 100 1.7 x 1011 0.00002 n 10,000 9.4 x 1013 0.14* 14 n 0.1 to 10,000 3V4 x 10 .0.022* Metal 23 2.6 x 1012 0.023 Total Sputtering 0.20

.Blistering He 23 4.7 x 1012 0.017 He 100 1.(7 x 1011 0;0019 D+ 23 6.Vx 1013 0.0023 T+ 23 .6.4 x 1015 0.0023 Total Blistering 0.024

* Includes both sides of the wall. -198- raainly in the form of oascades. These are initiated "by the primary recoil atoms v/hich can absorb several hundreds of keV of energy from the incident fast neutron. Typically, a cascade has a central vacancy-rich region (depleted zone), with interstitials residing in the form of clusters in the periphery. The excess vacancies can rearrange into more complex structures. Merkle's investigations have also revealed that recoils from neutron inelastic scattering i.e. (n, n ,"^ ) events can dominate damage production. In addition, sub-cascade formation is found to be prounounced, with upto six sub-clusters per cascade.

The net effect of radiation damage will of course be to reduce the life of the structural components. For example, if niobium were used, the life would be of the order of ten years or perhaps even less. The damaged components must evidently be replaced which raises the question of material for fresh fabrication. Again taking niobium as example, world resources of this element being rather limited, one must eventually think of reoycling, a process whioh would require dealing with material having residual radioactivity.* In other words, remelting, refining, casting, rolling etc. would all have to be done by remote operation.1 Fortunately, recycling may not be necessary for at leasjb a hundred years after fusion power comes into being so that some "coaling" of the components of at least the first generation reactors may be expected before they are taken up for refabrication.

$ The naturally occuring isotope of niobium is Wtr* with 100^ abundance. The radioactivity induced will include Nb92- 10.1 days, Nb94-- 2 x 10* years and Nb94*(isomeric state) - 6.6 min. -199-

6.7. Blanket Problems Including Tritium Breading

The blanket of a fusion reactor must not only absorb the fusion energy and transfer it to the power generation system, but must also cater to tritium breeding. In principle these two functions are separable, but in practice the two are considered together in the design and in fact one has to decide which aspect one is trying to optimize - tritium breeding or power'extraction. Breeding is based entirely on the following two neutron induced reactions in lithium. Li6 + n —^ He4 + T + 4.78 MeV „Li7 + n ^-> n« + He4 + T - 2.47 MeV (6.1) It is relevant to note in this context that naturally 6 7 occuring lithium has only the two isotopes Li and Li cited ^bove,, occuring in the ratio 7»5 : 92.5. World resources of lithium are'fairly 'plentiful if one takes congnizance of the lithium available in the sea in which case one can sustain a D-T programme for a long time.^4' (See also Table 6.4). Besides those cited above, there are several neutron reactions possible involving lithium isotopes as shown below:

Li6(n, n' 1 ) Li6 Li7(n, n» tf),11? Li6(n, n»ot) D -1.47 MeV Li7(n, 2n) L^6 -7.25 MeV Li6(n, 2nol) P -3.70 MeV Li7(n, 2no< ) D -8.72 MeV Li6(n,TO Li7 +7.25 MeV Li7(n, 7f ) Li8 +2.03 MeV Li6(n, p) He6 -2.73 MeV (Li8-» 2He4) (He6--> Li6) Li7 (n, D) He6 -7.76 MeV -200-

Among these, the Id'(n, a' V) Li reaction alone is important (from the point of energy deposition in the blanket).

Table 6.4 WORLD RESERVES OP LITRTUM^

From known resource* ?ro» undie- of Li ; • covered sources (gueastiraate)

Energy Available using lithium for "breedin g - 106 _ 1fl6 Q tritium (50(50 * efficiency 3 x 10 Q • 3 x 10 Q for fusion)

No. of years at 2.8 Q 1Q6 .-6 per annum

Note: 1 Q = 10 Btu » 1.06 x 10 J. Current world energy consumption per year is »*0.17 Q

c For breeding, the Li (n, (X) £ reaction is undoubtedly the more important of the two reactions in Eq.(6.1) but, for large neutron energies, the contribution to'tritium production by the inelastic scattering reaction Li (n, n'oC) T is expected to be significant. (See also Fig.6.^)/U) It must be clear from above that lithium in some form or other must form an integral part of the blanket , system. Designers accommodate ~this by incorporating lithium into the coolant fluid. The latter au»t Beet the following criteria: It must -201-

OpE—I—r I ! I illlj 1 I I Ultlj 1—I I I Mil) 1 I | 11| '£

Li(n,n'o)T

_ 6Li(n,a)T in 0.

THRESHOLD FOR

7Li(n, n'ooT 0.0! j i 11 mil i i » inn ± ' Him 0.01 0.1 I 10 00 NEUTRON ENERGY, MeV

Pig.6,8. /iicrgy dependence of the cross-section of the two important reactions in lithium pertaining to iritium breeding. (After Cairna et al. ref.14). -202-

(i) "be a good heat transfer fluid with adequate hydrodynamic properties even in the presence of large magnetic fields, (ii) he non corrosive with respect to "blanket materials and the materials of construction used in the power generation system, (iii) withstand high doses of radiation (iv) have adequate neutronic properties (i.e. nuclei other than Li in the coolant must have relatively low absorption crosssectaron for neutrons), and (v) "be of such a nature as to permit easy recovery of tritium.

Table 6.5 TRITIUM BREEDING- CALCULATIONS^15)

Total radial ThioTo^e.BS-of "blanket -co Material 21 39 ' :ltl0 84 96

Lithium metal

(a) 0.264 0.578 0.934 1.232 1.453 1.528 (T>) 0.254 0.503 0.668 0.722 .0.736 0.737

(a) - Total Tritium atoms produced per incident neutron (h) - Tritium atoms produced "by Li'(n, n'oC)T per incident neutron -203-

Understandably, molten Lithium has emerged as a popular choice especially in view of the experience already available with liquid sodium in fast "breeders. Table 6.5 shows typical results for breeding calculations for a (15 blanket employing molten lithium. ) . In passing, the •7 sizeable contribution from Id (n, n»o6)T may be noted. Several other calculations of a similar nature have been performed, and it emerges That breeding is improved if some neutron moderator like beryllium or graphite is employed in the blanket. At first sight it would appear that lithium itself could play the role of the moderator, especially as the average logarithmic decrement in neutron energy per collision is larger for Id than for C and Be. (As a result , the number of collisions required for thermaiization is smaller in the case of Id, the comparative figures being Id - 67, Be - 86, 0 - 114)* However it turns out that the moderating ratio i.e. ( j£lTC£/H'C&) (where 1ST is the number of nuclei per unit volume, and (y and (j^ are the scattering and absorption cross-sections respectively) is much higher for Be and C than Id on account of the large absorption in the latter. While some blanket designs envisage explicit moderator layers (of graphite usually), others incorporate the moderating nucleus in the coolant by choosing for the latter, the mixed salt LiF - BePg called flibe (LiP and BePg in "the ratio 2:1 - usual notation -

Li2BeF '). Besides having a built-in moderator, the molten salt also enables the overcoming of the MHD problem associated with lithium metal flow i.e.! the turbullanc© and the consequent pressure drop that arises when one tries to pump a conducting fluid at high velocities through a magnetic field environment. It haB been suggested that the -204-

MHD problem can "be circumvented "by either making the pipe itself out of a semi-conductor or insulator, or alternatively, coating the inside of the metal pipe with suitable insulator compatible with Li. Several materials like Sm203, ThOg, YgO^ have been suggested but detailed , studied on compatibility with lithium are yet to be carried out. The relatively low electrical conductivity of flibe eliminates the MED problem but in its place there arises another] This refers to the generation of an emf between the wall and the fluid on account of Hall effect. Though. LiF and BeFp a:re very stable compounds, the induced emf which can be as high as 4 volts, can cause an electrolytic decomposition of the coolant, accompanied by pijjie erosion. To some extent the problem can be controlled by proper choice of pipe material, dimensions, fluid velocity and the field environment. In passing it may be noted that some experience in molten salt technology is already available from the spade work done in connection with the Molten Salt; Breeder Reactors (MSBR). It may also be remarkedihat the mean radiation load on flibe in a fusion reactor will be more than 10-fold lower than what is expected;)Jia a MSBR. This comparison is, however, deceptive for while in a MSBR the radiation load will be relatively uniform throughout the core, in a fusion reactor blanket the dose will be very high near the vacuumi wall, being in fact ouch higher than that in a MSBR. Nevertheless, even this density is not likely to approach and maximum radiation density at which molten salts have been tested.^ '

The problem of tritium recovery requires to be tackled at two levels (i) recovery of unburnt tritium from -205-

the "ash" or unburat fuel, and (ii) recovery of the tritium bred in the blanket.

Two methods have been prop^edV(i 7 V' for recovering T from the ash, the first of which, is. schematically illustrated in Pig.6.9\ The principle involved is "cryogenic super fractionation" i.e. to first liquefy, the gaseous ash and recover tritium by fractional distillation. Helium poses no problem at all on accouat of its high volafcaltty, and can be eliminated even at the liquefier stage. An alternate proposal seeks to exploit the fact that hydrogen and its isotopes permeate most metals at elevated temperatures and that the permeation rates are different if or different isotopes. Thus it might be ekpected that a properly designed multistage permeation unit c/ould make a sharper cut distinction between hydrogen and other isotopes than would be possible with fractional distillation which separates molecules and not atoas. Preliminary studies, have indie axed that permeatioa cells using Pd or Pd-Ag are feasible...

The technique used for tritium recovery from blanket must naturally depend on the coolant. The system proposed for molten Li is schematically illustrated in Pig.6.10. Here the tritium produced diffuses into the coolant channel through the metallic walls, and is swept by the helium stream. Subsequently, tritium is recovered from helium essentially as discussed earlier. One problem likely to be faced in this method is the possible embrittlement of the metallic wall consequent to the dissolution of hydrogen. The problem is aggravated by the -206-

REACTOR DIVERTOR UQUEKIER -•He ASH

TO INJECTOR MAKE I UP —> • I • < FRACTIONATOR

VD2 V°2 H2.HD

Pig.6.9. Flo;, chart .illustrating the recovery of tritium from "aoh" by cryogenic fractionation.

r BLANKET

BO \ HELIUM PUMP ^

DRAG STREAM

f2 TO REACTOR

Pig,6.10. Schematic -illustration ot The recovery of tritium from the molten lithium in thJ blanket. •207-

fact that i3otope of hydrogen involved is tritium because the decay product of T is He which becomes immobilized in the metallic lattice contributing further to embrittlement.

In the case of the molten salt (LipBeF,), one saeks to exploit the fact that presumably, the tritium on generation, would combine immediately with free fluorine to form TP. The latter is carried away by a helium stream and passed through beds of KF where the following reaction takes place

TP + KF —>- H?2 *

The acid fluoride formed is then separated and tritium is eventually recovered from it.

6.8. Magnet Problems

As already indicated, MCTR's require big magnets capable of providing high fields over large volumes. The conceptual design of MCTR's has proceeded on the assumption that such magnets will be available, the assumption being based on extrapolation of current experience and technology. Although in principle the desired fields can be achieved with suitable scaled up versions of conventional wetter- cooled copper-coil magnets, it turns out that the electrical power required would be roughly equal to the entire useful output of the power plantl Fortunately, the possibility of using superconducting magnets provides a way out.

A superconducting msgnet is essentially similar to a copper coil magnet with the difference the coil is wound -208

out of a superconducting material. On account of the aero resistance of the superconductor, the coil can support very high current densities facilitating thereby the production of high magnetic fields. The principal advantage of the superconducting magnet over the corresponding normal maget is that joule heating is avoided in the windings, permitting the generation of high fields without excessive expenditure of power. Figure 6.11 which compares the various types of magnets in terms of their power consumption •x (for field volume <-o 500 cnr) clearly e*. Sablishes the superiority of the superconducting magnet. Another advantage is that except for start up, the magnet may "by suitable design, he operated in the persistent-current mode without drawing power from an external supply. The current can persist in the loop of the windings almost indefinitely. There are, however, some drawbacks. Fitstly, power is required for refrigeration, and this can be substantial (though not as much as for conventional magnets). Secondly, magnetic fields tend to destroy superconductivity when they exceed a critical value.

Now most superconducting elements exhibit "ideal"

superconductivity, and for these, the critical field (Hc ) required to destroy superconductivity is rather small being v» 1 kilo Gauss. Clearly such elements are not useful as magnet materials. Fortunately, certain alloys and intermetallic compounds can carry t*uper currents for high fields. In such substances, there exists a critical field He beyond which the system goes into a "mixed state" i.e. in which superconducting and normal regions coexist. The currents choose the path along the superconducting regions evaHsntM Possible Present Maximum Steady Field Possible

Superconducting Magnets ;opper Solenoid

i sro 2 Ton Iron Electromagnet

Ainico Permanent Magnet I- I i 10 100 1000 10,000 MAGNET POWER (kW) Fig.6.11. Log-Log.plot of magnetic field v/s power for different types of magnets, The field volume is 500 cm-5. -210-

and the maxerial is still without resistive loss until an upper critical field Hc2 is reached. Prominent among these so-called Type II superconductors are Nb-Ti alloys.

The current carrying capacity of a.superconductor is the most important property as far as magnet applications is concerned. It is expressed as a critical current density J which is the 'current it can sustain per unit cross section without becoming normal. In a Type II super

conductor, for fields H in the region Hc. *<" H < H£ 9 the flux can penetrate the sample and reside in the non superconducting regions* A ourrent J flowing through the sample will exert a ioreritz force ^ J x H on each flux line causing them to move and thereby generate heat unless +.he flux is pinned "by suitable metallurgical treatment. Even if pinning centres are available, flux lines can jump due to possible thermal "disturbances. To minimise such jumps, the superconductor is divided into a large number of fine filaments and embedded in a copper matrix, a technique known as stabilization. _ The .copper matrix provides mechanical support as well as efficient heat leak. (It may be noted in this context that superconductors are poor heat conductors). Figure 6.12 shows a wire made by Imperial Metal Industries containing 13255 Nb-Ti alloy fillaments in a copper matrix, with partitions made of high resistance Cu-Ni alloy.(18^

Fairly large superconducting magnets for high energy physics research have already been built, and the characteristics of a typical unit at the National Accelerator Laboratory, Batavia U.S.A. are given la -211-

Fig.6,12. Photograph of the cross section of a raultistrand wire for superconducting magnet. (After Sychev and Zenlcevich, ref.18). -212-

i'able 6.6.^ ; Figure 6.13 gives an idea of the gap that exists "between present available magnets and those required for fusion plants. The basic criterion governing magnet requirement for fusion reactors ±3 the cost of the magnet. In fact for steady state systems, the main cost of the reactor is expected to be associated with the magnet and associated support structure. Now the volume of superconductor required to generate a given field, increases in proportion to the surface area of the volume enclosed. It is advantageous therefore in optimising cost, to arrange for the confinement volume to be as large as possible, consistent with stability requirements. A rough estimate then leads to a field requirement un 100-150 Kg.

One important factor pertaining to long term performance of the superconducting magnet is the effect of radiation damage on the superconductor. This problem, already of interest to magnets used in high energy physics, acquires a special significance in the case of magnets associated with fusion systtems. The potential damage to magnet components can be divided into three categories M (i) reduction in T_, (ii) decrease in J_, and (iii) increased resistance of the copper stabilizer. A study of several Hh and V compounds showed that when exposed (at 60°C) to a fast neutron flux (>1 MeV), the transition temperature T drops to 1/lOth value of the unirradiated specimen for fluencies i^ 10.,.* •" Presumably a sinilar effect may be expected even for refrigerated specimens. Figure 6.14 shows a typical plot of the dependence of J„ on irradiation. ' Damage cannot perhaps be totally avoided but could be minimized by interposing appropriate -213-

TT i—i—r

PROJECTED FUSION MAGNETS

PRESENT SUPER COND. MAGNETS

' I. ' L wJ 0 30 60 90 120 150 180 FIEtD faG) Pig.6.13. Figure illuatrating the gap in *agnet technology. -214-

L ,19 20 10s1"7 100~8 10 10 FLUENCE (n/cm2iE>1MeV)

.6.14. Variation of critical current at 4.2°K and 40 KG as a function of fast neutron dose for Ifbli rault if ilament composite. Irradiation done at around 60°C. -215-

gable 6.6 CHARACTERISTIC OF NAL BUBBLE CHAMBER MAGHEE

Magnetic field induction 1.8 Tesla Inside dia. of winding 478 cm Outside dia. of winding 528 cm Length of winding 304 cm Stored energy 80 MJ Current , ! . 2000 A Current density 1606 A/oa Weight of windings 45 tonnes Cooler capacity 500 W Cost $2.5 x 106

radiation shields. ObvitJitrs-'-cahdidates for shielding materials are those which can slow down the neutrons and subsequently absorb>|hem, in addition to providing shielding,. Metallic hydrides with high dissociation temperature have been suggested, ammg them zrR,, YIL,, (21V

6.9. Power Generation

Power generation schemes proposed for fusion reactors generally tend to follow that adopted in liquid metal fast breeder reactor i.e. the heat is transferred from the coolant fluid to water via,aa;;appropriate sequence of heat exchangers, and the steam /produced is made to -216-

operate turbines. To optimise the proceteH, an intermediate potassium stage has also been proposed as sketched in Pig. 6.15'.

A direct energy conversion scheme has also been 22} suggested but specifically for mirror machines.(N ' The basic idea is to use the reaction

D + He5 _^> He4 + H where the energy is carried away entirely by charges particles, and to exploit the fact that in a mirror machine, particles tend to leak out at the ends. The required He must, by and large, be obtained from elsewhere, although 3 some use could be made of the HRe< resulting from possible D-D reactions (see Eqn.(1.1 )).

The direct conversion scheme is illustrated schematically in Fig.6.16, and envisages essentially four steps. First the particles escaping from the machine emerge into an expander chamber where the rotational motion of the charged particles is converted into a translations!, motion. This is accomplished by expanding the magnetic field in a fan-shaped geometry from the high mirror values of oa 100 Kg to about 500 G. Next the ions are separated from the electrons by employing a field folding which causes the (light) electrons to adiabatically follow the field lines and thereby get separated from the ions. The third step is io* collection, and is accomplished by electrostati­ cally decelerating and directing the ions on to collector elements maintained at appropriate potentials. This is the Li - K HEAT K TRANSFER Vapour r Pot. Vapour

STEAM TURBINE

Water Pig.6.15. Binary vapour cycle proposed for recovery of heat in fusion reactors. Heat extracted 'by lithium is first transferred to potassium which in turn transfers it to steam. CHARGE DECELERATION MAGNET REPARATION «. COLLECTION COIL ELECTRODE

TO MIRROR I ro MACHINE -~* 00 I

DC OUTPUT. Fig.6.16. ^^iJ^^&J^d^~3£ -219-

most difficult step "but work done to date suggests thai; collection efficiencies upto 90% are possible. The final step is conversion to a common potential, and employs a series of voltage reducers and Voltage multipliers as required. Observe that the system described corresponds essentially to a "Van de Graaff operated in reverse", i.e. high speed charged particles are made to pass through a decelerating electric field leading to a conversion of kinetic energy into electrostatic potential energy.

The principal attractive feature of direct conversion is that efficiency can be made almost arbitrarily high, the limiting factor being practical and economic rather than thermodynamic. Another important advantage of the He (d, p)H'- process is the absence of neutrons. This not pnly minimises hazards of'induced radioactivity, but also radiatioa damage.problems. Direct conversion can also be employed with D-T fuel, but in this case its role will obviously be secondary.

6.10. Possible Hazards Associated with Fusion Reactors

Understandably/ the first item deserving consideration under this heading is the hazard associated with radioactivity. The bulk of the radioactivity in a TNR lies in the tritium inventory and the neutron activation of the structural material. It is expected that the hazards due to human ingestion of such activity will be less than that associated with fission products, since they are not likely to be stored in bones like strontium or plutonium. Claims have been made ' that -220-

the hazard potential of a 5000 MV/tt) fusion reactor is equivalent to that of a 5 KW fission reactor but this has been disputed on the ground of being too speculative.^4'

One respect in which a TNR can be adjudged definitely superior is that whereas a fission reactor can undergo a possible undesirable power excursion due to built-in excess reactivity, a fusion reactor cannot have such an excursion. This because in a fusion reactor deviations from optimal operating conditions mostly quench the plasma, making reactor run aways impossible.

The closest possibility of a maximum credible accident exists not on the nuclear side but in the tritium system. A breach in the presaarized tritium system could cause undesirable discharge of activity into the environment, and a suitable containment is therefore required.

Another important non-nuclear hazard is that associated with the reaction of liquid alkali with water or air. This is a problem well known in sodium-cooled fast breederso

Closely related to the radioactivity hazard ic the energy released during the emanation of radiation, and the possibility that tins -might lead to vaporization and • dispersal i.e. the after-heat problem. It turns out that this is far leas severe as compared to fission reactors on account of the differences in the activity of the waste products. -221-

6.11. Laser Driven Systems An ICTR differs from a MCTR in two major respects: (i) there is no magnet and (ii) the euergy release is explosive I In one of the systems proposed, for example, one D-T pellet is fired every second, releasing 200 MJ of energy ( */* 100 lbs of TNTi) Apart from these two differences, the technological problems of ICTR's are not dissimilar to those of MCTR's.

Two concepts have been proposed for absorbing the energy released when the DT pellet is fired with the laser. One is the swirling-vortex conceptv( 21)' illustrated in Pig.6.17j. Here molten lithium .held in a spherical chamber is swirled with sufficient velocity as to form a free vortext around the vertical axis, extending from the top to well below the mid-plane. The frozen DT pellet is injected into the cavity, and ignited by the laser pulse when it reaches .the mid-plane. ?3nergy deposited in the lithium is removed by circulating the lithium as Illustrated. The geometry of the proposed scheme does not seen conducive for exploiting the implosion technique which requires isotropic irradiation. To attenuate the shock wave generated by the thermonuclear burn, the liquid is filled with gas bubbles, a process which enhances the compressibility of the medium and thereby facilitates dissipation of the shock wave before it reaches the chamber wall.

An alternate concept proposed at Los Alamos envisages an inner empty cavity where in the DT pellet is -223-

h fired, imploded if necessary (see Fig.6.10). The iiuaer

-..•all is thin and paroust a&& allows the passage of lithium to form a protective coating. Erery time a pellet is fired, the lithium layer ablates thereby draining away some of the energy of the shock wave and lessening its impact. Notwithstanding this, reverherations between the two walls of the blanket can be set up by the remnants of the shock wave. To minimize this, an intermediate structural wall has been suggested (see Fig.6.l6) which ia thick enough to damp the reverberations (and thereby prevent the motion of the inner wall) but thin enough as not to cause perturbations to the breeding. On account of the porocity of the inner wall, the lithium layer io restored by radial flow from the blanket after each pulse. The other technological problems of ICTR are similar to those of MCTR and therefore do not require separate discussion.

Questions may be raised as to the relative ?aerit3 of ICTR's and MGTR's particularly in relation to costs. Obviously it is too premature to compare their economics; even so, rough considerations suggest that costs will be comparable, the differences being determined largely by the relative cost of the laser system and the magnet sysxem. Since the former system is expected to be less expensive, the ICTR may turn out to have a small edge over MCTR, cost wise.

In terms of size, it is expected that individual ICTR«s can be made relatively "small", a typical unit having a power output of on 200 MWe. Thus a large power -223-

-Tn/UECTORYOf OEUTWM*TWTILM PELLET

LASER BEAM

UTHIUM INLET i

w %

Ik

VPMSSUREVtSJa'

AM BUSBIES.

tfTWLM OUTLET.

Pig.6.17. Illustration of the swirling vortex concept. (After Lubin and Traa3f ref.24). -224-

PcLLET INJECTKJN

POROUS, WETTED WALL

LITHIUM MAIN PRESSURE VESSEL BLANKET REGIONS INNER STRUCTURAL WALL AND FLOW BAFFLE

- TO STEAM GENERATOR

202-5 MW THERMAL PER UNIT

RECIRC PUM?

CONDENSATE PUMf

Fig.6.18. Schematic drawing of the laser-fusion reactor employing the wetted-wall concept. (After Booth, ref.4). -225-

station based on ICTR's will have several units operating in parallel. On the other hand, the basic Tokomak unit is expected to have a large thermal rating on account of the large plasma size.v( 25^' In this sense the fabrication and other related problem may be more severe for MCTE's. Actually it is not fair to push such intercomparisons too much since neither system has worked. It may even turn out that one of the sch»»es is a washout in which case question of comparison vill obviously-lose meaning!

6„12„ Concluding Reaarkg To sum up, the quest for fusion power is now rvt a sort of cross roads. Thanks to the arduous efforts of the past two decades, enough is known at present about the plasma to say with mature confidence (rather than with the misplaced enthusiasm that characterised the early days!) that controlled fusioa will most likely be achieved and that fusion reactors will be & reality. " ]£xaidable problems still remain, many of them technological. Ifowever, in the view of many, the largest hurdle is the lack of -*««aajl%ttent. Current investments in KB are relatively "meagre", the annual worldwide «xp«nditure baing tr> $200 million. It has been -argaed that for a break-through, a much more massive of fart i« aquired, rather like the "man on the moon" project. If this ia. done, then there is no question that fusion power will bo tamed within about a decav e and that fusion reactors can become a eotnmerical proposition by the turn of the century. The lively anti­ cipation has been. fnr*her heightened by the recent advent of the laser-implosion idea. (The* newness af this scheme -226-

_and the rapidity with which it has caught on can be appreciated from the fact tha;fc at the 1969 Culham Conference on Fusion Reactors, laser-induced fusion received no mention whereas "by early 1972 there was even a detailed report on a ICTR based station!)

And so there is a big IF. In 1969, Tuck remarked at the Culham Conference, "the fusion research organization itself shows certain signs of strain and old age new comers and new ideas in the field have to be turned away for lack of funds .... the financial "climate for scientific research has been perceptibly cooler every­ where and controlled fusion research is not getting the support it deserves". The early seventies saw no qualitative change in the situation, and as late as October 1974 the Economist declared, "In most countries fusion programmes; are controlled by various national atomic agencies. The fusion laboratories make up minor departments contributing nothing to the main work of the agencies, which is still concerned with promoting ordinary atomic fission reactors .... The present arrangement is the equivalent of giving the shipyards charge of aircraft developments". However, thanks to the increasingly vocal clamour of the fusion lobby. It is quite likely that there will be a "wind of change", especially in view of the recent oil crisis. The elevation of the fusion programme from a small, sporadic and half-hearted effort to a massive and aggressively organized goal-oriented mission is thus very much on the cards. -227- REFERENCES

1. R.Carruthers etai.', Culham Laboratory Report, CLM-R-85 (H.M.S.O., 1967):"'. 2. R.V.Werner et al.. USAEC Report UCRL-72883, Lawrence Liver- more Laboratory (1971).

3. R.W.Werner, B.Myers t-\PwB.Mohr,v J.D.Lee.: arid N;C.Chri8tofi2os, in Proceedings of the Nuclear Fusion Reactors Conference 1969 (UXAEA, Culham Laboratory for BNES, 1970), p.449. This volume hereafter abbreviated Culham Conference Proceedings. 4- L.A.Booth, USAEC Report LA-4858-MS, Vbl.X, Los Alamos Scientific Laboratory (;1972). 5. Atom, June 1975. 6. Quoted by E.F.Johnson, in The Chemistry of Fusion Technology, edited by D.M.Gruen (Plenum, New YorK, 1972;. 7. D.J.H.Wort, in Culham Conference Proceeding,-fp.517. 8. R.F.Post, Ann. Rev. Nucl. Sci. 20, ,509 (19700..- 9. G.M.McCraven and S.K.Brents, Culham Conference Proceedings, P.353.

1 0. A.P.Fraas, Nuclear Technology 22_f 1 (197.4) 11. R.V.Nandedkar, private coniiiiuni cat ion.,, 12. G.L.Kulcinski, R.G.Brpwn, R.G.Lbtt. and P.A.Sanger, Nuclear Technology 22, 20 (1974). 13. Kj.L.M^rkle, Ifuclear: Technology 22,£6 (1974). ; 14-. E.J.Cairns, F.A.Cafasso and Y.A.Mardrii-. :in.Chemistry of Fusion Technology, p.91• 15- W.B.Myers et al...quoted by J.P.Lee, in The Chemistry of Fusion Technology, p.51. 1 6. W.R.Grimes and S.Cantor, in The Chemistry of Fusion Technology, p.161. -228-

E.F.Johnson, Ibid, p.191. V.V.Sychev and Y.B.Zenkevich, Atomic Energy Review 11, 711 (.1973). A.R.Sweedler, D.Cox, D.G.Schweitzer and G.w.Webb, BHL-19264, (1974). D.M.Parkin and D.G.Schweitzer, Nuclear Technology 22, 108 (1974). G«G.Lihowtiz, in The Chemistry of Fusion Technology, p.321.. R.F.Post, Culham Conference Proceedi^Sigs, p.88. A.P.Fraas, USAEC Report ORNL-TM-32'31, Oak Ridge National Laboratory, (1971). * M.J.Lubin and A.P.Fraas, Sc. Am. 224., 21 (1974). R.Carruthers, Culham Conference Proceedings, p.337. -229

APPENDIX

In this appendix^ we present a brief glimpse of the world effort in controlled thermo nuclear research. Figure A-1 gives the pictorial distribution of this effort. Some of the details are spelt out below, having been extracted from a report prepared circa 1975 by A.N.Beloze- rov of IAEA (published ;n Nuclear Fusion J_4, 109 (1974)).

JAPAN

Research in nuclear fusion and plasma physics is financed by the Ministry of Education and the Atomic . Energy Commission (AEC) of Japan. The scientific policy of nuclear fusion research is determined by the Nuclear Fusion Subcommittee which is an organ of the Japan Science Council, Recently, the AEC of Japan established an advisory committee for research and development in nuclear fusion supposed to make recommendations on the program covering the period from 1975. to 1979 inclviding initiation of large national projects» Although the annual growth of the financial support of nuclear fusion research is about 20% (barely compensating price escalation), a 200 or 300% increase is expected for large AEC projects. The money for the first large project, the construction of a DC fly-wheel generator of 10p MW power output, was granted recently; it is scheduled for completion by 1974-1975, and will be located in the Institute of Plasma Physics, Nagoya Univer­ sity- In the Japan Atomic Energy Research Institute, a JFT-2A tokamak with major radius ^ 60 cm and tear-drop-1ype cross-section is presently under construction, The next stage will be the construction of a large-scale low-beta torus and some cusp-type devices. The construction of the -230-

OTHERS U.K X 16-5% 14-3%/"/\ V/FRG ) y^ 7-9i%__ USSR *-*-*^3% 35% U.S § 22% /

1963 1971

(a)

1950 1960 1970 (b) (a) -hows the percentage contribution to fusion research by the -najor nations, (b) indicates th* funding by the United States ^Q_catea the -231- large-scale machine will depend on the discussion of the advisory committee of AEC after the results from JFT-2A are obtained *

FEDERAL REPUBLIC OF GERMANY

Two institutes are engaged in fusion research, IPP at Garching with a scientific staff of about 24O persons and the Institut fur Plasmaphysik der Kernforschungaanlage at Julich, v/ith about 50 scientists. Both institutes are associated with EURATOM. Basic problems of plasma physics are investigated at Ruhr-Universitat Bochum and University of Stuttgart (Institute for Plasma Research). About 10$ of all the efforts in the field of nuclear fusion is devoted to technological problems. .The major project is Wendelstein-

VII (W-VII) stellarator (R = 200 cmr r = 36 cm). The project includes the construction of a current pulse generator of 1*4 GJ of useful energy per pulse to maintain a main magnetic field of 4 Tesla (EM ^135 MJ) for approxi­ mately 10s. The time schedule foresees the delivery of the huge generator and the coils of the main magnetic field in April 1974. Experiments with the ¥-711 device will be started in 1976. One of the objectives of the experiment is the production of plasma conditions typical of large -tokamaks to allow a comparison of tokamaks and stellarators at similar dimensions. Various heating methods including RF heating will also be investigated. For this purpose, the WEffA device is being constructed with a geometry similar to that of the Petula tokamak at Grenoble (constructed at Garching and re-assembled at CEN-Grenoble), The construction of a high-beta stellarator is based on the results obtained on -232-

ISAE. T. 1 (at IPP), 'Jhe high-temperature plasma will "be produced "by shock heating using a new technique,, developed in the ISAR II experiments- Conceptual studies of tokamaks including tokamaks with non-circular cross-sections are being performed with the aim of an auxiliary experiment for the large European project. Experiments on laser-produced plasmas are being continued in IPP in two directions (a) for filling magnetic confinement systems, e-.g, W-II, V/EGA and W-VII; (b) for the development of laser fusion reactors, lor these purposes, a 1-kJ, 1-ns iodine laser is being constructed.

UNION OF SOVIET SOCIALIST REPUBLIC '

Major efforts are going into the tokamak devices. Before putting into operation the tokamak model of T-10 (scheduled for the middle of 1975) various aspects of plasma behaviour are being studied on existing tokamak devices. At the Kurchatov Institute,, experiments on injections of neutral atoms are planned for T~4 and T-6, HP-heating experiments on TM-3... effects of non-circular shape of the cross-section of the plasma column on T-8 and T-9f feedback stabilization experiments,, ion heating experiments on TO-1 . Plasma heating at hybrid resonance frequencies are being studied on the Tokamak-FT-1 at the Ioffe Institute. The experiment on the adiabatic compression of plasma are carried out on the "Tuman-2" device- Methods of dynamic stabiliza­ tion by means of a HP-field in the megahertz range are studied at the Suchumi Physico-Technical Institute. The design of -nhe main version of the tokamak model of T-10 has been completed at tne Efremov Institute, which in future, will be the principal Institute for fusion-reactor design -233- studies. It is believed that scientific feasibility of a tokaraak-"based reactor will "be established by 1980-1982. The Kharkov Physico-Technical Institute and the Lebedev Physics Institute are continuing their research on Stellarators. The major results of the past year was the achievement of a true 3tellarator regime for the first time by using the method of ion-cyclotron heating at Kharkov and injection of plasma produced by a laser beam in the Lebedev. Institute, new stellarator systems, the torsatron "Saturn" and "Wint-20" are being investigated with the aim of designing an efficient spatial divertor. In the Lebedev Institute, the construction of a rev; stellarator L-2 is in progress, its parameters are R = 100 cm, r = 17.5 cm, field strength 20 KG, maximum shear v-v 0.2 - 0-4-

Basic plasma research has been carried out at the mirror machines PR-5, PR-6 and Ogra-1, Ogra-2. Some experimental evidence on the existence of the drift loss-cone instability- mode was obtained on PR-5 and PR-6. It was shown that instability can be stabilized if a small amount of a "warm" low-density plasma i3 injected.

Along with the stationary systems, pulsed thermo­ nuclear systems have been investigated. New methods for generating high-current relativistic electron beams have oeen developed at the Kurchatov Institute (Zavoisky division). Currents upto 200 kA and particle energies up to 1 MeV have been achieved. At the Lebedev Institute experiments on the spherical compression of targets have been carried out on the 9-laser-beam installation (1.3 kJ with a pulse duration of 30 ns or 0.6 kj or 2 ns). It is believed that a 10-30 fold compression has been reached. The feasibility of the -234- reactoi' based on the laser-produced plasma is still being questioned,, mainly because of the apparent low efficiency of such a multistage system, A high-density theta-pinch is being investigated at the Kurchatov Institute U3ing the "collapsing-linear" approach. For this purpose, a 50-MJ inductive energy storage is being developed to produce a field of 3 MG; 1000-fold compression temperature of — 1 keV -I Q •Z and densities of _~ 10 -* cm are expected.

UNTIED STATES OF AMERICA

On the basis of recent experimental success in the tolcamak devices, some alterations in the OIR-program have been made. They should result in a shortening of the overall time scale needed to complete a demonstration power plant, Thus,, the former estimate that a commercial fusion power plant can be built by the year 2000 has been advanced to 1995- The magnetic confinement program will go along in three major directions. About 60>b of all the research ( <-n US $50M) will be devoted to the low beta toroidal systems,, mainly tokamaks. It will include demonstration of confinement at reactor-like conditions and exploration of alternative toroidal configurations as well as the testing of some engineering featiires such as divertors and fuelling systems, design of D-T burning system for implementations earlier than originally planned. The rest of the effort will be divided equally between high-beta toroidal systems and mirror machines. The large experiments are generally being planned in such a way that they allow, first operation on hydrogen and then be converted into D-T burning experiments. -235

Pour major experiments are presently receiving top priority: PLT, ORMAC, SGYLLAG and 2-XII mirror machine. Plans for the construction of one or two new machines larger than PIT in the next two years are being discussed. Low-priority experiments such as ASTRON, super-conducting lEVITRON and IMP (Injection into Microwave plasma) are terminated. About $30M will be spent on the laser-produced plasmas- This research is sponsored by the AEC's Division of Military Applications. It includes the development of large neodynium glass laser systems. Chemical lasers which, because of their high efficiency, appear more feasible for CTR applications are also being developed. Theoretical studies will also be expanded in order to develop methods for the simulation of the behaviour of plasma confinement and reactor systems. For this purpose, construction of a large computer centre exclusively dedicated to CTR work is planned. Presently CDC-6600 is considered for this purpose. For future researqh, a computer equivalent to CDC 7600 is contemplated. The estimated cost is $10 M plus $2.5 M a year of operating costs.

UNITED KINGDOM

Two major developments have taken place in 1975; Firstj, the decision to increase the level of effort over a three-year period from 1973r-1975 (£17 M total expenditure and rise in the number of professionals from 160 to 200) by about 30^, and secondly the integration of 'the UK fusion research into a common research program of EEC and EURATOM. -236-

In 1975; studies of CLEO Tokamak have been continued at Gulhaa Laboratory, Stability of the plasma can be achieved for up to 100 ns, and first experiments on neutral injection heating have been carried out successfully, A large tokamak DITE is being built to investigate divertor, neiitral injection heating and bootstrap currents, Design studies for a large EURATOM tokamak machine operating with plasma currents upto 3 MA also have been performed at Culham Laboratory. She CLEO stellarator will D3 in operation in 1974 with the main aim of comparing it with tokamak' confinement„ The High-Beta Toroidal Experiment (HBTX) has achieved MHL-stability for values of beta of about 0.5. It is not clear, however, how HBTX can be incorporated into the fusion reactor. The construction of the Super-conducting Levitron has been completed, and initial physics experiments have .been performed with the plasmas produced by a fast- neutral-particle injector. The decision to start intensive research on laser-produced fusion will be made after review of the USA-USSR results in this field- Meanwhile small-scale work is being done in the development of high- power lasers.

AUSTRALIA

About 30-40 scientists are engaged in plasma research.- Low-beta toroidal confinement experiments have been carried out in the Liley torus devices. The. construction of a new device LT-4(R = 50 cmr r = 14 cm, maximum toroidal field 35 KG) will start this year. The Canberra homopolar generator (1,6 MA, 800 V for a part of a second), which is now under construction, will be used to -237- provide the energy for the toroidal field. High-beta work is done in co-operation with Los Alamos Laboratory.

PRANCE

Two approaches are bei^ig investigated within the CEA: magnetic confinement and laser-driven fusion. The latter is subsidized from military funds. The tokamak program includes the start of the experiments on Tokamak- TFR (R = 100 cm, r =' 20 cm, magnetic field 60 EG, plasma current 400 kA) whose construction has been completed in spring 1973. Further improvement of the TFR is planned for 1974 with the installation of neutron injection for additional plasma heating. The injection methods v/hich are being developed for this purpose will also be used for the construction of a neutral-injection source with long injection times of 10-20 s, for the Wendelstein-VII stellarat or at Garching. Two other toroidal machines are now under construction at Grenoble laboratory; PETULA of the tokamak type and.WEGA of the stellarator type, the latter being a joint project with PPL of Garching. Various heating methods will he studied on these devices, There are plans for closing down the research on the existing, three mirror, configurations in -1974.

EURATOM

Fusion research is based on the national programs of the member states. According to the investment program for the years 1974-75 which has been prepared in collaboration with the associated laboratories, a major -238- -•• effort, approximately 60$, would be directed towards the "low-beta closed configuration", mainly tokamaks, .15$ to the high-beta closed configurations, 10$ to heating and injection, and 5% to open systems.

Plans to start the design of a large tokamak (plasma current ^ 5- MA, field 30 - 60 KG),' as recommended in the final report of the European Working Group, has been approved by EURATOM. An international team (approximately 25 professionals and supporting st'aff) is now being called together for the preparatory'-phase. At the end of this stage (in two years), the final decision will be made of whether this project is to be continued or not= The estimated cost, of the project is $50 M. "

ITALY . '

Construction of .a tokamak-type facility (R = 83 cm, r = 21 cmt magnetic field 100 KG, current ^ 1 MA) is in progress. The plasma-diagnostics that is going to be used is also being organised.•. A new larger plasma-focus facility with an energy of-1:MJ is being built as a joint effort with Culham and Julich laboratories. This device should be ready by 1974. Experiments on the three linear machines of the Plasma Physics Laboratory in Milan are being continued, the main aim being the study of various methods of plasma heating, '

NETHERLANDS Pour major lines of research are being explored in the EURATOM-FOM institute: (1) toroidal screw-pinchP •239

(2) gas-blanket experiment for fusion-reactor designs, (3) interaction between plasma and R.F. fields (this program is "being de-emphasized now), (4) turbulent heating. In the Amsterdam division of the Institute, 'beam plasma interaction experiments with a 0,5 MY, 10 kA electron "beam are planned.

SWEDEN

Swedish fusion research is mainly carried out by the Division of Plasma Physics and Fusion Research at the Royal Institute of Technology in Stookholm. The main lines of research axe: (a) Study of the boundary region between plasmas and neutral-gas blankets for the densities in the impermeable range (of the order of 10 3 cm ), (b) study of confinement schemes based on purely polidal main fields including internal-ring devices with magnetically-shielded supports, (c) investigation of simpler and more flexible alternatives to minimum-average-B stabilization..