A confinement method with apparently more merit than the aforementioned is inertial confinement fusion, involving compression of a small fusion fuel pellet to high density and temperature by external laser or ion beams, Fig.4.1. The density — temperature conditions so achieved are expected to provide for a pulse of fusion energy before pellet disassembly. The incident laser or ion beam induces an
inward directed momentum of the outer layers of the pellet, thereby yielding a high density of material, the inertia of which confines the fuel against the fusion reaction explosive effect of disassembly for a sufficient time to allow enough fusion reactions to occur.

Laser or

Some critical characteristic features of inertially confined fusion can be described by the following. Consider, for this purpose, a d-t pellet at an advanced stage of compression with a plasma formed therefrom and fusion bum occurring. The tritium ion density Nt will, in the absence of leakage, decrease according to its bum rate

with the symbols here used as previously defined. In this context, we may further identify a triton mean-life, X,, by

with Nd = Nd,0 taken to be some suitable initial average deuterium population density and <Gv>dt is taken at some appropriate average temperature. Expecting the pellet disassembly to be rapid suggests that Ttu should also be small. Two possibilities of reducing Tfi, become evident from Eq.(4.3). One can enhance <Ov> by increasing the relative speed of the reactants which, however, is limited by the techniques of heat deposition in the fuel as well as by the more rapid disintegration at higher temperatures. The other option is to increase the fuel
density by several orders of magnitude. The reaction rate parameter <Ov>(lt is a maximum at an ion temperature of ~60 keV, and the density is a maximum at the onset of fusion bum which is also occurring at the time pellet disassembly begins.

A pellet, once compressed and with fusion reactions taking place, will evidently heat up further and hence tend toward disintegration. The speed of outward motion of the pellet atoms is, to a first approximation, given by the sonic speed vs which, in a d-t plasma, is given by

Here, у is the ratio of specific heats (7 = 5/3), Nj is the ion density, and m, the average ion mass.

As a characteristic expansion, we take the pellet’s spherical inflation from its initial radius Rb when the fusion bum began, to a size of radius 2Rb. During this process, the fuel density will have decreased by a factor of 23 and the rate of fusion energy release will have accordingly decreased by the factor (2 ) = 64; hence, most of the fusion reactions will have taken place during this initial stage of disassembly. Thus, the time for doubling of the pellet radius is taken as being representative of the inertial confinement time, Tlc, given by

Evidently, Tfu of Eq.(4.3) should be shorter than-or perhaps of the order of-Tic for sufficient fusion bum to take place so that, as a required initial condition, we must have

T fu ^ Tic

that is,

For the case of Nd,0 = Ntj0= Nii0/2, that is half of the compressed fuel density at the beginning of the fusion bum, we may therefore write the requirement as

ґ 1/2

‘ 4Ш,

Taking an average fusion fuel temperature of about Eth = 20 keV, we obtain by substitution,

Ni,0 Rb > 1024 cm2 • (4.8)

Some essential technical features of inertial confinement fusion may now be

qualitatively and quantitatively established. First, as will be shown in Ch. 11, the beam energy required to compress a pellet corresponds to the resultant heat content of the compressed pellet, and hence varies as RI. Existing laser beam powers are such that Rb needs to be kept in the range of millimetres or less. Second, the fuel density will evidently need to become very large, typically exceeding 103 times that of its equivalent solid density. Finally, the quantity of the initial fuel which bums up needs to be carefully specified since it relates to the overall energy viability of each fusion pulse as well as to the capacity of the surrounding medium to absorb the blast energy.

Thus, as an initial conclusion, we may assert that very high power drivers and very high compression is required for fusion energy achievement by inertial confinement. Further analysis of inertial confinement fusion is the subject of Ch. 11.

One of the simplest systems for magnetic containment is the pinch concept; here the plasma carries an electric current and is confined by the magnetic field induced by this current. As the current is increased, the larger magnetic field compresses the plasma and also raises its temperature by Joule-heating. Hence, confinement and heating is simultaneously provided. For that, extremely large currents (some 105 A) are needed thus rendering pinches to operate only in short pulses. The two principle configurations, denoted as z-pinch and 0-pinch, are sketched in Fig. 9.10.

V

b) 6 — pinch

Fig. 9.10: Representation of pinch geometries.

The confinement requirements are particularly obvious in the case of a z — pinch. In order that the plasma particles be retained, the electric currents must be high enough to generate a magnetic field of such strength that the magnetic energy density is able to balance the plasma pressure, that is

where T = Ті = Te has been assumed. The magnetic field induced by j can be found from Maxwell’s equation

VX —= j.

К

Using Stoke’s integral theorem, we then relate

= I

where ds is the path element along the circumference of the plasma column of cross section area A with dA denoting the oriented differential area element, and I defines the total electric current in the column. At the plasma surface, r = a, the path integral is

—2mlree = —Be(a) (9.49)

і к к

where e0 is the unit vector in the azimuthal direction.

Since В possesses only an azimuthal component, B0, the absolute value of the magnetic induction at the surface, is here given by

B(a) = ^~ . (9.50)

2na

Upon insertion of Eq. (9.50) in Eq.(9.46) and considering the particle numbers as referring to a plasma column of unit length, i. e. in the volume a27t X 1 m,

N*= N*+ Nl = a2n(Ni + Ne), (9.51)

we arrive at the requirement

2

=N*kT, (9.52)

8k

which is commonly referred to as the Bennett pinch condition.

Magnetic pinches are also troubled by instabilities. Two types of instabilities discussed here can occur in a cylindrical plasma carrying a large current along its axis.

One is the so-called ‘sausage’ instability, Fig. 9.11, which arises due to axial perturbations in the plasma column diameter and can be elucidated as follows. Consider the initial equilibrium state to be disturbed by some expansion of the plasma column to a larger radius. Since B(a) ~ 1/a, Eq. (9.50), the magnetic pressure at ai > a0 is weaker than that at the equilibrium surface (a0) and subsequently reinforces the expansion process. As these perturbations grow they make the plasma column look like a string of linked sausages and finally disrupt the plasma column. A restabilization of this pinch instability is possible by applying a sufficiently strong magnetic field along the axis.

Another characteristic instability in a cylindrical pinch is associated with

perturbations of the linear axial geometry, i. e., when the plasma column exhibits an axial curvature as illustrated in Fig. 9.12. It is seen therein that the magnetic field lines are closer together at the inside of a helical bend and hence the magnetic induction is larger there than at the outside of the bend. This difference between the magnetic pressure at the inside and outside of the bend causes any small disturbance of the straightness of the equilibrium plasma cylinder to grow until the plasma column scrapes on the surrounding wall. This unstable behaviour against bending is called a helical kink instability and can as well be controlled by an additional strong axial magnetic field.

Though a fusion device magnetically pinching the plasma is also seen to suffer instabilities which are mainly caused by the extreme electrical currents supplied, a 0-pinch can turn out to be remarkably stable. However, as for all open systems, linear pinches suffer from end leakage. To diminish end losses, a pinch usually features a magnetic mirror configuration, whereby also a minimum B — field structure is desirable. On the other hand, closed pinches can be realized when the plasma column is bent into a torus. Such a topology for magnetic confinement fusion will be discussed in the following chapter.

Problems

9.1 What assumptions are needed to make the previously stated magnetic confinement requirement, Eq. (4.12), consistent with Eq. (9.5)? Consider that the kinetic plasma pressure is largest at the centre and negligible at the plasma surface.

9.2 A magnetic mirror field, confining a fusion plasma with fuel particle densities Nd = Nt = 4xl019 m‘3 at Tj = 30 keV, varies from Btmx = 6 T at its throats to Вmin = 1 T at the mid-plane. Find:

(a) the number of ions escaping through the loss cones.

(b) an approximate value of the mirror confinement time.

9.3 Depict the trapping and loss domains of deuterium and tritium ions along their E0 = constant lines, akin to those shown in Fig. 9.4, for the mirror configuration of problem 9.2.

9.4 Attempt to sketch a loss cone distribution function accounting for the absence of particles with large Уц / vj_.

9.5 Discuss the development of the flute instability.

9.6 For each of the following open magnetic confinement fusion reactor concepts, draw a sketch of the concept, label the major components, and describe their purpose. Describe how fusion fuel ions are confined, and list what the energy and particle losses are in the system, and where they occur. Describe all the different electrical currents, magnetic fields, and their purpose. You may have to perform a literature search for some of the concepts.

(a) Magnetic Mirror

(b) Ying-Yang Coils

(c) Z-pinch

(d) Theta-Pinch

(e) Reversed Field Mirror

(f) Combined: Magnetic Mirror with Ying-Yang Coils

In addition to normal structural considerations, it is necessary to evaluate neutron-induced radiation damage and radioactivation effects in the selection of both structural components and coolants. As discussed previously, radiation damage occurs by atom displacement and by nuclear transmutation involving primarily those producing 4He; as expected the damage is most severe in the first — wall and associated structures on the side facing the fusion plasma. Atomic displacement rates and gas production rates are summarized in Table 13.3 for various materials placed in a neutron flux typical of the first-wall in a tokamak with a 1 MWm’2 neutron wall loading. The displacement rate is not strongly dependent on the type of material whereas the gas production rate is very sensitive to material choices. Nickel bearing alloys generally have a large ratio of gas-production and displacement rate. Lithium also possess a significant gas — production capacity but if this occurs in the liquid, pressure buildup and swelling are not a problem as it can be in solids. Neutron-induced transmutations in blanket materials also result in radioactivation which is most important with respect to reactor maintenance, and storage of reactor components. The level of radioactivation, along with other radioactivity aspects such as the tritium inventory, will be a key factor in determining the environmental impact of fusion reactors.

An illustration of the residual radioactivity of selected materials after a 2-year exposure is shown in Fig.13.10. The large variation in radioactive level and its effects with time for the various materials is a notable characteristic that must be considered. Inertial confinement fusion blanket designs using a thick liquid-metal

first-wall have a built-in advantage relative to minimizing activation of the chamber wall and structure. The falling liquid can reduce the neutron flux hitting this structure so that radioactivity levels are lowered by an order of magnitude or more relative to a dry wall.

Problems

13.1 Consider a neutron wall loading limit of 5 MWm’2 in a torus configuration with minor radius a = 2 m. If the plasma fuel ion densities are given by Na(r) = Nt(r) = N(r) of Eq. (6.54) where N(0) = 1020 m"3, what is the highest plasma temperature allowed?

13.2 Using the sputtering data for D+ at 100 eV bombarding Fe in Fig.13.6, evaluate К in Eq.(13.13). Sketch the sputtering-curve predicted by this equation, and discuss any differences with Fig.13.6. Compute the time it would require for a flux of 106 deuteronscm’^s’1 at 100 eV to sputter away 10% of the thickness of a 1 cm iron wall.

13.3 Consider a deuterium plasma at Te = 10 keV containing 1% oxygen. Estimate the emitted radiation power using a weighted sum of the powers from the individual species.

13.4 Estimate the percentage of iron impurity in a d-t plasma that would cause the ideal ignition temperature to double.

13.5 Evaluate impurity effects on the Lawson criterion.

13.6 Consider a 50:50% MCF device having Tj ~ Te, a plasma beta value of 0.2, |/ ~ 10’3 and a doubly charged-ion impurity concentration of 1% of N,. What is the modified ignition temperature (see problem 5.9)?

It is a common observation that matter and energy are closely related. For example, a mass of water flowing into the turbines of a hydro-electric plant leads to the generation of electricity; the rearrangement of hydrogen, oxygen, and carbon in chemical compounds in an internal combustion engine generates power to move a car; a neutron-induced splitting of a heavy nucleus produces heat to generate steam; two light nuclei may fuse and immediately break up with the reaction products possessing considerable kinetic energy. Each of these examples illustrates a transformation from one state of matter and energy to another in which an attendant release of energy has occurred.

These matter-energy transformations may be represented in various forms. For the hydro-electric process we may write

m(hi)-^m(h2) (1-1)

where m(hi) and m(h2) is a mass of water at an initial elevation hi and final elevation h2; the resultant energy E released can be evaluated by computing (mghi — mgh2), where g is the local acceleration due to gravity.

An example of an exothermic chemical reaction is suggested by the process

CH4 +202 -*2H20+C02 (1.2)

*

with an energy release of about 5 eV.

The case of neutron induced fission of a 235U nucleus is represented by

n+235t/->vn + £p; (1.3)

І

where n is a neutron, Pj is a particular reaction product, and n is the number of neutrons emitted in this particular process. Here, the total energy released possesses a slight dependence on the kinetic energy of the initiating neutron but

Appendix A provides equivalents of various physical quantities.

3

is typically close to 200 MeV.

The fusion reaction likely to be harnessed first is given by

2 H+2H —» n+4He (1.4)

with an energy release of 17.6 MeV. Accounting for the fact that the above species react as nuclei, we assign in a more compact notation the names deuteron, triton, and alpha to the reactants and reaction product, to give

d +1 —» n + a. (1.5)

The fundamental features of matter and energy transformation are thus evident. In the hydroelectric case, a mass of water has to be raised to a higher level of potential energy-performed by nature’s water cycle-and it subsequently attains a lower state with the difference in potential energy appearing as kinetic energy available to generate electricity. For the case of chemical combustion, an initial energetic state of the molecules corresponding to the ignition temperature of the fuel, has to be attained in order to induce a chemical reaction yielding thereupon new chemical compounds. The energy release thereby is due to the more tightly bound reaction product compounds with a slightly reduced total mass; such a mass defect is generally manifested in energy release-typically in the eV range for chemical reactions. In the case of fission, the initiating neutron needs to possess some finite kinetic energy in order to stimulate the rearrangement of nuclear structure; interestingly, the thermal motion of a neutron at room temperature is sufficient for the case involving nuclei such as 235U. For fusion to occur, the reacting nuclei must possess sufficient kinetic energy to overcome the electrostatic repulsion associated with their positive charges before nuclear fusion can take place; the alternative of fusion reactions at low temperature is also possible and will be discussed later. Again, in the case of fission and fusion, the reaction products emerge as more tightly bound nuclei and hence the corresponding mass defect determines the quantity of nuclear energy release-typically in the MeV range.

Evidently then, a more complete statement of the above processes is therefore provided by writing an expression containing both matter and energy terms in the form

Ein “t* A7 m ^ Eout 47 out (1-6)

with the masses measured in energy units, i. e. multiplied by the square of the speed of light. The corresponding process is suggested graphically in Fig. 1.1.

The depiction of Fig. 1.1 suggests some useful generalizations. Evidently, a measure of the effectiveness and potential viability for energy generation by such transformations involves microscopic and macroscopic details of matter-energy states before and after the process. In addition, it is also necessary to include considerations of the relative supply of the fuel, Mi„, the toxicity of reaction products, Mout, the magnitude of Eou, relative to Ein, as well as other technological, economic and ecological considerations. Additional issues may include availability of the required technology, deployment schedules, energy conversion losses, management and handling of the fuel and of its reaction products,

As a first fluidic description of a medium containing fusion fuel ions and sustaining fusion reactions, we consider a characterization which emphasizes particle mobility. Consider therefore an arbitrary volume V containing several time-varying particle populations Nj*(t), N2 (t), …, Nj*(t), …, each species characterized by some macroscopic kinetic property such as temperature. We take the volume’s surface to be non-reentrant and of total area A with its outward normal direction determined by the differential vector dA.

Our interest here is in the Nj (t) population which, in the case of a spatially varying number density Nj(r, t), is found from

N](t)=jNj(r, t)d3r. (6.3)

v

In general, the population Nj (t) can change with time because of various types of gain and loss reaction rates, R±j and because of inflow and outflow rates, F±j , across the total surface A. Hence, the rate equation for Nj*(t) is evidently

with the subscripts к and 1 enumerating different reaction types.

To begin, we now restrict ourselves to the case for which, in any fractional volume AV of V, the reaction contributions are zero or that the reaction gains are exactly canceled by the reaction losses; that is, we take

ад

к I

Additionally, we refer to the condition

F*+j = 0

which means there is no fueling by injection or other mechanisms, and we determine the outflow rate across the surface, accounting for global particle leakage, via the local particle current vector Jj(r, t) through

F. j = ij(r, t)-dk. (6.7)

A

As previously introduced, dA is an oriented surface element pointing outward. Obviously, particles leak out where J|A also points in this outward direction.

We now substitute Eqs.(6.3) and (6.5) — (6.7) into Eq.(6.4) to obtain for the space-time description of the j-type particle species

j-Nj(r, t)d3r = — J J j(r, t)-dA (6.8)

t V A

for the case specified by Eqs. (6.5) and (6.6). Then, in order to reduce both integrals to the same integration variable, we use Gauss’ Divergence Theorem

jjj(r. t) dA =jV-Jj(r, t)d3r (6.9)

Here, the partial derivative is used since the temporal change in Nj(r, t) is considered at fixed spatial coordinates, respectively. Since the volume V is arbitrary, we must therefore have

^-Nj(r, t) + V-Jj(r, t) = 0. (6.10b)

Э t 1

This important relation is known as the Continuity Equation and must hold everywhere in the volume of interest and for all times of relevance providing the assumptions imposed here hold. If, however, there were sources and sinks for the considered species j in the volume of interest-i. e. reactions which produce or consume j-type particles-or particle injection, the corresponding gain and loss rate densities would appear on the right hand side of Eq.(6.10b).

Though compact, Eq.(6.10b) however suffers from a severe shortcoming: it represents only one equation containing two unknown functions, the scalar particle density Nj(r, t) and the vector particle current Jj(r, t). Evidently, another relationship between these two quantities is necessary. As it turns out, this is often possible.

Many particle fluid media possess the property that the local particle current J(r, t) is proportional to the negative particle density gradient, — VN(r, t). This is often called a diffusion phenomenon and associated with the label Fick’s Law. Introducing a proportionality factor D, called the diffusion coefficient, we may write therefore everywhere in the medium

J j(r, t)=-DjVNj(r, t)

so that, by substitution into Eq.(6.10b) we get

Nj (r, t) + V • [- DjVNj (r, t)] = 0. (6.12)

Numerous reductions are now often applicable. If the medium is homogeneous and isotropic then the diffusion coefficient is a space-independent scalar, and we obtain

^-Nj(r, t)-DjV2 Nj(r, t) = 0. (6.13)

ot

Steady-state conditions in such a medium will yield Laplace’s Equation

V2 N j( r, t) = 0. (6.14)

The diffusion coefficient in Eq.(6.12), Dj, is clearly of importance in specifying the spatial variation of the particle density Nj(r, t). Considerable thought has gone into analytical characterizations of this parameter so that the diffusion processes be adequately incorporated. For example, for neutral particles in random thermal motion and in regions sufficiently far from boundaries, it has been found that, to a good approximation

Dj°cvjA, j (6.15)

where Vy is the mean speed of the j-type particles and A. j is the mean-free-path between scattering collisions among the j-type particles-and is much smaller than

(Vn/n)"1.

In contrast to neutral particles, charged particle motion in a magnetic field may involve considerable anisotropic diffusion governed by the local B-field requiring that a distinction between a direction which is parallel or perpendicular to the local В-field be made. Relationships with a good physical basis in classical diffusion are

where COg is the gyrofrequency and Tc is the mean collision time.

By analogy to the mean-time between fusion events Tfu, Eq.(4.3), we take herein

and recall that the Coulomb cross section Gs varies as v’4, Eq.(3.15).

Further, substituting for the averaged square velocity by the thermal energy E, h kT, or the kinetic temperature, respectively, and using the explicit

Thus, a higher temperature and an increasing magnetic field will reduce classical diffusion across the В-field lines. This conclusion must, however, be tempered by the recognition that collective oscillations in a plasma destroy some of the classical diffusion features. The plasmas of interest to nuclear fusion applications are severely affected by such collective processes through plasma instabilities which do not obey classical diffusion considerations.

The destabilizing oscillations induce turbulent phenomena which enhance diffusion across the magnetic field lines. Incorporation of these considerations leads to so-called Bohm diffusion characterized by

Db x ~~ (6.19)

D

thereby placing greater demands on magnetic confinement at higher temperature.

Finally, we add that the Continuity Equation for the particle density Nj(r, t), Eq.(6.13), translates into an equation for the mass density pj by using Pj(r, t) = Nj(r, t)mj,

^-Pj(r, t)-DjV2Pj(r, t) = 0 (6.20)

at

and is generally found to be of considerable utility, particularly in conjunction with other relationships as we will show next.

Some useful parameter estimates about inertial confinement fusion can be obtained by an analysis of selected particle kinetics and energy transfer processes. Consider, therefore, a spherically symmetric pressure wave converging towards the pellet centre. Suppose that ignition and bum conditions are attained when the radius of the compressed pellet is Rb and hold over a bum time tb, during which the pressure generated by the shock waves and the heating due to fusion reactions causes the pellet to expand to an extent where the density, and hence the fusion reaction rate, have decreased to insignificant levels.

At any time during this nuclear bum period, we take the total number of ions in the burning part of the pellet to be Nb. This total ion population decreases with time because of fusion reactions which occur at the rate

Here Rfu is the fusion reaction rate density with each fusion event destroying two ions; the integration is over the pellet bum volume Vb. Substituting for the fusion reaction rate density involving deuterium and tritium gives

= ~2j Nd(t)N,(t)<OV>dt d3r

Vb

The variables can now be separated and integrated to give

Nu л 1 4

= <U’5)

L nU>) 2 о

Here Ni>0 is the ion density at the beginning of the bum, t = 0, and Nj f is the fuel ion density at the end of the bum, t = xb. Integration and rearrangement of the terms yields for the bum time

where < OV >dt denotes the fusion reactivity parameter averaged over the bum period according to

____________ у ч

<(Tv>dt=—<CTv>dt(t)dt. (11.7)

TbJ0

It will be useful to introduce the symbol fb for the fraction of fuel burned during Tb,

Ni,0 — Nij

Thus

and we also use

Pb ~ Ni, omi (119)

where рь is the pellet density during the bum, m,- is the average ion mass, and the mass contributions of the electrons have been neglected, attributable to щ. being three orders of magnitude smaller than m,. Substitution then yields the explicit expression for Ть, Eq.(l 1.6), as

where Vis is the speed at which the core mass moves outward. The corresponding kinetic energy is of the order of the ion thermal energy, Eq. (2.19c), so that we may use

1 2 . Jtn vdis ■

and therefore we derive here

This specifies the conditions on the pellet density and pellet radius at the beginning of the fusion bum that are required for a specified bum fraction with the reaction occurring at some average temperature. A useful numerical value for

this parameter is obtained by taking < av >dt = <ov>dt (T = 20 keV) and an estimate for vdls also at this temperature, yielding for a 50% bum fraction

PbRb>3 g-cm"1 • (11.16)

For Rb ~ 1 mm this demands a density pb = 30 g-cm3 and is indeed very high compared to d-t liquid density of pe ~ 0.2 g-cm’3. Thus, the compression of the
initial fuel pellet by a factor of about 103 to 104-relative to liquid density-appears to be necessary for a satisfactory bum.

The hybrid reactor discussed above is characterized by a blanket which sustains both fissile fuel breeding as well as fission reactions. This latter function assigns some fission reactor characteristics to the blanket and consequently may impose similar safety considerations such as possible criticality or loss-of-coolant accidents and fission product release. In order to minimize these problems, it is possible to design a blanket which tailors the neutron spectrum in order to maximize fissile fuel breeding processes. The bred fuel could then be used in various companion fission reactors so that the fusion breeder reactor could be viewed as a "nuclear fuel factory" much as uranium mining and enrichment plants presently serve this function. Indeed, some additional fuel service features could be introduced if desired. For example, the fuel could be enriched in the fusion neutron-driven blanket to a desired level with a minimum of fission product accumulation-that is, rejuvenated in-situ while it is still retained in its cladding-for direct insertion into a fission reactor.

Another appealing extension can be conceived of by drawing upon the d-d rather than the d-t fusion reaction. Though the plasma conditions become more
demanding, the absence of tritium fuel breeding in the blanket and handling does render this fusion cycle very appealing and makes all of the fusion neutrons potentially available for breeding. The scheme is the following

We begin by recalling the d-d and d-h fusion reactions in addition to the base d-t reaction. If only deuterium fuel is supplied and if the reaction product tritium is consumed at its rate of production, then the overall fusion reaction cycle may be represented by

d + d —4 t + p

d + t —4 n + cc d + d —4 n + h 5d —4 2n + cc + p 4- h.

In the above processes, the charged products a and p will be, to a large extent, retained in the plasma and the two neutrons will enter the blanket and breed fissile fuel. While the energetic a and p serve only for plasma heating, the bred helium-3 (h) can either be recirculated in the fuel to provide added energy release via d-h reactions or it could be extracted and used as fuel for small fusion reactors optimized for the reaction

d 4- h —4 p 4- cc. (15.24)

The appeal in this kind of d-h satellite fusion reaction is that the fuels and reaction products are not radioactive and, additionally, neutrons are produced only diminutively by d-d side reactions so that the chamber walls would suffer less activation. Further, the energy associated with the charged particle reaction products would be suitable for transformation into electricity by direct conversion techniques.

These features suggest that small d-h fusion reactors might be placed near populated sites to provide radiologically cleaner and smaller size nuclear energy sources. One difficulty with the satellite approach, however, is that the amount of 3He bred by the hybrid is limited so that the satellite power would only be a fraction of that supplied by the hybrid-client reactor complex. Nevertheless, this could be attractive for specialized applications requiring small electrical plants.

Additionally, and as illustrated in Fig. 15.6, one may conceive of one large central parent d-d reactor simultaneously breeding fissile and fusile fuel for a distributed system of various fission and fusion satellite reactors.

Problems

15.1 If fissile fuel burning were to be incorporated as a fourth layer in Fig. 15.2, where would it be most effectively inserted based on neutron energy considerations?

15.2 Confirm the correctness of the Qfi / (1 — Cfi) term leading to Eq.(15.5).

15.3 Confirm that Eq.(15.21) follows from Eqs.(15.19).

15.4 Undertake a power balance analysis for the Symbiont/Satellite system of Fig. 15.6.

15.5 Estimate the d-h satellite to hybrid power ratio for the system described in Fig. 15.6.

The fusion reaction rate density expression of Eq.(2.23) is very restrictive since all particles were taken to possess a constant speed and their motion was assumed to be monodirectional. However, the general case of an ensemble of particles possessing a range of speeds and moving in various directions can be introduced by extending Eq.(2.23) to include a summation over all particle energies and all directions of motion. The integral calculus is ideally suited for this purpose requiring, however, that we redefine some terms. Letting therefore the particle densities be a function of velocity v endows them with a range of energies and range of directions; that is, we progress from simple particle densities which give the number of particles per unit volume, to distribution functions describing how many particles in a considered position interval move with a certain velocity, according to

Fig. 2.4: Fusion cross section for a deuterium beam incident on a tritium target.

The relative speed of two interacting particles at a point of interest is, by its usual definition, given as

Vr = Va — Vi •

(2.28)

By first impressions, this double integral appears formidable, particularly when it is realized that the vectors va and vb possess, in general, three components so that a total of six integrations are required. However, aside from any algebraic or numerical problems of evaluation, this integral contains two important physical considerations. First, the cross section Ofa(lva — vbl) must be known as a function of the relative speed of the two types of particles, and second, the distribution functions Fj(vj) must be known for both populations of particles.

We note however that Eq.(2.28) possesses all the properties of an averaging process in several dimensions; that is, it represents averaging the product oab(lva — vb)lva — vbl with two normalized weighting functions Fa(va) and Fb(vb) over all velocity components of va and vb. Such averaging yields the definition

< C7V >ab ~ J ^Oab^ya-yi>f[ya-ybFa(ya)Fb(vb)d3 Vad.3 Vb — (2.29)

This parameter, here named sigma-v (pronounced "sigma-vee"), is often also called the reaction rate parameter. Note the implicit dependence on temperature via the distribution functions Fa(va) and Fb(vb) so that <Ov>ab is a function of temperature.

The reaction rate density involving two distinct types of particles a and b, Eq.(2.28), is therefore written in compact form as

(2.30)

This sigma-v parameter for the case of d-t fusion under conditions in which both the deuterium and tritium ions possess a Maxwellian distribution, that is F()() —» M ()( ) in Eq.(2.29), and where both species possess the same temperature, is

depicted in Fig.2.5. Note that generally <Ov>ab will also be a function of space and time because the velocity distributions may also depend upon these variables.

We make two additional comments about the reaction rate density expression, Eq.(2.30). First, the assumption of Maxwellian distributions, i. e.

Appendix C provides a tabulation of this and other <ov>ab parameters.

F()() —» M()() as discussed in Sec.2.3, is very frequently made in tabulations of sigma-v; the reason for this is because many approaches to the attainment of fusion energy rely upon the achievement of plasma conditions that are close to thermodynamic equilibrium. For cases where equilibrium conditions do not exist, the appropriate distribution functions Fa(va) and Fb(vb) must be determined and used in Eq.(2.29). For example, in some experiments, deuterium beams are injected into a tritiated target or into a magnetically confined tritium plasma to cause fusion by "beam-target" interactions. In such cases, one substitutes Fd(vd) by a delta distribution function at the velocity vb(t) characteristic of the instantaneous velocity of the beam ions slowing down in the plasma. However, Ft(vt) for the plasma target could well be assumed to be Maxwellian at the temperature of the tritium plasma. Then the averaged product of (7 and vr is often called the beam-target reactivity < ov >d! for d-t fusion and is displayed in Fig.2.6; it appears to be a function of both the target temperature and the instantaneous energy of the slowing-down beam deuterons.

Secondly, Eq.(2.30) assumes that the а-type and b-type particles are different nuclear species; the case when they might be indistinguishable is treated in a subsequent chapter where, in addition to d-t fusion, the case of d-d fusion is also considered.

A demonstration of the occurrence of fusion reactions is readily accomplished in a simple experiment employing a small accelerator which bombards a tritiated target with deuterium ions of such a high energy that during slowing down in the target they pass through the most favourable energy range for fusion; consider, in particular, the tritium plasma target of Fig.2.6 for this purpose. The appearance of neutrons and alphas as reaction products at the proper energies is then the proof of fusion events. The objective of fusion energy research and development is, however, the attainment of a sufficiently high fusion reaction rate density under controlled conditions subject to the overriding requirement that the power produced be delivered under generally acceptable terms. This is a considerable challenge and to this end a variety of approaches have been and continue to be pursued.

Problems

2.1 Calculate the ratio of gravitational to electrical forces between a deuteron and a triton.

2.2 Determine the Coulomb barrier for the nuclear reactions d-t, d-h, and p — nB.

2.3 Confirm the correctness of Eqs.(2.17) and (2.19).

2.4 For Maxwellian distributed tritons at 9 keV, calculate

(a) the average kinetic energy,

(b) the average speed, and

(c) the kinetic energy derived from the average speed of (b). Compare the energies of (a) and (c), and explain any difference.

2.5 Transform M(v), Eq.(2.14), into M(E), Eq.(2.15), with the aid of the appropriate Jacobian.

2.6 Find M(E) from M(v) for the case of isotropy using spherical co­ordinates.

2.7 Consult appropriate sources to determine particle densities N (m’3), the corresponding energies kT (eV) and temperatures T (K) for the following plasmas: outer space, a flame, the ionosphere, commonly attainable laboratory discharges and values expected in a magnetically as well as inertially confined fusion reactor. Plot these domains on an N vs. kT plane.

A fusion energy system will contain energy in various forms. The relative magnitude of each form and an understanding of how changes in each energy component can be affected need to be considered and assessed. We begin with an overall system energy balance and subsequently examine selected details of energy forms and transformations.

8.1 System Energy Balance

A fundamental requirement of any power system is that it be a net energy producer. That is, during some time interval X, we require the overall net energy

Enet = Е*ош ‘ E*n > 0 (8.1)

where Ein* is the total energy externally supplied to sustain fusion reactions and associated processes and Eout* is the total recovered energy. Note that the asterisk notation is again used to indicate magnitudes referring to the entire reaction volume as opposed to the respective density expressions. While energy can exist in various forms-thermal, kinetic, radiation, electrical, magnetic, etc .-common units need to be employed in energy balance equations; for that, energy in electrical form is particularly useful.

In Fig.8.1, we depict a power plant’s energy flow in a simple general form. The energy eventually delivered to the reaction chamber differs from the provided input energy, Ein, by a factor T|in which, obviously, is the efficiency of converting the different forms of energy involved and of coupling specific energy forms into the fusion plasma. Analogously, the energy released from the fusion domain, which may consist of thermal energy as well as electromagnetic and neutron radiation, will have to be converted-for external use-into electricity with some overall efficiency Т|0ш typical for the employed energy conversion cycles.

Recall the connection between energy and power P0 = dE0 /dt. Indeed, since time variations are of dominant interest, particularly for pulsed systems, it is common in energy accounting to consider the time variations of power. Thus, the basic energy balance Eq.(8.1), may equally be written

We proceed now to identify individual power or energy components

It is useful to recognize that, unlike fission reactors which sustain a fission chain by particle feedback (neutrons) requiring only a critical mass, fusion reactors will need a critical plasma energy concentration. For that, energy has to be initially supplied to the plasma which, upon reaching the critical heat condition, will be capable of sustaining the fusion processes by self-heating due to fusion power deposition in the plasma. Note therefore, that the nuclear fusion chain is maintained by energy feedback.

The achievement of the mentioned critical energy concentration in the plasma is yet an ongoing task in fusion research. Depending on the fusion device and fuel cycle deployed, today’s fusion experiments are far from demonstrating energy self-sufficiency.

In general, the fusion plasma energy will be balanced by external (auxiliary) and internal (fusion power deposition) heating against the several losses due to radiation and other leakage. Considering the thermal energy content in the total plasma volume, that is we take here E = Eth*, we integrate its temporal variation

rate over a typical fusion bum time Tb to relate

tb, ч,

f Щ-dt = E*aux + Efu — El — Elad — f—dt (8.3)

о dt ‘ JoV

where Eaux* is the auxiliary heat supply, Efu represents the fusion energy production, En is the energy of fusion neutrons escaping from the plasma, and

Erad* denotes the radiation losses-all over time Tb; the last term of Eq.(8.3) accounts for energy leakage through plasma transport processes as suggested by Eq.(6.53). Inspection of Fig.8.1 suggests taking

E*aux ~ ‘Піп E*in (8.4)

with T)jn denoting the efficiency of input energy coupling to the plasma. Further, since the ratio

is a constant for the fusion cycle considered (see Table 7.1), we take

E*fu-E*n = fcE)u (8.5b)

with fc representing the charged particle fraction of fusion product energy (recall that fc, dt= 0.2 only). Commonly, the definition of the plasma fusion multiplication or, briefly called the plasma Q-value,

E fu

Пт Eu

Note that Qp is a measure for how efficiently an energy input to the plasma is converted into fusion energy. In the case of an energetically self-sufficient fusion plasma-later on we will call this state an ignited one-that is when the plasma meets its power demands by the fusion energy deposition alone and hence auxiliary heating is no longer required, the plasma Q-value approaches infinity as Ein* —> 0. The left hand side of Eq. (8.7) can be immediately integrated to give

‘■b

jdE*,h= Ел(ть)-Ел(0).

Obviously then, if the plasma is restored to the same energetic state after the bum, that is if Ел*(Ть) = Eth*(0), the fusion energy generated in this bum is found from Eq.(8.7) to balance the losses by

For the case of energy self-sufficiency, Qp —> «>, the fusion energy delivered to the plasma via the charged reaction products is seen to balance the total energy loss from the plasma. Considering a d-t plasma and collecting the relevant power densities previously derived, we use the explicit expressions given in Eqs.(2.21a),

(7.2b), (3.44) and (5.88), whereby we suggest Erad* to be due to bremsstrahlung

ld3rEth(r, t)_ EiM I т E(r>t) т E'(t)

with TE(r, t) representing the local energy confinement time.

As mentioned previously, the plasma state described by Eq.(8.10), that is when the internal fusion power deposition is capable of balancing the plasma power losses, is commonly referred to as ignition. An auxiliary energy supply is thus no longer needed.

For a fusion reactor operating in steady state the time integrals in Eq.(8.10) may be omitted thus leaving behind a global plasma power balance. Further, if ignition is assumed to occur everywhere throughout the plasma volume (e. g. in a homogeneous plasma), then we may disregard the volume integration as well and finally derive the local d-t fusion ignition condition

fed, Pd, ( Ni ,Ti)= Pbr( Ni, Ne>Te)+ Pc%( Ne, Te) + (8-12)

2 Те

where the thermal plasma energy density has been taken as

E, h,j = 1 NjkTj, j = i, e (8.13)

according to the average particle energy in a Maxwellian distribution, Eq.(2.19c), multiplied by the respective number density. Explicit expressions for the several power terms in Eq.(8.12) may be readily introduced via their definitions in previous sections and thus reveal the complex interrelation between the plasma density and its temperature as required for ignition.

The design of a muon-catalyzed reactor system is dominated by the need for an on-line accelerator to produce pi-mesons which are collected and then decay into the desired muons in a domain of interest. In this respect, the muon catalyzed fusion power flow pattern resembles that of an inertial confinement fusion system, Fig. 11.2, particularly with respect to the need for a significant recirculating power flow to the accelerator.

While a generally accepted design has yet to emerge-and largely awaits the development of suitable accelerators yielding a sufficiently intense beam of muons-we can conceive of a generalized schematic such as depicted in Fig.12.5. Here, light ions (p, d or t) of energy in excess of ~1 GeV will strike a low atomic mass number target with the transmitted ions either recirculated by magnetic forces or allowed to impact upon one or more successive targets of increasing atomic mass. The pi-meson will be emitted with a highly anisotropic directional distribution to be collected for decay into muons. The muons thus produced would be collected and focused onto a cylindrical fusion core consisting of liquid deuterium and tritium under very high pressure and at a temperature ~103 K. This cylindrical fusion core will be surrounded by a blanket which serves, variously and as required, the functions of (i) tritium breeding, (ii) fissile fuel breeding, and

(iii) energy removal.

The technology required to develop this conceptual system has many similarities to that involved in other fusion concepts; however, the pi on collector, the focusing of muons into the fusion core, and the efficient recovery of residual energy in the accelerator target are problems yet to be resolved.

Liquid