A more plausible approach to the confinement problem is to recognize that ions are known to be affected by electrostatic fields, so that confinement by such force effects can also be conceived.

One specific and interesting electrostatic approach to confinement involves a spherical metallic anode emitting deuterons towards the centre. These ions then pass through a spherical negatively charged grid designed to be largely transparent to these deuterons. The positive ions converge toward the centre and a positive space charge forms tending to reverse the motion of the ions and thereby establish a positive ion shell inside the hollow cathode. This internal positive ion shell is called a "virtual anode" in order to distinguish it from the "real anode" which produced these ions.

The metallic cathode, situated between the real and virtual anode, also emits electrons towards the centre. After passage through the virtual anode, the electrons similarly form a "virtual cathode" located further towards the centre. It is conceived that several such nested virtual cathodes and virtual anodes will form with the ion density increasing toward the centre. Finally, then, fusion reactions are expected to occur in these inner ion shells of increasing particle density.

However, in general, the electric fields required, the likelihood of discharge breakdown, and problems of geometrical restrictions have generally served to limit consideration of electrostatic confinement in the pursuit of fusion reaction rates suitable for an energetically viable system.

In the case that collisions are neglected, particles are trapped in a mirror when they do not appear in the loss cone in velocity space. However, plasma ions and electrons do suffer collisions, which can bring them randomly from the confinement region of velocity space into the loss cone. Upon entering the loss
cone, the particles escape immediately within one transit time t = L/v over the length L of the device. Due to their relatively small mass, electrons diffuse more rapidly both in velocity and coordinate space and hence are first to be scattered into the escape cone and then lost. This initially rapid loss rate causes the build­up of a positive electrostatic potential in the confined plasma which then consists of a surplus of ions having not yet scattered into the loss cone. Since this positive potential tends to retain the remaining electrons in the magnetic bottle, the overall plasma confinement time is governed by the ion escape time as characterized by the ion-ion collision time and can be shown, for ions encountering simultaneous multiple collisions, to be given by

(9.42)

at the plasma temperatures of interest to nuclear fusion, with Aj representing the ion’s atomic mass number.

Current Carrying Bars

The mirror confinement time Хм must also be determined by the size of the
loss cone, and consequently the approximation

(9.43)

can be shown to hold for large values of the so-called mirror ratio Binax / Bmm. By substituting, one obtains

(9.44)

where q, = Z, e has been replaced and C is a constant taken for typical fusion temperatures here to be

C= 1.78 x 1016 s

when Tj is measured in keV, N, in particles per m3 and Хм in seconds. Note that the mirror confinement time depends on the ion temperature and on the ratio Bmax/Bmin. but not on the actual magnitude of В or the plasma size. Evidently, a higher density will enhance the scattering into the loss cone and thus reduce Хм-

We add that the heuristic derivation of Хм presented here refers to a classical treatment in the sense that collective effects such as instabilities are suppressed. Therefore, the above scaling applies when stabilization of those collective perturbations has been provided and then appears to be in good agreement with particle containment times recorded in magnetic mirror device experiments.

From a neutronic point of view, the blanket must be designed to provide an adequate tritium breeding ratio to sustain a substantial volumetric power density, allowing for continuous power extraction via heat exchange with the coolant, and to serve as shielding for equipment, especially for the superconducting magnet coils in MCF, and personnel.

Tritium is produced by the neutron-induced reactions

jNdN, <ov>d[d3r

(13.22)

associated with isotropically distributed neutrons, Nn(vn), colliding with lithium nuclei at rest. Here N6 and N7 are the 6Li and 7Li atom densities in the blanket volume Vb, and On6 and <3n7 are their corresponding microscopic neutron absorption cross sections; Nn is the speed dependent density of neutrons in the blanket, vn is the neutron speed and Vc is the fusion core volume. We point out that the n(7Li, n, a)t reaction, Eq.(13.21b), is a threshold reaction and requires an incident neutron energy in excess of 2.47 MeV. For that, the neutron flux (Nnvn) has to be considered in the fast energy range, that is the second integral in the numerator of Eq.(13.22) yields zero in the thermal and epithermal neutron energy range.

In the absence of tritium from other sources, it is necessary to have С, > 1 in order to compensate for tritium transport losses during extraction and transfer as well as for its decay before injection into the fusion core. Indications for breeding capabilities can be obtained from upper limit estimates given in Table 13.1 for various blanket materials; as shown, adequate tritium breeding may be obtained if a neutron multiplier such as 9Be or Pb is added.

Early blanket concepts employed liquid lithium as a coolant, thereby providing adequate tritium breeding. However, other considerations like pumping power requirements, the effect of magnetic fields on a flowing metal, and materials compatibility forced the development of alternative designs with lithium added in other forms. This includes various solid lithium compounds, molten salt fluids (e. g. 2LiF + BeF, called FLIBE), and a lithium-lead eutectic,

Since the blanket is exposed to high energy neutrons entering from the fusing plasma, the neutron density is a maximum in the first wall domain and then attenuates rapidly, even if a reflector zone completes the blanket composition. A consequence of this is that energy deposition will similarly vary with the depth of blanket penetration, Fig.13.8. The general trend of an exponential fall-off from the plasma side to the blanket interior must be considered in designing the coolant flow pattern and also in calculations of breeding, radiation damage, and activation. A maximum power density of ~ 80 Wcm’3 occurs in the multiplier zone, while the average is ~ 15 Wcm’3. For comparison, power densities of < 100 Wcm’3 apply to light and heavy water fission reactors. The fusion blanket region should operate at a high average temperature (> 1000 K) in order to facilitate a reasonable thermodynamic conversion efficiency.

Safety aspects suggest considering helium as an appropriate coolant since it is an inert gas which reacts with neither the lithium, the beryllium neutron multiplier nor other structural material. Further, it offers the advantage that the bred tritium is conveniently transported out of the blanket with the helium coolant flow.

In assessing the energetic performance of a fusion reactor blanket, we refer to the internal energy flows illustrated in Fig. 13.9, from where it is evident that the energy removable from the blanket is

El = b(fn Efu + Erad) + 2 Ent (13‘23)

i

where a blanket coverage factor b depending on the specific blanket geometry is introduced, since a fusion reactor blanket will feature several channels through its structure, e. g. for injection tubes, diagnostic equipment, etc., and hence cannot completely envelope the fusion plasma. Further, in Eq.(13.23), Ent accounts for the total energy released by an t — type neutron-induced reaction. If exothermic, these reactions can then provide for multiplication of the energy of fusion neutrons having initially entered the blanket. To generalize such energy enhancement, it is convenient to account for it by the explicit blanket multiplication factor

Mb=——————- і—————————————— (13.24)

fnE*fu

allowing us to now rewrite Eq.(13.23) in the following form:

El = MbfnE*ju + bE*rad ■ (13.25)

Introducing specific power expressions based on reaction rate densities and the corresponding reaction Q-values, we find for Eq.(13.24)

2 RneQnedir

Mh=b + — f—£—————— (13.26a)

fn}RfuQfud3r

vc

or, respectively, for d-t fusion and assuming lithium as the only neutron reactive substance in the blanket, we obtain

Qn6 f j <7n6^6^n(vn)vndvnd3r + Qn7 j °7Nn(vn)vndvп(1Ъr

Mb =b +—- ^^—————————————————————————————— .

fn4,Qd,)NdN, <ov>dtd3r

Vc

(13.26b)

With the blanket composition and dimension known, and for a specified fusion plasma, Mb can be readily calculated and is seen to range from 1.3 to 1.8 for pure fusion blankets-those which do not contain fissionable material.

1. Introduction

Matter and energy are fundamental components of our physical world. These components manifest themselves in a variety of ways under different physical conditions and can be affected by a variety of processes. Our interest in this first chapter relates specifically to the fusion of light nuclides which forms the basis of energy release in stars and which is expected to be harnessed on earth.

Media in which fusion reactions occur consist mainly of interspersed charged particles which are affected by short and long range forces. The cumulative effect of these forces combined with the intractability of an analytical description of each individual particle suggests that various approaches be used in the determination of the macroscopic behaviour of an ensemble of moving and colliding particles.

5.3 Particle Motion

The motion of a single particle of mass m is described by Newton’s Law

d Vі / f.4

тТГ (6Л)

where Fj is the j-th force vector acting on the particle and v is its velocity. For example, an isolated particle of mass m and charge q moving in a gravitational field g, an electric field E, and a magnetic field B, has its space-time trajectory described by

d

m— = mg + ^E + q( v x В ) . (6.2)

dt

While Eq.(6.2) is indeed very useful for some applications, it suffers from an overriding restriction: it describes the motion of an isolated particle only and thus excludes any possible interaction with other particles. This is indeed a severe restriction for fusion energy applications because power density requirements demand that about 1020 fusile ions be contained in one m3; with such a large number of ions nearby each possessing a different velocity, it is evident that Coulomb interactions alone will lead to a most complex collection of time varying electrostatic forces acting on the particles. While in principle, one might specify a dynamical equation of the form of Eq.(6.1) for each particle, one would need perhaps 1020 force terms on the right hand side per unit volume; solving such equations for each of the interacting particles is, of course, totally unmanageable for computational purposes, due to the enormous number of these coupled equations and the lack of knowledge of individual initial conditions.

As an alternative to using an exceedingly large number of equations each containing many terms, it has been found that other approaches which are mathematically tractable provide, in selected applications, satisfactory agreement

with experiments and a useful predictive quality. One frequently used conceptual approach is to consider a plasma as a multicomponent interpenetrating low — density fluid and then employ suitable continuum mechanics methods. Another approach is to use probabilistic considerations for each group of particle species and then perform an analysis based on methods of sampling statistical physics.

While no one approach is generally applicable to all cases of conceivable interest, a careful choice of conceptual constructs and methodologies can lead to descriptions which are remarkably useful in characterizing selected space — velocity-time aspects of a particular species’ population in a fusion medium. It is important therefore to develop a good understanding of the imposed assumptions in order to recognize important restrictions of a particular conceptual development and its consequent mathematical description.

Both magnetic and inertial confinement fusion involve two key processes for the attainment of a viable fusion energy system:

1. heating and ionization of the fuel to high temperature to achieve a favourable fusion reaction rate density, and

2. confinement of the fuel for a sufficiently long time to yield a net energy gain.

The particle density in a magnetically confined plasma is expected to not significantly exceed 1021 m’3, which constitutes a very low density gas when compared to atmospheric particle densities of about 1025 m’3 at standard temperature and pressure. In contrast, inertial confinement typically requires fuels compressed to densities that are several orders of magnitude greater than solid-approaching number densities of 1031 to 1032 rn3-exceeding even densities found in stars. A major feature of this high density is that since the fusion reaction rate is proportional to the density squared, the inertial confinement time required for a net energy gain will be significantly smaller than those in magnetic confinement devices. These points can be put into perspective by comparing the Lawson criterion-like requirements for inertial and magnetic confinement. For magnetic confinement, magnetic field limitations typically restrict ion densities to the order of 1021 m’3 with a confinement time necessary for energy break-even of about 1 s. On the other hand, for inertial confinement, the compressed density can be 1031 m’3 over a time interval of the order of 10’9 s.

Interest in inertial confinement fusion energy emerged later than that in magnetic confinement fusion. Its relevance to power production became apparent when it was recognized that concentrated beams from powerful pulsed lasers could be used to initiate a compressive process in a small solid or liquid target pellet, possibly resulting in a sufficient number of fusion reactions to yield a net energy gain.

The sequence of events in inertial confinement fusion can be briefly described as follows. A small pellet, with a radius less than ~5 mm and containing a mixture of fuel atoms, is symmetrically struck by energetic pulses of electromagnetic radiation from laser beams or by high energy ion beams from an accelerator, Fig. 11.1a. Absorption of this energy below the surface of the pellet leads to local ionization and a plasma-corona formation, Fig. 11.1b. The important consequences of these processes are an outward directed mass transfer by ablation and-by a rocket-type reaction-an inward directed pressure-shock wave leading to compressing and heating of the target, Fig. 11.1c. A follow-up shock wave driven by the next laser or ion beam pulse will then propagate into an already compressed region where it travels faster than its predecessor. Subsequent shock waves can thus propagate even more quickly. Tuning the beam’s pulse repetition rate such that the shock waves arrive at the pellet’s core simultaneously will provide for an adequately compressed state possessing temperatures suitable for initiating a substantial fusion bum. With the temperature and fuel density sufficiently high, the fusion reactions will occur until the pellet disassembles in a micro-explosion due to its excessive energy content, Fig. 11.Id. The disassembly typically takes place in a time interval of about 10’8 s, corresponding to the propagation of a pressure wave across the pellet with sonic speed vs.

Experience with inertial confinement has shown the importance of several processes and phenomena. For example, it is essential that the incident laser or ion beam strike the pellet symmetrically and that efficient energy coupling between the beam and the target be attained. The inner core should reach a high density very quickly before thermal conductivity heats the central region causing an internal pressure build-up that opposes high compression. A substantial fraction of the nuclear fuel should also bum before pellet disintegration.

We note that laser beams can penetrate to deeper layers of the pellet when they possess a higher frequency. Hence, intensive short-wavelength lasers are sought as drivers for inertial confinement fusion, evidently with a reasonable efficiency also required. A phenomenon of concern is that very high energy electrons generated in the initial laser light absorption process will penetrate into the centre of the target before the arrival of the dominant pressure wave thereby causing an undesirable preheating of the central core region resulting in an outward force effect to retard compression. Accelerators, on the other hand, transfer the beam energy more directly to ions in the target and can therefore be significantly more efficient; this provides some appeal for the use of light or heavy ion accelerators for such purposes.

The preceding discussion stressed the energy multiplication potential of the fusion hybrid. Next, we consider the fissile fuel breeding capacity of such a reactor. Recall the schematic depiction of the d-t fusion-fission system as suggested in Fig. 15.4. The total fusion reaction rate is given by

Rd, = NdNt <<Jv>dt d (15.14)

with the integration carried out over the fusion core volume Vc. Similarly, we define the neutron-nucleus interaction of type і [і: a (absorption), c (capture), t
(tritium breeding)] involving nuclei j Ц: g, f, £ , x] of density Nj according to

/?;, = (15.15)

V,,v„

where ф is the neutron flux depending on vn and the integration is carried out over the entire neutron speed range and the blanket volume Vb.

A fissile fuel recycle fraction er, representing the fraction of bred fuel burning in situ in the blanket, is now introduced. Then, l-er is the fraction of bred fuel made available for external fission reactors. For every X units of fuel bred, only the fraction ЄгХ is retained and subsequently fissions in the blanket. For steady state operation, it will be seen below that the fissile breeding ratio is conveniently identified to equal l/er.

Our objective now is to find a suitable expression for the hybrid breeding ratio

in terms of lumped reactor physics parameters. To do this we must formulate reaction rate equations which describe the production and removal of the various nuclear species of interest in the hybrid reactor. The rate equations for the total populations of tritium Nt* in the system, and of fissile fuel Nf* and neutrons Nn* in the blanket are given by

— Rn(,l ‘ Rdl

= c. — Г?….

= ЬК, + X К ‘ +(Rf ])Rnf, a -{J-VS )Kg, a -(J-Vl )Kta ‘ К

(15.17c)

Here b represents the blanket coverage factor introduced in Sec. 13.7 and ту denotes the average number of neutrons emitted per neutron absorbed in j — nuclides. Obviously r| > 1 for neutron-multiplying and fissile nuclei, while the fertile materials feature r|g < 1 and T), < 1, respectively.

The case of steady-state operation, defined by dN; _ dN) _ dNn

v R

, r g, m ng, m

with Vg, m indicating the average number of neutrons released per neutron- multiplying reaction of type m [m: f, 2n, 3n] in fertile nuclei g. The set of Eqs. (15.19a) — (15.19c) can be manipulated to give the following explicit expression for the hybrid breeding ratio, Eq. (15.16):

,(15.21)

x lynf, a m=f,2n,3n, a,a

where we have assumed a blanket design such that b ~ 1 — Tjt. The first term on the right hand side of Eq. (15.21) resembles the breeding ratio of an "infinite"
fast breeder, but here its numerical value may be much larger because the neutron spectrum in the hybrid blanket driven by 14.1 MeV d-t fusion neutrons can peak at a higher energy. Indeed, for neutron energies up to 14 MeV, the energy dependence of T|f is given by

rjf~a]+a1E (15.22)

where ai is in the range 1 to 1.5 and 0<a2<0.2-depending upon the isotopes involved-for E in units of MeV. Employing a neutron multiplier, which obviously features T|x > 1, will provide for an increased breeding ratio. Further neutron gain takes place by fast fissions of fertile nuclei and inelastic scattering from them associated with vgif = 3, vg>2„ = 2 and vg,3„ = 3, respectively. All these neutron production mechanisms together are capable of maintaining a high neutron availability-for-breeding against parasitic absorption and leakage out of the blanket. Hence, the breeding ratio for such a fusion-fission hybrid can substantially exceed the breeding ratios obtainable with fast breeder fission reactors.

Indeed, for recent hybrid concepts designed to stress fuel production, it is estimated that a single hybrid can support up to 10 fission reactors each having a thermal power level equal to that of the hybrid; in contrast, the typical fast breeder may typically support about one companion fission reactor of comparable power. Thus, the roles of such systems would be quite different: the hybrid focuses upon fuel breeding whereas the fast-fission breeder emphasizes energy production with some breeding.

The power in a fusion reactor core is evidently governed by the fusion reaction rate. If only one type of fusion process occurs and if this process occurs at the

rate density Rfo with Q&, units of energy released per reaction then the fusion power generated in a unit volume is given by

With Rft, expressed in units of reacuons-m -s ana i^ft, in iviev per reaction, me units of the power density Рд, are MeV-m’^s"1 which can be converted to the more commonly used unit of Watt (W) by the conversion relationship 1 eV-s"1 = 1.6 X 10"19 W since 1 W = 1 J-s’1. For the case of a uniform power distribution the total

energy released during a time interval T in some volume V follows from Eq.(2.20) as

(2.21b)

That is, energy may be viewed as the area under the power curve while power may be interpreted as an instantaneous energy current.

The energy generated in a given fusion reaction, Qfu in Eq.(2.20), is the "Q — value" of the reaction, Eq.(1.15), and can be experimentally determined or extracted from existing tables; this part of the power expression is simple. However, the determination of the functional form of the reaction rate density Rfu is more difficult but must be specified if the fusion power Рд, is to be computed.

In order to determine an expression for the fusion reaction rate density, consider first the special case of two intersecting beams of monoenergetic particles of type a and type b possessing number densities Na and Nb, respectively, Fig. 2.3.

In a unit volume where the two beams intersect, the number of fusion events in a unit volume between the two types of particles, at a given time, is given by a proportionality relationship of the form

Rfu ж Na Nb Vr

where vr is the relative speed of the two sets of particles at the point of interest. This relation is based on a heuristic plausibility argument for binary interactions. Obviously, some idealizations are contained in Fig.2.3 and Eq.(2.22), such as particles of varying energy and direction of motion as well as the interaction of particles with others of their species not being accounted for, but will be considered in subsequent sections.

The proportionality relationship of Eq.(2.22) can be converted into an explicit equation by the introduction of a proportionality factor represented here by <7ab for a given vr:

The subscript ab and the functional dependence on vr, indicated in Gab(vr), is to emphasize that the magnitude of this parameter is specifically associated with the particular types of interacting particles and their relative speed. The common name for Oab(vr) is "cross section" and, since all the terms of Eq.(2.23) are already dimensionally specified, its units are those of an area. This parameter has been assigned the name "bam", abbreviated b, and defined as

lb = W’24 cm2 = 10’28 m2 • (2.24)

Figure 2.4 illustrates the cross section for a case of deuterium-tritium fusion. Note a maximum of a few bams in the vr = З X 106 ms’1 range in this figure.

A reaction statement for d-t fusion which incorporates both mass flow and energy flow may be written as

+ d + t —^ (t/r) —^ n + cc + Qdt (7.63)

where Ed, is the energy supplied to heat the deuterium and tritium to thermonuclear temperatures and to sustain their high kinetic energies against all energy loss mechanisms; (dt) is the short-lived intermediate nuclear state previously denoted by (5He)*. Energy viability evidently demands Edt < Qdt.

In establishing fusion reactor conditions it is the electrostatic repulsion between the fuel ions which demands intensive heating of the plasma to high temperatures in order that the Coulomb barrier between two fusing nuclei can be more easily penetrated or overcome providing for a substantial fusion reaction rate. One might therefore speculate about the appealing prospects of using some "agent x" which could neutralize the Coulomb repulsion and induce a fusion event by making it easier for the ions to enter the range of their strong nuclear forces of attraction. That is, this agent might serve the function of a "catalyst" similar to the common practice of using catalyzing compounds to enhance the rate of chemical reactions. In particular, this catalytic agent x must be released after the d-t fusion event and subsequently induce another fusion reaction. Taking the average energy cost of producing one agent x to be Ex and if each x catalyzes % fusion events, then we write for the entire sequence

(7.64)

With Qat a known constant, the fusion catalyst should possess an average energy cost of production Ex and catalyze % d-t events such that now Eq.(7.65) is satisfied.

Interestingly, a catalyzing agent with properties suggested above does exist. It is a subatomic particle commonly called the "muon" and represented by the symbol (X. This particle appears as a product in various types of high energy nuclear reactions and possesses the following properties:

charge: q^ = — І. бхІО’19 C (same charge as an electron)

mass: тц = 207 me (207 times as heavy as an electron)

lifetime:^ = 2.2xl0"6 s (it is unstable) .

The property that a muon has the charge of an electron but a much larger mass means that it can enter into an orbit of a hydrogen atom with a Bohr radius 207 times smaller than that for an electron; we suggest this comparison between a conventional hydrogen atom and a muonic hydrogen atom in Fig.7.7.

A consequence of a much tighter muon orbit around a hydrogen nucleus is that to another hydrogen ion or hydrogen atom, the muonic hydrogen atom appears like an oversized and overweight neutron. Hence, this "oversize" neutron might approach another hydrogen ion or atom more closely because of a reduction of the repulsive Coulomb forces. Then, when this conventional hydrogen and the muonic hydrogen are sufficiently close to "notice" the details of spatial charge variation, they are already close enough for nuclear forces of attraction to dominate and to bring about a fusion event.

Several additional points are important in this context of muon catalyzed d-t fusion. Reactions of the type depicted in Fig.7.8 have been experimentally confirmed in liquid hydrogen at temperatures in the 300 К to 900 К range; the implication therefore is that muon catalyzed fusion may be sustained in a temperature environment more like that of existing fission reactors. Then, an appropriate accelerator for muon production has to be associated close to the muon-fusion chamber suggesting a system configuration similar to that of an ion beam-sustained inertial confinement scheme. Finally, the muon mean life-time of

2.2 (Is is unaffected by whether it is bound to a nucleus or not and hence demands that the various muonic induced reactions generally proceed at a fast rate.

Problems

7.1 Use «7v>dt from Appendix C for 10 keY and 100 keY and calculate Rfu for a 50:50% and 25:75% mixture of deuterium and tritium for which Nd + N, = N = 1021 m’3.

7.2 With the branching of the d-d fusion reactions occurring with essentially equal probability, determine the deuterium destruction rate and the helium-3 production rate.

7.3

For d-d catalyzed fusion, what are the ratios of (a) Nh / Nd, (b) Nt / Nd, (c)R+n / R+a> (Ф R+P / R+a> and (e) R+a(d-t) / R+a(d-d-p)’

7.4 Formulate reaction rate expressions for d, t and h as suggested in Table 7.2.

7.5 Undertake an analysis similar to that leading to the results of Fig.7.3 for the case of F = 0.

7.6 Fission reactors possess power densities of about 107 Wm’3, For this power range, determine the required particle density for d-t fusion at 10 keV.

7.7 Consider a catalyzed deuterium reactor operating under the conditions Nj=Ne=1020 rn3, kT=10 keV. Calculate the equilibrium concentrations of all three species (d, t, and h).

7.8 Physics requirements for d-h fusion are clearly more demanding than for d-t. Some advantages often cited involve environmental, safety, and cost factors. Discuss how these factors are advantageous and whether or not the advantages outweigh the disadvantages. Do you foresee any advances in physics that might significantly affect any trade-offs? Discuss which confinement system might be best suited for burning d-h.

7.9 It has been suggested that running a “lean” deuterium mixture (e. g. 30 % d, 70 % h) could further reduce neutron production. However, this is a trade-off against power density. Suggest, using equations and sketches of graphs, how to determine an “optimum” mixture ratio. (You may assume that a 1000 MW — electric plant is desired and that this power level is fixed; describe how to find what mixture ratio would provide 1000 MWe for the least input power.)

Energetics, Concepts, Systems

A literal interpretation of the parameter is that it represents the average number of d-t fusions a muon can catalyze during its lifetime. Equation (12.21) could be evaluated if the muon injection rate density F^(t) is known as a function of time together with the time dependent concentration N^dt(t); the former is specified by accelerator operation but the latter can only be determined from a solution of the system of dynamical particle balances given by Eqs.(12.15) in combination with the energy dynamics in the reaction domain. To avoid having mathematical complexity obscure the physical features of the problem, we will assume steady-state operation. That is, we assume a constant fuel temperature

allowing therefore a constant reaction rate parameter and also take a constant injection rate, F^(t) = F^° during the operating time x of interest. Hence, after an initial transient during start-up, the density of the NMdt(t) molecular ions reaches a constant value, N0^, characteristic of steady-state. Then the muon recycle

Note that the reactor operating time x will invariably be much in excess of the muon mean lifetime so that x + хд = x.

By maintaining constant fuel and muon densities by appropriate feed rates, a constant N^d, implies that all the other intermediate particle densities will also be constant in time; that is, we will have in Eqs.(12.15)

dN fi _ dN і _ dN d _ dN ju _ dN ^ _ dN fjdt

dt dt dt dt dt dt Under these conditions, the system of linear algebraic equations for N^, N^, and N^dt, Eq.(12.15a) and Eqs.(12.15d) to (12.15f), can be solved to yield an explicit ratio N’^ / F^° as required for Eq.( 12.22). This gives

NU_

Вц)

and may be interpreted as a muon "residence unavailability" penalty. In compact form the muon recycling efficiency therefore reduces to

‘ 1 Л

a + Bnj

For the case of muon catalysis at liquid hydrogen conditions, Table 12.1, the muon recycle efficiency is calculated to be

Ад-34. (12.26)

That is, on average, one muon catalyzes some 34 d-t fusions during its mean lifetime of 2.2×1 O’6 s.

The energy multiplication assessment follows similarly from a substitution of Eq.(12.21) into Eq.(12.20); we take Qdt = 17.6 MeV and use an estimate of Ец ~ 3000 MeV to find

This provides a useful estimate of the energy viability of a muon catalyzed fusion system which, as is evident, is too low by perhaps a factor of thirty.

Some additional considerations may, however, be introduced which suggest possible increases in ME. Clearly Qdt is a constant and Ецс could not be further reduced except for possible altemative-and highly speculative-methods of muon production. Considerable research has in recent years been undertaken to determine if Хц, the average number of d-t fusions catalyzed by one muon, could be increased. Experiments have revealed that at specific temperatures and at significantly elevated pressures, values of 160 are possible-corresponding to an increase in the energy multiplication of Eq.(12.27) by a factor of about 3.5. However, while a definite dependence on medium temperature and composition has been established, it is not yet evident that an according optimization can be sufficient for energy viability.

There exists, however, another approach which can be summarized by the following. Supposing the fusion domain is surrounded by a neutron multiplication and breeding blanket domain, the function of which is to multiply the neutron by (n, xn) reactions and to breed both tritium and fissile fuel for companion fission reactors. (Fusion-fission hybrids and similar integrated systems will be discussed in greater detail in Chs.14 and 15.) The fission energy eventually thus generated might be considered a benefit. For example, if a fission

(12.28)

to yield a significant positive energy balance. Such a hybrid system, based on muon-catalyzed fusion, would have to be evaluated on the same basis as other fusion based hybrid concepts and will be further discussed. A potential advantage of this system could be the relative simplicity of the reaction chamber; however, accelerators which produce sufficient numbers of collimated muons at an acceptable average energy cost per muon represent a significant design uncertainty.