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A most spectacular display of fusion energy is associated with stars, where confinement comes about because of the gravitational pressure of an enormous mass. High density and temperature thereby result toward the stellar centre enabling the fusile ions to bum. While energy leakage and particle escape occurs from the star’s surface, the interior retains most of the reaction power and prevails against the occurrent radiation pressure through the deep gravitational potential wells, thus assuring stable confinement for times long enough to bum most of the stellar fusionable inventory.
Since fusionpowered stars possess dimensions and masses of such enormity, it is evident that confinement by gravity cannot be attained in our terrestrial environment.
In deriving the relative requirements of the magnetic field strength and the plasma pressure for effective confinementSec. 9.1we considered, in the fluid model, the case of magnetohydrodynamic equilibrium. However, we did not examine whether this equilibrium state is stable or not. In an equilibrium state all forces are balanced allowing thus for a steadystate solution of the set of magnetohydrodynamic equations discussed at the end of Sec. 6.3. The equilibrium is labeled stable if small perturbations are inherently damped and it is unstable if small deviations from the equilibrium state are amplified, that is, if perturbations propagate and grow with time; this then is called an instability.
The unperturbed state requires perfect thermodynamic equilibrium in which the plasma particles have Maxwellian velocity distributions, while the plasma density and the magnetic field is uniform. Note that in the magnetic confinement configurations of interest to nuclear fusion reactors these requirements are not met. In the example of a mirror field, the isotropy of the Maxwell distribution is significantly disturbed by the particles lost through the mirror throats, which had obviously a dominant Vrcomponent. Further, a mirrordevice will feature VB and VNj and thus be nonuniform. Though all forces can be balanced in a steady state, this state is not in perfect thermodynamic equilibrium and possesses so — called ‘free’ energy which can drive instabilities. Even periodic motions of the plasma fluid elements, e. g. plasma oscillations or, alternatively, waves, can thus be induced. An instability constitutes a motion which reduces the free energy and brings the plasma closer to perfect thermodynamic equilibrium. There exists a wide range and variety of possible plasma waves and instabilities, the discussion of which is far beyond the scope of this textbook. Hence we restrict ourselves to a few demonstrative examples of interest here.
The instability most relevant to mirror machines is the socalled flutetype instability. In the simple mirror geometry of Fig.9.3, the curvature of the magnetic fieldexcept for the endregionsis seen to be convex. Any outward perturbation of the confined plasma, i. e. a ripple on its boundary surface where all magnetically confined plasmas appear to have an energy density gradient, takes the plasma into regions of lower magnetic induction and lower kinetic pressure; hence, such a displacement to regions of reduced energy density will provide for free kinetic energy to let the perturbation grow. This can lead to flutes of plasma moving across magnetic field lines, Fig. 9.6, and result in particle loss from the containment region.
Fig. 9.6: Depiction of the flute instability. 
Further insight into the onset and mechanism of the flute instability is provided if we recall the drifts and forces associated with a socalled "bad" convex Вfield curvature, where the curvature drift, Eq.(5.61), will lead to charge separation occurring perpendicular to the magnetic field and the radius of curvature, Fig. 9.7. This polarization creates an azimuthal electric field causing an additional ExB drift, which transports both ions and electrons in the radially outward direction thus forming the flutelike bumps on the plasma column.
This type of instability can be avoided by generating a socalled ‘minimumB’ field configuration in which the field lines are (almost) everywhere concave into the plasma. Here the charged particle then senses an increasing Вfield in every direction and therefore finds itself in a magnetic well; the term minimumB is thus commonly used for such a magnetic topology.
The simplest means of producing such a minimumB field configuration is to locate four current carrying bars on the periphery of a magnetic device with their positions suggested in Fig. 9.8. These bars are called Ioffe bars with the current in adjacent bars flowing in opposite directions.
Another means of generating a minimumB magnetic field configuration is by using a coil having the shape of the seam of a baseball. If the coil of the baseball configuration is suitably flattened and oriented in opposition with another similar coil, one obtains again a minimumB configuration, called a "YinYang" coil configuration, Fig. 9.9.
Further examination of these minimumB configurations makes it clear that they all provide a central circular region for the plasma but that in two opposite directions the flattened fanshaped magnetic fields are open and thereby still form a magnetic mirror. This feature can be extended by adding such devices to the ends of mirror solenoidal fields to form more effective mirrors because the magnetic well can in principle be deeper, and further, the particles contained
Mirror devices are basically appropriate for steady state operation, in which the particle injection rate balances the diffusion leakage rate. The diffusion occurs dominantly through the open ends and constitutes also a diffusion in velocity space, sinceas previously mentionedthe velocity distribution in a mirror plasma is no longer Maxwellian due to the preferred loss of particles with large V / vj_; one rather deals with a socalled losscone distribution excluding all particles in the losscone as resulting from Eq.(9.38), Fig. 9.5. This deviation from the Maxwellian distribution drives socalled velocityspace instabilities, here specifically the ‘losscone’ instability, which can enhance the velocityspace diffusion into the losscone. It has been observed that such instabilities are less harmful to plasma confinement when the mirror device is short in dimension.
The first wall for a muon catalyzed fusion reaction will need to withstand pressures of the order of 108 Pa at moderate temperatures of about 103 K. Hence, this wall may be viewed as a thick cladding not unlike pressure tubes used in some fission reactors. However, while the design concept suggests some simplifications, significant problems remain to be studied; among these are the material requirements and the high pressure operation under conditions of high neutron influence.
A. A. Harms
McMaster University
K. F. Schoepf
University of Innsbruck
G. H. Miley
University of Illinois
D. R. Kingdon
McMaster University
Fusion energy is widely perceived as the ultimate terrestrial energy source. This appealing prospect has emerged by reason of scientific experiment, by the expectation of continuing scientific and technological progress, and by intriguing observations such as the following: life on earth is sustained as a consequence of fusion reactions in the sun; a mediumsized lake contains sufficient hydrogen fusion fuel to supply a nation with energy for centuries; the known diversity of fusion fuel cycles offers the eventual possibility of radiologically clean energy; fusion power appears essential for deepspace explorations; … However, between this set of appealing notions and current scientific understanding and available technologies, there exists a barrier of considerable proportions. Both a broadly sustained community commitment and a high level of motivation by its participants is required for the realization of this ultimate source of energy.
As active research participants in several areas (both technical and geographic) of the fusion energy enterprise, we have long been sensitive to emerging technological perspectives related to this unique form of energy. In addition, as educators, we have repeatedly sought a conceptual and didactic framework for fusion lectures which would integrate the fundamentals of fusion phenomena with the successful experiences of the past and provide a link to the broader promise of emerging developments; additionally, and at a more subjective level, we have also sought an instructional balance between the pragmatic nearterm educational role of a societal objective with the longterm inspirational value of a theme. Thus, while it is clearly essential to emphasize the well established concepts of magnetic and inertial confinement approaches to fusion, we believe it is also important to discuss topics such as spinpolarized fusion, advancedfuel fusion reactions, muoncatalyzed fusion, and other related and emerging concepts. A synthesis which includes these and similar topics will, in our view, impart a most desirable perspective not only to the next generation of fusion scientists and nuclear engineers, but also to other professionals concerned with energy for the long term.
The writing of this text has been pursued, on and off, for nearly two decades. In retrospect, this has provided us with good time to accommodate the two divergent developmental paths which have become solidly established in the fusion energy community: the process of sequential tokamak development towards a prototype and the need for a more fundamental and integrative research approach before costly design choices are made. Our belief is that we have herein accommodated both interests in a coherent instructional format and the amount
and level of material contained here allows for both avenues to be pursued.
In developing our subject we have found it useful to identify several distinct themes. The first is concerned with preliminary and introductory topics which relate to the basic and relevant physical processes associated with nuclear fusion. Then, we undertake an analysis of magnetically confined, inertially confined, and lowtemperature fusion energy concepts. Subsequently, we introduce the important blanket domains surrounding the fusion core and discuss synergetic fusionfission systems. Finally, we consider selected conceptual and technological subjects germane to the continuing development of fusion energy systems.
Our target group of interest is the senior undergraduate and beginning graduate university student in science or engineering. Familiarity with selected aspects of modem physics and a working knowledge of the differential calculus and vector algebra is assumed as a minimum prerequisite. In support of our pedagogical objectives, we have chosen to place considerable emphasis on the development of physically coherent and mathematically clear characterizations of the scientific and technological foundations of fusion energy specifically suitable for a first course on the subject. Of interest therefore are selected aspects of nuclear physics, electromagnetics, plasma physics, reaction dynamics, materials science, and engineering systems, all brought together to form an integrated perspective on nuclear fusion and its practical utilization. While the subject is of necessity broad, a focused pedagogical emphasis is consciously pursued: to identify and synthesize relevant physical concepts and their associated mathematical constmcts and thereby provide a learning experience appropriate for subsequent more specialized work in any of the several areas of fusion energy.
In the course of our involvement in teaching and research in fusion energy, we sense a deep debt of appreciation to many with whom we have been in contact on matters of fusion energy. This includes numerous participants in various specific fusion energy programs at national and international research centres and colleagues at various universities. A particular word of thanks to Dr.
A. P. Jackson (Chalk River Nuclear Laboratory, Canada), Prof. B. Lehnert (Royal Institute of Technology, Sweden), and Dr. G. Melese (General Atomics, USA) for their review of earlier drafts of this text. Additionally, we acknowledge those undergraduate and graduate students who, over the years, have passed on to us their comments on various versions of this work: A. Bennish, B. Bromley, B. Carroll, G. Cripps, B. Diacon, G. Gaboury, E. Hampton, X. Hani, T. Harms, S. Hassal, S. Ho, A. Hollen, M. Honey, J. Marczak, S. Mitchell, R. Ramon, P. Roberts, G. Sager, R. Scardovelli, A. Sguigna, P. Stroud, Y. Tan, D. Welch and J. Zielinski.
Our thanks to the several patient secretaries for typing the text and to the artists for the drawings. It is, however, with a particular sense of appreciation that we acknowledge two sterling coordinators for their accommodating disposition:
Jan Numberg (McMaster University) and Christine Stalker (University of Illinois).
To all we express our thanks.
A. A. H., K. F.S., G. H.M., D. R.K.
January 2000
We have established that the helical motion of a charged particle, guided by magnetic field lines as suggested in Fig. 5.5, involves a centripetal acceleration and therefore leads to the emission of radiation called cyclotron radiation and evidently involves an energy loss for the particle. We assess the associated power loss starting with the classical expression for the radiation emission rate of ions and electrons moving in an accelerating field which is known to exhibit the proportionality
Pcycx Ni q] af + Ne q] al = Neq] ae (5.82)
where a() denotes the respective accelerations and q( , the respective electrical charges with qj =  Zqe . Here, the single particle radiation as given by Eq.(3.40) has been multiplied by the particle number density in order to provide an expression for power density. Note also that the smaller mass of the electrons will ensure that with their attendant higher accelerationrecall = F/methe electrons will be the predominant contributors to this cyclotron power loss. Ion cyclotron radiation is hence neglected. For cyclical motion in a magnetic field, the acceleration of electrons is a constant given by
Further, knowing the electron gyrofrequency as (0g, e = lqeIB/nie with В as the controlling magnetic field, the relation of Eq. (5.41) allows us to write
vL _ я] В2
2
me
Finally, assuming a Maxwellian distribution of electrons, we may take from kinetic energy considerations or Eq.(2.19b)
where a constant of proportionality is explicitly introduced. With Ne in units of m"3, В in units of Tesla, and kT in units of eV, the constant is given by Acyc = 6.3xlO’20 J eV"1 Tesla"2 s_1 for Pcyc in units of W m"3. Thus, cyclotron radiation is most important at very high magnetic fields and high electron temperatures. Indeed, with electrons at a possibly very high energy, they may need to be subjected to a relativistic treatment.
Cyclotron radiation exists in the far infrared radiation spectrum with a
wavelength of 10’3 — 1СГ4 m and is therefore partially reabsorbed in a plasma. Further, the emitted radiation may be reflected from the surrounding wall in a magnetic confinement fusion device and thereby reenter the plasma. Hence, we choose to write the net cyclotron power finally lost from a plasma as
Pcyc = Acyc Ne в2 к Те V (5.88)
where If/ accounts for the complex processes of reflection and reabsorption of cyclotron radiation and is a dimensionless function involving several plasma parameters, including the magnetic field strength and the reflectivity of the surrounding wall. For a reasonable reflectivity of 90% and conditions expected in a fusion reactor, the range of l// could typically be from 10"4 (for low Te) to 10‘2 (for very high Te).
Problems
5.1 The maximum attainable magnetic field В is expected to be about 20 Tesla. For V = 0, estimate the associated gyrofrequency, cog, and gyroradius, rg, for electrons and protons given kTe = kTj = 5 keV.
5.2 Graphically depict the motion of an ion in a combined Efield and B — field. Display the results in an isometric representation.
5.3 (a) Determine the motion and position of a positive test charge in an electric field given by E = (E cos(cot), 0, 0) with initial velocity v(0) = (vx>0, 0, vz>0) and initial position at the origin.
(b) For a similar positive test charge in only a magnetic field given by В = (B cos(cot), 0, 0) with initial velocity v(0) = (vx0, vy>0, vz>0) and initial position at the origin, describe qualitatively and sketch the motion for 0 < t < 27t/co and co « cog.
5.4 Perform the integration suggested in Eq.(5.49).
5.5 Confirm the statement in Sec. 5.6 that integrating the solution of Eq. (5.56c), given appropriate initial conditions, over one gyration period yields a drift in the negativeV direction dependent upon vx.
5.6 As discussed at the end of Sec. 5.6, curvature drift and VBdrift will always occur together. Using the coordinate system of Fig. 5.10, approximatein analogy to Sec. 5.5 and 5.6the magnetic field strength B(y, z) = (0, By(y, z), Bz(y, z)) with a first order Taylorexpansion at the уaxis in order to account for both weak curvature and weak inhomogeneity (note that ЭВу/Эу = 9Bz/9z = 0 at the reference point). What simple condition relating the degree of curvature an of inhomogeneity is then required to satisfy Maxwell’s Eqs. (5.62) and (5.63)?
5.7 Demonstrate that the expression for the magnetic moment of an ion, Eq.(5.76), also follows from the basic definition of a current loop I encircling an area A (i. e., ц = IA), and also identify the units for the magnetic moment Ц.
5.8 Compare PnCyC with Pbr for a deuteriumtritium plasma of density Ne = N, = 1020 m’3 confined by a magnetic field of magnitude В = 6 T.
Consider the specific cases:
(i) Te= lOkeV; ул= 10’3
(ii) Te = 50 keV; yr= 10’2
and also compare Pbr + P%’c with P^ at the given conditions.
5.9 Consider a 50:50% MCF device having Tj = Te and a plasma beta value of 0.2, and for which ул ~ 10’3. Compute and compare the modified ignition temperature, i. e. the temperature at which Pbr + PTyC = С, л Рл, to T*ign of problem 3.6.
5.10 Calculate an expression for the power density of cyclotron radiation emitted from a dh fusion plasma of density N, + Ne and of kinetic temperature Т;~Те=Т by knowing that a particle of charge q and velocity v willaccording to classical electromagnetic theoryemit radiation at a power Prad q2  dv/dt 2.
(a) How much smaller is the cyclotron radiation power of ions than that of electrons?
(b) Compare the primary cyclotron radiation power density (that with no reabsorption) to the fusion energy release per unit time for the case of Nd = Nh = 2 x 1020 m’3, T = 80 keV and В = 6 Tesla, and discuss the importance of radiation reflecting walls in an MCF reactor.
(c) Plot PnCyC (accounting for reflection and reabsorption by the approximation
t//(T) ~ 10’4 2 [T(keV) / 1 keV]14) and Pbr as functions of T over the temperature range 10 to 120 keV.
An additional approach to confining fusion reactants, completely distinct from that of magnetically confined systems, is inertial confinement fusion which involves compressing a small fuel pellet to very high density by an intense pulse of energy. This compressive pulse of energy may be supplied by lasers or ion beams. We now investigate several issues fundamental to this approach to fusion.
The dominant power flow for steadystate hybrid operation is suggested in Fig.
15.2 and the corresponding station electrical output is given by
Pnet ~ Ць Ph, t Pin • (15.6)
Here Pbit* is the thermal power extracted from the blanket and converted into electrical form with efficiency гь while Pin is the input power supplied with efficiency Гщ to the device in order to sustain the fusion reactions.
Using an obvious definition for the blanket energy multiplication, we write
Pl, t = MbPn (15.7)
where Pn* is that component of Pdt* associated with the energy of the fusion — source neutrons entering that blanket; that is
p;=o.8p;(. (15.8)
The remaining 0.2 Pdt component associated with the alpha particles is assumed to be retained in the plasma.
giving for the total station power, Eq.(15.6), * * th, Pn * Pne, = rib Pn Mb———————— = ЦЬ Pn M fu, e 
Next, we define an effective fusioncomponent input power multiplication by
Note that upon introduction of the fusion plasma Qvalue, Eq. (8.6), which here for steady state is
the effective fusioncomponent input power multiplication is represented by
The significant result here is that the condition for energy viability of a hybrid is now
MbMfU, e>l (15.13)
and contrasts to a standalone fusion reactor for which, evidently, it is necessary to have MfUie > 1. Since the attainment of Mfux. > 1 is a demanding technical problem and since Mb > 1 seems readily possible, the hybrid would have the possible advantage that some critical plasma parameters could be relaxed when compared to a pure fusion system. This could be an important motivation for hybrid operation in the early development of fusion systems.
The important consequences of Eq.(15.10) for the hybrid are depicted in Fig. 15.5. Note that for Mb > 1, the energy multiplication of the fusion reactor, that is Mf^, can be less than unity and still provide for an energetically viable
overall system.
Fig. 15.5: Fusionfission hybrid electrical power as a function of the fusioncomponent input energy multiplication (Mfu e) and the blanket energy multiplication (Mb). Here, Pdt* = 2500 MW and ць = 0.3. 
It is most important to recognize that while the particles possess a range of velocities, speeds, and energies, the temperature T describes a particular distribution function and is a fixed parameter for a given thermal state; changing the temperature of the medium will alter the various moments of the function but its characteristic shape is retained, Fig.2.2.
Further, in a volume domain containing a mixture of particlesas in the case of a plasma containing electrons, various ion species, and neutralseach particle species may possess a different distribution function characterized by a different temperature. Then, however, the entire plasma is not in thermodynamic equilibrium. Indeed, in the presence of a magnetic field, even the same species may have a different temperature in, say, the direction parallel to the magnetic field lines than in the perpendicular direction. Several methods or devices used to obtain fusion energy involve plasmas that are just thatnot in thermodynamic equilibrium. Most that will be considered herein, however, are not so and thus we will rely on MaxwellBoltzmann distributions to characterize many of the plasmas that will be discussed in subsequent chapters.
Having a sufficiently accurate distribution function is of considerable utility.
For example, the most probable value % — that is the peak of the distributionis found by differentiating and finding the root of
Э M(Z)
In Eq. (2.17a), the subscript x is to suggest any one component of the vector v; hence v = 0.
Average values can similarly be found based upon the formal definition of
_ f
£ = 2————— . (2.18)
J M(Z№
Thus, for the three cases of interest here we get
Ух= jv* M(y)dx = 0 (i. e. v = 0) (2.19a)
and
E = J EM(E)dE = jkT (2.19c)
о
with the particles possessing three degrees of freedom.
The analysis leading to the depictions of Fig. 2.2 makes it clear that the temperature There in units of degrees Kelvin, Кis an essential characterization of a Maxwellian distribution; hence, the numerical value of T uniquely specifies an equilibrium distribution. It has also become common practice to multiply T by the Boltzmann constant к and to call this product the kinetic temperature, which is obviously expressed in units of energy, either Joules (J) or electron volts (eV) with the latter generally preferred. Using this product kT, a Maxwellian population at T = 11,609 К may be said to possess a kinetic temperature of 1 eV; similarly, a 3 keV plasma in thermodynamic equilibrium has an absolute temperature of 3.48ХІ07 K.
The convention of interchangeably using energy and temperature, wherein the adjective "kinetic" and Boltzmann’s constant in kT are commonly suppressed, may seem peculiar, but expressing a physical variable in related units is a very common practice. For example, travelers often use time as a measure of distance (s = vt) if the speed of transport is understood, test pilots often speak of a force of so many g’s (F = mg), and physicists often quote rest masses in units of energy (E = me2).
This convention of using the product kT leads to a number of uses which need to be distinguished; we note here several common cases: kT = (kinetic) temperature of a plasma; f kT = average energy of Maxwelliandistributed particles;
kT = most frequently occurring particle energy of Maxwellian distributed particles;
Our discussion of fusion bum thus far would seem to suggest that only simple collision considerations are relevant. However, fusion phenomena may include other intrinsic processes thereby increasing the fusion reaction rate, and a possible suppression of lessdesirable side reactions. We consider one such process next.
Neutrons and protons are known to possess an intrinsic spin. Quantum mechanical considerations characterize this spin as y2h with ti representing Planck’s constant. Then, when nucleons combine to form nuclei, the constituent spins add vectorially assigning a spin to the entire nucleus. The nuclides involved in dt fusion possess spins as follows in units of the Planck constant h: deuteron : 1 triton : Уг neutron : Уг alpha : 0 .
The direction of a nuclear spin is determined by reference to an external magnetic field with the convention that the (+) sign represents a direction parallel to the magnetic field, () refers to the antiparallel direction and (0) is for the transverse direction. We suggest these spins for a deuteron and a triton in Fig.7.6 where we also introduce the notation f 0 0 for the fraction of the respective nuclei oriented in the allowed directions; quantum mechanical considerations permit no
other directions.









It is known that spin conservation characterizes nuclear transformations. The maximum spin for 5He, produced momentarily as a compound nucleus when the reaction
г/ + г>(5Яе) >n + a (7.54)
occurs, is 3/2 in units of fi — occurring when the deuteron spin and triton spin are aligned. The microscopic fusion cross section for this process will be represented by Go and constitutes a maximum.
It is possible to write an expression for the dt fusion cross section Gdt in terms of G0 and the various fractional concentrations of deuterons and tritons in their allowed spin states, i. e., in terms of fd+, fd’, fd°, ft‘, and ft" as depicted in Fig.7.6. Referring to some specialized aspects of nuclear reaction theory, this cross section can be well represented by






and can be used to evaluate the extent to which Gdt approaches the maximum possible fusion crosssection, G0, for various mixtures of polarization of deuterons and tritons in a magnetic field.
Evidently, an equal fraction of fuel nuclei in each of their allowed spin states constitutes an overall completely depolarized state; such random polarization is defined by the ratios
f+ = L f’ = 1
J t 2 ’ J t 2
and the corresponding dt fusion cross section (Gdt)r, r is therefore given by
Thus, for this case of random spin polarization, the Gdt cross section equals 2/3 of the maximum possible value, G0
Consider now a parallel alignment of the deuterium and tritium nuclei. For this case we have
fd = l. fd = 0, fd = 0, f* = l, ft = 0 (7.58)
and hence
M+ + = {l + 0 + 0 + 0 + 0)oo=Oo ■ (7.59)
Comparison of this result with Eq.(7.57) indicates that a 50% increase in the dt fusion reaction rate density is achieved if the deuterons and tritons all possess a spin alignment in the direction of the magnetic field. By inspection, the same result occurs if the spins all point in the opposite direction, i. e.,
{ел)+ + = {о<к)__ ■ (7.60)
A further question of interest then is to consider a plasma for which the tritons spins are random and the deuterons are injected with a specified spin polarization. Setting therefore
f>i> fd=o, rd=o, /:=j, r=i (7.6i)
gives
M+, = [і + О + о + Щ) + оа0= }СТ0 (7.62)
as for Eq.(7.57).
Accelerators exist which could supply spin polarized deuterium or tritium ions into a plasma. Then, since the spin polarization can be sustained for time periods up to 10 s, a 50% increased fusion power density could be sustained. Additionally, it has been found that under conditions of polarized fusion, the neutron and alpha reaction products emerge with a preferred directional distribution requiring therefore special containmentwall considerations.
An assessment of the energy liberated in a nuclear reactionor in a sequence of reactionsrelative to the energy cost of causing that reaction, is fundamental for evaluating the attractiveness of any nuclear energy system. Recall that such a criterion was employed in the energy balance assessment for magnetic and inertial confinement fusion. We now seek to formulate a similar energy balance criterion for a muoncatalyzed dt system, i. e. a xdt system.
One unique feature of muoncatalyzed fusion is that muons have to be produced by appropriate accelerators and then directed into a liquid deuterium — tritium mixture. The accelerator beam energy requirements are such that the average energy cost of a muon is of the order of 3000 MeV. Then, since each dt fusion reaction catalyzed by a muon releases the usual 17.6 MeV of fusion energy, energy breakeven will require each muon to catalyze, on average, several hundred dt fusion reactions during its short lifetime of 2.2×10"6 s. Consequently, the rates at which the various processes of Fig. 12.2 occur are most important.
To establish a tractable formulation for this energy balance problem, we consider a unit volume of liquid deuterium and tritium into which muons are injected at a rate density Fig.12.3. These muons then initiate and sustain a complex reaction network as suggested in Fig. 12.2. The fusion energy so generated heats the fluidfuel mixture; this heat is transported with the moving fluid to heat exchangers for subsequent conversion. For the muonsustained reactions, it is known that an acceptable loss in accuracy results if the reduced reaction network of Fig. 12.4 is used instead of that of Fig. 12.2.
discussion of Sec. l2.2and vary with time according to the following system of rate equations: — K^jNfiNt Kfjd N у N d hfidt N ndt(l (12.15a) at 

^~ = — KtuN^NlKdt NД* N, + X» + F+l — F, at 
(12.15b) 

dNd. dt 
К fid N fi N d Kdt N fid Nt + Хц N fid F+d ~ Fd ■ Kfidt N fit N d 
(12.15c) 
dN fu _ „ . Aft N fit К fit N ft Nt Kdt N fid Nt Kfidt N fit N d at 
(12.15d) 

dN nd . Aft N fid К fid N ft Nd Kdt N fid N t at 
(12.15e) 

J Xfi N fidt Kfidt N fit N d Xfidt N fidt at 
(12.15f) 

dN П s w . A fidt N fidt at 
(12.15g) 

Щ2 = Хцл Nud, dOJ)+ Xu Nua at 
(12.15h) 

, ” Xfidt N fidti ^ ) XftNfta • at 
(12.15І) 
The densities of the various nuclear/atomic/molecular species in a considered reaction unit volume can then be formulated by inspectionbased on our 
Here, in Eq. (12.15b), the factor Kd. t is to account for the net reaction transfer from pd to pt, also shown in Fig. 12.4. These equations represent a socalled pointkinetics representation in the sense that any spatial effects can be
considered to be minimal.
By analogy to the energy viability analysis of magnetic and inertial systems, we consider a reactor chamber of volume Vr and an operating period x during which muon injection occurs. Then, the total energy supplied is
T T
El = V rFu (0 E^dt = Vr J F^iDdt (12.16)
о о
where Euc is the average energy cost of producing one muon. The total energy released as a consequence of muon injection can be written as
where now E^s is the average energy deposited in the medium per muon as the high energy muons are slowing down. Note the addition of the muon mean lifetime Тц to the injection period; this is necessary because a muon will continue
to catalyze nuclear reactions during its lifetime after injection has ceased.
Since the reaction energy Qdt is released whenever the (adt)fusion compound decays, we write for the fusion rate density in Eq.(12.17)
Here, the ratio ЕЦ5/ЕЦС accounts for the recoverable muon beam energy and is of the order of 1/10. Of greater interest therefore is the first term of Eq.(12.20) where Q* and E,,c are constants and the remaining factor defines the important parameter which represents the average number of dt fusion events catalyzed by one muon:
iriM
f^N^ftJdt
ZM = —T—————— • (1221)
J FM(t)dt
о
This parameter fa, called the muon recycle efficiency, is evidently of utmost importance and will be considered in detail next.