The basic relation which determines the motion of an individual charged particle of mass m and charge q in a combined electric and magnetic field is the equation

d

m— = qE + q(xB). (5.1)

dt

Here, v is the velocity vector of the particle at an arbitrary point in space and E and В are, respectively, the local electric field and magnetic flux density perceived by the particle; the magnetic flux density is commonly called the magnetic field. The last term in this equation is also known as the Lorentz force. We assume SI units so that the units of E are Newton/Coulomb while for В the units are Tesla; note that 1 Tesla (T) is defined as 1 Weber m"2 ( = 104 Gauss = 1 kg-s‘2-A’’).

While gravitational forces are of importance in stellar fusion, they are negligible compared to the externally generated magnetic field forces and the established electric fields associated with fusion energy devices of common interest; hence, gravitational force effects need not be included for our purposes.

The most widely known toroidal fusion device is the tokamak; this word is of Russian origin and is a contraction of toroidal, kamera (chamber), magnet and katuschka (coil). The essential features of a tokamak are suggested in Fig. 10.5, and the functions of the various components can be described as follows. The plasma torus is viewed as a single-winding secondary of a transformer. A current flow in the primary transformer winding will therefore induce a current in the plasma toms by transformer action. The resultant toroidal plasma current I provides for Ohmic heating and also generates the poloidal magnetic field B0. Further, the toroidal magnetic field coils generate the toroidal field B0 which is perpendicular to B0. These two magnetic fields combine by vector addition,

В = B^ +Be (10.9)

to provide the basis for the rotational transform in Fig. 10.3. Further, the stabilizing coils, Fig. 10.5, are to restrain the plasma from expanding as is the natural tendency of a current ring. Then, the total field vector В has to incorporate also this stationary field component. The expansion tendency arises from the difference between the poloidal magnetic pressure, B02/ 2|lo (Sec. 9.1), on the outside and the inside of the toms. Since B0 is larger on the inside, the associated higher magnetic pressure will steer the plasma ring towards an increased major radius. It is therefore seen that toroidal and poloidal magnetic fields alone cannot confine the plasma in a tokamak. Of additional need is a vertical magnetic field Bv, produced by the stabilizing coils shown in Fig. 10.5, in order to produce an inward directed j X Bv force which prohibits the outward expansion. If the toroidal plasma current is sufficiently strong, only a small Bv is required to stabilize the plasma. Hence, a tokamak may be characterized as a toroidal device featuring a large plasma current and a strong toroidal magnetic field such that

ВФ>Ве>Ву ■ (10.10)

The toroidal plasma current needed to supply the poloidal magnetic induction

is generated by a toroidal electric field E = (0,Еф,0) which is achieved here by the transformer’s temporally changing magnetic flux 4Vans that penetrates the hole in the toms.

I Stabilizing

The connection between the various physical field quantities involved is readily evident from the following. Since one of Maxwell’s Equations tells us

V x E= ,

dt

we conclude from Stoke’s Law,

that the work done per unit charge by the induced electric field over a closed path along the secondary transformer winding-which essentially is the plasma ring itself-is equal to the rate of change of magnetic flux, 4/(Tans, through the closed winding loop, i. e. through the hole in the toms. Hence, ds = 2к dR еф here and A=7tR2=constant, which has been assumed in Eq.(10.12). The induced toroidal electric field is then found from Eq.(10.12) to be of the magnitude
2nR dt

The flux change is achieved either by an air-core or an iron-core transformer as shown in Fig. 10.5. It follows then that a tokamak cannot operate in steady — state but only as a pulsed device unless non-inductive schemes of plasma current drive can be applied. The density of the plasma current driven by the induced toroidal electric field is subsequently found via Ohm’s Law which can be generalized for a conducting fluid in a B-field to be

T] j = E + VxB (10.14)

where T) is the specific resistivity of the plasma and V is the fluid velocity as defined and used in Eq.(6.21). Additional current terms can occur in Eq.(10.14), which, however, are of second order and therefore not itemized here. The resistivity of a thermal plasma is governed by collisional kinetic effects and can be shown from a plasma physics consideration to vary as

т)~{кТе)Ш • (Ю.15)

As the plasma is heated, the Coulomb cross section as vr’4 (Eq. 3.15) decreases and consequently the resistivity, which is proportional to <asvr>, drops accordingly.

Note that the induction of a plasma current suggests an easy way to heat the plasma; that is through Ohmic dissipation

Рон= i71 j2d’r , (10.16)

Volume

where the symbol OH refers to the common label of this effect. It is evident from the proportionality given in Eq.(10.15) that this heating method becomes less efficient at higher plasma temperatures (>1 keV) and will not suffice up to the required thermonuclear temperatures (~10 keV). For that, other methods must be employed of which some we discussed in Sec. 8.2.

Another effect associated with electric field induction in a plasma is noted here. The frequency of collisions between electrons and ions (compare with 1/Тц of Eq.(9.41)) is

(t«) 7 06 a, vr00 vi3 (10.17)

and shows that high-energy electrons undergo relatively few collisions and therefore predominantly carry the induced current. Consider now an electron from the high energy tail of its velocity distribution which moves in the direction opposite to E. Due to the low collision frequency, it will gain further energy making thus a collision with an ion even less likely. This in turn allows it to be further accelerated by E. This phenomenon is called ‘electron runaway’. If E, and hence the velocity gain of the electrons, is sufficiently large, the Coulomb cross section drops so quickly that these runaway electrons never encounter a collision and thus form a beam of accelerated electrons disengaged from the main part of

Closed Magnetic Systems the distribution.

We next consider a pulsed d-t fusion reactor for which, as for the previous case, tritium is supplied externally and ideal confinement in the fusion core exists. This pulsed bum mode can well be characterized by three stages of tritium injection, tritium bum, and tritium purging, Fig. 14.2. During injection and purging, some of the tritium will be lost by particle transport into the containment components while during the bum, tritium is destroyed by the fusion process. It is the bum stage that is of most interest to us.

As for the preceding continuous-bum model, we begin with a dynamical description during the bum time Ть — A rapid pulse implies a rapid change in plasma temperature so that sigma-v during the bum cannot be taken to be a constant and we will therefore represent this time dependence by <crv(t)>dt. The consequence of this is that the fuel inventory will vary similarly with time during the bum so that the time dependent reaction rate density Rdt is

R*(t) = Ndc(t)Ntc(t)<av(t)>dt. (14.15)

The dynamical equations for the fuel density in the core are simply

dNdc

-j^ = — Rds(t), (14.16)

at

and

dN

~-^- = — Rd,(t). (14.17)

dt

Note, here, the absence of any inflow-outflow terms during the bum time. The instantaneous d-t power in a unit volume at any time in the bum interval is

Р*(Ч = Ялтл = Nd c(t)Nt c(t)< ov(t) >dt Qdt. (14.18)

We cannot proceed further without a knowledge of the time dependence of <crv(t)>dt. Recalling the definition of this sigma-v parameter, Eq.(2.29) implies that <crv(t)>dt could be determined if the time variation of the deuterium and tritium velocity distributions were known. This may, in principle, be obtained from a detailed time dependent analysis of how the pulse-injection energy is transformed into fuel kinetic energy together with other concurrent energy

transformation processes. However, this analysis is both difficult and tedious, and falls outside the scope of our objectives here.

Still, even without a detailed knowledge of the pulse energetics, we can outline the dominant time variations of the tritium fuel. Consider then, at the beginning of a typical bum cycle, an injected energy pulse spread over a short time period relative to the fusion power pulse. Ionization occurs promptly and the kinetic energy of the fuel ions rises rapidly to initiate fusion reactions and thus triton destmction by fusion. The alpha particle fusion products may thereupon continue to heat the ions to sustain fusion power production. Eventually, a variety of power losses-leakage, radiation etc.-will cool the plasma until the power pulse can be considered terminated. We suggest some of the time variations for a typical pulse in Fig. 14.3.

Observations of natural and induced processes have shown that numerous types

of fusion reactions for which Q > 0 can be identified. The variables for different

*

reactions are the interacting nuclides , the reaction products which emerge, the Q-value of the reaction, and the dependence of the probability for the reaction to take place on the kinetic energy of the reactants. The fusion reaction most readily attainable under laboratory conditions and which is expected to be the first used for power generation purposes is the d-t reaction

d + t-^n + a + 17.6 MeV. (1.20)

Another most accessible fusion reaction involves deuterium nuclei as fuel:

p + t + 4.1 MeV n + h + 3.2 MeV

where h is chosen to represent the helium-3 nucleus (3He2+). This representation may appear somewhat unusual, but is seen to simplify notation in subsequent chapters. Equation (1.21) shows that d-d will fuse via two distinct reaction channels known to occur with almost equal probabilities at specific reaction conditions. Further, fuel deuterons may also fuse with two of the reaction products (tritons and helium-3) giving, in addition to the reactions of Eqs. (1.21) and (1.20),

d + h-^> p + a + 18.3 MeV . (1.22)

The above fusion reactions involve deuterons and the successively more massive light nuclides. Continuing along this pattern, a large number of reaction channels have been identified in specific cases of which d-6Li fusion is an example:

Appendix В displays the light-nuclide part of the Chart of the Nuclides.

1 Be + n + 3 A MeV

1 Li + p + 5.0 MeV p + a + t + 2.6 MeV 2a + 22.3 MeV h + a + n + 1.8 MeV

Here, each reaction channel possesses a unique probability of occurrence.

Fusion reactions involving the lightest nucleus, that is the proton, may occur according to the processes

■ h + a+ 4.0 MeV Г a+6Li + 2.1 MeV d + 2a + 0.6 MeV

p+uB -+3a + 8.7MeV (1.24c)

as well as others. Some reactions based on t and h are

t +1 —> 2n + a +11.3 MeV (1.25a)

h + h—>2p + a + 12.9 MeV (1.25b)

and

t + h —> n + p + a + 12.1 MeV . (1.25c)

Several features associated with fusion reactions need to be noted. First, the physical demonstration of a fusion reaction is not the only consideration determining its choice as a fuel in a fusion reactor. Other considerations include the difficulty of bringing about such reactions, the availability of fusion fuels, and the requirements for attaining a sufficient reaction rate density.

Another feature of the various fusion reactions listed above needs to be emphasized: in each case a different fraction of the reaction Q-value resides in the kinetic energy of the reaction products. Thus, a fusion reactor concept based on high-efficiency direct energy conversion of charged particles would appear particularly suitable for those reactions which are characterized by a high fraction of the Q-value residing in the kinetic energy of the charged particles. This is of particular interest because the neutrons appearing as fusion reaction products invariably induce radioactivity in the materials surrounding the fusion core.

Third, the fusion fuels are evidently the light nuclides displayed on the Chart of the Nuclides. In a subsequent chapter we will show that a subatomic short­lived particle called a muon and produced in special accelerators, may also play a role as a fusion reaction catalyst.

Most current fusion research and development activity is based on the expectation that the d-t reaction, Eq. (1.20), will be used for the first generation fusion reactors. While the world’s oceans as well as fresh water lakes and rivers contain an ample supply of deuterium with a particle density ratio of d/(p+d) ~ 1/6700, tritium is scarce; it is a radioactive beta emitter with a half life of 12.3
years, with the total steady state atmospheric and oceanic quantity of tritium produced by cosmic radiation estimated to be on the order of 50 kg. Since a 1000 MWt plant will bum about 250 g of tritium each operating day, a station inventory in excess of 10 kg will be required for every d-t based central-station fusion power plant so that other sources of tritium fuel are required.

The main source of tritium is expected to be its breeding by capture of the fusion neutron in lithium contained in a blanket surrounding the fusion core. The relevant reactions in 6Li and 7Li are

n+6Li—>t + a (1.26a)

and

п+7/л—> t + cc + n (1.26b)

with the latter possessing a high energy threshold Ethresh ~ 2.47 MeV. Lithium-6 and lithium-7 are naturally occurring stable isotopes existing with 7.5% and 92.5% abundance, respectively, and exist terrestrially in considerable quantity.

Additional sources of tritium may involve its extraction from the coolant and moderator of existing fission reactors, particularly heavy water reactors, where tritium is incidentally produced by neutron capture in deuterium via

n+2#—»3#. (1.27)

Of course, tritium could also be produced by placing lithium into control and shim rods of fission reactors.

Reaction (1.26b) is particularly interesting because the inelastically scattered neutron appearing at lower energy can continue to breed more tritium. Thus, in principle, it could be possible in such a system to produce more than 1 triton per neutron bom in the d-t reaction. Indeed, present concepts for d-t reactors generally assume a lithium-based blanket surrounding the fusion core that allows for tritium self-sufficiency. These and additional concepts will be discussed in subsequent chapters.

In addition to the particle-fluid characterization of a plasma, one may identify probabilistic methodologies which, in selected applications are most useful. We discuss here one such approach to the assessment of particle leakage from a fusion reaction volume.

Consider a volume V containing N j(t) particles of the j-type. This population changes with time because of gain and loss reaction rates, R +j and R _j, respectively, and also because of particle injection and leakage rates, F*+j and F*_j, Fig.6.2. The rate equation is evidently

^=(/?;,- — r-і)+(fIj — f:}) . (6.43)

Our interest is specifically in the particle leakage rate F _j and we wish to explore an approach which has no need for a detailed space distribution of the particle population; that is, we consider a probabilistic formulation in an integrated global sense for this arbitrary volume.

As our underlying physical basis, we adopt the proposition that the statistically expected fractional change in the population due to particle leakage is — AN*j/N*j in an interval of time At, and is only a function of time. This statement can be expressed in algebraic form as

where A, j(t) is a positive function of time to be considered subsequently. For now, we take the limit of At —> dt to write

(6.47)

However, this expression tells us little about the meaning of A-j(t) nor does it provide a suggestion on how to calculate it. This can be resolved if we introduce some probabilistic considerations.

Consider the probability density function fj(t) which describes the outcome that a particle j remains in the volume V over a time t until it escapes from the ensemble. Hence, this probability is given by the product of the following probabilities which also define fj(t):

L V о )_

Note that the substitution here follows directly from Eq.(6.44). Thus, the probability density function of interest is evidently

(6.49)

with the normalization J f j(t)dt = 1 as required.

о

With fj(t) now specified, we may compute some interesting time dependent quantities. Of particular interest is the particle mean residence time until leakage. Using x j for this quantity, we may compute it as

P.=

(6.53)

Problems 6.1 Specify the conditions which reduce Eq.(6.21) to its simplest form while still retaining electric and magnetic field effects. 6.2 Consider a particle density N(x, y, z, E, 0, ф) and formulate a reduced transport equation, Eq.(6.42), for each of the following cases: (a) neutral particles, (b) monoenergetic particles, (c) only the z-direction is relevant, and (d) both the source and collision terms are given. 6.3 Consider a plasma contained between two walls xQ units apart and effectively infinite in the у and z-directions. For the case that the plasma decays by diffusion to the walls and for separability of the form N(r, t) = R(r) T(t), determine N(r, t) throughout this plasma slab region for all time. Sketch this density function.

6.4 Confirm that for a cylindrically-shaped, magnetically confined plasma wherein radial diffusion is the dominant mode of transport, assuming a constant volumetric ion source, the steady state ion density will have the following relationship:

N(r)=N(0)jl-^, (6.54)

where N(0) is the ion density at r = 0, and a is the plasma minor radius. (Note that for these conditions, Eq.(6.10b) becomes VxJ(r) = S = constant.)

6.5 Given Eq.(6.54) for the ion density radial distribution, derive the average ion density.

In the discussion up to this point, little reference to the specifics of beams or target design has been made. As indicated, however, the requirements of pellet compression and beam-target coupling impose some very stringent demands on beam energy and the details of pellet composition.

High powered glass lasers have been used most frequently in inertial confinement fusion (ICF) research. The available beam intensity, focusing capability, state of technological development, and general availability are responsible for this popularity. Table 11.1 contains some properties of current lasers and, for comparison, also lists estimated requirements for actual fusion reactors.

Concerns about the eventual applicability of lasers to inertial confinement fusion has prompted considerable research on the potential role of ion

accelerators for such purposes. The prospects of enhanced localized beam energy deposition and control over beam energy while avoiding the previously mentioned undesired preheating by "hot" electrons produced in the laser light absorption process makes this alternative very promising. However, accelerators do introduce another set of problems among which are beam focussing for high- current accelerators as well as the need for large high-vacuum ion transport facilities. Table 11.2 lists some ion accelerator characteristics.

The composition of a pellet constitutes some interesting analysis and design problems. The existing types can be loosely grouped into three categories: (1) glass microballoons, (2) multiple shell pellets and (3) high-gain ion beam pellets.

Glass microballoons consist of thin walled glass shells containing a D2-T2 gas under high pressure, Fig. 11.3. The incident beam energy is deposited in the glass shell causing it to explode with part of its mass pushing inward and the remaining mass outward. Though microballoons are widely used in experiment, more efficient designs will eventually be needed for power plants.

Multiple-shell pellets contain an inner deuterium-tritium solid fuel core
surrounded by a high-Z inner pusher-tamper. Next is a thicker layer of low density gas surrounded by a pusher layer. Finally, a low-Z ablator material forms the outer layer, Fig. 11.3. This complex layer structure is designed for very specific functions. For example, the outer layer is to ablate quickly and completely when struck by the incident beam; the inner high-Z pusher-tamper is to shield the inner core region against preheating by hot electrons and X-rays.

Glass Microballoon Pellet

Low-Z Ablator Multiple Shell

Outer Pusher Pellet

Void

High-Z Pusher-tamper D-T Fuel

High-Gain Ion Beam Pellet

Fig. 11.3: Cross section of selected pellets for inertial confinement fusion.

More recently, heavy ion beam-pellets have been developed which depart in significant ways from the microballoon and layer shell design. In these designs, a vacuum sphere is surrounded by a D2-T2-DT fuel shell which is then surrounded by tamper-pusher materials, Fig. 11.3. These pellets are designed specifically for ion beam inertial confinement fusion with the thickness of the tamper-pusher materials carefully matched to the type and energy of the incident beams.

Major objectives of these designs are to optimize energy transfer, minimize hot electron production, and reduce requirements for symmetric beam energy deposition.

Research and developmental activity towards the realization of fusion reactors has emphasized, first, magnetic confinement fusion and, second, inertial confinement fusion; low temperature fusion is a distant third. Concurrently, variations on these three as well as alternative concepts continue to appear and lead to some interest and research support. We consider two such approaches: one in the "mechanical" forcing of fusing reactions-impact fusion-and the other involving the complete transformation of the mass of the interacting particles into energy-annihilation energy.

Impact fusion is based on the notion that a macroparticle containing hydrogen atoms will-when accelerated to sufficiently high speed and made to impact upon a hydrogenous target-lead to a number of fusion reactions. An estimate of the required speed can be obtained as follows. Consider a macroparticle of mass M and speed v hitting a target. The kinetic energy of the atoms in the projectile must be of the order of the Coulomb barrier, U0= U(R0) of Ch. 2, in order for fusion reactions to occur. For the N* atoms in the projectile of mass M, we should

Then, using U0 ~ 400 keV and ma ~ 3.3 x 10’27 kg for deuterium, yields a speed of v ~ 6200 km/s.

A speed of 6200 km/s is, by terrestrial standards, exceedingly high; recall that the speed of a bullet is < 1 km/s and the escape speed from the earth is ~ 11 km/s. Hence, the use of chemical explosives to attain such high speeds seems unlikely. One might, however, consider electromechanical means as suggested by the following.

Consider extending the principle of nuclear particle accelerators to macroparticles. Suppose a charge of Q is established on a projectile of mass M in a space characterized by a constant potential difference Ee per unit length. The force relation is evidently

M — = QEe (16.35a)

dt

and its integration using ds = v dt leads to

Here, L is the total path length. Currently attainable electric field strengths Ee, a reasonable length L, and assuming a constant Q/M suggest that speeds in the range of — 100 km/s are achievable.

A working device for such electromechanical acceleration is known as a rail gun or electromagnetic launcher, Fig. 16.6. As suggested therein, a current flows through a circuit of conducting rails and movable armature to which the projectile is attached. An increasing current flow generates a time varying magnetic field which, by its Lorentz force on the conductor and for the case of rigid rails, accelerates the projectile towards a stationary target.

Numerous other concepts have been suggested to aid in high speed attainment. Among them are speed multiplication by momentum conservation, electromagnetic energy focusing and the use of losses to generate high speed ablation.

Finally, we consider "fusion" energy by annihilation.

It is interesting to note that for either fission or fusion, the mass transformed into energy represents a small fraction of that of the interacting particles; typically

(16.36)

and is thus very small.

There exists, however, one type of reaction which converts all of its mass into energy: annihilation. Annihilation occurs whenever a particle meets its antiparticle. For example, a proton p and its antiproton p combine-without any external force effects-to yield

p + p —> 1456 MeV

with all subnuclear particles decaying very quickly to high energy gammas.

While the conversion of matter into energy is here complete and hence the energy release per initial mass of particles is a maximum, energy must still be

supplied: antiprotons p do not "exist" naturally and have to be produced in high energy accelerators.

Thus, human ingenuity continues to be at a premium in the attainment of this ultimate source of energy.

[1]

x

Fig. 5.7: Positive ion and electron drift in a combined uniform magnetic and electric field.

We now consider a generalization of the drift velocity caused by an arbitrary force F on a charged particle moving in an uniform В-field. To begin, we decompose the particle velocity into the components

v = Vsc+Vs (5.30)

where vgc is the velocity of guiding centre motion and vg is the velocity of gyration relative to the guiding centre. The equation of motion is now written as

m-^+m~[f=F + q(vgcx’B)+q(vgx B). (5.31)

Evidently the terms describing the circular motion, i. e. the second term on the left and the third on the right cancel one another according to Eq.(5.6). Further, in a static field exhibiting straight В-field lines the charged particle motion will be such that, averaged over one gyroperiod X, the total acceleration perpendicular to В must vanish; that is, the guiding centre is not accelerated in any direction perpendicular to B. Therefore, we average the transverse part of Eq.(5.31) over

[2] Note that from here on, the subscript kin, which was used in Eq. (4.11) for instruction and distinction purposes, is dropped.

Though a plasma is globally neutral, it may well acquire local charge variations which establish an electric potential and give rise to an electric field. The thermal motion of ions and electrons will therefore be influenced by the consequent force effects. An indication of the spatial extent of such an effect is represented by the Debye length. The following analysis yields a useful explicit expression for this important parameter.

Consider a dominant positive charge or electrode inserted in a plasma, Fig. 3.4. Due to the mobility of electrons, a negatively charged cloud will immediately form around this point with its density decreasing with distance. Similarly, a negative charge will also create a positive ion cloud spherically symmetric about it. Obviously, a plasma tends to shield itself from applied electric fields; that is, if an electrode is inserted into a plasma it will affect only its immediate surroundings. If the plasma were cold, one would observe as many charges in the surrounding cloud as are required to neutralize the inserted charge, Fig. 3.4. However, due to the finite plasma temperature, the plasma particles possess a substantial kinetic energy of thermal motion so that some-particularly those at the edge of the cloud-will escape from the shielding cloud by surmounting the electrostatic potential well which, as is known, decreases with increased distance r.

Evidently, we need to determine some characteristic shielding range and consider for that-over a small distance r from an inserted charge-some little
perturbation of the electric charge density in the plasma. Beyond this range, a uniform neutral plasma continues to reside. The local electric field E thus established is related to the local charge density pc by Maxwell’s First Equation

V’E = — .

£o

With this conservative field, a scalar potential function Ф is identified and related to the electric field by

E = — VO. (3.17)

By substitution in Eq.(3.16) and specializing for the case of interest, Fig. 3.4., Poisson’s Equation takes on the form

PC

£o which, upon introducing the definition

pc = iN і >

J=e, l

is written for a plasma containing only one species of ions as у 2^_4eNe +4iNi _

£0

in which qi and qe are the ion and electron charge, and N; and Ne are the local ion and electron densities, respectively. Determining the potential function Ф requires a knowledge of N; and Ne as functions of position or of Ф.

Fig. 3.4: Local charge variation in a plasma upon insertion of two dominant point charges.

The particle densities Nj, however, at thermodynamic equilibrium and in the presence of a potential energy Ф are known to depend upon Ф, the specific charge and the equilibrium temperature Tj according to the Boltzmann relation

with the factors Cj being determined from the evident affinity Ф—>0 as r—to represent the undisturbed background densities Nj(r—>°°). Equation (3.21a) may then be expanded by a Taylor series to give

This refers to the region where I qjO / kTj I « 1, which is actually the dominant contributor to the thickness of the shielding cloud. Retaining only the linear terms of the expansion and substituting into Eq. (3.20) yields

where we have used the charge-neutrality boundary condition 4eNe (r °°) + <hNi (r °°) = 0 •

The defining equation for the electrical potential function Ф, Eq. (3.22), is evidently of the form

, Ф У2Ф — — = 0 Xd

and for screening electrons by ions by

Хйі=

we realize the relation

(3.27)

/y*De Di

Taking Te = Tj = T and assuming the presence of singly-charged ions only, that is Nj(r—>°°) = Ne(r—>°o) = N/2, notably simplifies Eq. (3.25) to the expression

The important dependence of Xd is therefore

(3.28b)

with typical values of interest to thermonuclear fusion being in the range of about

1 pm to 1 mm.

The role of A*d may be interpreted as a "shielding length" parameter for a plasma and becomes evident by solving Eq.(3.22) for the boundary conditions

Ф(г = 0)= ф„ and Ф(г —>°°) = 0. (3.29)

which may be compared to the free-space potential given by

Thus, the potential Ф associated with the imposed electrostatic perturbation is attenuated in a plasma according to the magnitude of A, D and is commonly said to be shielded to the distance of the Debye length.

This A*d parameter is thus a useful concept and application of it relates to the elimination of the cross section singularity of Sec.3.2 as we will show next.

While the Lawson Criterion represents a reactor criterion, i. e. it refers to the energy viability of the entire plant, we can as well derive a criterion for the
energy viability of the fusion plasma. The latter may be deemed as energetically viable when external power has no longer to be delivered to it. This so-called ignited plasma state can be shown for a homogeneous plasma, as was also assumed for Eq.(8.28), with help of Eqs.(8.10) and (8.11) to be characterized by

ІсЛ Ге’ЦРьг + P;1)te> + 3NT (8.29)

where again Nj = Ne = N, Tj = Te = T was taken and the time integration was performed over a period equal to the global energy confinement time Te*. As in deriving Eq.(8.28), we can similarly substitute for the above power terms by their explicit density-temperature dependent expressions, take Nd = Nt = N/2 and solve

which represents a fusion plasma ignition criterion and does not contain any energy conversion efficiencies. It is displayed in its temperature dependence in Fig. 8.4. A contour plot similar to the Lawson Criterion becomes evident, however featuring quantitative differences. The minimum ignition temperature for N = 1020 m’3 is seen at T~ 30 keV and requires a product of plasma density and global energy confinement time of NTe* = 2.7 X 1020 m’3s and hence Te* = 2.7 seconds.

We note that the denominator in Eq.(8.30) can become negative for high temperatures where the cyclotron radiation terms takes over to dominate the plasma energetics. Such a regime is associated with a negative plasma energy balance and can therefore not be ignited. For that temperature range, Eq. (8.30) has no meaning. The more stringent ignition requirements with increasing T are evident from Fig. 8.4 where the ignition contours tend towards infinity as the plasma temperature approaches the critical value Tcrit, as indicated for the case of N = 1020 m‘3.

Another definition often used for classifying the fusion reactor operation is the scientific (energy) break-even which means that the total fusion energy production amounts to a magnitude equal to the effective plasma energy input,

i. e.

—^r=QP = l• (8-31)

Ліп Ein

This break-even condition can be analogously derived from Eq. (8.9) for Qp=l and is also demonstrated in Fig. 8.4. Recall that the ignition condition is associated with Qp —>

The performance of a high NTe* is not the only goal in fusion reactor research; as the preceding discussion suggests, it is additionally required for a

viable fusion plasma regime that, simultaneously with NTe*, the plasma temperature is established at a sufficiently high level. Hence, presently, the so — called triple product TNTe* is used most often for qualifying recent achievements of fusion experiments. This last ignition criterion is easily obtained by just multiplying Eq.(8.30) with T on both sides. Expectedly, it does not introduce a new characteristic, but rather exhibits a similar temperature dependence as in Fig. 8.4.

Problems

8.1 Reformulate the Lawson Criterion which incorporates the direct conversion of the alpha particle into electricity with 95% efficiency.

8.2 Discuss and compare the plasma energy kT ~ 15 keV associated with the minimum Nte* requirement given by the Lawson Criterion, with the recognition that a kinetic energy of Ek ~ 200 keV is required by hydrogen ions to overcome their Coulomb repulsion in order to fuse.

8.3 Develop Lawson Criteria for a d-d plasma (include both branches of the d-d reaction), and plot Nx vs T. Assume a fusion power to electrical conversion efficiency of 30%.

8.4 Develop Ignition Criteria for a d-t, d-h and d-d plasma, and plot Nx vs T for each. Assume alpha-particle heating of the d-t plasma, alpha-particle and proton heating of the d-h plasma, and triton, proton and h heating of the d-d plasma.

8.5 Derive-analogously to Eq. (8.28)-a more realistic MCF reactor criterion accounting also for cyclotron radiation losses Display graphically its temperature dependence, NexE* (T) for the two cases

(a) d-t: Nd = N, = Ne/2, T) = Te, |//Ne = 10’23 m3, В = 5 T, r|in = 0.5 and tlout=0.35,

(b) d-h: Nd = Nh = N,/2, T, = Te, |//N e= 1022 m3, В = 10 T, riin = 0.5 and

Bout-

Note: For the derivation of the criterion, use fractions of the ion density such that Nj = KjNi with j denoting the considered ion species. You should finally obtain:

ItCjZj

for j = a, b, impurities. Compare the plots from (a) and (b) with those that result if cyclotron radiation losses are neglected.

8.6 Discuss the physical differences between (i) the Lawson reactor criterion, (ii) the ignition criterion and (iii) the break-even condition. What fraction of the entire plasma loss power can be made up for by oc-particle heating in steady state operation, when Qp = 5?

Our previous discussion of magnetic confinement fusion (MCF), inertial confinement fusion (ICF) and muon catalyzed fusion makes it clear that each represents a different set of physical conditions requiring that the blanket design must be adapted to best suit each case.

The MCF plasma domain can be characterized by a low density gas (~1021 particlesm’3) at a high temperature (~108 K). These properties lead to some very specific design and operation requirements. Evidently, all fueling and diagnostic wall penetrations into the core must sustain a laboratory vacuum. Then, the need for a high plasma temperature requires that the plasma must be separated from the wall surface. Concurrently, the intense neutron flux striking the wall demands that its composition be highly resistant to neutron damage. Finally, provisions have to be made for the removal of the neutron and radiation energy deposited in the blanket. Figure 13.1 suggests some of the essential components and features of an MCF blanket system.

Different conditions apply to an ICF device. In this case, reactor operation involves target pellets on a ballistic trajectory which are struck by extremely intense pulses of ions or electro-magnetic radiation (lasers) when the pellet reaches a specific location. As a consequence of the resulting ablation, compression, fusion, and disintegration of the target, an intense pressure and radiation shock wave is generated which must be absorbed in the first wall. A favoured approach is to protect the wall by a liquid metal so that the debris and

radiation from the explosion are then absorbed in this layer, thus providing both shielding and a moving heat transfer fluid. We suggest this scheme in Fig.13.2.

Magnets and/or Other Auxiliary Equipment

Flowing Liquid Metal

Chamber Wall

In contrast to the above MCF and ICF, a muon catalyzed deuterium-tritium ((XDT) fusion reactor blanket domain may be depicted as suggested in Fig.13.3. Its dominant feature is a central channel containing a deuterium-tritium oxide mixture either in liquid or two-phase form. The reaction product alpha will be retained in the flow but the penetrating neutrons will enter the surrounding blanket region. The first wall may thus possess many of the properties of cladding associated with existing fission reactors.

Liquid or Two-phase D20, DTO, T

A feature common to all deuteron-triton burning fusion systems is the need to breed tritium. Hence, the blanket domain must contain a concentration of lithium, either as a pure substance or in compounds, so that the neutrons emitted in d-t fusion are captured in the lithium to produce tritium. Such a nuclear reaction linkage may well be represented by the following:

(13.1)

Should tritium be available from other sources, such as existing fission reactors, or if additional neutron multiplication is sustained by incorporating suitable (n, xn)-type materials in the blanket, then there may be an excess of
neutrons over that required for tritium breeding. These surplus neutrons could then be used to breed fissile fuel by capture in fertile nuclei. Such a system concept will be discussed in Ch. 15; for now we consider further details of the MCF, ICF and |lDT blankets.