The general fusion reaction, Eq.(1.7) and Fig. 1.2, may be more completely characterized by noting that an unstable intermediate state may be identified in nuclear reactions. That is, we should write

a + b —> (яЬ) —^ d + є + Qafj

where (ab) identifies a complex short-lived dynamic state which disintegrates into products d and e. The energetics are determined according to nucleon kinetics analysis, with nuclear excitation and subsequent gamma ray emission known to play a comparatively small role in fusion processes at the energies of interest envisaged for fusion reactors.

Two-body interactions can be examined from various perspectives. For example, Newton’s familiar law of gravitational attraction applies to any pair of masses ma and ть to yield a force

(2.2)

effective on particle a. Here, G is the universal gravitational constant and r = ra — rb is the displacement vector between the two interacting particles, while r denotes its absolute value. While this force expression is universal, a simple calculation will show that for nuclear masses of common interest, this force is significantly weaker than the electrostatic and nuclear forces associated with nuclides and hence can be neglected.

The important electrostatic force between two isolated particles of charge qa and qb separated by a distance r in free space is determined by Coulomb’s law, given by

(2.3)

for the electrostatic force felt by particle a; here £<, is the permittivity of free space and the factor An is extracted from the proportionality constant by reason of convention. This force-repulsive for like charges and attractive for unlike charges-is of considerable importance in fusion.

From the definition of work and the phenomenon of energy stored in a conservative field, the work done in moving a particle of charge qa from a sufficiently distant point to within a distance r of a stationary charge of magnitude qb, is the potential energy associated with the resultant charge configuration. Specifically, this is given by

Г

U(r)=jFc, a(r’)dr’

і dnEo (r’f

_ 1 ЯаЯь

dnEo r

subject to the restriction that the particle distance of separation r satisfies r > Ra + Rb where Ra and Rb are the equivalent radii of the two charged particles. For nuclides of like charge, the potential energy at approximately the distance of "contact" R0 = Ra + Rb, is called the Coulomb barrier and, in view of Eq.(2.4), is given by

1 <la<lb

4nEo(Ra + Rb)

On the basis of electrostatic force considerations only, this then is the minimum kinetic energy an incident particle would have to possess in order to overcome electrostatic repulsion and come close enough to another particle for the short — range nuclear forces of attraction to dominate. For deuterium ions, this energy can be calculated to be about 0.4 MeV, depending upon the precise value for R0. A useful approximation is R0 « Rp(Aa/3 + A{,/3) where Rp = (1.3-1.7)Xl015 m denotes the radius of a proton which cannot be assigned a definite edge for quantum mechanical reasons.

Consideration of quantum mechanical tunneling provides for a non-vanishing probability of penetrating the Coulomb barrier with energies less than U(R0). The probability for this penetration varies as

Pr( tunneling ) °С — exp — Y

Vr V Vr )

where vr is the relative speed of the moving particles and у is a constant. Thus,
even at very low energy, a nucleus possesses a small, though finite, probability of compound formation with another nucleus. This compound can decay into fusion products and hence, some fusion reactions will also occur at room temperature, though at an insignificant rate.

At sufficiently small distances, r < R0, the attractive strong nuclear force dominates and a compound nuclear state is formed. The kinetic energy of the initiating particles together with the resultant nuclear potential energy is then shared by all the nucleons. Nuclear stability considerations thereupon determine if and how the nucleus disintegrates. Figure 2.1 provides a graphical representation of these effects.

The use of only deuterium as a fusion fuel introduces several important considerations. For temperatures up to about 200 keV, the d-d reaction parameter sigma-v is substantially smaller (roughly two orders of magnitude) than it is for d-t fusion, Fig.7.5; hence, for equal particle densities and ion temperatures below about 200 keV, a d-d fusion reactor will possess a much smaller power density and hence will require a larger size for a specified total fusion power production. Note additionally that radiation losses tend to increase with higher temperature, thereby further contributing to problems of a viable power balance. Nevertheless, deuterium based fusion possesses some very appealing properties. For example, an important feature of the deuterium fueled reactor is that the products from d-d

fusion may fuse with the deuterium fuel and possibly even among themselves.

The primary d-d fusion reaction proceeds via two (almost) equally likely reaction channels

‘ t + p

h + n

where for notational simplicity, we use h = 3He. The bred tritium and helium-3 nuclei possess significant sigma-v parameters to fuse with the deuterium fuel, Fig.7.5. The reaction

d + t —> a + n (7.36)

will be dominant with

d + h —у (X + p

also taking place. In addition, there also exists the possibility of the reaction products to fuse among themselves to yield

t + t -» 2n + a (7.38a)

p + n + a

t + h^>

d + a.

Reactions involving reaction products are often called side reactions.

A complex system of linked reactions may possibly emerge. The following suggests a classification of d-d sustained reaction systems.

(a) PURE-D Mode

The idealized case of d-d fusion only is given by

d + d —> t + p (channel -1) (7.39a)

d + d —> h + n (channel — h) (7.39b)

which proceed at the rates

Rm = ~L<w >ddl, Qddt = 4.1 MeV (7.40a)

Rcuih = ~<(TV>dd, H. Qddh = 3.2 MeV (7.40b)

where the reaction Q-values are extracted from Table 7.1. Here, it is also essential to introduce the additional t and h subscript notation according to the channel designations of Eq.(7.39); values for <CTv>ddjt and <CTv>ddih are listed in Table C. l of Appendix C.

Further,

< ov >dd = < ov >ddl + < ov >dddl for which, to a very good approximation

< OV >dd, t °V >dd, h = у < OV >dd

at temperatures of common interest.

(b) SCAT-D Mode

The very large <CTv>dt parameter, Fig.7.5, suggests that for most of the temperature range shown, the bred tritium will be consumed almost immediately upon production while the bred helium-3 will not be burned so rapidly due to the smaller <ov>dh at these temperatures. This fusion reaction mode may therefore be represented as

(7.42a)

d + t —^ n + cc

d + d —> h + n (channel — h) .

Here the arrow suggests a reaction link. A summary reaction representation for this cycle is

5d^>2n + h + oc + p, QSCATD = 24.9 MeV (7.43)

providing Eqs.(7.42a) and (7.42b) occur at equal rates, i. e. Rddjt = Rdt so that

Ni

~Y«n>dd,,= NdN,<av>d, ■

This relationship implies

Nt _ 1 < >dd, t _ 1 < ov >dd Nd 2 <aw>dt 4 <ov>dt

and thus provides for triton fusion burn at a rate equal to its production rate. This fusion operation mode is often called the "semi-catalyzed-D cycle" (SCAT-D). By reference to Fig.7.5, the relative tritium concentration in the fusing plasma may therefore be small at low-to-medium temperatures but will increase for higher temperatures.

(c) CAT-D Mode

An examination of Fig.7.5 suggests that the fusing of helium-3 nuclei with deuterons is the next most likely nuclear reaction. This completes the catalysing process, hence the name CAT (catalyzed) cycle:

(channel -1)

d + t —> n + (X

d + d —> h + n (channel — h)

d + /i—>a + p

which is equivalent to

6d^>2n + 2a + 2p, QCATD = 43.2 MeV (7.47)

provided that the four reactions proceed at equal rates.

The exact sustainment of only one of the above specific d-d bum modes may in general be very difficult. The more general case is suggested in Table 7.2 where the connection reaction linkages will evidently vary with temperature and density.

The well known neutron-induced fission of a uranium-235 nucleus, commonly written as

n+ ^ vn+ P, + /*2 (12.1)

is known to proceed at room temperature because one of the reactants is a neutral particle and a uranium-235 nucleus possesses a substantial fission cross-section for thermal neutrons. The absence of Coulomb forces of repulsion and the presence of nuclear forces of attraction at short distances of separation (< R0) suggests a potential energy diagram as depicted in Fig. 12.la.

In contrast to the above, fusion of a deuterium ion with a tritium ion is represented by

d +1 —^ ті + ос (12.2)

but requires a high reactant temperature to allow a sufficient number of ions to overcome the Coulomb barrier, or to penetrate it by tunnelling. This will lead to substantial reaction rates and the consequent energy yield. The corresponding ion-ion fusion potential energy diagram is shown in Fig. 12.lb.

These two potential energy diagrams, Fig. 12.la and 12.1b, represent two conceivable extremes. A case between these extremes can be conceived of by the following conceptualization for deuteron-triton fusion. Consider a deuterium atom and a catalytic tritium nearby with both particles at low kinetic energy of relative motion, that is in a medium of low temperature. The catalytic tritium is taken to consist of the usual nucleus-а proton and two neutrons-but rather than its normal electron in a Bohr orbit, it contains a catalyst x in an orbit very close to the nucleus. This particle x is expected to possess an electric charge so as to

render the catalytic particle neutral. We add that this particle x may or may not be stable against radioactive decay and may or may not be in a stable tight orbit around the nucleus. As for any two approaching hydrogen atoms, here the deuterium atom and the catalytic tritium will tend to combine by hydrogen molecule formation, which accounts for the range of attraction outside R0 in Fig. 12.1c. To the neighbouring deuteron, the catalytic tritium will, during the lifetime of the catalytic state, appear like an oversize neutron; the two may thus form a compound ionic state where the deuteron and triton are close enough for nuclear forces of attraction to dominate and therefore render fusion at low temperature. In Fig. 12.1c, a "Coulomb sliver" occurs at the distance of the catalyst’s orbit and is expected to be thin enough to be penetrated on account of the nuclei’s available energy in the molecularly bound state.

b) ion-ion fusion at high temperature

C) neutral-neutral low temperature catalytic fusion

Fig. 12.1: Graphical depiction of low temperature fission, high temperature ion-ion fusion,
and low temperature catalytic fusion.

The sequence of events for this low temperature catalytic fusion event consist of three distinct stages:

1. catalytic atom formation:

x + t^xt (12.3a)

2. unstable intermediate formation:

xt + d^xtd (12.3b)

3. decay into fusion reaction products:

xtd^n + a + x. (12.3c)

These three stages may also be written in sequential form

x + t + d —^ xt + d —> xtd —У n + cc + x (12.3d)

and evidently possess some similarity to a fission process which, with a more detailed accounting of the process of reaction (12.1), may be written as

n+^U^U*<^36U + y (12.4)

^vn+ Px + P2 .

Indeed, we will show that branching reaction channels, each with their own probability, shown here in fission also apply to the catalytic reaction chain of Eq.(12.3d).

Low temperature fusion for which the catalyst x is a muon-recall our discussion of Sec. 7.7-has been experimentally demonstrated in liquid media at elevated pressures and in the temperature range 300 К to 900 K, formidable to muonic molecular formation. Evidently, if the process can be sustained as energetically and technologically favourable, then this novel approach might become a contender for a fusion device.

Some elementary aspects of muon physics can be described by the following. We begin with muon production. It is known that many high energy nuclear reactions yield the negative pi meson, 7t", as a reaction product,

(High Energy Reaction) -» n +■■■ — (12.5a)

This pion possesses a mean life of ~ 10’8 s and decays via

7г"-»р"+у^ (12.5b)

where Уц is the muon antineutrino. The negative muon p" decays with a mean life of 2.2 x 10"6 s according to

p —> e + Ve + (12.6)

where e" is an electron, v<? is an electron antineutrino, and vu is the muon

neutrino. It is common to dispense with the adjective "negative" for the muon and simply represent this particle by p rather than p’.

The initial kinetic energy of a produced muon depends upon the details of the initiating reaction but is typically about 200 MeV. In a dense liquid hydrogenous medium, this high energy subatomic particle slows down to about 2 keV in ~ 10’8 s and in another ~ 10"” s cascades down into a K-orbit around a deuteron or triton to form a muonic atom, pd or pt.

The details of the subsequent muon-nucleus, muon-atom and muon-molecule interactions are complex; for example, resonance phenomena involving muonic atoms and associated molecule formations have been identified suggesting the appearance of a variety of nuclear and atomic states. However, for present purposes, and in order to illustrate low temperature fusion, we incorporate these various processes in a collective dynamic characterization using macroscopic reaction parameters.

The simplest and most obvious method with which to provide confinement of a plasma is by a direct-contact with material walls, but is impossible for two fundamental reasons: the wall would cool the plasma and most wall materials would melt. We recall that the fusion plasma here requires a temperature of ~108 К while metals generally melt at a temperature below 5000 K.

Further, even for a plasma not in direct contact, there exist problems with a material wall. High temperature particles escaping from the plasma may strike the wall causing so-called "sputtered" wall atoms to enter the plasma. These particles will quickly become ionized by collisions with the background plasma and can appear as multiply-charged ions which are known as fusion plasma impurities. Then, as shown in the analysis leading to Eq. (3.44), the bremsstrahlung power losses increase with Z2 thus further cooling the plasma.

A current-carrying solenoid provides the simplest way to establish a cylindrically homogeneous magnetic field to contain a plasma. While such a device is relatively easy to construct and to operate, it suffers from a serious deficiency: leakage of plasma particles through the open ends constitutes a significant loss of fuel ions and thermal energy. A reduction of this end-leakage can be accomplished by establishing an increasing magnetic field at the two ends so that many of the charged particles are trapped because of the imposed constraints on particle motion with regards to conservation of energy and the magnetic moment as discussed in Secs. 5.7 and 5.8. This concept resulted in the name "magnetic mirror" and is illustrated in Fig. 9.3.

Some important features of mirror fields can be described and analyzed on the basis of our preceding discussion of ion motion in homogeneous and inhomogeneous magnetic fields.

As discussed in Ch. 5, a positively charged ion spirals in a counterclockwise pattern about the direction of magnetic field lines; this is suggested in Fig. 9.3 for the case of a non-zero speed component of the ion parallel to the direction of the В-field. As the ion approaches the ends, it is increasingly subjected to drifts due to the inhomogeneity of the mirror field. However, note that these drifts occur in azimuthal directions and the particles are still bound to their magnetic surfaces discussed above. Thus they would be well confined in relation to the radial direction (_LB), if not for an occurrent perpendicular diffusion due to collisional and/or turbulent transport, Eq.(6.19), resulting in particle migration out of the contained plasma region.

The more important particle losses, however, will appear in the direction parallel to B, that is through the ends of the solenoid. How effectively these losses can be reduced by squeezing the В-lines at the ends with a magnetic field strength increase, is discussed next.

The concept of trapping the ion in the magnetic "bottle" of a mirror field configuration can be formulated by drawing upon particle kinetic energy conservation and the constancy of the magnetic moment. As in Sec. 5.8, we write for an individual ion the energy in terms of the parallel and perpendicular velocity components, as established by the В-field lines, by

(9.27a)

With E0 a constant of motion and in the absence of other force effects, we can therefore depict the kinetic state of the ion anywhere in a collisionless plasma on

a)

Ion exists on the E0 = constant line

vj* axis

Trapping range (v2)max forV2

Introducing the constant magnetic moment p, Eq.(5.76), the energy conservation reads as = + (9.27b) Depending upon where in the mirror an ion exists, it will sense a different magnetic field-from a Bmm in the mid-plane to a in the mirror plane, Fig. 9.3. This range of magnetic field variation can be introduced into Eq.(9.27b) by recognizing that, with E0 a constant, vy will be a maximum when В is a minimum, Ео = 1тЫ L + fc (9-28) and therefore (уц)^ occurs at the central plane, Fig. 9.3. Conversely, vj| will be a minimum when В is a maximum. The condition for trapping of particles-that is stopping their parallel motion before or at the B^-plane and thus not allowing them to escape through the ends is, in the absence of collisional effects, therefore : 0. (9.29)

(9.34)

VI sin в or, after some trigonometric arranging

sin 2e = — l—. (9.35)

vl

With the help of this expression, it becomes evident that Eq.(9.33) translates into the following statement: particles traveling in the weak magnetic field region, that is about the mid-plane, in directions such that

will be trapped, whereas the others moving within the so-called ‘loss cone’ with a polar angle

will escape from the plasma region. In Eq.(9.36) only the positive sign of the square root is retained because 0 can vary only between 0 and K. Note that Eq.(9.37) actually defines two values of 0O which apparently relates to the two opposite loss cones thus including both the motions parallel and antiparallel to B. Denoting the one solution which is less than 71/2 with 0„, then the second solution is К — 0O.

t

vz

The fraction of an initially isotropic distribution of particles at the mid-plane, f(v)= f(v)/(47tv2), which gets lost through both mirror ends, i. e. particles in both velocity cones, is then

jf(y)d

r _ double cone________________

J loss oo

jf(y)d

0

= 7-cos0o.

Evidently, then cos 0O represents the fraction of the particle distribution which is trapped, that is

f, raP = c™e’ = (9.39)

V В max

where the ratio Bmax/Bmm is called the mirror ratio determining the effectiveness of confinement. As a consequence of the occurring end losses, a mirror plasma is never isotropic. Taking collisions into consideration, it is possible for particles which are in the confined region before collision, to be scattered into the loss cone.

So far we have discussed the confinement of charged particles by a magnetic mirror field which, however, could not explicitly explain why particles are reflected in the increased field of the mirrors. For that, we consider a particle moving outside the loss cone in velocity space. Its parallel motion can be stopped by the field strength of the mirror coils, that is the conserved total kinetic energy

E0 = E± + Et

= {m(vi) +0 (9.40)

z V fmax

= constant

is made up only by the perpendicular motion. If now, by some virtual means, the particle were displaced by an infinitesimal distance into the region of higher B, then it would gyrate faster and thus increase its perpendicular velocity to v_L>(v_L)max. This, however, would violate the energy conservation, Eq.(9.39), if no additional energy was transferred; it follows therefore, that the only directions the particle is allowed to move into-if it is displaced from the plane where it possesses (vx)max-are those associated with vx < (vx)max and thus making room again for E||. Since vx < (vx)max requires a reduced В-field, the particle can regain

E|| only by moving opposite to VB and is thus reflected and returned to the weaker field region. Obviously, the force FM derived in Sec. 5.7 as FM = (1/2mvx2/B)V||B, Eq.(5.72), causes this mirror reflection.

The selection of a method to protect an ICF chamber wall depends on a number of factors including the energy yield per pulse, the pulse repetition rate, and chamber vacuum requirements. In general, as the pulse rate increases, more protection is needed-such as thick liquid metal wall chambers; relatively large radius dry wall chambers could be considered for pulse energies less than 200 MJ.

Laser and heavy ion beam drivers require a high vacuum for good beam transmission; for example, lasers are limited to less than a few Torr (1 Torr ~

133.3 Pa) to prevent excessive scattering. In cases such as light ion beams, a higher pressure of the order of 5 to 50 Torr may be acceptable. If higher pressures are allowed, additional protection can be obtained by using a gas-fill in the chamber. The gas absorbs much of the radiation energy which is re-emitted and hits the chamber wall; this process significantly spreads out the time-width of the energy pulse thus reducing the shock effect.

Another important consideration for chambers with wetted walls, is the time required to clean-that is to purge-the chamber between pulses. This sets a limit for a maximum pulse rate. For example, it appears that about 1 s will be required to pump vapourized lithium out through an exhaust nozzle for a lithium-waterfall system. For such designs, a practical operating region could be a 1 Hz repetition rate with 100 MJ micro-explosions giving an average power of ~ 100 MW per chamber. Multiple chambers employing sequential switching of the laser beam from one to another might be used with a laser operating at a higher pulse rate.

Constants of motion are important because they may reduce the number of independent variables in an equation characterizing some dynamic property of particles. In the absence of other force fields, the total energy E0 of a charged particle in a magnetic field is made up solely of the kinetic energy, since the only acting force, i. e. the Lorentz force, does not possess a potential energy; hence

Eo = imvj +|mVj

is a constant so that

where, as before, Vjj and v± refer to velocity components parallel and perpendicular to the applied В-field direction.

Consider now the motion of a charged particle moving in a non-uniform B — field as described in the preceding section. Then, the parallel component of the force on this particle was seen to be given by the grad-B force, Eq. (5.72), as

Evidently, this relation holds only if

dt (5.80)

as the individual isolated charged particle moves in the spatially varying magnetic field. That is, the magnetic moment does not vary with time.

Note that the grad-B force of Eq. (5.72), used here for establishing Eq. (5.80), had been derived by approximating the radial magnetic field component according to Eq. (5.65). Hence, for the case considered, the magnetic moment |_t is conserved only in this approximation. From the definition of (I, Eq. (5.76), it is evident that (I is an invariant if В is constant. For magnetic fields slowly varying in space and/or in time, that is, relative changes in В are very small over a distance equal to the radius of gyration,

fv«i.

and/or within a period of gyration,

the magnetic moment is found to remain invariant in this first-order approximation. Consequently, (I is only approximately conserved and therefore called an adiabatic invariant.

The practical consequence of this is as it relates to the so-called magnetic mirror effect. As В increases toward the "throat" of the mirror region, vx must increase and, in view of Eq.(5.74), vM must decrease; thus, ions tend to decelerate along the axial direction as they move into the higher magnetic field of the mirror throat where vx can even become zero, if В is high enough. Note that Гц, which is given in Eq.(5.72) and which points opposite to VB, is still acting on the particles thus causing a reflection.

While alternate magnetic concepts may differ from tokamaks in geometry, size, time scales, input power requirements and technology, the principal objectives remain, that is the heating of a D-T plasma to fusion ignition and confining it sufficiently long to yield a net energy gain.

Out of many different designs proposed and discussed in the literature, one such concept is the so-called ‘bumpy torus’, which links a number of mirror sections end to end into a high-aspect-ratio torus, depicted in Fig. 10.19. As illustrated, several mirror coils are equally spaced in a toroidal array. The plasma contained by this magnetic configuration threads the bores of these axisymmetric coils and thus takes on the shape of a bumpy toroidal ‘sausage’. The magnetic field lines close on themselves and the plasma particles are confined in two ways: trapped particles reflect back and forth in individual mirror sections, and passing particles circulate around the major circumference of the bumpy torus plasma. To stabilize such a configuration, electron cyclotron resonance heating (ECRH) is applied to form an annular high-energy electron plasma in the central part of each mirror section. When the currents generated by these hot-electron rings are sufficient to provide a minimum-B configuration, they thus stabilize the toroidal core plasma. In such a reactor, which is called the Elmo (electrons with large magnetic orbits) Bumpy Torus, the hot-electron annuli would typically possess densities of about 1018 m"3 and temperatures Te>100 keV, while the core plasma exhibits a density in the order of 1020nT3 and a temperature of aboutl5 keV. It is thus possible that the beta-value of the confined bumpy torus plasma can become comparable to that of the annuli. The toroidal plasma appears to be macroscopically stable as long as its beta-value is smaller than, or at most, approximately equal to the beta of the annuli. To produce the stabilizing minimum-B property, the latter beta has to exceed a threshold value in the range 5-15% depending on the annular shape. It was experimentally shown that such hot-electron rings can produce beta’s up to 50% in steady-state operation. Hence, the core plasma-^ of an Elmo Bumpy Torus may also be established at substantially increased values in comparison to the tokamak. As a consequence, the fusion power density, as limited by the magnetic pressure, Eq. (4.14), is elevated or, for a given power density, the magnetic field requirements are significantly reduced. Another advantage over the tokamak is the large aspect ratio (~ 5-10 times greater) allowing for simpler engineering design and construction. Further, there is no need for power interruption as associated with pulsed operation; an Elmo Bumpy Torus can thus be operated in a steady-state mode.

Avoiding the production of hot-electron annuli, which is relatively inefficient by means of ECRH and leads to increased radiation losses, a toroidal minimum-B configuration can also be generated by toroidally linking modular coils of specific shape such that each already represents a minimum-B magnetic mirror. To introduce a rotational transform, the coils, which do not exhibit poloidal

symmetry, are rotated about the magnetic axis with respect to each adjacent coil.

A device combining the effect of a z-pinch with that of a 0-pinch-recall Fig. 9.10-will contain a plasma with currents in the axial as well as in the poloidal direction and thus generate a confining magnetic field consisting of helical field lines. Due to the form of the field lines, this configuration is called a screw — pinch. Though similar to the tokamak, it is operated at relatively high [3-20%, but features only very short periods of sufficient plasma confinement.

Another toroidal confinement concept, which has received great attention due to its improved stability against MHD modes, is the so-called Reversed Field Pinch (RFP). It is much like the tokamak: the plasma is confined by a combination of toroidal and poloidal magnetic fields with the latter generated by a toroidal plasma current induced by transformer action. The toroidal field Вф is established primarily by external coils. The essential difference, however, is that in the RFP the plasma currents parallel to the toroidal minor axis do not only produce the poloidal field, Be, but also diamagnetically alter the toroidal field such that Вф can change sign near the plasma boundary (field reversal). Further, the plasma current and Be in RFP’s are much stronger than in comparable tokamaks, whereas Вф is modest. This gives rise to strongly sheared magnetic field lines with their pitch increasing rapidly with greater radial distance. The Reversed Field Pinch configuration is produced by the high magnetic shear near the edge of the plasma which suppresses local MHD instabilities. The field reversal is suggested to emerge from a turbulent state as a self-organization mechanism.

In contrast to a tokamak, where the safety factor q has to meet the Kruskal — Shafranov stability criterion, q(r)> 1 everywhere and q(r=a) > 2.5 (see Sec. 10.1), an RFP features q(r)<l with a negative q(r—>a) consistent with the reversal of the toroidal field component in this edge region. The evolution from a tokamak plasma to an RFP requires the presence of an electrically-conducting shell just outside the toroidal plasma or of closely fitting external conductors for assisting the tokamak plasma to remain confined while reducing q and turning to the RFP configuration. MHD stability theory for RFP indicates the plasma-(3 limitation at the high value of —30%. Due to the high [3, the deployment of advanced fusion fuel cycles in these devices is conceivable. A fusion plasma system not constrained by the Kruskal-Shafranov criterion provides the profit that it can be heated ohmically to ignition, if the energy confinement is good. Experiments, however, have shown so far a TH lower than that for tokamaks of similar size. An obvious advantage over the tokamak is the elimination of the requirement of minimizing the aspect ratio, such as previously demanded by Eq. (10.49). Hence, simplified designs with good maintenance access are possible. A handicap common with tokamaks is that the RFP is a pulsed device as well.

The reactor concepts discussed so far are physically large, they employ complex technology, represent expensive designs, and possess only a relatively
low power density. Evidently, a high power density would be a desirable feature of a fusion reactor. This may be accomplished in compact power reactors which achieve the same total power as a conventional magnetic fusion device in a significantly smaller geometry. Among various designs proposed in this context, e. g. a compact RFP, we choose here to describe the spheromak reactor as a distinctive representative.

A spheromak is an advanced toroidal plasma containment device in which the confining magnetic configuration, as displayed in Fig. 10.20, is characterized by an extremely low aspect ratio and by the absence of external toroidal fields. With the minor plasma radius a ~ R0, this configuration appears almost like a sphere and, obviously, this has inspired the given name. An axial current flows through a field-reversed 0-pinch plasma to internally produce the toroidal field. Thus, both the poloidal and the toroidal magnetic field are self-generated. Only the steady magnetic-bottle field is provided externally by coils. Topologically, spheromaks are open confinement systems with the plasma, however, contained within a closed separatrix surface. The region inside this separatrix is-similar to the RFP — associated with q<l, while, at the plasma boundary, the safety factor is zero. The great advantage of this design is its geometric simplicity and compactness. Experiments with spheromak configurations to date could be operated for only short pulse lengths.

Most pulsed toroidal confinement systems suffer from extremely low periods during which the plasma can be stably contained, and from relatively large radiation power losses due to the impurities released by intense plasma wall interaction. Though there exists a number of confinement concepts which are not discussed here but nevertheless are interesting in their specific design and/or the physics to be applied, we conclude this chapter by recalling that-among the closed magnetic systems-tokamak and stellarator concepts are the most promising candidates to be utilized for future fusion reactor operation.

Problems

10.1 Find the drift velocity, as given in Eq. (10.5) for the case of a purely toroidal magnetic field, by combining the effects of the grad-B drift and the curvature drift, which are simultaneously present in a curved В-field, for the

Вг=0,Вв =-Вв(а) , Вф = R° Вф(RB) a R„ +r cost?

may be applied to a tokamak reactor, estimate the size of such a reactor having a

circular plasma cross-section if it is to achieve ignition and contains a 50:50% d-t fusion plasma with an average ion density Ni = 1020 m’3 at average temperature Ti=Te=30 keV producing 1000 MW of fusion power. Use the ignition criterion of Sec. 8.4.

10.3 Using j = VxB, show that for a tokamak the magnetic field is proportional to 1/R.

10.4 What perspectives do high-beta devices offer relative to low-beta tokamaks? How can a high-beta configuration be realized?

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

(a) Tokamaks

(b) Stellarators

(c) Spheromaks

(d) Reversed Field Pinch (it is a toroidal system)

10.6 List the favourable characteristics desirable for a future fusion power reactor.

10.7 Design an axisymmetric d-t tokamak reactor with circular cross-section capable of fusion plasma ignition demonstration assuming the empirical energy confinement time scaling

[‘] = 0 00338(/p [ma])° 85 (o[m])° 3 (r0 [m])1 2 (jV, [m’3 ]) (B, [tesla])0 2 (/с, л [mw])

where denotes the total fusion power in the plasma volume, and taking the

following fixed parameters:

ratio of first wall to plasma radius rw/a = 1.25 . 50:50% deuterium-tritium fuel mixture

. electron density Ne = 0.8 X 1020 m’3 . equal ion and electron temperature, T; = Te = 20 keV. plasma current Ip = 18 MA

toroidal magnetic field Bt = 6 Tesla. cyclotron radiation loss parameter |/ = 0.001 Specify the relevant plasma and reactor parameters allowing for ignition (i. e. so that Eq. (8.30) is met) and consistent with the constraints and requirements discussed in Sec. 10.4, in particular Eqs. (10.40b), (10.42) and (10.46), as well as with the engineering constraint of limiting the thermal power flux through the first wall by Pw < 1.5 MW m’2.

A detailed quantitative assessment of the energy multiplication capacity of a
hybrid requires a specific blanket design followed by a detailed neutronic analysis. However, an indication of this important energy gain can be obtained by employing a simplified lumped parameter characterization of the several processes occurring in the blanket and associated reactors.

We define Cb to be the average number of neutrons produced in the blanket by all neutron multiplication processes, expressed on a per fusion reaction basis. One of the neutrons so produced must be used for breeding tritium while the remaining Cb-1 neutrons could be used for fissile fuel breeding. Allowing for parasitic neutron captures and losses such as neutron leakage gives finally £b(Cb- 1) as the total number of fissile nuclei produced. Each of the bred fissile nuclei is able to eventually generate Qf, units of energy, which will take place-if the bred fuel is extracted and transported to client power plants-in associated fission reactors. Then, the total nuclear energy generated by the hybrid fusion-fission system must also include the breeding capacity of the medium in which these eventual fissions take place. Letting Cfi be the conversion ratio of these associated fission reactors, a total fission energy of Qn/(1-Cfl) is therefore eventually generated and attributed to each initial fissile breeding reaction in the fusion reactor blanket. The total energy generated per unit initial fusion reaction energy release Qft, defines an energy multiplication and is therefore given by

Here ME may well be called the energy multiplication capacity and is displayed in Fig.15.3. Clearly, the energy multiplication of a fusion reactor operating in tandem with a fission reactor can be substantial compared to that for a stand­alone fusion reactor (ME =1).

The fundamental reason for the remarkable energy multiplication of a fusion hybrid reactor operating in tandem with a fission system lies in the complementary nature of fusion and fission reactions. A d-t fusion reaction results in one neutron and a total energy release of 17.6 MeV while a fission reaction results in 2 to 3 neutrons and -200 MeV. Consequently, fission reactions can be viewed as energy "rich" and fusion reactions, by this yardstick, as energy
"poor". Further, a hybrid fusion reactor blanket may well regenerate the fuel it consumes, i. e. breed tritium and have some neutrons left over. In contrast, improving the performance of a fission breeder reactor-which produces abundant amounts of energy-demands an enhanced neutron population. Hence, a "neutron — for-energy" and "energy-for-neutron" exchange in the hybrid provides mutual advantages.

Another way of viewing fusion-fission energetics is to note that a neutron used for tritium breeding in support of a fusion reaction contributes basically

17.6 MeV of energy while a neutron used to support fission energy by fissile fuel breeding adds more than 10 times as much energy to the entire yield of the system. Thus, the high energy neutrons from a fusion reaction possess a greater capacity for neutron multiplication than thermal neutrons in a thermal reactor, reinforcing the "neutron rich / energy poor" view of fusion. This contrasts to the recognition that on a basis of "energy per initial fuel mass" involved, fusion is exceedingly more "energy rich".

The recognition that the Coulomb barrier for light ion fusion is in the 0.4 MeV range for the lightest known nuclides-p, d, and t-suggests that one approach to the attainment of frequent fusion reactions would be to heat a hydrogen gas up to a temperature for which a sufficient number of nuclei possess energies of relative motion in excess of U(Ro), Eq.(2.5). However, because of the tunneling effect, such excessive heating is not required and substantial rates of fusion reactions are achieved even when the average kinetic energy of relative motion of the reactant nuclei is in the tens-of-keV range. Note, however, that even providing such reduced conditions in a gas will be associated with heating up to temperatures near ~ 10s K, for which, in recognition of the ionization potential of hydrogen being only 13.6 eV, the gas will completely ionize. The result therefore is an electrostatically neutral medium of freely moving electrons and positive ions called a plasma. With the average ion energy in a fusion reactor plasma thus allowed to be substantially less than the Coulomb barrier, the energy released from fusion reactions must exceed-for a viable energy system-the total energy initially supplied to heat and ionize the gas, and to confine the plasma thus produced. In practice, this means that when a sufficiently high plasma temperature has been attained, we have to sustain this temperature and confine the ions long enough until the total fusion energy released exceeds the total energy supplied. In subsequent chapters, we will consider some details of the relevant phenomena and processes and identify parametric descriptions of energy balances.

On the basis of the above considerations then, it is apparent that selected aspects of the classical Kinetic Theory of Gases, augmented by electromagnetic force effects, can be used as a basis for the study of a plasma in which fusion reactions occur. Thus, for the case of N atoms of proton number Z„ complete ionization yields Nj ions and, in the case of charge neutrality, Z, Ni = Ne electrons; that is

N^Ni+ Ne= Ni + Zi Ni ■ (2.7)

For hydrogenic atoms, Z; = 1 and hence Ne = N;.

Often, the expression "Fourth State of Matter" is also assigned to such an assembly of globally neutral matter containing a sufficient number of charged particles so that the physical properties of the medium are substantially affected by electromagnetic interactions. Indeed, such a plasma may also exhibit collective behaviour somewhat like a viscous fluid and also possesses electrostatic characteristics of specified spatial dimension.

For a state of thermodynamic equilibrium, the Kinetic Theory of Gases asserts that the local pressure associated with the thermal motion of ions and electrons is given by

Pi = jN jiw,- v? (2.8a)

and

Pe = j Nemev2e (2.8b)

where the subscripts і and e refer to the ions and electrons respectively, and N( > refers to the subscript-indicated particle population densities; note that it is the average of the squared velocity which appears as the important factor. The average kinetic energy of the electrons and ions can be introduced by simple algebraic manipulation of the above equations:

= ^NiEi (2.9a)

and

= (2.9a)

Similar expressions may be written for the neutral particles in a plasma.

Accepting the kinetic theory of gases as a sufficiently accurate description allows for the use of well-known distribution functions. Implicit in this assumption is that the plasma under consideration is sufficiently close to thermodynamic equilibrium and that processes such as inelastic collisions, boundary effects and energy dependent removal of particles are of secondary importance.

The distribution functions for the particles of interest include dependencies on space, time and either velocity, speed or kinetic energy. Here we take a stationary ensemble of N* particles uniformly distributed in space-either neutrals, ions or electrons-allowing us to write

N(S)=N*M($) (2.10)

where £ is one of the independent characteristic variables of motion-v, v, or E — and M(Ej describes how the particles are distributed over the domain of this variable. Hence, N(EJ is the distribution function of the ensemble of relevant particles in ^-space and the symbol M() represents a normalized distribution function for the variable £ such that

J M( t, )d^ = 1 (2.11a)

with the integration performed over the entire definition range of the variable considered, i. e.

Jm ()d = jM(v)dv =Jm (E)dE =1.

—«> о 0

Note that the particle number of the ensemble is given by

]n(^=N )m(^=n •

For a gas or, for the case of interest here, a plasma in thermodynamic equilibrium and in the absence of any field force effect, its particles of mass m moving in a sufficiently large volume follow the Maxwell-Boltzmann velocity distribution function given by

Here к is the Boltzmann constant and T is the absolute temperature for the

Finally, the corresponding distribution of particles in kinetic energy space E is described by the Maxwell-Boltzmann distribution
0 < E < «>.	(2.15)
Velocity, vx
Fig. 2.2: Schematic depiction of the Maxwell-Boltzmann distribution functions for vx, v and E as independent variables; in each case, Ti < T2.