One of the most effective means of plasma confinement to date involves the use of magnetic fields. A particle of charge q and mass m located in a magnetic field of local flux density В is constrained to move according to the Lorentz force given by

dts

m— = g(vxB) (4.9)

dt

with v the velocity vector.

For the case of a solenoidal В-field produced by a helical electrical current with density j, Fig. 4.2a, ions and electrons will move (as will be shown in Ch. 5)-depending upon the initial particle velocity-either parallel or antiparallel to the В-field lines and spiral about them with a radius of gyro-motion of

(4.10)

Scattering reactions may, however, transport them out of these uniform spiral orbits with two consequences: they may be captured into another spiral orbit or they may scatter out of the magnetic field domain. In the absence of scattering they are essentially confined as far as directions perpendicular to В are concerned, traveling helically along В with an unaffected velocity component parallel to B. Subsequently, they will eventually depart from the region of interest.

Solenoidal fields belong to the oldest and most widely used magnetic confinement devices used in plasma physics research. The ion and electron densities may clearly be enhanced by increasing the magnetic field thereby also providing for a smaller radius of gyration. There exists, however, a dominant property which renders their use for fusion energy purposes most detrimental: for

solenoidal dimensions and magnetic fields generally achievable, fuel ion leakage through the ends is so great that these devices provide little prospect for use as fusion reactors.

b) Mirror B-field

If, however, the magnetic field strength is increased specifically at each end of the cylindrical region, i. e. the B-field lines appear to be substantially squeezed together at the ends, Fig. 4.2b, the number of leaking particles is considerably reduced. Such a squeezed field configuration is referred to as a magnetic mirror since it is able to reflect charged particles, as will be shown in Sec. 9.3. We note that ions and electrons possessing excessive motion along the magnetic axis will still penetrate the magnetic mirror throat.

The attractiveness of the mirror concept not withstanding, a magnetic mirror can thus not provide complete confinement and-as in all open-ended configurations-is associated with unacceptably high particle losses through the ends. Hence, to avoid end-leakage entirely, the obvious solution is to eliminate
the ends by turning a solenoidal field into a toroidal field, Fig. 4.2c. The resultant toroidal magnetic field topology has spawned several important fusion reactor concepts; the most widely pursued of such devices is known as the tokamak, which will be discussed in Ch. 10.

At first consideration, the charged particles could be viewed as simply spiraling around the circular field lines in Fig. 4.2c, not encountering an end through which to escape. Any losses would have to occur by scattering or diffusion across field lines in the radial direction causing leakage across the outer surface. Further, we mention that collective particle oscillations may occur, thereby destabilizing the plasma.

One important plasma confinement indicator is the ratio of kinetic particle pressure

Pkin= NikTi+ NekTe (4.11a)

to the magnetic pressure

<4»b)

with |i0 the permeability of free space. This ratio is defined as the beta parameter, P, and is a measure of how effectively the magnetic field constrains the thermal motion of the plasma particles. A high beta would be most desirable but it is also known that there exists a system-specific PmaX at which plasma oscillations start to destroy the confinement. That is, for confinement purposes, we require

Thus, the maximum plasma pressure is determined by available magnetic fields thereby introducing magnetic field technology as a limit on plasma confinement in toroids.

In order to assess the fusion energy production possible in such a magnetically confined, pressure-limited deuterium-tritium plasma, we introduce Eq.(4.12), with the equality sign, into the fusion power density expression

Pfu~ Nd Nt Gv >dt Qdt ^4 J 3)

to determine-for the case of Nd = Nt = N/2, Ni = Ne and T, = Te the magnetic pressure-limited fusion power density

which is displayed in Fig. 4.3 as a function of the plasma temperature.

Another overriding consideration of plasma confinement relates to the ratio of total energy supplied to bring about fusion reactions and the total energy generated from fusion reactions. Commercial viability demands

Е/и» Emppiy • (4-15)

Most toroidal confinement systems currently of interest are expected to operate

in a pulsed mode characterized by a bum time Xb. Assuming the fusion power to be constant during xb, or to represent the average power over that interval, the fusion energy generated during this time is given by

E fu= P fU’ Tb=<cTV >dt Nd NtQdtTb (4-16)

for d-t fusion. For perfect coupling of the energy supplied to an ensemble of deuterons, tritons, and electrons, we have

Esupply = jNdkTd+lNtkTt +1NekTe. (4.17)

One may therefore identify an ideal energy breakeven of Efu = ESUppiy as defined

by

< CTV >dt Nd N, Qdt tb = j NdkTd +1 N, kTt + f NekTe (4.18)

Imposing Nd = N, = N/2, Ni = Ne, and Td = T, = Te = T we may simplify the relation to write for an ideal breakeven condition:

Since Q* = 17.6 MeV and <Gv>d, is known as a function of kT, we readily compute the product
for kT = 12 keV. Note that for fusion bum to be assured over the period xb, it is required that the actual magnetic confinement time, Tmc > Xb. The incorporation of other energy/power losses as well as energy conversion efficiencies suggests that this ideal breakeven is an absolute minimum, and operational pulsed fusion systems need to possess a product much higher. For instance, the accelerated motion of charged particles inevitably is accompanied by electro-magnetic radiation emission, as described by Eq.(3.39), which constitutes power loss from a confined plasma.

There exist various versions of Eq. (4.19)-depending upon what energy/power terms and conversion processes are included; collectively they are known as Lawson criteria in recognition of John Lawson who first published such analyses.

Problems

4.1 Verify the expression for the sonic speed in a compressed homogeneous deuterium-tritium ICF pellet, given in Eq. (4.4), by evaluating the defining equation v2 = dp/dp with p denoting the mass density of the medium in which the sonic wave is propagating. Assuming the sound propagation is rapid so that heat transfer does not occur, the adiabatic equation of state, pp’Y = C, is applicable to the variations of pressure and density. Determine the constant C using the ideal gas law. To quantify y, which represents the ratio of specific heats at fixed pressure and volume, respectively, c/cy, use the relations cp — су = R (on a per mole basis) and су = fR/2 known from statistical thermodynamics, where R is the gas constant and f designates the number of degrees of freedom of motion (f = 3 for a monatomic gas, f = 5 for a diatomic gas,…).

4.2 Compute vs of Eq.(4.4) for typical hydrogen plasmas and compare to sound propagation in other media.

4.3 Re-examine the specification of a mean-time between fusions, Sec. 4.5, for the more general case of steady-state fusion reactions. How would you define, by analogy, the mean-free-path between fusions, X*?

4.4 If 10% bumup of deuterium and tritium is achieved in a compressed sphere with Rb = 0.25 mm and fulfilling the bum condition of Eq. (4.8) with Ni>0Rb = 5 x 10‘28 m’2, at what rate would these pellets have to be injected into the reactor chamber of a 5.5 GWt laser fusion power plant? The pellets contain deuterium and tritium in equal proportions. Power contributions from neutron — induced side reactions such as in lithium are to be ignored here.

4.5 Assess the minimum magnetic field strength required to confine a plasma having Nj = Ne = 1020 m"3 and Tj = 0.9 Te = 18 keV, when [3max = 8%.

4.6 Compute the maximum fusion power density of a magnetically confined d-h fusion plasma (Nd = Nh, Td = Th = Te, no impurities) as limited by an assumed upper value of attainable field strength, В = 15 Tesla, dependent on the plasma temperature. Superimpose the result for pmax = 2% in Fig. 4.3.

Closed magnetic field configurations are those in which the field lines do not enter or leave the plasma confinement region, and thus offer the advantage of having no ends from which the plasma might escape. The simplest such configuration is a torus. Fusion devices containing toroidal plasmas have emerged as the dominant experimental facilities. We discuss here several of their distinguishing features as they relate to a conceivable energy generating device, emphasizing in particular, the tokamak concept.

Tritium is of interest for nuclear fusion because, along with deuterium, it is a basic fuel for the most readily achievable fusion reaction. Since tritium does not occur naturally in sufficient quantities and is radioactive, its production and management creates some special dynamical features of relevance.

13.1 Tritium Properties

An essential role of a tritium nucleus is as a reactant in the fusion reaction

As previously used, h again represents the helium-3 nucleus; the maximum (3′ energy is 18 keV with an average of ~5 keV. This property of nuclear instability is responsible for two important characteristics of tritium: it is naturally scarce and where it does exist, it is a radioactive hazard.

This natural scarcity of tritium, with the only known inventory being some 20 kg in the oceans and atmosphere where it is produced by reactions initiated by cosmic radiation-combined with the recognition that kilogram quantities of tritium may be required for a commercial central station fusion reactor-means that tritium will need to be bred on a substantial scale. As indicated in Sec. 1.4 and Ch. 13, tritium breeding can occur by incidental neutron capture in the (heavy) water of fission reactors or by neutron capture in lithium in the blanket of a fusion reactor. The feature that every d-t fusion reaction produces one neutron, Eq. (14.1), means that the breeding by neutron induced reactions of one triton for every one destroyed would only be possible if no neutrons escaped or were lost to parasitic reactions. Since some neutron losses are unavoidable, a means of neutron multiplication is also essential for the breeding process.

An indication of the radiation hazard associated with tritium is suggested by calculating the decay rate of, say, 1 kg of tritium. From the definition of nuclear activity, Act, we have

(14.3)

where A, is the decay constant, Eq.(14.2). The total number of tritons N,*

associated with a given mass of tritium M, is given by

M, — N*mt, (14.4)

Translating this quantity into Curies, knowing that 1 Ci = 3.7 x 1010 dps (= 3.7 x 1010 Bq), the activity of 1 kg of tritium is equal to 107 Ci. To place this quantity into context, we add that only about 10’3 Ci of tritium activity can be handled without special licensing provisions. Evidently then, extreme care needs to be exercised in the management of large quantities of tritium.

Tritium, being a hydrogen isotope, can be readily transported by gaseous, liquid, and solid carriers. Its extraction is possible by catalytic exchange and cryogenic distillation processes. Two tritium transport characteristics have been found useful. For the case of a local tritium density gradient VN, in a diffusion medium, Fick’s rule of diffusion provides for a tritium current Jt given by

J,=-DtVN, (14.6)

where D, is the tritium diffusion coefficient. Also, tritium permeation through a barrier containing a differential tritium density yields a tritium flux ф, well represented by

Ф,~К,

V /

Here N,,i and Nt,2 are the upstream and downstream tritium densities, Z is the barrier thickness, and K, is the tritium permeation constant. Parameters such as D, and K, are strongly material and temperature dependent, and, additionally, the permeation constant is a function of surface conditions. Low temperature, multiple wall barriers appear to be the most promising means of tritium containment.

Conservation conditions are fundamental aids in the quantitative assessment of nuclear processes. Of paramount relevance here is the joint conservation of nucleon number and energy associated with an initial ensemble of interacting nuclear species of type a and type b which, upon a binary collisional interaction, yield two particles of type d and e:

a + b —» d + e. (1.7)

Note that the details of the highly transient intermediate processes are not listed— only the initial reactants and the final reaction products are shown.

An accounting of all participating nucleons is aided by the notation ^nuclear species named X containing^

A nucleons of which Z are protons J

and therefore yields the more complete statement for the reaction of Eq.(1.7) in a form which lists the number of nucleons involved in this nuclear rearrangement: £Xa + bXb^Xd+£X’. (1.9)

Recall that Aj is the sum of Zj protons and Nj neutrons in the nucleus, Aj = Zj + Nj. Nucleon number conservation therefore requires

Aa + Аь — Ad + Ae (1-Ю)

and, similarly for charge conservation we write

Za + Zb = Zd + Ze • (1-11)

Characterization of energy conservation for reaction (1.7) follows from the knowledge that the total energy E of an ensemble of particles is given by the sum of their kinetic energies Ek and their rest mass energies Er= me2; here m is the rest mass of the particle and c is the speed of light in free space. The total energy of an assembly of particles, for which we add the asterisk notation, is

therefore

E = ^(Ek, j + Er, j)=^(Ek, j + mjc2) • (1-12)

j j

For the nuclear reaction of Eq.(1.7) we write the total energy-which must be conserved-as

* *

Ebefore ~ Eafter

and hence

(Ek, a + maC ) + (Ekj? + тьс )-(Ek, d + mdc )+(Ek, e + mec ) ■

Rearrangement of these terms yields

(Ek, d + Ek, e)'(Ek, a + Ekj>)= [(ma + m)-(md + me)]c2 ■

This important equation relates the difference in kinetic energies-before and after the collision-to their corresponding differences in rest masses; thus, as shown, a change in kinetic energy is related to a change in rest masses. Particle rest masses have been measured to a very high degree of accuracy allowing therefore the ready evaluation of the right-hand part of this equation. This defines the Q-value of the reaction

Qab= [(ma + rm)-(md + me)]c2 = -[(md + me)-(ma + ть)]c2 ^

and represents the quantity of energy associated with the mass difference before and after the reaction. Hence, we may write more compactly

Qab = (-^)abC2 (115b)

where Am is the mass decrement (i. e. Am = mafter — mbefore) for the reaction. Evidently, Qab is positive if (Ат)аь < 0 and negative otherwise; the former case­involving a decrease of mass in the process-constitutes an exoergic reaction and the latter may be termed endoergic. Further, for the case of Qab < 0, the kinetic energy of the reaction-initiating particle must exceed this value before a reaction can be induced; that is, a threshold energy has to be overcome before the reaction will proceed.

Equation (1.15b) is a form of the famous relation

E = me2 (1.16)

and asserts-as first proposed by Einstein-that matter and energy are equivalent. As a consequence, we may assert that if processes occur which release energy of amount E, then a corresponding decrease in rest mass of amount (-Am) must have taken place.

The preceding analysis leads to a compact space-time description for the particle or mass density. Of particular interest to us now is a characterization of collective motion in a plasma. In such a description, the identity of individual particles is put aside and the plasma is characterized by the space-time changes of macroscopic variables such as bulk speed, temperature, and pressure defined in a fluidic context.

To begin with we take the Continuity Equation, Eq.(6.12), as a necessary equation for the particle density everywhere in space and time. That is, if another equation is developed in which Nj(r, t) appears as an independent function, both equations must hold simultaneously and any solution methodology is expected to involve both equations.

To be specific, we consider a space-time ensemble of Nj particles each of
mass rrij and charge qj to which we assign an average velocity Vj over an infinitesimal volume containing Nj. A simplified one-dimensional analogy is the case of freeway traffic with cars moving at different speeds, all together amounting to some average speed of traffic motion which, nonetheless, may vary with time and location.

In order to provide some continuity to our preceding analysis, we suppose that an electric field E and magnetic field В act on this moving space-time ensemble of Nj particles, referred to as a fluid element. Its equation of motion is

d у, / ч

Njmj~dt = NjqJE+ Ni4j{VixB)’ J = i’e■ (6.21)

Recall that Nj used here is determined by Eq.(6.12) everywhere in space and time.

The two terms on the right hand side possess an easily established interpretation; clearly, the second term c)Vj/c)t is simply the acceleration of the Nj ensemble at a fixed coordinate point; then, the first term incorporates the process of spatial variation in the ensemble velocity much as in our one-dimensional analogy the traffic in one section of the freeway may move faster than that in another section; the term convection is often used for this kind of fluid motion. We insert

Eq.(6.23b) into Eq.(6.21) and again use Pj = Nj mj to obtain a corresponding equation for the ensemble’s motion

= P-E + P-(V, XB)

where the mass density Pj satisfies the Continuity Equation, Eq.(6.20), and pjC(r, t) = Nj(r, t)qj is the respective charge density. Note that, because of electromagnetic interactions in a plasma, PjC, E and В are all related via Maxwell’s Equation which, simultaneously, must also hold with Eq.(6.24).

At this juncture, we recognize another process which is generally important in a hot plasma: the thermal motion of particles makes them randomly enter and exit the fluid element considered, leading to local pressure gradients which act as an additional force in Eq.(6.24). While, in general, this force can be obtained

wherein the local particle pressures are related to the respective mass density via the Equation of State

where C is a constant and 7j is the ratio of specific heats for constant pressure and constant volume (7j = Cp/Cv) referring to the j-th species.

An important point must be emphasized that concerns the electric and magnetic fields, E and B, when the above set of equations is evaluated for ions and electrons. As noted previously, it is required that Maxwell’s Equations be satisfied for the medium of interest. The complete set of resultant equations for Pi, Pe, Vj, Ve, pi, pe, E, В-consisting of 16 simultaneous scalar equations — represents what is known as a self-consistent description of the plasma fluid approximation by the so called magnetohydrodynamic (MHD) equations. The imposition of initial and boundary conditions clearly leads to a nontrivial problem description.

The Lawson criterion and general energy flow analyses of Ch. 8 can be applied to inertial confinement as well as to magnetic confinement. However, the parameters of interest and the nomenclature is different for each. We now re­examine the energy balance specifically for an inertial confinement fusion system, where one pulse is still taken as the characteristic time interval for which an energy balance is established.

For present purposes, we define the following three energy components necessary in a parametric analysis of an inertial confinement fusion system, Fig. 11.2:

Ebe* = energy contained in the laser or ion beam which triggers compression;

E, h = thermal energy of the compressed target ions and electrons following impingement of the beam;

Efu = fusion energy released during the associated bum time xb.

Not all of the beam energy will appear as thermal energy of the ions and electrons in the target; a fraction may be reflected or scattered and some energy is carried off with the ablated outer layer. Hence, a coupling efficiency T]c can be defined which relates Ebe and Eth by

Е*н= ЛсЕІе > 0 <т)с<1. (П.17)

A characteristic pellet energy multiplication Mp relates Efu* to Ebe* by

E*ju = МрЕІе > (П.18)

and for an energetically viable system, Mp has to substantially exceed 1. Note that T]e and Mp are design parameters of the system.

The overall energy flow for an electricity producing inertial confinement fusion reactor system is suggested in Fig. 11.2 for which the station electrical energy output is given by

Ena = Vju Efu — El • (11-19)

Here рь, is the efficiency of converting the fusion energy into electrical form and Ej„ is the circulating electrical energy component required to sustain the lasers or ion accelerators. The conversion of this electrical energy into beam energy is taken to occur with an efficiency T)in defined by

^ = % 0<Пт<1. (11.20)

Em

The station electrical energy output can be compactly written by defining an electrical energy multiplication as

Л fuE’fu

El

The essential requirement for a viable inertial confinement fusion system is therefore

Me> 1.

This energy viability criteria can be related to the several conversion efficiencies and the pellet multiplication already defined. A substitution of Eqs.(11.18), (11.20) and (11.21) into Eq.(11.23) yields

ЛыЛ/иМР> 1, (11.24)

and thus specifies the necessary pellet energy multiplication required. As currently envisioned, lasers are relatively inefficient with r|in ~ 0.06, while qfu ~ 1/3 for conventional energy conversion; this yields a requirement of Mp > 50. This demanding result can be reduced if the driver is more efficient; for example

for ion accelerators, T|jn ~ 0.3 may be possible, giving Mp >10. Further, the beam coupling efficiency enters via

Note that (Eft, / Eth ) is the ratio of fusion energy produced to the energy deposited in the pellet and hence, in analogy to Eq. (8.6), can be identified as the according pellet plasma Q-value which, upon introduction in Eq. (11.26), has to satisfy the requirement

for energy viability. Evidently, a very high coupling efficiency ric is desired. For example, for a laser with ric = 0.05 and the previous tiin and r)tu values, we require Qpp > 1000. One way to meet this requirement is to have a high fusion gain pellet which in turn implies a very high compression.

In some of the preceding chapters we have referred to specific fusion systems and devices which are currently under extensive development. Our interest here is to examine some special system concepts deserving further examination.

15.3 Direct Energy Conversion

By way of introducing some unique possibilities for fusion energy conversion, we consider the sequence of energy transformations in a fission reactor. We recall that a fission event produces energetic and massive fission products which transfer their kinetic energy by collision to the host atoms in the fuel. This heats the fuel to elevated temperatures with the resultant thermal energy then transported to the coolant which is subsequently used to produce steam under pressure. The expanding steam causes rotation of a steam turbine which is directly connected to an electrogenerator to produce electricity for a distribution network.

As previously shown, the majority of reaction products in advanced fusion cycles (d-h, d-d) are ions. The motion of these ions constitutes a current flow which could, in principle, be converted into electrical energy. Some fusion reactors, particularly magnetic mirrors, are especially well suited for such purposes because ions leaking through the mirror ends already possess desirable directional properties. Specifically then, the direct collection of charged particles represents a transformation of the kinetic energy thereof to electrostatic potential energy which can act to sustain a current through an external load, RL. This concept is illustrated by the idealized collector shown in Fig. 16.1 where an ion beam impinges on a single plate collector held at voltage V+.

For analytical purposes we consider an ion beam with an initial angle-energy distribution such that the differential current J(|J.0,E0) gives the number of ions with direction cosine ц0 (i. e., ц0 = cos 0) and energy E0 crossing the plane at x=0 per cm2 per second per unit direction cosine and per unit energy. Then, the total beam current Jb, in units of cm’V, is given by integration over all ion energies and direction cosines:

(16.1)

energy, are turned around by the potential V+ prior to reaching the collector and thus become so-called retrogrades. Under the assumption that space-charge effects do not distort the ion trajectories significantly, the current Jc reaching the collector is given by

oo fIc(E0)

jc= J KH0,E0)dpL0dE0 (16.2)

E„=qV* №=1

where Цс(Ео) is the smallest direction cosine that an ion of initial energy E0 can have and still be collected, and V+ is the plate voltage. For simplicity, consider a parallel beam, J(p.0, E0) so that Eq.(16.2) reduces to

Jc=]j{E0)dE0 (16.3)

qV+

and the possible power generated in sustaining a load is thus

Pc = qV+ ]j{E0)dE0. (16.4)

qV+

With q and J(E0) specified by the fusion device, the operating voltage Vop should be such that the power output is maximized:

This last term requires differentiation of the lower limit of the integral leading to

J-]j{E0)dE0=-qJ(qV + ). (16.7)

Substituting this expression in Eq.(16.6) gives the following integral conditions on Vgp as the maximum power output:

4V+J(qV+)= J J(E0)dE0 . (16.8)

чК

The maximum ion beam-to-electricity conversion thus occurs at the operating voltage V„ . The efficiency, given by the ratio of power output at V^p divided by the incident beam power, gives therefore

and assume a wide energy spread such that Ei < E2/2. The optimum voltage is then found from Eq. (16.8) to be E2/(2q), while the maximum efficiency is

% = ,, ■ Е, йЬ.. (16.11)

E2)

Thus, for the relatively narrow energy spread of Ei ~ E2/2, the efficiency is -67%, while at the other extreme, when Ei — 0, the efficiency decreases to 50%.

While the efficiency for a single plate is attractive, values exceeding 90% are possible with multiple-plate collectors, even with a wide range of beam energies. This is possible because the voltages on various plates can be set so as to efficiently intercept particles having different energies, Fig. 16.2. Efficiencies over 90% require 5 or more plates. However, a number of non-ideal effects not considered here, such as leakage currents, secondary electron emission currents and space charge effects can cause lower efficiencies in practice.

Direct collection requires extraction of a beam of charged particles from the fusion plasma. With a mirror-reactor, the escaping plasma, Fig. 16.3, is first magnetically expanded in order to convert vx energy to уц, and thus form a

B2 and A2. Conservation of the magnetic flux and particle flow provides the relations

BiAi = B2A2 (16.12a)

J1 Ai = J2 A2 (16.12b)

and the imposition of adiabatic invariance gives

After most of the particle energy is converted to Ец in the expander it is necessary to separate ions and electrons prior to collection. One technique is to sharply bend the magnetic field lines at the exit of the expander. Electrons will still be trapped on the lines while the ions, due to their larger momentum, will non-adiabatically cross field lines thus providing the desired separation. Separate collectors can be used for the ions and the electrons, which is particularly essential for the ions since they carry most of the energy.

A common approach to the achievement and sustainment of fusion reactions involves conditions in which the fuel mixture exists in a plasma state. Such a state of rapidly moving ions and electrons provides for extensive scattering due to Coulomb force effects. These are particularly important because they lead to kinetic energy variations amongst particles, and more importantly, to particle losses from the reaction region thereby affecting the energy viability of the plasma.

3.1 Collisional Processes

Collisions between atomic, nuclear, and subnuclear particles take many forms. The important process of fusion between light nuclides represents a "discrete" inelastic process of nucleon rearrangement in which the reactants lose their former identity. In contrast, Coulomb scattering among ions and electrons causes "continuous" changes in direction of motion and kinetic energy. All these phenomena occur in a plasma to a varying extent and are therefore important in all confinement devices.

There may also exist a need to describe other selected collisional events in a fully or partially ionized medium, requiring therefore that the distinguishing characteristics of various processes be identified. Among these we note atomic processes such as photo-ionization, electron impact excitation, fluorescence, charge transfer and recombination, among others. Nuclear processes include inelastic nuclear excitation, nuclear de-excitation and elastic scattering.

A commonly occurring and important type of collision in a plasma is charged particle scattering attributable to the mutual electrostatic force. Such Coulomb scattering can vary from the most frequently occurring small-angle "glancing" encounters due to long-range interactions, up to the least likely near "head-on" collisions. The Coulomb scattering probability for ions is much larger than that to undergo fusion. Note that the deflections encountered in scattering reactions may lead to significant bremsstrahlung radiation power losses which lower the plasma temperature.

In general, the complete analysis of charged particle scattering is physically complex and mathematically tedious. As a consequence, we chose here to employ selected reductions in order to convey some of the essential and dominant features of specific relevance for our purposes here.

While the discussion in Sec. 4.6 and the confinement requirement suggested there, Eq.(4.12), provide some essential conceptual information about magnetically confined fusion, a sufficiently high plasma temperature has to be attained for sufficient fusion reactions to occur, Sec. 2.5. To attain this state, a neutral gas is heated and thereby ionized, as the gas kinetic temperature surmounts the respective ionization energy potentials. Upon this plasma formation, the heating process is still to be continued in order that the plasma approach the 10 keV temperature range favourable for a reasonable <Gv>. When these plasma temperatures are reached, their sustainment has to be considered. Of
the three processes mentioned, only ionization is elementary while both heating and high temperature sustainment are far more difficult.

The initial phase of a bum cycle, that is the attainment of sufficiently high temperature, as well as subsequently the sustainment of those conditions, can be accomplished by means such as the following:

(a) Resistive heating: this process involves Ohmic heating due to an electric current in the plasma.

(b) Compression: mechanical and/or magnetic forces are used to compress the plasma adiabatically and thus raise its temperature.

(c) Electromagnetic wave heating: electromagnetic waves from lasers or radiofrequency generators may be used to deposit energy in the plasma.

(d) Beam injection: neutral particles or pellets are injected and deposit their energy by collisional effects.

(e) Internal heating: charged fusion products collisionally transfer most of their birth energy to the plasma ions and electrons.

Resistive heating is based on Ohmic energy dissipation effects. The power deposited in a unit volume of a plasma by this method is given by

Pres = m2 (8.14)

where I is the current density; the parameter T| is the plasma resistivity and possesses a dependence on a number of collisional effects and is of the form

Ц~кТ’3/2 ■ (8.15)

The feature that the resistivity of a plasma decreases with increasing temperature means that Ohmic heating becomes progressively less effective at higher temperatures. Above about 1 keV, supplementary heating must be employed unless-as is possible for some system concepts-massive currents are applicable.

Compression methods of heating can be classified into two broad categories. If it occurs very rapidly-on the scale of 10’6 s or less-then it is an implosion and complex gas dynamics and shock wave considerations need to be introduced. If the compression occurs over longer intervals compared to the speed of thermal energy transfer, but still short relative to radiation losses, then the compression is adiabatic and the necessary relation

pVr = constant (8.16)

holds. Here 7 is the relevant adiabatic gas coefficient, as in Eq.(6.26).

Since a plasma consists of an aggregation of numerous moving electrical charges, it is subject to collective interactions exhibiting typical resonance effects. Therefore, coupling of high power (high frequency) waves to the plasma appears to be an effective heating mechanism. A most favourable application of such is the electromagnetic coupling via waves having the ion cyclotron frequency COg, i, Eq. (5.12). The ion motion can thus be resonantly enhanced to high kinetic energies. The irradiation by those high frequency waves is usually performed at the frequency C0gi„ or the harmonics 2cogJ, or 3cogJ (30-100 MHz depending on the magnetic field strength). The absorption of the irradiated electromagnetic energy increases with higher ion temperature Tj and can be managed most effectively by minimizing the distance between the wave antenna and the plasma edge.

Aside from this so-called ion cyclotron resonance heating (ICRH), there is, of course, also the possibility of electron cyclotron resonance heating (ECRH) which will require frequencies in the range of 28-140 GHz. Note that the specific heating of a particular plasma species may lead to a significant difference in Te and Tj; it is the latter which is to be raised for a sufficient fusion reactivity.

Beam injection, particularly neutral atoms or macro particles, has proven to be an important and effective means of heating. The mechanism of energy transfer in the plasma initially involves the conversion of the high energy neutrals into high energy ions by charge exchange and impact ionization; the resultant fast ions subsequently transfer part of their energy to the plasma ions and electrons by Coulomb collisions.

The rate at which injected fast ions transfer their kinetic energy Ef to the plasma involves two distinct components. The average energy transfer rate to electrons is approximately of the form

dE

<—>f->e~ AeNeT’e’2Ef, (8.17)

dt

while the transfer rate to ions may be approximated by

dE

~ Ai Ni Ef/2 (8.18)

dt 1

with both Ae and A, constants for the respective species and specified beam particles. Thus, at a high ion beam energy, most of the energy transfer is to electrons whereas at lower energies the thermal ions get a larger share.

We note that fusion product heating will occur due to the same collisional effect, i. e. Coulomb scattering. Hence the same energy transfer rates, as given by Eqs.(8.17) and (8.18), apply to the slowing down of the charged fast fusion products in the plasma which in turn is heated. In d-t fusion, this fusion power deposition is called alpha-heating, attributable to the charged particles released from the nuclear reaction. Very high energetic fusion products, e. g. 15 MeV protons from d-3He fusions, can as well lose energy by nuclear elastic scattering, which will result in discrete energy transfers to the plasma ions.

In this chapter we have focused on the concept of low temperature fusion based on the mechanism of muon catalysis. Recent years have witnessed considerable discussion of “cold” fusion in electrolytic cells and other experimental devices. No clear and consistent characterizations of the associated processes have
emerged to date so that a pedagogical presentation must await further developments.

Problems

12.1 Formulate the complete reaction dynamical description for the 14- particle species of Fig. 12.2.

12.2 Undertake a dynamical analysis of the case depicted in Fig. 12.4 assuming that Я, ц = 0. Compare this characterization to a neutron induced fission chain.

12.3 Explore analytical solutions of the system of equations, Eq.(12.10), given suitable initial conditions.

12.4 For Хц = 34, find the average time between muon induced fusion events and compare this to the mean lifetime of the pdt atom. What does this suggest about the time scale of the formation of the pdt atom relative to its decay?

12.5 A particular upper bound for the muon recycle efficiency occurs in the absence of any muon sticking to the alpha particle. Examine the consequences of this limit for the case leading to Eq.(12.27).

Part IV Components, Integration, Extensions