The need for plasma confinement requires that some control on ion and electron motion be considered. While moving charges accordingly generate electromagnetic fields which additionally act on other moving particles, for the case of low densities, these induced fields-as well as the effect of collisions-may be neglected so that the trajectories of charged particles may be considered to be governed entirely by external field forces acting on them.

Bending a solenoidal coil around until its ends meet deforms an initially straight axial magnetic field into an axisymmetric toroidal field, as shown in Fig. 10.1. The toroidal field В is produced by simply passing a current through the coil wound around the torus. The torus is topologically defined by its major radius R0 and the minor radius a. The central line along the toroidal ring is called the minor axis, while the major axis is the one pointing out from the centre of the plane display of Fig. 10.1, i. e. going perpendicularly through the torus ring, Fig. 10.2.

Recalling our discussion of drift motion in Ch. 5, an examination of the toroidal configuration in Fig. 10.1 suggests that some complications do arise. The coils around the torus are more closely spaced on the inboard side than on the outboard side, creating therefore a radial variation in the magnetic field of the form

(10.1)

R

as shown in Fig. 10.2, with R denoting the distance from the major axis. Recall that a gradient in the magnetic field creates a grad-B force which drives the positive ions in one transverse direction and the negative electrons in the other. Further, the curvature drift drives them apart in the same way. The net result is a local charge separation, thereby generating a vertical electric field causing simultaneous ion and electron drift in an outward direction perpendicular to the major torus axis, Fig. 10.2.

To demonstrate these drifts we consider a simple magnetic field generated only from the current in the toroidal coils; then the field possesses only a component in the toroidal direction, еф, that is, we have

(Ю.2)

The magnitude of the toroidal field can be calculated by employing Eq.(9.50) with I now representing the total current threading the hole of the torus. This then gives

(10.3)

and proves the proportionality introduced in Eq. (10.1). Considering a charged particle moving along B, with speed % we note that it will be subjected to the curvature drift and, if it possesses also a velocity component perpendicular to B, to the VB drift as well and hence, according to Eqs.(5.51) and (5.61), its guiding centre drift velocity (ІЗ) will be

(10.4)

Curn

Coil

Introducing Eqs.(10.2) and (10.3) into Eq.(10.4) we find the drift motion in the simple toroidal field here to be described by

This equation is called the toroidal drift, the result of which is a polarization of the plasma with an electric field vector pointing in the negative z-direction. The resulting E-field unfortunately causes both the ions and electrons to move radially outwards due to the ExB drift with a velocity of

ExB

В B*(Ro) Ro

as shown in the detail of Fig. 10.2. Thus, the plasma in a simple toroidal field drifts outward until it strikes the surrounding wall. In this case, there is no radial equilibrium and hence practically no plasma confinement.

A common approach for reducing this outward drift resulting from charge polarization is the following. In addition to the toroidal magnetic field Вф produced by the toroidal coil windings, a poloidal magnetic field B9 is introduced. Depending upon the relative magnitude of these two B-field components, B9 and Вф, a helical path of particle migration results, as suggested in Fig. 10.3. Then the particles spend equal time in the upper and lower halves of the toroid thus canceling out the undesired charge separation when the particles have moved along a twisted field line through a completed poloidal rotation, 0 = 2k.

The poloidal angle made by a В-field line after it has traversed through a 271- revolution in the toroidal direction (ф) is called the rotational transform, and represented by l (iota). If l is very large, the revolving field lines become too tight making the plasma unstable against kink-type perturbations, Sec. 9.6. Therefore it is useful to define a safety factor by

2k

q = — (10.7)

i

which measures the field line pitch and it is seen from stability considerations to

require q>l in the plasma and q>2.5 at the edge of the plasma in order to suppress kink-type instabilities (Kruskal-Shafranov criterion). Thus, the rotational transform is constrained by

к2я (10.8)

which means that, following a magnetic field line, its pitch has to be such that the number of revolutions around the major torus axis exceeds the number of revolutions around the minor axis.

While the rotational transform does provide for some spatial mixing of the charged particles and so greatly reduces the electric field and, consequently, the transverse outward drift, another interesting feature occurs. The effect of the twisted magnetic field lines-each of which completely traces out a magnetic flux surface by its revolutions around the toroidal and poloidal axes-is to create a system of nested toroidal flux surfaces which guide ion motion. Note, that in order to avoid the undesired charge separation, the rotational transform must not be an integer multiple of 2it. In that case, the field lines would recombine after some trips around the torus and hence not cover the entire magnetic flux surface on which they lie. Further, note that q may vary from one magnetic flux surface, as defined in Eq.(9.24), to the next, and this leads to a shear in the magnetic field since the В-lines point in a different direction as one proceeds radially onto different flux surfaces. We illustrate this in Fig. 10.4. In typical tokamak plasmas q ranges from 1 near the centre of the plasma to values of 3-4 at the plasma edge. It is this shear which is effective against the growth of kink and drift perturbations. A perturbation aligned with B(r) will, at a point with increased minor radial distance r + dr, encounter field lines at a different angle which again will vary as the perturbation grows to another distance r + dr’. Any helically resonant instabilities are thus radially localized.

The poloidal magnetic field can be established in two ways. One is by arranging outside the plasma a set of poloidal coils which carry a current in the toroidal direction. Devices based on this principle are called stellarators. Others, known as tokamaks, employ a toroidal current flowing in the plasma; the latter happen to be relatively less complex devices and are therefore of greater current interest.

Tritium fuel cycle characteristics are largely determined by the mode of reactor operation and the source of tritium. We consider first a d-t fusion reactor operating at constant power for which tritium is supplied externally. Hence, the fuel leaks out of the containment vessel along with the fusion reaction products as suggested in Fig. 14.1. Our interest is now in the specification of the tritium inventory in the fusion core and the tritium inflow requirements in terms of some

гзішм/(Л ЬнЧ тшїпТ. гЗіиІБЗІ IfinoilBTOqO тоізбзі ohsd

зі оО поігіЛ

лоїовзі поігиі J-Ь mud гиоиліїлоз згії тої alouboiq поііовзі Ьпв but to woI4 : I. M.§ЇЧ

угігпзЬ зіві поіізбзі поігиі зівіг ^Ьвзіг згії гі Jnioq §піііБІг згіТ

(8.М) *<^>„7^ =

smulov-linu згії пі гпоіпі Ьпв гпоізІизЬ ’to isdmun згії зів 0_,И Ьпв э, ьИ sisriw гі угігпзЬ iswoq поігиі Іпвігпоз Ьзівізоггв згіТ лзгітвгіз поіізбзі поіги’і

уІІпзЬіуз

(*>*) й& <Ь< VO > = Й&ІЬ* =

noitomtesb-woilluo-woilni Ізиі згії Іхзп тоЫгпоЭ. Ьэ1ээ1§эп гпоіізвзі b-b ritiw

Івгії ог ЭЮЗ поігиі згії юі гэггэзо^

(01.М) лЯ-ьЧь = іьЯ-ь. Ч-ьЧ=2^

їо

ЬпБ

(П. М) • лЛ-Лі = йЯ-Л-Л = 2і^

lb

зів )1 Ьпв Ь1 Ьпв І ТІ.§іЧ пі bsloiqsb гзіві woJlJuo-woilni Ізиі згії зів ()Ч зізН зізгі гі увозЬ muilhl ;muilhl Ьпв типзІизЬ 1о гпоіізвії mud svilosqzsi згії ггзі гізит гі Іігпвіі пі гі muilhl згії зтії 1о гіі§пзІ згії ззпіг Ьз1ззІ§зп уМвГ1Ьги[

.зііі-іівгі гіі пвгії

поігиі згії ІІв, гпоі! изп юі niril YbBsilqo гі зюз поігиі Bmzfilq ЧЭМ пб ззпі2 поігиі пі bsouboiq zsriqlfi згіТ. ц,Я = ПЛ ззпзгі Ьпв тоїізізгії ЯвзІ Iliw гпоііизп —Ьпв, гпоі! ззІз Ьпв гпоі впігв^ згії riliw ізбізіпі уІІвпоігНІоз гпоіізвзі згії оі Ьзпііпоз Ibw ггзі то этот зів узгіі-з§івгіз зіііззіз іізгії оі sldfiludhllB Yd Ьз§впвт гі ПЛ зіві wollluo іізгіТ. поі1віи§і1поз Ьізії зіізп§вт 3vil33q23i гпоізІизЬ, гпоіізвзі зЬіг §пііззІ§зИ. поіізбііхз lohsvib yd.§.э, гпвзт Івтзіхз

gnibbiy гзІБі Івирз 1б bemud зів гпоіііі Ьпв (Si>i) — Л = ьЛ юі ь=,

ізузі iswoq згії Ьпв ^тоіпзупі muilhl этоз поігиі згії nsswlsd поііззппоз А

:(Q. M).p3 moil zwollol

(£і. М)

Thus, the operation of the fusion core which maximizes <av>dt also minimizes the tritium core inventory.

The tritium injection requirements for steady-state operation, dN, iC/dt = 0, are given by Eq.(14.11):

Ft = — = (14.14)

/, f, Q*

providing therefore, a rigid relationship between the variables of power, bum fraction, and tritium injection rate.

A detailed kinematics characterization of reaction (1.7) requires the specification of both the kinetic energy and the momentum of the initial state of the reactants a and b as well as-depending upon the reaction details desired-the appropriate
field forces which may act on the particles. However, some useful relations about the energies of the reaction products d and e can be obtained for the simple case in which the reacting particles possess negligible kinetic energies relative to the Q-value of the reaction, i. e. Eka + Ek, b « Qab, and in which the total energy liberated is shared by the two reaction products d and e in the form of their kinetic energy. Under these conditions, Eqs.(1.14) and (1.15a) give

Tmdvd +ттеу2е = Qab • (1.17a)

Then, restricting this analysis to the case that the centre of mass be at rest, Fig. 1.2, momentum conservation provides for

md’Vd = me’Ve ■ (1.17b)

Before Collision:

ma, Ag, Zg mb> Ab, Zjj

After Collision:

:k, d

Fig. 1.2: Kinematic depiction of a head-on nuclear fusion reaction with the centre of mass

at rest.

For the specific case of d-t fusion, Eq. (1.5), for which Q* = 17.6 MeV, the neutron and alpha particle kinetic energies are therefore found to be

Ek, n ~jQdt ~ 14-1 MeV, Екм =1<2л = Зі MeV. (U9)

Thus, an 80 — 20% energy partitioning occurs between the reaction products.

As a final characterization of a statistically varying population, we consider what is often called transport theory or a kinetic description. A compact derivation which leads to a defining equation for a particle density distribution function and which also includes collisional effects as well as source effects-that is, comprehensive transport of particles and their properties-is suggested in the following.

Consider an ensemble of N particles each characterized by a distinguishable velocity vector Vi and coordinate vector Гі at time t. For that we introduce a distribution function fN represented by

f n = f N{ri>vi;r2>v2;~-;ri, Vi;—;rN>VN;t)> (6.27)

which is able to specify the probability that at time t, the particle 1 is found about the coordinate Г] with the velocity vector V] while the position and velocity of particle 2 are about r2 and v2, and the remaining (N-2) particles are similarly found at distinct coordinate intervals in the position-velocity space. With N as the number of particles of interest, each characterized by three Cartesian space coordinates and three Cartesian velocity components, this phase-space is evidently a 6N-dimensional phase-space.

A fundamental principle in the description of particle distributions is known
as the Liouville Theorem; this theorem states that a volume element in the 6N- dimensional phase-space remains constant for the motion of particles obeying the Hamiltonian equation of motion. For the N-particle distribution function fN, Eq.(6.27), this theorem leads to the Liouville Equation

(6.28)

if particle creation and/or particle destruction does not occur. Performing the differentiation of Eq.(6.28), the explicit differentiation which fN must satisfy is evidently

(6.29)

This differential equation is generally the starting point for an analysis of a many particle system. Note that the acceleration term dv/dt can be replaced by F,/m, via Newton’s Law with Fj representing the total force acting on the i-th particle.

Equation (6.29) is now subject to integration over a suitable large set of coordinates Гі and velocities Vj in order to substantially reduce the dimensionality of the phase-space distribution function from fN to, say fr. Thus, for r « N, we obtain

(6.30)

with Ar a normalization constant generally chosen by convention, e. g. Ar = 1/(N — r)!, found via permutational calculus.

We will find it useful to separate the total force F, acting on the i-th particle

j *i, r= 1, 2, …, N — 1 ,

where we replaced Э/Эг and Э/Эу by the gradient symbols

Э, Э. Э

Bulk Particle Transport and

Vv = 3—* + 3—j + 3—-^ • (6.31c)

dyx a у y a vz

Equation (6.31a) is known as the Bogolyubov-Bom-Green-Kirkwood-Yvon hierarchy and closes only with the Liouville Equation, Eq.(6.28), since the equation for fr contains fr+i. To accommodate the cutting off of this sequence at a reasonably small r, a physical approximation for fr+i is therefore needed in terms of f i,…,f r in order to arrive at a solvable set of equations. From this set of r equations the r-1 equations may be used to eliminate f2,…,fr, leaving a single kinetic equation for the one-particle distribution function fb We now consider the case r = 1 which is described by Э f F6*’ 1 r

-3^ + V, • V, / ; + — • Vv, / ; =—- V,2-V, J2d3r2d3V2. (6.32)

dt m, m, J

Here, fi(ri, Vi, t) is the one-particle distribution function used to determine the probability of finding particle 1 at time t about the position ri with velocity Vi; note however, that herein we have lost the corresponding information about the other particles. Further, Eq.(6.32) contains the two-particle distribution

f2(ri, vi, r2,v2,t) which specifies the probability that at time t particle 1 moves with Vi about r, and, simultaneously, particle 2 will be found about r2 with velocity v2, with no such individual information about the remaining N-2 particles.

Writing the one-particle distributions

f,(ri, Mi, t)=f(l) (6.33a)

for particle 1,

f i(r2,v2,t) = f(2) (6.33b)

for particle 2, and expressing the two-particle distribution by cluster expansion as f2(rj, vi, r2,v2,t) = film + C(l,2) (6.33c)

with C representing the so-called pair correlation function relating kinetic variations of particles 1 and 2, most significantly within the close range of their interaction, we find upon substitution of these expansions the following kinetic equation:

+ v, • V, fil) + ^■ Vv, fil) + —( f F12 f(2)d3 r2 d3 v2 at m, m,

= -—[F12 — W, C(l,2)d3r2d3y2. m, J

The expression within brackets in the fourth term on the left-hand side represents the field force experienced by particle one due to the presence of the other particles and may be written as

j¥12f(2)d3r2d3v2 = F{ield • (6.35)

Actually, this is the particle contribution to the self-consistent field (for example:

an ion distribution generates an electromagnetic field which in turn maintains this distribution). The right-hand side of Eq.(6.34) is the collision term for which some approximation must be applied to expand it also as a function of f(l) and f(2). Since various physical models exist for approximating the collision term, we choose to represent it here simply by (5f(l)/5t)c. Thus the final kinetic equation for a one-particle distribution function is

Іr dt Jc

(6.36)

where F includes all external and field forces. Herein and subsequently we omit the subscript and f(r, v,t) will refer to the one-particle distribution function which is representative of any one of the considered N particles of the same species, however does not provide information which distinguishes the individual kinetic behaviour of the particles. For charged particles moving in an electric and magnetic field, the total force is evidently

F = q(E + vxB) (6.37)

incorporating self-consistent fields.

If particle production and/or destruction also take place and particles can leave and/or enter the reaction volume under consideration, then we account for these sources and sink processes on the right-hand side of Eq.(6.36) by an additional term (5f/5t)s to be specified according to the appropriate conditions.

In the statistical sense [f(r, v,t) d3r d3v] is interpreted as the probability of finding at time t a particle of the species of interest in the differential volume element d3r d3v about the point (r, v) in the 6-dimensional phase space, Fig.6.1. We may now conclude that the expected number of such particles in this differential volume element is

N(r,,t)d3rd3y = N* f(r, v,t)d3rd3v (6.38)

where N is the total number of particles in the entire volume at some reference time. Hence

6 expected number of particles in dr3 у about r and in dv about v at time t,

Obviously the normalization applied to Eq.(6.39) is

N(t)

N with the number density obtained by

N( r, t) = J N( r, v, t) d3 v = N * J /( r, v, t) d3 v.

The transport equation for a charged particle density distribution N(r, v,t) in an electric and magnetic field and subjected to collisional effects as well as to particle production/destruction is therefore finally written as

(6.42)

A solution analysis of this equation for N = N(r, v,t) requires the self-consistent specification of E and В (determined in conjunction with Maxwell’s Equations) everywhere, identification of the form of the collision and source/sink terms and the imposition of initial and boundary conditions. A reaction domain will generally contain more than one species of particles so that the determination of the overall collision kinetics requires several equations of the form of Eq.(6.42)~ one for each species. All of these may be coupled by several collision terms since collisions will invariably occur among the same and other species. Needless to say, the dimensionality of this problem is formidable. A simplification is possible by a reduction of the number of independent variables and elimination of some terms depending upon the problem of interest.

It is useful to note that the left-hand side of Eq.(6.42) constitutes, by our classification of processes, continuous changes in phase space while the right — hand parts are discontinuous changes. Thus, particle collisions, particle reactions and particle destructions are viewed as discontinuous-that is discrete processes — whereas particle acceleration can vary smoothly in the coordinate-velocity space. If only binary collisions are considered, the analysis is labeled “classical” or “neoclassical” if applied to toroidal geometry. As plasma wave-particle interactions become dominant such that the transport of mass, momentum or energy results therefrom, then the label “anomalous transport” is used.

Estimates of the beam energy required to compress a pellet to bum conditions can be obtained by starting with Eq.(11.17) in the form

Eth

Пс

and hence, from Eq.( 11.18)

Based on our previous considerations, the total thermal energy in the pellet of radius Rb at the beginning of the fusion bum is explicitly given by

E;h(0) = ^NeJTe(0) + lNuokTl(0))(^Ri)^4KNiJTLX (11.30)

where we have taken

Ne* = Nu, and Te(0)=Ti(0)=Tio. (11.31)

According to Eq. (11.28), the beam energy requirement is therefore

1

In order to avoid the explicit calculation of the reaction energy released during the fusion bum time, Tb, as suggested by Eq. (7.15), let us take here the

Inertial Confinement Fusion proportionality

Efu x NloVb^b

and combine this with Eqs. (11.29) and (11.30) to obtain Mp(4nNiokTio Rl) — rjcNl0-jitRlXb

Further, recalling the expression for the disassembly speed, Eq.( 11.11), and relating

(11.35)

we re-arrange Eq. (11.34) to isolate Rb as

Mr
(11.36)

The beam energy requirement now follows by substituting this expression for Rb in Eq. (11.32) to yield

M3P

Nlnt where Abe is a temperature dependent factor. The exponents here are significant since they indicate a beam energy sensitive to Nji0, t|c and Mp. For the case of a target at normal liquid density and with Mp = 100, and even for Г|с = 1, this energy is found to be of the order of 1012 J, which is inaccessibly large. (Note that relating this to the approximately 10’9 s during which this energy needs to be delivered would represent an input power greatly exceeding the total steady power capacity of all electric power plants in North America.) If, however, target compression increases the density by 103, then the energy requirement is reduced by 106. This illustrates the reason for the very strong interest in high compression of the target.

The principle of magnetic induction is based on the phenomenon that a time varying magnetic field will induce a current flow in an electrical conductor. Faraday’s law provides the important relation in the form

V = — n^-¥n (16.16)

dt

where n is the number of identical turns, V is the induced voltage across the conductor ends, and 4/m(t) is the spatially integrated magnetic flux crossing the coil area A

4>Jt) = m-dk. (16.17)

A

Here, B(t) is the time varying magnetic field and dA is the normal differential area. Evidently, a pulsed fusion reactor is necessary.

A schematic of how the magnetic field in a plasma could be coupled to an external circuit is suggested in Fig.16.4. The fundamental energy transformation mechanism is that any expansion of the confined plasma, working against the magnetic field, appears as an induced voltage across the coils, Eq.(16.16).

L———— 1

Fusion Energy

Plasma

чжшии

Г-ШШШЬ

—w————————

Load

Fig. 16.4: Depiction of magnetic coupling by compression-expansion of a long cylindrical
plasma. Part (a) represents the plasma-coil configuration while part (b) suggests its

equivalent circuit.

Two versions of conceivable systems are envisaged. In one, the plasma is first compressed magnetically to ignition with the expansion of the plasma against the magnetic field providing for energy transformation by induction. A
related approach is to move the fusion plasma through a linear bum chamber and plasma energy could be removed by having the plasmoid exit through an expanding section of magnetic field.

For either case, the analysis of electromagnetic coupling follows from the equivalent circuit of Fig. 16.4. The power induced in the secondary magnet coils of n turns is given by

p* =vi = — nl^^-i2 Rc (16.18)

dt

where dT’m/dt is the total rate of change in magnetic flux caused by the expanding plasma, I is the induced current, and I2RC represents Joule-heat losses in the coil. Neglecting Joule-losses, the energy transferred during an expansion

with the plasma radius varying from r2 to r3 in Fig. 16.4 follows by integration

з

£* = — nJ/J4V (16.19)

2

Consider now an isobaric expansion in which a constant magnet current I is required to maintain a constant field В and in which the magnetic field associated with the coils is

B = 4m-L (16.20)

with L as the plasma length. The magnetic flux in the region between the plasma and coil is

Vmj = n(R2-r2pj)B, (16.21)

where the subscript j signifies the radius position so that the corresponding

plasma volume is given by

Vpi = 7Cr2L. (16.22)

Combining these relations we find that the energy transferred to the secondary coil to be

E* =-п1(‘¥т3-Ч>т1) = ^{уp3-VРг) ■ (16.23)

To understand the origin of the transferred energy E*, we consider the various individual components of the expansion work. To simplify the analysis, we take p-the ratio of plasma to magnetic field pressures-as a constant. The expansion work qp+ done by the plasma pressure pp is then

3 2

qP = J PpdV = 0^-(y,3 — Vp2) (16.24)

where we have used Eqs. (9.10) and (9.11) to substitute for the plasma pressure. Flowever, the trapped field Bp in the plasma simultaneously expands and performs work against the confining field. This work is

3 2

4′ = }-&dv=1(,-^Tf(v^-v^ (16-25)

2 2^o 2Я>

where we have introduced the relation Bp2 = (1 — (3)B2 reflecting the condition that the confining field В has to balance the total of kinetic pressure, pp, and the trapped-magnetic-field energy density, Bp2/(2fi0), in the plasma. In addition, as the plasma and trapped field expand, it displaces the confining field originally contained in the volume between r3 and r2. The corresponding energy Ed must appear in the secondary coil along with pdV-work; this term is given by

Ed = y—{vP3-VP2)- (16.26)

The total energy transferred to the coil is then the sum of these three components:

E* = qP+ + ql+Ed (16.27)

which, upon substitution of Eqs.(16.24) through (16.26), agrees with Eq.(16.23).

Similar expressions can be derived for all the components in a cycle; expansion steps transfer energy to the coil, while compression steps require an input energy. Note, however, that the net output will be determined by the plasma-work terms Eq.( 16.24) alone because the energy associated with the trapped and displaced magnetic fields must cancel out in a complete cycle. That is, in order to return to the starting point, the displaced field energy transferred to the coils during expansion steps must be replaced during compression steps. For this reason, we will compute work terms based on the plasma term alone. It must be remembered, however, that the actual energy transferred during any given stage will exceed the plasma term; this becomes important in the actual sizing of coils and the evaluation of joule losses.

As illustrated in Fig. 16.5, a Carnot cycle consists of two isothermal and two adiabatic processes. The term "adiabatic" is used here in a dual sense: first, we assume no heat-transfer or radiation losses occur during the compression or expansion legs; second, since the Carnot cycle is by definition an equilibrium cycle, these stages use slow reversible processes where magnetic-field lines remain "frozen" in the plasma.

A stable plasma with an ideal beta~l confinement is assumed throughout. Furthermore, the number of plasma particles is taken to be constant, corresponding to negligible leakage and bumup; energy carried off by neutrons and radiation is omitted, and additional energy losses such as Joule-heating losses in the magnetic coils are also neglected.

The cycle begins at point 1, where a total of 2N plasma particles (Nj* = Ne* = N *) are assumed to be trapped in the magnetic field. The following steps then follow:

Stage 1 to 2: Slow-adiabatic compression to a maximum field value Brrax and temperature T^. A compression work input q. is required. In the present idealized cycle, any fusion input is neglected prior to

ignition at point 2.

Fusion stage. A constant temperature is maintained during the fusion bum (qf* = fusion-energy input to the plasma) by expanding the plasma. Thus, work q+ is performed against the confining field.

Slow-adiabatic expansion to Bmin, extracting work q+. It is assumed that the temperature and density are reduced so rapidly that fusion input can be neglected after point 3.

Cooling stage. Radiative cooling qr is achieved at constant temperature Tmin by simultaneously compressing the plasma (q. = work input).

During the fusion bum (stage 2 to 3), the plasma expansion work must just balance the fusion input so that isothermal conditions are maintained. To a good approximation the plasma pressure-temperature-volume relations can be taken from ideal gas laws with care taken to incorporate the trapped-field defined (3 parameter. This gives

* ‘ }

?/ = ?+ =J PpdVP

2

= 2N‘kTw In

However, since pressure balance requires that B2 «= V"1 during this stage, Eq.( 16.28) can be written as

During the slow-adiabatic compression (1 to 2) and expansion (3 to 4) stages, ideal gas laws apply so that

1 1

(16.31)

with у denoting the ratio of specific heats. However, because of the isothermal stages, T3 = T2 and T4 = Ti,

Bi _ B4 B2 Вз

Using this result in the relations for qf and qr, we obtain the expansion cycle efficiency

net work out

Bc = —

(16.33)

__ j 1 min

Tmax

As expected, this result corresponds to the classical Carnot efficiency. Indeed, since Tmin could be in the range of 10 eV while Тща* is on the order of 103 eV, a Carnot limit approaching 100% is indicated. While this is encouraging, a basic question remains: How close can an actual cycle approach this ideal limit? Additional losses enter by non-ideal effects which have been neglected here. Another problem is that, for various reasons, it may be impractical to follow a Carnot cycle; for example, internal combustion engines typically utilize an Otto cycle rather than a Carnot cycle. Here, the control necessary to achieve an isothermal fusion bum may be difficult. Indeed pulsed reactor conceptual design studies to date have assumed an isobaric bum.

Another important point to stress is that the expansion cycle efficiency only applies to that portion of the fusion energy that remains in the plasma. Thus neutron and radiation energy that is processed through a thermal cycle must be included in the overall efficiency using the fractional energy flows as weighting factors with each cycle efficiency. Advanced fusion fuels with a larger fraction of
energy retained in charged particles would more closely approach the electro­magnetic cycle efficiency, but even with p-nB, radiation losses from the plasma would probably involve significant energy flows and be processed through a blanket thermal cycle.

A practical consideration in designing devices to operate on expansion — compression cycles is the need for a large plasma chamber in order to accommodate a relatively large expansion ratio. Thus the blanket and magnetic coils in a magnetic confinement device must have large dimensions making them relatively costly. Application to inertial confinement fusion has the advantage that the chamber dimensions are large anyway, compared to the target, in order to accommodate the micro-explosion shock effects. All of these various considerations, fuel cycle, confinement approach, economic and complexity tradeoffs, must be factored into the decision to select a best energy conversion system for a given device.

Consider an isolated system of two ions of charge qa and qb, and possessing corresponding masses rria and гпь. The magnitude of the Coulomb force is given

by

1 ЧаЧь

4rt£o r2

where r is the distance of separation at any instant; evidently, as the charges move, this distance and also the direction of the force varies with time. The resultant trajectory of the particles a and b is suggested in Fig.3.1 for charged particle repulsion and attraction. In order to clearly specify a cross section for this process, it is useful to introduce the impact parameter r0 and the scattering angle 0S with respect to the initial and final asymptotic particle trajectories. In Fig. 3.1 the two colliding particles are shown in antiparallel motion because such a simplified situation always applies if the collision is analyzed in the centre of mass reference frame. Thus, the deflection angle 0S here indicated is identical to the scattering angle in the centre of mass system 0C, which is associated with the directional change of the relative velocity vr, to be subsequently introduced. Intuition based on physical grounds suggests a rigid relationship between the three parameters r0, 0C and vr.

For the case of azimuthally symmetric scattering, Fig. 3.2, and which here applies because of the specific form of Fc, every ion of mass mb and charge qb moving through the ring of area

dA = 2Tir0dr0 (3.2)

will scatter off particle а-which has a charge of the same sign-through an angle 0C into the conical solid angle element d£2* associated with the shadowed ring of Fig. 3.2; this conical solid angle element is evidently

dQ* =2Ksm(ec)dec. (3.3)

The number of ions which scatter into a solid angle element can change substantially with r0 and vr; it is therefore necessary to look for an angle dependent cross section 0(Q) which here, due to azimuthal symmetry, is only a function of 0c and yields a total scattering cross section CTS according to

4к к

Cs = ^Gs(ec)d2Q = ^(js(ec)d&

That is, as(0c) is a function of the conical solid angle £2* corresponding to a specified 0C and possesses units of bams per steradian (b/sr). Since das represents the differential cross sectional target area dA of Fig.3.2, and all ions entering this

area will depart through the solid angle element dQ*, we may write

N2tu r0dr0 = Nas (вс )dQ* (3.6)

with N denoting a particle number per unit area, which, upon insertion of Eq.(3.3) yields specifically

crs(ec) =

where 0 < 0C < K. Here, the standard absolute-value notation for the Jacobian of a transformation has been incorporated. The process discussed above refers to the so-called Rutherford scattering and CTs(0c) is known as the corresponding differential scattering cross section.

Case of Repulsion

Particle

Case of Attraction

where the parameters introduced are as follows:

0C = scattering angle for particle b off of particle a in the COM (Centre of Mass) system which relates to 0L in the LAB (Laboratory) system according to

cot( eL) = — csc( вс) + cot( вс) ; ma

mr = reduced mass of the two body system = (ma mb ) / (ma + mb ) ;

vr = relative speed of the two particles of interest

= I ve — vj. (3.9c)

The meaning of the impact parameter r0 is unchanged.

Equation (3.8) provides a useful connection between the impact parameter r0, the scattering angle 0C in the COM system, and the relative particle speed vr. For

Charged Particle Scattering simplicity, we may write

d6,=-dr0 r„

jsec2

(3.11)

and hence

Then, substitution of this expression in Eq.(3.7) gives

(3.14)

(3.15)

This explicit algebraic form is frequently called the Rutherford scattering cross section.

The more significant information about this charged particle interaction cross section is suggested in Fig. 3.3 and illustrates the following important result: the cross section approaches GS(0C) —> K2/4 for "head-on" collisions (i. e. r0 —» 0) and becomes arbitrarily large for increasingly smaller "glancing" angles (i. e. r0 —» °o).

An informative conclusion about the singularity as 0C —» 0 is that, for example, if there were to exist only two nonstationary charged particles in the universe then some deflection would occur with certainty regardless of how far apart they were; that is, the Coulomb 1/r2 force dependence may become infinitesimally small at sufficient separations but, in principle, it never vanishes. Identifying a "reasonable" length beyond which charge effects can be considered to be unimportant, or even be ignored, is provided by the Debye length concept to be discussed next.

It is often desirable to use easily understood and readily recognizable parameters as indicators of the merit of a particular system or process. A widely used formulation that provides a test of the energy balance for fusion devices is known as the Lawson Criterion. It is an algebraic relationship based on the assertion that the recoverable energy from a fusion reactor must exceed the energy which is supplied to sustain the fusion reaction; that is, Eout* > Ein* must hold during a representative time interval X which may be the cycling time for a pulsed fusion device or a typical period of time if steady-state operation is envisaged.

We may associate the recoverable energy from a fusion device with three forms of energy: the fusion reaction energy release, Efu, all radiation losses, Erad and the thermal motion of the particles, E, h, Fig. 8.2. Since no energy conversion can take place with 100% efficiency, the fraction of the energy that is recovered can be written as

Eout = Tlfi, E*fu + Лrad Erad + Лт Ел (8-19)

where the TJ’s are the respective conversion efficiencies for each energy component.

The energy supplied to the device in order to sustain the fusion reactions, E, n* in Fig.8.2, appears as the thermal motion of the particles and as radiation:

Лы E*n = Erad + E*th • (8.20)

Here Т|іп is the fraction of Ein* which is coupled to the plasma and amounts to Eth* and Erad*; that is, only the fraction Г|ІПЕІП* is actually deposited to heat and sustain the fuel at a constant temperature.

The energy viability statement, Eq.(8.1), can now be written as

* *

t?/H E}u + Лrad Erad + Л. Н Ел > ~ad + ~ (8.21)

Ліп

or

ЛіпЛош{е% + E*ad + E*h) > Emd + Eth (8.22)

where, for reasons of algebraic convenience, we have taken Tm as an average conversion efficiency

Лот = ——- у 1 = fU> rad> th • (8-23)

2jEe

l

Consider now the idealization that the system operates in a mode characterized by a constant power production during a global energy confinement

time, Те* such that the global energy terms, E, can be readily evaluated from the respective power densities by

E*t = TE<Pe(r)d. (8.24)

v

Also, for present purposes it is assumed that radiation losses are due to bremsstrahlung only so that Prad = Ръг. Equation (8.22) can therefore be expressed in terms of these steady power flows for time Те* to give

Ліп Лот J d3 r( t£* Pfu + т£> Ры + 3NT) > j d3 r( т£* Ры + 3NT) (8.25)

V V

where we have substituted for the thermal energy density according to Eq. (8.13), Eth(r) = j(NiTi+ NeTe) = 3NT (8.26)

assuming Ni = Ne = N and T, = Te = T. Further, we have suppressed Boltzmann’s constant к in our notation, since from now on we will always refer to T as kinetic temperature in units of eV or keV and hence already incorporate the constant k.

Next we introduce in Eq. (8.25) the density and temperature dependent expressions for Ре, and Pbr discussed in preceding chapters and-for reasons of a simplified demonstration of the parameter interrelation-we assume homogeneity throughout the plasma volume V to obtain

7^r<ov>ab QabTE.+AbrN2^TE. +3NT>AbrN24rxE.+3NT

f + Oab У

(8.27)

with the Kronecker-8 introduced in the fusion rate density in order to account for the case of indistinguishable reactants, i. e. when a = b. Here T(), Qab and Abr are constants, whereas <Gv>ab is a function of temperature (see Fig.7.5). Taking a 50:50% fuel mixture, i. e., Na= Nb = N/2, and solving the above inequality for the product NTe* yields the well-known Lawson Criterion for energy viability

3(1 ~ Ліп Лот)Т

<crv >ab(T)Qab .. A ГЕ

Ліп Лош — An^ g -(J-Ліп Лот )Abr V Т 1 ~r Oab )

representing a rough, but useful reactor criterion.

Thus, energy viability requires that the density N times the global energy confinement time, Те*, must exceed a particular function of ion temperature. With all constants known and for illustrative purposes taking T|in • T|out ~ 1/3, as was suggested by Lawson, it becomes possible to determine this lower bound of Nte* as a function of the plasma temperature as shown in Fig.8.3.

As indicated in Fig.8.3, the lower bound of Nte. attains a minimum value at a specific temperature. This is a useful result and defines the combination of particle densities, confinement times, and temperature needed for reaction energy break-even conditions. The extremum point in the case of d-t fusion, Nte. ~ 1020 m’3s at approximately 15 keV was commonly cited as an initial experimental objective. Note also that no particular fusion design was necessary in the derivation of this criterion.

Equation (8.28) does not contain all relevant processes. For example, cyclotron radiation emission is not included and it may be difficult to sustain the various operational parameters as constants during the time interval TE*. Nevertheless, Eq.(8.28) is a useful and widely employed criterion. For commercial power applications, it would be necessary to exceed the minimum Lawson limit by perhaps a factor of ten or better.

The structure immediately surrounding the fusion reaction chamber needs to serve several functions, among which are the following: (i) to sustain a sufficiently clean plasma domain, (ii) to recover energy from the emitted radiation and reaction products, (iii) to shield the surrounding structures and personnel, and (iv) to breed tritium required in the d-t reactor core. This fusion reactor blanket thus serves a most essential role and deserves close examination. While the fusion chamber is generally maintained at plasma conditions such that — at best-an energy self-sufficient reaction chain is established, note that it is indeed the adherent blanket where the neutron and radiation energy released from the plasma is deposited, and which will finally provide the energy transformable for external utilization, i. e. for driving electric generators in a power plant.