As an initial case, consider a domain in space for which В = 0 with only a constant electric field affecting the particle trajectory. Orienting the Cartesian coordinate system so that the z-axis points in the direction of E, Fig. 5.1, and imposing an arbitrary initial particle velocity at t = 0, referenced to the origin of the coordinate system, we simply write for Eq.(5.1)

d

m—=q £,k,

where к is the unit vector in the z-direction. This vector representation for the motion of a single charged particle in an electric field can be decomposed into its constituent components

d

dt dt

and these can be solved by inspection:

m

Integrating again with respect to time gives the position of the charged particle at any time t:

The corresponding trajectory for a positive ion is suggested in Fig. 5.2 with the particle located at the origin of the coordinate system at t = 0. This description therefore constitutes a parametric representation of a curve in Cartesian (x, y,z) space and corresponds to the trajectory of the charged particle under the action of an electric field only and for the initial condition specified.

The important feature to note is that the components of motion for this individual charged particle perpendicular to the E-field, that is vx and vy, do not change with time; however, the velocity component in the direction of the E — field, vz(t), is seen from Eq.(5.4) to linearly vary with time. The particle is accelerated in the direction of E for a positive charge and in the opposite

Individual Charge Trajectories direction for a negative charge.

In an axisymmetric case, the flux surfaces (Fig. 10.4) embracing the magnetic field lines are, to a good approximation, annular toroidal surfaces with r = constant. The charged particles follow the helical field lines resulting from the combination of B6 + B0 and hence move on the flux surfaces, except for excursions of the order of the gyroradius.

While most of the particles are free to spiral around the helical field lines as they encircle both the major and minor axis of the torus, there is a class of particles which appear to be trapped in a magnetic well formed by the field variation between the inboard and outboard side of the torus. Both the toroidal and poloidal fields are stronger on the inside than on the outside of the torus, which results in an overall field variation as illustrated in Fig. 10.6 and at length amounts to a sequence of magnetic mirrors. Some of the plasma particles exhibiting a lower vjj in comparison with the particles moving completely about the torus (known as passing particles), are trapped by these mirrors according to the effects discussed in Ch. 9. If the particle trajectory from a number of toroidal journeys is projected onto a transverse plane of the torus (ф = const), a kind of banana-shaped orbit results for trapped particles, i. e. for particles encountering the mirror reflection. This is displayed in Fig. 10.7, where the trajectory of an untrapped particle is also illustrated. Its guiding centre motion is seen on a curve not quite coinciding with the corresponding magnetic flux surface, which is due to first order drift motion across B.

Recalling from Chapter 9 that the fraction of particles trapped in a mirror field is

	(10.18) (10.19)
and assuming Be « Вф such that B-B. °e
1 Ra + rcosd

we find here the ratio

В min Routboard ^ B( Ґ Cl)

В max R/nboard В(г — ~Cl)

1

R„+ а _ 1-е 1 l + e

R„ — a

where Є = a/R0 is the inverse aspect ratio of the tokamak. Thus, the fraction of trapped particles in a tokamak is given by

For a tokamak reactor with a/R0 * 1/3, approximately 70% of the plasma particles appear to be trapped, and thus their enhanced radial diffusion across the confining magnetic field is viewed to be a significant process in tokamak particle leakage.

The trapped particles bounce back and forth between poloidal angles of ±9b, thus avoiding the inner torus region. In combination with the toroidal drift this bounce motion results in the poloidal cross sectional orbit shown in Fig. 10.7, from where the name ‘banana orbit’ becomes self evident.

Apparently, the poloidal rotation of the magnetic field lines tends to suppress the vertical drifts due to field curvature and VB for trapped particles as well; they would spend equal times in the upper and lower halves of the torus if there were no toroidal electric field which, however, is necessary for driving the plasma current. As a consequence of this E-field, the banana orbits are no longer symmetric. It is seen that an ion spends a longer time period in the lower half of

the torus drifting radially inward than in the upper half where it drifts radially outward. In total, the trapped particles should be subjected to a net inward drift towards smaller minor radii, which thus pinches the plasma column. This phenomenon was predicted by A. Ware and is since called the Ware-pinch effect.

Another effect associated with banana trapping is the enhanced particle leakage. As previously discussed, collisions can cause particles to jump across the confining magnetic flux surfaces and thereby determine the diffusion losses. While the maximum distance which an untrapped particle can be displaced as a result of a single collision is of the order of the gyration radius rg, a trapped particle can be subjected to a maximum excursion as great as the banana width Ar, rap indicated in Fig. 10.7. Since typically Arttap> rg and the co-efficient for diffusion perpendicular to the magnetic field, D, increases with r2g = 2 / CO2,

(Eq.(6.16b)) and Ar2ttap, respectively, the trapped particles may escape more rapidly from the tokamak plasma than the untrapped ones. Evidently, if the fraction of trapped particles is large, this leakage enhancement constitutes a substantial problem in tokamak confinement.

Consider now the case of a d-t fusion power device equipped with a tritium breeding blanket, the purpose of which is to breed tritium for the core. We will retain the steady-state operational mode description and also assume that the bred tritium is extracted from the blanket and supplied to the core on a time scale very short relative to the tritium decay half-life; hence, the decay of tritium need not be incorporated into the analysis. Figure 14.4 provides a schematic of the relevant particle flows and reaction rates.

The two important reaction rates are tritium destruction by fusion in the core

Rd, = Nu, cN,,c <w>dt d3r (14.19)

k:

and tritium breeding in the blanket

Rnu = J J4 (v„) — v„ [an6(v„) • N6 + an7 (v„) ■ N7 ]dvnd V. (14.20)

у. v

rb v n

Here Vc and Vb are the core and blanket volumes, Nn is the neutron density, vn is the neutron speed, N6 and N 7 are the lithium-6 and lithium-7 densities and (J„6 and o„7 are the corresponding neutron absorption cross sections for tritium breeding.

Replacement of the burned tritium requires

Rnl. t — Rdt ■

That is, the tritium production rate must be equal to or exceed the tritium destruction rate. In practice, tritium losses in the overall cycle by radioactive decay and transport into containment walls will occur so that for operational reasons R*n(>, > R*dt; this is incorporated in the tritium breeding ratio C, by requiring

*

C, = > 1 • (14.22)

Rdt

We will next consider some dynamical aspect of the tritium inventory in the blanket which incorporates tritium decay and transport.

Extraction

While a very small number of fusion reactions occur naturally under existing terrestrial conditions, the most spectacular steady state fusion processes occur in stellar media. Indeed, the formation of elements and the associated nuclear energy releases are conceived of as occurring in the burning of hydrogen during the gravitational collapse of a stellar proton gas; the initiating fusion process is

p + p-^>d + fi+ + v + 1.2 MeV (1.28)

where p+ represents a positron and v a neutrino. Then, the deuteron thus formed may react with a background proton according to

p + d h + 5.5 MeV. (1.29)

Subsequently, this helium-3 reaction product could fuse with another helium-3
nucleus to yield an alpha particle and two protons:

h + h—>a + 2p + 12.9 MeV. (1.30)

The next heavier element is beryllium, produced by

h + aWBe + 1.6 MeV (1.31)

and is an example of a rare helium-4 fusion reaction. Also, lithium may appear by

1Be + p-^1Li +0.06MeV. (132)

A progression towards increasingly heavier nuclides is thus evident. This process is known as nucleosynthesis and provides a characterization for the initial stages of formation of all known nuclides.

Closed fusion cycles have also been identified of which the Carbon cycle is particularly important:

nC + p—»13A + 1.9 MeV
13 A—»13C + p+ + v + 1.5 MeV

13 C + PW4N+ 7.6 MeV

14 A + p^X50 + 7.3 MeV

75 CM>I5A + ^ + v + 1.8 MeV
15 A + p^>nC+a + 5.0 MeV.

This sequence of linked reactions is graphically depicted in Fig. 1.3 and may be collectively represented by

4p^>a + 2p+ + 2V + 25.1 MeV (1.34)

if all the reactions of Eq. (1.33) proceed at identical rates. This relation suggests that protonium bums due to the catalytic action of the isotopes 12C, 13C, 13N, 14N, 15N, and 150.

Fig. 1.3: Graphical depiction of the Carbon fusion cycle.

The purpose of a fusion energy device is to initiate and sustain nuclear fusion reactions under acceptable operational conditions. An understanding of the reaction rates and the time dependence of bum processes is therefore of importance and will now be considered.

7.1 Elementary D-T Burn

Consider a deuterium-tritium ion population at thermonuclear temperatures in a suitable unit volume of interest. Regardless of what approach is used to heat and confine the ions, a quantity of energy Qdt is released whenever the event

d + t —^ n + ex (7.1)

occurs. As discussed in Sec. 2.5, these reactions proceed at the rate

Rdt(r>t) = JJNd(r>vdd)Nt(r>vt.^CTarflvd • v(|)|vd — td3 vd d3 v,

= N*d N*, JJ /d (r, vd, t) f t (r, v,, t)Udt ([vd — v,|)[Vd — v«I d3 vd d3 v,

,t)ft(r,t, t)(jdt(

Vd — vt|)|vd — Ytd3vd d3v,

\fd(T’yd>t)ft(r, vt, t)d3vdd3v, = Nd(r. t)N,(r, t)«JV>dl(r, t),

(7.2a)

in a unit volume where use has been made of Eqs.(6.40) and (6.41). Note that <av>dt appears correctly now as a function of position and time, arising from the averaging process over the particle distribution functions which explicitly depend upon these variables. Accordingly, the rate of energy-density release-that is the fusion power density-is given by

Pdi(r, t)= Rdt(r, t)Qdt = Nd(Г, t)N,(r, t)<CFv>dt(r, t)Qdt. (7.2b)

Recall that Qdt = 17.6 MeV and <CTv>dt is, of course, primarily a function of the ion temperature and is graphically displayed in Fig.2.5 with a tabulation listed in Appendix C.

Evidently then, the fusion power in a unit volume will change if the ion temperature changes with time or if the fusile ion densities vary with time. These variations are governed by the occurring reactions, by the injection rates, and by the leakage rates, all of which add to or subtract from the particle and energy

densities. Adopting our previous rate equation analysis of the form of Eq.(6.4) to that of particle density and energy density variations at a fixed point in space, i. e., г Ф r(t)-or for the case of uniform ion distributions-so that total derivatives can be used, we choose as an appropriate description for a particular ion density Nj(r, t) the general rate equation

^i={F+j-F-j)+{R+j — R. j). (7.3)

Here, as well as in the following, we omit the space-dependent notation in view of the fact that we are only considering the time dependence of the particle density. Further, F+J and F. j are the ion injection rate and leakage rate densities, with F. j equivalent to the term Vxjj(r, t) in the Continuity Equation, Eq.(6.10b), and R+j is the reaction rate density which adds to Nj(t) while R. j is the reaction rate density which removes particles from the population Nj(t) in the fixed unit volume.

For the special case of zero ion leakage and no ion injection, or if these two processes are exactly equal and hence cancel each other in Eq.(7.3), and for R+j=0, the instantaneous time dependence of the deuterium and tritium ions’ densities is therefore specified by the rate equations

= — Rdt(t) = — Nd(?) N, (t) < ov >dt (7.4a)

dt

and similarly

^±=-Nd(t)N,(t)<ow>d, ■ (7.4b)

dt

These equations can be combined in the form

4~(Nd + N,) = -2Nd(t)N,(t)«n>d, • (7-5)

dtv ‘

Suppose that a 50:50% ratio between the deuterium and tritium ions exists so

that

Ni (t) = Nd (t) + N,(t) = j N; (?) + j N; (?) (7.6)

is the total fuel ion density at an arbitrary time during the bum; this gives for Eq.(7.5) therefore

ёЫ±=.5а1>* iVf(?). (7.7)

dt 2

This differential equation is valid in the unit volume of interest and for a bum time under conditions of F+i — F., = 0. Bum times can be very short, of the order of 10‘8 s for inertially confined fusion, or of the order of seconds-and envisaged to be much longer-for magnetically confined systems.

The associated instantaneous fusion power density generated in this bum is obtained by multiplying the d-t reaction rate density with the energy release Qdt per fusion event and thus

Pd, U) = Nd (*) N, (t) < OV >dt Qdt = <aV>* Qd> Nf (t)

A determination of the time dependence of the bum requires a knowledge of how the fuel density Nj(t) at a fixed position in space varies during a bum time Хь — This variation is described by Eq.(7.7) for the case F+i — Fd = 0 and R+1 = 0, and represents a differential equation which is separable to give

jj<av>dl dt 0

for 0 < t < Ть, with Nj(0) = Nj?0 as an initial condition. The left part can be integrated so that

——— + — = -±i<crv>d[ dt (7.10a)

Ni(t) Ni,0 2j0

or

N,(t) =————— ^————— . (7.10b)

— + i<av>dl dt

Ni, o 0

If the ion temperature remains sufficiently constant during the bum-implying that, for example, the alpha particle reaction products transfer some of their kinetic energy to the plasma and so compensate for energy loss by bremsstrahlung radiation, or that external heating is supplied or that excess energy is removed-then <Gv>d, is also a constant and the resultant isothermal fuel ion density variation with time is simply

Ni(t) = —————————— (7.11)

—— + L2«n>dlt

Ni,0

Thus, the important consequence is that the bumup fraction is determined by the product of three factors: NU), <Gv>dt, and Хь — Figure 7.1 illustrates this variation of the bum fraction with the ion temperature T for several bum times.

The fusion power released in a fixed unit volume, Eq.(7.8), is now given by

Г “|2

and evidently tends to decrease more rapidly with time than the fuel ion density, Fig.7.2.

Finally, the total energy released in a unit volume at a fixed r during the isothermal fusion bum is given by

In general, with the ion temperature varying during the bum time, the more exact non-isothermal expression can be shown to be

The development of beam-target configurations, specifically the effort to achieve increased symmetric energy deposition over the surface of the pellet, has led to what is known as indirect drive. The distinction between this approach and that of direct drive-imparting pulses of energy from lasers or ion beams directly onto the pellet as previously discussed-is as follows.

For a system utilizing indirect drive, the target consists of both a fuel pellet — similar to those discussed in Sec. ll.5-and a small cylindrical cavity, inside which the pellet is located. This cylindrical vessel, known as a "hohlraum", is a few cm long, is made of a high-Z material such as gold or other metal, and has "windows" transparent to the driver on each end, Fig.11.4. Then, instead of requiring all the driver beams to impinge symmetrically on the pellet, as is necessary for direct drive pellet compression, the beams enter both ends of the hohlraum obliquely and ablate the inner surface of the cavity. The high-Z material of the hohlraum emits soft X-rays when so irradiated, and by focusing the driver beams to the appropriate points inside the cavity, a highly symmetric irradiation of the fuel pellet results-followed by the previously discussed stages of pellet compression and heating depicted in Fig. 11.1.

As with most methods of ICF pellet compression to date, indirect drive has only been examined using laser drivers, however, the same approach is believed to be applicable to ion beams. The crucial aspect of indirect drive is establishing the optimal "pointing" of the laser beams-the positioning and focusing of the beams on the cavity’s inner surface which results in symmetric irradiation of the fuel pellet by the emitted soft X-rays. Development has shown this not only to be possible, but symmetric energy deposition on the pellet surface is achieved with fewer complications than when all the driver beams must symmetrically impinge

directly on the pellet.

The other major advantages of indirect drive as compared to direct drive are the better ablation and subsequent compression achieved with X-rays as opposed to the visible light of lasers, and reduced instabilities during pellet compression. These characteristics and the demonstration of energy deposition levels over an entire pellet surface with <1% deviation from uniformity are very appealing. However, the reduced energy coupling from the beam to the pellet, pc in Eq.( 11.17), and the increased complexities of hohlraum manufacture-in addition to the pellets alone-when scaled up to a power plant-type system are disadvantages not be overlooked. Despite these drawbacks, the inclusion of the hohlraum concept and the use of indirect drive-pellet compression does appear necessary in the continuing development of inertially confined fusion systems.

The designer of an ICF power plant faces a number of key decisions including the choice of the driver and the choice of protection scheme for the reaction chamber wall. The selection of a driver will depend on advances in the technology associated with specific types of drivers and on the beam-target coupling efficiency that can be achieved with specific beams. Protection for the reaction chamber wall from radiation and pellet debris released in a microexplosion is a unique and challenging aspect of ICF reactor design. Various possible approaches have been proposed including a large radius chamber with a "dry" wall and various "wet" wall concepts such as a falling liquid metal veil, liquid metal jets, liquid metal droplet sprays or a thin surface layer of liquid metal. The latter concepts all allow a smaller, more compact chamber but face various problems such as the difficulty in quickly pumping out vaporized material between pulses. Other unique design issues relate to pellet manufacture, pellet handling and positioning in the chamber, protection of mirrors, focusing magnets for ion beams, and other beam transport elements.

Note in this context that it is the collisions with small deflections which are responsible for the total Coulomb scattering cross section as becoming very large.

A number of physical considerations suggest that 0min should be chosen in accordance with a maximum impact parameter beyond which Coulomb scattering is relatively small. In a vacuum, the Coulomb field from a dominant isolated charge extends to infinity, implying therefore a scattering "deflection" interaction
at any arbitrary impact parameter. In a plasma, however, the target charge would be surrounded by a particle cloud of opposite charge. The fields due to surrounding particles will effectively "screen-out" the Coulomb field of an arbitrary charge at some maximum impact parameter distance which corresponds to some minimum scattering angle. This distance may be specified as the radius of an imaginary sphere surrounding a target ion such that the plasma electrons will reduce the target ion’s Coulombic field by 1/e at the sphere’s surface. The Debye length, AD as defined by Eq.(3.23a), evidently corresponds to this distance and we take the maximum impact parameter as

fo, max Ad *

Thus, 0mm follows from the inversion of Eq.(3.10):

Figure 3.5 displays the Coulomb scattering cross section for the case of d-t interactions and for comparison also shows the fusion cross section; as suggested previously, it is evident that Coulomb scattering events occur orders of magnitude more frequently than fusion reactions.

The concept and development of fusion reactors based on the confinement of a high temperature plasma with magnetic fields has been approached from many perspectives. However, utilizing the effects of magnetic fields on charged particles has, in general, led to systems wherein the magnetic field lines are either open-as in the case of the magnetic mirror previously discussed-or closed-as for the case of toroidal devices. We first consider the former, following historically the development of magnetic confinement devices which began with open field line systems.

As suggested in Fig. 13.1, the first wall in a typical MCF reactor must directly face the fusion plasma. Hence, it will intercept the fusion neutrons, bremsstrahlung radiation and cyclotron radiation, as well as any plasma constituents which leak across the outer magnetic field lines. Therefore, the wall must maintain structural integrity against particle and electromagnetic radiation damage as well as against stresses induced by temperature gradients and pressure forces associated with vacuum requirements.

Radiation damage in a d-t fusion first wall is largely associated with the 14.1 MeV fusion neutrons. These neutrons have two important effects: i) knock-on collisions that displace nuclides from their normal lattice positions causing internal voids in the microstructure and ii) neutron capture reactions which result in 4He production and hence a build-up of helium pressure in the material lattice; the latter causes a volumetric swelling of the material since helium is relatively immobile and does not rapidly diffuse out of the structure except at very high temperatures. These (n, a) reactions typically have thresholds requiring neutrons of energy in excess of several MeV; thus, the swelling phenomenon is most pronounced in d-t fusion devices due to the high flux of 14 MeV neutrons.

Radiation damage effects may be contained by setting a limit on the neutron wall loading. The average loading, here represented by A„, is defined as

fnpA*Yr

An= ———————- (13.2)

A-w

where Рш(г) is the fusion reaction power per unit volume, Vc is the fusion core volume, Aw is the total wall area and fn is the fraction of the fusion energy carried by the neutrons; the commonly used units for this wall loading are MW m 2. For ICF, however, note that-due to the high density of the compressed pellet- neutrons can be absorbed to some amount in the dense fusion plasma. For reasons of simplicity, we neglect this affect here.

The above definition can be expressed in a more explicit form by using the appropriate equation for the power profile Pfu(r). Adopting the simplified geometry of an axisymmetric torus with circular cross section, and assuming only a radial dependence for the fusion power, gives for Eq.(13.2)

in the case of d-t fusion. Here 7 is the conversion factor of MeV/s to MW, a is the minor radius of the torus and R0 is its major radius. Note that it is the fuel ion densities as well as their temperature as a function of radius which enter as the determining space-dependent functions.

Limits on the neutron wall loading Л„ depend on the specific design and are most commonly set by radiation damage as it affects the component’s lifetime. Recent designs generally specify a range of 1 — 5 MW-m"2 for this parameter.

Additional considerations now need to be added. For example, the power associated with Л„ is not absorbed in the first wall itself since most of the neutrons are transmitted more deeply into the blanket with little attenuation. Thus, actual thermal wall loading must be evaluated separately based on (i) the incident bremsstrahlung and cyclotron radiation, (ii) direct neutron interactions, and (iii) the interactions associated with backscattered neutrons. Since the radiation is largely absorbed near the front surface of the wall, the surface temperature is strongly dependent on this power flow. Indeed, surface heat fluxes over 1 MW-m"2 may be difficult to transmit without exceeding surface temperature limits set by vaporization pressure considerations.

Consider now a single particle trajectory with E = 0 and В constant in the space of interest. The relevant governing equation now becomes

d

m—=q(x B) (5.6)

dt

and by the definition of a vector cross product, the force on the particle is here perpendicular to both v and B. Writing Eq.(5.6) in terms of its vector components, we find

— (v* і + wy j + vz k) = ~ [(vA і + V;j + V; k) x (Bx і + By j + Bz k)] (5.7)

where i, j and к are orthogonal unit vectors along the x, у and z axes, respectively.

Orienting this coordinate system so that the z-axis is parallel to the B-field, thereby imposing Bx = By = 0, Fig. 5.3, and equating the appropriate vector components of Eq.(5.7), we obtain the following set of differential equations together with their initial conditions:

and

^ = 0. vz(0)=vz, o• (5.8c)

at

The interesting feature here is that vx(t) and vy(t) are mutually coupled but that vz(t) is independent of the former. Hence, the z-component is therefore straightforward and based on Eq.(5.8c), we have

V;(f)= vz, o= V|| (5.9)

which is a constant with the parallel line subscripts referring to the direction of the В-field. Then with the reference z-coordinate as z(0) = z0, another integration gives

z(t)=Zo + V (5-Ю)

with the trajectory of the charged particle motion in the z-direction thus established.

To solve for the velocity and position components perpendicular to the B — field direction-that is vx(t), vy(t), x(t) and y(t)-we first uncouple Eqs.(5.8a) and (5.8b) by differentiation and substitution to obtain

d2 vy dt2

(5.11b)

where, upon recognizing that both Eq.(5.1 la) and Eq.(5.1 lb) describe a harmonic oscillation with the same frequency which results in a circular motion of the particle given suitable initial conditions, we have introduced the gyration frequency-sometimes also called the gyrofrequency or cyclotron frequency-as

Evidently, the following forms satisfy Eq.(5.11):

vx(t)= Ax cos(cogt) + Dx sin(0)gt),

Vy (t)= Ay cos( cog t) + Dy sinf cog t)

with Ax, Ay, Dx and Dy constants to be determined for the case of interest.

It is known that a plasma possesses diamagnetic properties. This means that the direction of the magnetic field generated by the moving charged particles is opposite to that of the externally imposed field. Therefore, not only must the equations resulting from Eq.(5.6) be satisfied but the moving charged particle representing in fact a current flow along its trajectory must generate a magnetic field in opposition to B. Some thought and analytical experimentation with Eqs.(5.13) yields velocity components for a positive ion as

vx(t)=v0cos(cOgt + ф) (5.14a)

and

y(t)= — V0 sin(cOgt + ф)

where v0 is a positive constant speed and ф is the initial phase angle. For a negatively charged particle, Eq.(5.14b) will change its sign, while Eq.(5.14a) remains unchanged. Then, with vx(t) and vy(t) thus determined, a further integration yields x(t) and y(t) as

x(t)= x0 + — sin(o)gt + $) (5.15a)

(Og

y(t)= y0 + sign(q)~ cos( (Ogt + ф) (5.15b)

(Og

describing thereby a circular orbit about the reference coordinate (x0, y0) for an arbitrary charge of magnitude q and sign designated by sign(q). This coordinate constitutes a so-called "guiding" centre which can, in view of Eq.(5.9), move at a constant speed in the z-direction. The gyration in З-dimensions would therefore appear as a helix of constant pitch.

From Eqs.(5.14) it follows that the particle speed in the plane perpendicular to the magnetic field is a conserved quantity, and hence we write

V0 = — y/v* + Vy = V_L > (5.16)

in order to denote the velocity component perpendicular to B. Evidently, if vy = 0,

the particle trajectory is purely circular, and the corresponding radius is obviously determined by

V-L _ V-L m COg |^| 3z

The label gyroradius-as a companion expression to the gyrofrequency £Og-is generally assigned to this term rg although the name Larmor radius is also often used.

From the above it is evident that a heavy ion will have a larger radius than a lighter ion at the same Vj_, and, by definition of £Ug, Eq.(5.12), an electron will possess a much higher angular frequency than a proton. Figure 5.4 depicts the circular trajectory of an ion and an electron as a projection of motion on to the x — y plane under the action of a uniform magnetic field В = Bzk. A useful rule to remember is that with В pointing into the plane of the page, an ion circles counterclockwise while electrons circle clockwise. Note that this is consistent with the diamagnetic property of a plasma by noting that, at the (x0,y0) coordinate, the self-generated В-field points in the opposite direction to the external magnetic field vector B.

B (into page)

With a constant velocity component parallel to the В-field and a time- invariant circular projection in the perpendicular plane, the particle trajectory in З-dimensions represents a helical pattern along the В-line as suggested in Fig.5.5; note that vz o = V|| solely determines the axial motion. The pitch of the helix is also suggested in Fig.5.5 and is given by

(Og

which is constant for our case. Thus, an isolated charged particle in a constant, homogeneous magnetic field traces out a helical trajectory of constant radius, frequency and pitch.

_L

Az

T

5.1 Combined Electric and Magnetic Field

Consider next the effect of a combined E and В field on the trajectory of a charged particle. With both fields uniform and constant throughout our space of

interest, the trajectory is expected to be more complex since it will combine features of the two trajectories of Figs. 5.2 and 5.5.

With the В-field and E-field possessing arbitrary directions, it is useful to orient the Cartesian coordinate systems so that the z-axis is parallel to В with E in the x-z plane, Fig. 5.6. The equation of motion, Eq.(5.1), is now expanded to give

m~dt^!x‘l+ Vy^+ v*k) = + £zk) + (vj+ Vj j+ vzk)x(fizk)] (5.20)

so that the velocity components must satisfy

^L = — Ex + — Bzvy = — Ex + 0)g VySign(q), (5.21a)

at m m m

~~ = — — BzVx = -cogVx signfq), (5.21b)

at m

and

dvz _ q dt m

Here, the gyrofrequency C0g, Eq.(5.12), has again appeared and the charge sign of the particle is indicated by sign (q), thereby specifying the direction of gyration. The velocity components vx and vy may be uncoupled by differentiation and substitution of Eqs.(5.21a) and (5.21b):

—~ = 0 + cog~sign(q) = — o)2 vx (5.22a)

dt2 dt

^-jL=-0)g^jLsign(q) dt2 dt

With Ex, B2, and Cflg as constants, this last expression may therefore be written as

(5.23)

where

(5.24)

and

The associated (x, y,z) coordinates follow from a further integration to yield

y = yo + —co^(0gt + <p )sign(q)- — t (5,29b)

COg Bz

z = zo + y\t+ ■ (5.29c)

These equations, Eqs.(5.28) and (5.29), show that the charged particle still retains its cyclic motion but that it drifts in the negative у-direction for Ex > 0, Fig. 5.7, and oppositely if Ex < 0. Hence, the guiding centre’s drift is in the direction perpendicular to both the В-field and E-field and in contrast to the preceding (E = 0, В =£ 0) case, ion and electron motion across the field lines now occur. Additionally, the pitch increases with time-Eqs.(5.28c) and (5.29c)-so that the trajectory resembles a slanted helix with increasing pitch.

О Bz-Field

Ex-Field [1]

where a vector product identity has been utilized for the last step. Note from Eq.(5.31) that the motion of the guiding centre parallel to the magnetic field В can only be affected by the parallel component of F. Then, since vg is perpendicular to B, the velocity component vgCju = уц and is readily seen to be given by

JgcwW — vgc, ii(0) + — [ F\dt m J

Of greater interest here is the perpendicular component vgCi.

Eq.(5.34) as

_ _ FxB

V? C l q B2

because (у д • В ) = 0. This identifies what is called the drift velocity due to the force F, i. e.

Vgc,± = V£>F — (5.37)

This generalization allows us to specify an expression if the additional force acting in a static magnetic field domain is an electric force F = qE, so that the drift velocity due to a stationary electric field is

vfl, r = ^. (5.38)

B2

The label "E X В drift" is frequently used in this context. Note that the drift is evidently independent of the mass and charge of the particle of interest, and consistent with the drift velocity term in Eq.(5.28b), i. e. (- Ex / Bz).

5.2 Spatially Varying Magnetic Field

Each of the preceding cases involved a В-field taken to be uniform throughout the space of interest. Such an idealization is, in practice, impossible although it
can be approached to some degree by using sufficiently large magnet or coil arrangements. In general, however, it often becomes necessary to incorporate a space dependent В-field which governs the particles’ motion. The associated equation of motion is hence written as

m— = q[vx B(r)l. (5.39)

dt

A comprehensive analysis of this equation for a most general case may obscure some important instructional aspects so that a more intuitive approach has merit; this approach also provides for a convenient association with the preceding analysis while still leading to some useful generalizations.

We consider therefore a dominant magnetic field in the z-direction characterized by an increasing strength in the у-direction; that is, a constant non­zero gradient of the magnetic field strength, VB, exists everywhere given by

VB = ^j. (5.40)

dy

Figure 5.8 provides a graphical depiction of this В-field showing also a test ion with specified initial velocity components.

From the previous analysis of the motion of a charged particle in a uniform В-field, it is known that the radius of gyration in the plane perpendicular to the local В-field is given by

»■« = — ==ГЬ (5-41)

ft), I qB

where В is the local magnetic field magnitude at the point where the charged particle’s guiding centre is located; thus, the gyroradius rg varies inversely with

B. Any motion in the z-direction will be additive and depends upon the initial vy component; for reasons of algebraic simplification and pictorial clarity, we may therefore take vy = 0.

Referring to Eq.(5.41) and Fig. 5.8, it is evident therefore that an isolated
charged test particle will trace out a trajectory whose radius of curvature becomes larger in the weaker magnetic field and smaller where the В-field is stronger. We can translate this effect into a process which tends to transform a circular path into a shifting cycloid-like path with the consequence that the particle trajectory now tends to cross magnetic field lines as suggested in Fig. 5.9.

Note that the radius of gyration, Eq.(5.41), is directly proportional to the mass of the particle and that the sign of the charge enters into the direction of the trajectory. This is also indicated in Fig. 5.9 for a positively and negatively charged particle, although not to scale. The feature that the gradient causes a drift of the positive ions in one direction and for electrons in the opposite direction has important consequences because the resulting charge separation introduces an electric field which, in a plasma, can contribute to plasma oscillations and instability, which we will further discuss in subsequent chapters.

With an intuitive approach of the gradient-B drift phenomenon thus suggested, we consider next an approximate though useful analytical representation. The first approximation to be introduced is that for the case of a weak magnetic field inhomogeneity over distances corresponding to a gyroradius, that is, IVBI/B « l/rg, only the first two terms of a Taylor expansion of Bz(y) are retained, i. e.

where y0 corresponds to the у-coordinate of the guiding centre. Extending to the vector form for Eq.(5.42), we write

Ъ(У)~В0 + (У-У0Щ — Bo

where now B0 = B(y0). We next expand the equation of motion, Eq.(5.39), and write

This latter expression is a general formulation since the у-axis was arbitrarily chosen due to the geometry of Sec. 5.3.

Finally, recalling Eqs.(5.36) and (5.37), we find the grad-B drift velocity as

vl В x VB

vD, vB=SI?’1^Ti——————- T~ (5-51)

2(og B2

and thus the positive ions and negative electrons drift across В-field lines in opposite directions in a non-uniform magnetic field as displayed in Fig. 5.9.

In subsequent chapters, we will discover that fusion devices which utilize magnetic fields for confinement of the reacting ions involve very complex magnetic field topologies. Not only are there electric fields and spatially varying magnetic fields present-thus giving rise to the drift velocities vD, E and vd, Vb just discussed-but there are additional complexities due to curvature in the magnetic field lines and gradients along these field lines. We thus now formulate expressions for the effective forces and resulting drift velocities due to these inhomogeneities in the В-fields, analogous to Eqs.(5.38) and (5.51).