Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 (2004)
6 (2004)630–633
000–000
J. Plasma Fusion

Ignition Condition for p-11B Reactor withLHD type Magnetic Field Configuration
WATANABE Tsuguhiro, MATSUMOTO Yutaka1, HISHIKI Masahito1,OIKAWA Shun-ichi1 and HOJO Hitoshi2
National Institute for Fusion Science, Toki, 509-5292, Japan
1Graduate School of Engineering, Hokkaido University, Sapporo, 060-8628, Japan
2Plasma Research Center, University of Tsukuba, Tsukuba, 305-8577, Japan(Received: 9 December 2003 / Accepted: 28 October 2004)

Abstract
Proton-Boron fusion reactor (p-11B reactor) might be able to be achieved by the combination of the LHD typemagnetic field configuration and the ion cyclotron range of frequency (ICRF) heating scheme of protons. The LHD
magnetic field has the excellent ability for the high energy particle confinements. This characteristic is studied bynumerical computations of high energetic particles in the LHD magnetic field where the strong ICRF field is applied. Itis shown by the Langevin equation analysis that the steady state distribution function of ICRF heated proton becomesto the quasilinear plateau distribution function (QPDF). The fusion reaction rate of p-11B is calculated for QPDF protons and studied the ignition condition. It is found that the ignition condition becomes possible to be satisfied if effec22
3
tive temperature of proton is of the order of 300 keV and nτ >~ 10 s/m .
Keywords:
LHD, ICRF heating, chaotic orbit, chaotic field line, p-11B reactor
promising [4-7]. High β plasma confinement is progressivein LHD experiments. Theoretical studies show the potentiality of high beta plasma confinement by the combination of theline-tying effect of diverter field lines and high magnetic
shear structure in peripheral regions [8].
In Sec. 2 we briefly summarize the ICRF heatingprocess in LHD. The steady state distribution function of theICRF-heated proton in LHD is evolved in Sec. 3. Evaluationof ignition condition is summarized in Sec. 4. Summary anddiscussions are in Sec. 5.

1. Introduction
Energy resources, whose consumption keeps increasing
year by year, should be transfer to more clean ones. ProtonBoron fusion reactor (p-11B reactor)[1,2] probably offers theultimate energy resource. The fuels (Proton and Boron) areubiquitous on the earth and fast neutrons are not generated.
Main reaction_ 8.7 MeV
p + 11B → 3α + Qα , Qα ∼(1)

is neutron-free. The side reaction

p + 11B → n + 11C − 2.8 MeV(2)

2. ICRF heating process in LHD

is an endothermal reaction and produces only slow neutrons
when energy of proton exceeds about 3 MeV or more.
A detailed study of the p-11B reactor has been done [3].
It has been, however, pointed out that the p-11B reactor could
not satisfy the ignition condition since the bremsstrahlung
power loss due to the large atomic number of boron (ZB = 5)
is too large.
But, the progress of LHD experimental and theoreticalstudies has brought a new prospect for the p-11B reactor.
Experiments and computational studies are establishing thehigh performance for production and confinement of highenergetic particles in LHD configurations. The ion cyclotron
range of frequency (ICRF) heating of LHD plasma is very

The quasilinear theory on the evolution of the plasma
distribution function under ICRF heating has been developedby Stix [9]. The average change in energy per ICRF resonance zone transit, ∆W⊥ is given by,∆W⊥ =Mp
22π
ER2F ∂ BM∂r v||e2Mp

2pe,

(3)

where Mp is the mass of a proton, e the elementary electriccharge, ERF the RF electric intensity, B the magnetic fieldintensity,rthe length along the magnetic field line and v|| theparticle speed parallel to the magnetic field line, respectively.
©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: wata@nifs.ac.jp1630

Watanabe T. et al., Ignition Condition for p-11B Reactor with LHD type Magnetic Field Configuration

If we substitute typical values of the ICRF experimentsof LHD as

ERF = 20 kV/m, B = 2.75 T,

1 ∂B= 0.3 m,
B ∂r

(4)

we get for 2 keV protons_ 3.48 keV,∆W⊥ ∼(5)where v|| is the thermal velocity of 2 keV proton and assumed
to be constant. In this case, the standard statistical theory pre_ 8.2 ×
dicts that protons are necessary the total (103/∆W⊥)2 ~4
10 events (almost equal to 4.6 s for 2 keV protons in LHD)for their average energy to reach to 1 MeV (= 103 keV) by
the ICRF heating. In the computer simulations of the ICRF
heating in LHD (total particle number = 726), however, weare observing average energies reaching to 0.7 MeV by theheating during 12.4 ms. In addition, the ICRF experiments ofLHD have been observed a smooth high energy proton tailreaching to 500 keV [6].
These results imply that a rapid heating process — “runaway ion heating process” — works in the ICRF heating ofLHD. That is, the accelerated ions are accelerated preferentially further more in the ICRF heating of LHD. The rapid
heating process is caused with the trapped and near trapped
particles whose magnetic mirror turning points coincide with
the ion cyclotron resonance pints. As they dwell for a considerably longer time near resonance, particles within this classgain much more energy than the quasilinear average changein eq. (3)[10]. For the realization of rapid heating process,non-particle-loss is necessary in the course of the transition
from passing particle to helical trapped mirror particle. Inaddition, a complete confinement of a helical trapped particle
is essential factor for rapid heating process. In the following,
we show that the LHD magnetic field configuration satisfies
these two requirements, the runaway ion heating process ispossible and the steady state distribution function do not
become the Maxwellian.
In LHD, the magnetic field is produced by the continuous helical coils (r/m = 2/10). There are 3 types of particles,
the passing, reflecting, and chaotic orbit particles according
as the initial pitch angles of particles. Passing particles andreflecting particles can form the closed drift surface and beconfined completely in collisionless case. Reflecting particles, which are trapped in the helical mirror region and are
possible to circulate around the magnetic axis, have shorter
periodic lengths compared to passing particles. Then adiabaticity of reflecting particles becomes stronger than that ofpassing particles. This fact realizes the good confinement ofreflecting particles and the high performance of ICRF heatingin LHD. Reflecting particles can be confined even in the outside the last close flux surface (LCFS).
Chaotic orbits particles, whose pitch angles are in therange of transition regions between passing particles andreflecting particles, have the finite but controllable lifetime.
Lifetimes of chaotic orbit particles become long (short) byincreasing (decreasing) magnetic field strength or by decreas-Fig. 1 ICRF heating process in LHD. Poincarè plots areshown in (v||, v⊥) at the poloidal section of φ = π /2, thatis the starting poloidal section of particles (initial conditions are shown by open circles : total 11 particles).
The chaotic orbit region has no degradation effect for
heating process in LHD. Particles are lost to the vacuum vessel wall along the diverter field lines with v|| <0 when E >~ 1 MeV.

ing (increasing) the particle energies. LHD has no loss-cone[11]. These nature are shown by Poincarè plot in (v||, v⊥)
plain of the ICRF heating process (Fig. 1). Dense plots
expanding in v⊥ direction are accumulated in small v⊥ region.
This means that the energy changing in each transitions of
the ICRF resonance region is relatively small and its direction is almost perpendicular to the magnetic field. On theother hands, ICRF-heated particles with large v⊥ show sparseplots expanding in reflection region. This means that theenergy changing in each transitions of the ICRF resonanceregion is large. The ICRF-heated particles can pass throughthe chaotic orbit region without the degradation effect. Highenergy ions can be heated further more by preferentiallyabsorbing the RF energy. Thus, we consider that the runaway
ion heating process occurs in ICRF heating of LHD plasma.

3. Steady state distribution function ofICRF-heated proton in LHD
The simplest model, which determines the steady state
distribution function of runaway ion heating process, can beassumed to be built up by the Langevin equation

dv= (−µ' + α'v2 − β'v4)v + R(t),dt(6)where v is the ion velocity, µ' the coefficient for friction
force mainly due to electron drag, α' the coefficient for therunaway ion heating process and β' the coefficient for a termination process of ion heating, respectively. R(t) is a random force due to resonant cyclotron heating process. The
Fokker-Planck equation corresponding to this Langevinequation is given by [12]2631

Watanabe T. et al., Ignition Condition for p-11B Reactor with LHD type Magnetic Field Configuration∂f∂ ⎧⎛∂⎞ ⎧=⎨ ⎜−g + D ⎟ f ⎨ ,∂ t ∂ v ⎩⎝∂v ⎠ ⎩where C = 1/1.9242060···. Effective temperature Teff for theQPDF is defined by(7)3np Tp_eff =2

where f is the distribution function of ion,g(v) = (−µ' + α 'v2 − β 'v4)v(8)

Tp_eff = T × 0.959735333668⋅⋅⋅.

(9)

∫(10)

⎛1
⎞
〈σ v〉 = d3vb d3vpσ ⎜ Mp(vp − vb)2⎟⎝2⎠∫∫|vp − vb| fp(vp)fb(vb),(11)

whereµ'Dβ'D=MpT,α'η≡⎛ Mp ⎞ 1=⎜,
⎟D ⎝ T ⎠ α23 ⎛ Mp ⎞ 1,⎜⎟
4 ⎝ T ⎠ β(12)np and T are the density and temperature of the proton. If thepopulation inverted distribution function is formed, the particle energy transfers to the RF energy. Then, the steady state
distribution function will be settled down to the formation ofthe plateau, even under the runaway ion heating process. The
condition of the plateau formation

dd2
f(E)
=f(E)
= 0,dEdE2
E=Eplateau
E=Eplateau(13)reduces toβ = 3α 2 ,Eplateau= α.
T(14)Then the parameter α determines the energy at the plateau.
We expect that the ICRF heating of LHD can sustain the highenergy proton to the optimal energy level (Eplateau = T). Thus,
it is assumed that α = 1 and β = 3. We call this distributionfunction as the quasilinear plateau distribution function(QPDF). Explicit form of QPDF for proton is given by⎛ Mp ⎞fp(E) = np × C ⎜⎟⎝ 2πT⎠η=(19)

1.58 × 1022〈σ v〉Pα=> 1,PB(Tp_eff /e)1/4(20)since PB is proportional to T 1/2
e_eff . Figure 2(a) shows η calculated by eq. (20). It can be seen from Fig. 2(a) that η for the_ 300 keV.
p-11B reactor can exceed unity around T p_eff ~The power loss due to the heat conduction and convection flow (PL) is also important for the p-11B reactor. PL ischaracterized by the energy confinement time τ as3/2⎡ ⎧ E ⎛ E ⎞ 2 1 ⎛ E ⎞ 3⎧⎤
× exp ⎢− ⎨ − ⎜ ⎟ + ⎜ ⎟ ⎨⎥ ,3 ⎝ T ⎠ ⎩⎦⎣ ⎩T ⎝ T ⎠Pα> 1,PB

where Pα = npnb 〈σ v〉 Qα is the heating power density by α
particles, and PB is the power density of bremsstrahlung loss
due to electrons. If eq. (19) is not satisfied, it is impossible tokeep burning in the p-11B reactor of which confinement performance is dramatically improved. In eq. (19), the electron
temperature is assumed to be determined by the power balance between the losses due to bremsstrahlung and heating
due to electron drug of protons. Since the high energetic tailis decreased very much in QPDF protons, it is assumed thatthe distribution function of the electron is also QPDF ofeffective temperature Te_eff. In this case, Te_eff is proportional
to T p1/2_eff and electron drug time τE becomes almost twice
compared with the case of Maxwellian distribution function.
The density ratio δ ≡ nb /np is one of an important factor
for the sustainment of p- 11 B burning. If δ is too large,
bremsstrahlung loss exceeds the fusion power density. If weassume that ne = (1 + 5δ)np and Tb = Tp/20, δ optimized tomaximize the value of η approximately becomes
0.1418588···. Then, the ignition condition given by eq. (19)for the p-11B reactor is reduced to3=(18)where σ is the p-11B fusion cross section and fb(vb) is the distribution function for boron assuming Maxwellian distribution with the temperature Tb.
The most fundamental condition necessary for a fusion
reactor to keep burning is given by the following,⎛ M ⎞ 3/2
⎡ ⎧ E 1 ⎛ E ⎞ 2 1 ⎛ E ⎞ 3 ⎧⎤f(E) = np C ⎜ p ⎟ × exp ⎢− ⎨ − ⎜ ⎟ + ⎜ ⎟ ⎨⎥ ,⎝ 2 πT ⎠β ⎝ T ⎠ ⎩⎦⎣ ⎩T α ⎝T⎠1
Mpv2,2(17)We have calculated the p-11B fusion reaction rate 〈σ v〉of protons with QPDF (eq. (11)) assuming β = 3.

where C' is a constant for normalization. This steady state
distribution function can be rewritten as a function of energy
E:

E=(16)

4. Ignition condition for p-11B reactor

The steady state solution of the Fokker-Planck equation (7) is
given by [12]⎛1⎞f(v) = C'exp ⎜dvg(v)⎟ ,⎝D⎠Mp(vx2 + vy2 + vz2)
f(v),2and this relation is reduced toand D is defined by the auto-correlation of random force R(t),〈R(t)R(t' )〉 = Dδ (t − t').

∫d3v

(15)

3632

Watanabe T. et al., Ignition Condition for p-11B Reactor with LHD type Magnetic Field Configuration

the steady state distribution function of the ICRF-heated protons becomes to QPDF given by eq. (15). Based on theseresults, the possibility of the p-11B reactor using the LHD
type magnetic field and the ICRF heating scheme are studied.
It is found that the ignition condition becomes possible to be22
3
satisfied in the case of Tp_eff ~ 300 keV and neτ >~ 10 s/m .
To prevent the dilution of fuels and to reduce the radiation loss, the α ash exhaust is very important for fusion reactor. The combinations of the LHD magnetic field and ICRF
heating scheme could realize the active α ash exhaust. An
electric power necessary for exhausting the α ash and holding the QPDF-distribution function of protons is a circulating
power of the fusion power plant, which is desirable to be insmall power level. Since the fusion-produced α particles arein population inversion, the excitation of the microinstability
can be expected. Because the ratio of cyclotron frequency
between the α particle and the proton is 1 : 2, there is a possibility that holding of the proton QPDF-distribution and a
rapid exhaust of the α ash are achieved simultaneously by
this microinstability. In this case, the circulating power of thefusion power plant can be greatly decreased. N.J. Fisch has
pointed out the possibility of α power channeling using ionBernstein waves in tokamak [13]. The detailed analysis of themicroinstability in p-11B reactor is an interesting problem inthe future.

Fig. 2 Ignition condition for p-11B reactor. (a) : η shows theratio of fusion power density P c and the radiation
power density P B by bremsstrahlung. Dot-dashedcurve shows that the ignition condition (η > 1) cannot
be satisfied by Maxwellian proton. (b) : n × τ values
necessary to keep the ignition condition (τ : energy
confinement time of core plasma). For the comparison, ignition conditions of the DT and D3 He reactors
are shown, also.

⎧3
⎧ 13
3
PL = ⎨ npTp_eff + nbTb + neTe_eff⎨ .
222⎩⎩τ

References[1] J.M. Dawson, UCLA Report (University of California,Los Angels 1976) PPG-273.
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part B (Academic Press, New York, 1980) p. 453.
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[4] K. Saito et al., Nucl. Fusion, 41. 1021 (2001).
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[7] Y. Matsumoto et al., Jpn. J. Appl. Phys. 43. 332 (2004).
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[10] T.H. Stix,Waves in Plasmas (AIP, New York 1992) p.
520.
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[13] N.J. Fisch, Phys. Plasmas 2, 2375 (1995).

(21)

If PL is considered, the ignition condition necessary for afusion reactor to keep burning can be written as

Pα> 1.
PB + PL(22)

In this case, neτ value for the p-11B reactor is shown in Fig.
2(b). This result indicates that the ignition condition (eq.
(22)) becomes possible when Te_eff is of the order of 300 keV
22
3and neτ >~ 10 s/m .

5. Summary and discussions
We show that the LHD magnetic field has excellent
ability for the high energy particles confinement by thenumerical computation of the ICRF-heated particles and that4633