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Simulation Study of Triton Confinementand Nuclear Reaction in the Deuterium

Plasma Experiment at LHD

M. Homma1, S. Murakami1, M. Isobe2, H. Tomita3 and K. Ogawa2

1Department of Nuclear Engineering, Kyoto University, Kyoto, Japan2National Institute for Fusion Science, Toki, Japan3Department of Quantum Engineering, Nagoya University, Nagoya, Japan

Corresponding Author: homma@p-grp.nucleng.kyoto-u.ac.jp

Abstract:Deuterium plasma experiments are planned in Large Helical Device (LHD). During deu-terium plasma discharges, 1MeV tritons are produced by D-D fusion reactions betweendeuterium beams and deuterium thermal plasmas. The motions of these energetic tritonsare complicated because of their large finite orbit effect and the three-dimensional mag-netic field configuration of LHD. The confinement of energetic tritons is investigated bythe Global NEoclassical Transport (GNET) code, which can solve the five-dimensional driftkinetic equation using Monte Carlo methods. We evaluate the velocity space distributionand particle loss fraction of the energetic tritons. The loss of the tritons is attributed totwo processes: prompt orbit loss and diffusive loss. The loss fraction of energetic tritonsincreases to 30% on a short time scale of approximately 10−5 s by prompt orbit loss andthen gradually increases to 90% on a slow-down time scale of approximately 10−1 s by dif-fusive loss for the assumed plasma parameters. The prompt loss fraction is also almostindependent of the plasma density and largely depends on the magnetic configuration. Fur-thermore, the amount of neutrons generated by D-T fusion reactions between tritons anddeuterium thermal ions is estimated.

1 Introduction

Energetic tritons of 1 MeV are produced by D(d,p)T fusion reactions during D-D dis-charges, and the secondary D-T fusion reactions between tritons and deutrium plasmasproduce 14 MeV neutrons. The confinement and slowing down of energetic tritons can beexperimentally investigated by measuring the production rate of 14 MeV neutrons. Theseexperiments have been performed in JT-60U[1] known as triton burn-up experiments.

In the Large Helical Device (LHD), experiments using deuterium plasmas are plannedto clarify the isotope effect on energy confinement or turbulent transport and to under-stand energetic ion confinement. Plasma confinement is expected to improve by use ofdeuterium plasmas. In the deuterium experiments, 1 MeV tritons are produced by fu-sion reactions between deuterium Neutral Beam Injection (NBI) beams and deuterium

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thermal ions. Understanding the behavior of energetic tritons would make it possible toexperimentally study energetic particle confinement in future reactors.

In helical systems such as LHD, however, motions of the energetic tritons are compli-cated since magnetic field configurations are inherently three dimensional. In addition,energetic tritons have large orbits that can easily become complicated. These complicatedmotions may lead to a significant loss of energetic tritons.

In this study, we investigate the confinement of energetic tritons for the LHD deu-terium plasma using the Global NEoclassical Transport (GNET) code, in which the driftkinetic equation of energetic particles is solved in five-dimensional phase space. The ve-locity distributions of energetic tritons are calculated over a range of minor radii, andwe present the characteristics of the triton distribution in velocity space. Next, we cal-culate the energy and particle loss fractions of tritons and investigate their dependenceon plasma parameters. Finally, we evaluate T(d,α)n nuclear reaction rates using the ob-tained velocity distribution of tritons and simulate triton burn-up in the LHD deuteriumplasmas.

2 Simulation Model

We apply the GNET code[2, 3], in which finite drift orbits and complex motions of trappedparticles are included, to solve the drift kinetic equation using Monte Carlo methods. Thedrift kinetic equation for tritons in five-dimensional phase space is described as follows:

∂fT∂t

+ (v‖ + vdr) ·∂fT∂x

+ v̇ · ∂fT∂v

= Ccoll(fT) + Lparticle(fT) + ST, (1)

where fT is the distribution function of tritons, v‖ is velocity parallel to the field line,vdr is the drift velocity, C

coll(fT) is a linear Coulomb collision operator, Lparticle(fT) is the

particle loss term from the last closed flux surface (LCFS), and ST is the source term ofthe tritons.

GNET is also applied to evaluate the source profile of the tritons solving the driftkinetic equation for NBI beam ions. Using the distribution functions of NBI deuteronsobtained by GNET, we calculate the D(d,p)T fusion reaction rates between deuteriumbeams and thermal deuterium ions. The averaged reaction rate of the D(d,p)T reactioncan be expressed as follows:

〈σDDv〉 =∫∫

fbeamD (vbeamD )f

thD (v

thD )σDD(EDD)|vbeamD − vthD |dvbeamD dvthD , (2)

where fD and vD represent distribution function and velocity of deuterons. The suffixbeam and th mean the NBI fast ion and Maxwellian thermal ion respectively. The crosssection σDD is given as follows[4]:

σDD(EDD) =

[(1.220 − 4.36 × 10−4EDD)2 + 1

]−1× 372

EDD

[exp

(46.097E

− 12

DD

)− 1

] , (3)

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where EDD is the deuteron kinetic energy for the relative velocity between the Maxwellplasma and the deuteron beam-ion.

Using the distribution functions of tritons obtained by GNET, we calculate the T(d,α)nfusion reaction rates between energetic tritons and thermal deuterium ions. Similarly, theaverage of reactivity for the T(d,α)n reaction can be expressed as follows:

〈σDTv〉 =∫∫

f thD (vthD )fT(vT)σDT(EDT)|vthD − vT|dvthD dvT, (4)

where vT is velocity of energetic tritons. The cross section for the reaction is given asfollows[4]:

σDT(EDT) =409 +

[(1.076 − 1.368 × 10−10EDT)2 + 1

]−1× 50200

EDT

[exp

(49.95E

− 12

DT

)− 1

] , (5)where EDT is the deuteron kinetic energy for the relative velocity between the Maxwelldeuterium plasma and the fast triton.

3 Results

Confinement of tritons in the D-D experiment on LHD is simulated assuming typical valuesfor the plasma parameters: core electron temperature Te(0) = 3.0 keV; edge electrontemperature Te(a) = 0.1 keV; core ion temperature Ti(0) = 3.0 keV; edge ion temperatureTi(a) = 0.1 keV; core electron density ne(0) = 0.8, 2.0, and 3.5 × 1019 m−3; edge electrondensity ne(a) = 0.1× 1019 m−3; magnetic field strength B0 = 2.75 T; magnetic axis majorradius Rax=3.60m; and beta value β = 0.23%. The radial profiles of plasma temperatureand density are given by

Te(r) = (Te(0) − Te(a))[1 −

(ra

)2]+ Te(a), (6)

ne(r) = (ne(0) − ne(a))[1 −

(ra

)8]+ ne(a). (7)

The bulk plasma is assumed to be a hydrogen-deuterium mixed plasma with equal amountsof each species.

Five NBI heating systems are installed in the LHD: three tangential injection beams(Eb = 180 keV) and two perpendicular beams (Eb = 40 keV). The injection energy ismuch higher for the tangential injection beams, and the fusion reactions between thetangential injection beams and the thermal ions are dominant in the LHD deuteriumplasma experiments. Assuming tangential NBI with energy Eb = 180 keV, we evaluatethe quantities per megawatt of heat power.

We evaluate the D-D fusion reaction rate applying the velocity distribution of deuteronbeams obtaind by the GNET code. In Fig. 1, the radial profiles of the evaluated triton

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0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1

D(d

,p)T

fus

ion

rate

s [1

07cm

-3s-

1 ]

r/a

ne=0.8×1019m-3

ne=2.0×1019m-3

ne=3.5×1019m-3

(a) Plasma density dependence

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1

D(d

,p)T

fus

ion

rate

s [1

07cm

-3s-

1 ]

r/a

Rax=3.50(m)Rax=3.60(m)Rax=3.75(m)

(b) Magnetic configuration dependence

FIG. 1: Radial birth profile of tritons with different (a) plasma density (ne(0) = 0.8, 2.0,and 3.5 × 1019 m−3) and (b) magnetic configurations (Rax = 3.50, 3.60, and 3.75 m).

production rate due to D-D fusion reactions in various density and magnetic configurationcases are shown. It is found that the triton production rate does not simply depend onthe density or the position of the magnetic axis because the population of energetic beamions depends on the beam ion birth and slowing-down process. Hence, interestingly, thehighest production rate occurs at ne(0) = 2.0 × 1019 m−3. The total production rate oftritons per 1 MW of heat power is approximately 3 × 1014 s−1 regardless of the plasmadensity or the magnetic configurations.

We apply the calculated source term to the GNET code and evaluate the triton dis-tribution in velocity and real spaces. First, we calculate at the typical case of densityne(0) = 2.0 × 1019 m−3 and magnetic configuration Rax = 3.60 m. The velocity distribu-tions at minor radii r/a = 0.2, 0.5, and 0.9 and the total are presented in Fig. 2. Herev‖ and v⊥ represent the parallel and perpendicular velocity components relative to themagnetic field direction, and they are normalized by the velocity of 1 MeV tritons v1MeV.

In Fig. 2(a), a relatively large number of tritons are seen in the region where |v⊥| �|v‖|. These are the deeply helically trapped particles, whose orbits are stable along helicalripples. However, the distribution is reduced in the neighboring region (|v⊥| > |v‖|).Particles in this region make a transition between passing and trapped particle orbits andbehave as stochastic particles. The stochastic behavior of the transition particles wouldenhance the radial diffusion of energetic particles. In Fig. 2(b) and (c), we see a smallor negligible distribution of deeply trapped particles. This is because helically trappedparticles drift to the region close to the plasma edge in those radial positions and areeasily lost by orbit loss.

In Fig. 3, typical orbits of the energetic tritons in the poloidal cross section at variouspitch angles are shown. The direction of the magnetic field is upward orthogonal to thepage, and the direction of ∇B is leftward parallel to the page. Starting at time t = 0 s atminor radius r/a = 0.5 and poloidal angle θ = 90◦, a triton with v‖ > 0 moves poloidally ina counterclockwise orbit. The stable motion of deeply trapped particles and the stochasticbehavior of the transition particles are seen in Fig. 3(a) and (b), respectively. Due to the

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-1 -0.5 0 0.5 1

v||/v1MeV

0

0.2

0.4

0.6

0.8

1

v ⊥/v

1MeV

18 19 20 21 22 23 24 25 26

(a) r/a = 0.20

-1 -0.5 0 0.5 1

v||/v1MeV

0

0.2

0.4

0.6

0.8

1

v ⊥/v

1MeV

14 16 18 20 22 24 26

(b) r/a = 0.50

-1 -0.5 0 0.5 1

v||/v1MeV

0

0.2

0.4

0.6

0.8

1

v ⊥/v

1MeV

6 8 10 12 14 16 18 20 22 24

(c) r/a = 0.90

-1 -0.5 0 0.5 1

v||/v1MeV

0

0.2

0.4

0.6

0.8

1

v ⊥/v

1MeV

14 16 18 20 22 24 26 28

(d) total

FIG. 2: Contours of the velocity distribution function of tritons at different normalizedminor radii: (a) r/a = 0.20 (near the magnetic axis), (b) r/a = 0.50, (c) r/a = 0.90(near the LCFS), and (d) total.

poloidal drift motion, the orbit of the passing particles with v‖ > 0 is shifted outward,while that with v‖ < 0 is shifted inward. This is the reason for the asymmetry of thevelocity distributions at r/a = 0.2 (Fig. 3(c)) and at r/a = 0.9 (Fig. 3(d)).

The lost triton distribution in velocity space is presented in Fig. 4. In the GNETcode, a particle is considered to be lost when it has reached the LCFS. The particle lossmechanism in fusion plasmas is classified into two major categories: prompt orbit lossand diffusive loss. A lot of tritons escape with almost all of their initial energy of 1 MeV.This is prompt orbit loss due to drift motion immediately after their birth. In the passingregion, we can notice the tritons that get partially thermalized before reaching the LCFS.This occurs because the passing particles moving near the LCFS undergo pitch-anglescatterings due to collisions with bulk ions. In the region with pitch angle θp ∼ 120◦,a large number of lost particles can be seen independent of their energy. This tendencyresults from stochastic diffusion of the transition energetic tritons, as mentioned above.

The temporal history of particle and energy loss fraction is presented in Fig. 5. Promptorbit loss normally occurs before a particle has completed its first orbit in the poloidaldirection, i.e., before t ∼ 10−5 s. In this calculation, 30% of the tritons generated escapeas a result of prompt orbit loss, and then diffusive loss becomes dominant. The totalparticle loss fraction, which is the sum of the prompt orbit and diffusive loss fractions,reaches 96% at t = 0.3 s with the energy loss fraction at 92%. Particle and energy lossfractions obtained in this calculation can be more or less overestimated because re-enteringparticles[5] are not considered in the GNET code. The energy loss fraction is a little lowerthan the particle loss fraction. This suggests that a triton loses some of its energy due tocollisions with the bulk plasma before reaching the LCFS.

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-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

z/a

r/a

(a) Helically trapped particle

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

z/a

r/a

(b) Transition particle

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

z/a

r/a

(c) Passing particle with v‖ > 0

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

z/a

r/a

(d) Passing particle with v‖ < 0

FIG. 3: Typical orbits of (a) helically trapped particle (θp = 92◦), (b) transition particle

(θp = 110◦), (c) passing particle with v‖ > 0 (θp = 10

◦), and (d) passing particle withv‖ < 0 (θp = 170

◦).

-1 -0.5 0 0.5 1

v||/v1MeV

0

0.2

0.4

0.6

0.8

1

v ⊥/v

1MeV

0

2

4

6

8

10

FIG. 4: Contours of the velocity distribution function of tritons that escaped from theplasma.

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0 10 20 30 40 50 60 70 80 90

100

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

part

icle

loss

fra

ctio

n [%

]

time [sec]

(a) Particle loss fraction

0 10 20 30 40 50 60 70 80 90

100

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

ener

gy lo

ss f

ract

ion

[%]

time [sec]

(b) Energy loss fraction

FIG. 5: Temporal history of (a) particle loss fraction and (b) energy loss fraction.

We investigate the dependence of particle loss fraction on the plasma parameters. InFig. 6(a), a comparison of the particle loss fractions for the three cases, ne(0) = 0.8, 2.0,and 3.5 × 1019 m−3, is presented for fixed ion and electron temperatures, Ti(0) = Te(0) =3.0 keV. In the case of ne(0) = 0.8 × 1019 m−3, the prompt orbit loss fraction is 30% andthe total loss fraction increases to 98% at t = 0.3 s. When ne(0) = 3.5 × 1019 m−3, 30%of all the tritons are lost from the confinement domain due to prompt orbit loss, and intotal 92% have entered the loss region by time t = 0.3 s. Changes in the plasma densityhave little effect on prompt orbit loss. For ne(0) = 3.5 × 1019 m−3, however, the diffusiveloss fraction is approximately 6% lower than that for the case ne(0) = 0.8 × 1019 m−3.The slowing-down time of energetic tritons due to collisions with background plasmasdepends inversely on plasma density. Hence, the slowing-down time is shorter for thehigher density case and the diffusive loss of tritons is reduced.

The loss dependence on the position of the magnetic axis is presented in Fig. 6(b).When Rax = 3.50 m, the prompt orbit loss fraction is 25% and the total loss fraction

0 10 20 30 40 50 60 70 80 90

100

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

part

icle

loss

fra

ctio

n [%

]

time [sec]

ne=0.8×1019m-3

ne=2.0×1019m-3

ne=3.5×1019m-3

(a) On plasma density

0 10 20 30 40 50 60 70 80 90

100

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

part

icle

loss

fra

ctio

n [%

]

time [sec]

Rax=3.50(m)Rax=3.60(m)Rax=3.75(m)

(b) On magnetic configurations

FIG. 6: Dependence of particle loss fraction on (a) plasma density (ne(0) = 0.8, 2.0, and3.5 × 1019 m−3) and (b) magnetic configurations (Rax = 3.50, 3.60, and 3.75 m).

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is 89% at t = 0.3 s In the magnetic configuration Rax = 3.75 m, which is the standardconfiguration in LHD, the prompt orbit loss rate is as high as 45%. When the magneticaxis is shifted outward, the particle orbit greatly deviates from the flux surface and theprompt orbit loss fraction increases.

We evaluate the secondary D-T reaction rate applying the obtained distribution func-tions of tritons. In Fig. 7, the radial profiles of the evaluated production rate of 14 MeVneutrons in various density and magnetic configuration cases are shown. The fusion ratespeak at approximately r/a = 0.2 because of the presence of well-confined trapped parti-cles. In these radial positions, energetic particles deeply trapped in helical ripples existas seen in Fig. 2(a). It is found that those trapped tritons significantly contribute tofusion reactions with background deuterons. In the magnetic configuration Rax = 3.50 m,however, the distribution has a flat peak near r/a = 0.2 because there are few trappedtritons in the radial position.

We calculate the total production rates of 14 MeV neutrons integrated over the plasmavolume. The value is 6.5 × 1010 s−1 for the typical plasma parameters; ne(0) = 2.0 ×1019 m−3 and Rax = 3.60 m. Figure 8(a) indicates the dependences of the neutron emissionrates on the plasma density. The production rate increases with the plasma densitybecause the fusion rate is proportional to the density of target deuterons. In Fig. 8(b),a comparison of the D-T neutron emission rates for the three configurations is presented.When the magnetic axis is shifted outward, the loss of energetic tritons increases as shownin Fig. 6(b). Therefore, the production rate is the lowest for the magnetic configurationRax = 3.75 m.

4 Conclusion

We have investigated the confinement of tritons in the LHD plasma during the D-Dexperiment using GNET, the five-dimensional drift kinetic equation solver. The tritonproduction rate has been evaluated by using the distribution function of NBI deuterons

0

2

4

6

8

10

12

14

0 0.2 0.4 0.6 0.8 1

T(d

,α)n

fus

ion

rate

s [1

03cm

-3s-

1 ]

r/a

ne=0.8×1019m-3

ne=2.0×1019m-3

ne=3.5×1019m-3

(a) Plasma density dependence

0

2

4

6

8

10

12

14

0 0.2 0.4 0.6 0.8 1

T(d

,α)n

fus

ion

rate

s [1

03cm

-3s-

1 ]

r/a

Rax=3.50(m)Rax=3.60(m)Rax=3.75(m)

(b) Magnetic configuration dependence

FIG. 7: Radial birth profile of neutrons with different (a) plasma density (ne(0) = 0.8, 2.0,and 3.5 × 1019 m−3) and (b) magnetic configurations (Rax = 3.50, 3.60, and 3.75 m).

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0

2

4

6

8

10

12

14

16

0.5 1 1.5 2 2.5 3 3.5 4neu

tron

pro

duct

ion

rate

[10

10s-

1 ]

electron density ne [1019m-3]

(a) On plasma density

0

2

4

6

8

10

12

14

16

3.50 3.60 3.75neu

tron

pro

duct

ion

rate

[10

10s-

1 ]

position of magnetic axis Rax [m]

(b) On magnetic configurations

FIG. 8: Dependence of total production rates of neutrons on (a) plasma density (ne(0) =0.8, 2.0, and 3.5 × 1019 m−3) and (b) magnetic configurations (Rax = 3.50, 3.60, and3.75 m).

and the dependency of the production rate on the plasma parameters has been calcu-lated. We have evaluated the velocity distribution of the tritons and analyzed the lossmechanisms of energetic tritons. Prompt orbit loss begins immediately after the birth oftritons, and later the diffusive loss, which is caused by stochastic behavior of the tran-sition particles and pitch-angle scatterings of the passing particles, is the primary causeof continuing particle loss. Furthermore, we have calculated the particle and energy lossfractions and investigated the dependence on plasma parameters. Finally, we have sim-ulated triton burn-up and estimated the amount of 14 MeV neutrons produced by D-Tfusion reactions between tritons and deuterium plasmas.

In the deuterium experiment plasma, beam-beam reactions, in which both reactingions are injected NBI deuterons, would also occur as well as beam-thermal reactions.Therefore, it is necessary to estimate the contribution of beam-beam fusion reactions formore precise prediction of triton burn-up ratios and neutron production rates.

References

[1] NISHITANI, T., et al., Plasma Phys. Control. Fusion 38 (1996) 355.

[2] MURAKAMI, S., et al., Nucl. Fusion 40 (2000) 693.

[3] MASAOKA, Y., et al., Nucl. Fusion 53 (2013) 093030.

[4] HUBA, J.D., NRL Plasma Formulary, Naval Research Laboratory, Washington, DC(2002).

[5] SEKI, R., et al., Plasma Fusion Res. 3 (2008) 016.