Studies of the fast ion confinement in the Large
Helical Device by using neutron measurement
and integrated codes
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公開日: 2021-12-16
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キーワード (En):作成者: NUGA, Hideo, SEKI, Ryosuke, OGAWA,
Kunihoro, KAMIO, Shuji, Fujiwara, Yutaka, OSAKABE,
Masaki, ISOBE, Mitsutaka, Nishitani, Takeo,
YOKOYAMA, Masayuki, LHD, Experiment Group
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Under consideration for publication in J. Plasma Phys.

1

Studies of the Fast Ion Confinement in LHD
by Using Neutron Measurement andIntegrated Codes
H. Nuga1 †, R. Seki1,2 , K. Ogawa1,2 , S. Kamio1 , Y. Fujiwara1 , M.
Osakabe1,2 , M. Isobe1,2 , T. Nishitani1 , M. Yokoyama1,2 , and LHD
Experiment Group1
1

National Institute for Fusion Sciences, National Institutes of Natural Sciences, Toki, Japan
2
The Graduate University for Advanced Studies, SOKENDAI, Toki, Japan(Received xx; revised xx; accepted xx)

The neutral beam (NB) fast ion confinement in the Large Helical Device (LHD) is studied
for several full field (Bt ∼ 2.75 T) magnetic configurations by the combination between
the neutron measurement and simulations. To investigate the NB fast ion confinement,
we have performed a series of short-pulse NB injection experiments. The experimentresults are analyzed by integrated code TASK3D-a. From this investigation, the eﬀective
particle diﬀusion coeﬃcient of the tangential and perpendicular NBs are approximately
Deﬀ ∼ 0.1m2 /s and Deﬀ ∼ 1m2 /s in the standard configuration. It is clarified that the
NB fast ion confinement improves when the plasmas are shifted inward. Moreover, it
is also found that the simulation, which considers the deuteron dilution eﬀect due tothe presence of impurity ions, can describe the neutron emission rate consistent to themeasurement.

1. Introduction
Generally, the temperature of the fusion plasmas is sustained by the external heating.
The heatings are achieved by the kinetic energy transfer from fast particles to the bulk
plasmas through the Coulomb collision. Additionally, in the future fusion power plant,
the fusion plasmas are mainly sustained by the fusion-born α particles, which have 3.5
MeV kinetic energy. Therefore, fast particles should be confined until their velocities are
decelerated to the plasma thermal velocity. For this reason, the fast particle confinement
is one of the most important issues(Heidbrink & Sadler 1994; Gorelenkov et al. 2014) for
magnetically confined fusion devices.
There are several diagnostics systems to investigate the fast particle physics(Gorelenkov
et al. 2014), such as neutral particle analyzer (NPA)(Medley & Roquemore 1998;
Osakabe et al. 2006; Kamio et al. 2019), fast ion Dα spectroscopy (FIDA)(Luo et al.
2004; Heidbrink et al. 2007), fast ion loss detector (FILD)(Darrow et al. 1995; Ogawa
et al. 2010), and neutron spectroscopy(Adams et al. 1993; Ishikawa et al. 2007). Because
the deuterium plasma experiment campaign began in the Large Helical Device (LHD)(Takeiri 2018a,b; Osakabe et al. 2018, 2017) from March 2017, the investigation of
fast ion behavior using neutron diagnostics(Isobe et al. 2018; Ogawa et al. 2019) andFIDA(Fujiwara et al. 2019) have been enabled in LHD.
In LHD deuterium plasmas, ∼ 2.45 MeV fast neutrons are yielded owing to thedeuterium-deuterium (DD) fusion reaction. Because of the high injection energy neutral† Email address for correspondence: nuga.hideo@nifs.ac.jp

2

H. Nuga, et. al.

beams up to Eb ∼ 180 keV, most of the neutrons are generated by the fusion reaction
between fast deuteron and thermal deuteron instead of the reaction between thermalions(Osakabe et al. 2017). This is the so-called “beam-thermal” fusion reaction. In suchcases, we can study the fast ion global confinement from the neutron emission rate
Sn . This is because the DD fusion reactivity increases monotonically with their relative
kinetic energy up to E ∼ 2 MeV. If the plasma temperature and density are constant in
time, the decrement of the neutron emission rate comes from the decrement of the fusioncross-section due to the fast ion deceleration and the decrement of the fast ion density.
Because the fast ion deceleration can be estimated by simulations, we can estimate thefast ion confinement from the neutron measurement without simulating the fast ion
transport, which is usually too complex to estimate accurately.
We have studied the eﬀective confinement time of NB fast ions in LHD using neutron
measurement in this paper. There are several previous works to investigate fast ion
confinement through the neutron diagnostics. Earliest works in tokamaks(Strachan et al.
1981; Heidbrink et al. 1988, 1991) showed that the classical collisional slowing down model
can explain the neutron emission within good agreement. The following investigation in
JT-60U tokamak(Tobita et al. 1994) reported the deviation from the classical collisionalslowing down model due to the toroidal field ripple for perpendicular NB. Anomalous
fast ion losses, which can not be explained by the toroidal field ripple, were observed
on TFTR(Ruskov et al. 1999). Moreover, fast ion confinement was investigated by using
neutron measurement also in various types of magnetically confinement device such as
the reversed field pinch devices(Fiksel et al. 2005; Anderson et al. 2014), the sphericaltokamak(Heidbrink et al. 2003), the tokamak(Zhang et al. 2012), and the stellarator andheliotron(Isobe et al. 2001; Nuga et al. 2019), In our previous work(Nuga et al. 2019),
we have reported the NB fast ion confinement time by using neutron measurement for
single magnetic configuration in LHD. In this paper, we investigate the dependence ofthe fast ion confinement on the magnetic configuration and discuss the contribution of
the charge exchange loss to the fast ion confinement especially in low electron density
plasmas.
The rest of this paper consists of the following. The simple instruction of the experimental scenario and experiment devices are provided in section 2. Simulation tools andmodels used in this paper are introduced in section 3. The results of the analyses toestimate the fast ion confinement time are shown in section 4 and section 5. The dilution
eﬀect of the deuteron density due to the presence of impurity ions is discussed in section
6. Section 7 concludes this paper.

2. Experimental setup
2.1. Experimental scenario
A series of experiments which use short pulse NB injection have been performed in
LHD to investigate the fast ion confinement time. The decay time of Sn , yielded by DD
fusion reaction has been observed after short pulse NBs have been turned oﬀ. In theprevious work(Nuga et al. 2019), we performed the short pulse NB experiment in single
magnetic configuration (Rax = 3.6 m, Bt = 2.75 T), where Rax and Bt indicate theposition of the magnetic axis and the strength of the magnetic field on the magnetic
axis. The direction of the magnetic field is counter clockwise in the top view. In thispaper, we have performed the short pulse NB experiment in five magnetic configurations,(Rax , Bt )=(3.53m, 2.80T), (3.55m, 2.79T), (3.60m, 2.75T), (3.75m, 2.64T), and (3.90m,

10
8
6 NB#4 perp
4 ENB~35keV
20neutron emission Te[keV]
rate[e14/s]
ne[1019m-3 ]port

PNB[MW]

Studies of the Fast Ion Confinement in LHD

3

NB#5 perp ENB~80keV
NB#1 co-NB#2 ctr-

NB#3 co-ENB~180keV

ENB~150keV

ENB~155keV

3
2

Te(ρ=0.5)1ne(ρ=0.5)

0
2
1
0
2.5

3

3.5

4
time[sec]

4.5

5

5.5

Figure 1. Typical waveform of this series of experiments (SN148442) is displayed. the NB port
through power in each beam, the plasma temperature and density on the position, where thenormalized minor radius ρ = 0.5, and the neutron emission rate are shown.

2.54T), respectively. Typical waveform of this series of experiments is shown in figure 1.
NB pulse width is ∼ 40 ms in each beam.
In this series of experiments, the fast ion confinement time, τc , is assumed to be:τc−1 = τn−1 − (τncl )−1 ,(2.1)

where τn is the neutron decay time after NB has been turned oﬀ and τncl is the e-folding
time of the neutron decay rate due to the classical fast ion deceleration. This is because
the reduction of the neutron emission rate is caused by the decrease of the fusion crosssection due to the deceleration and the decrease of the fast ion density due to the fast ion
loss. Because the contribution of the fast ion loss is ignored at the estimation of τncl , τc canbe estimated from τn and τncl . In this paper, τn can be obtained from the experimental
results and τncl can be obtained from the simulation results instead of the analyticalsolution given by the reference(Strachan et al. 1981). The method of the simulation will
be introduced in section 3.
2.2. Experiment Apparatus
In this series of experiments, two external heating systems, electron cyclotron heating(ECH) and NBI, are used. Plasmas are sustained by 3.5 MW ECH.
The top view of the NBI system in LHD(Takeiri et al. 2010) is shown in figure 2.
Typical injection energy and power in each beam is also displayed. In LHD, there arethree negative ion-based tangential NBs and two positive ion-based perpendicular NBs.
The direction of NB#1 and NB#3 are co-direction to the magnetic field and the directionof NB#2 is counter-direction to the magnetic field. It is noted that the injection energyof NB#4 is set to be untypically low energy, Eb ∼ 35 keV, to investigate the dependenceof the perpendicular NBs on the injection energy.
Experiments have been performed with five magnetic configurations as noted in sec.
2.1. The averaged plasma minor radius, which depends on the position of the magnetic
axis, is approximately 0.58 − 0.63 m.
LHD is equipped with several diagnostic systems(Kawahata et al. 2010). In thispaper, we have mainly used electron temperature and density profiles, the ion ratio
among proton, deuteron, and helium ion, and the time evolution of the neutron emission
rate. The Thomson scattering diagnostics is used for the measurement of the electron
density and temperature(Narihara et al. 2001; Yamada et al. 2010). A fast-response, wide4

H. Nuga, et. al.
(NB#5)
E~80keV, Pport~9MW(NB#2) E~150-160keV
Pport ~2MW

(NB#3) E~140-155keV
Pport ~1.5MW(NB#1)
E~180keV
Pport ~1.5MW(NB#4)
E~35-60keV
Pport ~2.5-7MW

Figure 2. Top view of NBI system in LHD. There are three tangential beams, NB#1, NB#2,
and NB#3, and there are two perpendicular beams, NB#4 and NB#5. Because the direction of
the magnetic field line is counter-clockwise, NB#1 and NB#3 is the co-direction to the magnetic
field line.

Measurement Data
3D equillibrium
VMEC
NB fast ion birth prof.
FIT3D
HFREYAMCNBINB heating, Sn
CONV_FIT3D
Figure 3. Flow of the simulations to obtain the time evolution of the neutron emission rate.

dynamics range Neutron Flux Monitor (NFM) (Isobe et al. 2018) is equipped in LHD tomeasure the neutron emission rate.

3. Simulation tools
Estimation of τncl is required to evaluate τc as shown in eq. 2.1. Although there is theanalytical expression of τncl (Strachan et al. 1981), we use the following simulations toinclude the plasma profiles, plasma volume, and NB birth profile for the estimation of
τncl .
In this paper, the neutron emission simulations have been performed by the
combination among VMEC(Hirshman & Whitson 1983), FIT3D (HFREYA andStudies of the Fast Ion Confinement in LHD

5

MCNBI)(Murakami et al. 1995; Sato et al. 2008; Vincenzi et al. 2016; Seki et al.
2019), and CONV_FIT3D(Nuga et al. 2019), which are included in the integrated
transport analysis suite TASK3D-a(Yokoyama et al. 2017). The flow of the simulations
are displayed in figure 3. Three-dimensional magnetic equilibrium is calculated by
VMEC code. According to the obtained magnetic configuration, FIT3D calculates NB
fast ion birth profile. At this step, NB fast ion orbits are followed for a few tens of
micro seconds by using Monte Carlo method to consider the NB fast ion prompt loss
eﬀect. In this process, the eﬀect of the birth profile broadening owing to the deviationof the NB fast ion drift orbit from the magnetic surface is also considered. Moreover, forpositive ion-based NBs, the full, half, and one third energy fractions of injected neutral
particles are assumed to be 0.78, 0.16, and 0.06, respectively. Although these values are
the typical values of perpendicular NBI systems in LHD, it is not reliable for NB#4.
This is because NB#4 has untypically low injection energy in the following analyses.
CONV_FIT3D solves the NB fast ion slowing down equation to obtain the energy
distribution of NB fast ion. Here, the fast ion transport across the magnetic surface
during slowing down process is neglected. In LHD, the beam-thermal fusion reaction isdominant instead of the reaction between thermal ions due to the insuﬃcient plasma
temperature. Additionally, in this paper, the fusion reaction between NB fast ions is also
negligible. Therefore, it is enough to consider the fusion reaction between NB fast ion
and thermal ion in this paper. Because it can be considered that the velocity distributionof thermal ion is isotropic, the velocity direction of NB fast ion is not necessary forthe estimation of the fusion reaction rate. For this reason, CONV_FIT3D, which canobtain the energy distribution of NB fast ion, can estimate the neutron emission rate.
The details of CONV_FIT3D are introduced in section 3.1.

3.1. CONV_FIT3D
CONV_FIT3D is a code solving the fast ion slowing down equation based on theclassical Coulomb collision theory. In the previous work(Nuga et al. 2019), we had chosen
the model which assumes NB fast ion velocity vEP is suﬃciently faster than thermal
ion velocity vth . However, in this paper, we have a perpendicular NB#4, which has
an injection energy of Eb = 35 keV. NB fast ions, whose kinetic energy is 35 keV,
can not satisfy the assumption vEP ≫ vth because typical temperature of this series ofexperiments is a few keV. Therefore, we have refined the slowing down equation based on
the eq. (30c) in reference (Karney 1986). This model assumes the background Maxwellian.
The refined equations are expressed as below:dE
2 2= − ZEP
E×
dt
τse
[
(
)
]
3/2 3/2∑ ni Z 2
3π 1/2 mEP Te1i1+g(ui )
1/2
n
m
E 3/2
e
i
4me
i
g(ui ) = erf(ui ) − ui erf ′ (ui )3/2
3(2π)3/2 ϵ20 mEP Te
τse =
1/2
n e4 me ln Λ
∫ eu
2
2

erf(u) =exp(−x )dx
π 0
2
erf ′ (u) = exp(−u2 ),
π

(3.1)
(3.2)
(3.3)
(3.4)
(3.5)

H. Nuga, et. al.

neutron emission rate[e14/s]

port

PNB[MW]

610
8
6 NB#4 perp
4 ENB~35keV
20SN148422

NB#5 perp ENB~80keV
NB#1 co-NB#2 ctr-

NB#3 co-ENB~180keV

ENB~150keV

ENB~155keV

5
simulation
experiment

4
3
2
1
0
2.5

3

3.5

4
time[sec]

4.5

5

5.5

Figure 4. Example of the neutron emission simulation assuming nD = ne and τc = ∞ isdisplayed.

√√
where τse is the Spitzer slowing down time on electron, ui = v/ 2vth,i and vth = Ti /mi .
In the following calculation, we assumed that the ion temperature Ti is equal to theelectron temperature Te . In actuality, however, it is expected that Te is greater than Ti
because the plasmas are sustained by ECH.
As noted above, in LHD, the beam-thermal fusion reaction is dominant. In this case,the fusion reaction rate can be expressed as:∫(3.6)
Rbt = nD ⟨σv⟩bt (Ti , EEP )fEP (v EP )dv EP ,where nD is the deuteron density in the bulk plasma. The beam-thermal reactivity ⟨σv⟩bt ,which is derived by Mikkelsen(Mikkelsen 1989), is implemented in CONV_FIT3D. The
fitting coeﬃcients used in the expression of ⟨σv⟩bt are chosen from reference(Bosch &Hale 1993).

4. Analysis results for tangential beams
In figure 4, the example of the simulated neutron emission rate is shown with themeasured neutron emission rate. The solid curve is the measured neutron emission
rate and the dashed curve is the simulated neutron emission rate. The simulation overestimates the neutron emission rate due to the assumptions that nD = ne and τc = ∞.
From measured and simulated neutron emission rates, we can estimate τn and τncl byusing the least mean square fitting. The fitting functions are f (t) = Snp,exp exp(−t/τn )for measured decay rate and f (t) = Snp,sim exp(−t/τncl ) for simulated decay rate, whereSnp,exp and Snp,sim are the peak values of the measured and simulated neutron emission
rate. The durations which is used for this fitting are defined as the times when the neutron
emission rates decay to Snp,exp /e and Snp,sim /e, respectively.
Similar experiments have been performed with the same magnetic configuration (Rax =3.53 m, Bt = 2.8045 T) and the diﬀerent plasma parameters to survey the dependenceof τn on τncl . It is noted that the control parameter is the electron density ne in theseexperiments. Additionally, because the plasma heating power is constant in each shot, theelectron temperature Te also changes with the electron density. Figure 5 is obtained fromexperiment and simulation results for three tangential NBs. Their injection energies arealso displayed in fig. 5. In this figure, x axis denotes the simulated neutron decay time τncl
and y axis denotes the experimental neutron decay time τn . The y error-bars shown in fig.
5 denote the standard deviation of τn . The solid guideline indicates y = x. If the NB fastτn : neutron decay time (exp) [sec]Studies of the Fast Ion Confinement in LHD
0.4

7

tangential
SN148417-SN148434: Raxis=3.53m, Bt=2.8045T (CCW)

0.3
0.2
Eb =180keV NB#1
=150keV NB#2
=155keV NB#3

0.1
0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

τcl
n : neutron decay time (simulation) [sec]Figure 5. The experimentally measured neutron decay times are plotted against the simulated
neutron decay time for tangential NBs in the configuration of Rax = 3.53 m. Dashed guide curveis τn = τc τncl /(τncl + τc ), τc = 0.53 s.

ions are confined perfectly, plots obey this guideline. Actually, because the NB fast ion
loss is not ignorable, plots separate from the guide line as shown in fig. 5. Additionally,in fig. 5, there is the dashed guide curve which indicates τn = 0.53τncl /(τncl + 0.53). The
reason for the value τc = 0.53 is explained later. From fig. 5, it is found that the diﬀerent
trend, which does not obey the dashed guide curve, appears in the τncl > 0.4 region. Thismeans that the dominant transport mechanism is diﬀerent between the τncl < 0.4 s region
and the τncl > 0.4 s region.
It is noted that high frequency (> 10 kHz) MHD instabilities triggered by NBI hardly
appeared in short τncl region. On the contrary, in long τncl region, NB fast ion driven
instabilities were observed in several discharges. Therefore, we can ignore the anomaloustransport due to fast ion driven instabilities in short τncl region. The following analyses
in long τncl region may include the anomalous transport. The low frequency (< 10 kHz)MHD instabilities excited by bulk plasma pressure were observed through this series ofexperiments. The typical amplitude of the low frequency magnetic fluctuation, which isdetected by magnetic probe set on the vacuum vessel, were δB/Bt < 2×10−5 . This value
is suﬃciently low as compared to the case that the low frequency magnetic fluctuation
plays important role to the NB fast ion loss(Ogawa et al. 2010). Therefore, it can beconsidered that NB fast ion loss due to the low frequency fluctuation can be negligible.
4.1. Long τncl region (τncl > 0.4 s)
In LHD plasmas, it is known that the neutral atom density in plasma core region has
negative ne dependence(Fujii et al. 2015; Goto et al. 2011, 2016) because the penetrationof neutral atoms into the plasma core increases with the decrease of the plasma density.
Therefore, it can be considered that the neutral atom density in plasma core region inτncl > 0.4 sec region is higher than that in τncl < 0.4 sec region. Here we focus on theτncl > 0.4 s region to investigate the contribution of the charge exchange loss to the fast
ion loss.
From fig. 5, it is found that the values of τn for NB#1, NB#2, and NB#3 areapproximately τn ∼ 0.25 s, 0.2 s, and 0.2 s at τncl = 1 s. Here we assume that the
separation from the dashed guide curve in long τncl region can be described only by the
charge exchange loss. In this assumption, the neutron decay time τn can be expressed as:−1τn−1 = (τncl )−1 + (τceﬀ )−1 + τcx,

where τceﬀ

(4.1)

is the eﬀective confinement time in short τncl region and τcx is the charge
exchange time. In this configuration, the value of τceﬀ , which will be obtained in section

8

H. Nuga, et. al.
60

80

150

180

60

80 100 120 140 160 180 200

14

1/<σ v>cx[s/m 3 *10 ]

35
2

1

0
0

20

40

beam energy [keV]
Figure 6. The dependence of the inverse of the charge exchange reactivity 1/ ⟨σv⟩cx on beamenergy is shown. Here, it is assumed that the velocity of neutral atom is negligible.

neutron decay time (exp) [sec]SN148417-SN148434: Raxis=3.53m, Bt=2.8045T (CCW)τeff
c =0.53[sec]
0.2

0.1

Te(ρ=0.5)=3.4 keV
ne(ρ=0.5)=0.92e19 m−3
NB#1 Eb =180keV
NB#2 =150keV
NB#3 =155keV

0

0

0.1
0.2
0.3
neutron decay time (simulation) [sec]0.4

Figure 7. The short τncl region (τncl < 0.4 s) of figure 5 is focused.

4.2, is τceﬀ = 0.53 s. For these assumptions and estimations, the charge exchange times for
NB#1, NB#2, and NB#3 are approximately τcx = 0.90 s, 0.47 s, and 0.47 s, respectively.
The dependence of the charge exchange reactivity on the beam energy is shownin fig. 6. Although fast ions in plasmas have a velocity distribution, we estimate thecharge exchange reactivity ⟨σv⟩cx in each beam from the beam injection energy. This isbecause that we have discussed the fast ion confinement through the measurement of
the neutron emission rate, which exponentially depends on the beam energy within therange of NB injection energy. Accordingly, the charge exchange time can be expressed−1
as τcx= n0 ⟨σv⟩cx (Eb ), where n0 is the neutral atom density. From this expression, we
can estimate the values of n0 in each beam. The neutral atom densities, which are felt
by fast ions from NB#1, NB#2, and NB#3, are n0 = 1.7 × 1014 m−3 , 1.9 × 1014 m−3 ,and 2.1 × 1014 m−3 , respectively. In the reference (Goto et al. 2016), the neutral atom
density in the LHD plasma, whose electron density is ∼ 2 × 1019 m−3 , is measured as
approximately 1014 m−3 . Because the plasmas in long τncl region have the electron densityof approximately ne ≲ 0.5×1019 m−3 , it can be expected that the neutral density is larger
than 1014 m−3 in these plasmas. Therefore, it can be considered that the estimation of
n0 is within the realistic value and the charge exchange loss is not negligible in long τncl
region.
4.2. Short τncl region (τncl < 0.4 s)
Here we focus on the region τncl < 0.4 s of fig. 5 to exclude the cases that the charge
exchange loss is significant. The dashed guide curve in figure 7 is obtained by the weighted
least mean square fitting in τn = τncl τceﬀ /(τncl + τceﬀ ) function in τncl < 0.4 s region. It isnoted that because three tangential NBs have a similar trend, they are fitted in a single

Diffusion coefficient [m2 /s]

Studies of the Fast Ion Confinement in LHD

9

0.3

0.2

0.1
Tangential NBs
0
3.5

3.6

3.7
3.8
Position of magnetic axis[m]3.9τn : neutron decay time (exp) [sec]Figure 8. The eﬀective particle diﬀusion coeﬃcients are plotted against the position of themagnetic axis.

0.05

perpendicular
SN148417-SN148434: Raxis=3.53m, Bt=2.8045T (CCW)

0.04
0.03
0.02
E b =35keV NB#4
=80keV NB#5

0.01
0
0

0.1

0.2

0.3

0.4

τcl
n : neutron decay time (simulation) [sec]Figure 9. The experimentally measured neutron decay times are plotted against the simulated
neutron decay time for perpendicular NBs in the configuration of Rax = 3.53 m. Dashed guidecurve is τn = τc τncl /(τncl + τc ), τc = 0.057 s.

curve. For this fitting, the eﬀective fast confinement time in this region is obtained asτceﬀ = 0.53 s.
Similar analyses are performed for diﬀerent configurations shown in section 2.1. To
compare the confinement among diﬀerent magnetic configurations, the eﬀective fast ion
confinement time τceﬀ is converted into the eﬀective particle diﬀusion coeﬃcient Deﬀ . The
relation between τceﬀ and Deﬀ is expressed as
Deﬀ =

a2
,
5.8τceﬀ(4.2)

where a is the plasma minor radius. Here it is assumed that Deﬀ is uniform in the plasma
and the plasma poloidal cross-section is not elongated. The Rax dependence of Deﬀ isshown in figure 8. It is found that the fast ion confinement becomes poor when the plasma
is shifted outward. Additionally, the absolute value of Deﬀ obtained in this paper is larger
than that obtained in tokamaks(Tobita et al. 1994; Heidbrink et al. 1988).

5. Analysis Results for perpendicular beams
Perpendicular beams are analyzed by a method similar to the tangential beams. Figure
9 shows the relation between τn and τncl for perpendicular NBs. The dashed guidecurve indicates τn = 0.057τncl /(τncl + 0.057). In figure 9, the diﬀerent trends appear at
approximately τncl = 0.1 sec.

H. Nuga, et. al.
τn : neutron decay time [sec]

10
0.05
0.04

perpendicular
SN148417-SN148434: Raxis=3.53m, Bt=2.8045T (CCW)τeff
c =0.057[sec]

0.03
0.02

NB#4 E b =35keV
NB#5 =80keV

0.01
0

0

0.02

0.04

0.06

0.08

0.1

τcln : neutron decay time (simulation) [sec]Diffusion coefficient [m2 /s]

Figure 10. The short τncl region (τncl < 0.1 s) of figure 9 is focused on.
4321Perpendicular NBs
0
3.5

3.6

3.7
3.8
Position of magnetic axis[m]3.9

Figure 11. The eﬀective particle diﬀusion coeﬃcients for perpendicular beams are plottedagainst the position of the magnetic axis.

5.1. Long τncl region (τncl > 0.1 s)
Similar to section 4.1, we investigate the contribution of the charge exchange loss
to the fast ion confinement. From fig. 9, the neutron decay time of NB#4 and NB#5
at τncl = 0.2 s are approximately τn = 0.025 s and 0.035 s, respectively. From thesevalues and eq. (4.1), the charge exchange times of NB#4 and NB#5 can be estimatedas τcx = 0.057 s and 0.17 s, respectively. The neutral atom densities evaluated from theτcx are n0 = 2.1 × 1014 m−3 , and 1.3 × 1014 m−3 , respectively. These values of density
are within the range of realistic value as noted in sec. 4.2. Therefore, it can be also
considered that the charge exchange loss are not negligible for perpendicular beams inlong τncl region.
5.2. Short τncl region (τncl < 0.1 s)
Owing to the above discussion, we focus on the τncl < 0.1 sec region to exclude the casesthat the charge exchange loss is significant. In figure 10, the short τncl region τncl < 0.1
sec of figure 9 is focused on. The guide curve indicates τceﬀ τncl /(τncl + τceﬀ ), τceﬀ = 0.057
sec. The value of τceﬀ is obtained by the weighted least mean square fitting in τn =
τceﬀ τncl /(τncl + τceﬀ ).
Similar to the discussion in section 4.2, the eﬀective particle diﬀusion coeﬃcient Deﬀ
for perpendicular beam is estimated for the five magnetic configurations. In figure 11,
the dependence of Deﬀ on Rax for perpendicular beams are shown. It is found that theperpendicular beams have a poorer confinement than the tangential beams. The same astangential beams, The confinement of the perpendicular beams becomes poor when theplasmas are shifted outward.
It is noted that the perpendicular beam injection energy is not uniform in eachconfiguration due to experimental constraints. In the cases of Rax = 3.75 m and 3.9m, the perpendicular beam injection energy is Eb = 60 keV. The data of Rax = 3.6 m
configuration consist of the data with Eb = 35 keV, 60 keV, and 80 keV. In the remaining

Studies of the Fast Ion Confinement in LHD

11[e14/s]

1

p,exp

2

Sn

SN148417-SN148434: Raxis=3.53m, Bt=2.8045T (CCW)
y=0.42x

NB#1
NB#2
NB#3
NB#4
NB#5

0
0

1

2

3

4

5

6

p,sim

F* Sn

[e14/s]Figure 12. The relation between Snp,exp and FSnp,sim is shown for the Rax = 3.53 m
configuration. The solid line indicates y = x and the dashed line indicates y = 0.42x.

configurations (Rax = 3.53 m and 3.55 m), the data consist of those with Eb = 35 keV
and 80 keV. These data are fitted in a single curve as shown in fig. 10.

6. Discussion regarding the deuteron dilution
In this paper, in order to estimate the fast ion confinement time, we have focused on theneutron decay time instead of the absolute value of the neutron emission rate itself. This isbecause that there are some ambiguities for estimating the absolute value of the neutron
emission rate. The fusion reaction rate in beam-thermal reaction dominant plasmas is
described as eq. 3.6. Our simulation over-estimates the neutron emission rate due to theassumptions of nD = ne and τc = ∞ as shown in fig. 4. In actual plasmas, however,deuterons are diluted due to the presence of impurity ions. Because the evaluation of the
impurity density profile is diﬃcult, the ambiguity can not be reduced to estimate theneutron emission rate. On the other hand, the estimation of the neutron decay time has
fewer ambiguities. This is because the fast ion slowing down process mainly depends onthe measurable values, which are electron density ne and temperature Te . Since we haveestimated the fast ion confinement time τc until the previous section, the diluted deuterondensity can be evaluated by the comparison between the peak values of measured andsimulated neutron emission rate.
The relation between two neutron emission rate peak values, measured peak value
Snp,exp and simulated peak value Snp,sim , can be expressed as(Nuga et al. 2019):
Snp,exp
Snp,sim
F≡

=

nD
F,
ne

τc 1 − exp(−t0 /[τncl τc /(τncl + τc )]),τncl + τc
1 − exp(−t0 /τncl )(6.1)

where t0 is the beam injection duration t0 ∼ 40 ms. Here we assume that the plasma
parameters and the fast ion birth rate are constant during beam injection duration.
Figure 12 shows the relation between Snp,exp and FSnp,sim for the Rax = 3.53 m
configuration. It is noted that, in this figure, data in long τncl region (τncl > 0.4 s fortangential beams and τncl > 0.1 s for perpendicular beams) are omitted. From eq. (6.1),
the slope of dashed line in fig. 12 indicates the ratio of the deuteron density to theelectron density nD /ne . The value is obtained by the weighted least mean square fitting
in y = αx, where α is the fitting parameter. For this reason, here we assume the deuterondensity ratio to the electron density is approximately nD /ne = 0.42.
It is noted that this fitting is performed by using only tangential NBs. This is because
the full energy fraction of perpendicular NBs is unclear. From fig. 12, plots of NB#5

12

H. Nuga, et. al.
0.2

nC /ne0.15

SN139593
SN139594
SN139595

0.1
0.05
0
-1

-0.5
0
0.5
ρ : normalized minor radius

1

Figure 13. The carbon density ratio to the electron density nC /ne is shown for three discharges.
The horizontal axis indicates the normalized minor radius ρ. The minus value of ρ means the
position inside the magnetic axis.

has a good agreement to the fitting line. This indicate that the assumption of the fullenergy fraction, 0.78, used in the simulation is valid. On the contrary, results of NB#4 areseparated from the fitting line. One of the reasons of this deviation is that the assumptionof the full energy fraction is invalid. If the deviation can be explained only by the fullenergy fraction, the fraction becomes 1.2. Therefore, it seems that there is the otherfactor separating the results from the fitting. In this paper, NB#4 has a low injectionenergy, whose value is out of typical range of injection energy. Because of this unusual
operation of NB#4, the Sn peak value estimation of NB#4 is not reliable.
The deuterium ion ratio and the helium ion ratio in this series of experiment areapproximately nD /(nH + nD ) ∼ 1 and (nD + nH )/(nD + nH + nHe ) ∼ 0.9. Here, we assume
that there are only three ion species, deuteron, helium ion, and carbon ion in plasmas.
This assumption indicates ne = nD + 2nHe + 6nC . We can estimate the typical ion ratiosfrom the deuteron ratio estimation, two measured ratios, and this assumption. These
ratios are nD /ne = 0.42, nHe /ne = 0.047, and nC /ne = 0.081. The eﬀective charge of theplasma, which consist of these ion ratios, is approximately Zeﬀ = 3.5. From references(Zhou et al. 2010; Huang et al. 2015), the typical value of the eﬀective charge in LHD
plasmas measured by EUV spectrometer is approximately Zeﬀ = 2 − 4. Therefore, thisestimation is within the range of the realistic value.
Additionally, the carbon density profile, nC , is measured by the charge exchange
spectroscopy of CVI using neutral beam(Chen et al. 2019; IDA et al. 2019) in LHD.
Unlike nD /(nH + nD ) and (nD + nH )/(nD + nH + nHe ), the carbon density ratio tothe electron density changes widely shot-by-shot as shown in figure 13. In most of thedischarges, the value of nC /ne has changed around ≲ 0.1 and it does not exceed 0.2.
Therefore, it can be considered that the value nD /ne = 0.42 is reasonable.
Here we can recalculate the neutron emission rate by using estimated τc and ion ratios.
Figure 14 shows the time evolutions of the measured and simulated neutron emission rate.
In this simulation, the dilution eﬀect of deuteron density and the fast ion loss eﬀect areincluded. As compared to fig. 4, it is found that the consistent result for the measured
neutron emission rate can be obtained by this simulation.

7. Conclusion
We have investigated the NB fast ion confinement in LHD deuterium plasmas in fivemagnetic configurations. The eﬀective confinement time of NB fast ion has been estimatedfrom the decay time of the neutron emission rate after short pulse NBs are turned oﬀ. It
can be considered that the decay of the neutron emission rate can be expressed by the
combination between the reduction of the fusion cross-section due to the fast ion slowing

neutron emission rate [e14/s]

Studies of the Fast Ion Confinement in LHD
NB#5
Eb ~80keV

1.6
1.2
0.8

simulation

measurement

2

13

NB#1
Eb ~180keV

NB#2
Eb~150keV

SN148422

NB#3
Eb ~155keV
NB#4

0.4 Eb ~35keV
0
2.5

3

3.5

4
time[sec]

4.5

5

5.5

Figure 14. Time evolutions of the measured and simulated neutron emission rate are shown.
The bulk deuteron density is assumed to be nD /ne = 0.42 and the fast ion confinement times
are assumed to be τc = 0.6s for tangential beams and τc = 0.06s for perpendicular beams.

down and the reduction of the fast ion density due to the fast ion loss. Therefore, thecomparison between the simulated neutron decay time τncl , which ignores the fast ion
loss, and the measured neutron decay time τn clarifies the fast ion confinement time.
As shown in figure 5 and figure 9, the relation between τn and τncl has two trends.
In short τncl region, τn obeys the curve, f (τncl ) = τc τncl /(τncl + τc ), where τc is constant.
In other words, this region is the high electron density region. On the other hand, τnseparates from the guide curve as τncl increases. This indicates that the dominant fast
ion loss mechanism is diﬀerent in each region. As shown in section 4 and section 5, inlong τncl region, the charge exchange loss can not be negligible because the low electron
density plasmas include many neutral particles. To reduce the charge exchange loss inthe estimation of τc , we have focused on the short τncl region. Here the short τncl region
indicate τncl < 0.4 sec for tangential beams and τncl < 0.1 sec for perpendicular beams,
respectively. In this region, we can estimate the eﬀective confinement time τceﬀ by theweighted least mean square fitting in f (τncl ). In addition, similar analyses have been
performed for the other magnetic configurations. In figures 8 and 11, the eﬀective NB
fast ion diﬀusion coeﬃcient Deﬀ is plotted against the position of the magnetic axis. It
is clear that the tangential NBs have good confinement rather than the perpendicular
NBs. Additionally, inward shifted plasmas have a good confinement of NB fast ions inboth injection directions.
The dependence of the neo-classical diﬀusion coeﬃcient on the position of magnetic
axis in LHD have been discussed numerically in the reference(Murakami et al. 2002).
This reference showed that the neo-classical diﬀusion coeﬃcient in 1/ν regime decreases
as the plasma shifting inward. Because the neo-classical diﬀusion is dominated by trapped
particles in usual, it can be considered that the dependence of Deﬀ on Rax for perpendicular beam ions has same trend to the reference result. For tangential beam ions, it
can be roughly explained that the orbit deviation from the magnetic surface becomeslarger as the plasma shifting outward because the toroidal curvature component of themagnetic field, B1,0 (poloidal mode number = 1, toroidal mode number = 0), increases
as the plasma shifting outward. Obviously, however, the sensitivity of Deﬀ to Rax is
diﬀerent from the reference results. Since the detailed discussion of this issue requires
orbit simulations, this is the future work.
We have focused on the decay time of the neutron emission rate to estimate τceﬀ insteadof the absolute value of the neutron emission rate Sn . The absolute value of Sn linearly
depends on the bulk deuteron density, which is diﬃcult to measure in the plasma coreregion. In actual plasmas, the deuteron density is diluted by the presence of the impurity
ions. This dilution eﬀect can be estimated by the obtained τceﬀ and by the diﬀerence of14

H. Nuga, et. al.

the peak value of the measured and simulated neutron emission rate. From figure 12, theratio of the deuteron density to the electron density is roughly estimated as nD /ne = 0.42.
This estimation has no significant contradictions with impurity measurements as shownin section 6. According to the obtained τc and nD /ne , we can simulate the consistentneutron emission rate to the measured neutron emission rate.
Although we have estimated quantitatively the fast ion confinement time τc in LHD
plasmas, the transport mechanism itself is still unclear. The clarification of the transportmechanism dominant in the short τncl region is the future work.

Acknowledgments
This work is supported partly by LHD project budgets (NIFS07KLPH004 and
NIFS14KNTT025). It is performed on “Plasma Simulator” (FUJITSU FX100) of NIFS
with the support and under the auspices of the NIFS Collaboration Research program(NIFS18KNSR007).

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