Indirect energy transfer channel between
fast ions via nuclear elastic scatteringobserved on the large helical device
Cite as: Phys. Plasmas 29, 092502 (2022); https://doi.org/10.1063/5.0097720
Submitted: 01 May 2022 • Accepted: 11 August 2022 • Published Online: 12 September 2022
H. Matsuura,

K. Kimura,

D. Umezaki, et al.

Phys. Plasmas 29, 092502 (2022); https://doi.org/10.1063/5.0097720© 2022 Author(s).

29, 092502

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Indirect energy transfer channel between fast ions
via nuclear elastic scattering observed on the large
helical device
Cite as: Phys. Plasmas 29, 092502 (2022); doi: 10.1063/5.0097720
Submitted: 1 May 2022 . Accepted: 11 August 2022 .
Published Online: 12 September 2022
H. Matsuura,1,a)K. Kimura,1
D. Umezaki,12,32,3M. Goto,N. Tamura,
M. Osakabe,2,3

K. Ogawa,2,3
T. Nishitani,4

M. Isobe,2,3
Y. Kawamoto,2and S. Sugiyama5

T. Oishi,2,3

AFFILIATIONS
1

Department of Applied Quantum Physics and Nuclear Engineering, Kyushu University, 744 Motooka, Fukuoka 819-0395, Japan

2

National Institute for Fusion Science, National Institutes of Natural Sciences, 322-6 Oroshi-cho, Toki 509-5292, Japan3

Department of Fusion Science, The Graduate University for Advanced Studies, Sokendai, 322-6 Oroshi-cho, Toki 509-5292, Japan
Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan

4

5

National Institutes for Quantum and Radiological Science and Technology, Rokkasho, Aomori 039–3212, Japana)

Author to whom correspondence should be addressed: matsuura@kyudai.jp

ABSTRACTAn energy transfer phenomenon between energetic ions, which cannot be explained only considering the Coulomb scattering process, wasobserved on a large helical device (LHD). This phenomenon often occurs in fusion reactivity enhancement and fast-ion slowing-down process that can be observed as a delay in the decay time of the D(d,n)3He neutron generation rate. The transferred energy required to inducesuch a reactivity enhancement or delay in the fast-ion slowing-down time (neutron decay time) was examined based on the
BoltzmannFokkerPlanck analysis in which a discrete energy transfer process, called nuclear elastic scattering (NES), is included. It was
shown that even though the cross section of the NES is smaller than that of the Coulomb scattering, enough knock-on population appears in
the energetic region in ion distribution function to induce the observable NES effects; thus, enough energy is transferred from beam ions tofast component of bulk ion distribution function indirectly and the transferred energy per unit time via NES is comparable to the Coulomb
scattering rate. This study analytically demonstrates that the observed phenomena on LHD can be explained smoothly by considering thealternative indirect energy transfer channel between energetic ions, which can be comparable with the one via Coulomb scattering.
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0097720

I. INTRODUCTION
In a thermonuclear fusion reactor, fusion born non-thermal ionsmust sustain the bulk plasma pressure through collisional power transfer, i.e., nuclear burning. Energetic ions play an essential role in determining important physical information for nuclear burning. The
Coulomb scattering process is the dominant energy transfer processthat takes place between charged particles in a thermonuclear fusion
reactor. However, nuclear elastic scattering (NES) contributes to theenergy transfer process and can bring certain modiﬁcations to theslowing-down behavior as the ion energy increases.1–3 The cross section of the NES is deﬁned by subtracting the Coulomb scattering contribution from the experimentally measured one.4 Therefore, theactual energy transfer mechanism can be accurately reproduced only
after the NES contribution is incorporated into the analysis in addition

Phys. Plasmas 29, 092502 (2022); doi: 10.1063/5.0097720
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to the Coulomb contribution. NES induces a modiﬁcation of the ion
velocity distribution functions. When the ion velocity distributionfunction is modiﬁed from Maxwellian, the emission spectrum of thefusion products changes its shape from that of a Gaussian distribution.5 A non-Gaussian energetic component in the T(d,n)a neutron
emission spectrum was observed in JET.6,7 This component is createdby T(d,n)a reactions between the energetic recoil (knock-on) components in the deuteron and triton distribution functions owing to theenergy transfer phenomena via NES. The NES effect on the fast-ion
slowing-down process has also been observed in the large helicaldevice (LHD).8 The enhancement of the D(d,n)3He reactivity due to
NES was also observed.9 The NES occurs in a high-temperatureplasma operation. The NES effect may be negligible in several stages inthe DT fusion reactor operation; however, in some other stages, e.g., in29, 092502-1

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D3He plasma, its effect can inﬂuence nuclear burning.2,3 Therefore, itis essential to understand the NES features in burning plasma and validate the numerical approaches to develop a thermonuclear fusion
reactor.
NES is a non-Coulomb scattering process in which a large fraction of the ion energy is transferred during a single scattering event. As
a result of this, the energetic ions lose their energy faster (transferredenergy per unit time increases) than when only the Coulomb scattering process2,3,10 is considered. In contrast, energetic ions are continuously produced via NES process of energetic ions owing to theaccumulation of discrete high-energy transfer processes from energetic
to thermal ions. Through this process, the number of ions in theenergy range where recoil ions appear is enhanced, and the slowingdown time is delayed.8 The fast protons do not much interact with fast
deuterons via NES and do not provide distortion of the fast deuteron
distribution function directly; rather, the fast protons interact withbulk deuterons via NES and create a knock-on population in the fast
deuteron distribution function. As a result, fast proton energy is transferred to fast component of the deuteron distribution function indirectly. Some recoil ions also lose their energy through the Coulomb
scattering process. Others may lose their energy through NES and create recoil ions again in a signiﬁcantly higher energy range. Thus, energetic ions can deliver their energy through NES to the lower energy
ions distinctively from Coulomb scattering. Therefore, it is essential tounderstand these processes to achieve stable and efﬁcient nuclear
burning in future fusion reactors. However, the quantitative experimental conﬁrmation has not been extensively studied. This study
attempted to understand these processes by observing the delay in theslowing-down time of energetic ions that appear as a delay in the neutron decay time of the neutron generation rate.
The LHD in the National Institute for Fusion Science is equippedwith negative-ion neutral beam injectors (NBI) with beam energies
ranging from 160 to 190 keV.11,12 According to the general elastic scattering theory, a proton of 180 keV can transfer a maximum energy of
160 keV to a stationary deuteron in a single elastic scattering interaction determined depending on the scattering angle in the center ofmass (CM) systems. The most probable energy transferred is approximately 80 keV, as the target ion is nearly stationary. This study focuseson the deuterons injected by the positive-NBI with an energy of5060 keV to observe the energy transfer process via NES induced byprotons ranging between 160 and 190 keV. If a meaningful energy
transfer due to a large-angle scattering occurs as previously predicted,
the slowing-down time of the 5060 keV deuterons, which appears asa delay in the neutron decay time, would increase. In the experiment,
the decay time of the neutron generation rate produced by the beam
deuterium is observed.
This study evaluates the transferred energy per unit time via NES
and quantitatively examines the phenomena observed in the LHD.
The Fokker–Planck (FP) equation was used to analyze the time evolution of the velocity distribution function through the Coulomb scattering process. The FP equation was introduced to treat small-angle
scattering processes, such as Coulomb scattering. The Boltzmann
equation was used to examine the time variation of the velocity distribution function. The BoltzmannFokkerPlanck (BFP) model10,13
was developed, where small-angle Coulomb and large-angle scattering(NES) processes were considered. Based on the BFP simulation, theamount of indirect transferred energy from energetic protons to

Phys. Plasmas 29, 092502 (2022); doi: 10.1063/5.0097720
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ARTICLE

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slowing-down deuterons through NES was evaluated. The transferredenergy evaluated is ensured by comparing the BFP simulations with
the DD reactivity enhancement and the neutron decay time observedin the LHD, and the transferred energy via NES is also compared withthe one through Coulomb scattering. The measured data wereexplained by considering the knock-on fast ion population that is produced by the indirect transferred energy via NES. A numerical prediction of the NES effect, which is expected to appear in future fusion
devices, is also demonstrated.
II. EXPERIMENTAL APPARATUS
Figure 1 summarizes the performances of NBIs, i.e., injected particle species, approximate port-through beam powers and energies, inthe LHD for speciﬁc shots (a) #172066 and #172068, (b) #164881 and#164874 used in this study. Here, the port-through power is deﬁned asa sum of deposition and shine-through powers. In the experiment,
deuterium plasma was heated using NBIs and electron cyclotron resonance heating (ECH). Three negative-hydrogen (H) beam (NBI#1, 2,
and 3) and two positive-deuterium (D) beam (NBI#4 and 5) injectors
were used for shots #177066 and #177068, and one negative-hydrogen(H) beam (NBI#1), two negative-deuteron (D) beam (NBI#2 and 3)
and two positive-deuterium (D) beam (NBI#4 and 5) injectors for

FIG. 1. Schematics of the neutral beam injector patterns in large helical device withapproximate beam energies and port-through powers for typical shots (a) #172066
and #172068, and (b) #164874 and #164881.

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ARTICLEshots #164874 and #164881. NBI#1, 2, and 3 are the injectors that are
directed tangentially toward the axis of the toroidal magnetic ﬁeld,
with the negative-ion sources providing fast hydrogens or deuteriumat energies of approximately 170190 keV with a port-through powerof approximately 24 MW (tangential injectors). NBI#4 and 5 areperpendicular to the magnetic axis, with positive-ion sources ofapproximately 5060 keV deuterium with a port-through power ofapproximately 48 MW. High-purity (enriched) H was prepared intangential injectors for this experiment. The deuterium atomic ratio
contained in the hydrogen beam as an impurity was suppressed to beless than 1 ppm. The neutron generation rates were estimated to be
the average value of the three 235U ﬁssion chambers (FCs). One of the
ﬁssion chambers was placed on the magnetic axis, and the remaining
two ﬁssion chambers were placed on different sides of the LHD vacuum vessel. A magnetic axis Rax ¼ 3.6 m and a toroidal magnetic ﬁeld
Bt ¼ 2.75 T (counterclockwise direction) were maintained throughout
the experiment.
III. ANALYSIS MODEL
In the BFP model, a uniform deuterium plasma accompanied bydeuterium and hydrogen beam injections is assumed. Three kinds ofion species were considered, i.e., ﬁeld deuteron with Maxwellian
distribution function, fast deuteron, beam proton, and electrons(see Table I). The fast deuterons are divided into energetic components owing to beam injection (beam deuteron) and recoil (knock-on)
deuteron created by NES. The neutron generation by the D(d,n)3He
reaction between the fast and ﬁeld deuterons is considered. The timedependent BFP equation for fast deuteron (proton), i.e., k ¼ fast
deuteron or beam proton, is given by!
 Coulomb 
X @fk NES 1 @dfk X @fk
v 3 fkþþ 2¼
v @v 2sk
dt
@t j
@t i
c ðvÞ
i
j
þ Sk ðt; vÞ  Lk ðt; vÞ;(1)where fk ðt; vÞ is the time-dependent isotropic velocity distributionfunction of the species k. The ﬁrst term on the right-hand side of Eq.
(1) represents the effect of the Coulomb collisions.14 The summation
is taken over all background species, i.e., j ¼ ﬁeld deuteron, fast deuteron, beam proton, and electron. Thus, the Coulomb collision term isnonlinear retaining collisions between ions of the same species. The
second term accounts for the NES of species k.10 This study considers
NES between (1) fast deuteron and ﬁeld deuteron, (2) fast deuteronand fast deuteron, (3) beam proton and ﬁeld deuteron, and (4) beam
proton and fast deuteron, as (k, i) ¼ (fast deuteron, ﬁeld deuteron),scitation.org/journal/php(fast deuteron, fast deuteron), (beam proton, ﬁeld deuteron), and(beam proton, fast deuteron). The second term accounts for the NES,


@fk
@t

NES
i

1
1
ð
ð
2p 0
0
¼ 2 v fk0 ðv Þ vi fi ðvi ÞPðv0 ! vjvi Þ
v
0



v0 ð
þvi

0
2

v0r rNES ðv0 r Þdv0 r dvi dv0

jv0 vi j
vþv
1
ð
ði
2p
fk ðvÞ vi fi ðvi Þ
vr2 rNES ðvr Þdvr dvi ;
v
0

(2)

jvvi jwhere v0 r ¼ jv 0  v i j and vr ¼ jv  v i j, k ¼ beam proton (or fast deuteron), and k0 ¼ beam proton (or sum of fast and ﬁeld deuterons). The
ﬁrst term in Eq. (2) represents the rate of scattering ions into the velocity interval dv due to NES from the overall velocity space. The second
term in Eq. (2) denotes the rate of scattering out of ions from thevelocity interval dv due to NES. Suppose the scattering event of ion
species k which has the velocity v 0 by ion species i which has the velocity vi . If the incident ion is scattered through an angle u, thencos u ¼

jvj2  jV C j2 þ jVj2:
2jV C jjVj

(3)Here, v and V represent the velocities of ion species k after the scattering in the laboratory and the center of mass (CM) systems, respectively, V C the velocity of center of mass and u the angle between V C
and V. The NES cross sections are taken from the work of Cullen andPerkins15 (see Fig. 2). For the deuterium beam, it was shown that if thetarget ion species are triton and 3He, the angular distribution of NES
is, to a certain extent, backward-peaked.14,15 In this case, the assumption of isotropic scattering may underestimate the NES effect on
plasma properties. However, for proton or deuteron targets, the isotropic scattering assumption would be a good approximation.15,16 This
study assumed isotropic scattering for NES in the center of mass

TABLE I. Charged-particle species considered in BoltzmannFokkerPlanck
simulation.

Species
ElectronField deuteron
Fast deuteron
Beam proton

Distribution functions
Maxwellian
MaxwellianfD ðvÞ ð fDbeam ðvÞ þ fDrecoil ðvÞÞfp ðvÞð fpbeam ðvÞÞPhys. Plasmas 29, 092502 (2022); doi: 10.1063/5.0097720
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Field particle
Energetic particle

FIG. 2. Cross sections of deuteron-deuteron and proton-deuteron NES as a function of the relative energy of reactants.

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ARTICLEsystems. In this case, the probability distribution function P (i.e., probability that the injected beam ion which has the speed v0 is scatteredinto the speed region v owing to the NES with background ion, which
has speed vi ) can be written as follows:
8>< jvj dv for jV  Vj  v  jV þ Vj;
CCPðv0 ! vjvi Þdv ¼ 2jV C jjVj>
:0otherwise:(4)
The third term on the right-hand side of Eq. (1) represents the diffusion in velocity space due to thermal conduction. In this study, thevelocity dependence of the particle and energy conﬁnement times wasincorporated into the analysis using the dimensionless parameter c following the Bittoni’s treatment17 to reﬂect the particle and energy losses
during particle motion in a three-dimensional magnetic conﬁnement
conﬁguration to a uniform plasma model. Bittoni’s treatment isgiven by8kkk>when v < vth;< CcðpÞ scðpÞ
 c
k
v
(5)
scðpÞ ðvÞ ¼
kkk
when v  vth:>: CcðpÞ scðpÞ vkth
kis the thermal velocity of ion species k, and the coefﬁcient
Here, vthk
Cc ðor Cpk Þ is determined so that the velocity-integrated energy (or particle) loss rate becomes ð3=2Þnk Ti =skc ðor nk =sp Þ for each ion species. The
asterisk in Eq. (5) implies that the variable is a function of the velocity.
Considering both energy loss mechanisms due to thermal conduction
and particles transport loss from the plasma, the conﬁnement time for
conduction loss time skc is determined using the global energy conﬁnement time sE to satisfy the following relation for each species:ðð1=2Þmk vk2
ð3=2Þnk Ti ð3=2Þnk Ti
¼
þ
vk:
(6)
fk ðvk Þd~
sE
skc
sk
p ðvÞ

The high exponent c chosen ensures a rapid increment of energy andparticle conﬁnement times for energetic particles compared with thethermal components. As discussed in Ref. 10, as in the usual fusion
plasma, the c would not be an inﬂuential parameter. This study
assumed c ¼ 6 throughout the simulation. The c dependencies on the
NES effects are discussed in Sec. IV. The particle conﬁnement time for
protons (tangential injection to the magnetic ﬁeld) is assumed as 0.5 s,referencing the work of Nuga et al.18 The conﬁnement time for deuterons (vertical injection to the magnetic ﬁeld) is determined to be consistent with the present experiment.
The source [SDðpÞ ðvÞ] and loss [LDðpÞ ðvÞ] terms for beam deuteron and proton are described as follows:SDðpÞ ðt; vÞ  LDðpÞ ðt; vÞ ¼SDðpÞfDðpÞ ðt; vÞdðv  vDbeamÞ:ðpÞsp ðvÞ
4pv2

(7)

beam
is the speed of the beam-injected deuteron (proton). The
Here, vDðpÞ
beam injection rate SD ðSp Þ is determined using beam injection energy
NBI
NBI
EDðpÞ
and power PDðpÞ
, and plasma volume V as
NBI
NBI
NBI
SDðpÞ ¼ PDðpÞ =EDðpÞ V. The plasma volume was taken as 30 m3
throughout the simulations. The reaction rate coefﬁcient of theD(d,n)3He fusion reaction between fast and ﬁeld deuterons is given by

Phys. Plasmas 29, 092502 (2022); doi: 10.1063/5.0097720
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hrviDðd;nÞ3 He ðtÞ ¼

scitation.org/journal/php

ð4p
dvD vD2 1ðt; vD ÞfDfield ðt; vD Þ
field
nfast
n
D
D

(8)

with

1ðt; vD Þ ¼

2

ð

6
2p
dvDfast vDfast fDfast ðt; vDfast Þ6
4
vD

fast
vD
ðþvD

3
7
dvr vr2 rDD ðvr Þ7
5:(9)fast v j
jvD
DHere, vr is the relative velocity of the beam and ﬁeld deuterons, i.e.,vr ¼ jv fast  v field j. The D(d,n)3He fusion cross sections are takenfrom the work of Bosch and Hale.19
The transferred power from charged particle species k to j is evaluated using both velocity distribution functions obtained as follows:
ð  
dEk
Coulomb
4pv2 fk ðt; vÞVdv;(10)
Pk!j ðtÞ ¼ 
dt j
where ðdEk =dtÞj is the energy loss rate of species k due to the
Coulomb collisions with species j. The energy loss rate is given by
2
3
1
 
ðv
ð
Zk2 Zj2 e4 ln K 6mk
dEk¼
fj ðt; xÞx2 dx  v fj ðt; xÞxdx7
4
5:
dt j
mj
e20 mk v
0v(11)
The transferred power from ion species k to i through NES is givenby NESð1@fk
NES
mk v 2
ðtÞ ¼ 4pv2 Vdv:
(12)
Pk!i
2
@t i
IV. RESULTS AND DISCUSSION
A. Observation of attenuation feature of the neutron
generation rate
In the experiment, the deuterium plasma was heated using ECH
and NBIs (a pair of deuterium NBI#5 and one (or two) high-purity
hydrogen NBI#1 (NBI#1 þ 3) beams). Figure 3 shows a typical waveform pattern of a deuterium plasma discharge, i.e., # 172068, including
the time evolutions of (a) the neutral beam heating powers for injectors NBI#1 (port-through and deposition powers), and #5 (portthrough power), (b) the electron temperature at the radius r ¼ 0 anda/2 (a is a minor radius of the LHD plasma), (c) the electron density at
the radius r ¼ 0 and a/2, and (d) the neutron generation rate measured
by the ﬁssion chamber. After 4 s at the beginning of plasma discharge,
deuterium beam injection (NBI#5) was terminated, whereas thehydrogen beams NBI#1 remained injected. After terminating NBI#5
at 4 s, the neutron generation rate by the D(d,n)3He reaction gradually
decreased. The authors focused on the decay time sn deﬁned as thetime after which the fast-thermal D(d,n)3He reaction rate is reduced
to 1/e. The time evolutions of the normalized neutron generation ratefor (i) w/o hydrogen beams, (ii) with NBI#1, and (iii) with NBI#þ3
are plotted in Fig. 4. The neutron generation rates were normalized by
the values when deuterium beam was terminated. The decay times
when NBI#1 or NBI#1 þ 3 (or w/o NBI#1) remain were estimated byleast squares ﬁtting and the errors were estimated by changing
the time limits from 50 to 100 ms as 44.3 6 0.8 and 46.8 6 0.47

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FIG. 4. Attenuation processes of the normalized neutron generation rates when (i)no hydrogen beams are injected (#172066), along with the shot when (ii) NBI#1
remained injected (#172068) and (iii) NBI#1 þ 3 remained injected (#164881) as afunction of the time after NBI#5 was terminated.

B. Evaluation of transferred power via NES
and comparison with experiment

FIG. 3. Time evolution of the deuterium plasma discharge (shot no. #172068). (a)neutral beam heating powers for injectors NBI#1, and 5; (b) electron temperaturesat r ¼ 0 and a/2; (c) electron density at r ¼ 0 and a/2; and (d) neutron generation
rate detected by the ﬁssion chamber (FC).

(35.2 6 0.13) ms, respectively. The injection power, time-averaged
electron temperature, density from 4.0–4.1 s, and decay time for shots(i) and (ii) are summarized in Table II along with (iii) #164881. In the
shot #164881, after 4.3 s at the beginning of plasma discharge, deuterium beam injection (NBI#5) was terminated, whereas the high-purity
hydrogen beams NBI#1 þ 3 remained injected. The ion temperature isapproximately 2 keV in all shots. Since the theoretical scaling of theslowing-down time is ss / Te3=2 =ne , no valid reason exists under the
background plasma conditions to explain the difference in the neutron
decay time.

Phys. Plasmas 29, 092502 (2022); doi: 10.1063/5.0097720
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At ﬁrst, we consider the situation when high-purity beam is
injected into deuterium plasma to examine the net transferred power
to create the knock-on tail in a deuteron distribution function. Figure
5 shows the time evolution of the (a) proton and (b) deuteron velocity
distribution functions after a high-purity hydrogen beam is injected
into the deuterium plasma. Here, t ¼ 0 represents the time when ahigh-purity hydrogen beam begins to be injected. The hydrogen beamenergy is taken as 180 keV and power as 2.0 MW (deposition power),corresponding to case (i) shown in Figs. 3 and 4. The proton distribution function forms gradually after the high-purity hydrogen beam
injection. The energetic protons continuously knock the bulk deuterons to the energetic region; thus, the knock-on tail gradually forms inthe deuteron distribution function. Figures 6(a) and 6(b) show thepower balance of the fast (knock-on) component in the deuteron distribution function as a function of elapsed time after the high-purity
hydrogen beam injection. Here, PpNES! fast d is the power transferredfrom the proton to fast (knock-on) deuterons through NES; PpCoulomb! fast d ,Coulomb
Coulomb
Pfast
d ! bulk d , and Pfast d ! bulk e are the transferred power from proton
to fast deuterons, fast deuterons to bulk deuteron, and fast deuterons
to bulk electrons through Coulomb scattering, respectively. The power
lost from the plasma is plotted with a blue dotted line. The plasma
condition is the same as those in Fig. 5. The parameter n in Fig. 6(b) is
the ratio of the sum of the heating power densities (solid lines), i.e.,
Coulomb
PpNES! fast d þ Pp ! fast d , to the sum of lost power densities (dotted lines),
Coulomb
Coulombi.e., Pfast
d ! bulk d þ Pfast d ! bulk e þ loss. The transferred power fromprotons to fast deuterons due to NES PpNES! fast d gradually increases andalmost reaches an equilibrium value at 2 s after the high-purity hydrogen beam injection. The power is estimated at approximately 1.8 kW.
It is known that most of the proton beam power is transferred to bulk

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TABLE II. Plasma and beam conditions for the large helical device experiments.

H-beam pattern(i) w/o NBI(ii) NBI#1(iii) NBI#1 þ 3

H-beam deposition power (MW)Te(0) (keV)
Te(a/2) (keV)
ne (m3)
sn (ms)0
7.9
4.0
0.79  1019
35.2 6 0.13

2.0
6.5
4.2
0.83  1019
44.3 6 0.8

3.6
7.1
4.2
0.93  1019
46.8 6 0.47

FIG. 5. Time evolutions of (a) proton and (b) deuteron distribution functions
after high-purity hydrogen beam was injected with 180 keV energy and 2 MW
power.

Phys. Plasmas 29, 092502 (2022); doi: 10.1063/5.0097720
Published under an exclusive license by AIP Publishingions and electrons via Coulomb collisions, i.e., the transferred power
to ions (or electrons) is approximately 0.16 (or 0.56) MW in the present simulations; the fraction of transferred power from protons tobulk deuterons via NES to via Coulomb scattering is less than 2% at
most. However, it should be noted that the cross section of theD(d,n)3He reaction of the recoil deuteron with a 100 keV energy range
is more than four orders larger compared with the typical bulk deuterons. In general thermonuclear fusion plasma, it should be recognizedthat the small number of fast deuterons sustains the fusion power(neutron) production.
In the experiments shown in Figs. 3 and 4, the hydrogen beamwith 50-keV energy and 4.5-MW port-through power, i.e., NBI#5, wasinjected superimposing on the high-purity hydrogen beam. Almost
50% of hydrogen port-through power penetrated the plasma in theexperiment. In general numerical simulations, the fraction of thetransparent to port-through (injection) power tends to be larger forthe vertical beam injection than for the tangential beam injection. In
the present simulation, it is assumed that the deposition power of thevertical beam injection (#NBI5) is almost 50%60% of the portthrough power, i.e., 2 MW. Figure 7 shows the time evolution of (a)proton and (b) deuteron distribution functions after the deuterium
beam was turned off, along with (c) deuteron distribution functions
when NES was neglected. In the simulation, the conﬁnement time for
deuterons is assumed as 0.04 s. The discussion for the conﬁnement
time is made later (in Fig. 8). The recoil ions by NES are produced
mainly along the proton beamline, which is injected tangentially to thetoroidal magnetic axis; thus, the electron temperature at r ¼ 0 (centerof the plasma) is essential for the recoil ion production (knock-on tailformation). A plasma condition shown in Fig. 4(ii) was selected.
During 300 ms after terminating the deuterium beam, the proton distribution function is kept in almost the same shape. In contrast, thebeam component in the deuteron distribution function rapidly decays,
leaving the residual knock-on tail component. The proton distributionfunction does not change its shape signiﬁcantly over time because
the hydrogen beam is continuously injected with 180-keV energy and
2-MW power after terminating the deuterium beam injection. In thesimulation, the transferred powers due to NES between fast deuteronand ﬁeld deuteron, fast deuteron and fast deuteron, beam proton andfast deuteron were insigniﬁcant in current discussion, because the fast
deuteron population is much smaller compared with fast protons.
From the proton and deuteron distribution functions shown in
Fig. 7, the transferred energy per unit time from protons to fast deuterons (which are about to slow down) can be evaluated by Coulomb
scattering and NES processes at each time step. Figure 8 shows theresults as a function of the elapsed time after terminating the29, 092502-6

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FIG. 6. Time evolution of (a) heating and lost power balanced for knock-on deuteron component and (b) ratio of received and lost powers after high-purity hydrogen beam was injected with 180 keV energy and 2 MW power.

deuterium beam. The green line represents the time evolution of thetransferred energy per unit time from protons to slowing-down fast
(beam plus knock-on) deuterons through Coulomb scattering. The
red line represents the transferred energy through NES (the dottedline represents the total value of the green and red lines). The energy
transferred from protons to the fast (beam plus knock-on) component
in the deuteron distribution function through Coulomb scattering
decreased rapidly. This is because the number of fast deuterons produced by external beam injection (NBI#5) reduces immediately after
deuterium beam termination [Fig. 7(b)]. This reduction is because of
the slowing-down and loss process of deuterons from the plasma. In
contrast, the energy transferred from the protons to fast deuterons
through NES is nearly constant for 100 ms. This is because proton andbulk deuteron distribution functions (Maxwell component) do not

Phys. Plasmas 29, 092502 (2022); doi: 10.1063/5.0097720
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FIG. 7. Time evolutions of (a) proton and deuteron distribution functions when (b) NES is considered and (c) neglected, during 300 ms period after the deuterium beam was terminated.

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FIG. 8. Time evolution of the transferred power from protons to energetic deuteronsvia (1) Coulomb scattering and (2) NES, along with the total value (dotted line). The
plasma condition is the same as shown in Fig. 4(ii).

change their shapes throughout 100 ms, as discussed previously. Theenergy is transferred through NES from energetic protons to the bulk
deuterons. However, the energy is apparently transferred from energetic protons to fast deuterons since the bulk deuterons that received
energy are scattered to the energetic region. This energy transfer channel is called “an indirect energy transfer channel.” The right-side axis
in Fig. 8 represents the ratio of the energy transferred through NES
per unit of time to the total energy transferred. The fraction gradually
increased and reached 50% at approximately 60 ms after terminating
the deuterium beam. Although the NES cross section of protons with180 keV hydrogen beam energy is signiﬁcantly smaller than that of the
Coulomb one, the NES processes occur between energetic protons andbulk deuterons (the Coulomb scattering process occurs between energetic protons and fast deuterons). The magnitude of the transferredenergies through Coulomb scattering and NES processes could
become comparable because the bulk component of the deuteron distribution function is several orders of magnitude larger than the fast
deuteron component. Thus, we could observe the phenomena inducedby NES in the LHD.
Additional BFP simulations were made by changing the highpurity hydrogen beam injection powers and deuteron conﬁnement
times, and for each plasma condition, the decay times of the neutron
generation rates were evaluated. The results are shown in Fig. 9, where
the neutron decay times estimated through the BFP simulation areplotted as a function of the hydrogen beam power for several deuteronconﬁnement times. The solid lines represent the neutron decay time
when the NES effect is considered, whereas the dotted (black) lines
represent the decay time when the NES effect is neglected. The dashed(green) lines indicate the decay time when the hydrogen beam was notinjected. The background plasma conditions were chosen by roughly
averaging the experimental condition shown in Table II, i.e., Ti¼ 2 keV, Te ¼ 7 keV, ne ¼ 0.8  1019 m3, ENBI ¼ 180 keV, and sp¼ 0.5 s. The ion temperature and electron density are almost radially

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FIG. 9. Neutron decay times when (1) high-purity hydrogen beam is injected, i.e.,
NES considered (solid line) and neglected (black dotted line), and (2) high-purity
hydrogen beam is not injected (green dashed line) as a function of the high-purity
hydrogen beam injection powers. The observed experimental values are plotted as
red squares.

ﬂat, so we used radial average values in the simulation. Since the proton beam was injected tangentially to the magnetic axis and most
knock-on tail would be generated around the central region, we chosethe average electron temperature around the central region. The redsquares represent the decay time observed in the LHD experiments
shown in Table II. During the neutron decay process shown in Fig. 4,
the plasma temperature (or density) is ﬂuctuated about 8% (or 10%)from the averaged value at the maximum. The sensitivities of theplasma parameters on the neutron decay time are estimated using the
BFP model for the experimental condition shown in Table II-ii(sDp ¼ 40 ms case in Fig. 9), and the results are summarized in Table
III. The ﬂuctuations of the neutron decay time caused by the plasma
temperature or density can be estimated less than 1.8%. Since the particle loss from the plasma is inﬂuential in the current plasma condition, the sensitivities may be smaller compared with the previous
prediction not considering the particle loss from the plasma.20,21 The
errors for the least squares ﬁtting shown in Table II are also estimatedless than 1.6%; in addition, counting errors in FC are considered lessthan 0.4%;12 thus, we have drawn error bars roughly within 63% in
the neutron decay times observed in the experiment in Fig. 9. The neutron decay time can be increased in a non-equilibrium state by both
TABLE III. Sensitivity of the plasma parameters on neutron decay time sn (ms).

DTe or Dne

10%

0

þ10%

Ti sensitivity
Te sensitivity
ne sensitivity

43.1 (0.5%)
43.0 (0.7%)
43.9 (þ1.4%)

43.3
43.3
43.3

43.4 (þ0.2%)
43.5 (þ0.5%)
42.6 (1.6%)

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FIG. 10. Comparison of the neutron decay process between observed data (seeTable II and Fig. 4) and BFP simulations when deuteron conﬁnement time isassume as 39 ms.

Coulomb scattering and NES processes, even if the beam energy is inapproximately 180 keV, as discussed previously. The BFP simulation
model explains the experimental data within a realistic range if thedeuteron conﬁnement time is set to around 40 ms. In this case, theenergy transferred per unit time from protons to fast deuterons
through NES is approximately 1.8 kW, which is the same order withthe one through the Coulomb scattering, i.e., approximately 3–8 kW.
Consequently, a delay in the deuteron slowing-down time (delay inthe neutron decay time) would be induced and we can observe thephenomena in LHD. In Fig. 10, the neutron decay times evaluated by
the BFP simulations for the plasma conditions shown in Table II arecompared with the observed data. Here, the deuteron conﬁnement
time is set to 39 ms. The dependency of the c value deﬁned in Eq. (5)is examined. Figure 11 shows the decay times of the neutron generation rates for different c values. The solid lines represent the neutron
decay time when the NES effect is considered, whereas the dotted(black) lines represent the decay time when the NES effect is neglected.
The dashed (green) lines indicate the decay time when the hydrogen
beam was not injected. The decay time increases as the c value
increases. This is because the higher c value ensures a large conﬁnement time for the energetic region. The c dependency on the NES
effect (deviation between solid and dotted line) is quantiﬁed, i.e., thedeviation for c ¼ 4, 6, and 8 are 2.1%, 3.6%, and 4.9% when 3.6 MWhydrogen beam power, respectively.
The NES effect on the D(d,n)3He reactivity for several plasma
parameters is predicted for the future fusion device, e.g., JT60-SA.
Figure 12 shows the enhanced neutron generation rate due to NES for
several (a) electron temperatures, (b) electron densities, and (c) beam
injection energies. The default plasma parameters are chosen referencing the shot conditions reported in Ref. 9 (#164874), i.e., Ti ¼ 2 keV,
Te ¼ 12 keV, ne ¼ 1019 m3, ENBI ¼ 170 keV, and beam deposition
power PNBI ¼ 7 MW. In this experiment, approximately one order of

Phys. Plasmas 29, 092502 (2022); doi: 10.1063/5.0097720
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FIG. 11. Neutron decay times when (1) high-purity hydrogen beam is injected, i.e.,
NES considered (solid line) and neglected (black dotted line), and (2) high-purity
hydrogen beam is not injected (green dashed line) as a function of the high-purity
hydrogen injection powers for several c values.

neutron generation rate was observed as a result of the high-purity
hydrogen beam injection. In a brief calculation, the neutron generationin Maxwellian plasma with the above plasma parameter is approximately 1011 m3 s1. As shown in Fig. 12(a), the BFP simulation also
reproduces approximately one order of the enhancement in the neutron generations rate for the plasma condition of #164874 in Ref. 9,i.e., Te ¼ 12 keV, which is consistent with the experimental observation. For high electron temperature, the slowing-down of fast ions is
weakened; however, energetic components of proton and deuteron
distribution functions grow large. Thus, the enhancement of fusion
reactivity appears larger for the high electron temperature range.
Similarly, the slowing-down of energetic ions is weakened by decreasing electron (ion) density. However, the resulting magnitude of theknock-on tail is not signiﬁcantly inﬂuenced by the electron (ion) density because the number of recoil ions increases in proportion to theincreases in the ion density as shown in Fig. 12(b). It is worth notingthat the density inﬂuences the neutron decay time and the absolutevalue of the neutron generation rate; thus, an experiment with lowdensity plasma would be proﬁtable to observe the NES effect since thecompetitive slowing-down process due to Coulomb scattering
decreases in low-density plasma. One important parameter would be
the beam injection energy. The transferred energy in a single NES
event is increased with an increase in relative velocity. The NES effectbecomes essentially signiﬁcant with increasing relative velocity. As
shown in Fig. 12(c), when the beam injection energy increases to
500 keV, the magnitude of the enhancement in the neutron generation
rate reaches almost one order larger than that of 170 keV energy. In
the LHD experiment, the high-purity hydrogen beam (approximately
170–180 keV) was used to distinguish the recoil deuterons from beaminjected protons. This is because the recoil component is smaller thanthe beam component. The NES effect for the deuterium beam may be

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reactor. The c dependency on the neutron generation rate is quantiﬁed. The neutron generation rate when c ¼ 4 (or 8) for Ti ¼ 2 keV, Te¼ 6 keV, ne ¼ 1019  m3, ENBI ¼ 170 keV, and beam deposition
power PNBI ¼ 7 MW is 2.82 (or 3.22)  1011 m3 s1; the difference
from the neutron generation rate (3.07  1011 m3 s1) when c ¼ 6[yellow lines in Figs. 12(a)–12(c)] is evaluated as 8.1 (4.9)%. The
deviation of the neutron generation rate due to c value is insigniﬁcantcompared with the magnitude of the enhancement due to the knockon tail formation.
V. CONCLUDING REMARKSIn this study, the transferred power was estimated using NES
between energetic ions to adapt to the data observed in LHD, i.e., thedelay process of the neutron decay time induced by the delay in the
deuteron slowing-down process. By considering an indirect energy
transfer channel induced by recoil ions, which receive the energy fromfast ions via NES, the reactivity enhancement observed in the LHD
can be explained. The BFP simulations reproduced the experimental
data with adjustment parameters (fast deuteron conﬁnement time,and its velocity dependency, i.e., c parameter). It is shown that thetransferred energy due to NES per unit time could reach the sameorder as the one due to Coulomb scattering, even if the beam injectionenergy is 180 keV.
Isotropic plasma was assumed throughout the simulations, andthe transferred power due to NES was roughly evaluated. In the LHD
experiments, the directions of deuterium and hydrogen beams to thetoroidal magnetic ﬁeld were different, which might induce the difference in the scattering process compared with the isotropic model. The
average relative speed between the tangential and vertical injection
beams becomes larger than that of the isotropic one; thus, theCoulomb scattering frequency tends to decrease, and the NES frequency increases slightly. In the comparison between observed dataand simulations, errors caused by the radial proﬁles of plasma density
and temperature might be included. For detailed evaluation, moresophisticated simulations considering the anisotropy of the plasma
would be necessary. Currently, the analysis using the particle simulation code considering the NES phenomena is developing.22 The discussion including anisotropic effects will be made in the subsequent
analysis. The NES effect becomes more signiﬁcant as the beam injection energy increases. In such a case, the fraction of the transferredenergy via NES increases, and the energy range where recoil ionsappear is shifted to the high-energy side. Conﬁrmation of the scalingof the NES with injection energy will require further experiments withhigh energy NBI.
FIG. 12. Time evolution of enhanced D(d,n)3He neutron generation rates due toknock-on tail formation in deuteron distribution function after high-purity hydrogen
beam is injected for several (a) electron temperatures, (b) electron densities, and(c) hydrogen beam injection energies.

observed in the experiment with higher beam injection energy. The
NES effect essentially becomes important for beam-injected and further energetic fusion-born ions. Therefore, further conﬁrmation using
such large beam injection energies would be necessary to predict the
NES effects induced by fusion-born energetic ions in the future

Phys. Plasmas 29, 092502 (2022); doi: 10.1063/5.0097720
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ACKNOWLEDGMENTS
The authors would like to thank the cooperative program of No.
NIFS20KLPH047 and the LHD experimental group for theircontributions. The ﬁrst author is grateful to Emeritus Professor Y. Nakao
for his discussions and encouragement over the course of many years.
AUTHOR DECLARATIONSConflict of Interest
The authors have no conﬂicts to disclose.

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Author Contributions
Hideaki Matsuura: Conceptualization (lead); Formal analysis (lead);
Investigation (equal); Project administration (lead); Writing – originaldraft (lead). Masaki Osakabe: Conceptualization (equal); Data curation(equal); Funding acquisition (equal); Investigation (equal); Project
administration (equal). Takeo Nishitani: Conceptualization (equal);
Data curation (equal); Investigation (equal); Project administration(equal); Validation (equal). Shota Sugiyama: Conceptualization (equal);
Investigation (equal); Validation (equal). Kento Kimura: Investigation(equal); Validation (equal). Daisuke Umezaki: Investigation (equal);
Validation (equal). Kunihiro Ogawa: Conceptualization (supporting);
Data curation (equal); Funding acquisition (equal); Investigation(equal); Project administration (equal); Validation (equal); Writing –review & editing (equal). Mitsutaka Isobe: Conceptualization (equal);
Data curation (equal); Funding acquisition (equal); Investigation(equal); Project administration (equal); Validation (equal); Writing –review & editing (equal). Yasuko Kawamoto: Data curation (equal);
Investigation (equal); Validation (equal). Tetsutarou Oishi: Data curation (equal); Investigation (equal). Motoshi Goto: Data curation(equal); Investigation (equal). Naoki Tamura: Data curation (equal);
Investigation (equal).
DATA AVAILABILITY
The LHD data can be accessed from the LHD data repository at
https://www-lhd.nifs.ac.jp/pub/Repository_en.html.23
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