J. Plasma Fusion Res. SERIES, Vol.7 (2006) 98-101

Ion Channeling by Nuclear Elastic Scattering and Its Effecton Neutral Beam Injection Heating in Thermonuclear Plasmas
MATSUURA Hideaki and NAKAO Yasuyuki
Department of Applied Quantum Physics and Nuclear Engineering,
Kyushu University, Fukuoka 812-8581, Japan（Received: 4 October 2004 / Accepted: 22 November 2005）

Abstract
Distortion in the fuel-ion (deuteron and triton) velocity distribution functions due to nuclear elastic scattering (NES) by beam ion during neutral-beam-injection (NBI) plasma heating operation is examined bysolving the Boltzmann-Fokker-Planck (BFP) equations for deuteron, triton and beam ion, simultaneouslywith the Fokker-Planck equation for α-particle in the deuterium-tritium (DT) thermonuclear plasmas. Using
the obtained distribution functions, enhancement in the T(d,n) 4 He fusion reaction rate coeﬃcient between
background deuteron and triton are evaluated. It is shown that the enhancement becomes appreciable whenbeam energy is larger than 〜700 keV.
Keywords:
nuclear elastic scattering, NBI plasma heating, Boltzmann-Fokker-Planck equation,
T(d,n)4 He fusion reaction rate coeﬃcient
1．Introduction
It is well known that for suprathermal ions, theNuclear Elastic Scattering (NES) contributes to theslowing-down process. In conceptual designs of nextgeneration fusion devices, use of beam energy morethan 1 MeV is considered. In this case, the NES eﬀects
on slowing down of beam particles may not be ignorantcompared with those due to Coulomb collisions. The
Coulomb scattering process is characterized by manysmall-energy-transfer events. On the other hand, the
NES is a non-Coulombic, large-energy-transfer (LET)
scattering process. Devany & Stein [1] first pointed
out the necessity to take into account the nuclear-forces
contribution to ion-ion scattering in high-temperatureplasmas. In order to investigate the NES eﬀect, many
analysis formulations to describe the discrete nature of
LET scattering have been developed [2-11].
The global NES eﬀect on plasma burn-characteristics was first estimated by Nakao et al. [4] by using the energy loss rate [1]. On the basis of the multigroup analysis method [5], more accurate estimation
were made by Galambos et al. [6] and Nakao et al.
[7], accounting for the LET knocking-up of backgroundions from thermal to higher energy range. A knockon tail formation in deuteron (triton) distribution functions owing to the NES by α-particles in DT plasmas
and its eﬀect on the emitted-neutron spectrum were examined by Fisher [12] and Ballabio [13]. By explic-

itly separating the continuous slowing-down (FokkerPlanck collision) and the LET scattering (Boltzmann
collision) terms, we have developed the BoltzmannFokker-Planck (BFP) analysis model [8,9].
We derived the energy loss rate of fast ion due to
NES, including thermal motion eﬀect of backgroundions, and using the derived expression we roughly estimated NES eﬀect on fractional NBI power deposition
to ions in DT plasmas [10]. Since the NES is a discrete energy transfer process, however, the NES eﬀect
should be examined considering the shape of the velocity distribution function of both beam and backgroundions. Recently by solving the BFP equation for beam
ion on the assumption that background deuteron and triton are Maxwellian, we more accurately examined thefractional NBI power deposition to ions in DT plasmas[11]. In this paper we solve the BFP equation not only
for beam ion, but also for background deuteron and triton. Considering the distortion in background deuteronand triton distribution functions, we further investigate
the NES eﬀect on the T(d,n)4 He fusion reaction rate coeﬃcient between background deuteron and triton during NBI plasma heating operations.
2．Analysis model
To facilitate the analysis we only consider the nuC 2006 by The Japan Society of Plasma

Science and Nuclear Fusion Research
Corresponding author’s e-mail: mat@nucl.kyushu-u.ac.jp

98Ion Channeling by Nuclear Elastic Scattering and Its Eﬀect on Neutral Beam Injection Heating in Thermonuclear Plasmas

clear elastic collision between beam ion and background fuel ions (D and T). We solve the following
BFP equations for D, T and injected beam ion simultaneously with the Fokker-Planck equation for α :C  
NES



∂ fa
1 ∂ v3 fa
∂ fa
+
+ 2
∂t
∂t i
v ∂v 2τC∗ (v)i+ S a (v) − La (v) = 0,

S beam (v)− Lbeam (v) =where S NBI is NBI rate per unit volume and vNBI is
speed corresponding to injected beam energy E NBI . We
express the injection rate by using the beam energy
and plasma Q-value, i.e. S NBI = P f /(QE NBI ). HereP f represents fusion power produced by T(d,n)4 He reaction, i.e. P f = nD nT σvDT E f (E f = 17.6 MeV).
The confinement time of beam ion may be written asτ∗P (v) = C P τP Max[1, v/vth ]γ . The coeﬃcient C P isdetermined so that the velocity-integrated particle loss
rate becomesn/τP . In present calculations we assume
that the loss of energetic ions is smaller compared withthat of thermal ones, thus large γ value, e.g. γ = 6,
is chosen. In this assumption, the influence of the γvalues on the fusion reactivity is small. Consideringboth energy loss mechanisms due to thermal conduction and particle transport, the global energy confinement time is defined as 1/τ E = 1/τC + 1/τP . For background deuteron and triton, we assume that the particle loss is compensated by adequate fuelling method,i.e. S D(T ) (v) − LD(T ) (v) = 0. In this treatment we may
overestimate the power supplied by fueling (they are injected to keep the fuel-ion density constant). Since in 〜
10- keV temperature range, however, both powers supplied by fueling and lost by particle loss are smaller thanthose by NBI and α-heating, the influence of the fueling and particle loss terms on the fuel-ion distributionfunctions is negligible.
From the obtained deuteron and triton distribution functions, we can evaluate the T(d,n)4 He fusion reactivity asσvT (d,n)4 He = 8π2 dvD vD fD dvT vT fT(1)

where fa is the velocity distribution function of speciesa (a = D, T, beam − ion). The first term in Eq. (1) represents the eﬀect of the Coulomb collisions with background ions and electron. The second term accounts
for the nuclear elastic collision between beam ion andbackground ions:
NES


 ∞
2π ∞ 
∂ fa= 2
va fa (va )vi fi (vi )P(va → va |vi )
∂t i
va 0
0
 va +vi
v2
r σNES (vr )dvr dvi dva|va −vi |

−

 2π
va

i

 va +vi|va −vi |→ ∞fa (va )vi fi (vi )0

v2r σNES (vr )dvr dvi ,
→

→

(2)
→

where vr = | v a − v i | and vr = | v a − v i |, subscript i
represents background ion species, i.e. D, T for beamion and beam-ion for D and T. We have introduced theprobability distribution function P (i.e. probability thatthe injected beam ion which has the speed va is scattered into the speed region va owing to the NES withbackground ion, i.e. deuteron or triton, which has speedvi ). If we assume that the NES is isotropic in the CM
system, then
⎧
⎪
2va
⎪
⎪
, for vmin ≤ va ≤ vmax⎪
⎨ 2vmax − v2min,
P(va → va |vi ) = ⎪
⎪
⎪
⎪
⎩
0
, otherwise
(3)
where
vmax(min) =

mi

2
v2
a + vima + mi+ (−)

2 2
m2a v2
a + mi v i

ma + mi

S NBI
fbeam (v), (5)δ(v−vNBI )− ∗τP (v)
4πv2

 vD +vT|vD −vT |dvr v2r σDT (vr ) .

(6)

To estimate the degree of the reactivity enhancement
due to tail (non-Maxwellian component) formation by
NES, we compare the obtained fusion reaction rate coeﬃcient with the one when both deuteron and triton distributions are Maxwellian. For this purpose, we must
identify the temperature of bulk (Maxwellian) component in deuteron and triton distribution functions. We
determine the bulk temperature T bulk by comparing thebulk component of the obtained distribution functions
with Maxwellian by mean of the least squares. To quantitatively estimate the reactivity ehnacement, we introduce the decrement parameter;⎞⎛⎟⎟⎟⎜⎜⎜ σvT (d,n)4 He⎟⎟⎟⎠ × 100 [%],−1
η = ⎜⎜⎜⎝(7)
σvTMaxwell4(d,n) He.

(4)
The third term in Eq. (1) represents the diﬀusion invelocity space due to thermal conduction. The typical energy-loss time due to thermal conduction may be
written as τC∗ (v) = CC τC Max[1, v/vth ]γ Here the coeﬃcient CC is determined so that the velocity-integrated
energy loss rate becomes (3/2)nT/τC . By adjusting
the parameters γ, we can simulate various loss mechanisms due to thermal conduction. For beam ion, particle source (S beam (v)) and loss (Lbeam (v)) terms can be
written as

is the T(d,n)4 He fusion reaction
where, σvTMaxwell(d,n)4 He
99

Ion Channeling by Nuclear Elastic Scattering and Its Eﬀect on Neutral Beam Injection Heating in Thermonuclear Plasmas

rate coeﬃcient when both deuteron and triton distribution functions are Maxwellian of temperature T bulk . In
this paper the NES cross-sections are taken from thework of Perkins and Cullen [14], and the T(d,n)4 He fusion cross sections are taken from the work of Duane[15].

tons via NES by injected beam ions. The suprathermal
component in deuteron distribution function is somewhat larger than that in the triton distribution function.
This is because that the D-D NES cross-sections arelarger than those of D-T ones.
In Fig. 2 (a) the enhancement parameter η is presented as a function of beam-injection energies for several background densities. In this calculation, the electron temperature T e = 10 keV, confinement times τ E =(1/2)τP = 3.0 sec and Q = 1 are assumed. According to the tail formation in both deuteron and triton distribution functions, the T(d,n)4 He fusion reaction rate
coeﬃcient between background deuteron and triton increases from the values when the NES is neglected.
In Fig. 2 (b), the deuteron distribution functions whennD = nT = (1/2)ne = 0.5 × 1019 and 5 × 1019 m−3 are
exhibited. For large background densities, the bulk temperature is more easily increases, thus the degree of theincrement in the high-energy component in the velocity region where fusion cross-sections have a peak (〜100 keV deuterium energy) becomes relatively small.

3．Results and discussion
We consider the DT plasma in which mono-energetic
deuterium beam is injected. In Fig. 1 we first show
the steady-state (a) deuteron and (b) triton distributionfunctions when 1 MeV (2 MeV) mono-energetic beam
is injected. The solid lines indicate the present calculations and the dotted lines denote Maxwellian of temperature T bulk . In this calculation, the electron temperature T e = 10 keV, fuel-ion and electron densities
nD = nT = (1/2)ne = 1019 m−3 , confinement times
τE = (1/2)τP = 3.0 sec and Q = 1 are assumed. We canfind that suprathermal (non-Maxwellian) components
appear in both deuteron and triton distribution functions, owing to the recoils of thermal deuterons and tri-

JE = (1/2)JP = 3.0sec

D

ENBI =1MeV
Q =1

Te =10keV
Q =1

2

Te =10keV
nD = nT =1019m-3

ENBI [keV]nD = nT =ENBI =1MeV
Q =1ENBI = 1MeV

2

Q =1

Te =10keV

Te = 10keV

nD = nT =1019m-3

Fig. 2 (a) Enhancement parameters and (b) deuteron
distribution functions for several background densities.

Fig. 1 (a) Deuteron and (b) Triton distribution functions
when 1 MeV and 2 MeV mono-energetic deuterium beam is injected into DT plasmas.

100Ion Channeling by Nuclear Elastic Scattering and Its Eﬀect on Neutral Beam Injection Heating in Thermonuclear Plasmas

This is the reason why the enhancement parameter issmall for large background densities.
We next show the enhancement parameter η for several confinement times in Fig. 3 (a). The electron temperature T e = 10 keV, densities n D = nT = (1/2)ne =1019 m−3 and Q = 1 are assumed. In Fig. 3 (b), thedeuteron distribution functions when τ E = (1/2)τP =3.0 and 1.5 sec are presented. When the confinement
times are small, the bulk temperature does not so easily
increase, thus the degree of the increment in the highenergy component in the velocity region where fusioncross-sections have a peak (〜100 keV deuterium energy) becomes relatively large. The enhancement inT(d,n)4 He reaction rate coeﬃcient becomes large forsmall confinement times.
We have shown that the enhancement in theT(d,n)4 He fusion reaction rate coeﬃcient due to NES

JE =(1/2)JP = 1.5sec

becomes appreciable when beam energy is larger than〜700 keV, and the degree of the enhancement is influenced by the plasma parameters. In this paper we only
consider the T(d,n)4 He reaction rate coeﬃcient between
background deuteron and triton. When the tail is created in the background triton distribution function, thefusion reaction between injected beam and backgroundtriton would also be influenced. Throughout the calculations Q = 1 has been assumed. The NES eﬀect
becomes smaller for large Q values. If we consider the
NBI heating during plasma startup operation for small
Q values, however, more significant NES eﬀect on thefusion reaction rate coeﬃcient may appear. To clarify
the NES eﬀect in startup operations, time-dependent
startup simulation considering electron power-balance
would be necessary.
Acknowledgements
This work was supported by a Grant-in-Aid for Scientific Research of the Ministry of Education, Culture,
Sports, Science and Technology of Japan.

Q =1

D

[1]
[2]
[3]

Te =10keV
nD = nT =1019m-3

[4]
[5]

ENBI [keV]
[6]
[7]
[8]

JE =(1/2)JP =1.5sec
3[9]

ENBI =1MeV
Q =1

[10]
[11]

Te =10keV
nD = nT =1019m-3

[12]
[13]
[14]
Fig. 3 (a) Enhancement parameters and (b) deuteron
distribution functions for several conﬁnement
times.

[15]

101

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