Proceedings of ITC/ISHW2007

Effect of nuclear elastic scattering on slowing down of ICRF
resonated ions and plasma heating characteristics
Atsushi Fujimaru, Hideaki Matsuura, Yasuyuki Nakao
Department of Applied Quantum Physics and Nuclear Engineering, Kyushu University,744 Motooka, Fukuoka 819-0395, Japan
In scattering collisions between light nuclei, effect of nuclear force becomes appreciable when the colliding
particle energy is higher than ~ several 100keV. A deuterium-tritium (DT) plasma accompanied with ion-cyclotron
range of frequencies (ICRF) heating at the second harmonic of deuteron is considered, and the nuclear elastic
scattering (NES) effect on the energy deposition of resonated deuteron is examined on the basis of thetwo-dimensional Boltzmann-Fokker-Planck (BFP) equation for a resonated deuteron. It is shown that the fraction ofresonated deuteron energy deposited to ions increases by about 10 % owing to the NES effect.
Keyword: nuclear elastic scattering, the fraction of resonated deuteron energy deposited to ions,DT plasma, ICRF heating

1. Inroduction
In scattering collisions between light ions, maininteraction is Coulomb scattering. When the colliding
particle energy is higher than ~ several 100keV, however,
the effect of nuclear force on scattering process of lightion becomes appreciable. The cross section obtained bysubtracting purely Coulomic component from themeasured scattering cross section is defined as nuclear
elastic scattering (NES) cross section [1,2]. It is necessary
to evaluate the NES effect on slowing down process ofhigh energy ions for comprehension of a characteristic ofnuclear burn plasma.
In recent ICRF heating experiments on Large
Helical Device (LHD), highly-energetic trapped ions( ≥ ~1MeV) were observed [3]. In the presence of neutralbeam injection (NBI) heating, it was found that thefraction of beam-ion energy deposited to ions increases
owing to the effect of NES [4,5]. The NES effect canappear also in plasma under ICRF heating because thehighly-energetic resonated ion loses its energy via
collisional process. It is important to grasp the NES effecton ICRF plasma heating.
In this paper, we consider a deuterium-tritium (DT)plasma accompanied with ICRF heating at the second
harmonic of deuteron, and examine the NES effect on theenergy deposition of resonated deuteron. Thetwo-dimensional
Boltzmann-Fokker-Planck(BFP)
equation is solved to obtain the velocity distributionfunction of resonated deuteron. Using the obtained
distribution function, the fraction of resonated deuteronenergy deposited to bulk ions is evaluated.

author’s e-mail:fujisin @nucl.kyushu-u.ac.jp

2. Analysis ModelIn this paper to facilitate the analysis triton- andelectron-distribution functions are assumed to be
Maxwellian at the same temperature. We solve thefollowing BFP equation [4-8] for deuteron:ColNES ∂f  ∂f ∑j  ∂tD  + ∑i  ∂tD ji ∂f + D  ∂t RF−f1 ∂  v3fD  − Dv 2 ∂v  2τ C* (v)  τ P* (v)loss ∂f −  D  + S (v) = 0, ∂t  R(1)where f D(v,μ) is the velocity distribution function ofdeuteron (μis the direction cosine between the velocityof deuteron and the external magnetic field). TheτC* (v)and τP* (v) stand for the typical energy-loss time dueto thermal conduction and particle-loss time due toparticle transport respectively [5]. It is assumed that they
are followed Bittoni’s treatment [9].
The first term in Eq. (1) represents the effect of
the Coulomb collisions with bulk charged particles, i.e., j= D, T, α-particle and electron [10,11].
The second term accounts for the NES with bulk
deuteron and triton, i.e., i= D, T [4-8]. In this paper, theNES cross sections are taken from Cullen and Perkins[2].
The third term in Eq. (1) represents the effect of RFdiffusion. The quasi-linear diffusion in velocity spaceowing to ICRF injection [12] can be written as

Proceedings of ITC/ISHW2007 ∂f D  ∂t 

RF=∂f 1 ∂  v⊥ D D .
v⊥ ∂v⊥ ∂v⊥ (2)Here v ⊥ represents vertical velocity component to theexternal magnetic field and, D represents diffusion
coefficient due to ICRF second harmonics [12-15], andcan be written as2k v k v  ECD = 2D J1  ⊥ ⊥  + − J 3  ⊥ ⊥  .
vth  ωcD  E+  ωcD S0δ (v − v fuel ),4πv 2(8)where, S0 can be written as

S 0 = nD nT < σv > DT +nD

τP,

(9)

(3)

and vfuel is the speed of the fueled particle, which is muchsmaller than the thermal velocity (nearly zero).

(4)

3. Results and Discussion
In Fig. 1(a) the velocity distribution function ofdeuteron, and (b) the Maxwellian at the same temperature

Here E-/ E+ can be shown as following form [11].

E−L≈− .
E+RS (v ) =From the equation of dispersion relation of cold plasmas,
the parameter of Bessel function can be written as
12  RL  2v 1− µ2 ,=c  R + L 
ωcDk ⊥ v⊥(a)(5)

where c represents the speed of light, and ωcD denotes
the cyclotron frequency of deuteron. The coefficient CD isdetermined from the ICRF absorbed power, PICRF, which
is described as∂f∂f PICRF = −2πmD ∫∫ Dvth2 v v 2 1 − µ 2 D − vµ D  dvdµ.
∂v
∂µ 

()

(6)
The forth and fifth term in Eq. (1) represent thediffusion in velocity space due to thermal conduction andparticle transport respectively.
The sixth term in Eq. (1) denotes the effect ofparticle loss due to T(d,n)4He reaction.
lossn f  2m  ∂f D  = T D  T 
2π  mD  ∂t  R×(b)

3/ 2∞
 m
1
dvT vT exp − T vT2 ∫v0
 mD v+ vT× ∫ dvr vr2σ DT (vr ).

(7)v −vTHere vT and vr represent the velocity of triton and therelative velocity between deuteron and triton respectively.
In addition, σ DT (v r ) denotes the cross section ofT(d,n)4He reaction, and is provided by Duane [16].
The seventh term in Eq. (1) represents particle
source. The particle losses due to fusion reaction andparticle transport are compensated by some appropriate
fueling method. The source term S(v) can be written as
following form.

Fig. 1(a) The velocity distribution function of deuteron,and (b) Maxwellian.

are shown. In this calculation the bulk temperature
Ti=Te=10keV, ion and electron densities ne =2nD=2nT=4×1019m-3, energy and particle confinement
time τE=(1/2)τP =3sec , and ICRF power absorbed bydeuteron 40MW are assumed. Here vth represents thethermal velocity of deuteron, and v⊥ (v//) denotes thevertical (parallel) velocity component to the external

Proceedings of ITC/ISHW2007
magnetic field. As a result of the ICRF heating, a
non-Maxwellian tail is formed in high energy region,
because of the vertical velocity component increased dueto the ion-cyclotron resonance. From the obtained
deuteron distribution function, the transferred energy
from high-energy resonated deuteron to bulk ions via
NES is estimated asNES1 ∂f PDNESmD v 2  D →i = − ∑ ∫2
 ∂t  ii = D ,T(10)dv × V p .

Where Vp represents plasma volume, and throughout thecalculations Vp =800m3 is assumed.
The integrand in Eq. (10) is shown in Fig. 2. In thiscase the bulk temperature Ti=Te=10keV, ion and electron

to bulk ions and electrons via Coulomb scattering, and (b)to bulk ions via NES, the loss powers from plasma due to(c) T(d,n)4He reaction, and (d) Transport and Conduction,(e) the ICRF absorbed power, (f) the transferred powerfrom alpha particle to deuteron via Coulomb scattering
are shown. In this case the bulk temperature Ti=Te=10keV,
ion and electron densities ne=2nD=2nT=4×1019m-3, energy
and particle confinement time τE=(1/2)τP =3sec , andICRF power absorbed by deuteron 40MW are assumed.
About 70% of resonated deuteron energy is deposited toplasma, and about 30% is lost from plasma. Besides, it isfound that the transferred power to bulk ions via NES issmaller than that via Coulomb scattering relatively. By
using the plasma heating powers due to Coulomb
scattering and NES, the fraction of plasma heating power
deposited to ions is calculated. The plasma heating power
due to Coulomb scattering and NES are written
respectively as PDC→ j ( i.e., j= D, T, e ) and PDNES( i.e., i=→i
D, T ), the fractional power deposition to ions can bewritten asCoulomb + NESFion∑P=∑PNES
D →i+ ∑ PDC→iNESD →i+i = D ,Ti = D ,Ti = D ,T

∑P

C
D→ j.

(11)

j = D ,T ,eIf NES is not included, the fraction is written as∑P=∑PC
D →iCoulomb
Fion

i = D ,TC
D→ j.

(12)

j = D ,T ,eFig. 2 The deposited energy to bulk ions via NES.

densities ne = 2nD=2nT=4×1019m-3, energy and particle
confinement time τE=(1/2)τP =3sec , and ICRF power
absorbed by deuteron 40MW are assumed. It is found that
deuterons which have the large vertical velocity
component lose their energy due to NES. In consequence,the deposited energy from high energy resonateddeuteron to bulk ions increases relatively compared withthe case that only Coulomb scattering is considered.
In Table. 1 the transferred powers from deuteron (a)Table. 1 The transferred, heat and loss power.
（τE=3.0sec、Te=Ti=10keV、ne=2nD(T)=4.0×1019m-3、B=5.0T、Vp=800m3）

(a)

(b)

D→→D D→→T

D→→e

D→→D

D→→T

5.7

21.0

0.43

0.08

2.9(c)

(d)

0.36

12.3

(e)

(f)

40.0

2.3

(e)+(f)(a)+(b)+(c)+(d)

error

42.3

42.9

1.4%

[ MW ]To express numerically the NES effect on the fractional
power deposition to ions, we introduce the following
enhancement parameters:
Coulomb + NES
 Fion− 1 × 100[％].
CoulombFionξ = (13)

By using this parameter, the NES effect on the fractional
power deposition to ions is evaluated. In the case shownin Table. 1, for example, the fractional power deposition
to ions is calculated from Eq. (11) and (12)Coulomb+NES
Coulombas Fion= 0.29, so, the= 0.30, and Fionenhancement parameter ξis estimated as ξ = 3.4[％].
In Fig. 3 the ξ value is plotted as a function ofplasma temperature. In this case the ion and electron
densities ne=2nD=2nT=4×1019m-3, energy and particle
confinement time τE=(1/2)τP =3sec, and ICRF power
absorbed by deuteron 40MW (solid line), and 60MW
(dotted line) are assumed. For T=15keV temperature, theξ value reaches 12% and 5% for ICRF power absorbedby deuteron 60, 40 MW, respectively. It is found that theenhancement parameter ξ increases with increasing
PICRF. This is because the high-energy component indeuteron distribution function becomes relatively large

Proceedings of ITC/ISHW2007
and the transferred energy from resonated deuteron tobulk ions via NES increases for high PICRF. It should be
noted that the ξ value decreases at the low- andhigh-temperature range. The reason would be that in
low-temperature plasmas the slowing down of energetic
ions due to Coulomb scattering (mainly) by electrons isintensified; thus the high-energy component in deuterium
velocity distribution function becomes relatively small,which causes a reduction in the transferred power fromresonated deuteron to ions via NES. On the other hand at
the high-temperature range, relative velocity between
energetic resonated deuteron and bulk ions becomes
small, and the contribution of NES is reduced comparedwith that of Coulomb scattering.

Fig. 3 The ξ value as a function of plasma
temperature

4. Conclusion
It is revealed that the fraction of ICRF heating
power deposited to bulk ions increases owing to the NES
effect. In the case of the bulk temperature Ti=Te=15keV,ion and electron densities ne=2nD=2nT=4×1019m-3, energy
and particle confinement time τE=(1/2)τP =3sec, andICRF power absorbed by deuteron 60MW (40MW), theenhancement parameter ξ reaches almost 12% (5%).

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