Proceedings of ITC/ISHW2007

Modification of α-particle emission spectrum and its effect onplasma heating characteristics in non-Maxwellian
deuterium-tritium plasmas
Hideaki Matsuura and Yasuyuki Nakao
Department of Applied Quantum Physics and Nuclear Engineering,
Kyushu University, Motooka 744, Fukuoka 819-0395, JapanThe α-particle emission spectrum in a tail-created (non-Maxwellian) deuterium-tritium (DT) burning plasma is
examined by solving the Boltzmann-Fokker-Planck (BFP) equations for deuteron, triton and α-particle
simultaneously. It is shown that owing to the existence of energetic component in fuel-ion energy distributionfunctions due to neutral-beam injection (NBI) and/or nuclear elastic scattering (NES), the generation rate of thehigh-energy (>3.52MeV) α-particle increases significantly compared with the case of Gaussian distribution. The
influence of the broadened energy spectrum on the α-heating characteristics is discussed.
Keywords: neutral beam injection heating, knock-on tail formation, α-particle emission spectrum,
nuclear elastic scattering, Boltzmann-Fokker-Planck equation

1. Inroduction
Energetic ions in burning plasma have important roles
in various stages of fusion-reactor operations. Theenergetic deuterons produced by neutral-beam-injection(NBI) heating and/or ion-cyclotron resonance frequency(ICRF) heating[1,2] create a non-Maxwellian tail indeuteron and triton velocity distribution functions. The
nuclear elastic scattering[3,4] (NES) of injected-beam
and/or energetic α-particle by thermal ion also causes aknock-on tail formation in fuel-ion distributionfunctions[5-8]. In magnetically-confined deuterium-tritium(DT) plasmas, the resulting modification of the neutron
emission spectrum was computed, and its application to theplasma diagnostics was proposed[9]. By observing thedeviation of the neutron emission spectrum from Gaussian
distribution, the knock-on tail formation in fuel-ion
velocity distribution functions was experimentally
ascertained[10,11]. A similar modification of the emission
spectrum would also be observed for fusion-produced
α-particle. This modification may influence the plasma
confinement condition via α-heating power, the fractionalenergy deposition to ions and turbulent transport process ofenergetic α-particle, and/or α-particle diagnostics. It is
hence important to grasp accurately the modification of
α-particle spectrum in reactor plasmas.
In this paper, we consider a DT plasma accompaniedwith injection of a mono-energetic deuterium beam. On thebasis of the Boltzmann-Fokker-Planck (BFP) model[12,13],
the modification of the α-particle emission spectrum isevaluated simultaneously considering the distortion of
author’s e-mail:mat@nucl.kyushu-u.ac.jp

deuteron, triton and α-particle distribution functions. It isshown that the modification is significantly influenced by
the tail formations in both deuteron and triton distributionfunctions due to beam injection and NES. The effect of themodification on plasma heating characteristics (fractionalenergy deposition to ions) is discussed.

2. Analysis Model
2.1 Boltzmann-Fokker-Planck model
The BFP equation for ion species a (a= D, T andα-particle) is written asCNES⎛ ∂f ⎞⎛ ∂f ⎞∑j ⎜⎝ ∂ta ⎟⎠ + ∑i ⎜⎝ ∂ta ⎟⎠i+1 ∂ ⎛ v3fa ⎞⎜⎟v 2 ∂v ⎜⎝ 2τ c* (v ) ⎟⎠+ S a (v ) − L a (v ) = 0 ,(1)where f a (v ) is the velocity distribution function of thespecies a. The first term in the left-hand side of Eq.(1)represents the effect of the Coulomb collision[14]. Thesummation is taken over all background species, i.e. j=
D, T, α-particle and electron. The collision term is hence
non-linear, retaining collisions between ions of the samespecies. The second term accounts for the NES of species aby background ions. We consider NES between 1)
α-particle and D and 2) α and T, i.e., (a,i) = (D,α), (T,α),(α,D) and (α,T) . The NES cross-sections are taken fromthe work of Cullen and Perkins[4].
The third term in the left-hand side of Eq.(1)represents the diffusion in velocity space due to thermal

Proceedings of ITC/ISHW2007
conduction. To incorporate the unknown loss mechanismof energetic ions into the analysis, we simulate thevelocity-dependence of the energy-loss due to thermal
conduction and the particle-loss time (see Ref.5-7).
The source ( Sa (v ) ) and loss ( La (v ) ) terms takedifferent form for every ion species. For deuteron, thesource and loss terms are described so that the fueling,beam-injection, transport loss and the loss due to T(d,n)4He
reaction are balancing [5-7];

S D (v) − L D (v) =+SD
δ (v − v Dfueling )24πv

S NBIf (v )
δ (v − v DNBI ) − ς D f D − D*.
2τ p (v)
4πv

(2)

24πv
ς D (T ) (v D (T ) )∫D(T )n DnT

× f D (T ) (v D (T ) )dv D (T ) ,(3)with

ς D(T) =2πv D (T ) ∫v T(D) f T ( D ) (v T(D) )⎡ v D +v T⎤× ⎢ ∫ dv r v r2 σ DT (v r )⎥dv T(D) .
⎢⎣ v D −v T⎥⎦(4)

The T(d,n)4He fusion cross has been taken from the workof Bosch[16].
For triton the NBI injection term has not been
included in Eq.(2), and the source and loss terms aredescribed so that the fueling rate, transport loss and the loss
due to T(d,n)4He reaction are balancing [5-7].
For α-particle, the source and loss terms are written as

S α (v) − L α (v) = S α (v ) −f α (v ),τ *p(v)S α (v) =(5)where NBI and fueling rate in Eq.(2) are replaced by the(dE / dv ) dN α,
4πv 2dE(6)where N α (E ) represents the α-particle generation rate,
which is described in the next section.
2.2 neutron and α-particle emission spectrums
The α-particle (neutron) emission energy spectrums
are written as

dN α (n )Here v Dfueling indicates the speed of the fueled deuteron,
which is much smaller than the thermal speed, i.e. nearly
equal to zero. The fueling rate S D is determined so thatthe deuteron density is kept constant, i.e. S D = n D / τ p+ n D n T σv DT − S NBI . The S NBI is the NBI rate per unit
volume and v DNBI is the speed corresponding to injectedbeam energy E NBI . We express the injection rate S NBI
using the beam energy E NBI and injection power PNBI ,i.e. S NBI = PNBI /(E NBI V ) . Here V represents theplasma volume. Referring to the design parameter for
ITER[15], we assume V = 800m 3 . The T(d,n)4He reaction
rate coefficient is written asσv DT =α-particle generation rate due to T(d,n)4He reaction,dE×rr(E ) = ∫ ∫ ∫ f D (v D ) f T (v T )r rdσδ (E − E α (n ) )v r dv D dv D dΩ ,dΩ(7)where Eα (n ) is the α-particle (neutron) energy in thelaboratory system;m α (n )1(Q + K )E α (n ) = m n (α )V 2 +2mα + mn+ V cosϕ2m n m α(Q + K ) ,mn + mα(8)where m α (n ) is the α-particle (neutron) mass, V is thecentre-of-mass velocity of the colliding particles, ϕ is theangle between the centre-of-mass velocity and theα-particle (neutron) velocity in the centre-of-mass frameand K represents the relative energy given by

K =r 2
1 mnmα rv D − vT .
2 mn + mα(9)Using the α-particle emission spectrum, the source term of
BFP equation for α-particle, i.e. Eq.(6), is determined. Bymeans of the computational iterative method, both theenergy spectrum and the deuteron, triton and α-particle
velocity distribution functions are consistently obtained.
The differential cross sections of the T(d,n)4He reaction aretaken from the work of Drosg [17].

3. Results and Discussion
In Figure 1 we first show the (a) deuteron and (b)triton distribution functions as a function of deuteron or
triton energy when 10, 40 and 100 MW NBI heating are
made. In the calculations, the ion and electron densitiesn e = 2n D = 2n T = 4 × 1019 m -3 , electron temperature
Te = 20keV , energy and particle confinement timesandbeam-injectionenergyτ E = (1 / 2)τ p = 3 sec
E NBI = 1MeV are assumed. The dotted lines in bothProceedings of ITC/ISHW2007
1810PNBI=100MW
4010-3-1(4πv/mD) f D [m keV ](a)16101510Maxwellian

1410w/o NBI

131012Te = 20keV

11ENBI = 1.0MeV

1010010123101010Deuteron Energy [keV]

410181017-110-3(4πv/mT) f T [m keV ]1010PNBI=100MW
4010(b)

1615Maxwellian

1410-2w/o NBI

131012Te = 20keV

11ENBI = 1.0MeV

1010010123101010Triton Energy [keV]

410(dNn/dE)normalized [m-3s-1keV-1]

1710

heating is applied, which is well agreed with the previous
calculation[9] and experiment[11]. The dotted line denotes
a Gaussian distribution corresponding to the case when noNBI heating is made. It is found that the neutron emission
spectrum is broadened toward both low and high energy
directions, and fraction of the neutron with both more andless than 14MeV energy increases with increasing NBI
powers. We next show the normalized α-particle emission
spectrum in Fig.3 as a function of the α-particle energy.
The calculation conditions are the same as the ones in
Fig.1 and 2. As was seen in Fig.2, the broadness of theα-particle emission spectrum also tends to be conspicuousfor large NBI heating powers. For example, when 40-MW
NBI heating is made, the fraction of the generation rate ofα-particle with 5-MeV birth energy is almost 50 times
larger than the case when no NBI injection is made and isalmost 104 times larger than the value for Gaussian
distribution.

Fig.1 (a) Deuteron and (b) triton distribution functions when1019 -3-3
P =100MW
10 ne=4×10 m
NBI-4 T =20keV40e1010-510-610-710-810w/o NBI-910Gaussian-1010101214161820neutron energy [MeV]

10, 40 and 100 MW NBI heatings are made. The dotted lines
denote Maxwellian when no NBI heating is made.

Fig.2 The neutron emission spectrum as a function ofneutron energy in the laboratory system.
-2(dNα/dE)normalized [m-3s-1keV-1]

deuteron and triton distribution functions denoteMaxwellian at 20keV temperature. The bold lines
represent the distribution functions when no NBI heatingis made. It is found that the non-Maxwellian tails due toNBI are formed less than 1-MeV energy range in thedeuteron distribution function. The relative intensity of thetail is increased by NBI with increasing beam-injection
powers. The knock-on tails due to NES of injected beamand α-particle are also observed in the triton distributionfunction and in the energy range above 1-MeV in thedeuteron distribution function. We also find that the bulk
temperature increases due to large NBI powers.
In Figure 2 the normalized neutron emission
spectrum in deuterium-tritium plasmas is exhibited as afunction of neutron energy in the laboratory system. The
calculation conditions are the same as the ones in Fig.1.
The bold line expresses the neutron spectrum when no NBI

1019 -3-3
P =100MW
10 ne=4×10 m
NBI-4
40
10 Te=20keV
10-510-610-710-810w/o NBI
Gaussian
-9
10-10
100
2
468α-particle energy [MeV]Fig.3 The α-particle emission spectrum as a functionof α-particle energy in the laboratory system.

Proceedings of ITC/ISHW2007
181017ENBI=0.2MeV
1.02.0-3-1(4πv/mD) f D [m keV ]1016101510Maxwellian

1410w/o NBI

1310PNBI = 40MW

1210Te = 20keV

1110010123101010Deuteron Energy [keV]410Fig.4 The deuteron distribution function when NBI
heating is made with 0.2, 1.0 and 2.0-MeV beaminjection energies.

(dNα/dE)normalized [m-3s-1keV-1]-21019 -3
E = 0.2MeV-3NBI10 ne=4×10 m0.4-4 T =20keV0.8e10
1.2-510
1.6-62.010-710w/o NBI-810Gaussian-9PNBI= 40MW
10-101002468by α-particle with energy above 4MeV to total α-heating
power almost reaches to 13.3 (15.2)%, which is roughly
1.5 times larger than the value when Gaussian distribution
is assumed for the spectrum, i.e. 8.7 (10.0) %.
The transport processes of α-particle in the fusion
devices, e.g. ripple and orbit losses[18,19], tend to be
influenced by α-particle’s energy. The increment in thefraction of the energetic (≥3.5MeV) α-particle may cause
the enhancement of the α-particle loss[18]. On the contrary,
the initial kinetic energy of fuel ions (before the T(d,n)4He
reaction occurs) must be transformed into a part of theemitted α-particle and neutron energies. Thus if the correct
α-particle spectrum is not adopted into a calculation, wemay underestimate the α-heating power to some extent.
In the fusion devices, the α-particle diagnostics using
γ-ray generating nuclear reaction, e.g. 9Be(α,nγ)12C
reaction, has been developed[20]. The influence of thebroadened emission spectrum on the diagnostics methodshould be examined. Further detailed investigation for thecorrelation between the α-particle spectrum and burning
plasma characteristics would be required.

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α-particle energy [MeV]Fig.5 The α-particle emission spectrum as a function of
α-particle energy in the laboratory system.

In Fig.4 we next show the deuteron distributionfunction for 0.2, 1.0 and 2.0 MeV beam-injection energies.
In this case, the NBI heating power is taken as 40MW, andother plasma parameters are the same as the ones in
Fig.1-3. It is shown that the fraction of the energetic
deuteron increases with increasing beam-injection energy.
In Fig.5 the normalized α-particle emission spectrum isshown. The calculation conditions are the same as the onesin Fig.4. As a result of the increment in the energetic
component of deuteron distribution function (as shown inFig.4), the fraction of the energetic (>4MeV) α-particle tototal generation rate also increases.
The α-particle emission spectrum in the presence ofNBI heating has been evaluated. When electron densitiesn e = 4 × 1019 m -3 , electron temperature Te = 20keV ,NBI power PNBI = 40(60) MW and energy E NBI= 1MeV are assumed, the fraction of the power carried

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