J. Plasma Fusion Res. SERIES, Vol. 9 (2010)

Anisotropic Fusion-product Generation and Emission Spectra
in Beam-injected DT Plasmas
Hideaki MATSUURA and Yasuyuki NAKAO
Department of Applied Quantum Physics and Nuclear Engineering,
Kyushu University, Motooka 744, Fukuoka 819-0395, Japan(Received: 25 October 2009 / Accepted: 26 January 2010)

The double differential emission spectra for fusion-produced alpha-particle and neutron as a function ofemission energy and angle relative to the direction of beam injection are examined on the basis of the one
dimensional (1D) Boltzmann-Fokker-Planck (BFP) model in beam-injected (non-Maxwellian) deuterium-tritium(DT) plasmas. It is shown that owing to the existence of energetic component in fuel-ion energy distributionfunctions due to mono-directional neutral-beam injection (NBI) and/or nuclear elastic scattering (NES), the angular
distributions of the emission spectra have highly anisotropic and broadened configuration compared with Gaussian
distribution. The shape of the spectra and the fraction of the energetic ( >3.5 MeV alpha-particle and >14.1 MeV
neutron) components to the total generation rate are quantitatively evaluated for various plasma conditions.
Keywords: neutral beam injection heating, knock-on tail formation, alpha-particle emission spectrum,
nuclear elastic scattering, Boltzmann-Fokker-Planck equation
behavior in the burning plasmas, it would be required toaccurately grasp the characteristics of fusion-producedalpha-particle source. In this paper, the 2-dimensional (2D)
fusion source, i.e. alpha-particle double differential
emission spectrum, as well as neutron double differential
spectrum as a function of energy and emitting direction(the angle between the direction of alpha-particle (neutron)
emission and that of beam injection) are quantitatively
evaluated for various plasma parameters.
We consider a DT plasma accompanied with injectionof a mono-energetic and mono-directional deuterium beam.
To facilitate the analysis, on the basis of the 1-D
Boltzmann-Fokker-Planck (BFP) model [10-11,13], thevelocity distribution functions of deuteron and triton are
first derived simultaneously considering the broadened
alpha-particle emission spectrum in the energy space.
Using the obtained deuteron and triton distributionfunctions, we examine the T(d,n)4He fusion reaction
between the mono-directional deuteron distribution andisotropic background triton distribution function. The
double differential emission spectra of alpha-particle andneutron due to T(d,n)4He fusion reaction are evaluated as afunction of birth energy and emission angle relative to thebeam-injection direction. It is shown that the modificationof the spectra is influenced by the tail formation in both
deuteron and triton distribution functions due to beam
injection and NES, and the fractions of energetic ( > 3.52
MeV ) alpha-particle and energetic ( > 15 MeV) neutron
significantly increase or decrease due to the modificationof the emission spectra depending on the emission
directions.

1. Introduction

Energetic ions in a burning plasma have importantroles in various stages of fusion-reactor operations. The
non-Maxwellian tail is created by neutral-beam-injection(NBI) and/or ion cyclotron range of frequencies (ICRF)
heating. The knock-on tail is also formed due to nuclear
elastic scattering (NES) of fast ions[1-4]. The non-thermal
fusion has been examined[5,6] for a number of decades
and suitable models to predict the non-thermal fusion
yields have been developed[7]. In conceptual designs of
next-generation fusion devices, however, use of deuterium
beam-injection energy much higher than the current
experiments, e.g. 1 MeV, is considered[8]. In such a case,the effects of NES on fractional beam-energy deposition toions[9,10] and T(d,n)4He reaction rate coefficient[11] canbe appreciable. The effect on the modification of thefusion-product emission spectrum can also be appreciable.
We have investigated the modification of alpha-particle
emission spectrum assuming isotropic beam injection[12](ion velocity distribution functions, α-particle and neutron
emission spectra have been treated in one-dimensional
velocity space assuming uniform plasmas). In the actual
case, however, external beam is injected in a specificdirection, so the emission spectra of α-particle and neutron
would have anisotropic distributions. In such a case,asymmetric α-particle generation with energy above 3.5
MeV may influence the plasma confinement andsustainment properties. The detailed estimation for thealpha-particle behaviors in burning plasma would be
important.
For the purpose of analyzing the alpha-particle

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

48©2010 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchH. Matsuura and Y. Nakao, Anisotropic Fusion-Product Generation and Emission Spectra in Beam-Injected DT Plasmas

2. Analysis Model+

2.1 Boltzmann-Fokker-Planck model
The BFP equation for ion species a (a= D, T and
α-particle) is written as
CNES⎛ ∂f ⎞⎛ ∂f ⎞∑j ⎜⎝ ∂ta ⎟⎠ + ∑i ⎜⎝ ∂ta ⎟⎠
i

+

SD =

(1)

C c ( P )τ c ( P )⎧γ⎩C c ( P )τ c ( P ) (v/v th )when v < v thwhen v ≥ v thσv DT =

τp

+ n D n T σv DT − S NBI .

(4)

2
4π
v ς D (v D )∫D
n Dn T× f D (v D )dv D ,
with

ςD =

(5)

2π
v T f T (v T )vD ∫⎡ v D +v T⎤× ⎢ ∫ dv r v r2σ DT (v r )⎥dv T .
⎣⎢ v D −v T⎦⎥

(6)

The T(d,n)4He fusion cross has been taken from the workof Bosch[16].
For triton the NBI injection term is not included inEq.(3), and the source and loss terms are described so thatthe fueling rate, transport loss and the loss due to T(d,n)4He
reaction are balancing [10-11,13].
For alpha-particle, the source and loss terms are
written as, (2)The high exponent γ chosen ensures rapid increment ofboth confinement times in higher energy range comparedwith the one in the thermal energy range, and thus
energetic particle itself and its energy are hardly evacuatedcompared with that of thermal particle. In this paper
throughout the calculations γ = 4 is assumed. As was
discussed in Ref.10, in usual plasma condition, the γ would
not be a significant parameter as far as we choose
sufficiently large values, e.g. γ ≥ 4 . Considering energy
loss mechanisms both due to thermal conduction andparticles transport loss from plasma, the global energy
confinement time is defined as 1 / τ E = 1 / τ C + 1 / τ P .
(see Ref.10-11,13).
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 each other[10-11];

S D (v) − L D (v) =nD

The S NBI is the NBI rate per unit volume and v DNBI is the
speed corresponding to injected beam energy E NBI . We
express the injection rate S NBI using the beam energy
andinjection
power
,
i.e.
E NBI
PNBI
S NBI = PNBI /(E NBIV ) . Here V represents the plasma
volume. Referring to the design parameter for ITER[8], we
assume V = 800 m 3 .
The T(d,n)4He reaction rate coefficient is written as

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]. The
summation is taken over all background species, i.e. j=
D, T, alpha-particle and electron. The collision term ishence non-linear, retaining collisions between ions of thesame species. The second term accounts for the NES of
species a by background ions[10-11,13]. 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 aretaken from the work of Perkins and Cullen[2].
The third term in the left-hand side of Eq.(1)represents the diffusion in velocity space due to thermal
conduction. To incorporate the unknown loss mechanismof energetic ion into the analysis, following Bittoni’s
treatment[15], we simulate the velocity dependence of theenergy loss due to thermal conduction τ c* (v ) and theparticle-loss time τ *p (v ) by using a dimensionlessparameter γ, i.e.

τ c*( p ) (v ) = ⎨(3)

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.

1 ∂ ⎛ v3fa ⎞⎟⎜v 2 ∂v ⎜⎝ 2τ c* (v ) ⎟⎠+ S a (v ) − L a (v ) = 0 ,S NBI
f (v )
δ (v − v DNBI ) − ς D f D − D*.
2
τ p (v)
4πv

S α (v) − L α (v) = S α (v ) −f α (v ),τ *p (v)(7)

where NBI and fueling rate in Eq.(3) are replaced by thealpha-particle generation rate due to T(d,n)4He reaction,

S α (v) =d 2N α(dE / dv )dΩ lab ,2∫
dEd Ω lab
4πv

(8)where N α (E ) represents the alpha-particle generation
rate, which is described in the next section, and Ωlab is aunit vector in the direction of emission of the particle in thelaboratory system.

SDδ (v − v Dfueling )
2
4πv

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H. Matsuura and Y. Nakao, Anisotropic Fusion-Product Generation and Emission Spectra in Beam-Injected DT Plasmas

2.2 alpha-particle and neutron emission spectra

times τ E = (1 / 2)τ p = 3 sec and beam-injection energy
E NBI = 1 MeV are assumed. The electron density has
different values according to the NBI heating power as
6.6 − 7.0 ×1019 m -3 . The dotted line in deuteron
distribution function denotes Maxwellian distribution at
20 keV temperature. The bold line represents the
distribution function when no NBI heating is made. We
can see that the non-Maxwellian tail due to NBI is formedin the energy range below 1 MeV in the distributionfunctions. The relative intensity of the tail is increased by
NBI with increasing beam-injection powers. The NES is a
non-Coulombic, large-angle scattering process, and a large
fraction of the fast-ion energy is transferred in a single

The double differential alpha-particle (neutron)
emission energy spectrum is written as

d 2 N α (n )rr dσ
(E , θlab ) = ∫ ∫ ∫ f D (v D ) f T (vT )dEdΩlabdΩr r× δ (E − E α (n ) )δ (Ωlab − Ωα (n ) )vr dv D dv D dΩ , (9)where Eα (n ) is the α-particle (neutron) energy in thelaboratory system[17];m n (α )1
E α (n ) = m α (n )Vc2 +(Q + E r )2
mα + mn+ V c cos θ c

2m n m α(Q + E r ) ,mn + mα(10)where m α (n ) is the α-particle (neutron) mass, Vc is thecentre-of-mass velocity of the colliding particles, θc isthe angle between the centre-of-mass velocity and the
α-particle (neutron) velocity in the centre-of-mass frame,
Q is the reaction Q -value, and Er represents therelative energy given by

Er =

r 2
1 m DmT r
v D − vT .
2 m D + mT

(11)

The θlab represents the angle between the direction ofemitted α-particle (neutron) and that of beam injection in
the laboratory system, and Ωα (n ) is a unit vector in the
direction of emission of the alpha-particle (neutron) in thelaboratory system, which is determined using the classicalrrkinematics as a function of v D , vT and θc . From thealpha-particle emission spectrum, the source term of BFPequation for alpha-particle, i.e. Eq.(8), is determined. Bymeans of the computational iterative method, the energy
spectrum and the deuteron, triton and alpha-particle
velocity distribution functions are consistently obtained.
Throughout the calculations the differential T(d,n)4He
cross section is assumed to be isotropic in thecentre-of-mass frame. As was described previously, in thispaper, although the energy distribution of the deuteron
distribution function is taken into account, the beam
-deuteron is assumed to have mono-directional distribution.
The background triton has been assumed to be isotropic
distribution as derived by solving Eq.(1).

Fig.1 Deuteron distribution functions when 20, 40 and 100
MW NBI heatings are made. The dotted line denotes
Maxwellian when no NBI heating is made.

3. Results and Discussion
In Fig.1 we first show the deuteron distributionfunctions as a function of deuteron energy when 10, 40 and
100 MW NBI heating are made. In the calculations,electron temperature Te = 20keV , the ion densitiesn D = n T = 3 × 1019 m -3 , energy and particle confinement

Fig.2 Deuteron distribution function when NBI
heating is made with 0.2, 1.0 and 2.0-MeV beaminjection energies with 40 MW power.

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H. Matsuura and Y. Nakao, Anisotropic Fusion-Product Generation and Emission Spectra in Beam-Injected DT Plasmas

event. Owing to the NES of energetic alpha-particle bylow-energy fuel ion, the energetic component, i.e.
knock-on tail, is created in the fuel-ion distributionfunction. The knock-on tail due to NES of alpha-particle isobserved in the energy range above 1 MeV in the
distribution functions. In Fig.2 the deuteron distributionfunction are shown for several beam-injection energies.
The NBI heating power is taken as 40MW, and otherFig.5 Comparison between double differential spectrum
for alpha-particle (solid lines) and the spectrum derived
in 1D calculation (dotted line). The thin line represents
the spectrum obtained by integrating the double
differential spectrum with respect to θlab . PNBI=40 MW,
ENBI=1.0 MeV, Te=20 keV and ne=6.7×1019 m-3 areassumed.
Fig.3 The double differential emission spectrum foralpha-particle as a function of alpha-particle energy
and emission angle relative to the beam-injection
direction in the laboratory system. PNBI=40MW,
ENBI=1.0 MeV, Te=20 keV, ne=6.7×1019 m-3 andτE=(1/2) τP=3.0 sec are assumed.

Fig.6 Comparison between double differential spectrum
for neutron (solid lines) and the spectrum derived in 1D
calculation (dotted line). The thin line represents the
spectrum obtained by integrating the double differential
spectrum with respect to θlab . PNBI=40 MW, ENBI=1.0
MeV, Te=20 keV and ne=6.7× 1019 m-3 are assumed.
Fig.4 The double differential emission spectrum forplasma parameters are the same as those in Fig.1. The
electron density has different values as 6.6 − 7.0 × 1019 m-3
depending on the beam-injection energy E NBI . It isshown that the fraction of the energetic deuteron increases
with increasing beam-injection energy.
In Fig.3 (Fig.4) the normalized double differential
spectrum for fusion-born alpha-particle (neutron) is
exhibited as a function of alpha-particle (neutron) energy

neutron as a function of neutron energy and emission
angle relative to the beam-injection direction in thelaboratory system. PNBI=40 MW, ENBI=1.0 MeV,
Te=20 keV, ne=6.7×1019 m-3 and τE=(1/2) τP=3.0 sec
are assumed.

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H. Matsuura and Y. Nakao, Anisotropic Fusion-Product Generation and Emission Spectra in Beam-Injected DT Plasmas

component significantly increases. On the contrary in the
opposite direction relative to injected beam direction, i.e.
θlab = π , the fraction of the energetic alpha-particle(neutron) significantly decreases. Since injected beamenergy is much higher than the mean energy of thebulk-ion component, the center-of-mass velocity has
almost the same value as the injected-beam velocity. The
alpha-particle (neutron) is produced almost isotropic
direction in the center-of-mass frame, and hence thealpha-particle (neutron) which is emitted toward beaminjection direction, i.e. θlab = 0 , has considerably higher
energy compared with the one which is emitted toward the
opposite direction, i.e. θlab = π , in the laboratory system.
In Fig.5 (Fig.6) the energy spectra of alpha-particle(neutron) in the θlab = 0 and π directions are shown aswell as the spectrum derived in the 1D calculation. The
calculation condition is the same as those in Fig.3 and 4.
We can see the spectrum has highly anisotropic
distributions depending on the emission angle. The thinline represents the spectrum obtained by integrating the
double differential spectrum with respect to θlab . It seems
that the shape of the integrated spectrum is almost close to
the one derived in the 1D calculation. To quantitatively
estimate the distortion of the spectrum, the following
parameter is introduced;

and emission angle relative to the beam-injection directionin the laboratory system. The NBI power is taken as 40MW,
and other calculation parameters are the same as those in
Fig.1. It is found that the alpha-particle and neutron
emission spectra are broadened toward both low and highenergy regions, and fraction of the alpha-particle (neutron)born with more than 3.5 MeV (14 MeV) energy increases
compared with the Gaussian distribution function. We canalso find that in the beam-injection direction, i.e. θlab = 0 ,the fraction of the energetic alpha-particle (neutron)⎛ d 2N j ⎞⎜⎟dE∫⎜ dEd Ω ⎟lab
E⎠
F>Ecut (θ lab ) = ∞cut ⎝ 2,⎛ d Nj ⎞∫0 ⎜⎜ dEdΩlab ⎟⎟dE⎝⎠∞

Fig.7 The alpha-particle emission spectra in the
beam-injection direction (θlab=0) for various electron
temperatures as a function of alpha-particle energy in thelaboratory system. PNBI= 40 MW, ENBI= 1.0 MeV are
assumed.

(12)which implies the fraction of the alpha-particle (neutron)
generation with the energy above Ecut to total generationin the specific ( θlab ) direction. For example, the cut-offenergy Ecut can be chosen as Ecut = 4 MeV(15 MeV) for alpha-particle (neutron). The fractions
when the modification is considered F>Ecut and when
Gauss distribution is assumed F Gauss are evaluated, and> E cut
the increment in the fraction due to the modification, i.e.
, is estimated. When electron temperature
F>Ecut F>Gauss
Ecut
Te = 20(10) keV , electron density n e = 6.7(6.2) × 1019 m -3 ,NBI power PNBI = 40 MW and energy E NBI = 1 MeV(see Fig.5), if we only look at the alpha-particles which isemitted in θlab = 0 direction, the fraction of thealpha-particle generation with energy above 4 MeV to total
θlab = 0 alpha-particle component reaches 25.1 (10.1)%,
which is roughly 3.3 (5.6) times larger than the value when
Gaussian distribution is assumed, i.e. 7.7 (1.8) %. If welook at the θlab = π direction, the fraction of thealpha-particle generation with energy above 4 MeV to total
θlab = π alpha-particle component is 0.64 (0.11)% ,which is roughly 1/12 (1/17) times larger than the value
when Gaussian distribution is assumed. For neutron in thesame plasma condition (see Fig.6), the fraction of theFig.8 The alpha-particle emission spectra in the
beam-injection direction (θlab=0) for several PNBI as afunction of alpha-particle energy in the laboratory
system. Te= 10 keV, ENBI= 1.0 MeV are assumed.

52

H. Matsuura and Y. Nakao, Anisotropic Fusion-Product Generation and Emission Spectra in Beam-Injected DT Plasmas

energyandparticleconfinement
times,
beam-injectionenergyE NBI
τ E = (1 / 2)τ p = 3 sec
and
power
are
assumed,the= 1 MeV
PNBI = 40 MW
alpha-particle loss increased several percent comparedwith the case when mono-energetic (3.5 MeV)
alpha-particle source is assumed[18]. In the estimation,isotropic alpha-particle generation was assumed. If we
could consider the anisotropic alpha-particle emission
spectrum derived in this paper, the influence on theconfinement would be further enhanced depending on the
direction of the beam-injection. Further detailed studies
for modification of the double differential emission spectra,
and as a next step issue, investigation of the effect of thespectrum modification on the alpha-particle behaviors inburning plasma would be required for various plasma
conditions.

neutron generation with energy above 15 MeV to total
θlab = 0 neutron component reaches 3.3 (2.3)%, which isroughly 13.0 (1100.0) times larger than the value when
Gaussian distribution is assumed for the neutron emission
spectrum, i.e. 0.26 (0.0021) %.
In Fig.7 the θlab = 0 component of thealpha-particle double differential spectrum is shown for
two cases of electron temperature, i.e. Te = 10 keV and
20 keV . The other parameters are same as those in Fig.5.
In low temperature range, the Gaussian distribution has
narrow half width, and hence broadness of the spectrum
becomes relatively small. In the same manner, thedependence of the spectrum on PNBI is also examinedand presented in Fig.8. We can see the energetic , i.e. > 4
MeV, component increases with increasing PNBI as waspreviously denoted in Fig.1.
In this paper the double differential emission
spectra for alpha-particle and neutron produced byT(d,n)4He reaction have been investigated. Throughout thecalculations, the following assumptions have been made:(a) deuteron has mono-directional distribution function, (b)background triton has isotropic distribution function and(c) the differential T(d,n)4He cross section is isotopic in thecenter-of-mass system. In actual plasmas accompaniedwith NBI heating, the deuteron distribution function has
2D configuration and triton distribution may be also
anisotropic distribution function. In such a case, if we
could reflect the exact 2D velocity distribution function in
the calculation, the F>E value might be somewhat
cut
smaller compared with the present calculations. In
ENBI < 1 MeV range, the anisotropy of the differential
T(d,n)4He cross section in the center-of-mass frame would
not be significant so much. An anisotropic suprathermal
component in the fuel-ion distribution would also becreated by ICRF heating. By improving the present modelso that we can include the 2D distribution function effect,
the double differential emission spectra for fusion products
in ICRF-heated plasma can also be estimated.
If the alpha-particle emission spectrum has ananisotropic distribution, the alpha-particle velocity
distribution function itself may also have an anisotropic
configuration. The transport processes, e.g. first-orbit loss,of alpha-particle in the fusion devices would be affectedby the particle energy. The fractional energy deposition
from alpha-particle to bulk ions and electrons via
collisions is also changed depending on the relative
velocity between alpha and bulk particles. The
modification of the alpha-particle emission spectrum, e.g.
increment in the number of energetic (≥3.5 MeV)
alpha-particle, may influence the alpha-heating
characteristics. We performed considerably roughestimation for the effect of alpha-particle spectrum
modification on the transit alpha-particle loss, andobtained the result that when electron temperature
Te = 20keV , the electron density n e = 4.2 × 1019 m -3 ,4. References
J. J. Devany and M. L. Stein, Nucl. Sci. Eng. 46, 323(1971).
[2] S. T. Perkins and D. E. Cullen, Nucl. Sci. Eng., 20, 77(1981).
[3] L. Ballabio, G. Gorini, J. Källne, Phys. Rev. E, 55, 3358
(1997).
[4] J. Källne, L. Ballabio, J. Frenje, S. Conroy, G. Ericsson,
M. Tardocchi, E. Traneus, et al., Phys. Rev. Let, 85, 1246
(2000).
[5] R. J.Hawryluk, Rev. Mod. Phys. 70, 537 (1998).
[6] JET Team (prepared by P.R. Thomas), Nucl. Fusion 39,
1619 (1999).
[7] R. V. Budny, M. G. Bell, A. C. Janos, M. G. Bell, A. C.
Janos, D. L. Jassby, L. C. Johnson, D. K. Mansfield, D. C.
McCune, H. Redi, J. F. Schivell, G. Taylor, T. B. Terpstra,
M. C. Zarnstorff, S. J. Zweben, Nucl. Fusion 35, 1497
(1995).
[8] R. Aymar, Fusion Eng. Des., 55, 107 (2001).
[9] H. Matsuura, Y. Nakao, K. Kudo, Nucl. Fusion, 39, 145(1999).
[10] H. Matsuura, Y. Nakao, Phys. Plasmas, 13, 062507 (2006).
[11] H. Matsuura, Y. Nakao, Phys. Plasmas, 14, 054504 (2007).
[12] H. Matsuura, Y. Nakao, Phys. Plasmas, 16, 042507 (2009).
[13] M. Nakamura, Y. Nakao, V. T. Voronchev, K. Hegi,
H. Matsuura, O. Mitarai, J. Phys. Soc. Jpn. 75, 024801(2006).
[14] M. N. Rosenbluth, W. M. MacDonald, D. L. Judd, Phys.
Rev. 107, 1 (1957).
[15] E. Bittoni, J. G. Cordey, M. Cox, Nucl. Fusion 29, 931(1980).
[16] H. -S. Bosch and G. Hale, Nucl. Fusion 32, 611 (1992).
[17] G. Lehner and F. Pohl, Z.Phys., 207, 83 (1967).
[18] T. Fukue, H. Matsuura, Y. Nakao, to be published in Proc.
APFA/APTTC2009, Aomori, Japan, 2009.
[1]53