J. Plasma Fusion Res. SERIES, Vol. 9 (2010)
Tomohiro FUKUE, Hideaki MATSUURA and Yasuyuki NAKAO

Department of Applied Quantum Physics and Nuclear Engineering, Kyushu University,Motooka, Fukuoka 190395, apan(Received: 20 November 2009 / Accepted: 21 February 2010)

The effect of fusion emission spectrum modification on prompt alphaparticle loss is evaluated for theITERlike deuteriumtritium plasma accompanied with neutralbeaminjection (NBI) heating. It is shown thatalphaparticle loss increases by several percent from the value when monoenergetic alpha source is assumed in themonotonic magnetic shear equilibrium.
Keywords: alpha particle, prompt loss, ITER, spectrum modification, neutral beam injection

evaluate the effect of spectrum modification on promptalphaparticle loss using derived alphaparticle emission
spectrum and energy dependence of alphaparticle loss
fraction.

In magnetic confinement fusion reactors,understanding of fusion alphaparticle behavior is one of
the most important issues from the viewpoint ofevaluating plasma heating characteristics and heat
deposition on the first wall, and hence numericalcalculations were made in order to clarify alphaparticle
behavior in various reactor systems [13]. In thesesimulations, monoenergetic alphaparticle source wasassumed [1]. However, when the fuelion has thermal
distribution, alphaparticle emission spectrum takes aform of Gaussian. In addition, it is known that fusion
emission spectrum spreads from Gaussian owing to thenonMawellian tail formation in fuelion velocity
distribution function by NBI or largeenergytransfer
scattering of alphaparticles [4]. Energetic particle has alarge Larmor radius and banana width. This makes
firstorbit loss and bananadrift diffusion larger [5]. Thisimplies that alpha particle confinement property depends
on its birth energy. Hence alphaparticle losses may beaffected by the alpha source spectrum.
In this paper, assuming ITERlike fusion reactor, theeffect of emission spectrum modification on alphaparticle confinement is evaluated by comparing thealphaparticle loss with the one when delta function (orGaussian) is assumed for emission spectrum. At first, we
calculate fuelion distribution function by solvingBoltmannFokkerPlanck (BFP) equation and derive thealphaparticle emission spectrum. Secondly, we eamine
energy dependence of alphaparticle loss fraction. Theguiding center code ORBIT [6] is used for the calculation.
In the current study, we look at only the alphaparticle
loss in its firstbounce time, i.e. prompt loss. Finally, we α The fuelion velocity distribution function inburning plasma accompanied with NBI heating isobtained by solving the following BFP equation for ion
species  ( = D, T, and alphaparticle), ∂ ()  ∂ () ∑  ∂   + ∑  ∂  3()1 ∂+ 2
 +  ( ) −  () = 0 ∂  2τ * () (1)where   ( ) is the velocity distribution function of ion
species . The first term in lefthand side of Eq (1) isFokkerPlanck term, which represents oulombinteraction between ion species  and background spacies (  = D, T, alpha, and electron). The second term
represents nuclear elastic scattering (NES) of species  bybackground ion , (,  ) = ( , α ), ( , α ), (α ,  ) and (α ,  ) .
The third term represents the diffusion in velocity space
due to thermal conduction. The fourth term is source term,and the fifth term is loss term. Figure 1 shows thedeuteron distribution function obtained by solving BFPequation (solid line). In this calculation, the ion densities =  = 2 × 1019 m −3 , electron and ion temperature =  = 20keV , energy confinement times τ  = 3.0sec ,NBIenergy = 1MeV,andNBIpower = 40MW are assumed. The dotted line in Fig.1 is552©2010 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchT. Fukue et al., Effect of Emission Spectrum Modification on Alpha-Particle Confinement in Beam-Injected DT Fusion Plasma

Maxwellian at the same temperature.
The alphaparticle emission spectrum is described as0.0014
19

where σ /   is the differential cross section ofT( , ) 4 He reaction. The 17.6MeV energy produced byT(d,n)α reaction is divided between alphaparticle andneutron. The fraction of the energy provided foralphaparticle depends on velocities of both deuteron andtriton before the reaction. α represents thealphaparticle energy in the laboratory system which iswritten as3

1(4πv/mD)D [m keV ]161410Te = 20 keV

1

ENBI = 1MeVPNBI = 40MW

with
2

103

100.0002
3000

4000

5000

 [T]

Plasma parameters
MSRS
5.3

5.3

  [MA]

11.6

15.4

 [cm]a [cm]

634

634

198

198

For the purpose of computing testparticle orbit,
ITERlike profiles of magnetic field, plasma pressure andcurrent density is determined from safety factor profile,flux surface and TF ripple profile which are taken fromprevious works [710]. The major plasma parameters usedin the calculation are given in Table I.
The equilibrium flux surfaces are geometricallydetermined using the parametric dependence of thecylindrical coordinates,  =  ( , θ ) and  =  ( ,θ )[7]. Here,  and  represent the spatial variables of thecylindrical coordinate system [ ,  , φ ] ( φ is thetoroidal angle ),  is the flux surface radius in the equatorial
plane containing the magnetic axis and θ is the poloidalangle. The parametric dependences of  and  aredescribed as1010.0004

 α 

1810PNBI = 40 MW0.0006

Table I

1010 010ENBI = 1 MeV

Fig.2 Alphaparticle emission spectrum as a function ofalphaparticle in the laboratory system. The electron
temperature  = 20keV , energy confinement timeτ  = 3.0sec , and beaminjection energy = 1MeV are assumed.

where α (  ) is the alphaparticle (or neutron) mass, is the centerofmass velocity of the colliding particles,θ  is the angle between the centerofmass velocity andthe alphaparticle velocity in the centerofmass frame, is the reaction value, and  represents the relative
energy. Figure 2 shows the calculated alphaemission
spectrum. The calculation conditions are the same as
those in Fig.1. The dotted line is Gaussian at the sameparameters. Here, these two spectra in Fig.2 arenormalized so that ∫ ( α /  )  = 1 . The alphaemission spectrum is broadened toward both low and highenergy regions. The fraction of emission rate above
3.52MeV energy is larger than that below 3.52MeV. Thisis because the kinetic energy carried by deuteron andtriton before the T(d,n)α reaction is transferred to thekinetic energy of fusionproduced alphaparticle (andneutron).

100.0008

alpha Energy [keV](3)120.0010 τE = 3.0 sec

0.00002000


2α 1α = α 2 +( +  ) +  cos θ ( +  ) ,2
α + α + 

10Gaussian

Ti = Te = 20 keV

1(2)3 σ= ∫∫∫   (|  |)  (|  |)δ (  −  )     ,3nd = nt = 2×10 m

0.0012(dNα/dE) [cm s keV ]α()


410Deutron Energy [keV]
Fig.1 Deuteron distribution function for electron temperature = 20keV , energy confinement time τ  = 3.0sec ,and beaminjection energy  = 1MeV .
553

( ,θ ) =  0 +   , ( , θ ) = κ  sin θ 0 =  +  ( ) +  cos θ ,  = − Λ ( )sin 2 θ ,Λ ( ) = δ  ( /  )2 ,(4)(5)(6)(7)(8)

T. Fukue et al., Effect of Emission Spectrum Modification on Alpha-Particle Confinement in Beam-Injected DT Fusion Plasma

where δ  is triangularity, κ is ellipticity and  ( )=  0 (1 − ( / ) 2 ) is Shafranovshift. The triangularityand ellipticity are taken as δ  = 0.54 and κ = 1.94
[8].
The analytic form for TF ripple is written as [2]4δ ( ,  ) = δ 0 exp[((  − min (  )) +   ) / ω ] . (9)22 0.5q

Here  is the ellipticity of ripple contours, ω is thescale length of the ripples, δ 0 is the minimum value ofripple field, and min is the radius at which value ofripple field equal δ 0 . The ripple data field is determinedby referring to the consulting previous work [9], = 0.09 , ω = 0.384 m , δ 0 = 5.0 × 10−6 , min
2= 5.22 − 0.0443 . The number of toroidal field coils istaken as 18. Figure 3 shows the equilibrium (doted lines)
and maximum TF ripple (solid lines) contours.

3210.0
Fig.4

300

0.1%

Z [cm]

 () =

0.01%

100

100
200r/a

0.6

0.8

1.0

Safety factor profile used in the calculation.

(the number of lost testalphaparticle)
× 100 [%] .
(total number of testalphaparticle)(10) (ψ ) =  (ψ ) = 20.0(0.9(1 −ψ ) + 0.1) .

300

Fig.3

0.4

Background ion (D and T) density  =  = 2 × 1019 m −3
has been assumed to be radially constant. The
temperature profiles of ion and electron was assumed tobe04000.2alphaparticle loss  (  ) as

1%

200

Monotonic Shear
Reversed Shear

5500

600700R [cm]

800(11)

The radial profile of alphaparticle generation rate wasgiven by temperature and density profiles.
Figure 5 shows the  (  ) as a function ofalphaparticle energy for two cases (a) monotonic shear,(b) reversed shear. It is shown that  (  ) depends
strongly on alphaparticle energy. lphaparticle emission
spectrum is shown in the same energy scale as acomparison. The solid lines in Figs.5 (a) and (b) are fitting
curves of  (  ) obtained by means of the leastsquare
method. In Fig.5 (a), exponent is 1.46 and in Fig.5 (b),exponent is 1.95. The alphaparticle loss fraction is larger
in RS. In the RS mode, safetyfactor is larger than MS
mode, and this implies that poloidal magnetic field issmaller than the toroidal magnetic field. This makes
bananawidth of trapped alphaparticle larger. s the result,the confinement region will be smaller and alphaparticle
loss will be larger.

900

TF ripple contours (solid lines) and flux surfaces(dotted lines).

The safety factor profile is an important factor that
affects ripple loss. In this study, we assume two cases ofoperation. First case; safety factor has monotonic radial
profile, i.e., monotonic shear (MS). Second case; q has
high central value and nonmonotonic radial profile, i.e.,reversed shear (RS). Figure 4 shows the safety factor
profiles used in this study. The data was taken fromRef.[10].
For the purpose of calculating alphaparticle loss, weuse the guidingcenter code RBIT. s a first step, wecalculate only the prompt alphaparticle loss. The
calculation follows 15000 alphaparticle orbit for 20
toroidal transit time, i.e. ≈ 1.6 × 10−4 msec. Thealphaparticle which reaches the last flux surface isregarded as lost one. In this paper we define the      α To evaluate the effect of spectrum modification on thealphaparticle loss, alphaparticle loss fraction ξ which

554

T. Fukue et al., Effect of Emission Spectrum Modification on Alpha-Particle Confinement in Beam-Injected DT Fusion Plasma36(a) MS(b) RSemission spectrumemission spectrum
4F(E) [%]

F(E) [%]

2

3 1.46

FMS(E) = 1.27×10 E

10013 1.95

FRS(E) = 1.49×10 E

2234
alpha Energy [MeV]

005

12345

alpha Energy [MeV]Fig.5 rompt loss fraction of alphaparticle  (  ) vs alpha energy for two cases: (a) monotonic shear, (b)reversed shear. The solid lines are fitting curves of  (  ) obtained by means of the leastsquare
method.

When reversed shear is assumed, the ξ when thecalculated spectrum is used reaches 1.87%, which isroughly 5.8% larger than the value when deltafunction isassumed for the alphaparticle emission spectrum, 1.77%.

takes into account the spreading of alphaparticle emission
spectrum is defined. The ξ is written as follows.

ξ=∫ α (  )  (  )
∫ α (  )

.

(12)

The effect of alphaparticle emission spectrum
modification on prompt alphaparticle loss has been
evaluated. It is shown that prompt alphaparticle loss
increases several percent from the value whenmonoenergetic alpha source is assumed. For thiscalculation, we used Eq.(11) to express the ion
temperature profile. The alphaparticle loss depends onradial profile of alphaparticle source which determinedby the ion temperature profile. lphaparticle generation
rate and temperature profile is changed by intermittent
transport or transient MHD phenomena. If ion
temperature profile was more flat, the ξ parameter
would further increases. In this study, we have consideredalphaparticle loss only in the first bounce time. In thenext step, the calculation for full slowingdown time isnecessary for more detailed analysis. In such a calculation,
since pitch angle scattering frequency is in inverse
proportion to alpha energy, the effect of alphaparticle
emission spectrum modification on alphaparticle loss
may be weakened to some extent. Throughout thecalculations isotropic beam injection has been assumed(ion velocity distribution functions, alphaparticle
emission spectrum have been treated in onedimensional
velocity space). In the actual case, however, externalbeam is injected in a specific direction, so the emission
spectrum of alphaparticle would have anisotropicdistribution. In such a case, the fraction of energetic

We evaluate the ξ for the following threeconditions. First case; the calculated emission spectrum isused. Second case; Gaussian distribution is assumed foremission spectrum. Third case; deltafunction is assumed
for emission spectrum. In Table II, we show the ξ forthree conditions of calculation. The ξ values for
Gaussian and calculated spectra are larger than that for
deltafunction spectrum. This is due to the fact that  (  )shape is not liner but exponentially increasing withalphaparticle energy. In addition, when the calculated
spectrum is used, fraction of alphaparticle which is bornwith the energy above 3.52 MeV is larger than that with theenergy lower than 3.52 MeV.
When monotonic shear is assumed, the ξ when thecalculated spectrum is used reaches 0.81%, which isroughly 3.5% larger than the value when deltafunction isassumed for the alphaparticle emission spectrum, 0.78%.
Table IIspectrum
calculated
Gauss distributiondelta function

The ξ parametersMS

0.81 %
0.79 %
0.78 %

RS

1.87 %
1.83 %
1.77 %555

T. Fukue et al., Effect of Emission Spectrum Modification on Alpha-Particle Confinement in Beam-Injected DT Fusion Plasmaalphaparticle which is emitted toward beam injected
direction increases. The alphaparticle loss fraction
depends on its emission angle relative to the direction oftoroidal magnetic field. Hence the alphaparticle loss
property would be affected by beam injected direction.
Further detailed studies for the effect of emission
spectrum modification on alpha particle loss would berequired.

The author is grateful to Dr. R. B. White forpermitting the use of ORBIT code.

K. Tobita, ., , 1585 (1995).
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[1]
[2]
[3]
[4]



556