23P148-P
Calculation Method for Alpha-Particle Transport in Degenerate Plasmaand Its Application to Degeneracy Diagnostics
縮退プラズマ中でのアルファ粒子輸送計算法の検討と縮退度診断法への応用
Ryota Mizoguchi1, Yasuyuki Nakao1, Hideaki Matsuura1, and Tomoyuki Johzaki2
溝口亮太1、中尾安幸1、松浦秀明1、城崎知至2
1

Department of Applied Quantum Physics and Nuclear Engineering, Kyushu University,744 Motooka, Fukuoka 819-0395, Japan
2
Institute of Laser Engineering, Osaka University, 2-6 Yamadaoka, Suita, Osaka 565-0871, Japan
九州大学大学院工学府エネルギー量子工学専攻 〒819-0395 福岡市西区元岡 744
大阪大学レーザーエネルギー学研究センター 〒565-0871 吹田市山田丘 2-6A kinetic model to properly calculate the transport of -particles in highly compressed plasma isformulated. The effect of electron degeneracy including Pauli blocking is fully incorporated. Aftershowing outline of the model together with some result of test calculation, the code is applied toexamine applicability of a method proposed to diagnose electron degeneracy in compressed DT fuel forfast ignition. The method uses the reaction 9Be (, n )12C governed by fusion-produced energetic
-particles; in the previous proposal, behavior of -particles was calculated by assuming infinite plasma.
We show that the proposed method would be applicable to compressed pellets with areal densities
larger than 0.3 g/cm2.

1. Introduction
In superdense plasmas as realized by laser
implosion, the electrons should be in Fermi
degenerate state. One of the consequences ofelectron degeneracy is reduction in the stopping
power of plasma for energetic charged particles.
Accordingly, their ranges are lengthened than in
the case of non-degenerate electron plasma. The
electron degeneracy might have influences on theignition condition in the fast ignition scheme
where the implosion is tailored so as to keep theisentrope parameter close to 1.
There have been transport codes for energetic
ions in degenerate electron plasmas.[1] However,in these previous codes, an important item was
not included. Since transition of the energy stateof plasma electrons due to -e scattering collisions,for example, is limited by Pauli’s exclusion
principle, scattering between an electron and otherparticles is restricted―Pauli blocking. Thus,
recently we have developed transport code forenergetic ions in which the effect of electron
degeneracy including Pauli blocking is fully
incorporated. In the present paper first we show
the outline of this model together with someresult of test calculation.
The extent of the degree of electron degeneracy
in compressed DT fuel for fast ignition is a matterof interest. So, next using the developed code we
examine applicability a method previously
proposed to diagnose electron degeneracy incompressed for fast ignition. [2] The proposed
diagnostics method considers a DT fuel admixed

with a small amount of 9Be and uses thefollowing reaction governed by fusion-produced
energetic -particles:α  9 Be  1 2C*[2  ; 0]  n ;C  1 2C  γ ( E  4.44MeV )12 *(1)

The DT/9Be pellet is compressed to highdensities by implosions such as tailored for fastignition, but is not subjected any heating laser. In
such a case, the fuel would not be ignited, and mostof the nuclear reactions would occur around themaximum compression. Thus, nuclear reaction
products are expected to carry information about
the compressed state of the fuel such as the degreeof electron degeneracy. In [2], however, it wasassumed that -particles slow down in an infinite
plasma; leakage from the pellet was not accountedfor. In the present study, we calculate the transportof 3.52 MeV -particles in degenerate plasma, andapply it to the degeneracy diagnostics.
2. Transport Equation
The transport of energetic charged particles inplasmas is described by the Boltzmann-FokkerPlank (BFP) equation. The BFP equation for thecase of non-degenerate electrons was written as
follows:
1  v t  Ω   R  r , E , Ω, t j
S j r , E  r , E, Ω, t   j Tj r , E   1   2    r , E, Ω, t 
E   dE d Ω  NES E  E , Ω  Ω, t  r , E, Ω, t   Qr , E , Ω, t (2)Deposited energy per unit path [-]3. Energy Deposition
We solved the BFP equation for a sphericallysymmetric case where 3.5-MeV source -particles
are injected into the central region (r / R < 0.05) of acompressed ‘stationary’ DT plasma sphere. Figure
1 shows profile of deposited energy by -particles
per unit path length. It is confirmed that the widthof the energy deposition region becomes broader
than in the case of non-degenerate background.

14to electrons
to ions
total

12
10
8

degeneracy ignored
degeneracy considered6=0.54

 =429g/cm
kT=0.4keV

3

2
00.0

10

10-3

infinite plasma2R = 1.0 [g/cm ]2
R = 0.5 [g/cm ]2
R = 0.1 [g/cm ]-4

kT=0.4keV

10-5

0.1

110

kT/Ef [-]Fig.2. Reaction probability

Experimentally P-Be is determined as the ratioof 4.44-MeV -ray yield to the DT neutron yield.
Thus, if these yields are measured and the plasma
temperatures are given by another measurement, wecan assess the degeneracy using the P-Be-curve.
We also estimated the yield of 4.44-MeV -rays
emitted from compressed DT/ 9Be pellets. In thecase that R = 0.5 g/cm2, g/cm3, kT = 0.4,～
1.0 keV and nBe /ni = 0.1, for example, the yield wasestimated to be 105～108, which seems enough to bedetected.

R=0.3g/cm

20.2

0.4

r/R [-]

Fig.1. Energy deposited per unit path

4. Reaction Probability
Assuming DT/ 9Be pellets compressed to various
areal densities, we calculated the probability P-Bethat 9Be reaction occurs during the transport offusion-born -particles as a function of thedegeneracy parameter  (≡ kTe / E Fermi). The
probability is calculated using (r, E), the-particle flux integrated over the angular space:P  Be depends on . The more the range of - particles islengthened, the more they leak from the pellet.
Therefore, when the pellet is small, the probability
P-Be is reduced.

Pe [-]

where andQ stand for respectively the angular
flux and source of energetic particles,; R and NES
are macroscopic removal and macroscopic transfer(differential) cross sections; S and T represent thecoulombic stopping power and the angulardeflection coefficient, respectively.
The electron degeneracy effect should beincorporated in FP terms, i.e. the terms including Sand T. To do this, we get the FP terms back to theBoltzmann integral form. After incorporating Pauliblocking, we recover the FP form suitable to treat
small-angle scattering.

 nBe r   Be E  r, E dEdV
 nD r nT r    v  dV(3)Figure 2 shows the calculated probability. The
fractional number density of 9Be admixture wasfixed to nBe/ni = 0.1. We can see that P-Be strongly

5. Concluding Remarks
We have calculated the transport of energetic
-particles in degenerate plasma and confirmed theeffect such as range lengthening due to electron
degeneracy.
Using the transport code, we also examined theapplicability of previously proposed method todiagnose electron degeneracy in compressed DTfuel. It was shown that the method would beapplicable if the areal density of compressed pellet
is larger than 0.3 g/cm3.
References[1] G. Kamelander, Atomkernenergie, 48 (1986) 231;
T. Johzaki, et al., Proc. of 1996 Int. Conf. on Plasma
Phys. (1997) 2038.
[2] Y.Nakao, et al., Fusion Sci. Technol., 56(2009) 391.