23P149-P
Effect of Electron Degeneracy on Ion Distribution Function inUltra-High Density Plasmas
超高密度プラズマ中のイオン分布関数に対する電子縮退の影響
Kohei Sugita, Hideaki Matsuura, and Yasuyuki Nakao
杉田昂平、松浦秀明、中尾安幸
Department of Applied Quantum Physics and Nuclear Engineering, Kyushu University
744 Motooka, Fukuoka 819-0395, 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
In ultra-high density plasmas as realized in inertial confinement fusion, the electron degeneracy can arise
and it is thought to affect the energy distribution of coexisting ions through Coulombic ion-electroninteraction. We evaluate these effects by developing and solving the model equation for the distributionfunction of ions coexisting with degenerate electrons. It is shown that the ion distribution maintains aMaxwellian form at a temperature equal to that of degenerate electrons.

1. Introduction
In inertial confinement fusion, the fuel iscompressed to ultra-high density as 1000 times thesolid density and the wave nature of electron
becomes conspicuous in such plasmas. Sinceelectron is fermion, the energy transition isrestricted by Pauli’s exclusion principle and suchsituation is called “electron degeneracy”. The
degree of degeneracy can be evaluated by thedegeneracy parameter Θ  kTe / E F (EF is Fermienergy); the smaller it is, the stronger thedegeneracy becomes. The consequences of electron
degeneracy are as follows:(a) The electron distribution function becomes tofollow the Fermi-Dirac statistics.
(b) Scattering between an electron and otherparticle is restricted (Pauli blocking).
In addition, these events can also affect the energy
distribution of coexisting ions through Coulombic
ion-electron interaction. These effects can beevaluated by calculating the distribution function ofions, but there has been no proper equation inwhich are perfectly incorporated the above two
effects due to electron degeneracy. Thus wedevelop the model equation to describe thedistribution function of ions coexisting withdegenerate electrons and estimate the magnitude ofthese effects by solving it.
2. Derivation of the equation followed by the ion
distribution function
We start from the balance equation consisting of
the scattering collision term (in- and out-scattering
rates), the loss rate due to nuclear reactions and the
independent source. Usually, small-angle Coulomb
scattering term is written in the Fokker-Planck (FP)form, but the FP term is not suitable for describing

individual scattering; it is difficult to incorporate
Pauli blocking, which is effect on the individual
scattering, into the final form of the FP term.
Therefore we get back to the Boltzmann integral [1]and incorporate Pauli blocking into it. After that,
we recover the FP-like form for small-angle
Coulomb scattering. Instead of verbosity v we use
energy E as an independent variable and adopt theflux (E) defined by (E) = v f (E), where f (E) is
the ion energy distribution function. The final formof the equation in steady-state is n  E    E  ( E )j ejNIjabsj
2S j ( E ) ( E )   2 D j ( E ) ( E )j Ej E





(1)

    n j NIj E  E  E dE  Q E ,j ewhere n j is the number density of “background”
species j. On the left-hand side, the first and second
terms are respectively the removal rate due tolarge-angle scattering and the loss rate due toabsorption. The first and second terms on theright-hand side are the expansions of the smallangle scattering term. The third term represents thelarge-angle in-scattering rate. The last term is theindependent source. The (differential) crosssections in Eq. (1) are the averages over the thermal
motion of target j [2] and the symbol “NI ”
represents “elastic nuclear plus interferencescattering”. Additionally Sj and Dj are quantities
defined during processing the small-angle Coulomb
scattering term; the former is the Coulombic
stopping power while the latter is the energy
dispersion coefficient. The effects of electron
degeneracy are incorporated in these coefficients.

10341033

Te = 0.5 keV1032

 = 0.1

1031

1030

1029

1028

1027

1026

1025

10The event like (a) can be explained as follows.
When the electron degeneracy arises, the number ofelectrons in lower energy region, which aresusceptible to Coulomb interaction, gets smaller
and Coulomb interaction they undergo getsweakened. Therefore the energy being transferred
to ions from the electrons gets smaller than that in
the case of non-degenerate plasmas. As a result, theion distribution function spreads toward
high-energy side in order to weaken Coulombinteraction that the ions undergo, and the ion
temperature gets higher, because the distribution ofions is determined so that the energy flows between
the ions and the electrons are canceled each other.
Meanwhile the event like (b) happens for thefollowing reasons. Pauli blocking mainly restricts
energy gaining of the electrons because of higher
probability of electron occupation in thelower energy region, and the energy moving toelectrons from the ions lessens. In other words, theenergy transferred to ions from electrons relatively
becomes larger. This effect counteracts the result of(a), so the ion temperature does not change from theresult without considering the electron degeneracy.

3.5

degeneracy ignored
degeneracy partially considered(Pauli blocking ignored)
degeneracy fully considered
-3

10-2

10-1

100101102

Deuteron Energy [keV]

Fig.1. Deuteron distribution function

From the calculated distribution function weevaluated the ion temperature:
22 0 E f E dE E .
33  f E dE

2.5
2.0
1.5
1.0
0.5
0.0

0.1

110

 [-]

T

degeneracy ignored
degeneracy partially considered(Pauli blocking ignored)
degeneracy fully considered

3.0Ion Temperature [keV]-3-1

Distribution Function [m keV ]3. Results and discussion
Supposing steady-state DT plasmas at various
electron temperatures Te and degeneracy parameters
, and fixing the distribution function of electrons,
we solved Eq. (1). Figure 1 presents, as an example,
the deuteron distribution function when Te = 0.5keV,
 = 0.1. The solid curve shows the distributionfunction calculated by fully considering the effectof electron degeneracy, while the dashed curve isobtained by partially considering the degeneracy
effect, that is the electron distribution function
based on the Fermi-Dirac statistics was used but
Pauli blocking was ignored. The dotted curve(almost falling on the solid one) is the result in thecase ignoring the electron degeneracy. It can beseen that each ion distribution function forms theMaxwellian distribution. However, the ion
temperature gets higher when the degeneracy effectis patially considered.

(2)0Figure 2 presents  dependency of the ion
temperature when Te = 0.5keV. In the case where
the electron degeneracy is ignored, the ion
temperature is equal to the electron temperature.
As for how to incorporate the electron
degeneracy, we obtain the following results:(a) The ion temperature gets higher when thedegeneracy effect is partially included (ignoringPauli blocking), and this is significant at low
region.
(b) However, the result of (a) disappears when weconsider Pauli blocking.

Fig.2. Ion temperature

4. ConclusionBy properly incorporating the effect of electrondegeneracy, we have found the following results;
the ion distribution function maintains Maxwellian
form and the ion temperature becomes equal to theelectron temperature even if electrons are indegenerate state. This is because two effects ofelectron degeneracy counteract each other.
References[1] For example, F. Reif: Fundamentals of Statisticaland Thermal Physics (McGraw-Hill, New York,1965).
[2] Y. Nakao, K. Tsukida and V. T. Voronchev: Phys. Rev.
D. 84, (2011) 063016.