J. Plasma Fusion Res. SERIES, Vol.7 (2006) 127-130

Energy Distribution Function of Fast Neutral Atoms and Neutron
Production Rate in Inertial Electrostatic Confinement Device
YOSHINAGA Sho, MATSUURA Hideaki, NAKAO Yasuyuki and KUDO KazuhikoDepartment of Applied Quantum Physics and Nuclear Engineering, Kyushu University, Fukuoka 812-8581, Japan（Received: 4 October 2004 / Accepted: 28 May 2005）

Abstract
Fast-neutral energy distribution function in inertial electrostatic confinement (IEC) plasmas is studied bysolving the Boltzmann equation for fast neutral, which is produced by several kinds of charge-exchange reactions, for various presumed ion distribution functions. From the obtained fast-neutral distribution functions,the Doppler-shift spectrum (energy spectrum) of fast-neutral in IEC devices is evaluated. By comparing thespectra between the present calculation and previous experiment [1,2], the broadness of the ion distributionfunction in the energy space is estimated.
Keywords:
IEC, charge-exchange, fast-neutral, energy distribution function, Doppler shift spectrum
1．Introduction
The inertial electrostatic confinement (IEC) is a concept for electrostatically confining high-energy fuelions in potential well [3-7]. In ideal IEC plasmas, theions converge toward the center of the device. The IEC
fusion system has intrinsic potential for earlier practical use of fusion as a compact and economical neutrons/protons source. So far, neutrons/protons morethan 108 ns−1 produced by D(d,n)3 He (3 He(d,p)4 He) fusion reactions have been observed on several devices.
To improve further the device performance, it is important to understand the physics of the IEC plasmas.
In the device several kinds of ion-neutral collisionsoccur, e.g. elastic-scattering, ionization, charge exchange in parallel with the fusion reaction. Especially,
the charge exchange reaction (CX) changes accelerated ions to fast-neutrals which can cause fusion reaction with background gas. In previous researches, it
has been shown that fusion reaction between the fastneutral and background gas is comparable with thosebetween ion and back ground gas [8,9].
We have examined correlation between the ion energy distribution function and neutron production ratein spherical IEC devices [10-12], and have shown thatneutron production rate between ion and backgroundgas is sensitively aﬀected by the shape of ion energy distribution function itself. In order to identify the shape ofthe energy distribution function of fast neutral in typical
IEC devices, the Doppler-shift wavelength of the hydrogen specific Hα line has also been measured [1,2]. The
fast-neutral distribution function is strongly influenced

by the shape of the ion distribution function. In this paper, we solve the Boltzmann equation for excited fastneutral for various presumed ion distribution functions,and using the obtained fast-neutral distribution function
we evaluate the Doppler-shift spectrum (energy spectrum) in the IEC devices. By comparing the spectra between the present calculation and previous experiment[1,2], we estimate the broadness of the ion distributionfunction in the energy space.
2．Analysis model
2.1 Collisional processes

We consider the following collisional processes.
H+ (E) + H2 → H∗ (E) + H+2(1)H+2 (E) + H2 → H∗ (E/2) + H(E/2) + H+2(2)H+3 (E) + H2 → H∗ (E/3) + H2 (2E/3) + H+2 ,(3)H∗ + H2 → H+ + H2 + e,(4)∗∗H + H2 → H + H2(5)where H∗ is fast excited neutrals and H2 is backgroundgas. The eqs. (1)-(3) represent the CX reactions inwhich the fast H∗ neutrals are produced. Here E is kinetic energy of the particles. We assume that the fastneutral which is produced by CX reaction has the same
velocity as the ion before the reaction. Reaction (4) isthe ionization reaction, which has high cross-sectionsin 1 − 100 keV energy range. The eq. (5) indicates theelastic-scattering.
C 2006 by The Japan Society of Plasma

Corresponding author’s e-mail: syo-y@nucl.kyushu-u.ac.jp

Science and Nuclear Fusion Research127

Energy Distribution Function of Fast Neutral Atoms and Neutron Production Rate in Inertial Electrostatic Confinement Device

vH2 . The background gas distribution function f H2 isassumed to be Maxwellian at 0.1 eV temperature. Anion which has velocity v H ∗ changed to the fast excitedneutral which has velocity v H ∗ . The second term on
the right side in eq. (8) represents the eﬀect of the fastexcited H∗ neutral loss by ionization reaction with background gas (eq. (4)).
Ionization
∞∂ fH ∗
2π=−fH ∗ (vH ∗ )vH2 fH2 (vH2 )∂t LossvH ∗02.2 Ion distribution function

In order to investigate the fast-neutral energy distribution function for various shapes of ion energy distribution functions, we introduce the following modelfunction.
  E − ξ |q φ | 2  L 2 a a 0−(6)fa (E, L) = ca exp −αa qa φ0
βa L0awhere qa is charge of the particles of species a , total energy E = 1/2mav2 + qφ and angular momentum
L = marv⊥ respectively. Here rcat is the cathode radius.
The subscript a represents particle species (i.e. H+ ,√H2 + ), φ0 the grid voltage and L0a = rcat ma qa φ0 . By adjusting the αa , βa and ξa , we can simulate the broadnessof the distributions in the energy, angular momentum
direction, and the position of energy peak. The coefficient ca is determined so that the density at the cathode na is equal to na cat . Following Thorsons’s treatment[9,13], we relate na cat to the measured cathode current
Ia meas byna cat =Iameas11,21 − γ2 1 + δ 4πqa rcat
2qa φ0 /mavH ∗ +vH2|vH ∗ −vH2 |(10)
The third term on the right side in eq. (8) is the Boltzmann collision integral (eq. (5)),
Elastic
∞∞∂ fH ∗2π= 2v H ∗ fH ∗ (v H ∗ )vH2 fH2 (vH2 )∂tvH ∗ 00P(v H ∗ → vH ∗ |vH2 )(7)−where γ represents the transparency factor of the innergrid [4], δ is the number of secondary electrons emitted
from the grid due to ion impact [13]. Throughout thecalculation γ = 0.95, δ = 1 are assumed.

∂ fH ∗∂ fH ∗+µ∂t∂r ∂ f ∗ CX  ∂ f ∗ Ionization  ∂ f ∗ ElasticH
H
H−+. (8)
=∂t a
∂t
∂t
Most of ions can be considered moving along the direction µ = ±1 (µ is direction cosine). So, we assume high convergence of the ions towards the centerof sphere (i.e. βa = 0.05), and only consider the particle of µ ≈ ±1, i.e. partial diﬀerential term about µ isignored. The second term on the left side in eq. (8) is
the streaming term for fH ∗ .
The first term on the right side in eq. (8) represents
the eﬀect of H∗ generation by CX between Ha + ion andH2 gas (eqs. (1-3)).
CX ∞2π∂ fH ∗=fHa+ (vH ∗ )vH2 fH2 (vH2 )∂t avH ∗0av2r σa CX (vr )dvr dvH2 ,vH ∗ −vH2|vH ∗ −vH2 |v2r σElastic (v r )dvr dvH2 dvH ∗∞vH ∗ +vH2vH2 fH2 (vH2 )0|vH ∗ −vH2 |v2r σElastic (vr )(11)where P(v H ∗ → vH ∗ |vH2 ) is the probability function thatthe H∗ which has the speed v H is scattered into thespeed region vH + ∆vH due to the elastic scattering with
the H2 which has speed vH2 . If we assume that elasticscattering is isotropic in the center of mass system,⎧
⎪
2vH ∗22⎪⎪∗≤v+v0≤v⎪ v2∗ +v2HH∗
H2 .
⎨H2HP(vH ∗ → vH ∗ |vH2 ) = ⎪⎪⎪⎪⎩ 0(otherwise)(12)
IonizationCX)andCross-sections of CX (σa ), ionization (σelastic-scattering(σ Elastic ) are taken from the work ofTabata and Shirai [14]. We substitute the ion energy distribution function into eq. (9) and solve the Boltzmann
equation to obtain the fast neutral energy distributionfunction. In the region within the spherical cathode, thepotential structure is assumed to be zero, while outsideof the spherical cathode, Child-Langmuir radial potential is assumed.

The fast excited neutral energy distribution functionfH ∗ is determined by solving the Boltzmann equation.

|vH ∗ −vH2 |2πfH ∗ (vH ∗ )vH ∗dvr dvH2 ,2.3 Boltzmann equationvH ∗ +vH2

v2r σIonization (vr )dvr dvH2 .

3．Results and discussion
The radial profile of the fast-neutral energy distribution function is presented in Fig. 1. In this calculation,
grid voltage φ0 = 10 kV, total cathode current I meas =30 mA, cathode radius rcat = 0.05 m, anode radiusrano = 0.25 m, background gas pressure = 0.67 Pa,
αa + = 0.1 and ξa = 0.2 are assumed. It is foundthat fast-neutrals are distributed all over the device almost uniformly. This is because the mean free path of(9)where vr is the relative velocity between the fast excited neutral velocity v H ∗ and background gas velocity
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Energy Distribution Function of Fast Neutral Atoms and Neutron Production Rate in Inertial Electrostatic Confinement Device

Fig. 1 Radial proﬁle of fast-neutral distribution function.
Fig. 2 Doppler-shift spectrum of fast excited neutral,
αH+ = αH 2+ = αH 3+ = 0.1 and ξa = 0.2.

the fast-neutrals is longer than that of ions. There arehigh energy peaks in the fast neutral energy distribution function, which is caused by CX of background gas
with the high-energy and the high-concentration ions in
the cathode region.
In previous experiment [1], single light channel (µ ≈±1) Doppler-shift spectrum was measured in IEC devices of grid voltage φ0 = 10 kV and background gas
pressure = 0.67 Pa. The accelerated ion in the device consist of three kinds of hydrogen molecules, i.e.
H+ (20 %), H2 + (60 %), and H3 + (20 %), and the maximum energy is considered as ∼ 2 keV. The Dopplershift spectrum [1] is shown by white circles in Fig. 2.
In the same condition with the experiment [1], we evaluate the Doppler shift spectrum (energy spectrum) fromthe obtained fast-neutral distribution function, and theresult is shown by solid line in Fig. 2. In this case, wechoose the same broadness of the distribution functionin the energy space as previous calculations [11-13], i.e.
αH + = αH2 + = αH3 + = 0.1, ξa = 0.2. In Fig. 2 it is difficult to distinguish the three peaks of the intensity. InFig. 3 we next show the calculation when α H + = 0.1,
αH2+ = 0.075 and αH3+ = 0.05 are assumed. In this calculation, the eﬀect of the mass on the broadness of theion distribution function in the energy space is included.
It is found that three peaks produced by the CX reactions (eqs. (1)-(3)) appear in almost the same energy
range compared with the experiments [1]. (We changedξH2 + = 0.225, ξH3 + = 0.25 respectively to adjust energypeaks.) The highest peak around 0.9 nm (0.8 keV) energy range appear owing to the CX reaction caused by
H+3 (eq. (3)). Since the cross section of the ionization
reaction eq. (4) increases with increasing energy of fast
excited neutral H∗ , the excited neutral easily disappear
from the device via ionization reaction. This is the reason why the highest peak in Fig. 3 is sustained by theCX reaction caused by H3 + , even though 60 % of ionsare occupied by H2 + .

Fig. 3 Doppler-shift spectrum of fast excited neutral,
αH+ = 0.1, αH 2+ = 0.075, αH 3+ = 0.05 and ξH+ = 0.2,
ξ H 2+ = 0.225, ξH 3+ = 0.25.

4．Concluding remarks
The energy distribution function of fast excited neutral has been derived by solving the Boltzmann equation. It has been shown that the experimentallyobtained Doppler-shift spectrum (energy spectrum) offast-neutral can be reproduced by our model using theαH + = 0.1, αH2 + = 0.075 and αH3 + = 0.05 parameters.
In this paper, we assumed that potential structure iszero within the spherical cathode. In actual devices,
however, virtual anode may appear by the space chargeof high-concentrated ions within the spherical cathode [13]. To incorporate this eﬀect into the analysis,
the Boltzmann equation must be solved simultaneouslywith the Poisson equation.
Acknowledgements
We wish to acknowledge partial support for thiswork from the Kyushu Electric Power Co. and the Com-129

Energy Distribution Function of Fast Neutral Atoms and Neutron Production Rate in Inertial Electrostatic Confinement Device

putational Cooperative program of the National Institute for Fusion Science.

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