Download Distortion of bulk-electron distribution function and its effect on core

The fifth International Conference on Inertial Fusion Sciences and Applications (IFSA2007)
IOP Publishing
Journal of Physics: Conference Series 112 (2008) 022066
doi:10.1088/1742-6596/112/2/022066
Distortion of bulk-electron distribution function and
its effect on core heating in fast ignition plasmas
Hideaki Matsuura, Yasuyuki Nakao, Masanori Katsube,
Tomoyuki Johzaki∗
Department of Applied Quantum Physics and Nuclear Engineering,
Kyushu University, Motooka, Fukuoka 819-0395, Japan
∗
Institute of Laser Engineering, Osaka University, Suita, Osaka 565-0871, Japan
E-mail: [email protected]
Abstract. A possibility of distortion in bulk-electron velocity distribution function due to
injection of intense laser-induced (LI) electrons is examined. The Fokker-Planck (FP) equations
for LI-fast and bulk electrons are simultaneously solved consistently considering the time
evolution of Rosenbluth potentials in two-dimensional (2D) velocity space. The FP simulation
shows that an asymmetric distortion of bulk-electron distribution function becomes appreciable
when intensity of the fast electron relative to bulk plasma density is increased, and that in this
case the bulk-ion heating tends to be enhanced (accelerated) to some extent.
1. Introduction
To realize the energy production in a fast ignition scheme, complete clarification of the energytransfer process from laser-induced (LI) fast electron to dense core plasma is necessary. A part of
the energies carried by the fast electrons may be transferred to core plasma via electromagnetic
forces induced by energetic electrons themselves. In a dense plasma, however, collisional
interaction is dominant for energy-transfer process from fast electron to bulk plasma, and a
slowing-down component (≤MeV energy) in LI-fast electron particularly contributes to the
local plasma heating. The shape of the velocity distribution functions of both slowing-down
and bulk electron would determine an important physics of plasma heating efficiency. So far,
energy distribution function for fast electron has been investigated using sophisticated simulation
codes[1]. In most of the previous studies, however, the deposited power from fast electron to bulk
plasma has been estimated without considering the distortion of the bulk-electron distribution
function. In this paper, we focus our attention on the Coulombic collisional interaction as a
possible drive force for the distortion of bulk electron. We examine whether a distortion of the
bulk-electron distribution function from Maxwellian due to collisions with LI-fast electron can
be appreciable or not and whether the distortion can be effective on the bulk plasma heating or
not.
2. Analysis Model
An intense electron-beam injection in Maxwellian plasma is considered. The LI-fast and bulk
electron distribution functions fa (vk , v⊥ ) are obtained by simultaneously solving the following
Fokker-Planck (FP) equations;
c 2008 IOP Publishing Ltd
1
The fifth International Conference on Inertial Fusion Sciences and Applications (IFSA2007)
IOP Publishing
Journal of Physics: Conference Series 112 (2008) 022066
doi:10.1088/1742-6596/112/2/022066
∂
∂fa
∂fa
∂fa
=−
+ uk
+F
∂τ
∂ξ
∂uk
∂uk
"
!
1 ∂
∂H
fa −
∂uk
v⊥ ∂u⊥
#
"
∂ ∂ 2 G ∂fa
∂ 2 G ∂fa
1 ∂
+
+
+
u⊥
2
∂uk ∂uk ∂uk ∂uk ∂u⊥ ∂u⊥
u⊥ ∂u⊥
∂H
u⊥
fa
∂u⊥
∂ 2 G ∂fa
∂ 2 G ∂fa
+ 2
∂uk ∂u⊥ ∂uk ∂u⊥ ∂u⊥
!#
.
(1)
The subscript a represents LI-fast and bulk electron. The right-hand side of Eq.(1) is the 2D
FP collision term. The Rosenbluth potentials H and G are computed using the LI-fast and bulk
electron distribution functions at each time steps;
H=
X Z b 2 m a Z
b
Za
mb
1X
fb (u0 )
dv 0 , G =
0
|u − u |
2 b
Z
fb (u0 )|u − u0 |dv 0 ,
(2)
where subscript b represents the ion, LI-fast and bulk electron (the collision between LI-fast
electrons is neglected). Here uk and u⊥ denote parallel and vertical velocity components relative
to the electron-injected (x-axis) direction.
The variables are normalized so that u ≡ v/v 0 ,
p
ξ ≡ x/τ0 v0 and τ ≡ t/τ0 , where v0 = 2T0 /me , τ0 = 4π20 me 2 v04 /e4 lnΛ. Here T0 = 2keV and
the Coulomb logalithm lnΛ ' 4 are assumed. The bulk ion distribution function is assumed to
be kept in Maxwellian. The time evolution of the temperature of ion distribution function is
roughly estimated from d((3/2)nT )/dt = P LI→i + Pbulk→i . The transferred power Pa→b from
species a to b is evaluated using the distribution functions f a and fb as
Z
1
∂fa Coulomb
dv .
(3)
ma v 2
2
∂t b
The second term in the left-hand side of Eq.(1) represents the effect of the streaming of
electrons toward fast-electron-injected (x-axis) direction. The third term in the left-hand side
represents the influence of the electric field. The normalized electric force F is written as
F ≡ qEk τ0 /me v0 . In this paper we roughly estimate the electric field as
Pa→b = −
Ek = qη
Z
fbulk (uk , u⊥ )vk dv ,
(4)
where η represents the plasma resistivity and is obtained assuming the collision with ions in
√
3/2
Maxwellian, i.e. η = πe2 me lnΛ/(4π0 )2 Te .
To look at the distortion of bulk electron distribution function, we consider an intense
electron-beam injection into an uniform 1keV deuterium-tritium (DT) Maxwellian plasma. As
a boundary condition, the LI-fast-electron distribution function at x = 0 is fixed as
fLI
me
me 2
= c exp −
(v − vpeak )2 −
v
2Tk k
2T⊥ ⊥
!
,
(5)
during the calculation. The coefficient c is obtained from the fast-electron density, and the
broadness of the fast-electron distribution function toward v k direction can be determined by
parameter Tk . (Throughout the calculations T⊥ is fixed at 1keV.) In this paper we choose the
vpeak as electron velocity corresponding to 100keV energy.
Generally LI-fast electron in the typical fast ignition plasma has much higher energy. As a
result of the relativistic effect, however, the relative velocity between ∼MeV and 1keV electron
is not significantly different from the one between 100keV and 1keV electron. Furthermore
because the Rutherford cross section is inversely proportional to fourth powers of the relative
velocity, much lower (∼keV) electron has more important role for the distortion of bulk-electron
distribution. As a first step, we estimate the presence of the bulk-electron distortion without
considering a contribution of the relativistically high-energy electrons.
2
The fifth International Conference on Inertial Fusion Sciences and Applications (IFSA2007)
IOP Publishing
Journal of Physics: Conference Series 112 (2008) 022066
doi:10.1088/1742-6596/112/2/022066
3. Results and Discussion
In Figure 1 we first show the time evolution of spatial profiles of (a) ion temperature and (b)
transferred power from LI-fast electron to ion P LI→i . In this case, Tk is taken as 10keV, bulk
ion and electron density 1030 m−3 and LI-fast-electron density 1028 m−3 are assumed (the LIfast-electron intensity is roughly estimated as ∼ 10 19 W/cm2 at x = 0). The LI-fast electrons
percolate into the target plasma and ion temperature gradually increases due to the collisional
power deposition from fast electrons to ions.
Figure 1. Time evolution of spatial profiles of (a) ion temperature and (b) deposited power
from LI-fast electrons to ions PLI→i , when FP equations for LI-fast and bulk electron are solved.
Figure 2. Time evolution of spatial profiles of (a) ion temperature and (b) deposited power
from LI-fast electrons to ions PLI→i , when bulk electron is assumed to be Maxwellian.
To look at the effect of the distortion of the bulk electron distribution function, we also
performed the simulation assuming a Maxwellian for bulk-electron distribution function. The
result is presented in Fig.2. The calculation conditions are the same as in Fig.1. From the
comparison between Fig.1 and Fig.2, it is found that in the region x ≥ 6µm the transferred power
from fast electron to ion when we assume Maxwellian for bulk electron distribution function is
somewhat small compared with the case when FP equation for bulk electron is solved (see
Fig.1(b) and Fig.2(b)). The reduced heating power causes the reduction in the degree of the
ion-temperature increment (see Fig.1(a) and Fig.2(a)).
In Figure 3 the deviation of bulk-electron distribution function from Maxwellian, i.e.
non−Maxwellian
Maxwellian is exhibited for (a) t=400fsec and (b) 1psec at x = 5µm. To
∆febulk ≡ fbulk
−fbulk
estimate the deviation, the effective temperature of the bulk-electron distribution was obtained
3
The fifth International Conference on Inertial Fusion Sciences and Applications (IFSA2007)
IOP Publishing
Journal of Physics: Conference Series 112 (2008) 022066
doi:10.1088/1742-6596/112/2/022066
by comparing the thermal component of the calculated bulk-electron distribution function with
a Maxwellian by mean of the least squares fitting[2]. It is found that the bulk electrons are
dragged toward the fast-electron-injected direction. The degree of the distortion is notable
especially during the beginning of the heating, and gradually calms down. At the beginning
of the heating, as a results of the asymmetric distortion of bulk-electron distribution function
(acceleration toward laser-injected direction), slowing down of fast electron tends to be slightly
enhanced. After the slowing-down of LI-electron distribution function is sufficiently progressed,
collisional interaction between LI-fast and backgroud plasma relatively decreases compared with
the Maxwellian plasma, and spatial propagation of the fast electron is hastened.
Figure 3. Deviation of bulk-electron distribution function from Maxwellian, ∆f ebulk ≡
non−Maxwellian
Maxwellian , at (a) t=400fsec and (b) 1psec at x = 5µm.
fbulk
− fbulk
Owing to an intense electron beam injection, an appreciable distortion in bulk-electron
distribution function can be induced. The resulting bulk-ion-heating enhancement (acceleration)
can be seen to some extent.
In actual fast ignition plasmas, fast electron is generated with much broadened (higher)
energy spectrum. So far in Fig.1-3, T k has been taken as 10keV at t=0. If a fraction of low
energy component in LI-fast-electron distribution function increases (T k ≥10keV), the collisional
interaction between LI-fast electron and bulk plasma is further intensified. In such a case the
effect of the distortion of bulk electron distribution function on plasma heating may be more
conspicuous. In this paper a contribution of the relativistically high-energy electrons has been
neglected. For such high energy electrons, the Coulomb energy deposition takes place mainly
to bulk electrons, and ions are heated by energy exchange with the bulk electrons. As far as we
look at the distortion in ∼ 1030 m−3 density range, influence of the MeV-energy electrons would
be small. Throughout the calculation, time variation of the bulk-ion spatial profile has not been
taken into account. To carry out the more accurate evaluation, spatial diffusion (power-loss
mechanism) for bulk ions should also be included. When electron beam passes through much
lower density region, the distortion may be affected more strongly by the electric field[3]. In
such a case more accurate treatment for electric field, e.g. coupling with the Poisson equation,
would be required.
References
[1] See, for example, Sentoku Y, Mima K, Ruhl H, Toyama Y, Kodama R and Cowan T E 2004 Phys. Plasmas
11 3083-3087, Yokota T, Nakao Y, Johzaki T and Mima K 2006 Phys. Plasmas 13 022702 1-9
[2] Matsuura H, Nakao Y 2007 Phys. Plasmas 14 054504 1-4
[3] Kemp A J, Sentoku Y, Sotnikov V, Wilks S C 2006 Phys. Rev. Lett. 97 235001 1-4
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