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Interaction of heavy-ion beams with target plasma
Ryouji Takahashi, Toshihiko Hotta, and Toshio Okada
Citation: AIP Conference Proceedings 369, 1138 (1996);
View online: https://doi.org/10.1063/1.50399
View Table of Contents: http://aip.scitation.org/toc/apc/369/1
Published by the American Institute of Physics
Interaction of Heavy-Ion Beams with
Target Plasma
Ryouji Takahashi, Toshihiko Hotta, Toshio Okada
Faculty of Technology, Tokyo University of Agriculture and Technology,
2-24-16 Naka- machi, Koganei-shi, Tokyo 184.
Abstract. Heavy-ion beams have been considered as a driver for inertial confinement fusion. The interaction of the ion beam with a target plasma is of central
importance. A number of studies on the heavy-ion beam interaction with target
plasma are found in the literature 1-4. Reported are the results of a numerical
study of an effective charge and nonlinear corrections to the stopping power for
heavy ions in plasma. It is important for heavy-ion stopping that charge states of
projectile ions are decided. The effective charge states of beam ions are supposed
as Betz formula. It is shown that the stopping power is increased and the stopping range is shortened according to nonlinear effects. The heavy-ion stopping is
simulated at the various temperatures of the target plasma. The target heating
of the ion beam and radiation effects are investigated.Energy is decreasing by the
varying of the electric and magnetic fields in the target. It is considered to be a
collective effect of stopping power. Though it is very small, in fact, there is the
energy loss. Here it is shown that by using the PIC code.
Effective charge theory
One t e r m that we have not satisfactorily defined thus far for use in the
Bethe model is Zeff, is the effective charge of the projectile ion. T h e effective
charge of an ion is usually inferred by comparing the stopping power of a
higher-Z ion to t h a t of a proton. Any deviation from a Z 2 dependence is
attributed to an effective charge. Betz, Brown, Moak, Steward, and Ziegler
have published "least-squares fits" to the effective charge. We have found that
the concise expression given by Brown and Moak (Eq.(1)) contributes to a
good agreement between computed and experimental stopping powers when
used in conjunction with our set of average ionization potentials and shell
corrections.
Zeff = Z0(1 -- 1.034exp(-137.04~/(Zo)°'ng))
(I)
© 1996AmericanInstituteof Physics
1138
where Zeff is effective charge, Z0 atomic number, 13=Vp/c, Vp is ion velocity
and c light velocity.
Linear Vlasov theory
The Vlasov-Poisson equations without external fields read
Of + v . Of + 0¢~ . -Of- = 0 .
o~
~
0~
The electrostatic potential is given by
(I)l(r,t) = Zeff Jf
(2)
Ov
3 exp(ik. (r - Vpt))
k
k2~(k, k • Vp)
(3)
I
The Linear stopping power is calculated by using asymptotic expansion
-
where Z = 3Z~'
(4)
in
Nm = no~3D, no is the density of plasma ~D Deby length.
Nonlinear Vlasov theory
During the collision, the election will be displaced by ~ due to the force of
the ion: ~ + w2~ = f/m, where f(t,b) is the force by the ion. The equation of
motion can be rewritten the form
¢(t)
1
tt
I
-
TruMp J-z~
dtf(t, b) sin[wp(t - [)]
(5)
with the force
F(t,b) =
z
~ff
e~ (v~t + ~)ex + (b + %)%
[(-~vt + ¢x)2 + (b + ~u)213/2'
(6)
The energy transfer is
2
1
dtF(t, b) exp(iwpt)
.
(7)
Nonlinear stopping power is calculated by using the Bessel function and the
Bark.s term
-
~
- 2~v~= In
IZI
+ ~
1139
in
IZl
- 2.4
.
(8)
Particle-in-cell simulations
At first, by determining the distribution of particles and the initial ion velocity, electric and magnetic fields are calculated by Maxwell equations.
VxE(x,t)
V × E(x,t)
eoV.E(x,t)
V.B(x,t)
=
:
=
=
-°B(x,t)
#0(e0°E(x,t) +j(x,t))
p(x,t)
0
(9)
The particle motion and electric and magnetic fields are calculated by using
Eq.(9) and difference methods. Thus the energy loss of the projectile particle is determined. Where numerical value is using the normalization. The
normalization unit is shown in table 1. Initial conditions are shown in table 2.
Table 1: Normalization units
space c/~op
velocity
c
time
dencity
no
mass
me
charge
q
current
47rnoqc
electric field
(47mom~ca)1/2
magnetic field (47rnom~c2)l/2
no(c/wp)Zmc2/2
energy
Table 2: Initial conditions
heavy-ion beam
species
Bi
atmic number
83
continue time
10[ns]
beam density
3.0 × 1015[cm-~]"''
energy of one particle
30[MeV]
target
species
Li
atomic number
3
1.0 x lO~2[cm-3]
plasma density
plasma temperature
lO00[eVl
1140
ISOIXI
I
.
.
.
.
I
'
'
'
I
'
},=
ct~
0~-I
0
5xlO "i
IxlO -~
i
~
0
Fig.1 The linear and nonlinear calcuIation ['or stopping power with constant temeprature T=1000(eV]. The
solid line, numerical evaluation of the
linear stopping power Eq.(4); dashed
line, numerical evaluation of the nonlinear stopping power Eq.(8).
.
,
,
.
,
.
.
[
i
5 ~10-1
Distance (cm)
Distance (cm)
E
r
r
l x l 0 -~
Fig.2 The solid line is the result of
the effective charge by taking the [inear
stopping power.The dashed line is the
result of the effective charge by taking
nonlinear stopping power.
,
~88.6
I3424
~
238&~
M 13423
2~$8.2
c~
13422
2388
[
]
200
Time
t
~
i
I
1
[
r
0.00
Time
(1/co p)
Pig.3 The stopping power by PIC code.
Energy loss is normalized.
(1/co p)
Fig.4 The stopping power by PIG code.
Energy loss is not normalized.
1141
L-i
Discussion
In summary, we have calculated HIB stopping power in consideration of the nonlinear
and collective effects.
According to Fig.l, it is shown that the stopping power is increased and the stopping
range is shortened according to nonlinear effects.
According to Fig.3, there is the energy loss by collective stopping power, but it is shown
that energy loss of the collective stopping power is not too large.
References
1.
2.
3.
4.
E.Nardi, E.Peleg and Z.Zinamon, Phy.
E.Nardi, Z.Zinamon, Phys. Rev. Lett.
Th.Peter and J.Meyer-ter-Vehn, Phys.
Th.Peter and JMeyer-ter-Vehn, Phys.
Fluids 21, 574 (1978).
49, 1251 (1982).
Rev. A43, 1998 (1991).
Rev. A43, 2015 (1991).
1142
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