J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)322–325
000–000

Two-Dimensional Kinetic Model for Fast Electron Transport inCompressed Core Plasmas
YOKOTA Tomohiro, NAKAO Yasuyuki, JOHZAKI Tomoyuki1 and MIMA Kunioki1
Department of Applied Quantum Physics and Nuclear Engineering, Graduate School of Engineering,
Kyushu University, Fukuoka 812-8581, Japan
1
Institute of Laser Engineering, Osaka University, Osaka 565-0971, Japan(Received: 9 December 2003 / Accepted: 20 March 2004)

Abstract
A Fokker-Planck type two-dimensional transport code for relativistic electrons in dense plasmas has beendeveloped. Being written in cylindrical coordinates with axial symmetry, the code can be used for analyzing the coreplasma heating in spherical targets in fast ignition scheme. The feature of energy deposition of beam electrons injected
into cylindrical D-T plasma is examined. The contribution of self-generated electromagnetic field to core heating
profile is estimated.
Keywords:
fast ignition, relativistic electron beam, energy deposition, Fokker-Planck calculation, pinch by magnetic field,
CIP-IDO scheme

1. Introductionmulti-dimensional (at least two-dimensional (2-D) in
coordinate space) calculations are indispensable.
In this paper, we first extend our 1-D kinetic transportmodel to 2-D one; the code is written in cylindricalcoordinates with axial symmetry. Next, we examine thefeature of energy deposition of fast electron beam injected
into stationary D-T plasmas. We also estimate thecontributions of the self-generated electric and magnetic fields
to the energy deposition profile.

In the so-called fast ignition (FI) approach to inertial
confinement fusion, an high intensity short-pulse laser isfocused on a pre-compressed pellet to generate a beam ofrelativistic electrons which should heat the fuel to ignition
temperature. [1] Thus, the study of electron propagation andenergy deposition in dense plasma is essential in this scheme.
The propagation of fast electrons to the dense coreregion, as well as laser-plasma interaction, has been actively
studied by means of particle simulations. On the other hand,little is reported about the energy deposition process in thedense core plasma. Since the collisions between the fastelectrons and the bulk plasma particles become significant in
the dense core region, a Fokker-Planck (FP) type calculation
would be appropriate for analyzing the core plasma heating.
Recently we developed a FP type transport model thatcan calculate the time- and space-dependent energy deposition
rate of fast electrons in dense plasma. [2] The collision terms
includes (a) short-range (b < λD) binary collisions with theplasma particles and (b) long-range (b > λD) collective effect,
i.e. collisions between screened particles. The effect of selfgenerated electric field is taken into account by assuming acurrent neutrality condition. So far, however, the code has
been written for one-dimensional (1-D) planer coordinate
system; the effect of magnetic field has not been included.
For more realistic examination of the core heating process,A relativistic FP-type transport equation is necessary for
analyzing the behavior of fast electrons. A general form of
the relativistic FP collision term was derived by Braams andKarney. [3] Their collision term, however, is too complicatedto be used for numerical study. To reduce it to a simple andtractable form, Nakashima and Takabe [4] assumed that ⏐u –uj⏐≈ ⏐u⏐, where u ≡ p/me , uj ≡ pj /mj ( j = e, i), p (or pj) is
the momentum and me (or mi) is electron (or ion) rest mass.
In this study, we also adopt the same approximation as
in ref. 4; the collision term is reduced to a simple (linearized)form. Furthermore, we extend it so as to include the longrange collective effect. [2]In the present analysis, we consider the 2-D (r-z)
cylindrical coordinate system (Fig. 1). Then, the relativistic
FP-type transport equation is expressed asCorresponding author’s e-mail: yokota@nucl.kyushu-u.ac.jp©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

2. Analysis model and calculation scheme1322

Yokota T. et al., Two-Dimensional Kinetic Model for Fast Electron Transport in Compressed Core Plasmas

1− µ 2∂f∂fp ⎡ ∂f
∂f ⎤sin ω++ 1 − µ 2 cos ω−⎢µ⎥∂t γ me ⎢⎣ ∂z∂rr∂ω ⎥⎦∂fsin ω ∂f− Fr+ ⎡ Fz µ + Fr 1 − µ 2 cos ω ⎤⎣⎦ ∂pp 1 − µ 2 ∂ω
1 ∂f+ ⎡ Fz ⎛⎝1 − µ 2⎞⎠ − Fr 1 − µ 2 cos ω ⎤⎣⎦ p ∂µ⎛ Y n Y n ⎞ m2 ∂ ⎛ 2 ⎞= ⎜ e e + i i ⎟ e2γ f⎠mi ⎠ p ∂p ⎝⎝ me1∂f ⎫m ⎡ ∂ ⎧γ ∂2 f ⎤+ ⎛⎝Ye ne + Yi ni⎞⎠ 3e ⎢ ⎨γ ⎛⎝1 − µ 2⎞⎠⎬+⎥ +S2p ⎣ ∂µ ⎩∂µ ⎭ 1 − µ 2 ∂ω 2 ⎦(1)
withFig. 1 2-D cylindrical coordinates system.
2⎛Z e ⎞Yj = 4π ⎜ j ⎟ lnΛ ( j = e, i ) ,⎜ 4πε ⎟0 ⎠⎝(2)⎛p 1 − µ 2 cos ω ⎞Bθ ⎟ ,Fz = (− e ) ⎜ Ez +⎟⎜
γ me
⎠
⎝(3)

⎛⎞pµBθ ⎟ ,Fr = (−e ) ⎜ Er −γ me ⎠⎝(4)

2

The self-generated electric and magnetic fields E and B
are evaluated by means of the generalized Ohm’s law,
Ampere-Maxwell equation, Faraday’s law and the equation
for total current density J = Jf + Jb (Jf : the current density of
the fast electron, Jb : the current density of the bulk electron).
Neglecting some terms which are not essential for the present
case, we obtain

E = − η Jf +where f(r, z, p, µ, ω, t) is the distribution function of fastelectrons, γ is the Lorentz factor, ne (or ni) is the number
density of the bulk electrons (or ions), S(r, z, p, µ, ω, t) is thesource term, µ is the directional cosine of the momentum
vector p relative to z axis and ω is the angle between theplanes formed by p and ẑ vectors and by the ẑ and r̂ vectors;ẑ and r̂ are unit vectors in the z and r directions, respectively.
The Coulomb logarithm lnΛ is extended so as to include
contributions not only from short-range binary collisions but
also from long-range collective effect [2]; i.e.

lnΛ = lnΛbinary + lnΛcollective ,lnΛ binary =⎛η⎞∂B= −∇ × ⎜ ∇ × B⎟ + ∇ × ⎛⎝η J f⎞⎠ ,∂t
⎝ µ0
⎠

(8)

(9)

where η is the plasma resistivity, and µ0 is the permeabilityof free space. Assuming rotational symmetry, we evaluate thefields Ez (r, z), Er (r, z) and Bθ (r, z).
The energy deposition rate of fast electron consists ofthe energy loss rates PB + PC due to binary collisions andcollective effect, and the Joule heating rate PJ through returncurrent, i.e. Pdep = PB + PC + PJ . These rates are evaluated by(5)PB + PC =1 ⎧⎪11⎛ τ ⎞+ ⎜⎨ln
⎟2 ⎩⎪ 2τ min 8 ⎝ τ + 1 ⎠−

η
∇×B ,
µ0

⎡⎛ Y n Y n ⎞ m 2 ∂ ⎛ 2 ⎞ ⎤ 3
γ f ⎥ dp , (10)− ∫ me c 2 (γ − 1) ⎢⎜ e e + i i ⎟ e2
mi ⎠ p ∂p ⎝ ⎠ ⎦⎣⎝ me⎫⎪ln 2 + 1 − ln 2 ⎬ , (6)(τ + 1)⎭⎪(2τ + 1)
21 ⎛ 2 v2 ⎞ 1
lnΛ collective = ln ⎜⎟−2 ⎜⎝ 3 λ D2 ωp2 ⎟⎠ 222⎞
PJ = η ⎛⎝ J Return,r + J Return, z⎠ ,

(7)

(11)

where JReturn, r (or JReturn, z) is the component of return current
density in the r (or z) direction. The return current density is
calculated bywith

τmin = (∆DB/2)/λD , τ = γ – 1 ,J Return = − Jf = − e ∫where v is the speed of the fast electron, λD is the Debye
length, ωp is the plasma frequency of the bulk electrons and∆DB is the reduced de Broglie wave length.

pf d 3p .
γ me(12)In the multi-dimensional FP calculation, the amount ofcalculations becomes enormous. Therefore, we have to select2323

Yokota T. et al., Two-Dimensional Kinetic Model for Fast Electron Transport in Compressed Core Plasmas

or develop an adequate calculation scheme, which saves amemory and a time for calculations and fulfills a good
accuracy of calculations. As for the numerical solution forthe present case, the fully implicit scheme is used for t. For
the space (r, z) and the momentum ( p) variables, the 3-D“explicit” cubic-interpolated propagation (CIP) scheme isadopted. The angular variables µ and ω are treated by meansof 2-D “implicit” interpolated differential operator (IDO)
scheme. By using the CIP-IDO scheme [5], which is basedon Hermite interpolations of the profile for the dependent
variables over a local area, accurate calculations are possiblein spite of a relatively small number of calculation-meshes,and hence the shortening of a calculation time can beexpected comparing with other schemes.
In the present calculation, the reflecting boundary
condition is applied on the beam axis (r = 0), while the freeboundary condition is used for the other spatial boundaries.
The upper and lower limits of the momentum p space aretreated by the free boundary condition. For the angularboundaries ( µ = ±1; ω = 0, π ), the reflecting boundary
condition is used. Here, we assume that the distributionfunction f on an arbitrary coordinate (z, r, p, µ) is symmetricwith respect to the r-z plane, i.e. f (z, r, p, µ, ω, t) = f (z, r, p,µ, –ω, t), and hence make calculations only for the half regionof ω (i.e. 0 ≤ ω ≤ π).

Fig. 2

Spatial profile of energy deposition rate at t = 0.4 ps. (atE0 = 1 MeV, I = 1020 W/cm3, ρ = 300 g/cm3, T = 0.5 keV)(a) with electric and magnetic fields, (b) with electricfield and without magnetic field and (c) without electricand magnetic fields.

Fig. 3

Spatial profile of the self-generated electric field at t =0.4 ps. (a) with magnetic field and (b) without magnetic
field.

Fig. 4

Spatial profile of the self-generated magnetic field at t =0.4 ps.

3. Results and discussion
To examine the energy deposition process of fastelectrons in dense plasmas, we consider a stationary D-T
cylindrical plasma ( ρ = 300 g/cm 3, T = 0.5 keV), andcalculate the transport and slowing down of mono-energetic(1 MeV) Gaussian (the half width at half maximum is 5 µm,
the peak intensity is 1020 W/cm2) relativistic electron beam
injected steadily into it. The source beam is directed forward( µ ≈ 1) and isotropic with respect to ω. In this calculation,
the size of a unit space cell is 1 µm × 1 µm. The momentum
variable p is divided into 15 meshes, and the angular variablesµ and ω are divided into 11 meshes and 16 meshes,
respectively.
Figure 2 shows the spatial profile of energy deposition
rate at t = 0.4 ps. Here, the calculations were made for thefollowing three cases, i.e. Case (a) considering the electricand magnetic fields, Case (b) neglecting the effect of magnetic
field and Case (c) neglecting the both fields. Figures 3 and 4
show the spatial profiles of self-generated electric field andmagnetic field at the same time-step, respectively. The selfgenerated magnetic field grows about 250 Tesla at maximum(Fig. 4). Because of this strong magnetic field, the beamelectrons are pinched toward the beam axis. Consequently,
the energy deposition rate and the self-generated electric field
are enhanced around the beam axis (see Fig. 2(a) and Fig.
3(a)). It is also shown that the self-generated electric fieldshortens the penetration of the beam electrons into the denseplasma. These effects would be favorable from the viewpointsof the efficient core plasma heating.

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Yokota T. et al., Two-Dimensional Kinetic Model for Fast Electron Transport in Compressed Core Plasmas

4. Concluding remarks

References

We have developed the 2-D FP-type transport code in
cylindrical geometry and calculated the energy deposition
profile of fast electron beams injected into a stationary dense
D-T plasma. The code is possible to calculate precisely thetime- and space-dependent heating rate (energy depositionrate) in dense plasma spheres. It has been confirmed that theself-generated electric field shortens the penetration of thebeam electrons into the dense plasma, and that the selfgenerated magnetic field pinches the beam electrons,enhancing the energy deposition around the beam axis.

[1] M. Tabak et al., Phys. Plasmas 1, 1626 (1994).
[2] T. Johzaki et al., Fusion Sci. Technol. 43, 428 (2003); Y.
Nakao et al., Proc. of 19th IAEA Fusion Energy Conf.
(Lyon, 2002), IAEA-CSP-19/CD, IF/P-08 (2003).
[3] B.J. Braams and C.F.F. Karney, Phys. Fluids B1, 7(1989).
[4] K. Nakashima and H. Takabe, Phys. Plasmas 9, 1505
(2002).
[5] for example, T. Aoki et. al., Comput. Phys. Comm. 102,132 (1997).

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