Simulation on interactions of X-ray and charged particles with first wall for IFE reactor

Fusion Engineering and Design 73 (2005) 95–103

Simulation on interactions of X-ray and charged particles with first wall for IFE reactor Hiroyuki Furukawa a,∗ , Yasuji Kozaki b , Keiji Yamamoto b , Tomoyuki Johzaki b , Kunioki Mima b a b

Institute for Laser Technology, Osaka University, 2-6 Yamada-oka, Suita, Osaka 565-0871, Japan Institute of Laser Engineering, Osaka University, 2-6 Yamada-oka, Suita, Osaka 565-0871, Japan Received 14 July 2004; received in revised form 3 December 2004; accepted 1 February 2005 Available online 25 May 2005

Abstract We have developed a simulation code to treat behaviors of wall ablation by X-rays, ␣ particles and charged particles from fusion burning targets. The absorption of energy of ␣ particles and charged particles by ablated wall plasmas was estimated using the stopping power obtained by dielectric functions of plasmas and Bethe formulas. We show that ablated wall material (Pb) moves as clump. We clarify that the screening effects of charged particles expanding from fusion burning by ablated wall material (Pb) decrease the ablation depth of liquid Pb compared with that of no screening effects. © 2005 Elsevier B.V. All rights reserved. Keywords: Simulation; Ablation; First wall; IFE reactor; Screening effects

1. Introduction Design studies of IFE reactor plants show that it is necessary to achieve pulse repetition rate of more than 3–10 Hz for economically attractive power plants. This pulse repetition rate is limited by the time of restoring chamber environment, as the residual material at the time of the next target injection may disturb the trajectory of the injected target and the laser beam propaga∗ Corresponding author. Tel.: +81 6 6879 8739; fax: +81 6 6878 1568. E-mail address: [email protected] (H. Furukawa).

0920-3796/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fusengdes.2005.02.001

tion. Then the pulse repetition rate of the IFE reactor chamber concept is limited by evacuation speed of materials that is ablated from first wall with the X-rays, ␣ particles and charged particles produced by nuclear reactions and heating target materials produced by fusion burning. It is very important to analyze interactions of X-rays, ␣ particles and charged particles with the first wall in order to study evacuation processes of ablated materials and to show that the condition ablated materials not to disturb target injection and laser irradiation. In this study, the material of the first wall of IFE reactor chamber is set to liquid Pb. We obtained profiles and

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energy spectra of X-rays, ␣ particles and charged particles using high-gain target implosion simulation [1,2]. We have developed a simulation code on ablation by X-rays, ␣ particles and charged particles with the consideration of absorption of energy of charged particles by ablated materials (plasmas) [3–7]. The absorption of energy of charged particles by ablated plasmas, in this paper we define ‘the screening effects’, is estimated using the stopping power obtained by dielectric functions of plasmas [8] and Bethe formulas [9,10]. We clarify that the screening effects is very important effect on the estimation of the ablation depth of liquid Pb. In Section 2, profiles and energy spectra of X-rays, ␣ particles and charged particles obtained by high-gain target implosion simulation are mentioned. In Section 3, basic equations of simulation code on interactions of X-ray and charged particles with liquid metals are described. In Section 4, results and discussions are mentioned. Fig. 1. The high-gain target design.

2. Profiles and energy spectra of X-rays, ␣ particles and charged particles We obtained profiles and energy spectra of X-rays, ␣ particles and charged particles using high-gain target implosion simulation [1,2]. Fig. 1 shows the high-gain target design [11,12] used implosion simulation. Here, CH density is 1.0 g/cm3 , DT density is 0.213 g/cm3 and DT gas pressure is 1 atm. Fig. 2 shows laser pulse shape used high-gain target implosion simulation. The wavelength of laser is 0.35 ␮m, total irradiated energy is 3.0 MJ, prepulse duration is 20 ns, maximum intensity of prepulse is 1013 W/cm2 , energy of prepulse is 80 kJ and mainpulse duration time is 14 ns, maximum intensity of mainpulse is 8 × 1014 W/cm2 , energy of mainpulse is 2.92 MJ (3 MJ to 80 kJ). The simulation is performed by spherically one-dimensional Lagrangian radiation hydro code ILESTA [1] coupled with Fokker–Plank code [2]. In this code, the fluid dynamics is treated by one fluid and two temperature approximations, and the transport of radiation is treated by the flux-limited multi-group diffusion model, and the transport of ␣ particles is obtained by Fokker–Plank equation [2]. Neutrons are free escaped. Fig. 3 shows the plasma spatial profile obtained by implosion simulation at the time 54 ns. In Fig. 3, ρ(z, t) is mass density, v(z, t) is velocity. Note that

when laser pulse is irradiated on the target surface, we set time equal to 0. Fig. 4 shows spectra of X-ray obtained by implosion simulation using flux-limited multi-group diffusion approximation. Using this profile, we calculate profiles and spectra of charged particles on the wall. In this calculation, we assume that: (1) carbons and hydrogens are equal velocities from time = 54 ns to the time reach to the wall. Deuteriums and tritiums are equal velocities; (2) between ␣ particles, carbons, hydrogens, deuteriums and tritiums,

Fig. 2. The laser pulse shape used high-gain target implosion simulation.

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Fig. 3. The plasma spatial profile obtained by implosion simulation.

there are no interactions each other from time = 54 ns to the time reach to the wall; (3) energies are distributed according to mass ratio. Fig. 5 shows spectra of ␣ particles and charged particles under assumptions mentioned above. Fig. 6 shows profiles of X-ray, ␣ particles and charged particles on the wall under assumptions mentioned above. Fig. 6(a) shows those for the case of chamber radius is 2 m, (b) shows for the case of chamber radius is 4 m and (c) shows for the case of chamber radius is 8 m. Table 1 shows the irradiated energy composition on the wall. Note the fact that the wall is irradiated by high energy ␣ particles is especially important for designs of laser fusion reactors.

Fig. 5. Spectra of ␣ particles and charged particles.

3. Newly developed simulation code ACORE (Ablation COde for REactor) In order to estimate temperature increases and ablation processes of liquid materials due to X-ray, ␣ particles and charged particles irradiation, we have developed an ablation simulation code. In this paper, this code is named ‘ACORE (Ablation COde for REactor)’. The absorption of energy of charged particles by ablated wall plasmas is estimated using the stopping power obtained by dielectric functions of plasmas and Bethe formulas. We set the geometry of the problem is planar one-dimensional for simplicity. Fig. 7 shows the schematic diagram of ACORE. Basic equations of ACORE are as follows [3–8].

Table 1 The irradiated energy composition on the wall

Fig. 4. Spectra of X-ray obtained by implosion simulation.

X-ray ␣ Particle Carbon Hydrogen Deuterium Tritium

Particle number

Peak energy

Energy (MJ)

2.95 × 1019 1.66 × 1020 1.66 × 1020 2.93 × 1020 2.93 × 1020

20 keV 2.70 MeV 1.08 MeV 0.09 MeV 0.12 MeV 0.18 MeV

3.8 10.6 37.0 3.1 8.1 12.1

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Fig. 6. Profiles of X-ray, ␣ particles and charged particles on the wall. For the case of chamber radius is: (a) 2 m; (b) 4 m; (c) 8 m.

3.1. Energy deposition of X-rays by liquid materials Total intensity of X-rays is defined as the integral on the energy:
∞ I(z, t) = I(z, ν, t) dν, (1) 0

where z is depth into material along by axis z, t is time and ν is the X-ray frequency.

Each intensity is related to absorption coefficient α: d I(z, ν, t) = −α(ν)I(z, ν, t). dz

(2)

Solution of Eq. (2) is: I(z, ν, t) = I(zv , ν, t) exp{−α(ν) × (z − zv )} where zv the vaporization front.

(3)

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Fig. 7. The schematic diagram of ACORE.

Heat quantity Qx is obtained as follows: Qx (z, t) = −

d I(z, t) = dz




∞

α(ν)I(z, ν, t) dν.

3.3. Equation of energy in liquid, U (4)

0

3.2. Energy deposition of charged particles Kinetic energy of charged particles Ek is related to stopping power W as follows: ∂Ek (z, t) = −W(Ek (z, t), ρ(z, t), T (z, t)) ∂z

(5)

where T(z, t) is temperature. Heat quantity Qp is obtained as follows:
Qp (z, t) = − 0

∞

Nk ∂Ek (z, t) dk, S ∂z

(6)

where Nk is the particle number of which energy is Ek , and S is the irradiated area.

kB ∂U(z, t) = ∂t 2mc(z, t)



∂ ∂z



∂T (z, t) κ ∂z  + Qx (z, t) + Qp (z, t) ,



zv < z < ∞

(7)

U(z, t) ≡ 21 nkB T (z, t) + ∆(z, t)

(8)

where kB is the Boltzmann constant, m is the mass of an atom, c the specific heat, κ the thermal conductivity, n is the number density of material and ∆ is internal energy of liquid. The critical value of U, namely Ucrt is determined as follows:   1 Lv Ucrt (z, t) = n(z, t)kB Tvap + (9) 2 c(z, t) where Tvap is vaporiztion temperature. If U > Ucrt , ablation will occur.

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Boundary condition of equation of thermal conduction is as follows:  ∂ ∂T (z, t)  Lv Zv (t) = κ (10) ∂t ∂z  z=zv

where Lv the latent heat of vaporization. 3.4. Equation of motion of vaporization front

W(Ek (z, t), ρ(z, t), T (z, t)) =

(11)

P0 in Eq. (11) is related to vaporization pressure Pvap as Clausius relation:   −Lv = nvap kB Tv Pvap = P0 exp (12) kB T v where nvap is number density of vapor. 3.5. Equation of fluid dynamics of plasmas Ablated plasmas behave according to fluid equations: ∂ρ(z, t) ∂ρ(z, t)v(z, t) + = 0, (13) ∂t ∂z  ρ(z, t)

∂v(z, t) ∂v(z, t) + v(z, t) ∂t ∂z

 =−

∂ P(z, t), ∂z (14)

∂ ∂ε(z, t) + {ε(z, t)v(z, t) + P(z, t)v(z, t)} ∂t ∂z    kB ∂ ∂T (z, t) = κ 2mc(z, t) ∂z ∂z  + Qx (z, t) + Qp (z, t) ,

(15)

P(z, t) = Pe (z, t) + Pi (z, t),

(16)

Pe (z, t) = ne (z, t)kB T (z, t),

(17)

Pi (z, t) = ni (z, t)kB T (z, t),

(18)

ε(z, t) =

1 2 2 ρ(z, t)v(z, t)

+ U(z, t)

3.6. Absorption of energy of charged particles by plasmas The stopping power of ablated plasmas is [8]:

Zv moves according to Eq. (11):    ∂ P0 −Lv kB T v Zv (t) = , exp ∂t nkB Tv m kB T v Tv = T (zv , t).

where ρ(z, t) is mass density, v(z, t) is velocity and ε(z, t) is total energy of plasmas. We set the ionization degree of ablated plasmas is unity for simplicity.

(19)

4πne Z02 e4 L. m e v2

(20)

For the case of slow velocities, the stopping number L is estimated using the dielectric function of plasmas:
1 me v kv 1 L=− dk 2 Im (21) 3 n e2 k ε(k, kv) (2π) e where ε the dielectric function of plasmas, Z0 the electric charge of charged particles, v = (2Ek /m)1/2 the velocity of incident charged particles, ne is electron number density, k is wave number and me is the mass of an electron. For the case of fast velocities, the stopping number L is estimated by Bethe formulas:   1.123 × me v2 L = ln (22) hωpe /2π where ωpe is electron plasma frequency.

4. Results and discussions We use X-ray absorption coefficients of liquid Pb in [13], and stopping power in [14]. We estimate stopping power of Pb plasmas for hydrogen, ␣ particle and carbon using Eqs. (20)–(22). For example, Fig. 8 shows stopping power of liquid Pb and Pb plasmas in which plasma density is solid density, temperature 1 eV and average ionization degree is unity. Fig. 8(a) shows that for carbon, (b) for hydrogen and (c) for ␣ particle. We make a table of stopping powers as a function of plasma density and temperature to use in ACORE. Note that the stopping power due to bound electrons is neglected, and we assume the ionization degree equals unity for simplicity. In future work, we would like to estimate the stopping power due to bound electrons, and the ionization degree as a function of plasma density and plasma temperature.

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Fig. 9. Energy depositions by X-rays and ␣ particles.

Fig. 8. Stopping power of liquid Pb and Pb plasmas for: (a) carbon; (b) hydrogen; (c) ␣ particle.

We obtain profiles and energy spectra of X-rays, ␣ particles and charged particles in Section 2. Fig. 9 shows energy depositions by X-rays and ␣ particles of which profiles and energy spectra obtained in Section 2 for the cases of chamber radius 2, 4 and 8 m. Note that in this case, energy depositions by X-rays are very small compared with those by ␣ particles because X-

ray absorption coefficients of liquid Pb are very small [13]. Due to Fig. 9, we can estimate the stopping range of ␣ particles is about 10 ␮m. Therefore, behaviors of ␣ particles determine the ablation depth of liquid Pb for the case radius 2 m. In this way, the scenario of design of laser fusion reactor chamber is very affected by behaviors of ␣ particles. For the case radius 2 m, energies of charged particles, which reach on the wall Pb after ␣ particles, are almost absorbed by ablated Pb plasmas by ␣ particles, not wall Pb. This is ‘screening effect’. This fact is cleared up by this research for the first time. Profiles and energy spectra mentioned above are putted into ACORE to simulate ablation processes of liquid Pb in the first wall of reactor chamber. The radius of the wall is set to be 4 m, and initial temperature of the Pb wall is set to 823 K. Fig. 10 shows the number density profiles ablated Pb at the time ablation starts (after the peak time of ␣ intensity). Note that the time is that from laser irradiated on target. The horizontal axis stands for the distance, and the vertical axis stands for the number density of ablated plasmas. Note that initially the surface of liquid exists at z = 0. Because of very large gradient of pressure of ablated plasmas, plasmas begin to blow with very fast velocities and the number densities decrease suddenly within 0.5 ns. Fig. 11(a) shows the number density profile and the temperature profile of ablated Pb at the time 1382.2 ns (the time intensity of charged particles is reduced) and (b) the same at 2982.1 ns (the time of passed the pulse

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Fig. 10. Number density profiles at the time ablation starts.

of charged particles). As shown in Fig. 11, ablated plasmas move as clumps. Note that this fact is very important for the chamber design, especially the analysis of evacuation in chamber. Fig. 12 shows the ablation depth with plasma screening effects and that without screening effects. Note that the ablation depth by ␣ particles is negligible small for

Fig. 12. Ablation depth with plasma screening effects and that without screening effects.

4 m chamber radius case. As shown in Fig. 12, absorption of energy by ablated plasmas is very important for estimation of ablation depth. For 4 m chamber radius case, after evaporation (ablation) starts, large parts of charged particles are absorbed by ablated plasma (Pb), and small parts of charged particles are absorbed by Pb wall. This is ‘screening effect’. For 4 m chamber radius case, ablation depth is very affected by behaviors of carbon particles. 5. Concluding remarks We have developed an ablation simulation code named ACORE. We show that ablated wall material (Pb) moves as clumps. We clarify that the screening effects of energies of charged particles expanding from directly driven targets by ablated wall material (Pb) decrease the ablation depth of liquid Pb compared with that of no screening effects. In future work, we would like to estimate the stopping power due to bound electrons, and the ionization degree as a function of plasma density and plasma temperature. For the calculation of ionization degree, there are many complicated problems such as equation of state [15], etc. And we would like to add radiation transport analysis [15] in ACORE. References

Fig. 11. Number density profile and temperature profile at the time: (a) 1382.2 ns; (b) 2982.1 ns.

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