INVESTIGATION OF THE DEGENERACY EFFECT IN FAST IGNITION FOR HETEROGENEOUS FUEL

Transcription

1 INVESTIGATION OF THE DEGENERACY EFFECT IN FAST IGNITION FOR HETEROGENEOUS FUEL M. MAHDAVI 1, B. KALEJI 1 Sciences Faculty, Department of Physics, University of Mazandaran P. O. Box , Babolsar, m.mahdavi@umz.ac.ir Received May 0, 013 In a fast ignition regime, with the conditions of high-density and low-temperature, the degenerate plasma can be obtained during the compression phase. In this paper, the effect of degeneracy in the energy transmitted from ions to electron collisions and in the radiation emissions from electrons is studied. The fuel ignition, propagating burn and the target energy gain for a typical fuel are calculated using the governing equations for ions and electrons in two-nodes model, the hot spot with D/T fuel in node 1 and the cold fuel surrounding ignitor with D/ 3 He in node by evaluating the degeneracy effect. In the obtained extremum ignition conditions, the target gain, minimum ignition temperature and the areal density of the hot spot are obtained as 145, 5 kev and 5.4 g/cm, respectively. Key words: Fast ignition; Degenerate plasma; Bremsstrahlung losses; Fermi-Dirac distribution; Heterogeneous fuel. 1. INTRODUCTION In inertial confinement fusion (ICF), the pellet of fuel is compressed to a high density by symmetric spherical implosion of a thin spherical shell by radiation. The ignition condition is obtained in a high density and high temperature hot spot that ignites the burning of the remaining cold fuel. The D/T fuel is the main fuel in ICF due to the high reaction rate at a relatively low temperature. Because of the production of the energetic neutrons in D/T fusion reaction, the advanced fusion fuels are considered. These fuels, such as D/ 3 He and P/ 11 B produce few or no neutrons, but they require a higher temperature for ignition than the D/T mixture, and the Bremsstrahlung losses in this regime might be greater than the fusion power, which makes self-burning of advanced fuels unlikely [1]. To overcome the Bremsstrahlung radiation, the electron temperature must be lower than the ion temperature. If the energy transmitted from ions to electrons is minimized, the loss of energy by Bremsstrahlung radiation will also be minimized. In this condition, the temperature of electrons is lower than the ions' temperature. At high density, Rom. Journ. Phys., Vol. 59, Nos. 1, P , Bucharest, 014

2 Investigation of the Degeneracy Effect in Fast Ignition 107 when the electron temperature is lower than the Fermi temperature, 3 KT F = / m(3 π n e), the plasma will be degenerate. In a degenerate plasma, the Pauli exclusion principle prevents some transitions of radiation. So, the most part of the relevant energy transfers' mechanisms involving electrons are affected and the ignition temperature will be decreased. The energy gain in a degenerate plasma is smaller than in the classical cases, because the high compressed energy is needed []. P/ 11 B and D/ 3 He fuels have to be very highly compressed to reduce ignition temperature, but the target gain remains negligible [3, 4]. However, the degeneracy is a necessary condition for having the burning wave in advanced fuel, while it is an advantage for D/T fuel. To increase the target gain, the fast ignition scheme has been proposed by the development of Peta watt laser technology [5]. In the fast ignition scheme, at first, the fuel core plasma is compressed to high-density 3 ( ρ 300 g / cm ) by spherical implosion with a laser intensity of about Wcm - [6]. Then, in near maximum density, the ultra intense Pico second laser pulse (~ 10 0 W cm - ) is penetrated in plasma through a channel; for example a cone of gold. The transferred energy to energetic electrons will be deposited to a small region of fuel which is called the hot spot. Finally, the hot spot starts to ignite and the burning wave propagates to the cold fuel surrounding the hot spot [7]. In this paper, the final stage is studied. This scheme presents advantages because the compression and ignition processes are separated, and there is no need for the formation of the hot spot in the first phase. So, the energy needed for compression is minimized and the hydrodynamic instabilities are prevented, and it may also produce a higher gain with less driver energy [8]. In this study, the heterogeneous fuel is considered for two nodes in the fast ignition regime. D/T fuel is chosen in node 1 to start burning in low temperature and D/ 3 He fuel is chosen in node (a large region of fuel) to reduce the neutron content in fusion medium. The ignition temperature for D/ 3 He, is high [9], Whereas with the burning of fuel in node 1, the ignition condition for node can be prepared well. In this study, the Thomas-Fermi and the ideal gas models are employed for the electron and the ion equations of state, respectively [10]. The electron thermal conduction, radiation effect (Bremsstrahlung, Compton scattering and inverse Compton scattering), ionelectron collision, fusion product heating and the external fast electron heating are considered in the energy conservation equations. Finally, the ignition conditions and the target gain will be calculated considering the appropriate distribution function for ions and electrons in the energy conservation equations.. PHYSICAL MODEL In this model a heterogeneous fuel is considered in two nodes for the fast ignition scheme with deuterium-tritium fuel in the first node and deuterium-helium in the second one (Fig. 1). A small portion of the compressed target, node 1, is heated by a PW laser to start the burning of fuel as an ignitor. The second node

3 108 M. Mahdavi, B. Kaleji 3 defines the volume surrounded ignitor. The fusion burning wave is launched from the ignitor to the rest of the precompressed plasma, node. The governing equations for ions and electrons in the ignitor and the volume surrounded ignitor can be written as; de e 1 1 ( ) 1 = f η Pf +η dpign + Pie PbV Pc Pcs Phe Pme (1) dt V V de i 1 1 ( ) 1 = f (1 η ) Pf + (1 η d) Pign Pie PbV Pc Pmi () dt V V de e 1 1 ( ) = (1 f) η Pf + Pie PbV Pc Pcs Phe Pme + dt V V (3) ' [ fpv+ P + P + P + P ] b c cs he me ignitor de i 1 1 ( ) = (1 f)(1 η) Pf Pie PbV Pc Pmi + dt V V ' + [ fpv+ P + P ] b c mi ignitor (4) The energy of electrons, E e, in a degenerate plasma is defined by [11]: E 3 5π T e = N T T F e e F P f, is the power of fusing D/T particles according to the f and η parameters. f is the fraction of the fusion energy that is deposited to node 1 and (1-f) fraction of the fusion energy is deposited to node which depends on the node sizes and the alpha particle range and, is the fraction of the fusion energy which goes to the electrons and (1-η), is the fraction of the energy that goes to ions which depends on temperature and density []. P ign, is the energy of the external ion beam that is deposited partially to electrons and partially to ions according to the η d parameter. P b, P c, P cs, P ie, P he, P m, are the Bremsstrahlung radiation, inverse Compton scattering, Compton scattering, ion-electron collision power, electron heat conduction and the mechanical expansion losses (see appendix A), respectively. The total number of particles of species k, N k, are governed by the equation; (5) dn dt k 6 j 1 = a k N j() N 1 j( ) σv (6) j V j= 1 Here, V is the volume of the heated plasma, < σν > j is the Maxwell averaged j reaction rate of the reaction j and a k k is the number of particles of species k(d, T,

4 4 Investigation of the Degeneracy Effect in Fast Ignition 109 P, n, 3 He, 4 He) created or destroyed in the reaction j (appendix B). The subscripts 1 and at N j(1) and N j() refer to the first and second reacting nuclei, respectively. If the electron temperature is lower than the Fermi-temperature, (7.1 kev), the electrons of the plasma are degenerate. In this case, the electrons' distribution function is Fermi-Dirac. Then, the specific emissivity is given by [1], b max 3 6 π e mi m ( ) ( ) 3 e Z 3 ion f e ve f i vi e i 3me c π π v b b 56 j( ν ) = ( ) ( ) min d v d v db (7) b, v are the impact parameter and relative velocity of ion and electron, respectively. The distribution ions and electrons obey the Maxwell-Boltzmann and the Fermi- Dirac distribution, respectively. By integrating over ν, the Bremsstrahlung radiation power is calculated as; 3 b ( / ) ( ) 0 P kev cm s = j ν dν (8) One of important radiative process that generates high energy photons, is the inverse Compton scattering. In this process, relativistic electrons transfer part of their kinetic energy to low energy photons there by creating high energy photons. The energy losses by inverse Compton scattering at high electron temperatures are given by, ( ) ( ) 8 T e kev T r kev PC ( kev s) = 4E rn ec r π e where 8/3π r e is the Thomson cross-section and r e, is the classical electron radius. E r, is the total radiation energy due to the Bremsstrahlung radiation. One of the parameters that is determined by the radiation processes in plasma, is optical depth [13]. The Bremsstrahlung optical depth is defined as ratio of the radiation energy density of hot photons of a black body to the Bremsstrahlung radiation energy density. When the optical depth is larger than 1, the plasma is optically thick and the black body radiation spectrum dominates the number of hot photons. Whereas if the optical depth is smaller than 1, the plasma is optically thin and the number of hot photons in the Bremsstrahlung spectrum is larger than in the black body radiation spectrum. For the optically thick plasma, the mean free path of the Bremsstrahlung photons is smaller than the plasma radius and the radiation can be treated as a black body radiation. In this case, the inverse Compton effect can significantly reduce the electron temperature. (9)

5 110 M. Mahdavi, B. Kaleji 5 3. BURNING WAVE IN FUSION PLASMA It is necessary for ignition that the energy deposited to plasma be larger than radiation losses (due to Bremsstrahlung radiation, Compton scattering, inverse Compton scattering, heat conduction and ion-electron collision), (P dep -P loss 0). The initial conditions are calculated by solving this equation in t=0 for D/T plasma in node 1. The extremum initial physical conditions are obtained in T ig = 5 kev, ρr = 5.4 g/cm for ρ = 10 4 g/cm 3 and the ratio of tritium to deuterium, x = 0.5 (Fig. ). It can be seen that with increasing in tritium content, the ignition temperature will be reduced, because the reaction rate of D/T in low temperature is high. We can compare these ignition conditions to ignition conditions in Ref. [14], that T ig =50 kev and ρr=4 g/cm in D/ 3 He/ 11 B fuel. With chance of heterogeneous fuel as D/T in node 1 and D/ 3 He, the ignition condition is much lower. In Fig. 3, the initial condition is compared in degenerate and classical plasma. Degeneracy causes reducing the ignition temperature. In lower density, ~ 100 g/cm 3, the initial temperature and confinement parameter are gained at T ig = 9.7 kev and ρr =7.7 g/cm. In physical conditions, the electron temperature, 3 kev, is lower than the Fermi energy, 7.1 kev. In this case, the plasma is degenerate. This means that the heating mechanisms deposit the energy to the ions in the plasma, not to the electrons. So, the ion temperatures are much higher than the electron temperatures in a target. In Fig. 4, the ion and electron temperatures are shown during the burning of fuel. In t = 1.08 ps the energy that goes to node from node 1, provided the ignition condition in node. After this time, the whole fuel is burning. Also in Fig. 4, the electron temperature plotted from the beginning to t = 1.08 ps. It can be seen that in this time, the electron temperature remains smaller than the Fermi temperature and plasma is degenerate during the burning in node 1. The difference between ion and the electron temperature remains high for a long time to develop the fusion burning wave. Then, the energy from collisional phenomena goes from the ions to the electrons. When electrons are degenerate, the energy exchange term between ions and electrons and all the radiation phenomena in plasma is different from the classical one due to the Pauli exclusion principle [15]. The ion-electron energy transition power has been compared in degenerate and classical plasma (Fig. 5). When plasma is degenerate, the Pauli exclusion principle causes a reduction in collision. The different lost powers during times are compared in simulated fuel pellet in Fig. 6. It is shown that the Bremsstrahlung radiation is the second time energy losing parameter. In Fig. 7, the number of photons obtained from the Bremsstrahlung radiation spectrum and the black body radiation spectrum, as a function of the electron temperature, are compared. First, at temperatures lower than 4 kev, the plasma is thick and, the distribution of photons can be considered the Planck distribution. But in T e > 4 kev, the plasma will be thin and, the radiation distribution can be considered as the Bremsstrahlung

6 6 Investigation of the Degeneracy Effect in Fast Ignition 111 radiation spectrum. In other words, with increasing the electron temperature about, 4 kev, at a time of about 4 ps, the Bremsstrahlung radiation is maximum and, the plasma will slowly become thin. 4. TARGET GAIN The main parameter in inertial confinement fusion that must be calculated is the target gain. The target gain is defined as the ratio of the fusion energy to the total driver energy including the compression energy for the main fuel, E com, and the ignition energy, E ign [16]. be fusion G = (10) E + E That comp ign b = - ρr(gcm ) R(gcm - ρ ) + 6 (11) The compression energy is given by; E com (kj) = 330 M /3 α cρ (1) where M c (g) and α are the mass of the fuel and the isentrope parameter (α = P C /P deg =5T/T F ). In Fig. 8, the energy gain of the target is plotted versus tritium to deuterium ratio for different density values in node 1. So the optimum ratio for the best value for the gain can be found. The maximum energy gain is 145 in these physical conditions. It is shown that when the density of fuel increases, the energy gain decreases. This is because the energy for the comparison of fuel is high. Also, in Fig. 9, the target gain versus the helium to deuterium ratio is calculated in node. The maximum gain is finally obtained in y = CONCLUSIONS In the degenerate plasma, the loss of energy would be minimized due to the Pauli's exclusion principle. The ignition temperature decreases when the compressed plasma density is increased. In addition, the target fusion energy gain, especially in aneutronic advanced fuel, also decreases with the increasing of the density. In fast ignition schemes, the energy driver is minimized since the driver energy will be deposited to a small part of the fuel (hot spot) and it isn't necessary to ignite all of the fuel at first. These conditions are necessary to get fusion in advanced fuel. For the heterogeneous fuel in the fast ignition regime, the D/T

7 11 M. Mahdavi, B. Kaleji 7 plasma is considered in the hot spot (node 1) for reducing ignition temperature, and the advanced fuel, D/ 3 He, is considered in node, for limiting the increasing tritium content in medium that forms most of the fuel masses. Considering the ignition condition for the D/T plasma in node 1, the initial temperature, confinement parameter, density number of electrons, the radius of hat spot, the speed of sound and the target gain are calculated as; 5 kev, 5.4 g/cm, ~ 10 8 cm -3, 5 µm, ~ 10 8 cm/s and 145 respectively. Radiation Process APPENDIX A Compton Scattering The photons coming from Bremsstrahlung radiation is scattered by Compton scattering and the photon energy is decreased. The Compton scattering power is calculated by [17]: the effective Compton cross-section, function is given by: E P (kev/s) = n σ c E (A1) r cs e C, eff r m ec σ C, eff, depends on the electron distribution σ 1 C, eff = C(T) σ C ( εγ (1- β cos θ ))(1 - β cos θ ) p dpd E(p)- Ω (A) µ 1 + exp( ) kt C(T) is the normalization constant of the Fermi-Dirac distribution function. Ion-Electron Collision The energy loss of the particle per unit length in degenerate plasma is given by []; where L is 6 4 dk 4πZ e = n el dl m v e f (u, z) L = v/v F udu 3 i π z dz (z + χ f (u,z)) + f (u,z) r χ i (A3) (A4)

8 8 Investigation of the Degeneracy Effect in Fast Ignition 113 f r (u,z) and f i (u,z), are the real part and the imaginary part of dielectric function. z =k/k F, u = w + iγħ/kν F, χ = e /πħν F and ν F (k F ) is the Fermi velocity (wave vector). The frequency of ion-electron collision is defined as; (dk/dl) ν e,i = ν (A5) K Finally, the electron-ion collision power is obtained by: Heat Conduction 3 3 P ie (kevcm /s) = e,i N iti ν (A6) In the fusion plasma, the fusion burning wave is preceded by the heat conduction of electrons and the energy of the hot spot is transported to colder fuel causing the ablation of the solid. The electron heat conduction is not inhibited, but this lost process remains negligible. The heat conduction is calculated according to the Spitzer formulation [15]; P (kev/s) = he 13 T 7 0 r0 ln Λ (A7) ln Λ, r 0 and T 0, are the Coulomb logarithm, spark radius and the maximum temperature of electron. Mechanical Expansion The other process that it causes losing the energy of electrons and ions, is the mechanical expansion. The burning wave is propagated by the speed of sound. The speed of sound in a degenerate plasma is; nt i F C s = ρ Then the expansion work for ion and electrons is obtained by: P mi,e (kev/s) = N i,eti,e 4 R (t)c s V (A8) 1 π (A9) Nuclear Reactions Appendix B The most nuclear fusion reactions with considering the released energy of reaction and their reaction rate, are considered following;

9 114 M. Mahdavi, B. Kaleji 9 D + D 3 He + n MeV D + D P + T MeV D + T 4 He + n MeV D + 3 He 4 He + P MeV (B1) (B) (B3) (B4) 6. FIGURES Fig. 1 The assembly of fuel with D/T fuel in node 1 and D/ 3 He fuel in node. Fig. Ignition condition for D/T fuel in node 1 for different ratios of tritium to deuterium.

10 10 Investigation of the Degeneracy Effect in Fast Ignition 115 Fig. 3 Ignition condition for D/T fuel for different densities in degenerate ρ=10 4 g/cm 3 (dashed line) and classical plasma ρ = 10 g/cm 3 (solid line). Fig. 4 Time evolution of the ion and electron temperatures.

11 116 M. Mahdavi, B. Kaleji 11 Fig. 5 The ion-electron collision term in degenerate and classical plasma. Fig. 6 The lost powers during time in fuel pellet.

12 1 Investigation of the Degeneracy Effect in Fast Ignition 117 Fig. 7 Optical depth of plasma, τ B, as a function of the electron temperature. Fig. 8 Target gain versus ratio of tritium to deuterium particle numbers in the node 1.

13 118 M. Mahdavi, B. Kaleji 13 Fig. 9 Target gain versus ratio of helium to deuterium particle numbers in node. REFERENCES 1. S. Son, N. J. Fisch, Phys. Letters A. 39, 76 (004).. S. Son, N. J. Fisch, Phys. Letters A. 356, 65 (006). 3. M. Mahdavi, F. Khodadadi, J. Fusion Energ. 31, (01). 4. M. Mahdavi, F. Khodadadi, Eur. Phys. J. D. 66, 0 (01). 5. M. Tabak, Phys. of Plasmas. 1, 166 (1994). 6. S. Atzeni, Plasma Phys. Control. Fusion. 51, 1409 (009). 7. C. Benedetti, P. Londrillo and et al., Nuclear Instruments and Methods in Physics Research A. 606, 89 (009). 8. S. Atzeni, A. Schiavi and et al., Phys. of Plasmas. 15, (008). 9. M. Mahdavi, F. Khodadadi, J. Fusion. Energ. 31, (01). 10. P. T. Leon, Sh. Eliezer, J. M. Martinez-Val, Phys. Letters A. 343, 181 (005). 11. S. Atzani, J. Meyer-Ter-Vehn, Inertial Fusion, AMS & Clarendon Press Oxford M. Mahdavi, B. Kaleji, T. Koohrokhi, Mod. Phys. Letters B (010). 13. M. Mahdavi, B. Kaleji, Plasma Phys. Control Fusion (009). 14. M. Mahdavi, S. Rohaninejad, J. Fusion. Energ (01). 15. J. M. Martinez-Vala, S. Eliezer, Nucl. Fusion (1998). 16. J. Badziak, S. Jablonski, J. Wolowski, Plasma Phys. Control. Fusion. 49 B651 (007). 17. M. Mahdavi, B. Kaleji, Mod. Phys. Lett. A (011).