1. INTRODUCTION
One of the approaches to solve the energy problem is the well-known inertial confinement fusion (ICF) driven by high power lasers. The physics of ICF is based on compressing and igniting rather than confining the fuel (Nuckolls et al., Reference Nuckolls, Wood, Thiessen and Zimmermann1972; Atzeni & Meyer-Ter-Vehn, Reference Atzeni and Meyer-Ter-Vehn2004; Velarde & Carpintero-Santamaria, Reference Velarde and Carpintero-Santamaria2007). In order to ignite the fuel with less energy it was suggested to separate the drivers that compress and ignite the target (Basov et al., Reference Basov, Guskov and Feoktistov1992; Tabak et al., Reference Tabak, Hammer, Glinsky, Kruer, Wilks, Woodworth, Campbell, Perry and Mason1994). First the fuel is compressed to high density, then a second short pulse driver heats and ignites a small part of the fuel, the “hot spot” or “igniter”, while the α-particles created in the nuclear interaction heat and burn the rest of the target. This idea is called fast ignition. The fast ignition problem is that the short laser pulse does not penetrate directly into the compressed target; therefore many schemes have been suggested (Guskov, Reference Guskov2013) to solve this issue. Presently from all the known fast ignition schemes the simplest fast ignition seems to be an “extra shock” wave (Betti et al., Reference Betti, Zhou, Anderson, Perkins, Theobald and Sokolov2007; Eliezer & Martinez Val, Reference Eliezer and Martinez Val2011).
We suggested recently a novel shock wave ignition scheme (Eliezer et al., Reference Eliezer, Nissim, Pinhasi, Raicher and Martinez Val2014a) where the ignition shock wave is generated in a pre-compressed target by the pondermotive force of a high irradiance laser pulse. The shock wave velocity in this scheme is in the intermediate domain between the relativistic and non-relativistic hydrodynamics. Here, this fast ignition scheme is further developed and analyzed for deuterium-tritium (DT) and proton–boron11 (pB11) fuels.
The interaction of a high power laser with a planar target creates a one-dimensional (1D) shock wave (Fortov & Lomonosov, Reference Fortov and Lomonosov2010; Eliezer, Reference Eliezer, McKenna, Neely, Bingham and Jaroszynski2013). The theoretical basis for laser induced shock waves analyzed and measured experimentally so far is based on plasma ablation. For laser intensities 1012 W/cm2 < I L < 1016 W/cm2 and nanoseconds pulse duration a hot plasma is created. This plasma exerts a high pressure on the surrounding material, leading to the formation of an intense shock wave moving into the interior of the target (Eliezer, Reference Eliezer2002). In this paper we are interested in laser irradiances I L > 1021 W/cm2. Shock waves induced by lasers with irradiances in this regime are described by relativistic hydrodynamics (Landau & Lifshitz, Reference Landau and Lifshitz1987). Relativistic shock waves were first analyzed by Taub (Reference Taub1948). Relativistic shocks may be a new route for fast ignition (Eliezer, Reference Eliezer2012) and these shocks may be of importance in intense stellar explosions or in collisions of extremely high energy nuclear particles.
The shock wave created in a 1D target by the ponderomotive forceFootnote 1 induced by very high laser irradiance, considered in this paper, is summarized schematically in Figure 1. In this domain of laser intensities the pondermotive force accelerates the electrons forward, so that the charge separation field forms a double layer (DL), in which the ions are accelerated forward. Figure 1a displays the capacitor model for laser irradiances I L, where the ponderomotive force dominates the interaction; Figure 1b shows the system of the negative and positive layers DL, n e and n i are the electron and ion densities, accordingly, E x is the electric field, λDL is the distance between the positive and negative DL charges, and δ is the solid density skin depth of the foil. The DL is geometrically followed by neutral plasma where the electric field decays within a skin depth and a shock wave is created. The shock wave description in the laboratory frame of reference is given in Figure 1c. This DL acts as a piston driving a shock wave (Naumova et al., Reference Naumova, Schlegel, Tikhonchuk, Labaune, Sokolov and Mourou2009; Eliezer et al., Reference Eliezer, Nissim, Raicher and Martinez Val2014b; Reference Eliezer, Nissim, Martinez Val, Mima and Hora2014c), moving in the unperturbed plasma. This model is supported in the literature by particle in cell simulation (Esirkepov et al., Reference Esirkepov, Borghesi, Bulanov, Mourou and Tajima2004; Naumova et al., Reference Naumova, Schlegel, Tikhonchuk, Labaune, Sokolov and Mourou2009) and independently by hydrodynamic two fluid simulations (Hora et al., Reference Hora, Lalousis and Eliezer1984; Hora, Reference Hora1991; Lalousis et al., Reference Lalousis, Foldes and Hora2012; Reference Lalousis, Hora, Eliezer, Martinez Val, Moustaizis, Miley and Mourou2013). Here, it is proposed to use the above shock wave as igniter for pre-compressed fuel in the framework of fast ignition. Fast ignition of DT and pB11 fuels using ultra-intense short pulse laser was suggested and elaborated by Hora and collaborators (Hora et al., Reference Hora, Lalousis and Moustaizis2014; Lalousis et al., Reference Lalousis, Hora and Moustaizis2014). Their approach is based on impact of plasma blocks, generated by laser pulses shorter than picosecond and powers in the range of petawatt approaching exawatt, with solid targets. Recently, it was suggested that the above approach based on plasma blocks combined with ultra-high magnetic fields may reduce the required laser pulse power to lower values (Hora et al., Reference Hora, Lalousis and Moustaizis2014; Reference Hora, Lalousis, Giuffrida, Margarone, Korn, Eliezer, Miley, Moustaizis and Mourou2015). Fast ignition of DT with ultra-intense laser pulse induced by accelerated protons by the DL described above was considered by Naumova et al., Reference Naumova, Schlegel, Tikhonchuk, Labaune, Sokolov and Mourou2009.
(a) The capacitor model for laser irradiances I L where the ponderomotive force dominates the interaction. (b) n e and n i are the electron and ion densities, accordingly, E x is the electric field, λDL is the distance between the positive and negative DL charges, and δ is the solid density skin depth of the foil. The shock wave description in the laboratory frame of reference is given in (c).
Relativistic or non-relativistic (Zeldovich & Raizer, Reference Zeldovich and Raizer1966) shock wave is described by five variables: The density ρ, the pressure P, the energy density e, the shock wave velocity u s and the particle flow velocity u p, assuming that we know the initial condition of the target (ρ0, P 0, e 0, and, the particle flow velocity u 0) before the shock arrival. The four equations relating the shock wave variables are the three Hugoniot relations describing the conservation laws of energy, momentum, and particles and the equation of state (EOS) connecting the thermodynamic variables of the state under consideration. The fifth equation necessary to solve the problem is obtained in a model where the pressure is induced by the laser ponderomotive force and its strength is a function of the laser pulse parameters. These equations for the relativistic and non-relativistic case are given in Appendix A (Eliezer et al., Reference Eliezer, Nissim, Raicher and Martinez Val2014b).
An ignition criterion for density-dimension product of the hot spot is calculated for non-equilibrium conditions where the electrons and ions temperatures are different, characteristic for fast ignition schemes. Electron and ion relaxation times relevant for these conditions are given in Appendix B. The equation describing the equality between nuclear fusion and energy losses is solved and the solution describes a surface in the 3D space of ρ·R-T e-T i, where ρ and R are the igniter density and dimension, accordingly, T e and T i are the electron and ion temperatures. The emission of bremsstrahlung into the hydrodynamics was well included in the computations of block ignition by Chu (Reference Chu1972) and the subsequent numerous computations (Lalousis et al., Reference Lalousis, Hora, Eliezer, Martinez Val, Moustaizis, Miley and Mourou2013; Reference Lalousis, Hora and Moustaizis2014). In this paper, we solve the ignition criterion for the general case where the electron and ion temperatures are not necessarily equal and the solution is not dependent on a specific fast ignition scheme. The energy balance equation for DT plasma was considered in the literature for the DT case with equal electron and ion temperatures (Guskov et al., Reference Guskov, Krokhin and Rozanov1976; Lindl, Reference Lindl, Caruso and Sindoni1988; Takabe et al., Reference Takabe, Mima and Nakai1989; Rozanov et al., Reference Rozanov, Verdon, Decroisette, Guskov, Lindl, Nishihara and Takabe1995). This is in contrast to the genuine two-fluid hydrodynamics with explicit appearance of the internal electric fields in plasmas (Lalousis & Hora, Reference Lalousis and Hora1983; Hora et al., Reference Hora, Lalousis and Eliezer1984; Eliezer et al., Reference Eliezer, Hora, Kolka, Green and Szichman1995).
In the second part of this paper our model for fast ignition is described, based on two temperature laser induced shock wave in the intermediate domain between relativistic and non-relativistic shock waves. The laser parameters required for fast ignition of DT and pB11 fuels are calculated.
Section 2 describes the ignition criterion and Section 3 presents a two-temperature model for laser induced shock waves. Application of this model for DT and pB11 is presented in Section 4. The paper is concluded in Section 5.
2. THE IGNITION CRITERION
We analyze the nuclear fusion reactions
$$\eqalign{& {A_1} + {A_2} \to {A_3} + {A_4} + {E_{\rm f}} \cr & {E_{\rm f}} = {E_{\rm \alpha}} + {E_{{\rm others}}}} $$E f is the fusion energy in each reaction, E α is α-particles energy usually deposited in part into the ignition domain and E others is the energy contained in the other particles and practically not contained in the ignition volume under consideration. The ignition fusion power W f [erg/(cm3·s)] is given by
$${W_{\rm f}}\left[ {\displaystyle{{{\rm erg}} \over {{\rm c}{{\rm m}^3} \cdot {\rm s}}}} \right] = {n_1}{n_2} \lt {\rm \sigma} v{ \gt _{12}}{E_{\rm \alpha}} $$where n 1 and n 2 are the appropriate densities of particles A 1 and A 2, σ is the cross-section of reaction (1), <σv>12 is the fusion rate of this reaction and E α is α-particles energy.
In general, not all of the α-fusion energy is deposited into the igniter. We define by f α the fraction of the α-particles created and deposited into the igniter domain, while (1–f α) is the escape fraction to the surrounding cold fuel. The value of f α can be approximated by (Guskov & Rozanov, Reference Guskov, Rozanov, Velarde, Ronen and Martinez Val1993)
$$\eqalign{ {\,f_{\rm \alpha} } & = \left\{ {\matrix{{\displaystyle{3 \over 2}{x_{\rm \alpha} } - \displaystyle{4 \over 5}x_{\rm \alpha} ^2 } & {{x_{\rm \alpha} } \lt \displaystyle{1 \over 2}} \cr {1 - \displaystyle{1 \over {4{x_{\rm \alpha} }}} + \displaystyle{1 \over {160{x_{\rm \alpha} }^3 }}} & {{x_{\rm \alpha} } \ge \displaystyle{1 \over 2}} \cr } } \right. \cr {x_{\rm \alpha} }(\rm \tau ) & = \displaystyle{R \over {{R_\rm \alpha }}}} $$The igniter dimension R in our model is taken to be
$$R = \left( {\displaystyle{{{u_{\rm s}}} \over c} - \displaystyle{{{u_{\rm p}}} \over c}} \right)c{{\rm \rm \tau} _{\rm L}}$$where u s and u p are the shock wave velocity and the particle velocity accordingly, τL is the laser pulse duration that causes ignition and c is the speed of light. The velocities u s and u p depend on the laser and fuel parameters, as it is shown in Appendices A and C. The α range R α is approximated for DT (Atzeni & Meyer-Ter-Vehn, Reference Atzeni and Meyer-Ter-Vehn2004) and for pB11 by (Eliezer & Martinez Val, Reference Eliezer and Martinez Val1998) by:
$$\eqalign{& DT\,{\rm fusion\!:} \quad {R_{\rm \alpha}} [{\rm cm}] = \displaystyle{1 \over {{\rm \kappa} {{\rm \rho} _0}}}\left[ {\displaystyle{{1.5 \times {{10}^{ - 2}}{T_{\rm e}}{{({\rm keV})}^{5/4}}} \over {1 + 8.2 \times {{10}^{ - 3}}{T_{\rm e}}{{({\rm keV})}^{5/4}}}}} \right] \cr & pB11\,{\rm fusion\!:}\;\; {R_{\rm \alpha}} \left[ {{\rm cm}} \right] = \displaystyle{a \over {\rm \rho}} {\left( {\displaystyle{{{T_{\rm e}}} \over c}} \right)^b} \cr & \left\{ \matrix{{T_{\rm e}} \lt 50\,{\rm keV}:a = 0.25,b = 0.79,\,c = 10\,{\rm keV} \hfill \cr {T_{\rm e}} \ge 50\,{\rm keV}:a = 1.1,b = 0.31,\,c = 100\,{\rm keV} \hfill} \right.} $$The initial density of the pre-compressed target is ρ0 and κ is the shock wave compression during the fast ignition process.
The equation describing the ignition requirement is given by
$${W_{\rm f}} - \sum {W\left( {{\rm losses}} \right)} \ge 0$$The power density losses, W (losses), include the power densities of the mechanical work (W m), bremsstrahlung radiation (W B), and the heat wave transport by electrons (W he). The ignition criteria are derived by explicitly writing Eq. (6) in the following way
$${\,f_{\rm \alpha}} {W_{\rm f}} - {W_{\rm B}} - {W_{{\rm he}}} - {W_{\rm m}} \ge 0$$The solution of the equality of Eq. (7) describes a surface in the 3D space of ρ·R-T e-T i. The ignition criterion (7) is solved for the general case where the electron and ion temperatures are not necessarily equal, extending previous studies for the DT case with equal electron and ion temperatures (Guskov et al., Reference Guskov, Krokhin and Rozanov1976; Lindl, Reference Lindl, Caruso and Sindoni1988; Takabe et al., Reference Takabe, Mima and Nakai1989; Rozanov et al., Reference Rozanov, Verdon, Decroisette, Guskov, Lindl, Nishihara and Takabe1995).
The bremsstrahlung power density losses W B are given by
$$\eqalign{ {W_{\rm B}}\left[ {\displaystyle{{{\rm erg}} \over {{\rm c}{{\rm m}^3} \cdot {\rm s}}}} \right] &= 1.5 \times {10^{ - 25}}{n_{\rm e}} \cr & \quad \sum\limits_{k = 1,2} {{n_k}{Z_k}^2 {T_{\rm e}}{{({\rm eV})}^{1/2}}\left( {1 + \displaystyle{{2{T_{\rm e}}({\rm eV})} \over {500,000}}} \right)}} $$Z k is the charge number of particle k, n e, and n k are the electron density and the ion densities of particle k, accordingly. The second term in the right-hand side of Eq. (8) stems from the relativistic corrections to the bremsstrahlung losses. The heat conduction losses from the igniter domain are approximated by (Chu, Reference Chu1972; Rozanov et al., Reference Rozanov, Verdon, Decroisette, Guskov, Lindl, Nishihara and Takabe1995):
$${W_{{\rm he}}}\left[ {\displaystyle{{{\rm erg}} \over {{\rm c}{{\rm m}^3} \cdot {\rm s}}}} \right] = \displaystyle{{{K_{\rm e}}{T_{\rm e}}} \over {{R^2}}} = \displaystyle{{3.11 \times {{10}^9}{T_{\rm e}}{{({\rm eV})}^{7/2}}} \over {{R^2}\ln {\rm \Lambda}}} $$The plasma logarithmic term lnΛ is
$$\ln {\rm \Lambda} = 24 - \ln \left[ {\displaystyle{{{n_{\rm e}}^{1/2}} \over {{T_{\rm e}}({\rm eV})}}} \right]$$In the solution of Eq. (7), a constant value lnΛ = 3.5 was taken, appropriate for very high densities where the plasma is strongly coupled.
The mechanical expansion power losses are estimated by (Eliezer & Martinez Val, Reference Eliezer and Martinez Val1998)
$$\eqalign{& {W_{\rm m}} = \displaystyle{{2{\rm \rho} {{\rm v}_{\rm m}}^3} \over R} \cr & {W_{\rm m}}\left[ {\displaystyle{{{\rm erg}} \over {{\rm c}{{\rm m}^3} \cdot {\rm s}}}} \right] = \displaystyle{2 \over {{{\rm \rho} ^{1/2}}R}}{\left( {\displaystyle{{{k_{\rm B}}{T_{\rm e}}{n_{\rm e}} + {k_{\rm B}}{T_{\rm i}}{n_{\rm i}}} \over {{{\rm \beta} _{\rm m}}}}} \right)^{3/2}}} $$The flow expansion velocity from the hot spot to the cold environment velocity, v m, is consistent with an isochoric pre-compression of the target. As the pressure in the hot region P is higher than the pressure in the cold region, a shock wave will develop in the cold target according to the momentum conservation law
$$P \approx {{\rm \beta} _{\rm m}}{\rm \rho} {v_{\rm m}}^2 $$Using the ideal gas EOS
$$P = {k_{\rm B}}{T_{\rm e}}{n_{\rm e}} + {k_{\rm B}}{T_{\rm i}}{n_{\rm i}}$$and Eq. (12) we get the second of Eq. (11). For cases of an ionized pre-compressed target considered here the value βm = 4/3 was used.
Substituting Eqs. (8), (9), (11) into Eq. (7) and multiplying by R 2 we get the following quadratic equation in ρ·R
$$a({T_{\rm e}},{T_{\rm i}}){({\rm \rho} \cdot R)^2} + b({T_{\rm e}},{T_{\rm i}})({\rm \rho} \cdot R) + c({T_{\rm e}}) \ge 0$$The solution of this inequality gives the ignition criterion as described in the domain above the surface in the 3D space of ρ R-T e-T i. In the following, Eq. (14) is written explicitly and solved for two cases: The DT and pB11 nuclear fusion fuel.
The DT case
$$D + T \to n + {\rm \alpha} + 17,589{\rm keV}$$Equal density numbers for deuterium and tritium n D and n T, accordingly are assumed.
$${n_{\rm e}}[{\rm c}{{\rm m}^{ - 3}}] = {n_{\rm i}}[{\rm c}{{\rm m}^{ - 3}}] = \left( {\displaystyle{{\rm \rho} \over {2.5{m_{\rm p}}}}} \right) = 2.39 \times {10^{23}}{\rm \rho} $$where m p is the proton mass. The fusion power W f [erg/(cm3·s)] for DT is given in Eq. (2) with number density–mass density relations from (16), yielding
$${W_{{\rm f},{\rm DT}}}\left[ {\displaystyle{{{\rm erg}} \over {{\rm c}{{\rm m}^3} \cdot {\rm s}}}} \right] = 8.07 \times {10^{40}} \!\!\lt \!{\rm \sigma} v{ \gt \,_{{\rm DT}}}\,{{\rm \rho} ^2}$$<σv>DT is the reactivity of the DT reaction fitted in the domain of ion temperatures 1 keV < T i < 100 keV by (Bosch & Hale, Reference Bosch and Hale1992)
$$\eqalign{ \lt {\rm \sigma} v{ \gt _{{\rm DT}}}\left[ {\displaystyle{{{\rm c}{{\rm m}^3}} \over {\rm s}}} \right] = & \;6.4341 \times {10^{ - 14}}{{\rm \zeta} ^{ - 5/6}}{\left( {\displaystyle{{6.661} \over {{T_{\rm i}}^{1/3}}}} \right)^2} \cr & \quad \exp \left[ { - 19.983{{\left( {\displaystyle{{\rm \zeta} \over {{T_{\rm i}}}}} \right)}^{1/3}}} \right] \cr {\rm \zeta} = & \;1 - \displaystyle{{15.136{T_i} + 4.6064{T_{\rm i}}^2 - 0.10675{T_{\rm i}}^3} \over {1000 + 75.189{T_{\rm i}} + 13.5{T_{\rm i}}^2 + 0.01366{T_{\rm i}}^3}} \cr & {T_{\rm i}}\,{\rm in\,keV}} $$The electron bremsstrahlung power per unit volume loss, including relativistic corrections, W B is given by
$${W_{\rm B}}\left[ {\displaystyle{{{\rm erg}} \over {{\rm c}{{\rm m}^3} \cdot {\rm s}}}} \right] = 8.58 \times {10^{21}}{{\rm \rho} ^2}{T_{\rm e}}{({\rm eV})^{0.5}}\left( {1 + \displaystyle{{2{T_{\rm e}}({\rm eV})} \over {0.511 \times {{10}^6}}}} \right)$$The mechanical expansion power loss is estimated for DT using Eq. (11)
$${W_{\rm m}}\left[ {\displaystyle{{{\rm erg}} \over {{\rm c}{{\rm m}^3} \cdot {\rm s}}}} \right] = 1.02 \times {10^{18}}{\left[ {{T_{\rm e}}({\rm eV}) + {T_{\rm i}}({\rm eV})} \right]^{1.5}}\left( {\displaystyle{{\rm \rho} \over R}} \right)$$Substituting the following equations (17) for W f, (19) for W B, (9) for W he, and (20) for W m into Eq. (7) and multiplying by R 2 we get a quadratic inequality in ρ·R as given by Eq. (14) with
$$\eqalign{{Y_{{\rm DT}}} \equiv & \;a({T_{\rm e}},{T_{\rm i}}){({\rm \rho} R)^2} + b({T_{\rm e}},{T_{\rm i}})({\rm \rho} R) + c({T_{\rm e}}) \ge 0 \cr a({T_{\rm e}},{T_{\rm i}}) = & \;8.07 \times {10^{40}} \lt\! {\rm \sigma} v \gt \cr & \quad\!\!\!\!\! - {\rm 8}{\rm. 63} \times {\rm 1}{{\rm 0}^{21}}{T_{\rm e}}{({\rm eV})^{1/2}}\left( {1 + \displaystyle{{2{T_{\rm e}}({\rm eV})} \over {500,000}}} \right) \cr b({T_{\rm e}},{T_{\rm i}}) = & - 1.02 \times {10^{18}}{\left[ {{T_{\rm e}}({\rm eV}) + {T_{\rm i}}({\rm eV})} \right]^{1.5}} \cr c({T_{\rm e}}) = & - \displaystyle{{3.11 \times {{10}^9}{T_{\rm e}}{{({\rm eV})}^{7/2}}} \over {\ln {\rm \Lambda}}}} $$The solution of Eq. (21) with <σv>DT from (18) and with f α = 1, lnΛ = 3.5 is given in Figure 2. Contours of equal ρ·R as a function of ions and electrons temperatures for DT are displayed. It is seen that for temperatures T i, T e in the range 10–50 keV ρ·R < 1.
2.1. The pB11 Case
$$p + {}^{11}B \to 3{\rm \alpha} + 8,700\,{\rm keV}$$In this case, due to the ions high temperatures required for fusion and also bremsstrahlung losses of the electrons, the boron to proton number density ratio, ε, is usually less than 0.4. In the following terms Eq. (14) is expressed as a function of the ratio ε:
$$\eqalign{\rm \varepsilon &= \displaystyle{{{n_{\rm B}}} \over {{n_{\rm p}}}}; \quad {n_{\rm i}} = {n_{\rm B}} + {n_{\rm p}} = (1 + {\rm \varepsilon} ){n_{\rm p}} \cr {\rm \rho} &= m_{\rm i}n_{\rm i}; \quad m_{\rm i} = \displaystyle{{(1 + 11\rm \varepsilon )} \over {(1 + \rm \varepsilon )}}{m_{\rm p}} } $$n B, n p, and n i are the appropriate number densities (cm−3) of the boron11, protons, and ions. The electron density n e is related to the ion density n i for a neutral plasma with ionization Z i by
$$\eqalign{& {n_{\rm e}} = \sum\limits_{k \;=\; {\rm p},{\rm B}} {{Z_k}{n_k}} = {n_{\rm p}} + 5{n_{\rm B}} = \left( {\displaystyle{{1 + 5{\rm \varepsilon}} \over {1 + {\rm \varepsilon}}}} \right){n_i} \cr & {n_{\rm e}} = \left( {\displaystyle{{1 + 5{\rm \varepsilon}} \over {1 + 11{\rm \varepsilon}}}} \right)\left( {\displaystyle{{\rm \rho} \over {{m_{\rm p}}}}} \right) \cr & {n_{\rm i}} = \left( {\displaystyle{{1 + {\rm \varepsilon}} \over {1 + 11{\rm \varepsilon}}}} \right)\left( {\displaystyle{{\rm \rho} \over {{m_{\rm p}}}}} \right)} $$The following relations involving ε are also important for our calculations
$$\eqalign{& \sum\limits_{k \;=\; {\rm p},{\rm B}} {{Z_k}^2 {n_k}} = \left( {\displaystyle{{1 + 25{\rm \varepsilon}} \over {1 + {\rm \varepsilon}}}} \right){n_{\rm i}} \cr & \sum\limits_{k \;=\; {\rm p},{\rm B}} {\displaystyle{{{Z_k}^2 {n_k}} \over {{m_k}}} \;=\;} \left( {1 + \displaystyle{{25} \over {11}}{\rm \varepsilon}} \right)\displaystyle{1 \over {(1 + {\rm \varepsilon} )}}{n_{\rm i}}} $$The fusion power W f [erg/(cm3·s)] for pB11 is given in Eq. (2) with number density–mass density relations from (24), yielding
$${W_{{\rm f},{\rm pB}11}}\left[ {\displaystyle{{{\rm erg}} \over {{\rm c}{{\rm m}^3} \cdot {\rm s}}}} \right] = 4.99 \times {10^{42}}\displaystyle{{{\rm \varepsilon} {{\rm \rho} ^2} \lt \!{\rm \sigma} v{ \gt \;_{{\rm pB}11}}} \over {{{(1 + 11{\rm \varepsilon} )}^2}}}$$<σv>pB11 is the reactivity of the p11B reaction fitted in the domain of ion temperatures 50 keV < T i < 500 keV by (Nevins & Swain, Reference Nevins and Swain2000)
$$\eqalign{& \lt\!{\rm \sigma} v{ \gt _{{\rm DT}}}\left[ {\displaystyle{{{\rm c}{{\rm m}^3}} \over {\rm s}}} \right] = 6.4341 \times {10^{ - 14}}{{\rm \zeta} ^{ - 5/6}}{\left( {\displaystyle{{6.661} \over {{T_{\rm i}}^{1/3}}}} \right)^2}\cr &\qquad\qquad\qquad\hskip 16.5pt \exp \left[ { - 19.983{{\left( {\displaystyle{{\rm \zeta} \over {{T_{\rm i}}}}} \right)}^{1/3}}} \right] \cr & {\rm \zeta} = 1 - \displaystyle{{15.136{T_{\rm i}} + 4.6064{T_{\rm i}}^2 - 0.10675{T_{\rm i}}^3} \over {1000 + 75.189{T_{\rm i}} + 13.5{T_{\rm i}}^2 + 0.01366{T_{\rm i}}^3}} \cr & {T_i}\,{\rm in\,keV}} $$The electron bremsstrahlung power per unit volume loss, including relativistic corrections, W B is given by
$$\eqalign{{W_{\rm B}}\left[ {\displaystyle{{{\rm erg}} \over {{\rm c}{{\rm m}^3} \cdot {\rm s}}}} \right] = \;& 5.35 \times {10^{22}}\displaystyle{{(1 + 5{\rm \varepsilon} )(1 + 25{\rm \varepsilon} )} \over {{{(1 + 11{\rm \varepsilon} )}^2}}} \times \cr & {{\rm \rho} ^2}{T_{\rm e}}{({\rm eV})^{0.5}}\left( {1 + \displaystyle{{2{T_{\rm e}}({\rm eV})} \over {0.511 \times {{10}^6}}}} \right)} $$The mechanical expansion power loss is estimated for pB11 by (Eliezer & Martinez Val, Reference Eliezer and Martinez Val1998)
$${W_{\rm m}}\left[ {\displaystyle{{{\rm erg}} \over {{\rm c}{{\rm m}^3} \cdot {\rm s}}}} \right] = 1.86 \times {10^{18}}\left( {\displaystyle{{\rm \rho} \over R}} \right){\left[ {\displaystyle{{(1 + 5{\rm \varepsilon} ){T_{\rm e}}({\rm eV}) + (1 + {\rm \varepsilon} ){T_{\rm i}}({\rm eV})} \over {(1 + 11{\rm \varepsilon} )}}} \right]^{3/2}}$$Substituting the following equations (26) for W f, (28) for W B, (9) for W he, and (29) for W m into Eq. (7) and multiplying by R 2 we get a quadratic inequality in ρ·R as given by Eq. (14) with
$$\eqalign{{Y_{{\rm pB}11}} \equiv & \;a({T_{\rm e}},{T_{\rm i}}){({\rm \rho} R)^2} + b({T_{\rm e}},{T_{\rm i}})(\rho R) + c({T_{\rm e}}) \ge 0 \cr a({T_{\rm e}},{T_{\rm i}}) = & \;{\,f_{\rm \alpha}} 4.99 \times {10^{42}}\displaystyle{{{\rm \varepsilon} \lt \!\sigma v{ \gt _{{\rm pB}11}}} \over {{{(1 + 11{\rm \varepsilon} )}^2}}} \cr &- 5.35 \times {10^{22}}\displaystyle{{(1 + 5{\rm \varepsilon} )(1 + 25{\rm \varepsilon} )} \over {{{(1 + 11{\rm \varepsilon} )}^2}}} \cr & \quad \times {T_{\rm e}}{({\rm eV})^{0.5}}\left( {1 + \displaystyle{{2{T_{\rm e}}({\rm eV)}} \over {0.511 \times {{10}^6}}}} \right) \cr \cr b({T_{\rm e}},{T_{\rm i}}) = & - 1.86 \times {10^{18}}{\left[ {\displaystyle{{\left( {1 + 5{\rm \varepsilon}} \right){T_{\rm e}}({\rm eV}) + (1 + {\rm \varepsilon} ){T_{\rm i}}({\rm eV})} \over {(1 + 11{\rm \varepsilon} )}}} \right]^{3/2}} \cr c({T_{\rm e}}) = & - \displaystyle{{3.11 \times {{10}^9}{T_{\rm e}}{{({\rm eV})}^{7/2}}} \over {\ln {\rm \Lambda}}}} $$The solution of Eq. (30) and <σv>pB11 from (27) with f α = 1, lnΛ = 3.5, and ε = 0.33, is given in Figure 3. It is seen that temperature ranges where Eq. (30) has real solutions ρ·R < 20 are 100–600 keV for the ions and 10–100 keV for the electrons. Moreover, the values of ρ·R are larger by about two orders of magnitude than for DT, requiring larger pre-compression and igniter size, and as it is shown below; larger energy laser.
3. THE IGNITER PERFORMANCE DRIVEN BY LASER INDUCED TWO TEMPERATURES SHOCK WAVES
As described in Section 1, in the framework of the piston model, a shock wave is generated in the target, with different ions and electrons temperatures. Following Chu (Reference Chu1972) and Eliezer and Martinez-Val (Reference Eliezer and Martinez Val1998), the igniter performance can be analyzed by an energy balance, dependent on the ions and electrons temperatures:
$$\eqalign{& \left( {\displaystyle{3 \over 2}} \right)\displaystyle{d \over {dt}}({n_{\rm e}}{k_{\rm B}}{T_{\rm e}}) = {{\rm \eta} _{\rm d}}{W_{\rm d}} + {W_{{\rm ie}}} - {W_{\rm B}} + {\,f_{\rm \alpha}} {{\rm \eta} _{\rm f}}{W_{\rm f}} \cr & \left( {\displaystyle{3 \over 2}} \right)\displaystyle{d \over {dt}}({n_{\rm i}}{k_{\rm B}}{T_{\rm i}}) = (1 - {{\rm \eta} _{\rm d}}){W_{\rm d}} - {W_{{\rm ie}}} + {\,f_{\rm \alpha}} (1 - {{\rm \eta} _{\rm f}}){W_{\rm f}}} $$A schematic view of this two temperatures shock model is given in Figure 4. Note that during the time the shock wave is driven by the laser-piston the heat wave and mechanical losses do not exist. kB is the Boltzmann's constant. W d [erg/(cm3·s)] is the power density deposited by the driver (induced by the laser-piston), ηd is the fraction of the driver energy deposited in the electrons inside the shocked volume, (1−ηd) gives the fraction of the driver energy deposited in the ions inside the shocked volume.
A schematic picture of the two temperature shock wave creation.
The deposition power density W d is dependent on the laser intensity I L and pulse time duration τL and on the shock compression κ = ρ/ρ0 (see Appendix A):
$${I_{\rm L}}\left[ {\displaystyle{W \over {{\rm c}{{\rm m}^2}}}} \right] = 4 \times {10^3}\displaystyle{{{W_{\rm d}}\left[ {\displaystyle{{{\rm erg}} \over {{\rm c}{{\rm m}^3} \cdot {\rm s}}}} \right]{{\rm \tau} _{\rm L}}\left[ {\rm s} \right]} \over {\rm \kappa}} $$The relation between the deposition power density and the laser parameters, as well as the particle u p and shock u s velocities are obtained from the Hugoniot–Rankine equations and presented in Appendix C. The size of the igniter is determined by the laser and the fuel parameters. The igniter is modeled as a cylinder with length R = l s, the shock dimension, defined in Eq. (4) and diameter 2R L = 3l s, larger than the shock dimension to justify1D approximation.
As described in Section 2, the shock dimension is used in the calculation of f α, the fraction of the α-particles (created in the DT or pB11 fusion) energy deposited inside the shocked volume.
Assuming that λi and λe are the appropriate mean free paths of the ions and electrons in plasma, one gets for ηd
$$\eqalign{& {{\rm \eta} _{\rm d}} = \displaystyle{{{{\rm \lambda} _{\rm i}}} \over {{{\rm \lambda} _{\rm i}} + {{\rm \lambda} _{\rm e}}}} \cr & {{\rm \lambda} _{\rm i}}[{\rm cm}] = \left( {\displaystyle{{3 \times {{10}^{23}}} \over {{n_{\rm i}}}}} \right)\left( {\displaystyle{{{m_{\rm p}}} \over {{m_{\rm i}}}}} \right){E_{\rm i}}[{\rm MeV}] \cr & {{\rm \lambda} _{\rm e}}[{\rm cm}] = \left( {\displaystyle{{5 \times {{10}^{22}}} \over {{n_{\rm e}}\ln {\rm \Lambda}}}} \right){T_{\rm e}}{[{\rm keV}]^{3/2}}{E_{\rm i}}[{\rm MeV}] \cr & {E_{\rm i}} = \displaystyle{1 \over 2}{m_{\rm i}}{u_{\rm p}}^2 = 1250({\rm MeV}){\left( {\displaystyle{{{u_{\rm p}}} \over c}} \right)^2}} $$It is important to mention that although λi and λe can in general be larger than the time dependent shocked domain u p·t, the charged particles that heat the shocked area have a velocity u p and therefore are not moving faster than the shock wave since u s > u p. Thus the shock wave moves into a cold domain not yet heated by the driver energy.
W ie [erg/(cm3·s)] is the ion–electron exchange power density given by
$$\eqalign{ {W_{{\rm ie}}}\left[ {\displaystyle{{{\rm erg}} \over {{\rm c}{{\rm m}^3} \cdot {\rm s}}}} \right] &= 2.70 \times {10^{ - 22}}{n_{\rm e}} \cr & \quad \left( {\displaystyle{{{T_{\rm i}}(E) - {T_{\rm e}}({\rm keV})} \over {{T_{\rm e}}{{({\rm keV})}^{1.5}}}}} \right)\sum\limits_k {\displaystyle{{{Z_k}^2 {n_k}} \over {{m_k}}}\ln {\Lambda _{{\rm e}k}}}} $$W B [erg/(cm3·s)], the electron bremsstrahlung power density losses and W f [erg/(cm3·s)], the fusion power density created in the shocked volume, were defined in Section 2. ηf is the energy fraction that is deposited in the electrons by the α-particles created in the fusion under consideration and (1−ηf) describes the energy fraction that is deposited in the ions by these α-particles. The function ηf for DT fusion was taken from Chu (Reference Chu1972):
$${{\rm \eta} _{\rm f}} = \displaystyle{{32} \over {32 + {T_{\rm e}}({\rm keV})}}$$For proton–boron fusion, the function ηf was taken from Eliezer and Martinez-Val (Reference Eliezer and Martinez Val1998):
$${{\rm \eta} _{\rm f}} = \displaystyle{{150} \over {150 + T_{\rm e}^{1.5} ({\rm keV})}}$$The time dependent temperatures equations are coupled to equations for the number densities of the ions species. For DT these equations read:
$$\displaystyle{{{\rm d}{n_{\rm D}}} \over {{\rm d}t}} = \displaystyle{{{\rm d}{n_{\rm T}}} \over {{\rm d}t}} = - \displaystyle{{{\rm d}{n_{\rm \alpha}}} \over {{\rm d}t}} = - {n_{\rm D}}{n_{\rm T}} \lt \!{\rm \sigma} v{ \gt _{{\rm DT}}}$$where, n D, n T, and n α are number densities of the deuterons, protons, and α-particles.
For proton–boron fusion the time evolution of the number densities of the ions species is:
$$\eqalign{& \displaystyle{{{\rm d}{n_{\rm p}}} \over {{\rm d}t}} = \displaystyle{{{\rm d}{n_{\rm B}}} \over {{\rm d}t}} = - {n_{\rm p}}{n_{\rm B}} \lt \!{\rm \sigma} v{ \gt _{{\rm pB}}} \cr & \displaystyle{{{\rm d}{n_\alpha}} \over {{\rm d}t}} = 3{n_{\rm p}}{n_{\rm B}} \lt \!{\rm \sigma} v{ \gt _{{\rm pB}}}} $$Given the laser and the fuel parameters, the ions and the electrons temperatures and densities time evolution can be obtained.
In the scheme of shock fast ignition considered here, the product ρ·R = ρ·l s is determined by the laser and fuel parameters. Following the criterion described in Section 2, ignition occurs if at some time t ig during the laser pulse duration, temperatures values are obtained such that a solution of energy balance (14) exists with T i(t ig), T e(t ig). In the framework of the model presented here, if such a solution exists, after time t ig, the surrounding fuel burns due to the released fusion energy in the igniter. Therefore, the product ρ·R[T i(t), T e(t)] is solved from Eq. (14) and ignition occurs if:
$${\rm \rho} \cdot R\left[ {{T_{\rm i}}({t_{{\rm ig}}}),\,\,{T_{\rm e}}({t_{{\rm ig}}})} \right] \lt {\rm \rho} \cdot {l_{\rm s}}$$4. ELECTRON AND ION TEMPERATURES CALCULATIONS FOR DT AND PB11
In this section solutions of the two temperatures shock model for DT and pB11 cases are presented, for laser parameters for which ignition is obtained during the laser pulse.
Figures 5–7 display the results of the two temperatures model given in Eqs. (31, 38) for DT pre-compressed to density ρ0 = 600 g/cm3. The fast ignition shock generated by irradiation with laser intensity of 7.5 × 1022 W/cm2, 1 ps pulse duration, and energy 3.67 kJ induces a compression of κ = 4. The laser energy density deposition in this case is W d = 7.5 × 1031 erg/cm3s and the shock length, the igniter dimension in our model, is l s = 0.833 µm. Therefore, in this case ρ·R = 0.2 g/cm2. For ignition, it is required that due to the shock wave energy deposition, to reach temperatures values where Eq. (14) has a solution ρ R ≤ 0.2 g/cm2. The electrons and ions temperatures, T e(t), T i(t) as a function of time, and ρ·R, obtained by solving Eq. (14) given T e(t), T i(t) are shown in Figure 5. It is seen that the ion temperature increase to about 68 keV and the electron temperature reaches 42 keV by the end of the laser pulse (Fig. 5b). There is no real solution of Eq. (14) for the product ρ·R for times less than about 0.1 ps (Fig. 5b). As the temperatures rise, Eq. (14) has real solutions, decreasing to ρ·R = 0.2 g/cm2 at about 0.3 ps, when the ions and electrons temperatures are slightly higher than 10 keV and ignition is set on. At this time fusion energy deposition rises and exceeds the shock wave energy deposition. The different terms in Eq. (31), deposition, fusion, radiation losses, and electron–ion exchange energy densities, as a function of time are given in Figure 6. The fraction of absorbed α-particles in the hot spot as a function of time is described in Figure 7. It is seen that as the time evolves and the temperatures increase, a larger fraction of α-particles leave the igniter region heating up the surroundings.
Proton–boron fusion requires higher temperatures to reach ignition compared with DT, mainly due to the higher radiation losses, implying larger size and more dense igniters. These constrains require larger laser energies, making the shock based fast ignition scheme presented here for pB11 fusion impracticable. The temperatures and the ρ·R product as a function of time for a pB11 case that may reach ignition according to the criterion considered here are displayed in Figure 8. However, in this case the igniter size is larger, l s = 4.32 µm, the pre-compressed fuel density is ρ0 = 4800 g/cm3, and the product required for ignition is ρ·R = 8.3 g/cm2. Therefore, the laser pulse generating the fast shock ignition should have an intensity of 1.61 × 1025 W/cm2, pulse duration of 1 ps, and energy of 21 MJ. The laser power of such system is in the exawatt range. Although such super-laser system is planned at University of Texas, it is still years away from actual development.
5. SUMMARY AND DISCUSSION
In this paper a general fuel ignition criterion for two temperatures plasma is presented. This criterion was considered for DT and pB11 pre-compressed plasma. Two temperatures plasma are important for schemes of nuclear fast ignition fusion, based on short pulse lasers. Here we applied the ignition criterion to a fast ignition scheme based on intense short pulse laser generated shock wave. The ignition criterion stems from energy balance between fusion energy and radiation, expansion and heat conduction losses, and is described by a surface in the 3D space defined by the electron and ion temperatures T e, T i, and the plasma density times the hot spot dimension, ρ·R. For appropriate fusion ion temperatures, namely T i larger than 10 keV for DT, and T i larger than 150 keV for pB11, the value of ρ·R required for ignition for pB11 is larger by a factor of 50 or more than for DT, depending on the electron temperature. Furthermore, following the ignition criterion described here, pB11 fusion is practically not possible if the electron and ion temperatures are equal, see also Kouhi et al. (Reference Kouhi, Ghoraneviss, Malekynia, Hora, Miley, Sari, Azizi and Razavipour2011). To eliminate these problems for pB11 fusion, application of non-thermal conversion of the laser energy by picosecond laser pulses to ultrahigh acceleration of plasma blocks (Chu, Reference Chu1972) to initiate the fusion reaction in solid density DT or pB11 fuel (Lalousis et al., Reference Lalousis, Hora, Eliezer, Martinez Val, Moustaizis, Miley and Mourou2013; Reference Lalousis, Hora and Moustaizis2014; Hora et al., Reference Hora, Lalousis, Giuffrida, Margarone, Korn, Eliezer, Miley, Moustaizis and Mourou2015).
Our suggested model for fast ignition based on two temperature laser induced shock wave in the intermediate domain between relativistic and non-relativistic shock waves is summarized. The size of the hot spot is dependent on the laser pulse intensity and time duration. Scaling laws between ρ R and the laser and fuel parameters are presented in Appendix A.
The laser parameters for fast ignition for DT and pB11 are calculated. For DT case one needs 3 kJ energy, 1 ps laser pulse duration to ignite a pre-compressed target at about 600 g/cm3. These laser parameters are of the same order as other fast ignition schemes (Naumova et al., Reference Naumova, Schlegel, Tikhonchuk, Labaune, Sokolov and Mourou2009). However, this scheme is not practical for pB11 ignition since it requires a laser with more than three orders of magnitude energy for the same pulse duration.
APPENDIX
APPENDIX A: RELATIVISTIC AND NON-RELATIVISTIC SHOCK WAVE RANKINE–HUGONIOT EQUATIONS
The relativistic shock wave Hugoniot equations in the laboratory frame of reference are given by the following equations
$$\eqalign{& ({\rm i})\;\displaystyle{{{u_{{\rm p1}}}} \over c} = \sqrt {\displaystyle{{\left( {{P_1} - {P_0}} \right)\left( {{e_1} - {e_0}} \right)} \over {\left( {{e_0} + {P_1}} \right)\left( {{e_1} + {P_0}} \right)}}} \cr & ({\rm ii})\;\displaystyle{{{u_{\rm s}}} \over c} = \sqrt {\displaystyle{{\left( {{P_1} - {P_0}} \right)\left( {{e_1} + {P_0}} \right)} \over {\left( {{e_1} - {e_0}} \right)\left( {{e_0} + {P_1}} \right)}}} \cr & ({\rm iii})\;\displaystyle{{{{\left( {{e_1} + {P_1}} \right)}^2}} \over {{\rm \rho _1}^2 }} - \displaystyle{{{{\left( {{e_0} + {P_0}} \right)}^2}} \over {{\rm \rho _0}^2 }} \cr & \qquad = \left( {{P_1} - {P_0}} \right)\left[ {\displaystyle{{\left( {{e_0} + {P_0}} \right)} \over {{\rm \rho _0}^2 }} + \displaystyle{{\left( {{e_1} + {P_1}} \right)} \over {{\rm \rho _1}^2 }}} \right]}$$P, e, and ρ are the pressure, energy density and mass density accordingly, the subscripts 0 and 1 denote the domains before and after the shock arrival, u s is the shock wave velocity and u p1 is the particle flow velocity in the laboratory frame of reference and c is the speed of light. We have assumed that in the laboratory the target is initially at rest, u p0 = 0. The EOS taken here in order to calculate the shock wave parameters is the ideal gas EOS
$${e_j} = {{\rm \rho} _j}{c^2} + \displaystyle{{{P_j}} \over {\Gamma - 1}}; \;j{\rm \;=\; 0,1}{\rm.}$$where Γ is the specific heat ratio. We have to solve Eqs (A1) and (A2) together with our piston model equation (Esirkepov et al., Reference Esirkepov, Borghesi, Bulanov, Mourou and Tajima2004; Eliezer et al., Reference Eliezer, Nissim, Raicher and Martinez Val2014b)
$${P_1} = \displaystyle{{2{I_{\rm L}}} \over c}\left( {\displaystyle{{1 - {\rm \beta}} \over {1 + {\rm \beta}}}} \right);{\rm \beta} \equiv \displaystyle{{{u_{{\rm p1}}}} \over c}$$We have five equations given in (A1), (A2), and (A3) with five unknowns: u s, u p1, P 1, ρ1, and e1 assuming that we know I L, ρ0, P 0, e 0, and Γ. We take ideal gas EOS with Γ = 5/3. The calculations are conveniently done in the dimensionless units defined by
$${{\rm \Pi} _{\rm L}} \equiv \displaystyle{{{I_L}} \over {{{\rm \rho} _0}{c^3}}};{\rm \kappa} \equiv \displaystyle{{{{\rm \rho} _1}} \over {{{\rm \rho} _0}}};{{\rm \kappa} _0} \equiv \displaystyle{{\Gamma + 1} \over {\Gamma - 1}};{\rm \Pi} = \displaystyle{{{P_1}} \over {{{\rm \rho} _0}{c^2}}};{{\rm \Pi} _0} = \displaystyle{{{P_0}} \over {{{\rm \rho} _0}{c^2}}}$$It is important to emphasize that if we take P 0 = 0 then we get only the κ > 4 solutions (Eliezer et al., Reference Eliezer, Nissim, Pinhasi, Raicher and Martinez Val2014a), therefore in order to see the behavior at the transition between relativistic and non-relativistic domain one has to take P 0 ≠ 0! In our numerical estimations we take P 0 = 1 bar = 106 in cgs units. For example, the Hugoniot Eq. (A1) together with the EOS Eq. (A2) yield
$$\eqalign{ \displaystyle{{P_0} \over {P_1}} = \displaystyle{{{{\rm \Pi} _0}} \over {\rm \Pi}} & = {\bf 0} \cr & \qquad \Rightarrow \left\{ \matrix{{\rm \Pi} = - B\left( {{{\rm \Pi} _0} = 0} \right) = \displaystyle{{{{(\Gamma - 1)}^2}} \over \Gamma} {\rm \kappa (\kappa} - {{\rm \kappa} _0}{\rm )} \hfill \cr {\rm \kappa} \equiv \displaystyle{{{{\rm \rho} _1}} \over {{{\rm \rho} _0}}} \ge {{\rm \kappa} _0} \hfill} \right.}$$
$$\eqalign{& \displaystyle{{{P_0}} \over {{P_1}}} = \displaystyle{{{{\rm \Pi} _0}} \over {\rm \Pi}} \ne {\bf 0} \Rightarrow \left\{ \matrix{{{\rm \Pi} ^2} + B{\rm \Pi} + C = 0 \hfill \cr {\rm \kappa} \equiv \displaystyle{{{{\rm \rho} _1}} \over {{{\rm \rho} _0}}} \ge 1 \hfill} \right. \cr & {\rm \Pi} = \left( {\displaystyle{1 \over 2}} \right)\left( { - B \pm \sqrt {{B^2} - 4C}} \right) \cr & B = \displaystyle{{{{(\Gamma - 1)}^2}} \over \Gamma} ({{\rm \kappa} _0}{\rm \kappa} - {{\rm \kappa} ^2}) + \,{{\rm \Pi} _0}(\Gamma - 1)(1 - {{\rm \kappa} ^2}) \cr & {\rm C =} \displaystyle{{{{(\Gamma - 1)}^2}} \over \Gamma} ({\rm \kappa} - {{\rm \kappa} _0}{{\rm \kappa} ^2}) \;{{\rm \Pi} _0} - {{\rm \kappa} ^2}{{\rm \Pi} _0}^2}$$The compression κ as a function of the dimensionless pressure Π = P/(ρ0c2) is given in Figure 9 for κ0 = 4 (Γ = 5/3). Although P 0/P 1 is extremely small one cannot neglect it in the very near vicinity of κ0 and in this domain one has to solve numerically Eq. (A6). Furthermore, in order to see the transition between the relativistic and non-relativistic approximation one has to solve the relativistic equations in order to see the transition effects like the one shown in Figure 9.
Compressibility as a function of the normalized shock pressure.
For convenience we write the non-relativistic Hugoniot equations for the ideal gas EOS:
$$\eqalign{& ({\rm i}) \;{u_{{\rm p1}}} = {\left[ {{P_1} - {P_0}} \right]^{1/2}}{\left( {\displaystyle{1 \over {{{\rm \rho} _0}}} - \displaystyle{1 \over {{{\rm \rho} _1}}}} \right)^{1/2}} \cr & ({\rm ii}) \;{u_{\rm s}} = \left( {\displaystyle{1 \over {{{\rm \rho} _0}}}} \right)\displaystyle{{{{\left[ {{P_1} - {P_0}} \right]}^{1/2}}} \over {{{\left( {\displaystyle{1 \over {{{\rm \rho} _0}}} - \displaystyle{1 \over {{{\rm \rho} _1}}}} \right)}^{1/2}}}} \cr & ({\rm iii}) \;{E_1} - {E_0} = \left( {\displaystyle{1 \over 2}} \right)\left[ {{P_1} + {P_0}} \right]\left( {\displaystyle{1 \over {{{\rm \rho} _0}}} - \displaystyle{1 \over {{{\rm \rho} _1}}}} \right) \cr & \left. \matrix{({\rm iv})\hfill \cr ({\rm v}) \hfill} \right\} \;{E_j} = \left( {\displaystyle{1 \over {\Gamma - 1}}} \right)\left( {\displaystyle{{{P_j}} \over {{{\rm \rho} _j}}}} \right) \,{\rm for}\, j{\rm \,= 0,1}}$$These equations are obtained from the relativistic Eq. (A1) by using e = ρc2 + ρE, P, and ρE are much smaller than ρc2 and u/c ≪ 1 where u stands for the velocities under consideration.
In the transition domain, between relativistic and non-relativistic shock waves we have (see Fig. 2)
$${10^{ - 9}} \le {\rm \Pi} \le {10^{ - 3}} \Leftrightarrow {\rm \kappa} = \displaystyle{{\rm \rho} \over {{{\rm \rho} _0}}} = 4.00$$In the domain defined by Eq. (A8) we have u s/c ≪ 1 and u p/c ≪ 1 and the non-relativistic Eq. (A7) are satisfied. Therefore in this transition domain the particle and shock wave velocities and the dimensionless shock wave pressure Π, normalized by ρ0c2, are obtained from the non-relativistic equations, namely
$$\eqalign{& \displaystyle{{{u_{\rm p}}} \over c} = \sqrt {\displaystyle{3 \over 4}{\rm \Pi}} ;\displaystyle{{{u_{\rm s}}} \over c} = \sqrt {\displaystyle{4 \over 3}{\rm \Pi}} \cr & {\rm \Pi} = \left( {\displaystyle{{{u_{\rm s}}} \over c}} \right)\left( {\displaystyle{{{u_{\rm p}}} \over c}} \right) \cr & \displaystyle{{{u_s}} \over c} - \left( {\displaystyle{{{u_p}} \over c}} \right) = \displaystyle{1 \over 3}\left( {\displaystyle{{{u_p}} \over c}} \right)}$$For the intermediate domain, relativistic to non-relativistic case where κ = 4, the shock wave fast ignition driver W d is given by
$${W_{\rm d}}\left[ {\displaystyle{{{\rm erg}} \over {{\rm c}{{\rm m}^3} \cdot {\rm s}}}} \right] = \displaystyle{1 \over 2}\left( {\displaystyle{{{\rm \kappa} {{\rm \rho} _0}{u_{\rm p}}^2} \over {{{\rm \tau} _L}}}} \right)$$This relation is further elaborated in Appendix C.
$$\eqalign{& \displaystyle{{{u_{\rm p}}} \over c} = \sqrt {\displaystyle{{2{w_{\rm d}}{\tau _{\rm L}}} \over {{\rm \kappa} {{\rm \rho} _0}{c^2}}}} \cr & {\rm \Pi} = \left( {\displaystyle{8 \over 3}} \right)\displaystyle{{{w_{\rm d}}{\tau _{\rm L}}} \over {{\rm \kappa} {{\rm \rho} _0}{c^2}}}}$$In this domain we also have the following relation between the dimensionless laser irradiance and the shock pressure
$$\displaystyle{{{I_{\rm L}}} \over {{{\rm \rho}_0}{c^3}}} \equiv {{\rm \Pi} _{\rm L}} \simeq \displaystyle{{\rm \Pi} \over 2}$$
$${I_{\rm L}}\left[ {\displaystyle{W \over {{\rm c}{{\rm m}^2}}}} \right] = 4 \times {10^3}\displaystyle{{{W_{\rm d}}\left[ {\displaystyle{{{\rm erg}} \over {{\rm c}{{\rm m}^3} \cdot {\rm s}}}} \right]{{\rm \tau} _{\rm L}}\left[ {\rm s} \right]} \over {\rm \kappa}}$$The shock wave length domain l s = R is related to the deposition power density by:
$$\eqalign{& {\ell _{\rm s}} = ({u_{\rm s}} - {u_{\rm p}}){{\rm \tau} _{\rm L}} = \displaystyle{{{u_{\rm p}}{{\rm \tau} _{\rm L}}} \over 3} = \displaystyle{{{{\rm \tau} _{\rm L}}} \over 3}\sqrt {\displaystyle{{2{W_{\rm d}}{{\rm \tau} _{\rm L}}} \over {{\rm \kappa} {{\rm \rho} _0}}}} \cr & {\rm \rho} R = {\rm \rho} {\ell _{\rm s}} = \sqrt {\displaystyle{{2{W_{\rm d}}{\rm \kappa} {{\rm \rho} _0}{{\rm \tau} _{\rm L}}^3} \over 9}}}$$The laser cross-section area S is given by
$$\eqalign{& {R_{\rm L}} = \displaystyle{3 \over 2}{\ell _{\rm s}} \cr & S = {\rm \pi} {R_{\rm L}}^2 = \displaystyle{{9{\rm \pi}} \over 4}{\ell _{\rm s}}^2 = \displaystyle{{\rm \pi} \over 2}\displaystyle{{{W_{\rm d}}{\tau _{\rm L}}^3} \over {{\rm \kappa} {{\rm \rho} _0}}}}$$Therefore the laser energy W L and the irradiance I L have the following scaling law
$$\eqalign{& {W_{\rm L}}\left[ J \right] = {I_{\rm L}}\left[ {\displaystyle{W \over {{\rm c}{{\rm m}^2}}}} \right]{{\rm \tau} _{\rm L}}\left[ s \right]S\left[ {{\rm c}{{\rm m}^2}} \right] \cr & {W_{\rm L}}\left[ J \right] = 2{\rm \pi} \times {10^3}\displaystyle{{{W_{\rm d}}{{\left[ {\displaystyle{{{\rm erg}} \over {{\rm c}{{\rm m}^3} \cdot {\rm s}}}} \right]}^2}{{\rm \tau} _{\rm L}}{{\left[ s \right]}^5}} \over {{{\rm \kappa} ^2}{{\rm \rho} _0}}} \cr & {I_{\rm L}}\left[ {\displaystyle{W \over {{\rm c}{{\rm m}^2}}}} \right] = \displaystyle{{4 \times {{10}^3}{W_{\rm d}}{{\rm \tau} _{\rm L}}} \over {\rm \kappa}}}$$The density times hot spot dimension scaling law is:
$${\rm \rho} R = {\rm \kappa} {{\rm \rho} _0}{\ell _{\rm s}} \propto {W_{\rm d}}^{0.5} {{\rm \kappa} ^{0.5}}{{\rm \rho} _0}^{0.5} {{\rm \tau} _{\rm L}}^{1.5}$$Finally, for a constant ρ·R the relevant scaling laws are
$$\eqalign{& {W_{\rm L}} \propto \displaystyle{1 \over {{{\rm \tau} _{\rm L}}{{\rm \rho} _0}^3 {{\rm \kappa} ^3}}} \cr & {I_{\rm L}} \propto \displaystyle{1 \over {{{\rm \tau} _{\rm L}}^2 {{\rm \rho} _0}{{\rm \kappa} ^2}}} \cr & }$$For example, the following table summarizes the dimensionless numerical values of our laser and shock wave parameters
Shock wave parameters relevant for fast ignition
For example, if the pre-compressed target has a mass density of 250 g/cm3 and the laser pulse duration is 1 ps then for the dimensionless laser irradiance of ΠL = 6.65 × 10−5 we get
$$\eqalign{{{\rm \Pi} _{\rm L}} &= 6.65 \times {10^{ - 5}}; {{\rm \tau} _{\rm L}} = 1\,ps; \cr & \quad {{\rm \rho} _0} = 250\left[ {\displaystyle{{\rm g} \over {{\rm c}{{\rm m}^3}}}} \right] \Rightarrow {\rm \rho = \kappa} {{\rm \rho} _0} = {10^3}\left[ {\displaystyle{{\rm g} \over {{\rm c}{{\rm m}^3}}}} \right] \cr {I_{\rm L}} =\; & {{\rm \rho} _0}{c^3}{{\rm \Pi} _{\rm L}} = 4.5 \times {10^{22}}\left[ {\displaystyle{W \over {{\rm c}{{\rm m}^2}}}} \right] \cr {\ell _{\rm s}} = & \;({u_{\rm s}} - {u_{\rm p}}){\tau _L} \simeq 1\,\,{\rm \mu m}; \cr & S = {\rm \pi} {R_{\rm L}}^2 = {\rm \pi (}1.5{\ell _{\rm s}}{{\rm )}^2} = 7.04 \times {10^{ - 8}}[{\rm c}{{\rm m}^2}] \cr {W_{\rm L}}[J] = &\; {I_{\rm L}}S{{\rm \tau} _{\rm L}} = 3.2\,\,{\rm kJ}}$$APPENDIX B: ELECTRON AND ION RELAXATION TIMES
The relaxation rates for electrons and for ions accordingly ν ee, ν ii, or equivalently the times τee, τii, that a species of particles reaches equilibrium, due to Coulomb collisions between electron–electron (ee) and separately ion–ion are given by (Eliezer, Reference Eliezer2012):
$$\eqalign{& \displaystyle{1 \over {{{\rm \nu} _{{\rm ee}}}}} = {{\rm \tau} _{{\rm ee}}}{\rm \,=\,} \left( {\displaystyle{{3\sqrt 6} \over {8{\rm \pi}}}} \right)\displaystyle{{\sqrt {{m_{\rm e}}} {{\left( {{k_{\rm B}}{T_{\rm e}}} \right)}^{3/2}}} \over {{e^4}{n_{\rm e}}\ln {\Lambda _{{\rm ee}}}}} \cr & \displaystyle{1 \over {{{\rm \nu} _{{\rm ii}}}}} = {{\rm \tau} _{{\rm ii}}} = \displaystyle{1 \over {{Z^4}}}\left\{ {\left( {\displaystyle{{3\sqrt 6} \over {8{\rm \pi}}}} \right)\displaystyle{{\sqrt {{m_{\rm i}}} {{\left( {{k_{\rm B}}{T_{\rm i}}} \right)}^{3/2}}} \over {{e^4}{n_{\rm i}}\ln {\Lambda _{{\rm ii}}}}}} \right\}}$$Numerically, Eq. (A20) can be written
$$\eqalign{& {{\rm \tau} _{{\rm ee}}}[s] = 1.07 \times {10^{10}}\left( {{{\displaystyle{1 \over {{n_{\rm e}}\ln {\rm \Lambda}}}} _{{\rm ee}}}} \right){[{T_{\rm e}}({\rm keV})]^{3/2}} \cr & {{\rm \tau} _{{\rm ii}}}[s] = 4.58 \times {10^{11}}\left( {\displaystyle{1 \over {{n_{\rm i}}\ln {\Lambda _{{\rm ii}}}}}} \right){[{T_{\rm i}}({\rm keV})]^{3/2}}}$$Regarding the fast ignition of a pre-compressed target with n e[cm−3] = 1026 and a laser pulse duration of 1 ps we have as order of magnitude estimates:
$$\eqalign{& {{\rm \tau} _{{\rm ee}}}({\rm DT}) \sim {10^{ - 17}}[s](1\,{\rm keV}) - {10^{ - 15}}[s](20\,{\rm keV}) \cr & {{\rm \tau} _{{\rm ii}}}({\rm DT}) \sim 4 \times {10^{ - 16}}[s](1\,{\rm keV}) - 4 \times {10^{ - 14}}[s](20\,{\rm keV}) \cr & {{\rm \tau} _{{\rm ee}}}(\,p{}^{11}B) \sim {10^{ - 17}}[s](1\,{\rm keV}) - 3 \times {10^{ - 14}}[s](50\,{\rm keV}); \cr & {{\rm \tau} _{{\rm ii}}}(\,p{}^{11}B) \sim \displaystyle{1 \over {{Z^3}}}\left\{ {{{10}^{ - 15}}[s](1\,{\rm keV}) - {{10}^{ - 12}}(200\,{\rm keV})[s]} \right\}}$$For the above numerical estimations we took lnΛ = 10.
APPENDIX C: SHOCK WAVE INDUCED TWO TEMPERATURES MODEL
We use the non-relativistic shock wave equations to justify our two temperature model equations. In this case the kinetic energy W k, the potential (internal) energy W p and their appropriate power densities are related to the particle flow velocity u p by:
$$\eqalign{& \displaystyle{{{W_{\rm k}}} \over V}\left[ {\displaystyle{{{\rm erg}} \over {{\rm c}{{\rm m}^3}}}} \right] = ({\rm \gamma} - 1){\rm \rho} {c^2}\left( {\displaystyle{t \over {{{\rm \tau} _{\rm L}}}}} \right) = \displaystyle{1 \over 2}{\rm \rho} {u_{\rm p}}^2 \left( {\displaystyle{t \over {{{\rm \tau} _{\rm L}}}}} \right) \cr & {W_{\rm d}} = \displaystyle{{\rm d} \over {{\rm d}t}}\left( {\displaystyle{{{W_{\rm k}}} \over V}} \right) = \displaystyle{1 \over 2}\left( {\displaystyle{{{\rm \rho} {u_{\rm p}}^2} \over {{{\rm \tau} _{\rm L}}}}} \right)}$$
$$\eqalign{& {U_{\rm p}}\left[ {\displaystyle{{{\rm erg}} \over {{\rm c}{{\rm m}^3}}}} \right] = \displaystyle{{{W_{\rm p}}} \over V} = \displaystyle{3 \over 2}n{k_{\rm B}}T \cr & \displaystyle{{\rm d} \over {{\rm d}t}}\left( {\displaystyle{{{W_{\rm p}}} \over V}} \right) = \displaystyle{3 \over 2}{k_{\rm B}}\displaystyle{{\rm d} \over {{\rm d}t}}(nT) = \displaystyle{3 \over 2}n{k_{\rm B}}\displaystyle{{{\rm d}T} \over {{\rm d}t}}}$$At this stage we have one fluid with temperature and number density T and n, accordingly and neglect all losses or fusion energy creation. In the second equation in (A24) we have assumed that n does not change in time.
The 1D non-relativistic Hugoniot equations in the laboratory frame of reference, assuming a fluid initially at rest, are the following mass, momentum, and energy conservations
$$\eqalign{& {{\rm \rho} _0}{u_{\rm s}} = {\rm \rho} ({u_{\rm s}} - {u_{\rm p}}) \cr & {{\rm \rho} _0}{u_{\rm s}}{u_{\rm p}} = P - {P_0} \cr & {{\rm \rho} _0}{u_{\rm s}}\left( {E - {E_0} + \displaystyle{1 \over 2}{u_{\rm p}}^2} \right) = P{u_{\rm p}}}$$The conservation energy of the third equation of (A25) can be written as
$$\eqalign{& {\rm Piston work} = {\rm Internal energy + Kinetic energy} \cr & {\rm Piston work} = \int {PdV = P(S{u_{\rm p}}t)} \cr & {\rm Internal energy} = {W_{\rm p}} = ({{\rm \rho} _0}S{u_{\rm s}}t)(E - {E_0}) \cr & {\rm Kinetic energy} = {W_{\rm K}} = ({{\rm \rho} _0}S{u_{\rm s}}t)\left( {\displaystyle{1 \over 2}{u_{\rm p}}^2} \right)}$$S is the cross-section area of the piston and E is the internal energy per unit mass. Since the initial pressure P 0 is negligible relative to the shock pressure P in our cases we get from the second and third equations of (A25)
$$\left. \matrix{P{u_{\rm p}} = {{\rm \rho} _0}{u_{\rm s}}{u_{\rm p}}^2 \hfill \cr P{u_{\rm p}} = {{\rm \rho} _0}{u_{\rm s}}\left( {E - {E_0} + \displaystyle{1 \over 2}{u_{\rm p}}^2} \right) \hfill} \right\} \Rightarrow {\rm \rho (}E - {E_0}{\rm )} = \displaystyle{1 \over 2}{\rm \rho} {u_{\rm p}}^2$$Inserting Eqs (A27) and (A26) into (A23) and (A24) we get for time independent density n,
$$\displaystyle{3 \over 2}n{k_{\rm B}}\displaystyle{{{\rm d}T} \over {{\rm d}t}} = {W_{\rm d}}$$During the time development of the temperature we have to take into account losses and the extra input energy due to nuclear fusion, namely
$$\displaystyle{3 \over 2}n{k_{\rm B}}\displaystyle{{{\rm d}T} \over {{\rm d}t}} = {W_{\rm d}} + {\,f_{\rm \alpha}} {W_{\rm f}} - \sum {{\rm losses}}$$Since we have two temperatures (at least) T e and T i we generalize Eq. (A29) to
$$\eqalign{& \displaystyle{3 \over 2}{n_{\rm e}}{k_{\rm B}}\displaystyle{{{\rm d}{T_{\rm e}}} \over {{\rm d}t}} = {{\rm \eta} _{\rm d}}{W_{\rm d}} + {\,f_{\rm \alpha}} {{\rm \eta} _{\rm f}}{W_{\rm f}} - \sum\limits_{\rm e} {{\rm losses}} \cr & \displaystyle{3 \over 2}{n_{\rm i}}{k_{\rm B}}\displaystyle{{{\rm d}{T_{\rm i}}} \over {{\rm d}t}} = (1 - {{\rm {\rm \eta}} _{\rm d}}){W_{\rm d}} + {\,f_{\rm \alpha}} (1 - {{\rm \eta} _{\rm f}}){W_{\rm f}} - \sum\limits_{\rm i} {{\rm losses}}}$$This model is schematically described in Figure 4.
