Heating a solid material with a laser-accelerated charged particle beam is an effective way to generate a powerful plane shock wave with a pressure of several hundred or even thousands of megabars (Mbar)^1–3 in a laboratory experiment. This is due to the fact that the energy flux density of such a beam is close to the intensity of the laser pulse that produces it. At the same time, in contrast to laser radiation, charged particles transmit their energy in Coulomb collisions and are thus able to heat a substance with a density that significantly exceeds the critical plasma density. The continual growth in the energies available at modern laser facilities with terawatt and petawatt power outputs allows us to consider laser-accelerated charged particle beams as an effective tool for generating the super-powerful shock waves in sufficiently large volumes of matter that meet the needs of such important applications as inertial confinement fusion (ICF), studies of the equations of state (EOS) of materials, and laboratory astrophysics. It should be noted that the above comments apply primarily to laser-accelerated electron beams, since the efficiency of laser energy conversion into fast electron energy is significantly (two to three times) larger than the efficiency of conversion into fast ion energy.

The potential capabilities of shock wave generation by heating a solid material with a petawatt flux of laser-accelerated fast electrons go beyond those of the method based on the impact of laser-accelerated pellets,^4,5 which has been shown to be capable of generating a plane shock wave with the record pressure of 740 Mbar.⁴ This pressure is close to the maximum achievable with the impact method, since the collisional mechanism of laser radiation absorption is limited by the value of the coupling parameter Iλ² ≈ 2 × 10¹⁴ W μm²/cm² (where I and λ are respectively the intensity and wavelength of the laser radiation). The record pressure was determined by the limiting intensity for the third Nd-laser harmonic of 10¹⁵ W/cm². The method based on direct heating of a solid material by laser-accelerated fast electrons is designed to use intensities exceeding the colli nsional absorption limit, for which a significant fraction of laser energy is transformed into the energy of fast electrons. It was shown in Ref. 6 that the use of a laser pulse with an intensity of I ≈ 10¹⁹ W/cm²–10²¹ W/cm² to heat a solid by fast electrons can generate a plane shock wave with a pressure of several tens of gigabars (Gbar). Higher pressures of up to several hundred Gbar can be achieved in the laboratory only in the case when a spherical target is imploded under the impact of a terawatt laser pulse. Therefore, in, for example, EOS studies, heating by laser-accelerated fast electrons offers great promise, bearing in mind the complicated diagnostic methods that are required when a spherical configuration is used compared with the traditional methods that can be applied in experiments with a plane shock wave.

This work is devoted to further study of the properties of shock waves driven by heating solid materials with laser-accelerated electron beams. In Ref. 6, it was shown that with this heating method, radiation energy loss is the main factor limiting the temperature of the plasma that is produced. In the compressed material behind the shock wave front, radiation energy loss plays an even greater role than in the heated region. The extreme compression of matter due to radiation cooling in a shock wave driven by laser-accelerated fast electrons is studied using both computational and theoretical approaches. This increase in compression due to radiation cooling is well known in the context of laminar flows in Z pinches^7–10 and in ICF target^10,11 implosions. It should be noted that the radiation cooling effect is a central problem in astrophysics and laboratory astrophysics, particularly in the context of accretion physics¹² and radiative laboratory shock physics.¹³

In this paper, the effect of radiation cooling behind the front of a powerful shock wave is considered. It is shown that this cooling leads to compression of matter in the peripheral region of the shock wave to a density several times greater than that at its front. First, in Sec. II, the features of the radiation cooling effect in a powerful shock wave driven by heating a dense material with laser-accelerated electrons are discussed. Then, in Sec. III, the results of numerical calculations are presented and discussed. Finally, Sec. IV presents the conclusions of the study.

Using the expressions (1) and (2) for the energy and the mass range of fast electrons, it is easy to find that the characteristic time of thermodynamic state evolution t_h increases with both the laser intensity and wavelength. In the nonrelativistic case, t_h grows as $t_h\sim I_L^{1/3}{\lambda }^{4/3}$, and in the relativistic case, it grows as $t_h\sim I_L^{1/6}\lambda$. In the case of an aluminum plasma at I_L = 10¹⁸ W/cm², with λ = 1.06 µm (the first harmonic of the Nd laser), which corresponds to ε_h = 200 keV, and at a conversion efficiency η = 0.2, the ablation loading time is about 130 ps.

The scales of pressure P_h and temperature T_h during the quasistatic heating depend only on the flux heating energy density I_h and are independent of the heating particle energy ε_h. This is a fundamental difference between material heating and shock wave generation driven by a charged particle beam and those driven by a laser pulse. The pressure and temperature of a laser-heated material depend on the energy of a light quantum through the critical plasma density ρ_cr, which is the scale of density in the region of absorption of radiation with the given quantum energy:^18,19$T_L\sim {(I_L/{\rho }_{\text{cr}})}^{2/3}$ and $P_L\sim {\rho }_{\text{cr}}^{1/3}{I}_L^{2/3}$ (where ${\rho }_{\text{cr}}\approx 1.83\times 10^{-3}A/Z{\lambda }_{\mu }^2$ g/cm³, with λ_μ being the laser radiation wavelength measured in units of μm).

There now follows a discussion of plasma heating by a beam of laser-accelerated fast electrons from the point of view of the establishment of the thermodynamic state of the resulting plasma. For a deceleration length of a subrelativistic fast electron of about 100 µm–200 µm, the time taken for its energy to be transferred to plasma electrons is about 1 ps. This is much shorter than the hydrodynamic time of the problem, which is about 100 ps–200 ps, and thus a fast electron transfers its energy to the plasma with stationary distributions of thermodynamic parameters. In turn, the energy transfer time is much longer than the plasma relaxation times, i.e., the electron–electron and electron–ion energy relaxation times, which for a plasma density in the heating region of several g/cm³ and a temperature of several keV are 0.0001 ps and 0.1 ps, respectively. The collisional ionization time is about 0.001 ps and involves the participation of plasma electrons with a Maxwellian spectrum. The main recombination mechanism under the conditions of the problem under consideration is triple collision. The recombination rate in a triple collision exceeds the photorecombination rate by more than 100 times, which means that a Saha equilibrium in terms of the ionization composition is established. Thus, heating of the material by laser-accelerated fast electrons occurs while the local thermodynamic equilibrium of the plasma is maintained.

The bulk of numerical calculations were performed using the one-dimensional two-temperature hydrodynamic code DIANA,²¹ supplemented by a module of energy transfer by fast electrons.²² The DIANA code takes into account all the main relaxation and transport processes in the plasma and the real equations of state. The energy transfer by intrinsic plasma radiation is considered under the approximation of volumetric losses due to the bremsstrahlung mechanism. The cooling function operates through a threshold frequency for photons: all photons with frequency above the threshold value leave the plasma and take away their energy, while for photons with frequency below the threshold, the energy is deposited locally. This code has been used successfully in investigations of the effect of energy transfer by fast electrons on the gain of direct-drive ICF targets.^22,23 The energy transfer from fast electrons in Coulomb collisions is calculated using a model of the stopping power of the plasma electrons adjusted for scattering by the plasma ions.^2,16,17 The controlling calculations were performed using the RADIAN numerical code,^24,25 which solves the one-dimensional equations of two-temperature gas dynamics together with the radiation transfer equation. When solving the radiative transfer equation, this code uses the multigroup approximation, the method of characteristics, the quasidiffusion method, and averaging over photon energies to effectively reduce the dimensions of the equation. These calculations showed that taking radiation transfer into account led to insignificant deviations of the thermodynamic parameters at a level of 5%–10% from the DIANA code results. Calculations using the DIANA code were performed in the range of fast electron energy flux density I_h = 2 × 10¹⁶ W/cm²–2 × 10¹⁹ W/cm², which, with the supposed efficiency of laser energy conversion into fast electron energy of η = 0.2, corresponds to a range of variation in laser intensity of I_L = 10¹⁷ W/cm²–10²⁰ W/cm². The initial energy of fast electrons was chosen according to the scaling (1) for the first harmonic of the Nd-laser radiation, and varied from a nonrelativistic energy ε_h = 100 keV to a relativistic energy ε_h = 3.8 MeV.

Figures 1–3 show the results of a numerical calculation of shock wave propagation in aluminum under a fast electron energy flux density of I_h = 2 × 10¹⁷ W/cm² and an initial energy ε_h = 200 keV (which corresponds to an intensity of the first harmonic of Nd-laser radiation I_L = 10¹⁸ W/cm² at a conversion efficiency η = 0.2). The mass range of fast electrons with the energy indicated above is about μ_h ≈ 0.028 g/cm². The mass range varies slightly with increasing temperature and decreasing density in the heated region, in accordance with the change in the Coulomb logarithm. Figure 1 shows the distributions of pressure, electron temperature, and density over the mass coordinate from the boundary of the half-space, on which the source of fast electrons was set, to the shock wave front at different moments in time. The mass coordinates of the edge of the half-space and the inner boundary of the heated region are respectively 0.54 g/cm² and about 0.512 g/cm². The time evolution of the thermodynamic parameter distributions is shown in the region heated by fast electrons and the shock wave region. The pressure has a fairly uniform distribution behind the front of the shock wave. Moreover, it varies nonmonotonically with time. From the initial time, the pressure behind the shock wave front reaches a value of about 10 Gbar at about 120 ps, after which it decreases to a minimum of about 6.5 Gbar at about 200 ps, then increases again, reaching a second maximum of about 7.5 Gbar at about 500 ps, and further decreases monotonically. The temperature distribution has a minimum and the density distribution has a maximum in the peripheral region of the shock wave, the positions of which fall at the same value of the mass coordinate. The maximum density in the radiation-cooled region reaches a value of about 45 g/cm³ at about 500 ps.

Figure 2 shows the time dependences of the numerically calculated values of the maximum pressure and maximum density, together with the time dependence of the pressure in the heated region, calculated on the basis of the analytical solutions (3)–(6) with a fraction of radiation energy loss of 40%. In the initial period up to 120 ps–150 ps, when the initial stage of shock wave formation occurs, the maximum pressure is determined by the pressure in the heated region. The solution (3) for the stage of quasistatic heating is in good agreement with the numerically calculated dependence. The pressure increases with time in a nearly linear manner and reaches a value of about 10 Gbar by a time of about 130 ps–150 ps. The ablation loading time t_h is about 150 ps, and the maximum value of the numerically calculated pressure is reached at t = 130 ps. During the ablation loading period, the temperature behind the shock wave is about 1 keV and the length of shock wave propagation is about 0.002 cm. The corresponding averaged mean free path of thermal photons is 0.01 cm, which is five times greater than the length of shock wave propagation. All of these values are in good agreement with the estimates in Sec. II, made on the basis of the expressions (3)–(9). As a result, the whole shock wave is transparent, and the growth in density begins from the wave front and reaches its maximum in the peripheral back region. At subsequent moments in time, the shock wave propagates under conditions of expansion of the heated region as a whole and ablation of matter at a depth exceeding the mass range of fast electrons. This minimizes the numerically calculated pressure dependence at t = 180 ps. The steady motion of the shock wave ends at a time of about 400 ps, after which the pressure decreases monotonically with time. The pressure in the steady-state shock wave is 30%–40% higher than the pressure from the analytical solution (4) in the heated region, which is a consequence of hydrodynamic energy transfer to a substance of higher density. At the end of the period of steady-state movement of the shock wave, the maximum value of the density is reached in its peripheral part. Subsequently, the maximum density decreases in accordance with the decrease in pressure.

Figure 3 shows the profiles of pressure, density, electron temperature, and plasma emissivity at t = 500 ps, corresponding to the attainment of maximum density values in the peripheral part of the shock wave. An increase in density in the peripheral region of the shock wave occurs near the maximum value of the plasma emissivity. The maximum compression to a density of 45 g/cm³ occurs in the region that is cooled to the maximum extent owing to radiation energy losses under the action of the maximum pressure developed in the shock wave. The degree of compression due to radiation cooling increases with laser pulse intensity, owing to increases both in pressure and in radiation energy loss. In the calculation for impact of radiation of the first harmonic of the Nd laser with intensity I_L = 10¹⁹ W/cm² (ε_h = 1.2 MeV) and a conversion efficiency from laser energy to fast electron energy of η = 0.2, the maximum density reaches a value of 65 g/cm³.

Numerical simulations conducted for a Cu target show an increase in the maximum density in the peripheral area of the shock wave compared with the result for an aluminum target. In the case of impact of a laser beam of intensity 10¹⁸ W/cm² (ε_h = 200 keV and η = 0.2), the maximum density reaches a value of about 58 g/cm³, whereas for a laser intensity 10¹⁹ W/cm² and a particle energy of 1.2 MeV, it reaches a value of 66 g/cm³. The saturation of the increase in density in the peripheral region of the shock wave with an increase in atomic number is associated with saturation of the increase in radiation energy losses: the increase in emissivity is compensated by a decrease in the transparency of the radiation region. In particular, in the numerical simulation of a Cu target with a laser intensity of 10¹⁸ W/cm², the time of ablation loading t_h is about 60 ps and the temperature T_h is about 0.5 keV. Both of these values are approximately half of those obtained from the calculations for an Al target, in accordance with (4) and (5). The corresponding averaged mean free path of thermal photons l_r in the Cu plasma, with its density approximately three times that of the Al plasma, is about 0.000 06 cm. During the period of ablation loading, the length of shock wave propagation is about 0.0005 cm, which is eight times larger than l_r. So, in contrast to the case of an Al target, where all of the shock wave is transparent and the growth in density begins from the wave front and reaches its maximum in the peripheral back region, the main part of the shock wave in Cu target is opaque, and a growth in density occurs only in the narrow peripheral region. It should be noted that to achieve higher densities, the intensity of the fast electron flux needs to be increased while the initial energies of the particles are kept the same. This has not yet been achieved in current experiments, although there is now the possibility of using laser acceleration as a powerful source of fast electrons.

The calculation results that have been presented here allow one to determine the experimental conditions for EOS studies. For a laser pulse with an intensity of 10¹⁸ W/cm², the pulse duration corresponding to the maximum density values is about 100 ps. To generate a nearly plane shock wave, the radius of the laser beam must be at least two lengths of thermal expansion of the heating region, i.e., about 100 µm. This means that the energy of the laser pulse in this case should be about 4 kJ–5 kJ.

Shock waves generated by heating solid materials using fast electrons may be of interest for laboratory studies of the physics of radiative shocks. The generation of such waves is due to the work done by a dense, high-temperature piston. As a result, a stratified strongly collisional plasma structure is formed (with a Coulomb logarithm of about 10), with inhomogeneous distributions of thermodynamic parameters. The region of strong radiation, located in the peripheral region of the shock wave at the interface with the piston, is relatively narrow, with a mass that does not exceed 10% of the mass of the piston. Electron thermal conductivity has a negligible effect on the thermodynamic state of this region. The radiation region, in the form of a plasma structure created by heating, is largely transparent to radiation. The optical thickness of the region heated by fast electrons (the high-temperature piston) is 0.0001. The optical thickness in the region of strong radiation is about 0.1 for mid-Z materials such as Al and about 0.5 for high-Z materials such as Cu. However, the optical thickness in the entire region covered by the shock wave can vary from fractions of unity for mid-Z materials up to several multiples of unity for high-Z materials. In the conditions of the problem under consideration, the Péclet number with respect to the electron thermal conductivity (the ratio of the hydrodynamic velocity to the velocity of the wave front of electron heat conduction) for the radiation region is quite large for both Al and Cu targets, with values of about 200–300. The Péclet number with respect to radiation is different for different regions of the stratified plasma structure. In the region of strong radiation, its value is about 10⁶ for an Al target and 10⁵ for a Cu target. The cooling parameter (the ratio of cooling time to hydrodynamic time) is about 0.01 for mid-Z materials and about 0.1 for high-Z ones. Plasma structures with such properties, created by heating a material with a petawatt flux of laser-accelerated fast electrons, are of likely interest with regard to laboratory astrophysics in connection with the effect of radiation cooling as controlled by changes in the laser pulse intensity and by appropriate selection of the target material.

Heating a solid material with a beam of laser-accelerated fast electrons can generate shock waves with properties particularly suitable for laboratory experiments. The pressure behind the front of such a shock wave can reach several hundreds and even thousands of Mbar at a temperature of several keV. Such a high temperature causes a high rate of radiation energy loss in a dense plasma, which is then partially transparent to intrinsic radiation. As a result, the conditions arise for strong radiation cooling and consequently increased plasma compression in the peripheral part of the shock wave. The theoretical and computational studies reported here show that under the impact of radiation of the first harmonic of a Nd laser with intensity 10¹⁷ W/cm²–10¹⁹ W/cm², the density of an aluminum plasma in the peripheral region of a fast-electron-driven shock wave can reach 30 g/cm³–60 g/cm³, while a density of 50 g/cm³–70 g/cm³ can be reached for a copper plasma. With an increase in the atomic number of the irradiated material, saturation of the increase in density occurs in the peripheral region of the shock wave, which is associated with saturation of the growth of radiation energy losses, where the increase in emissivity is compensated by a decrease in the transparency of the radiation region. This approach offers the possibility of investigating the states of materials at pressures of several Gbar, temperatures of several keV, and densities of several tens of g/cm³ and thus represents a new avenue for laboratory studies of material EOS. Investigation of the increase in density in the peripheral region of the shock wave, which is determined by radiative energy losses, is of great interest for establishing the optical properties of materials with high atomic number. In addition, the dependence of the increase in density on the degree of ionization makes it possible to study the kinetics of ionization of matter at ultrahigh pressures, which is a topic of great importance for physics at high energy densities. Such experiments can be performed using a sub-nanosecond laser pulse with an energy of 1 kJ–10 kJ.