Extreme matter compression caused by radiation cooling effect in gigabar shock wave driven by laser-accelerated fast electrons

Heating a solid with laser-accelerated fast electrons is unique way for a laboratory experiment to generate a plane powerful shock wave with a pressure of several hundred or even thousands of Mbar. Behind the front of such a powerful shock wave, dense plasma is heated to a temperature of several keV. Then, a high rate of radiation energy loss occurs even in low-$Z$ plasmas. The effect of strong compression of matter due to radiation cooling in a gigabar shock wave driven by fast electrons is found in computational and theoretical researches. It is shown that the effect of radiation cooling leads to the compression of matter in the peripheral region of shock wave to a density several times larger than the density at its front. Heating a solid by a petawatt flux of laser-accelerated fast electrons allows one to surpass the gigabar pressure level of a plane shock wave, which is the maximum level for the impact of laser-accelerated pellets. Higher pressure about 100 Gbar can be achieved under laboratory conditions only when a spherical target is imploded under the action of a terawatt laser pulse.

(Dated: 16 February 2021) Heating a solid with laser-accelerated fast electrons is unique way for a laboratory experiment to generate a plane powerful shock wave with a pressure of several hundred or even thousands of Mbar. Behind the front of such a powerful shock wave, dense plasma is heated to a temperature of several keV. Then, a high rate of radiation energy loss occurs even in low-Z plasmas. The effect of strong compression of matter due to radiation cooling in a gigabar shock wave driven by fast electrons is found in computational and theoretical researches. It is shown that the effect of radiation cooling leads to the compression of matter in the peripheral region of shock wave to a density several times larger than the density at its front. Heating a solid by a petawatt flux of laser-accelerated fast electrons allows one to surpass the gigabar pressure level of a plane shock wave, which is the maximum level for the impact of laser-accelerated pellets. Higher pressure about 100 Gbar can be achieved under laboratory conditions only when a spherical target is imploded under the action of a terawatt laser pulse.

I. INTRODUCTION
Heating a substance with laser-accelerated charged particle beam is an effective way to generate a plane powerful shock wave with a pressure of several hundred or even thousands of Mbar 1-3 in a laboratory experiment. This is due to the fact that the energy flux density of such beam is close to the intensity of laser pulse that produces it. At the same time, in contrast to laser radiation, charged particles transmit their energy in Coulomb collisions and are able to heat a dense substance with a density that significantly exceeds the critical plasma density. The permanent energy growth of modern laser facilities with terawatt and petawatt power allows us to consider laser-accelerated charged particle beam as an effective tool for generating the super-powerful shock waves in sufficiently large volume of matter that meet the needs of such important applications as inertial confinement fusion (ICF), study of matter equation of state (EOS) and laboratory astrophysics. The above primarily applies to laser-accelerated electron beam, since the efficiency of laser energy conversion into fast electron energy is significantly (2-3 times) larger than into fast ions.
Heating a solid by a petawatt flux of laser-accelerated fast electrons is the most effective method for generating a plane shock wave with the extreme pressure for a laboratory experiment. Its potential capabilities exceed ones of the method of impact of laser-accelerated pellets 4,5 which, in a modern experiment provided the generation of a plane shock wave with a record pressure of 740 Mbar 4 . This record pressure is close to the maximum achievable one when using the impact method, since the collisional mechanism of laser radiation absorption is limited by the value of the coupling parameter Iλ 2 ≈ 2 · 10 14 W µm 2 /cm 2 (I and λ are, respectively, the intensity and wavelength of laser radiation). The record pressure was determined by the limiting intensity for the third Nd-laser harmonic of 10 15 W/cm 2 .
The method of direct heating of a solid by laser-accelerated fast electrons is designed to use intensities exceeding the collisional absorption limit, when a significant fraction of laser energy is transformed into the energy of fast electrons. It was shown in 6 that the use of a laser pulse with an intensity of I ≈ 10 19 − 10 21 W/cm 2 to heat a solid by fast electrons can provide the generation of plane shock wave with a pressure of several tens of Gbar. A higher pressure, up to several hundred Gbar in a laboratory experiment, can be achieved only in the case when a spherical target is imploded under the impact of a terawatt laser pulse. Therefore, to study, for example, the equation of state of matter, the method of heating of solid by laser-accelerated fast electrons is very promising, bearing in mind that diagnostics in a spherical experiment turns out to be a more difficult task in comparison with those traditional methods that can be applied in experiments with a plane shock wave.
This work is devoted to further study of the properties of a shock wave driven by heating a substance with laser-accelerated electron beam. In 6 , it was shown that in this heating method, the radiation energy loss is the main factor limiting the temperature of produced plasma. In the compressed material behind the shock wave front, radiation energy loss plays an even greater role than in the heated region. The effect of extreme matter compression due to radiation cooling in a shock wave driven by laser-accelerated fast electrons is found on the basis of computational and theoretical researches. The effect of matter compression increase due to radiation cooling is well known in relation to laminar flows under Z-pinch [7][8][9][10] and ICF target 10,11 implosions. It should be noted that the radiation cooling effect is a central problem in astrophysics and laboratory astrophysics, in particular, in the sections of accretion physics 12 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 the radiation cooling leads to compression of matter in the peripheral region of shock wave to a density several times larger than the density at its front. First it is discussed the features of the radiation cooling effect in a powerful shock wave driven by heating a dense substance with laser-accelerated electrons. Then the results of numerical calculations are presented and discussed.

LASER-ACCELERATED FAST ELECTRONS
It is considered the generation and propagation of shock wave driven by heating the boundary region of semi-space with monoenergetic laser-accelerated fast electron flow. The dimensional parameters of the problem are the energy flux density of fast electrons I h , the mass range of heating particles in heated substance µ h , which is a function of the initial fast electron energy ε h and the density of substance ρ 0 . According to numerous experiments and theoretical models, the laser energy conversion into fast electron energy η = I h /I L lies in the range 0.1-0.3. Despite this relatively low conversion efficiency, laser-accelerated electrons are the undisputed record holder for energy flux density among charged particle beams of any other laboratory origin, taking into account the laser intensity 10 20 -10 21 W/cm 2 achieved in the modern experiment. The energy ε h increases with laser intensity I L and wavelength λ and can reach ultrarelativistic values. The dependence ε h (I L , λ) is given by well-known scalings 14,15 , which combine the data from numerous experiments and theoretical models where I L (19) and λ µ are measured in 10 19 W/cm 2 and µm.
The calculated results were obtained for the case of impact of laser irradiation on aluminum, which is often used as a reference material in EOS experiments. For estimates, an approximation formulas are used for mass ranges of non-relativistic and relativistic electrons, which are calculated in accordance with the data of 2,16,17 for aluminum plasma with an ion charge Z = 11: where energy ε h is measured in MeV.
The scales of physical quantities of the problem are determined by the thermodynamic parameters of a region heated by fast electrons. In plane geometry, the mass of heated layer remains constant and equal to the fast electron mass range, despite the increase in the heated layer temperature and the decrease in its density due to thermal expansion. Under these conditions, the thermodynamic state of heated layer is described by the solutions of Ref. 1 for a period of quasi-static heating, when the motion of the substance can be ignored and for the subsequent period of thermal expansion of the layer with a constant mass In these expressions t h is the duration of quasi-static heating period or the time of ablation loading (following to the notation of Ref. 1 ), during which isothermal rarefaction wave propagates from the outer surface of semi-space to the inner boundary of the heated layer: is the isothermal sound speed in the heated region, T h and P h are temperature and pressure that are reached at the end of the quasi-static heating period I h is flux energy density of fast electrons, µ h is mass range of the electron with energy ε h , constant, m p is the proton mass, A and Z are the atomic number and ion charge, γ is specific heats ratio.
Using expressions (1) and (2) for energy and mass range of fast electron, it is easy to get that the characteristic time of thermodynamic state evolution t h increases with both the laser intensity and wavelength. In the non-relativistic case t h grows as t h ∼ I From the point of view of achieving an extreme state of matter, the most interesting is the initial period of shock wave propagation, when a quasi-static heating occurs. Then, the characteristics of shock wave can be determined in the approximation of a uniform pressure distribution in the heated region and in the region involved into motion by the shock wave. Then, the velocity of shock wave D SW and the temperature T SW behind its front are expressed in terms of thermodynamic parameters of the heated region as The scales of pressure P h and temperature T h during the quasi-static heating depend only on the flux heating energy density I h and do not depend on the heating particle energy ε h . This is a fundamental difference between substance heating and shock wave generation driven by charged particle beam and laser pulse. The pressure and temperature of laserheated substance depend on the energy of a light quantum through the value of critical plasma density ρ cr , which is scale of density in the region of absorption of radiation with the given quantum energy 18,19 : At the same values of I h and I L , the pressure of plasma heated by fast electrons is, approximately, by factor (ρ 0 /ρ cr ) 1/3 larger and the temperature, in contrary, by factor (ρ 0 /ρ cr ) 2/3 smaller in comparison with the case of laser heating. However, in a shock wave, the temperature ratio is reversed. Indeed, in the approximation of a uniform distribution of pressure, the ratio of temperatures in shock waves driven by fast electron and laser radiation is The temperature in shock wave driven by powerful laser pulse is, as usual, several tens of eV, whereas the temperature in shock wave driven by fast electron beam can reach the value of several keV. Such a high temperature is a distinctive feature of the shock wave driven by fast electrons. Generation of shock wave with a Gbar-pressure and keV-temperature is a record opportunity for a laboratory experiment.
Such a large temperature of dense plasma is the reason for the intense bremsstrahlung emission and the associated effect of increasing of matter compression in the transparent plasma of shock wave. Thermal conductivity takes place at the ablation boundary (in the region of a shock wave piston) and has almost no smoothing effect in a shock wave region. Thus, the conditions arise for radiation cooling and, as a result, for increase in the compression of matter in shock wave. Simple estimates relating to the quasi-static heating confirm this fact. The averaged mean free path of thermal radiation in the region involved into shock wave could be estimated as 20 where σ = 1.03 · 10 17 J · cm −2 · s −1 · keV −4 is Stefan-Boltzmann constant, W r is emissivity of plasma electrons 20 temperature T and density ρ are measured in keV and g/cm 3 , respectively.
Below it is considered an example that corresponds to irradiation of aluminum target with  (8) and (9), for the radiation mean free path we get the value l r ≈ 0.03 cm, which is about 10 times larger the size of shock wave region l SW = D SW t h ≈ 0.004 cm. Then, for the considered example, the flux energy density of thermal radiation carried away, q r = W r l SW , is about of 10 17 W/cm 2 , which is half of fast electron energy flux density. In the approximation of adiabatic compression of a substance in a radiation-cooled region the increase in density in this region compared to the density at the shock wave front can be estimated as where ρ SW (0) = ρ 0 (γ + 1) / (γ − 1). This estimate for q r /q h = 0.5 gives an excess density in the radiation-cooled region compared to the density at the shock wave front of about 2.8.
This means that a density of about 30 g/cm 3 can be achieved in an aluminum target.  of higher density. To the end of the period of formation of the steady-state movement of shock wave, maximum density values is achieved in its peripheral part. Subsequently, the maximum density value decreases in accordance with a decrease in pressure. Figure 3 shows the profiles of pressure, density, electron temperature, and plasma emissivity at t = 500 ps, corresponding to the achievement of maximum density values in the peripheral part of shock wave. An increase in density in the peripheral region of shock wave takes place near the maximum of plasma emissivity. The maximum compression to a density of 45 g/cm 3 occurs in the region that is cooled to the maximum extent due to radiation energy losses under the

IV. CONCLUSION
Heating a solid with a beam of laser-accelerated fast electrons can provide the generation of a shock wave, which has characteristics unique for a laboratory experiment. The pressure behind the front of such a wave can reach several hundred 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 partially transparent to intrinsic radiation. As a result, conditions arise of strong radiation cooling and, as a consequence, of increasing in plasma compression in the peripheral part of shock wave. The performed theoretical and computational studies show that under the impact of radiation of the first harmonic of Ndlaser with intensity 10 17 -10 19 W/cm 2 , the density of aluminum plasma in the peripheral region of fast-electron-driven shock wave can reach 30-60 g/cm 3 red and 50-70 g/cm 3 for copper plasma. With an increase in the atomic number of a substance, saturation of the increase in density occurs in the peripheral region of the shock wave, that is associated with saturation of the growth of radiation energy losses, at which the increase in emissivity is compensated by a decrease in the transparency of the radiation region. Investigation of the state of a substance at a pressure of several Gbar, temperature of several keV, and density of several tens of g/cm 3 represents a new section of the EOS study in a laboratory experiment.
Investigation of the effect of increasing the 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 a high atomic number. In addition, the dependence of the effect on the degree of ionization makes it possible to study the kinetics of ionization of matter at ultrahigh pressures, which is one of the fundamental questions of modern physics of high energy densities. Such experiments can be performed using a sub-nanosecond laser pulse with energy of 1-10 kJ.