Extreme matter compression caused by radiation cooling effect in gigabar shock wave driven by laser-accelerated fast electrons
aa r X i v : . [ phy s i c s . p l a s m - p h ] F e b Extreme matter compression caused by radiation cooling effect in gigabar shockwave driven by laser-accelerated fast electrons
Gus’kov, S. Yu., Kuchugov, P. A.,
1, 2, a) and Vergunova, G. A. P.N. Lebedev Physical Institute of Russian Academy of Sciences Keldysh Institute of Applied Mathematics of Russian Academy of Sciences (Dated: 16 February 2021)
Heating a solid with laser-accelerated fast electrons is unique way for a laboratoryexperiment to generate a plane powerful shock wave with a pressure of several hundredor even thousands of Mbar. Behind the front of such a powerful shock wave, denseplasma is heated to a temperature of several keV. Then, a high rate of radiationenergy loss occurs even in low- Z plasmas. The effect of strong compression of matterdue to radiation cooling in a gigabar shock wave driven by fast electrons is foundin computational and theoretical researches. It is shown that the effect of radiationcooling leads to the compression of matter in the peripheral region of shock waveto a density several times larger than the density at its front. Heating a solid bya petawatt flux of laser-accelerated fast electrons allows one to surpass the gigabarpressure level of a plane shock wave, which is the maximum level for the impactof laser-accelerated pellets. Higher pressure about 100 Gbar can be achieved underlaboratory conditions only when a spherical target is imploded under the action of aterawatt laser pulse. a) Corresponding author: Kuchugov P.A., [email protected] . INTRODUCTION Heating a substance with laser-accelerated charged particle beam is an effective way togenerate a plane powerful shock wave with a pressure of several hundred or even thousandsof Mbar in a laboratory experiment. This is due to the fact that the energy flux densityof 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 collisionsand are able to heat a dense substance with a density that significantly exceeds the criticalplasma density. The permanent energy growth of modern laser facilities with terawatt andpetawatt power allows us to consider laser-accelerated charged particle beam as an effectivetool for generating the super-powerful shock waves in sufficiently large volume of matterthat meet the needs of such important applications as inertial confinement fusion (ICF),study of matter equation of state (EOS) and laboratory astrophysics. The above primarilyapplies to laser-accelerated electron beam, since the efficiency of laser energy conversion intofast 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 effectivemethod for generating a plane shock wave with the extreme pressure for a laboratory ex-periment. Its potential capabilities exceed ones of the method of impact of laser-acceleratedpellets which, in a modern experiment provided the generation of a plane shock wave witha record pressure of 740 Mbar . This record pressure is close to the maximum achievableone when using the impact method, since the collisional mechanism of laser radiation ab-sorption is limited by the value of the coupling parameter Iλ ≈ · W µ m /cm ( I and λ are, respectively, the intensity and wavelength of laser radiation). The record pressurewas determined by the limiting intensity for the third Nd-laser harmonic of 10 W/cm .The method of direct heating of a solid by laser-accelerated fast electrons is designed touse intensities exceeding the collisional absorption limit, when a significant fraction of laserenergy is transformed into the energy of fast electrons. It was shown in that the use of alaser pulse with an intensity of I ≈ − W/cm to heat a solid by fast electrons canprovide the generation of plane shock wave with a pressure of several tens of Gbar. A higherpressure, up to several hundred Gbar in a laboratory experiment, can be achieved only inthe 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 of2olid by laser-accelerated fast electrons is very promising, bearing in mind that diagnosticsin a spherical experiment turns out to be a more difficult task in comparison with thosetraditional 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 heatinga substance with laser-accelerated electron beam. In , it was shown that in this heatingmethod, the radiation energy loss is the main factor limiting the temperature of producedplasma. In the compressed material behind the shock wave front, radiation energy loss playsan even greater role than in the heated region. The effect of extreme matter compressiondue to radiation cooling in a shock wave driven by laser-accelerated fast electrons is foundon the basis of computational and theoretical researches. The effect of matter compressionincrease due to radiation cooling is well known in relation to laminar flows under Z -pinch and ICF target implosions. It should be noted that the radiation cooling effect is acentral problem in astrophysics and laboratory astrophysics, in particular, in the sectionsof accretion physics and radiative laboratory shock physics . In this paper the effect ofradiation cooling behind the front of a powerful shock wave is considered. It is shown thatthe radiation cooling leads to compression of matter in the peripheral region of shock wave toa density several times larger than the density at its front. First it is discussed the featuresof the radiation cooling effect in a powerful shock wave driven by heating a dense substancewith laser-accelerated electrons. Then the results of numerical calculations are presentedand discussed. II. THE FEATURES OF SHOCK WAVE DRIVEN BYLASER-ACCELERATED FAST ELECTRONS
It is considered the generation and propagation of shock wave driven by heating theboundary region of semi-space with monoenergetic laser-accelerated fast electron flow. Thedimensional parameters of the problem are the energy flux density of fast electrons I h , themass range of heating particles in heated substance µ h , which is a function of the initial fastelectron energy ε h and the density of substance ρ . According to numerous experiments andtheoretical models, the laser energy conversion into fast electron energy η = I h /I L lies in therange 0.1-0.3. Despite this relatively low conversion efficiency, laser-accelerated electrons arethe undisputed record holder for energy flux density among charged particle beams of any3ther laboratory origin, taking into account the laser intensity 10 –10 W/cm achievedin 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-knownscalings , which combine the data from numerous experiments and theoretical models ε h [MeV] = . (cid:0) I L (19) λ µ (cid:1) / , I L (19) λ µ < . , . (cid:0) I L (19) λ µ (cid:1) / , I L (19) λ µ > . , (1)where I L (19) and λ µ are measured in 10 W/cm and µ m.The calculated results were obtained for the case of impact of laser irradiation on alu-minum, which is often used as a reference material in EOS experiments. For estimates, anapproximation formulas are used for mass ranges of non-relativistic and relativistic electrons,which are calculated in accordance with the data of for aluminum plasma with an ioncharge Z = 11: µ [g/cm ] ≈ . ε h , ε h < . ε h , ε h > ε h is measured in MeV.The scales of physical quantities of the problem are determined by the thermodynamicparameters of a region heated by fast electrons. In plane geometry, the mass of heatedlayer remains constant and equal to the fast electron mass range, despite the increase in theheated layer temperature and the decrease in its density due to thermal expansion. Underthese conditions, the thermodynamic state of heated layer is described by the solutions ofRef. for a period of quasi-static heating, when the motion of the substance can be ignored ρ = ρ , T = T h tt h , P = P h tt h , ≤ t ≤ t h (3)and for the subsequent period of thermal expansion of the layer with a constant mass ρ = ρ (cid:18) t h t (cid:19) / , T = T h tt h , P = P h (cid:18) t h t (cid:19) / , t ≥ t h . (4)In these expressions t h is the duration of quasi-static heating period or the time of abla-tion loading (following to the notation of Ref. ), during which isothermal rarefaction wavepropagates from the outer surface of semi-space to the inner boundary of the heated layer: t h ≡ µ ρ c T = 12 (cid:20)
92 ( γ − (cid:21) / µρ / I / h . (5)4 T = ( C V ( γ − T h ) / is the isothermal sound speed in the heated region, T h and P h aretemperature and pressure that are reached at the end of the quasi-static heating period T h = 1 C V (cid:20)
916 ( γ − (cid:21) / (cid:18) I h ρ (cid:19) / , P h = (cid:20) γ − (cid:21) − / ρ (cid:18) I h ρ (cid:19) / , (6) I h is flux energy density of fast electrons, µ h is mass range of the electron with energy ε h , C V = ( Z + 1) k B / ( A ( γ − m p ) is specific heat at constant volume, k B is the Boltzmann’sconstant, m p is the proton mass, A and Z are the atomic number and ion charge, γ is specificheats ratio.Using expressions (1) and (2) for energy and mass range of fast electron, it is easy toget that the characteristic time of thermodynamic state evolution t h increases with both thelaser intensity and wavelength. In the non-relativistic case t h grows as t h ∼ I / L λ / andin the relativistic case it grows as t h ∼ I / L λ . In aluminum plasma at I L = 10 W/cm , λ = 1 . µ m (the 1st harmonic of the Nd-laser), that corresponds to ε h = 200 keV, and atthe conversion η = 0 .
2, the ablation loading time is about 130 ps.From the point of view of achieving an extreme state of matter, the most interestingis 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 uniformpressure distribution in the heated region and in the region involved into motion by theshock wave. Then, the velocity of shock wave D SW and the temperature T SW behind itsfront are expressed in terms of thermodynamic parameters of the heated region as D SW ≈ (cid:18) γ + 12 (cid:19) / c T , T SW ≈ P h C V ρ SW = (cid:18) γ − γ + 1 (cid:19) T h . (7)The scales of pressure P h and temperature T h during the quasi-static heating dependonly 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 generationdriven by charged particle beam and laser pulse. The pressure and temperature of laser-heated substance depend on the energy of a light quantum through the value of criticalplasma density ρ cr , which is scale of density in the region of absorption of radiation with thegiven quantum energy : T L ∼ ( I L /ρ cr ) / , P L ∼ ρ / cr I / L ( ρ cr ≈ . · − A/Zλ µ g/cm , λ µ is laser radiation wavelength measured in µ m).At the same values of I h and I L , the pressure of plasma heated by fast electrons is, approx-imately, by factor ( ρ /ρ cr ) / larger and the temperature, in contrary, by factor ( ρ /ρ cr ) / T SW ( h ) /T SW ( L ) = P h /P L = ( ρ /ρ cr ) / . The temperature in shock wave driven by powerfullaser pulse is, as usual, several tens of eV, whereas the temperature in shock wave drivenby fast electron beam can reach the value of several keV. Such a high temperature is a dis-tinctive feature of the shock wave driven by fast electrons. Generation of shock wave witha 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 emis-sion and the associated effect of increasing of matter compression in the transparent plasmaof shock wave. Thermal conductivity takes place at the ablation boundary (in the regionof 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 compressionof matter in shock wave. Simple estimates relating to the quasi-static heating confirm thisfact. The averaged mean free path of thermal radiation in the region involved into shockwave could be estimated as l r = 4 σT W r (8)where σ = 1 . · J · cm − · s − · keV − is Stefan-Boltzmann constant, W r is emissivityof plasma electrons W r = 1 . · (cid:18) ZA (cid:19) ZT / ρ , J · cm − · s − (9)temperature T and density ρ are measured in keV and g/cm , respectively.Below it is considered an example that corresponds to irradiation of aluminum target withthe 1st harmonic Nd-laser radiation with an intensity of 10 W/cm ( ε h = 200 keV) at theconversion η = 0 .
2, which corresponds to the fast electron energy flux density 2 · W/cm .According to (2), the mass range of fast electron with energy of 200 keV in an aluminumplasma with a charge Z = 11 is about of 0.032 g/cm . Further, according to (5)-(7) at γ = 5 /
3, the duration t h of quasi-static heating and the average temperature for this periodbehind the shock wave front are about 130 ps and 1.5 keV. Substituting T = 1 . ρ ≈ ρ = 10 . and Z = 11 in (8) and (9), for the radiation mean free path weget the value l r ≈ .
03 cm, which is about 10 times larger the size of shock wave region l SW = D SW t h ≈ .
004 cm. Then, for the considered example, the flux energy density of6hermal radiation carried away, q r = W r l SW , is about of 10 W/cm , which is half of fastelectron energy flux density. In the approximation of adiabatic compression of a substancein a radiation-cooled region the increase in density in this region compared to the densityat the shock wave front can be estimated as ρρ SW (0) ≈ (cid:18) − q r q h (cid:19) − / ( γ − (10)where ρ SW (0) = ρ ( γ + 1) / ( γ − q r /q h = 0 . can be achieved in an aluminum target.Below it is discussed plasma heating by a beam of laser-accelerated fast electrons fromthe point of view of the features of the formation of the thermodynamic state of the resultingplasma. With a deceleration length of a sub-relativistic fast electron near 100-200 µ m, thetime for transferring its energy to plasma electrons is about 1 ps. This time is much shorterthan the hydrodynamic time of the problem, which is about 100-200 ps; a fast electrontransfers its energy to a plasma with stationary distributions of thermodynamic parameters.In turn, the energy transfer time is much longer than the plasma relaxation times – theelectron-electron and electron-ion energy relaxation times, which for a plasma density inthe heating region of several g/cm and a temperature of several keV are 0.0001 ps and0.1 ps, respectively. The collisional ionization time is about 0.001 ps and occurs with theparticipation of plasma electrons with a Maxwellian spectrum. The main recombinationmechanism under the conditions of the problem under consideration is triple collisions. Therecombination rate in triple collisions exceeds the photorecombination rate by more than100 times, which means the establishment of the Saha equilibrium in terms of the ionizationcomposition. Thus, the heating of the substance by laser-accelerated fast electrons occurswhile maintaining the local thermodynamic equilibrium of the plasma. III. NUMERICAL RESULTS AND DISCUSSION
The bulk of numerical calculations was performed using the one-dimensional two-temperature hydrodynamic code DIANA , supplemented by a module of energy transferby fast electrons . DIANA code takes into account all the main relaxation and transportprocesses in the plasma and the real equations of states. The energy transfer by intrinsic7lasma radiation is considered in approximation of volumetric losses due to bremsstrahlungmechanism. The cooling function operates by using threshold frequency for photons – allphotons with frequency above threshold value leave the plasma and take away their en-ergy, in opposite case the energy is locally deposited. This code was successfully used forinvestigation of the effect of energy transfer by fast electrons on the gain of direct-driveICF targets . The energy transfer from fast electrons in Coulomb collisions is calculatedusing the model of the stopping power of the plasma electrons adjusted for scattering by theplasma ions . The controlling calculations were performed using RADIAN numericalcode , which solves one-dimensional equations of two-temperature gas dynamics togetherwith the radiation transfer equation. When solving the radiative transfer equation, themulti-group approximation, the method of characteristics, quasi-diffusion method and av-eraging over photon energies are used to effectively reduce dimension of equation. Thesecalculations showed that taking radiation transfer into account led to insignificant devia-tions of thermodynamic parameters at the level of 5-10% from the DIANA code results.Calculations using the DIANA code were performed in the range of fast electron energyflux density I h = 2 · –2 · W/cm , which, with the supposed laser energy conversioninto fast electron energy η = 0 .
2, corresponds to the range of variation in laser intensity I L = 10 –10 W/cm . The initial energy of fast electrons was chosen according to scal-ing (1) for the 1st harmonic of the Nd-laser radiation, and varied from a non-relativisticenergy ε h = 100 keV to a relativistic energy ε h = 3 . I h = 2 · W/cm and an initial energy ε h = 200 keV (corre-sponds to the intensity of the 1st harmonic of Nd-laser radiation I L = 10 W/cm atthe conversion η = 0 . µ ≈ .
028 g/cm . The mass range varies slightly with increasing temperatureand decreasing density in the heated region in accordance with the change in the Coulomblogarithm. Figures 1 show the distributions by the mass coordinate, respectively, of pres-sure, electron temperature, and density from the boundary of the semi-space, on which thesource of fast electrons was set, to the shock wave front at the different time moments. Themass coordinates of the semi-space edge and the inner boundary of the heated region are,respectively, 0.54 g/cm and about of 0.512 g/cm . The figures show the time evolution ofthermodynamic parameter distributions in the region heated by fast electrons and shock8ave region. The pressure has a fairly uniform distribution behind the front of shock wave.Moreover, the pressure varies non-monotonically with time. At the initial time, the pressurebehind the shock wave front reaches a value of about 10 Gbar at a time of about 120 ps,after which it decreases to a minimum of about 6.5 Gbar at a time of about 200 ps, thenincreases again reaching a second maximum of about 7.5 Gbar at a time of about 500 ps,and further decreases monotonously. The temperature distribution has a minimum, andthe density distribution has a maximum in the peripheral region of the shock wave, whosepositions fall on the same mass coordinate value. The maximum density in the radiation-cooled region reaches the value of about 45 g/cm at the time moment of about 500 ps.Figure 2 shows the time dependencies of the numerically calculated values of maximumpressure and maximum density. This figure also shows the time dependence of the pressurein the heated region, calculated on the basis of the analytical solution (3)-(6) with a fractionof radiation energy loss of 40%. In the initial period of time up to 120-150 ps, when theinitial stage of shock wave formation takes place, the maximum pressure is determined bythe pressure in the heated region. Solution (3) at the stage of quasi-static heating is in goodagreement with the numerically calculated dependence. The pressure increases with timeaccording to a law close to the linear one and reaches the values of about 10 Gbar by timemoments of about 130-150 ps. The ablation loading time t h is about 150 ps, the maximumof the numerically calculated pressure is reached at t = 130 ps. During the ablation loadingperiod the temperature behind the shock wave is about 1 keV and length of shock wavepropagation is about 0.002 cm. Corresponding averaged mean free path of thermal photonsis 0.01 cm, that is 5 times larger than the length of shock wave propagation. All thesevalues are in a good agreement with the estimations of Section II made on the base ofexpressions (3)–(9). As a result the whole shock wave is transparent and the growth ofthe density begins from the wave front and reaches the maximum in the peripheral backregion. At subsequent time moments, the shock wave propagates under the conditions ofexpansion of the heated region as a whole and ablation of matter at a depth exceeding themass range of fast electrons. This minimizes the numerically calculated pressure dependenceat t = 180 ps. The formation of the steady motion of shock wave ends at a time of about400 ps, after which the pressure decreases monotonically with time. The pressure in thesteady-state shock wave is 30-40% higher than the pressure of the analytical solution (4) inthe heated region, which is a consequence of hydrodynamic energy transfer to a substance9 IG. 1. Profiles of pressure (a), electron temperature (b) and density (c) over mass coordinateat various time moments: 20 ps (curves 1), 100 ps (2), 200 ps (3), 500 ps (4), 1 ns (5), 1.5 ns(6), 2 ns (7), obtained in the calculation under the impact of fast electron energy flux density I h = 2 · W/cm with initial energy of fast electrons ε h = 200 keV. of higher density. To the end of the period of formation of the steady-state movement ofshock wave, maximum density values is achieved in its peripheral part. Subsequently, themaximum density value decreases in accordance with a decrease in pressure. Figure 3 showsthe 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 shockwave. An increase in density in the peripheral region of shock wave takes place near themaximum of plasma emissivity. The maximum compression to a density of 45 g/cm occursin the region that is cooled to the maximum extent due to radiation energy losses under the10 IG. 2. Time dependencies of the numerically calculated maximal values of pressure (curve 1) anddensity (2), and also a pressure in heated region according to analytical expressions (3)-(6) underthe impact of fast electrons energy flux density I h = 2 · W/cm with initial energy of fastelectrons ε h = 200 keV. action of the maximum pressure developed in the shock wave. The degree of compressiondue to radiation cooling increases with laser pulse intensity, both due to an increase inpressure as well as an increase in radiation energy loss. In the calculation with impact ofradiation of 1st harmonic of Nd-laser of the intensity I L = 10 W/cm ( ε h = 1 . η = 0 . .Numerical simulations conducted for Cu target show the increase of the maximal densityin peripheral area of the shock wave compared with an aluminum target. In the case ofimpact of the laser beam of the intensity 10 W/cm ( ε h = 200 keV, η = 0 .
2) the maximaldensity reaches the value about 58 g/cm , for the laser intensity 10 W/cm and the particlesenergy of 1.2 MeV it achieves the value of 66 g/cm . The saturation of the increase indensity in the peripheral region of the shock wave with an increase in the atomic numberis associated with saturation of the increase in radiation energy losses – the increase inemissivity is compensated by a decrease in the transparency of the radiation region. Inparticular, in the numerical simulation of Cu target with the laser intensity 10 W/cm IG. 3. Profiles of electron temperature (1), density (2), pressure (3) and plasma emissivity (4) attime moment 500 ps, which corresponds to the achievement of maximal density in the peripheralregion of shock wave. These data are obtained in calculation under the impact of fast electronenergy flux density I h = 2 · W/cm with initial energy of fast electrons ε h = 200 keV. Shortvertical line indicates the inner boundary of heated region. the time of ablation loading t h is about 60 ps and the temperature T h is about of 0.5 keV.The both values are approximately 2 times less, than in the case of Al target calculations,in accordance with (4) and (5). Corresponding averaged mean free path of thermal photons l r in the Cu-plasma with the density of, approximately, 3 times larger compared with Al-plasma is about 0.00006 cm. During the period of ablation loading the length of shock wavepropagation is about 0.0005 cm that is 8 times larger than l r . So, in contrast to the caseof Al target, where whole shock wave is transparent and the growth of the density beginsfrom the wave front and reaches the maximum in the peripheral back region, the main partof shock wave in Cu target is opaque and growth of the density occurs only in the narrowperipheral region. It should be noted, that for achieving higher densities the intensity offast electron flux should be greater for the same initial energies of the particles but it isnot supported yet by modern experiments, which now have a possibility to use the laseracceleration as a most powerful source of fast electrons.The given calculation results allow one to determine the experimental conditions for12tudying the equation of state of matter. For a laser pulse with an intensity of 10 W/cm ,the pulse duration corresponding to the attainment of the maximum density values is about100 ps. To generate a near plane shock wave, the radius of the laser beam must be at leasttwo lengths of thermal expansion of the heating region, i.e. about 100 microns. This meansthat the energy of the laser pulse in this case should be about 4-5 kJ.A shock wave generated by heating the solid by fast electrons may be of interest forstudying the phenomena of radiative laboratory shock physics. The generation of such awave occurs due to the work of a dense, high-temperature piston. As a result, a stratifiedobject of strongly collisional plasma is formed (the Coulomb logarithm is about 10) withinhomogeneous 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 arelatively narrow region, the mass of which does not exceed 10% of the mass of the piston.Electron thermal conductivity has an negligible effect on the thermodynamic state of thisregion. The region of radiation, as a plasma object formed by heating, is largely transparentto radiation. The optical thickness of the region heated by fast electrons (high-temperaturepiston) 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 opticalthickness in the entire region covered by the shock wave can vary from fractions of a unitfor mid-Z materials up to several units for high-Z materials. In the considered conditionsof the problem, the P´eclet number with respect to the electron thermal conductivity (theratio of the hydrodynamic velocity to the velocity of the wave front of the electron heatconduction) for the radiation region is quite large for both Al and Cu targets, about 200-300. P´eclet number in relation to radiation is different for different regions of discussedstratified object. In the region of strong radiation its value is about 10 for Al target and10 for Cu target. The cooling parameter (ratio of cooling time and hydro time) is about0.01 for mid-Z materials and about 0.1 for high-Z ones. Plasma objects with this kindof properties, formed by heating a substance with a petawatt flux of laser-accelerated fastelectrons, can be of interest to laboratory astrophysics in connection with the controlled(due to a change in the laser pulse intensity and the selection of the target material) effectof radiation cooling. 13 V. CONCLUSION
Heating a solid with a beam of laser-accelerated fast electrons can provide the generationof a shock wave, which has characteristics unique for a laboratory experiment. The pressurebehind the front of such a wave can reach several hundred and even thousands of Mbarat a temperature of several keV. Such a high temperature causes a high rate of radiationenergy loss in a dense plasma, which is partially transparent to intrinsic radiation. As aresult, conditions arise of strong radiation cooling and, as a consequence, of increasing inplasma compression in the peripheral part of shock wave. The performed theoretical andcomputational studies show that under the impact of radiation of the first harmonic of Nd-laser with intensity 10 –10 W/cm , the density of aluminum plasma in the peripheralregion of fast-electron-driven shock wave can reach 30-60 g/cm red and 50-70 g/cm forcopper plasma. With an increase in the atomic number of a substance, saturation of theincrease in density occurs in the peripheral region of the shock wave, that is associated withsaturation of the growth of radiation energy losses, at which the increase in emissivity iscompensated by a decrease in the transparency of the radiation region. Investigation of thestate of a substance at a pressure of several Gbar, temperature of several keV, and density ofseveral tens of g/cm 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 shockwave, which is determined by radiative energy losses, is of great interest for establishing theoptical properties of materials with a high atomic number. In addition, the dependence ofthe effect on the degree of ionization makes it possible to study the kinetics of ionization ofmatter at ultrahigh pressures, which is one of the fundamental questions of modern physicsof high energy densities. Such experiments can be performed using a sub-nanosecond laserpulse with energy of 1-10 kJ. REFERENCES S. Gus’kov, X. Ribeyre, M. Touati, J.-L. Feugeas, P. Nicola¨ı, and V. Tikhonchuk,“Ablation Pressure Driven by an Energetic Electron Beam in a Dense Plasma,”Physical Review Letters (2012), 10.1103/physrevlett.109.255004.14
X. Ribeyre, S. G. kov, J.-L. Feugeas, P. Nicola¨ı, and V. T. Tikhonchuk, “Denseplasma heating and Gbar shock formation by a high intensity flux of energetic electrons,”Physics of Plasmas , 062705 (2013). S. Y. Gus’kov, “On the possibility of laboratory shock wave studies of the equationof state of a material at gigabar pressures with beams of laser-accelerated particles,”JETP Letters , 71–74 (2014). R. Cauble, D. W. Phillion, T. J. Hoover, N. C. Holmes, J. D. Kilkenny, andR. W. Lee, “Demonstration of 0.75 Gbar planar shocks in x-ray driven colliding foils,”Physical Review Letters , 2102–2105 (1993). M. Karasik, J. L. Weaver, Y. Aglitskiy, T. Watari, Y. Arikawa, T. Sakaiya, J. Oh, A. L.Velikovich, S. T. Zalesak, J. W. Bates, S. P. Obenschain, A. J. Schmitt, M. Murakami,and H. Azechi, “Acceleration to high velocities and heating by impact using Nike KrFlaser,” Physics of Plasmas , 056317 (2010). S. Y. Gus’kov, N. P. Zaretskii, and P. A. Kuchugov, “Features and limiting char-acteristics of the heating of a substance by a laser-accelerated fast electron beam,”JETP Letters , 135–138 (2020). R. S. Pease, “Equilibrium Characteristics of a PinchedGas Discharge Cooled by Bremsstrahlung Radiation,”Proceedings of the Physical Society. Section B , 11–23 (1957). V. V. Vikhrev, “Contraction of Z-pinch as a result of losses to radiation,” Pis’ma Zh. Eksp.Teor. Fiz , 104–107 (1978). L. Bernal and H. Bruzzone, “Radiative collapses in z-pinches with axial mass losses,”Plasma Physics and Controlled Fusion , 223–231 (2002). R. S. Craxton, K. S. Anderson, T. R. Boehly, V. N. Goncharov, D. R. Harding, J. P.Knauer, R. L. McCrory, P. W. McKenty, D. D. Meyerhofer, J. F. Myatt, A. J. Schmitt,J. D. Sethian, R. W. Short, S. Skupsky, W. Theobald, W. L. Kruer, K. Tanaka, R. Betti,T. J. B. Collins, J. A. Delettrez, S. X. Hu, J. A. Marozas, A. V. Maximov, D. T.Michel, P. B. Radha, S. P. Regan, T. C. Sangster, W. Seka, A. A. Solodov, J. M.Soures, C. Stoeckl, and J. D. Zuegel, “Direct-drive inertial confinement fusion: A re-view,” Physics of Plasmas , 110501 (2015). I. E. Golovkin, J. J. MacFarlane, P. R. Woodruff, J. E. Bailey, G. Rochau,K. Peterson, T. A. Mehlhorn, and R. C. Mancini, “Spectroscopic analysis and15LTE radiative cooling effects in ICF capsule implosions with mid- dopants,”Journal of Quantitative Spectroscopy and Radiative Transfer , 199–208 (2006). J. M. Blondin and D. F. Cioffi, “The growth of density perturbations in radiative shocks,”The Astrophysical Journal , 853 (1989). J. Laming, “Relationship between oscillatory thermal instabil-ity and dynamical thin-shell overstability of radiative shocks,”Physical Review E (2004), 10.1103/physreve.70.057402. F. N. Beg, A. R. Bell, A. E. Dangor, C. N. Danson, A. P. Fews, M. E. Glinsky, B. A.Hammel, P. Lee, P. A. Norreys, and M. Tatarakis, “A study of picosecond laser–solidinteractions up to 1019 w cm-2,” Physics of Plasmas , 447–457 (1997). M. G. Haines, M. S. Wei, F. N. Beg, and R. B. Stephens,“Hot-electron temperature and laser-light absorption in fast ignition,”Physical Review Letters (2009), 10.1103/physrevlett.102.045008. S. Atzeni, A. Schiavi, and J. R. Davies, “Stopping and scattering of rel-ativistic electron beams in dense plasmas and requirements for fast ignition,”Plasma Physics and Controlled Fusion , 015016 (2008). J. Honrubia and J. M. ter Vehn, “Three-dimensional fast electron transport for ignition-scale inertial fusion capsules,” Nuclear Fusion , L25–L28 (2006). Y. V. Afanasiev and S. Y. Gus’kov, “Nuclear Fusion by Inertial Confinement. A Com-prehensive Treatise,” (CRC Press, 1993) Chap. Energy Transfer to the Plasma in LaserTargets, pp. 99–119. J. Lindl, “Development of the indirect-drive approach to inertial confinement fusion andthe target physics basis for ignition and gain,” Physics of Plasmas , 3933–4024 (1995). Y. B. Zel’dovich and Y. P. Raizer,
Physics of shock waves and high-temperature hydrody-namic phenomena , edited by W. D. Hayes and R. F. Probstein (Academic Press, 1967). N. V. Zmitrenko, V. Y. Karpov, A. P. Fadeev, I. I. Shelaputin, and G. V. Shpatakovskaya,“Description of the physical processes in the DIANA program for calculations of problemsof laser fusion,” Voprosy Atomnoy Nauki i Tekhniki (VANT). Series Methods and Softwarefor Numerical Solution of Problems of Mathematical Physics , 34–37 (1983). S. Y. Gus’kov, P. A. Kuchugov, R. A. Yakhin, and N. V. Zmitrenko, “Effect of ‘wandering’and other features of energy transfer by fast electrons in a direct-drive inertial confinementfusion target,” Plasma Physics and Controlled Fusion , 055003 (2019).16 S. Y. Gus’kov, P. A. Kuchugov, R. A. Yakhin, and N. V. Zmitrenko,“Effect of fast electrons on the gain of a direct-drive laser fusion target,”Plasma Physics and Controlled Fusion , 105014 (2019). G. A. Vergunova and V. B. Rozanov, “Influence of intrinsic X-ray emission on the processesin low-density laser targets,” Laser and Particle Beams , 579–583 (1999). V. B. Rozanov and G. A. Vergunova, “Investigation of compression of indirect-drive targets under conditions of the NIF facility using one-dimensional modelling,”Quantum Electronics50