Unraveling Resistive Versus Collisional Contributions to Relativistic Electron Beam Stopping Power in Cold-Solid and in Warm-Dense Plasmas B

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Unraveling resistive versus collisional contributions to relativistic electron beam stopping power in cold-solid and in warm-dense plasmas B. Vauzour,1, 2 A. Debayle,3, 4 X. Vaisseau,1 S. Hulin,1 H.-P. Schlenvoigt,5 D. Batani,1, 6 S.D. Baton,5 J.J. Honrubia,3 Ph. Nicola¨ı,1 F.N. Beg,7 R. Benocci,6 S. Chawla,7 M. Coury,8 F. Dorchies,1 C. Fourment,1 E. d’Humières,1 L.C. Jarrot,7 P. McKenna,8 Y.J. Rhee,9 V.T. Tikhonchuk,1 L. Volpe,6 V. Yahia,5 and J.J. Santos1, a) 1)Univ. Bordeaux, CNRS, CEA, CELIA (Centre Lasers Intenses et Applications), UMR 5107, F-33405 Talence, France 2)Laboratoire d’Optique Appliquée,ENSTA-CNRS-Ecole Polytechnique, UMR 7639, 91761 Palaiseau, France 3)ETSI Aeronáuticos, Universidad Politécnica de Madrid, Madrid, Spain 4)CEA, DAM, DIF, F-91297 Arpajon, France 5)LULI, Ecole Polytechnique CNRS/CEA/UPMC, 91128 Palaiseau Cedex, France 6)Dipartimento di Fisica, Universitàdi Milano-Bicocca, Milano 20126, Italy 7)University of California, San Diego, La Jolla, California 92093, USA 8)SUPA, Department of Physics, University of Strathclyde, Glasgow G4 0NG, United Kingdom 9)Korea Atomic Energy Research Institute (KAERI), Daejon 305-600, Korea (Dated: 18 February 2014) We present results on laser-driven relativistic electron beam propagation through aluminum samples, which are either solid and cold, or compressed and heated by laser-induced shock. A full numerical description of fast electron generation and transport is found to reproduce the experimental absolute Kα yield and spot size measurements for varying target thickness, and to sequentially quantify the collisional and resistive electron stopping powers. The results demonstrate that both stopping mechanisms are enhanced in compressed Al samples and are attributed to the increase in the medium density and resistivity, respectively. For the achieved 10 2 time- and space-averaged electronic current density, ⟨jh⟩ ∼ 8 × 10 A/cm in the samples, the collisional and resistive stopping powers in warm and compressed Al are estimated to be 1.5 keV/µm and 0.8 keV/µm, respectively. By contrast, for cold and solid Al, the corresponding estimated values are 1.1 keV/µm and 0.6 keV/µm. Prospective numerical simulations involving higher jh show that the resistive stopping power can reach the same level as the collisional one. In addition to the effects of compression, the effect of the transient behavior of the resistivity of Al during REB transport becomes progressively more dominant, and for 12 2 a significantly high current density, jh ∼ 10 A/cm , cancels the difference in the electron resistive stopping power (or the total stopping power in units of areal density) between solid and compressed samples. Analytical 14 2 calculations extend the analysis up to jh = 10 A/cm (representative of the full-scale fast ignition scenario of inertial confinement fusion), where a very rapid transition to the Spitzer resistivity regime saturates the resistive stopping power, averaged over the electron beam duration, to values of ∼ 1 keV/µm.

PACS numbers: 52.50.-b,52.38.Kd, 52.65.-y, 52.70.La

I. INTRODUCTION the critical density of the plasma corona surrounding the compressed core or at the tip of a cone embedded within 3 Fast Ignition (FI)1 is an alternative approach to the the imploded capsule . Due to the substantial reduction conventional ignition scheme of Inertial Confinement Fu- of the laser energy requirement compared to conventional sion (ICF) aiming at the development of an efficient en- ICF, higher gains are expected with a significant relax- ergy production engine. In this two-step ignition sce- ation of the compression quality constraints. nario, a spherical pellet containing a Deuterium-Tritium At its source, the REB transports a total kinetic energy (DT) mixture is adiabatically compressed by high inten- & 30% of the laser energy. A mean kinetic energy of 1- sity (I ∼ 1014−15 W.cm−2) long (∼ ns) laser pulses with 2 MeV is ideal to provide an efficient coupling to the core. a total energy of the order2 of ∼ 250 kJ. During DT For this, the REB has to propagate through a highly stagnation, a relativistic electron beam (REB) is used inhomogeneous plasma: from a low-density (few tenths to create an ignition spark that initiates the propagation of g/cm3) plasma near the electron source, to a high- of a thermonuclear burn in the compressed core. This temperature (∼ 300 eV) and high-density (∼ 400 g/cm3) REB is generated by a high intensity (∼ 1020 W/cm2), plasma close to the compressed core. In order to isochor- high energy (100 kJ), short (∼ 10 ps) laser pulse, close to ically heat the compressed DT to thermonuclear conditions (∼ 10 keV), the REB has to deliver an energy of ∼ 20 kJ in less than 20 ps in a region of diameter less than 40 µm4. Due to the large associated current den- a)Electronic mail: [email protected] sity, ∼ 1014 A/cm2, the REB transport is critically de- 2 pendent on both collisional and collective mechanisms, simulations, show that the total REB stopping power such as important energy losses5–9, divergence10,11 and is higher in compressed materials due to an increase filamentation12–14, which could prevent the thermonu- of both the density and the resistivity of the sample clear ignition. The success of FI therefore relies on under- material. Detailed analytical estimates confirm this standing and characterizing such mechanisms. In partic- behavior. ular, the collisional and resistive stopping powers should The paper is organized as follows. Section II describes be accurately predicted, considering their important in- the experimental setup, providing details of the laser syn- fluence over the ∼ 100 µm stand-off distance separating chronization and the target hydrodynamics presented in the REB source from the compressed core15. Section III. The codes used to simulate the REB gen- Most experiments carried out so far exploring REB eration and transport in our experimental conditions are transport have involved cold and solid targets7,9,10,16. described in Section IV. The experimental results are Only a few studies have explored the warm dense matter presented in Section V, and systematically compared to regime using laser-induced planar17–20 or cylindrical21,22 results from the simulation codes. The benchmarked runs compression of the targets. In the case of a planar com- are used to estimate the REB energy losses and stopping pression geometry, pioneering works17,18 were performed power in Section VI-A, which are confirmed by the calcu- with a laser beam intensity not exceeding 1017 W/cm2. lations of a rigid-beam analytical model in Section VI-B. However, this intensity regime is not relevant for FI, as The main conclusions of this work are summarized in intensities of ∼ 1019−20 W/cm2 are required to generate Section VII. a sufficiently energetic electron beam to trigger the fusion reactions. Although more recent experiments used laser pulses at the higher peak intensities19,20, an accu- II. EXPERIMENTAL SETUP AND TARGET DESIGN rate quantification of the REB stopping power was not Figure possible owing to the tracer layers signal enhancement by The experiment was carried out at the PICO2000 facil- 1 electron refluxing. Concerning the cylindrical compres- ity, LULI laboratory (Ecole´ Polytechnique, France), us- sion geometry, the compressed regions had a small radial ing a dual-laser beam configuration: a long pulse (LP) extent with significant consequences on the REB guiding laser beam to drive the target compression and an intense and ruling out the possibility of a precise stopping power short pulse (SP) laser beam to generate the REB inside characterization as a function of the target material only the same target (see Fig. 1). The 5 × 5 mm2 surface-area (density, temperature, resistivity, etc.). targets were made of aluminum (Al) of varying thickness For the experiment described here, as in (10, 20, 40 or 60 µm), also called the propagation layer, references17–20, we adopt a planar compression ge- embedded between different K-shell fluorescence tracer ometry with the REB injection in the opposite direction layers: a 5 µm silver (Ag) layer, located at the front side to the compressing shock propagation. Unlike in (SP side), used to characterize the REB source (i.e. close references17,18, in which electron transport was explored to its generation zone) and two successive 10 µm layers in room-temperature insulators, we use a metal as of tin (Sn) and copper (Cu), placed at the rear side (LP the sample material: Aluminum samples of variable side), to characterize the fraction of the REB transmit- thickness, embedded in a multi-layer foil target struc- ted through the Al sample. To improve the compression ture, were shock-compressed by a factor of 2. This quality and also to prevent direct laser ablation/heating produced temperatures, upon sample compression, of of the Cu tracer, a 15 µm polypropylene (PP) layer was 2-3 eV, not far short of the material Fermi temperature added on the rear side of the target. Finally, a 5 µm (TF ∼ 11 eV). The changes in density and temperature, Al layer, coating the front surface of the target, enabled increased respectively by factors ∼ 2 and ∼ 100, cause the generation of the fast electrons without damaging the an increase in the material resistivity by more than one Ag tracer layer. The LP laser beam was focused on the order of magnitude23,24. As the target areal density rear side PP layer with an intensity of 3 × 1013 W/cm2 crossed by the REB is invariant during planar com- at λ = 0.53 µm. The pulse temporal profile was flat-top pression, it is, in principle, possible to experimentally with a 4.5 ns duration in order to homogeneously drive discriminate collisional from resistive contributions to the compression in the longitudinal direction. Coupling the REB energy losses by systematically comparing a Phase Zone Plate (PZP) with the focusing lens enabled results from compressed to solid samples of the same the production of a flat-top focal spot spatial profile of initial thickness. Although the material and plasma 400 µm diameter, thus creating a large and homogeneous parameters explored in this study are still far from compressed area around the REB propagation axis. The those typical of a real FI scenario, the comparison of SP laser beam was focused on the Al front layer and de- REB transport in warm-dense versus cold-solid matter livered 35 J pulses of 1.5 ps duration Full Width at Half using relativistic laser intensities is new. In particular, Maximum (FWHM) at λ = 1.06 µm. The focal spot it provides evidence of non-negligible collective effects was Gaussian-shaped with a diameter of 10 µm (FWHM) on REB energy losses, which are highly dependent on containing 25% of the total energy, producing a peak the temperature of the background medium25. The intensity of 3 × 1019 W/cm2. Due to Amplified Spon- experimental results presented, supported by numerical taneous Emission (ASE), each pulse was preceded, ap- 3

proximately 1.1 ns before, by a pedestal of 1012 W/cm2. Initial Al sample thickness [ µm] 10 20 40 60 This was previously characterized via a comparison of the Delay τcomp(SP/LP ) [ ns] 2.5 3.1 4.4 5.7 Al pre-plasma density profile interferometric measurements Effective thickness Lcomp[ µm] 6.3 12.2 23.6 35.1 Al 2 at the targets front side with 2D radiative hydrodynamic Areal density ρLcomp [mg/cm ] 2.6 5.3 10.6 15.9 19 simulations . Delay τsolid(SP/LP ) [ ns] 1.9 1.9 1.9 1.9 Al In order to characterize the fast electron population, Effective thickness Lsolid[ µm] 10 20 40 60 Al 2 K-shell fluorescence radiation, emitted as the fast elec- Areal density ρLsolid [mg/cm ] 2.7 5.4 11.8 16.2 tron pass through the different tracer layers, were col- lected by several diagnostics surrounding the target. A TABLE I. Summary of the different used time delays τ be- X-ray imaging device26, composed of a spherical quartz tween SP and LP laser beams in the cases of compressed (top) crystal (2131) in Bragg incidence (θ = 88.9◦) and or solid (bottom) targets. The final (i.e. at the moment of Bragg the electron injection inside the target) thickness as well as positioned at 210 mm from the rear side of the target, the final areal density of the Al sample, computed by CHIC ∼ enabled imaging of the Cu-Kα emission ( 8 keV) onto hydrodynamic simulations, are also shown. a CCD camera located ∼ 2.2 m away27. The magnifi- cation of the system was thus ∼ 10.5, and the spatial resolution was ∆x ∼ 20 µm. Two different X-ray spec- 25 SOP trace obtained in the case of a target with a 20 µm trometers were used to investigate REB energy losses . Al sample irradiated at LP full power (right panel). The The first one was a Cauchois-type hard X-ray Transmis- 28 time needed for the shock to transit the target is esti- sion Crystal Spectrometer (TCS) using a curved quartz mated to be 3.1 ns. The results are obtained with an crystal (2d = 0.6884 nm) to measure the absolute Sn- accuracy of 0.1 ns, evaluated from the standard devia- ∼ ∼ ( 25 keV) and Ag-Kα ( 22 keV) yields from the target tion of the laser jitter. The shock velocity, inferred from front side. The second one, a HOPG (2d = 0.6714 nm) 29 these measurements, is estimated to be 14-15 µm/ ns. Figure reflection flat crystal spectrometer located at the target On-axis density and temperature profiles, computed 3 rear side, was employed to measure the absolute Cu-Kα ∼ by the hydrodynamic simulations, are presented in Fig. 3 yields ( 8 keV). Finally, a Streaked Optical Pyrome- for the case of a 60 µm Al sample target at times just be- try (SOP) diagnostic, using a streak camera coupled to fore (i.e. 50 ps) the REB injection. Using a short delay, a visible radiation imaging device, was used during LP- τ = 1.85 ns (top panel), enables the injection of the REB only laser shots to measure the shock breakout time at just before the shock reaches the Al-sample. Therefore, the front side of the target relative to the start of the LP the REB propagates through cold (Te = 0.03 eV) and irradiation at the rear side (see the setup in Fig. 1). 3 solid Al (ρ = ρ0 = 2.7 g/cm ). By contrast, the use of a long delay, τ = 5.65 ns for the 60 µm-Al sample target (bottom panel), enables the shock to almost reach III. LASER BEAMS SYNCHRONIZATION AND the Ag/Al interface, effectively compressing the whole HYDRODYNAMIC CHARACTERIZATION OF TARGET Al sample. In this latter case, the REB crosses a ho- COMPRESSION mogeneous warm (Te ∼ 2 − 3 eV) and dense Al sample Figure (ρ ∼ 2ρ0). Note that for both cases, the front sides of 2 We introduced a time delay τ between the SP and LP the target (i.e. the front Al and Ag layers) are in similar laser beams. The delay was selected depending on the conditions: the Al front layer is partially ablated due to desired state of the target (i.e. solid or compressed) and the SP beam ASE intensity pedestal and the Ag tracer secondly on the thickness of the propagation layer (Al layer is kept intact. As can be seen on the left figure (solid sample). Indeed, in the case of compressed targets, the case), the LP beam is used to introduce similar conditions REB has to be generated when the shock has almost (density and temperature) to the compressed case inside completely compressed the Al sample without perturb- the rear side Sn, Cu and PP layers. More particularly, ing the Ag tracer. In order to determine the values of τ it enables the generation of a long PP plasma tail (few for each target, 2D radiative-hydrodynamic simulations hundreds of microns) that partly screens the rear side of the target compression have been performed using the electrostatic field as well as slows down and confines the code CHIC30. These simulations were benchmarked by fast electrons, preventing them recirculating inside the SOP measurements of shock chronometry using specific different K-shell tracers9,31. Therefore, the measured K- targets without the front Al and Ag layers. The signals shell fluorescence signals are produced by only one REB have been obtained by imaging the front side of the tar- transit, facilitating the quantification of the REB stop- get, radiating at the shock breakout, onto the slit of a ping power inside the Al samples. S20 streak camera. An example measurement is shown By virtue of the target areal density conservation, the in Fig. 2. The temporal trace of the LP laser beam, used planar compression enables experimental investigation of as a reference, is plotted on the left panel. Note that the importance of the resistive energy losses on the REB due to the absence of the laser beam amplification stages propagation by unequivocally decorrelating them from (so that the streak camera does not saturate) the tem- the collisional energy losses. More precisely, as illus- poral trace is ramp-like. This is a top-hat-like profile at trated in Table I, the Al sample is compressed to approx- full amplification. This time reference is compared to the imately twice its initial density ρ0 while its length is re- 4

duced by the same factor. Consequently, its areal density density. The maximum density starts at x = 18 µm and ρLAl along the REB propagation axis is maintained con- stretches as far as the end of the simulation box. The spa- stant with the compression. As the electron collisional tial and time resolutions are ∆x = ∆y = c∆t = λ0/60. stopping power is to first approximation proportional to Each cell contains 2 atoms of aluminum and 26 electrons. the density of the crossed medium, the collisional energy Collisions as well as ionizations are taken into account. losses, integrated over the sample length, are approxi- The initial ionization level is self-consistently calculated mately the same for a given initial sample thickness in using the Thomas-Fermi model. Absorbing conditions both compressed and solid Al. The corollary fails re- are applied to the fields, while escaping particles are rein- garding the resistive energy losses which should increase jected with their initial temperature. The simulation box −8 2 with the sample resistivity η (ηsolid ∼ 3 × 10 Ω m → size is 80 × 140 µm . Figure −7 ηcompressed ∼ 5 × 10 Ω m, see Fig. 4) mainly due to the The p-polarized laser beam interacts in oblique inci- 5 ◦ temperature raise from 0.03 eV to ∼ 2-3 eV. At the value dence (θL = 45 with respect to the target surface nor- reached, close to the Fermi temperature (TF ∼ 11 eV), mal). Its temporal profile is assumed to be Gaussian the resistivity is mostly governed by the electron-electron shaped with a peak intensity of 3 × 1019 W/cm2. To collisions and close to its maximum. For the full temper- avoid any influence of the simulation box rear side34 , ature range scanned during compression, below TF , the the laser pulse duration is reduced to 0.5 ps (full width resistivity depends on both density and temperature, re- at half maximum). In order to illustrate the absorption spectively with negative and positive powers (see Section region and the electron beam ballistic, the magnetostatic IV-B)23,24. However, it is the bigger change in temper- field is plotted in Fig. 5 when the laser maximum inten- ature that determines the resistivity change between a sity reaches the absorption region. The fast electrons Figure solid and a compressed sample. are mainly generated around the laser/plasma bound- 4 ary, that separates the strong magnetostatic field in the underdense plasma region from the Weibel induced mag- IV. REB GENERATION AND TRANSPORT netic fields in the overdense region35. The mechanism of SIMULATIONS acceleration, not yet fully understood, is probably a mixture of the Brunel heating36 and the J × B heating37–39. The simulation of both laser/plasma interaction and In the overdense plasma, the magnetic field filaments are 12 electron beam transport through dense matter involve attributed to a resistive filamentation instability , with different time and spatial scales, resulting in different nu- the characteristic scale length of the order of the beam merical resolutions. While the laser/plasma interaction skin depth. The REB source parameters, such as the requires a fine description of the plasma skin depth (of angular and the kinetic energy distribution functions, several nm), which can be obtained by resolving the ki- are extracted at 21 ≤ x ≤ 22 µm, i.e. approximately netic Vlasov equations, the electronic transport demands ∼ 1.5 µm beyond the absorption region, to ensure that a description of the solid background whether classical or the electron source is not perturbed by the evanescent quantical. Hence, the latter relies on a fluid description waves and the Weibel induced magnetic fields. with appropriate equations of state of the background The computed kinetic energy distribution is tempo- species. An alternative approach to describe the whole rally averaged over the entire simulation time and fitted experiment is to use two different codes. A 2D cartesian over the range 0.01 < ε < 20 MeV by the following nor- Particle-in-Cell (PIC) simulation32 has been performed malized function: to calculate the electron distribution function produced { ( ) by the SP interaction with the ASE-induced preplasma. a E0 for 0.01 < ε < 5 MeV Then, within some simplifications, the resulting electron f(ε) = ε ( ) (1) N exp − ε for 5 ≤ ε < 20 MeV, function has been used as the initial beam condition in a T 2D axisymmetric hybrid code of electron transport33. where E0 = 1.7 keV, a = 1.6, T = 3 MeV and N = 1×10−5 correspond to the fitting parameters. This function is plotted on Fig. 6 (red solid line) and compared A. Particle-In-Cell simulations of the REB source to the one extracted from the particle-in-cell simulation (gray solid line). Note that particular attention is paid The PIC simulation, set up to model the interaction of to accurately fit the low energy part of the spectrum, the SP beam on the front-Al-layer, assumes a similar elec- especially for electron energy ranging from ∼ 30 keV to tron density profile for all targets. The density profile, ∼ 5 MeV where the Sn and Ag K-shell ionization cross produced by the laser amplified spontaneous emission sections reach local maxima [see, for example, the Sn K- (ASE) pedestal, was measured by side-on interferometry shell ionization cross section on Fig. 6 (blue solid line with in a previous experiment19 and successfully reproduced squares)]. As reported a few times in the literature40,41, with the CHIC code (see Fig. 3). We thus use this profile this part of the spectrum is best described by a decreasing as an initial condition in the PIC simulation. For numer- power-law function, defined by the parameter a, rather ical cost reasons, the density is limited to the maximum than the commonly used Maxwellian functions (repre- 2 2 density ne = 80nc, where nc = meε0ω0/e is the critical sented by the red dashed lines). This precaution en- 5

abled us to accurately reproduce the experimental ab- tial density and temperature on-axis profiles of the differ- solute Kα yields as it will be demonstrated below. As a ent targets are extracted, at the REB injection time, from result, the electron mean kinetic energy is estimated to the hydrodynamic simulations (as in the examples pre- be ∼ 190 keV, a value much smaller than that given by sented above in Fig. 3, for the case of targets composed Figure ponderomotive scaling (∼ 0.5 − 1 MeV). of a 60 µm Al-sample). The total simulation time is set 6 The mean angular dispersion of the electron source is to 8 ps with temporal and spatial resolutions of 1 fs and fitted by the following angular distribution function11: 0.5 µm, respectively. Eight different cases, corresponding to the different initial thicknesses of the Al sample (10, [ ] 20, 40 or 60 µm) and of its state (solid or compressed), − − 2 (θ θr) are simulated. Almost 3×106 macro-particles are used to f(θ, r) = exp 2 , (2) 14 ∆θ0 simulate the propagation of ∼ 4 × 10 electrons through the target. The collisions as well as the ionization pro- ◦ ◦ where θr = arctan[tan(30 )y/y0] + 18 is the mean ra- cesses are taken into account. The electrical resistivity dial angle with y0 = 20 µm the initial REB radius, of each material composing the target is computed using ∼ ◦ 42 2 and ∆θ0 55 the dispersion angle . As shown in the classical Drude model: η = meν/e ne, where e and Fig. 5 and measured in the fitting formula of θr, the me are the electron charge and rest mass, respectively, ∼ ◦ electron beam propagates with an angle of 18 from ne the background electron density and ν the harmonic the target normal direction. This relativistic electron mean of the electron collision frequency, given by the propagation angle is a typical feature of oblique inci- Eidmann-Chimier model23,24: dent laser interaction with an overdense plasma. In- deed, the electron beam direction is, at the lowest order, −2 −2 −2 −2 defined by the well-known transverse momentum con- ν = (νe−ph + νe−e) + νc + νsp . (3) servation of the electrons in the boosted frame43. As a result, the electron propagation angle α can be esti- 2 2 √ The terms νe−ph = 2g0e Ti/4πε0vF ~ and νe−e = 44 γ−1 2 ~ mated by sin α = sin θL γ+1 . Hence, low energetic g1Te / TF refer to the electron-phonon and electron- electrons would propagate in the direction normal to the electron collision frequencies, respectively. Te, Ti and 2 target, while the ultra-relativistic electrons would propa- TF = mevF /2 are the electron, ion and Fermi temper- 2 1/3 gate along the laser incident wave direction. Considering ature, respectively. vF = ~(3π ne) /me corresponds ◦ the mean electron energy of 190 keV and θL = 45 , the to the Fermi velocity. The parameter g0 is calculated associated mean propagation angle would be of the order knowing the metal conductivity at the temperature Ti. ◦ of α ∼ 20 , a value consistent with the propagation an- The parameter g1 ∼ 1 − 10 is approximately known gle measured in the simulation. Finally, the laser-to-fast for several metals. In the case of aluminum g0 = 1.25 electron conversion efficiency, calculated by temporally and g1 = 1. The electron-electron collision frequency and spatially integrating the electron energy flux is esti- becomes important for an electron temperature higher mated to be 30 ± 10%. than a decimal fraction of the Fermi temperature. The increase of the electron collision rate with the electron temperature is saturated above the Fermi temperature. B. Hybrid simulations of REB transport Indeed, the mean time of flight between two collisions cannot be shorter than the time of flight over the in- 1/3 The kinetic and angular distribution functions as well teratomic distance r0 = (3/4πni) . Consequently, as the laser-to-fast electron conversion efficiency com- the collision frequency has to be limited to the maxi- 1/2 puted above are used as input parameters for the 2D mum rate νc = (kBTe/me) /r0. For electron tempera- axisymmetric hybrid simulations of the REB transport ture well above the Fermi temperature, the electron-ion using the code developed by J.J. Honrubia33. Given the collisions in a classical non-degenerated plasma become small propagation angle value reported in the PIC simu- the dominant effect and the resistivity is given by the ∗ 4 1/2 3/2 lation, the oblique beam propagation is neglected in the Spitzer√ formula: νsp = g2Z e ne ln Λ/me Te , with 2 ∗ hybrid simulation. This simplification is justified since, g2 = 1/ 2π(8ϵ0) . Z is the ionization state and ln Λ in our case, this parameter doesn’t change the simulation the Coulomb logarithm results much. The REB transport is computed through The electrical resistivities of aluminum, used in the the entire target with either compressed or solid sam- simulations for the solid (blue dashed line) and com- ples. The hybrid method neglects the high frequency pressed (red solid line) targets, are displayed on Fig. 4. effects, hence enabling a simplification of Maxwell equa- As discussed above, the compression of the Al sample tions by neglecting the Poisson equation and the displace- causes its resistivity to increase by a factor ∼ 17. ment current in the Maxwell-Ampere equation. The fast The Kα radiation, generated by the different tracer electrons composing the injected beam, with the asso- layers (Ag, Sn and Cu), is calculated using the ciated current density jh, are modeled kinetically while K-shell ionization cross section formula introduced 46 the return current, je formed by the background elec- by Hombourger as well as the semi-empirical for- 12,33,45 trons, is described as an inertialess fluid . The ini- mula for the Kα relaxation probability introduced by 6

47 Bambynek et al. . It enables reproduction of the Kα sig- source, remains roughly constant with the Al sample ef- nal yields and size for comparison with the experimental fective thickness LAl. This indicates only small shot-to- data. shot variations of the REB source even if a small differ- Two probes are set on the Al sample boundaries (i.e. ence can be noticed experimentally between the solid and at the Ag/Al and Al/Sn interfaces) in order to evalu- compressed targets. The Cu- and Sn-Kα yields, linked to ate the resistive and collisional energy losses of the REB, the REB population that has crossed the Al sample, are integrated over the samples thickness. both decreasing with the Al sample effective thickness. This is the signature of a progressive energy loss of the REB as it propagates deeper inside the target. Besides, V. COMPARISON OF THE EXPERIMENTAL RESULTS the same results show that this behavior is even more WITH SIMULATIONS pronounced for compressed targets, reflecting a more important slowing down of the REB in this case. Figure The Sn-K :Ag-K ratio is plotted in Fig. 9. This ratio 8 The experimental results are systematically compared α α removes any shot-to-shot variations of the REB source, to the hydrid transport simulations via the K emis- α especially the observed difference between compressed sion, assuming a 40% laser-to-fast electron conversion and solid targets (top right panel of Fig. 8). Besides, efficiency. This value best reproduces the experimen- the Sn and Ag-K yields considered here are measured tal absolute K yields. A first comparison concerns the α α with the same spectrometer (same crystal) and the re- Cu-K spot size measured with the 2D Cu-K imager. α α spective photon energies, 25 and 22 keV, are also signifi- Typical images obtained during the experiment are pre- cantly close so that one can fairly assume the two tracers sented on Fig. 7 (left) for the solid and compressed cases. are sensitive to the same electron energies. Their ratio By measuring the mean diameter of the each Cu-Kα is therefore associated to the fraction of the REB popu- spots (averaged FWHM over the two main axis of the lation, with energies above ∼ 30 keV, that has crossed spot) it has been possible to infer the REB shape and the Al sample. Moreover, because it is plotted as a func- its radial spreading evolution for growing sample effec- tion of the areal density ρL , any observed differences tive thickness L (i.e. the real sample length crossed by Al Al between the solid and the compressed cases should be the REB). The results, deconvoluted from the imagery related to resistive energy losses (the collisional losses system response (i.e. the spatial resolution), are plotted being the same, in both the compressed and solid Al as a function of the effective thickness of the Al sam- samples, for a given areal density). The fact that the ple (Fig. 7; see also Table I for values of L ) and com- Al ratio decreases with increasing areal density confirms the pared to the simulations. Good agreement is found be- results presented in Fig. 8. However, no striking differ- tween the simulations (open symbols) and the experimen- ence can be found between the compressed and the solid tal data (full symbols). The REB divergence is estimated ◦ data, indicating that in our interaction regime the re- to 22 ± 6 (half-angle) in concordance with previous resistive energy losses are possibly too weak, compared to sults obtained on solid targets with an equivalent laser collisional losses, to be experimentally observable. Fur- beam intensity26,48. Furthermore, the increase of the Al thermore, these results suggest that the higher decreasing temperature and density seems to not influence the REB rates as a function of L recorded for the compressed divergence, which remains roughly the same for solid and Al targets on the absolute Sn- and Cu-K yields (bottom compressed targets for our interaction conditions. The α panels of Fig. 8), are mainly due to higher collisional ef- REB divergence appears to essentially depend on the fects associated with the sample density increased by the Figure laser-matter interaction and generation processes35. shock-compression. Figure 7 The absolute yields measured with the TCS (Sn- and 9 Ag-Kα) and HOPG (Cu-Kα) x-ray spectrometers are plotted in Fig. 8 as a function of the final thickness of the Al sample and compared to the simulation results. VI. ESTIMATE OF THE REB ENERGY LOSSES AND STOPPING POWER These yields are obtained by temporally and spatially integrating the Kα emission from the fluorescent tracer layers. Fairly good agreement is found between experi- A. Simulations mental (full symbols) and numerical (empty symbols) re- Figure sults especially for the Sn- and Ag-Kα yields. Nonethe- In summary, our experimental results, confirmed by 10 less, the number of measured Kα photons is not well simulations, indicate that the REB energy deposition is reproduced numerically in the case of Cu-Kα emission, higher in the case of compressed targets for a given Al although the general trend is. The factor ∼ 3 disagree- sample thickness. The energy deposition seems to be ment could partly result from the observed anomalous mainly governed by collisional losses. The resistive en- broadening of the emission lines, as it can be seen on the ergy losses, in turn, are either approximately the same example spectrum shown in Fig. 1. This behavior hasn’t in compressed and solid targets, and/or too weak to be been fully explained but it could be due to aging or dete- experimentally observable. In order to understand these rioration of the crystal. Concerning the data trends, we results, simulated 2D maps of the volumic density of col- first observe that the Ag-Kα, a diagnostic of the REB lisional and resistive energy, deposited by the REB inside 7

the targets, are plotted in Fig. 10 c), d), e) and f), for the lisional energy deposition is clearly higher in the case of case of 60 µm Al initial sample thickness. These maps compressed Al. At r = 20 µm, the resistive energy de- correspond to the transverse section of the target, where position decreases progressively as a function of depth, the x and y axis represent the REB propagation axis and following the decrease of the mean current density from 11 2 its associated radius, respectively. For every map, the ⟨jh⟩ = 1.6 × 10 A/cm at the sample front surface (i. 10 2 continuous blue countour line represents the radius Half e. at 0 on the plot) down to ⟨jh⟩ = 5.6 × 10 A/cm at Width at Half Maximum (HWHM) for each x position 30 µm depth in the Al sample, and become less impor- and the white dashed lines correspond to the Al sample tant than the collisional energy deposition: at ∼ 5 µm for boundaries. As illustrated in the panels c) and d), the the compressed Al sample and ∼ 20 µm for the solid one. highest collisional energy deposition over the Al-samples The progressive decrease of ⟨jh⟩ with increasing depth occurs in a zone of radius & 35 µm at the left bound- is accompanied by an increasing difference between the ary, growing up to ∼ 65 µm or to ∼ 90 µm at the right resistive energy losses in solid and compressed Al. These boundaries of the compressed or solid samples respec- effects are even more pronounced for larger radii, as de- tively, even if multiple scattering events induce energy picted in the right panel, where at r = 50 µm the mean deposition over larger radii. The collisional energy depo- current density at the sample front surface is as low as sition is clearly higher inside compressed targets due to 2.8 × 1010 A/cm2. The resistive energy deposition is now the Al density increase (ρ ≈ 2ρ0). Unlike the collisional well below the collisional one and differences between energy deposition, the resistive energy deposition [panels the compressed and solid cases are clearly observable. e) and f)] takes place over a narrower, quite well delimited It is evident that the value of ⟨jh⟩ at the entrance of region around the REB propagation axis (y . 50 µm), the Al sample can play a major role on the observed ex- corresponding to the region where the electron current perimental differences in resistive losses between a solid density jh [panels a) and b)] is larger. For compressed and a compressed target. According to this, one could targets, the resistive energy deposition occurs over larger expect to clearly distinguish higher resistive losses for radii due to the Al sample resistivity rise in the region compressed targets when the mean current density ⟨jh⟩ under shock compression (see Fig. 4). Nonetheless, these is between ∼ 1010 A/cm2 and ∼ 1011 A/cm2. For the additional resistive energy losses for the compressed tar- present experimental conditions, ⟨jh⟩ has been estimated gets remain very weak and thus not sufficient to be de- to ∼ 8 × 1010 A/cm2 by temporally and spatially aver- tectable experimentally. This is confirmed by the coher- aging the current density over a radius of 30 µm around ence, in terms of radial extent, between the plots of the the REB propagation axis. Figure final electronic temperature Te [g) and h)] and the den- An estimate of the total REB energy losses is obtained 12 sity of energy deposited by collisions [c) and d)], and by by integrating the energy deposited by the two mecha- the fact that the values of Te achieved in both the com- nisms over the whole Al sample. The results are pre- pressed and solid Al samples as a function of Al-sample sented in Fig. 12 (left panel) where the integrated resis- ∼ depth are roughly the same (Te 100 eV). Besides, the tive and collisional energy losses are plotted as a function preponderant part of the energy losses occurs over the of the final thickness of the Al sample for both the solid front Al and Ag layers in the first tens of microns, i.e. and compressed cases. The obtained values have been close to the SP laser interaction region, where the REB weighted by the energy of the laser beam in the present ∼ 12 2 current density is maximum (jh 10 A/cm ). In this experiment (35 J). The collisional losses are clearly pre- region, the resistive energy losses are even larger than the dominant in our laser interaction regime, while the resis- collisional losses as expected from other studies involving tive energy losses contribute to approximately one third 7,49 Figure solid Al targets . of the total energy losses. In addition, these results sug- 11 gest that both collisional and resistive energy losses are The role played by the current density jh on the REB losses is detailed in Fig. 11, where lineouts of deposited enhanced in compressed targets, as it has been observed energy density (collisional and resistive) inside the Al above (Fig. 10 and Fig. 11), due to the growth of the den- sample are plotted as a function of the target depth for sity and the resistivity of the Al sample, respectively. the compressed and solid cases. The lineouts are pre- The mean collisional and resistive stopping powers can sented for various radii r: 5 µm, 20 µm and 50 µm where now be estimated by averaging the energy losses over the REB duration, the REB spectrum and 35 µm of crossed the mean current density ⟨jh⟩, averaged temporally at the Al sample front surface and radially for r±5 µm, has been Al sample. The collisional stopping power, ranging from ∼ estimated to be 2.6 × 1011 A/cm2, 1.6 × 1011 A/cm2 and solid and cold to warm ( 2 eV) and dense (2ρ0) Al, is 2.8 × 1010 A/cm2, respectively. At r = 5 µm, i.e. close to thus approximately 1.1-1.5 keV/µm and the resistive one the REB propagation axis, the mean current density is 0.6-0.8 keV/µm. high enough to result in the resistive energy deposition, A growth of the mean REB current density can be 2 proportional to jh, being higher than the collisional one, explored through hybrid simulations, in particular to proportional to jh. Nonetheless, at such current density prospect changes to the resistive losses. A simple way the resistive energy deposition seems not to depend on consists in increasing the REB total energy, assuming the sample state as this remains the same for both the an invariance on the fast electron distribution function compressed and solid targets. On the contrary, the col- and on the efficiency of the SP laser energy conversion 8 into fast electrons. Full integrated 3D simulations including the effect of higher laser energy on the REB source The resistive stopping power, averaged over the beam (spectrum and total energy) are far beyond the scope duration τL, can be reasonably approximated by the fol- of our objectives in this paper. The rise in the REB lowing expression: energy results in an increased number of fast electrons ⟨ ⟩ ∫ and therefore in a higher j , in virtue of the energy flux dε 1 τL h = eEdt , (4) conservation equation between the laser pulse and the dx t τL 0 REB: the beam density is proportional to the laser intensity. With regard to this, we carried out simulations where E corresponds to the electrostatic field generated for two other SP laser beam energies : ESP = 70 and by the electron beam. For a REB duration larger than 350 J. The results are presented in Fig. 12 (middle and the establishment time of the beam neutralization (∼ right panels) where the integrated energy losses inside the few fs) and shorter than the neutralization duration (few Al sample have also been weighted by the corresponding tens of ps), the complete beam neutralization reduces to 50,51 energy of the SP laser beam. As illustrated in the differ- the Ohm’s law : jh = −je = −E/η, where je rep- ent panels (and consistent with the mean electron-beam resents the return current composed of the background energy invariance), the rate of collisional loss is indepen- thermal electrons. Therefore the plasma electron heat- dent of the SP laser beam energy for both the solid and ing, described by the energy conservation equation, is compressed cases. This behavior changes for the rate of given by: resistive losses, which increases with the SP laser beam ∂T energy (and the electronic current density). Thus for C e = j .E = ηj2 , (5) 11 2 e e h ESP = 70 J, yielding ⟨jh⟩ = 1.5 × 10 A/cm at the ∂t sample front surface, the resistive losses become compa- where the ionization, the energy deposition by direct col- rable with the collisional losses inside Al over the first lisions and the heat conduction are neglected, and C 20 µm of propagation. Besides, the difference in resistive e is the heat capacity of the plasma electrons at constant losses between the compressed and the solid cases tends volume. It is worth noting that the ionization and the to fade progressively as the laser energy becomes more contribution of the direct collisions in the plasma heating important. The difference is almost nil when the energy are not negligible. However this simplification does not of the laser increases to E = 350 J and ⟨j ⟩ reaches SP h change the tendencies shown in this section. 1012 A/cm2. For such high electronic current densities, Assuming that the REB current density is large enough the characteristic heating time up to the Fermi temper- (j > 1010 A/cm2) to heat the medium to a tempera- ature, of the order of τ ∼ 3n T /2ηj2 ≈ 0.2 ps, is neg- h F e F h ture far beyond the Fermi temperature (T ≫ T ) it is ligible compared to the REB duration. The medium is e F possible to separate the collision frequency [Eq.(3)], and thus rapidly heated to temperatures where the resistivity, hence the resistivity, into four distinct regions, each one mainly governed by electron-ion collisions, decreases con- − corresponding to a temperature range where a given col- siderably according to η ∝ T 3/2 independently of the e lision process dominates, i.e. electron-phonon (ν − ), target density (Spitzer regime). Any initial difference in e ph electron-electron (ν − ) and electron-ion (ν ) collisions temperature, caused by the compression, is thus quickly e e sp as well as the saturation regime (νc). Hence, the resis- erased. For such REB current density it would then not tivity can be expressed as follows52: be possible, in the context of our experimental method, to discriminate between the resistive and collisional losses  by comparing yields from solid and compressed targets, ≤ ≤  ηe−(ph ) for T0 Te T1 as they will no longer depend on the propagation medium  2  Te  η1 for T1 ≤ Te ≤ T2 density. T1 ( ) 1 η = 2 , (6) Te  η2 for T2 ≤ Te ≤ T3  T2  ( )− 3  2  Te B. Analytical estimates of resistive stopping power η3 for T3 ≤ Te T3

In this section we introduce a simple analytical where formula which enables the resistive stopping power as a function of the REB current density jh and of the 2 laser beam duration τL to be estimated. This formula ηe−ph = meνe−ph/e ne (7) is useful to predict the ranges for which the resistive 2 η = m ν − (T )/e n (8) stopping power is comparable or even higher than the 1 e e e 1 e 2 collisional stopping power. In addition, it can predict the η2 = meνc(T2)/e ne (9) 2 (jh, τL)-values for which the resistive stopping power, η3 = meνsp(T3)/e ne. (10) averaged over τL, is signiﬁcantly sensitive to the initial conditions, e.g. the initial temperature and density of As the resistivity must be continuous, the transition tem- the propagation medium. peratures can be deduced from the above equations: 9

relation, using Eq. (14), leads to the resistive electron stopping power: 2 ~ 1/2 ≪ T1 = (2g0e TiTF /4πε0g1vF ) TF (11)  ( ) ⟨ ⟩ 2 1/2 2/3  3ene T2 T0 T2 = (TF ~/g1m r0) ∼ TF (12) | | − + Tef − T2 for Tef ≥ T2 e dε 2 jh τL ( 2 2T2 ) ∗ 4 1/2 = T 2 2 , ≫  3ene ef T0 T3 = (g2Z nee r0 ln Λ) TF , (13) dx t − for Tef < T2 2|jh|τL 2T2 2T2 In the following calculation, we assume the Coulomb log- (24) arithm is equal to 4. where Tef = Te(τL) is obtained using Eq.(15). In a similar way, the heat capacity is separated into In the case of a compressed target (T0 > T1) the REB two regions: doesn’t experience the temperature domain where the e- { ph collisions occurs. Consequently, the equations must 3 Te ne for Te ≤ T2 be modified in order to take this effect into account. The 2 T2 Ce = 3 , (14) temporal evolution of the plasma electron temperature ne for Te > T2 2 therefore reads: Within this simplification, for a degenerated state, Te < T2 ∝ TF , the heat capacity is proportional  ( ) t to n T /T , consistent with a free fermion gas where  T0 exp for t ≤ t2 e e F  τ1 2  ( ) Ce = π Tene/(2TF ). For temperatures higher than T2, − ′ 2 comp t t2 T (t) = T2 1 + for t2 ≤ t ≤ t3 , (25) the plasma electron gas can be considered as ideal and e  ( 2τ2 )  ′ 2/5 C = 3n /2.  t−t e e 5 3 ≥ T3 1 + 2 τ for t t3 For an initial temperature T0 smaller than T1 (solid 3 Al sample case) the integration of the energy conserva- (√ ) ′ ′ ′ − tion equation (5) provides the temporal evolution of the with t2 = τ1 ln(T2/T0), and t3 = t2 + 2τ2 T3/T2 1 , 50 plasma electron temperature : the other parameters remaining unchanged.  ( )1/2 Calculations using equation (24) are presented in  2t  T0 1 + for 0 ≤ t ≤ t1 Fig. 13. The results are plotted for REB current den-  ( τ0 )   t−t1 sity and duration (jh, τL) for a range of conditions T1 exp for t1 ≤ t ≤ t2 solid τ1 found in present short-pulse laser-plasma experiments T (t) = ( )2 , (15) e − 12 2  t t2 (j . 10 A/cm , τ ∼ 100 fs) to the full-scale FI con-  T2 1 + for t2 ≤ t ≤ t3 h L  2τ2 . 14 2 ∼  ( )2/5 ditions (jh 10 A/cm , τL 10 ps), and of course  −  5 t t3 ≥ T3 1 + 2 τ for t t3 including the parameters of the experiment described in 3 11 2 this paper (jh . 10 A/cm , τL ∼ 1 ps). Figure where the characteristic heating times are: In the top and middle panels the results correspond- 13 ing to a cold and solid Al (i.e. T0 = Ti = 0.03 eV and 3n T 2 3 τ = e 0 (16) ρ = ρ0 = 2.7 g/cm ) and to a warm and compressed Al 0 2 3 2ηe−phjhT2 (i.e. Te = Ti = 2 eV and ρ = 2ρ0 = 5.4 g/cm ) are 2 η − T plotted, respectively. Both present a similar behavior, τ = τ e ph 1 (17) 1 0 η T 2 but with larger values in the case of a compressed tar- 1 0 get. Hence, for small current densities, such that the 2 η1 T2 final temperature is smaller than the Fermi temperature, τ2 = τ1 2 (18) η2 T1 the averaged resistive stopping power is negligible. The η2 T3 strong increase of the resistive stopping power, associated τ3 = τ2 , (19) with the strong electrical resistivity increase, occurs when η3 T2 the electron plasma becomes classical, Te ≥ TF . For very and the transition times are: high currents, such that the plasma resistivity enters the ( ) t = T 2/T 2 − 1 τ /2 (20) Spitzer regime, the averaged resistive stopping power de- 1 1 0 0 creases. Its maximum culminates at several keV/ µm, for t2 = t1 + τ1 ln (T2/T1) (21) ∼ (√ ) a well defined set of parameters (jh, τL), where Tef TF , i.e. the final resistivity corresponds to its maximum value t3 = t2 + 2τ2 T3/T2 − 1 . (22) (see Fig. 4). Making use of Eqs. (5) and (4), the average electron We note that the resistive stopping power values ob- stopping power can be rewritten as: tained with this simple model for the parameters of the ⟨ ⟩ ∫ present experiment (indicated by the full blue circles in dε e Tef Fig. 13), of the order of 0.4 keV/ µm for cold-solid and = | | CedTe, (23) of 0.6 keV/ µm for warm-compressed propagation me- dx jh τL T t 0 dia, are in good agreement with those estimated in Sec- where Tef is the final electron temperature, which is con- tion VI A by comparing the Sn-Kα:Ag-Kα ratio from sidered here to be Te(τL). An integration of the previous both absolute yield data and simulation results: 0.5 and 10

0.8 keV/ µm, for solid and compressed Al-sample targets, compression factor, the resistive stopping power in the respectively. compressed target is just enhanced by 50 %. The ratio between the resistive electron stopping power For a small compression factor, the difference between in compressed and solid aluminum is presented in the solid and compressed targets is significant provided the bottom panel of Fig. 13, as a function of the beam cur- final electron temperature is smaller or comparable to rent density and the beam duration. Again, two regions the Fermi temperature, i.e., provided the electron current are well defined. For small current densities, the resistive density and pulse duration verify the inequality given in stopping power is much higher for the compressed target Eq. (27). For indication, the equality is plotted as a while the difference tends to disappear for high current red dashed line in the top and middle panel of Fig. 13, densities. The transition between the two regions coin- for the case of the compressed target. The difference cides with the end of the electron quantum state in the between resistive stopping power in compressed and in case of the compressed target. solid targets is noteworthy for parameters (jh, τL) on The aforementioned picture can be understood as fol- the left of the dashed red line. lows. The electron temperature evolution, defined by Eq. On the other hand, for the kind of plasma explored in (15), can be roughly simplified into two heating steps. our experiment, the resistive stopping power is compa- Indeed, the heating time-scales τ2 and τ3, respectively rable with the collisional⟨ ⟩ stopping power provided that the former reaches dε ∼ 1 keV/µm. The isocontours associated with a plasma resistivity ruled by the electron- ⟨ ⟩ dx t ion collisions in the saturation and Spitzer regimes, are dε of dx t, ranging from 0.5 to 7 keV/ µm, are presented negligible compared to τ1 dominated by the e-ph colli- in the same figure (top and middle panels): A small sions. Hence, during the first heating step, the plasma is window exists such that the resistive stopping power heated from the initial temperature, T0, to the transition is comparable with the collisional one, while being no- temperature T1, in a time equal to t1. Given the fast ticeably higher in the compressed case than in the solid heating rate once the electron-electron collision appears, case. For τL = 1 ps (common to Nd:Yag facilities) this the plasma is instantaneously heated to the characteristic window corresponds to 1011 . j < 1012 A/cm2, slightly ≤ h temperature T3. For small laser beam duration, τL t1, above the regime explored in the present experiment. the resistive electron stopping power is entirely determined by the e-ph collisions. Using Eqs. (15), and (24), In particular, the analytical estimation for the FI it can be written as: 14 2 conditions (jh = 10 A/cm , τL = 10 ps) yields a ⟨ ⟩ 2 time-averaged resistive stopping power not exceeding ∼ dε emeg0Tijh = eηe−phjh = . (26) 1 keV/ µm. The calculation was made for a warm-dense 1 5 4 dx 3 3 t 2ε03 3 π 3 ~ ne metal, which in the FI scenario can be representative of the cone tip material heated by thermal or compression This formula holds as long as the current density and waves. However we have to recall that the ionization beam duration verify the relation: as well as collisional losses are not taken into account ( ) 2 2 in this model. These simplifications could introduce a 2 3ne T1 T0 slight underestimation of the resistive stopping power, in j τL ≤ − . (27) h 13 2 4ηe−ph T2 T2 particular when jh > 10 A/cm . For higher beam current or longer beam duration, the resistive stopping power is determined by the Ohmic heat- VII. CONCLUSIONS ing in the Spitzer regime, and is written as:

⟨ ⟩ ( ) 2 ( ) 3 In conclusion, we performed an experiment in order to dε 3 5 5 2 n T 5 − 1 5 e 3 5 fully characterize the effects of background density and = eη3 jh . (28) dx t 2 3 τL temperature rise, driven by laser-compression, on a relativistic regime of REB energy transport. Our experimen- For small beam currents or short pulses, Eq. (26), the tal setup was designed to conserve the areal density of the electron resistive stopping power increases with the ion Al propagation layer during the compression. In these temperature and decreases with the plasma electron den- specific conditions, the energy losses due to direct colli- sity. In the previous example (Fig. 13), the (initial) tem- sions of REB electrons with the background material can perature of the compressed target is higher by nearly two be separated from the energy losses due to the resistive orders of magnitude than the (initial) temperature of the effect. This difference was hardly visible in the experi- solid target, while the electron density is higher only by ment due to the weak but non-negligible contribution of a factor of 2. The corresponding stopping power is thus the collective stopping power. To quantify the latter, we increased by a factor of ∼ 25 relative to the case of a cold- performed 3D REB-generation PIC simulations followed solid target. By contrast, the resistive stopping power in by 2D axis-symmetric REB-transport hybrid simulations, the Spitzer regime, Eq. (28), slowly increases with the accounting for the complex collective effects such as the 3/5 electron plasma density (∝ ne ), and decreases with the electrostatic field ionization, the collisional ionization and pulse duration and beam current. Hence, given the small the three body recombination far from a local thermody- 11 namic equilibrium. The simulations reproduced Kα ab- CeSViMa and from GENCI-CINES (Grants No. 2011- solute yield and size measurements. We showed that the 056129 and No. 2012-056129). M.C. participated in compression tends to increase the collisional and resistive this work thanks to funding from EPSRC (Grant number REB stopping powers, due to the growth in density and EJ/J003832/1). The authors gratefully acknowledge L. resistivity, respectively. The effects of the transient be- Gremillet for fruitful discussions. havior of Al resistivity during REB transport sum upon 1M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, the effects of compression. Although the collisional losses J. Woodworth, E. M. Campbell, M. D. Perry and R. J. Mason, are clearly the dominant process in our experimental in- Phys. Plasmas, 1, 1626 (1994) 10 2 2 teraction conditions (⟨jh⟩ ∼ 8 × 10 A/cm , at the Al- S. Atzeni, J. R. Davies, L. Hallo, J. J. Honrubia, P. H. Maire, sample front surface) the resistive processes are respon- M. Olazabal-Loum, J. L. Feugeas, X. Ribeyre, A. Schiavi, G. sible for about one third of the total energy losses. Schurtz, J. Breil and Ph. Nicola¨ı,Nucl. Fusion, 49, 055008 (2009) 3R. Kodama, P. A. Norreys, K. Mima, A. E. Dangor, R. G. We underline that the benchmarked hybrid code and Evans, H. Fujita, Y. Kitagawa, K. Krushelnick, T. Miyakoshi, the analytical calculations both predict a range for the N. Miyanaga, T. Norimatsu, S. J. Rose, T. Shozaki, K. Shige- 11 12 2 REB current density, 10 . jh < 10 A/cm for pulse mori, A. Sunahara, M. Tampo, K. A. Tanaka, Y. Toyama, T. Yamanaka and M. Zepf, Nature, 412, 798 (2001) duration τL = 1 ps (slightly above our experimental con- 4S. Atzeni, Phys. Plasmas, 6, 3316 (1999) ditions), where the resistive stopping power is compara- 5V. T. Tikhonchuk, Phys. Plasmas, 9, 1416 (2002) ble to the collisional one. Interestingly, a posterior sim- 6A. J. Kemp, Y. Sentoku, V. Sotnikov, and S. C. Wilks, Phys. ilar experiment using the same geometry and targets, Rev. Lett., 97, 235001 (2006) with a SP intensity above 1020 W/cm2, has produced 7J. J. Santos, A. Debayle, Ph. Nicola¨ı,V. Tikhonchuk, M. Man- data consistent with our predictions: a measurable differ- clossi, D. Batani, A. Guemnie-Tafo, J. Faure, V. Malka and J. J. 49 Honrubia, Phys. Plasmas, 14, 103107 (2007) ence between compressed and solid data . When jh ap- 8L. Volpe, D. Batani, A. Morace and J. J. Santos, Phys. Plasmas 12 2 proaches 10 A/cm , the Al sample is rapidly (∼ 0.2 ps, 20, 013104 (2013) 9 significantly smaller than τL) heated to a temperature L. Volpe, D. Batani, G. Birindelli, A. Morace, P. Carpeggiani, above the Fermi temperature where resistivity is gov- M. H. Xu, F. Liu, Y. Zhang, Z. Zhang, X. X. Lin, F. Liu, S. erned by electron-ion collisions (Spitzer regime). Conse- J. Wang, P. F. Zhu, L. M. Meng, Z. H. Wang, Y. T. Li, Z. M. Sheng, Z. Y. Wei, J. Zhang, J. J. Santos and C. Spindloe, Phys. quently, the material resistivity rapidly drops indepen- Plasmas 20, 033105 (2013) dently of its initial density/temperature (solid or com- 10J. S. Green, V. M. Ovchinnikov, R. G. Evans, K. U. Akli, H. pressed sample), saturating the time-averaged resistive Azechi, F. N. Beg, C. Bellei, R. R. Freeman, H. Habara, R. stopping power. While not as accurate as the transport Heathcote, M. H. Key, J. A. King, K. L. Lancaster, N. C. Lopes, simulations, the analytical model helps to clearly under- T. Ma, A. J. MacKinnon, K. Markey, A. McPhee, Z. Najmudin, P. Nilson, R. Onofrei, R. Stephens, K. Takeda, K. A. Tanaka, stand the saturation mechanism. In particular, the ana- W. Theobald, T. Tanimoto, J. Waugh, L. Van Woerkom, N. C. lytical expansion of our investigation to the fast ignition Woolsey, M. Zepf, J. R. Davies, and P. A. Norreys , Phys. Rev. conditions predicts that the resistive stopping power will Lett., 100, 015003 (2008) not undermine the energy transport to the target dense 11A. Debayle, J. J. Honrubia, E. d’Humières,and V. T. Tikhonchuk , Phys. Rev. E, 82, 036405 (2010) core. 12L. Gremillet, G. Bonnaud and F. Amiranoff, Phys. Plasmas, 9, 941 (2002) 13M. Manclossi, J. J. Santos, D. Batani, J. Faure, A. Debayle, V. T. ACKNOWLEDGMENTS Tikhonchuk, and V. Malka, Phys. Rev. Lett. 96, 125002 (2006) 14J.J Honrubia and J. Meyer-ter-Vehn, Nuclear Fusion 46, L25 (2006) The work presented here was performed under financial 15H. D. Shay, P. Amendt, D. Clark, D. Ho, M. Key, J. Koning, M. support from the Conseil Régionald’Aquitaine through Marinak, D. Strozzi and M. Tabak, Phys. Plasmas 19, 092706 (2012) project PETRA 2008 13 04 005, and both the French 16E. Martinolli, M. Koenig, S. D. Baton, J. J. Santos, F. Amiranoff, National Agency for Research (ANR) and the competi- D. Batani, E. Perelli-Cippo, F. Scianitti, L. Gremillet, R. Mlizzi, tiveness cluster Alpha - Route des Lasers through project A. Decoster, C. Rousseaux, T. A. Hall, M. H. Key, R. Snavely, TERRE ANR-2011-BS04-014. This study has been car- A. J. MacKinnon, R. R. Freeman, J. A. King, R. Stephens, D. ried out in the frame of the Investments for the future Neely, and R. J. Clarke, Phys. Rev. E, 73, 046402 (2006) 17T. A. Hall, S. Ellwi, D. Batani, A. Bernardinello, V. Masella, Programme IdEx Bordeaux LAPHIA (ANR-10-IDEX- M. Koenig, A. Benuzzi, J. Krishnan, F. Pisani, A. Djaoui, P. 03-02). The authors acknowledge LULI for giving access Norreys, D. Neely, S. Rose, M. H. Key, and P. Fews, Phys. Rev. to the LULI2000 facility in the context of the preparatory Lett. 81, 1003 (1998) phase of the HiPER project (Work Package 10: experi- 18D. Batani, J. R. Davies, A. Bernardinello, F. Pisani, M. Koenig, mental validation program), as well as the support of the T. A. Hall, S. Ellwi, P. Norreys, S. Rose, A. Djaoui, and D. Neely, Phys. Rev. E 61, 5725 (2000) LULI2000 engineering staff in designing the setup imple- 19J. J. Santos, D. Batani, P. McKenna, S. D. Baton, F. Dorchies, mentation and during the experimental run. Alphanov is A. Dubrouil, C. Fourment, S. Hulin, Ph. Nicola¨ı,M. Veltcheva, P. also gratefully acknowledged for the laser cutting of the Carpeggiani, M. N. Quinn, E. Brambrink, M. Koenig, M. Rabec targets. The numerical study was supported by Grant Le Glohaec, Ch. Spindloe and M. Tolley, Plasma Phys. Controll. Fusion, 51, 014005 (2009) No. ENE2009-11668 of the Spanish Ministry of Educa- 20J. J. Santos, D. Batani, P. McKenna, S. D. Baton, F. Dorchies, tion and Research and by the European Science Foun- A. Dubrouil, C. Fourment, S. Hulin, E. d’Humières,Ph. Nicola¨ı, dation SILMI program, and used HPC resources from L. Gremillet, A. Debayle, J. J. Honrubia, P. Carpeggiani, M. 12

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FIG. 1. (color online) Schematics of the experimental setup. The target design is detailed in the inset.

FIG. 2. (color online) Example of SOP measurements obtained by comparing the temporal trace of the LP laser beam (reference low energy ﬂux shot without target; left panel) and the radiative emission following shock breakout at the front side of a 20 µm Al-sample target (without the front Al and Ag layers ) and using the LP laser at full power (right panel).

FIG. 3. (color online) On-axis density (red solid lines) and temperature (blue dashed lines) profiles obtained with 2D CHIC hydrodynamic simulations for the case of a 60 µm Al sample target either solid (top panel) or compressed (bottom panel). The dotted green line locates the shock front position. The red and green arrows represent the SP (i.e. REB) and LP (shock) directions, respectively. Their tips position indicate approximately the critical density surface for the SP laser and the shock front. The colored areas above the density profiles delimit the different material distribution in the target.

FIG. 4. (color online) Evolution of the aluminum electrical resistivity as a function of the background electron temperature Te, for a solid (blue dashed line) or compressed (red solid line) target. Concerning the solid case, Ti is kept constant and equal to 0.03 eV. In contrast, for the compressed case Te is set equal to Ti, meaning that we assume a thermodynamic equilibrium in the Al sample. The initial resistivities (prior to the REB injection) are pointed out by the open circles for the two target states. These two curves have been calculated following the Eidmann-Chimier model23,24 described in the next section.

FIG. 5. (color online) Magnetostatic ﬁeld e|Bz|/meω0 in logarithmic scale at the time when the maximum of the laser ﬁeld reaches the absorption region. The white dashed line refers to the preplasma/target boundary.

FIG. 6. (color online) Simulated (PIC) REB distribution function (gray line) and its associated fits: the red solid line corresponds to the fit f(ε) given by (1), and the two red dashed lines to the often used sum of exponential decreasing fits. The blue solid line with squares represents the Sn K-shell ionization cross section.

FIG. 7. (color online) (left) Set of typical Cu-Kα images obtained during the experiment. (right) Evolution of the Cu-Kα spot size as a function of the eﬀective thickness of the Al sample (see also Table.I).

FIG. 8. (color online) Evolution of the experimental (full symbols) and simulated (empty symbols) absolute Ag-Kα (top right), Cu-Kα (bottom left) and Sn-Kα (bottom right) yields as a function of the ﬁnal thickness of the solid (blue) or compressed (red) Al sample.

FIG. 9. (color online) Comparison of the experimental and simulated evolution of the Sn-Kα:Ag-Kα ratio as a function of the Al sample areal density ρLAl. .

FIG. 10. (color online) 2D maps of the electronic current density at t = 1.5 ps (top), the ﬁnal volumic density of collisional and resistive energy deposited inside the target (middle), and the ﬁnal electronic temperature (bottom) computed by the hybrid simulations. The results are presented for solid (left) and compressed (right) target composed initially of a 60 µm Al sample. All the color values are in logarithmic scales. The white dashed lines correspond to the Al sample boundaries. The continuous blue countour lines represent the radius HWHM for each longitudinal position. The red arrows represent the SP laser position.

FIG. 11. (color online) Volumic density of energy deposited, by resistive (orange) or collisional dragging (green), inside the compressed (dashed line) or solid (solid line) Al sample as a function of target depth (relative to the REB injection surface) and for diﬀerent radii ⟨r⟩ from the REB propagation axis: 5 µm (left), 20 µm (middle) and 50 µm (right). The mean current densities labeling the plots correspond to the time-averaged and r ± 5 µm averaged values at the sample front surface (corresponding to 0 on the plots).

FIG. 12. (color online) Estimates of collisional (green squares) and resistive (orange triangles) REB energy losses inside the Al sample as a function of its effective thickness (open and full symbols respectively for compressed and solid samples), assuming different SP laser beam energies: 35 (left panel: parameters of the present experiment), 70 (middle panel) and 350 J (right panel). For the three cases we assumed a 40% energy conversion efficiency from the laser pulse to the REB. The energy loss values are normalized to the respective laser energy. 14

FIG. 13. (color online) Resistive electron stopping power [ keV/ µm] calculated according to Eqs. (15) and (24) against the ∗ 3 current density jh and the beam duration τL for cold Al at solid density: Z = 3, ρ = 2.7 g/cm , T0 = Ti = 0.03 eV (Top panel) ∗ 3 and for compressed Al: Z = 3, ρ = 5.4 g/cm , T0 = Ti = 2 eV (Middle panel). The ratio of resisitive stopping power between compressed and solid Al is presented in the Bottom panel. The black lines represent the isocontours. The dashed red lines in the top and middle panels correspond to the equality of Eq. (27). The full blue circles indicate the parameters of the present 10 2 experiment (i.e. jh ≈ 8 × 10 A/cm and τL ∼ 1 ps). Ag-Kα Ag-Kβ + Sn-Kα Cu-Kα Cu-Kβ

HOPG LP laser beam E [keV] 8 9 E [keV] 22 25 2D Cu-Kα TCS crystal imager

Cu (10µm) Sn (10µm) Fast electron tracer Fast electron tracer at the rear side PP (15µm) at the rear side Ablator

Ag (5µm) 22.5° Cu-Kα spot Tracer of the fast electron source

50µm Al (5µm) REB LP laser beam generation layer compressing the targ et 250J, 4.5ns, 0.53µm CCD focal spot:400µm (flat-top) 13 2 SOP 3x10 W/cm

Al sample with variable 45° initial thickness SP laser beam (10, 20, 40 and 60µm) generating REB 50µm REB propagation layer SP laser beam 35J, 1.5ps, 1.06µm focal spot: 10µm (FWHM) 3x1019W/cm2

l llll lll lll

l lll llll l

ll ll ll τ

ρ h

h Electrical resistivity η [Ω.m] T T i i =0.03 eV (ρ =0.03 eV =T e (ρ=5.4g/cm Cold and solid and Cold Al sample Al 0 =2.7g/cm T Warm and compressed and Warm 3 =2ρ e [eV] 0 ) 3 Al sample Al ) Spitzer regime Spitzer ∝T e -3/2

gg g

εgg

εg

εg 10µm 40µm Solid 50µm 23.6µm (40µm) 23.6µm 6.3µm (10µm) 6.3µm Compressed

Diameter (FWHM) [µm] 100 120 140 20 40 60 80 Effective thickness of thEffective thickness 0 02 040 30 20 10 0 Simulations: Compress Simulations: Solid Simulations: Compress Experiment: Solid Experiment: e Al sample [µm] sample e Al 06 70 60 50 ed ed Absolut Cu-Kα yield [ph/sr] 10 10 10 Effective thickness of thEffective thickness 10 11 12 02 04 30 20 10 0 Compressed Compressed Solid Solid Cu-Kα Experiment Simulations 06 70 60 50 0 e Al sample [µm] sample e Al

Absolut Sn-Kα yield [ph/sr] Absolut Ag-Kα yield [ph/sr] 10 10 10 10 10 10 Effective thickness ofthEffective thickness Effective thickness of thEffective thickness 10 11 12 10 11 9 02 04 30 20 10 0 02 04 30 20 10 0 Ag-Kα Sn-Kα 06 70 60 50 0 06 70 60 50 0 e Al sample [µm] sample e Al e Al sample [µm] sample e Al

1 α α 11ρ1 Solid Compressed 12 12 a) 11.5 b) 11.5 250 250 11 11 Electronic 200 10.5 200 10.5 current density 150 10 150 10 r [µm] at t=1.5ps 9.5 r [µm] 9.5 -2 100 100 [A.cm ] 9 9 50 50 8.5 8.5 8 8 20 40 60 80 20 40 60 80 x [µm] x [µm] 8 8 c) 7.5 d) 7.5 Final volumic 250 250 density 7 7 200 of collisional 6.5 200 6.5 6 6 energy 150 150 r [µm] 5.5 r [µm] 5.5 deposited 100 5 100 5 [J.cm-3] 50 4.5 50 4.5 4 4 3.5 20 40 60 80 20 40 60 80 3.5 x [µm] x [µm] 8 8 Final volumic e) 7.5 f) 7.5 250 250 7 density 7 6.5 200 6.5 of resistive 200 6 6 energy 150 150

r [µm] 5.5

r [µm] 5.5 deposited 100 5 100 5 -3 [J.cm ] 4.5 4.5 50 50 4 4 3.5 3.5 20 40 60 80 20 40 60 80 x [µm] x [µm] 4 4 g) h) 250 3 250 3 200 200 Electronic 2 2 temperature 150 150 r [µm] 1 r [µm] 1 [eV] 100 100 50 0 50 0

-1 -1 20 40 60 80 20 40 60 80 x [µm] x [µm] at r=5µm at r=20µm at r =50µm ] 8 ] 8 ] 8 3 10 3 10 3 10

10 2 7 7 =2.8x10 Acm 10 10 107 osited energy[J/cm osited energy[J/cm osited energy[J/cm 11 2 =2.6x10 Acm 11 2 6 6 =1.6x10 Acm 10 10 106

5 5 10 10 105

Volumicdensity of dep 0 5 10 15 20 25 30 Volumicdensity of dep 0 5 10 15 20 25 30 Volumicdensity of dep 0 5 10 15 20 25 30 Depth in Al [µm] Depth in Al [µm] Depth in Al [µm]

Solid Solid Collisional losses Resistive losses Compressed Compressed

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