Radiation drive temperature measurements in aluminium via radiation-driven shock waves: Modeling using self-similar solutions
aa r X i v : . [ phy s i c s . p l a s m - ph ] F e b Radiation drive temperature measurements in aluminium viaradiation-driven shock waves: Modeling using self-similarsolutions
Shay I. Heizler, ∗ Tomer Shussman, and Moshe Fraenkel Department of Physics, Nuclear Research Center-Negev,P.O. Box 9001, Beer-Sheva 8419001, Israel Department of Plasma Physics, Soreq Nuclear Research Center, Yavne 8180000, Israel
Abstract
We study the phenomena of radiative-driven shock waves using a semi-analytic model based onself similar solutions of the radiative hydrodynamic problem. The relation between the hohlraumdrive temperature T Rad and the resulting ablative shock D S is a well-known method for the es-timation of the drive temperature. However, the various studies yield different scaling relationsbetween T Rad and D S , based on different simulations. In [T. Shussman and S.I. Heizler, Phys.Plas., 22, 082109 (2015)] we have derived full analytic solutions for the subsonic heat wave, thatinclude both the ablation and the shock wave regions. Using this self-similar approach we derivehere the T Rad ( D S ) relation for aluminium, using the detailed Hugoniot relations and includingtransport effects. By our semi-analytic model, we find a spread of ≈ T Rad ( D S ) curve,as a function of the temperature profile’s duration and its temporal profile. Our model agrees withthe various experiments and the simulations data, explaining the difference between the variousscaling relations that appear in the literature. ∗ [email protected] . INTRODUCTION Radiative heat waves (Marshak waves) are a basic phenomena in high energy densityphysics (HEDP) laboratory astrophysics [1], and in modeling of astrophysics phenomena(e.g. supernova) [3, 4]. Specifically, once a drive laser or other energy source is applied toa sample, a radiative subsonic heat wave generates an ablative shock wave, propagating inthe material in front of the heat wave. This is the basic physical process which occurs insidethe walls of a hohlraum used to convert laser light into x-rays in the indirect drive approachof inertial confinement fusion (ICF) [1, 2].The heat conduction mechanism in these high temperatures (100-300eV and higher) andopaque regions is radiation heat conduction, rather than the electron heat conduction. Thisradiation-dominated heat conduction mechanism occurs even though the radiation energy(or the radiation heat-capacity) and the radiation pressure themselves are negligible relativeto the material energy and pressure. The wave propagates mainly through absorption andblack-body emission processes (Thomson scattering is negligible in the range of 100-500eV,compared to opacity). Although the equation which correctly describes the photons motionis the Boltzmann equation for radiation [7], when the radiation is close to local thermody-namic equilibrium (LTE), the angular distribution of the photons is close to be isotropic anddiffusion approximation yields a very good description of the exact behavior. The frequencydistribution is close to a Planckian with the same temperature of the material. In thiscase, the governing equation is replaced by a simple single temperature conduction diffusiveequation, where the diffusion equation is determined by the Rosseland mean opacity [4–6].Roughly, Marshak waves can be subdivided to supersonic waves, and subsonic waves.When the wave propagates faster than the sound velocity of the material, the materialhydrodynamics motion is negligible and the Marshak wave is considered to be supersonic .When the wave propagates slower than the sound velocity, hydrodynamics should be takeninto account and the radiation conduction equation is solved as part of the energy conserva-tion equation of the hydrodynamics system of equations [7]. This is the subsonic
Marshakwave. In this case, a strong ablation occurs, causing the heated surface to rapidly expandbackwards. Due to momentum conservation, a strong shock wave starts to propagate fromthe heat wave front (the ablation front), and it propagates faster than the heat front itself.Marshak offered a self-similar solution to the supersonic region, in the case that the2aterial’s opacity and heat capacity can be described through simple power-laws [8]. Later,a self-similar solution for the subsonic case was introduced, for the hydrodynamics equationscoupled to the radiation conduction equation [9–11]. Many solutions based on self-similarsolutions or perturbation theories were offered, backed also by direct simulations [5, 12–14]. Those solutions were recently used to analyze both qualitatively and quantitativelysupersonic Marshak wave experiments [15, 16]. We note that the self-similar solutions includeonly the heat wave region itself, but not the shock region, since the whole subsonic motionis not self-similar altogether.Recently, Shussman et al. offered a full self-similar solution to the subsonic problemfor a general power-law dependency of the temperature boundary condition (BC), basedon patching two self similar solutions, each valid for a different region of the problem [17].Shussman et al. used the Pakula & Sigel self similar solution [9–11] for the heat region,which determines a power-law time-dependent pressure BC for the shock region, and strongshock Hugoniot relations for the other BC. Since the full solution is composed of two regionswith different physical regimes: heat region ( ≈ ≈ − analytical re-visit of the experi-ments aimed for the evaluation of radiation temperature drive in hohlraums, by measure-ment of the shock velocity inside a wedged well characterized aluminium sample attachedto the hohlraum wall, first presented by Kauffman et al. [19, 20]. The relation between thehohlraum radiation temperature and the shock velocity, is a scaling analytical fit to fullradiation-hydrodynamic simulations. Based on newly performed simulations, this scalingrelation was argued to be a non-universal, and specifically to depend on the laser pulseduration [21, 22]. In this study we analytically examine the sensitivity of the radiation tem-perature to shock velocity scaling relation, to the different parameters of the temperatureprofile, such as the temperature profile’s duration and its temporal shape, not just qualita-tively but in fully quantitative manner. We use Shussman et al. analytical model [17] forthe ablation shock region, and take advantage of the binary EOS model which is of mostimportance in modeling of these experiments [18].3he present paper is structured in the following manner: first, in Sec. II, we will re-view previous experiments and scaling-laws offered to determine the radiation temperaturethrough the shock velocity measurement in aluminium. In Sec. III we derive the self-similarsemi-analytic equations that determine the explicit dependency of the resulting drive temper-ature in the different profiles’ parameters. Next, in Sec. IV the model results are presented,showing the dependency of the results on the various parameters, and reproducing the ex-periments and simulations presented in the literature. The model results will explain thedifferences between the various experiments’ results. A short discussion is presented in Sec.V. II. THE SHOCK-WAVE MEASUREMENTS EXPERIMENTS AND THE DIF-FERENT SCALING LAWS
The experimental method for the evaluation of the radiation drive temperature by mea-suring the shock velocity was proposed by Hatchett et al. [23–25] and performed by Livermoregroups [19, 20], and right after that by the German group [26, 27]. A schematic diagram ofthe American experiments can be seen in Fig. 1(a), where high energy laser beams enter ahohlraum to generate a high-temperature x-ray cavity of 100-300eV.The laser energy is absorbed in the high- Z material hohlraum walls, and generates softx-rays which undergo thermalization inside the hohlraum. The hohlraum walls are made ofhigh- Z optically thick materials (usually gold) to achieve a large laser to x-ray conversionefficiently. Since the hohlraum walls have a finite opacity, a nonlinear radiative heat wave isgenerated and quickly becomes subsonic. A diagnostic hole is covered by a wedged samplemade of reference-material which should be well characterized (by means of opacity andEOS) so aluminium is the natural choice. Although aluminium is less opaque than goldto x-rays due to its relatively low- Z , it is opaque enough so that the heat wave inside thealuminium is subsonic as well. The high energy ablates the inner surface of the hohlraumwalls and the aluminium wedge, yielding an ablative density profile. As a consequence (dueto conservation of momentum), an ablative (radiation-driven) shock wave is propagatingin front of the heat wave. The wedged shape allows to temporally resolve the position ofthe shock (Fig. 1(b)), and the shock velocity is determined from the slope. In the Germanversion experiments, a series of targets with different thicknesses were used instead of the4 a) (b)FIG. 1. (a) Typical experimental setup of evaluation hohlraum radiation temperature throughthe measurement of the shock wave velocity in aluminium wedge. (b) The shock velocity can besolved from the shock position, which is measured as a function of time, due to the varying samplethickness. Reproduced with permission from Phys. Rev. Lett. 73, 2320 (1994). Copyright 1994American Physical Society. . wedge, so the shock velocity can be determined with somewhat less accuracy and temporalresolution [27].Using the well-known material properties of aluminium, a scaling fit can be determinedby calibrating exact simulations, to formulate the relation between the incoming radiationtemperature and the out-going shock velocity [19, 27]. Kauffman’s scaling was fitted to theNova experiments [19, 20]: T KauffmanRad = 0 . · D . S (1)Where the shock velocity D S is measured in km/sec, and the drive radiation temperature T Rad is in heV (= 100eV). The power-law form is supported by self-similar analysis [23]. Thisscaling relation was calibrated for relatively high-temperature, 200 < T
Rad < − . t [nsec]00.250.50.751 N o r m a li ze d T R Nova 2.2nsec
FIG. 2. Different radiation temperature profiles that were used in the various experiments. Nova2.2nsec ≈ ≈ < T Rad < T EidmannRad = 0 . · D . S (2)A decade ago, there was a renewed interest in these experiments and Kauffman’s scalingrelation, due to the works by Li et al. [21, 22]. Following direct full simulations, they claimthat Kauffman’s scaling relation is not universal, but rather is only correct for Nova longlaser pulses ( ≈ ≈ T LiRad = 0 . · D . S (3)Li et al. claim that the difference between the two scaling relations is due to the differenttemperature profile’s duration, while the dependency on the temporal shape is negligible. We will examine these two claims carefully in this study .Quite recently, a new scaling-relation was offed by Mishra et al., based on new simula-tions [31] using a modified version of the widely-used MULTI code [32]. The simulationswere carried out using a constant radiation temperature boundary condition with a long ulse of 3nsec, and temperature range of 100 < T Rad < T Rad ( D S ) curve in T Rad ≈ T Rad ≈ † : T MishraRad = 0 . · D . S (4)which surprisingly is closer to Li short-pulse scaling relation Eq. 3 rather than to Kauffman’slong-pulse scaling relation Eq. 1. This fact will also be explained through our analytic model.Mishra et al. offer scaling laws fits for high- Z material as well, however, the fit is affectedagain, mostly by the high-temperature range 250 < T Rad < T R ∼ t ) after its rise-time,whereas the longer pulses (2-2.5nsec) have two (flat) steps structure [19, 20]. The SG-IIIpulses rises as T R ∼ t . − . after short rise-time, while The SG-II typical pulse rises slower, T R ∼ t . − . [21]. Some pulses, like the one that was used in the Back’s et al. supersonic † The exact power in the scaling law which appears in [31] is 0.65. However, it does not fit Ref. [31] ownsimulations data, so we assume they have used two-digits round. The exact value that fits their simulationsdata (keeping the pre-factor unchanged) is 0.653, as in Eq. 4. T R ∼ t − . (1-2.5nsec) after its first1nsec rise-time. In this study we exploit the general power-law solution of [17, 18]to study the sensitivity of the T R ( D S ) curves to the properties of the temperatureprofiles’ parameters. It should be noted that although most of the works investigating the radiation drivetemperature used aluminium, another group measured the shock-velocity in quartz [36].The empirical scaling law ( T quartzRad = 0 . D . S ) is of course different than the aluminiumscaling-laws. We shall not discuss this work here, since the focus of the present work is toexplain the differences between the different scaling-laws using aluminium. III. THE (SEMI-) ANALYTIC MODEL
In this section we present the semi-analytic model for estimating the T R ( D S ) curves. Thederivation is presented for aluminium, with a couple of delicate issues that have to be donecarefully for yielding accurate quantitative results: The detailed EOS, and the calibrationof opacity factors in order to include transport effects. The procedure is as follows: • Solving semi-analytically the ablative heat region as a function of the surface temper-ature BC ( T W ) which produces an analytic expression for the ablation pressure (seeSec. III A). This procedure involves the use of an opacity factor which is calibratedfrom an exact Monte-Carlo simulations (see Sec. III C). • Solving semi-analytically the shock region as a function of the ablation pressure whichproduces an analytic expression for the resulting shock velocity D S (see Sec. III B).This procedure involves the use of the detailed EOS of aluminium (see Sec. III D). • Determining the drive temperature T R for the given D S from the surface tempera-ture T W using the self-similar solution of the flux from the heat region solution (seeSec. III E).The final analytic expressions of the derivation are summarized in Sec. III F.8 . The ablative heat region To establish a self-similar solution of the ablative heat region, one must assume a power-law relation of the Rosseland mean opacity κ R and the internal energy e , as a function ofthe temperature T and the density ρ [9, 17]. We use Hammer & Rosen notations, where thetemperature has units of heV ≡ / cm units [5]:1 κ R = gκ ( t Pulse ) T α ρ − λ (5a) e = f T β ρ − µ (5b) κ ( t Pulse ) is a unitless factor which multiplies the nominal opacity. We use it here to calibratethe diffusion approximation solution to the exact transport (Boltzmann) solution using IMCsimulations (the calibration is found to be a temperature profile’s duration t Pulse dependent,see Sec. III C). This is due to the fact that aluminium is not an extremely-opaque material,so the diffusion solution yields a too fast heat-wave (unlike gold, in which case the diffusionapproximation yields an excellent transport solution).In addition, we assume an ideal gas-like EOS using an adiabatic factor γ , again usingHammer & Rosen notations (the index 1 denotes the heat-region): P ( ρ, T ) = r ρe ( ρ, T ) ≡ ( γ − ρe ( ρ, T ) (6)where γ ≡ ( r +1) is the ideal gas parameter in the ablation region. The different parametersfor aluminium, which is the material of the shock waves experiments, are given in Table I.The EOS analytical parameters β , µ , f and r for the heat region are taken from [37],while the opacity parameters α , λ and g are fitted to the up-to-date opacity code CRSTAtables for the range of 1 − T W ( t ) = T t τ (7)where in general T W ( t ) is the inner surface temperature of the sample (in heV) and t isthe time (in nsec). In this study T W is the inner surface of the aluminium wedge whichis attached to the hohlraum hole (see Fig. 1(a), the surface at which the x-rays from thecavity hit the wedge in Fig. 1(b)). The hohlraum temperature temporal profiles, measured9 ABLE I.
Power law fits for the opacity and EOS of aluminium in the temperaturerange of − . Physical Quantity Numerical Value f .
04 [MJ/g] β . µ g / / cm ] α . λ . r ≡ ( γ −
1) 0 . by the XRD/TGS diagnostics in the various experiments, set the limits of validity of τ forour investigations to be − . τ . x F , as well as the total energy stored in the heat region (which isalmost equal to the total energy, since the energy in the shock region is negligible): P F ( t ) = p ( τ ) κ P ω ( t Pulse ) T P ω t τ S ( τ ) ≡ P ( τ ) t τ S ( τ ) κ P ω ( t Pulse ) [Mbar] (8a) E W ( t ) = e ( τ ) κ E ω ( t Pulse ) T E ω t E ω ( τ ) (cid:20) hJmm (cid:21) (8b)The different powers are determined from dimensional analysis while the pre-factors aredetermined by solving the dimensionless ODE, as derived in details in [17]. In [17] theprocedure is derived for gold parameters, whereas here we present the equivalent results foraluminium, using Table, I. The powers for the ablative pressure and energy are: τ S ( τ ) = − µ + (4 + α + βλ ) τ − (4 + α ) µ τ λ − µ (9a) P ω = − − µ λ − µ ≈ − .
422 (9b) P ω = 4 + α + βλ − (4 + α ) µ λ − µ ≈ .
184 (9c) E ω = 2 − µ λ − µ ≈ − .
422 (9d) E ω = 8 + 2 α + 2 β + 3 λβ − α ) µ λ − µ ≈ .
784 (9e)10 a) τ p SimulationSelf-similar solution (b) τ e SimulationSelf-similar solution
FIG. 3. (Color online) (a) The pressure parameter as a function of the temperature power depen-dence τ , to be used in Eq. 8a. (b) The energy parameter as a function of the temperature powerdependence τ , to be used in Eq. 8b. E ω = 2 + 2 λ − µ + (2(4 + α + β ) + 3 βλ ) τ − α ) µ τ λ − µ (9f)The constants pre-factors p ( τ ) and e ( τ ) are determined from the solution of the dimen-sionless ODE, and are presented in the red curves in Fig. 3, as a function of the temperatureBC τ . In addition, we have performed direct simulations for validating the numerical con-stant (in black curves). The simulations were performed using a one-dimensional radiative-hydrodynamics code, which couples Lagrangian hydrodynamics with implicit LTE diffusionradiative conduction scheme, in an operator-split method. The hydrodynamics code usesexplicit hydrodynamics using Richtmyer’s artificial viscosity and Courant’s criterion for atime-step. In the diffusion conduction scheme, the time-step is defined dynamically suchthat the temperature will not change in each cell by more than 5% between time steps (formore details regarding the radiative-hydrodynamics code, see [15–18]). In both schemes,we have used a converged constant space intervals. The matching between the self-similarsolution and the simulations is very good. 11 . The shock region The database for the sock region is simply the EOS (heat conduction is negligible in thisregion). Again, we assume an ideal-gas EOS (the index 2 denotes the shock-region): P ( ρ, T ) = r ρe ( ρ, T ) ≡ ( γ − ρe ( ρ, T ) (10)where γ ≡ ( r + 1) is the ideal gas parameter in the shock region. Notice that we use abinary EOS following [18] using r = r . As opposed to r , the determination of r is morecomplex, as it is a function of the shock velocity – r ( D S ), following the detailed HugoniotEOS data of aluminium [40, 41]. The values of r ( D S ) are discussed in Sec. III D.Following [17, 18], we take the ablation pressure achieved from the heat region, Eq. 8a asa BC for the shock region: P ( t ) = P t τ S (11)where P = p ( τ ) κ P ω ( t Pulse ) T P ω and τ S is defined by Eq. 9a. Both P and τ S , are knownfunctions of τ , which determines the shape of the temperature profile (Eq. 7). The secondBCs are taken to be the strong shock limit of the Hugoniot relations. Shussman et al.present self-similar solution for the particle and shock velocities, located in the shock-frontposition x S of this form: u S ( t ) = u ( τ S ( τ ) , r ( D S )) P ( τ ) t τ S ( τ )2 [km / sec] (12a) D S ( t ) = r ( D S ( t )) + 22 u S ( t ) (12b)The powers in Eqs. 12 are determined by a dimensional analysis, and the pre-factor u ( τ S ( τ ) , r ( D S )) by the solution of the dimensionless ODE and is given in Fig. 4. Wecan see in Fig. 4(a) the dependency of u on both r and τ (via τ S ). The dependency of u on τ decreases for τ > .
2. Plotting u ( D S ) explicitly in Fig. 4(b) (through the r ( D S )functional form) discovers that except for low shock velocities of D S < u ( D S ) increases slowly. We have also performed a simple fit to the exact curves of u ( D S )in Fig. 4(b), which assumes a separation of variables, f ( D S ) and g ( τ ). Such a “universalsolution” would make an easy-to-use resource to researchers for future work. The fit hasthe form: u ( D S ) ≈ f ( D S ) · g ( τ ) = (cid:18) . − D S + 9 . . (cid:19) · . τ + 0 . . (13)12 a) r u ( τ S ( τ ) , r ) τ =-0.05 τ =0 τ =0.05 τ =0.1 τ =0.15 τ =0.2 τ =0.25 τ =0.3 (b) D S [km/sec]345678 u ( τ S ( τ ) , r ) τ =-0.05 τ =0 τ =0.05 τ =0.1 τ =0.15 τ =0.2 τ =0.25 τ =0.3 Exact ODEFit
FIG. 4. (Color online) (a) The velocity parameter as a function of the EOS parameter r ( D S ) fordifferent τ (via τ S ), to be used in Eq. 12. (b) The velocity parameter as a function of the shockvelocity D S for different τ (solid curves). A simple fit for these curves is sown in the dotted curvesthrough Eq. 13 with a maximal error of 10% (above D S = 30km/sec its maximal error is 2.5%). The fit of Eq. 13 is shown in the dotted curves in Fig. 4(b), and has an accuracy of 10%(above D S = 30km/sec its maximal error is 2.5%). Such an error represents a maximalerror of 2eV in the T Rad ( D S ) curves (which is negligible for any practical use), that areshown later in Sec. IV (Fig. 11). A fit for the u ( r ) curves that are shown in Fig. 4(a) canalso be derived by just substituting the simple analytic relation of r ( D S ) (see later Eqs. 20and 17) in the fit of Eq. 13. We note that in this work we’ve used the exact values of u ( D S )self-similar solutions and not the approximated fit.Finally, substituting Eq. 8a in Eqs. 12 yields the dependency of the out-going shockvelocity in the shock region, as a function of the surface temperature of the heat region (thefinal analytical relation between D S and the drive temperature T Rad will be discussed inSec. III E.): D S ( t Pulse ) = r ( D S ) + 22 u ( τ S ( τ ) , r ( D S )) q p ( τ ) κ P ω ( t Pulse ) T Pω t τ S ( τ )2 Pulse (14)Note that Eqs. 14 and 12 are nonlinear in D S due to the dependency of r ( D S ). This relationis calculated from the up-to-date detailed data of Hugoniot relation for Al [40, 41], and willbe discussed in Sec. III D. 13 . Calibration of the opacity factor κ ( t Pulse ) There is still one delicate issue, concerning the self-similar solution of the heat region, fordetermining the ablative pressure and the stored energy. In Eq. 8 we specify the dependencyof the physical parameters in κ , an arbitrary opacity multiplier of the nominal opacity inEq. 5a.We use this parameter as a tool to set radiative transport effects corrections, due to therelatively low-opacity of aluminium ( Z = 13). Although the heat wave in aluminium in theseexperiments in fully subsonic, and it generates the strong-shock limit in the shock region,it is still more optically-thin than the heat wave in high- Z materials, such as gold. In high- Z materials, the heat wave is well modeled using the LTE diffusion approximation. In the(nominal-opacity) aluminium case, diffusion yields a heat front which is too fast comparing tothe exact transport solution. We test this difference by a series of gray implicit-Monte-Carlo(IMC) [42] simulations that we set in our transport code with nominal opacities and EOS, asin Eq. 5 and 6. The IMC simulations have used one-dimensional radiative-hydrodynamicscode, which couples Lagrangian hydrodynamics (same code that was used in Sec. III A)with 1D IMC (Fleck & Cummings) scheme [42] in operator-split method, while the diffusioncalculations couple the hydrodynamics to implicit LTE diffusion radiative conduction instead(again, for more details regarding the radiative-hydrodynamics code, see [15–18]).In Fig. 5(a) the black curves are the temperature profiles (both material and radiation)of the exact transport solution using gray IMC code with BC of T Rad = 100eV in t = 2nsec,along with LTE diffusion approximation solution (red curve), whereas in Fig. 5(b) we presentthe pressure curves. In Fig. 6 we present the heat front position x F as a function of timeusing BC of T Rad = 100eV. We can see that in both Figures, the diffusion approximationyields a too fast heat wave, as expected, and a too high ablation pressure. A possible solutionis to use Flux-Limited diffusion [15], however, the nonlinear diffusion coefficient prevents aself-similar solution, which is detrimental for this study.Thus, we take advantage of the possibility to include an opacity factor multiplier in theframe of a self similar solution [17], to calibrate the LTE diffusion solution to the exacttransport behavior. Again, for high- Z materials, where LTE yields excellent transport so-lution, we set κ = 1. For Al ( Z = 13) we have performed a set of LTE diffusion simulationusing different opacity multipliers in the range 1 κ
2. We can see, for example in14 a) m [kg/cm ]020406080100 T [ e V ] IMC (exact) T R IMC (exact) T m κ =1 κ =1.25 κ =1.5 κ =2 t =2nsec (b) m [kg/cm ]0123456 P [ M b a r] IMC (exact) κ =1 κ =1.25 κ =1.5 κ =2 t =2nsec FIG. 5. (a) The temperature profiles (both material and radiation) using exact IMC simulation ofheat wave in aluminium using T Rad = 100eV in t = 2nsec, and with LTE diffusion approximationusing different opacity multipliers κ . (b) Same with the pressure profiles. t [nsec]00.511.52 m F [ kg / c m ] IMC (exact) κ =1 κ =1.25 κ =1.5 κ =2 FIG. 6. The heat front position x F as a function of the time using exact IMC simulation of heatwave in aluminium using T Rad = 100eV, and with LTE diffusion approximation using differentopacity multipliers κ . Fig. 5 that in t = 2 nsec , the LTE diffusion approximation using κ = 1 .
25, matches boththe correct heat front and ablation pressure (also in Fig. 6, where the green curve is closerto the black IMC curve). For t = 1 nsec , the most appropriate value for κ slightly varies to κ = 1 . κ and t Pulse : κ ( t Pulse ) ≈ . − . t Pulse . (15)15t is important to note that Eq. 15 was found by simulations to be almost universal for thedrive temperature range of 100 T Rad D. The EOS parameter r ( D S ) The detailed EOS for aluminium, and the detailed Hugoniot relations D S ( u p ) (often calledalso U s − U p ) are not the main interest of this study. However, these relations are extremelyimportant for yielding good quantitative results using the self similar solution in the shockregion (see Sec. III B). Specifically, we are interested in the relatively high shock velocityregimes - 10 D S D S ( u p ) curve is often approximated by the linearrelation: D S = c + Su p . (16)We have used the values for Al from [40] with an accuracy of ≈ −
3% (see also [41]): D S [km / sec] = .
448 + 1 . u p u p . / sec6 .
511 + 1 . u p u p > . / sec (17)The relation D S ( u p ) for aluminium, Eq. 17, is plotted in Fig. 7(a). We can notice the changeof the slope at 6 . / sec.However, The strong shock limit contradicts Eq. 16 (or Eq. 17): V S V = r r + 2 = γ − γ + 1 (18a) D S = r + 22 u p = γ + 12 u p (18b)i.e., the strong-shock relation yields S = ( γ + 1) / c = 0. Note that when u p → ∞ , c is negligible, and Eq. 16 tends to Eqs. 18. In our case, the shock is not strong enough forthe constant c to be negligible. Thus, we define a functional form of S ′ ( D S ) in the followingway: S ′ ( D S ) ≡ D S u p = c + Su p u p , (19)which sets an EOS parameter that is a function of D S : r ′ ( D S ) = γ ′ − S ′ ( D S ) − . (20)16 a) u p [km/sec]01020304050 D s [ k m / s ec ] (b)
20 40 60 80 100 D s [km/sec] r FIG. 7. (Color online) (a) The detailed Hugoniot EOS relation of aluminium, taken from [40]. (b)The effective EOS parameter that is taken in Eq. 14, assuming strong shock relation (Eq. 18a)using Eq. 20.
Eq. 20 with aluminium parameters (Eq. 17) is presented in Fig. 7(b) in logarithmic scale.We can see the decrease of r with D S which softens at D S > . r D S E. The hohlraum temperature T Rad
Eq. 14 presents the nonlinear relation between the inner surface temperature (via T W = T t τ ) and the out-going shock velocity. However, we need to associate it to the hohlraumdrive temperature, which is higher. We can recognize three different radiation tempera-tures [34, 35]: The drive temperature T Rad , which is the temperature that characterized theincident flux toward the hohlraum’s wall, the wall surface temperature T W as mentionedbefore, and the temperature of the emitted flux T obs , that an x-ray detector would measure,which is approximately the temperature 1mfp inside the sample (see also in [15, 16]). Wenote also that studies which have simultaneously measured the hohlraum temperature byboth the shock velocity (which should represent T Rad ) and the radiated flux (which shouldrepresents T obs ) methods, have yielded different temperatures (see Table I in [27]), with T Rad > T obs in most measurements.We are interested in T Rad which characterizes the incident time-dependent flux F inc (0 , t ) on17he inner surface of the sample. Since this study is restricted to LTE diffusion approximation,we follow the Marshak boundary condition (an angular-integrated approximated version ofthe Milne boundary condition), which is defined by an integral over the incident flux [7, 34,35]: F inc (0 , t ) = 12 Z I ( µ, , t ) µdµ ≡ σ sb T ( t ) (21)were I ( µ, , t ) is the specific intensity on the inner surface of the sample, µ is the cosineof the photons direction with respect to the sample’s axis and σ sb is the Stefan-Boltzmannconstant. In the diffusion approximation, the specific intensity is a sum of its first twomoments: I ( µ, , t ) ≈ cE (0 , t ) + 3 µF (0 , t ) (22)where E (0 , t ) is the energy density, and F (0 , t ) is the radiation flux, which are defined as: E (0 , t ) = 12 c Z − I ( µ, , t ) dµ ≡ a Rad T W ( t ) (23a) F (0 , t ) = 12 Z − I ( µ, , t ) µdµ ≡ ˙ E W ( t ) (23b) c is the speed of light and a Rad is the radiation constant ( a Rad = 4 σ sb /c ). We note that E W ( t ) was derived explicitly by the self-similar solution - Eq. 8b. Substituting Eq. 22 inEq. 21, using the definitions of Eqs. 23 yields: F inc (0 , t ) = c E (0 , t ) + 12 F (0 , t ) . (24)Using the definitions of E (0 , t ) and F (0 , t ), Eqs. 23, yields the relation between T Rad and T W (with the help of the self-similar Eq. 8b): σ sb T ( t ) = σ sb T W ( t ) + ˙ E W ( t ) / F. Final equations
We summarize briefly the final procedure for yielding T Rad ( D s ): • First, we use Eq. 14 to find the relation T ( D s ), using the opacity factor calibrationEq. 15 and the Hugoniot relations of Al Eq. 20. We use the coefficients p ( τ ) and u ( τ s ( τ ) , r ( D S )) directly from Figs. 3(a) and 4(a).18 a) (b) FIG. 8. (Color online) (a) Typical temporal temperature profiles in different duration that werestudied theoretically in this work, normalized to the wall temperature T W in end of the pulse t Pulse .The wall temperature are in solid curves, while we added (dotted curves) for each specified T W the matching radiation drive temperature T Rad that represents the incident flux to the wall byEq. 26c (in a typical wall temperature of 200eV). (b) Normalized temporal temperature profiles,wall temperature T W in solid curves and drive temperatures T Rad in dashed to T ( t Pulse = 1nsec). T Rad temporal behavior matches approximately to T W with τ T Rad ≈ τ − . • Second, we use Eq. 7 to find T W ( D s ). • Finally, we use Eq. 25 with Eq. 8b to find T Rad ( D s ). We use Fig. 3(b) for the constant e ( τ ).The final equations are: T ( t Pulse ) = r ( D S ) + 2 · u ( τ S ( τ ) , r ( D S )) q p ( τ ) κ P ω ( t Pulse ) D S Pω t − τ S ( τ ) Pω Pulse (26a) T W ( t Pulse ) = T ( t Pulse ) t τ Pulse (26b) T Rad ( t Pulse ) = (cid:20)(cid:18) σ sb T W ( t Pulse ) + e ( τ ) E ω ( τ )2 κ E ω ( t Pulse ) T E ω t E ω ( τ ) − (cid:19) (cid:30) σ sb (cid:21) / (26c)In Fig. 8(a) we plot different typical temperature profiles that we have studied theoret-ically in this work. The pulses have different time duration (long and short pulses) anddifferent temporal shapes, via τ , and normalized to the wall temperature T W at the end of19he pulse t Pulse . We can see that as τ decreases, the profiles rises more rapidly. The solidcurves represent the profiles of the wall temperatures T W , while we add the matching drivetemperature T Rad (the temperature of the incident flux to the wall) for each pulse in dottedcurves. The relation between T W and T Rad in Fig. 8(a) is via Eq. 26c (and 8b), using the wallenergy in a representative wall temperature of 200eV (thus, the drive temperature is higherat the end of the pulse). We note that the temporal profile of T Rad slightly differs fromthe temporal profile of T W . In Fig. 8(b), we plot normalized temperature profiles of both T W and T Rad (each one is normalized to its own value at the end of the pulse) in intervalsof 0.05. We can see that the temporal behavior of T Rad nicely matches approximately to τ T Rad ≈ τ − .
05 for all τ . IV. RESULTS
First, we plot (Fig. 9) the hydrodynamic profiles for a typical example case, using asurface temperature of T = 200eV at 1nsec, using different temporal profiles, τ = 0, 0 . κ = 1 . t = 1nsec), andfor the shock region we take r = 2 / D S = 50km/sec (see Fig. 7(b)).The different regions can be seen clearly, as the heat wave region creates sharp tem-perature profile, and the material ablates rapidly. This creates strong ablation pressures( ≈ − τ yields different shock velocities, with the same maximal T W . Lower τ yields a faster heat wave, a stronger pressure profile, and faster particle (and shock) veloci-ties, due to more net energy stored in the sample. Our model therefore, predicts a sensitivityto the temporal profile. We will examine this later versus the experimental results.Using Eqs. 26, we demonstrate in Fig. 10 the dependency of both the temperature pulse’sduration t pulse and the temporal profile, via τ , for a given typical out-going shock velocityof 50km/sec. The important result of Fig. 10 is that there is clearly a gap of ≈ t pulse (exactly the 10eV that Li et al. claim tobe the difference between Kauffman’s and their results), and ≈ for a given shock velocity of 50km/sec . This deviation is of course20 m [ gr/cm ] T [ e V ] m [ gr/cm ] ρ [ g r / cc ] m [ gr/cm ] P [ M b a r ] m [ gr/cm ] u [ k m / s e c ] τ = 0 τ = 0 . τ = 0 . FIG. 9. (Color online) The hydrodynamic profiles of the temperature, the density, the pressureand the velocity as a function of the Lagrangian coordinate, for different boundary conditions - τ ,using the binary EOS model (The velocity profile is zoomed on the shock region). We used r = 0 . κ = 1 . t Pulse = 1nsec) for the heat wave region, and r = 0 .
67 forthe shock region (typical value of Eq. 20).(a) t pulse [nsec]170180190200210220230 T R a d [ e V ] τ =-0.05 τ =0.05 τ =0 τ =0.1 τ =0.2 τ =0.3 τ =0.15 τ =0.25 (b) -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 τ T R a d [ e V ] t pulse =1sec t pulse =1.5sec t pulse =2sec t pulse =2.5sec t pulse =3sec FIG. 10. (Color online) (a) The drive temperature as a function of temperature profile’s duration t pulse for various temporal shapes ( τ ). (b) The drive temperature as a function of the temporalshape ( τ ) for various values of temperature profile’s duration t pulse . In both figures we can see thatthe curves saturates for τ > .
15 and t pulse > t pulse or τ . This demonstrates the strength of our model, which leads to a non universal T Rad ( D S )relation, supporting the different fits to the experiments reviewed in Sec. II. Nevertheless,we can see that the T Rad deviation saturates for τ > .
15 and t pulse > T Rad ( D S ) for different values of − . τ .
3, for short pulse (Fig. 11(a))and long pulse (Fig. 11(b)) experiments. We have added Kauffman’s scaling relation (inred), as well as Li’s (green), Eidmann’s (orange) and Mishra’s (magenta). First, we can seethat for a given D S there is a range of possible T Rad with a spread of few dozens of eVs, asa function of τ , similarly to Fig. 10. In addition, longer pulses yield higher T Rad than shortpulses, for a given D S and τ . Moreover, the difference between short pulses (Fig. 11(a))and long pulses (Fig. 11(b)) is ≈ τ > .
1, thespread decreases dramatically, and T Rad ( D S ) becomes “universal”, and saturates with τ .Second, Fig. 11(a) (short pulse) shows that Kauffman’s scaling law is at the upper limitof the possible blue curves fan, whereas in Fig. 11(b) (long pulse) it is right in the middleof the blue curves, matching τ ≈ . − .
1. This is due to the fact that Kauffman’s scalinghave used long pulses for calibrating the scaling law [19, 20], as Li et al. pointed out [21].Also, vise versa, in Fig. 11(b) (long pulse) Li’s scaling law lies at the lower part of thepossible blue curves fan, whereas in Fig. 11(a) (short pulse) it is right in the middle of theblue curves, matching τ ≈ . − .
15. This is due to the fact that Li’s scaling have used short pulses for calibrating the scaling law [21]. As mentioned in Sec. III F, τ T Rad ≈ τ − . τ T Rad ≈ − .
05, and Li’s for τ T Rad ≈ . − .
1, This matches the temporal profiles in the experiments quite well (seeFig. 2). The simple semi-analytic model reproduces and explains the difference betweenKauffman’s and Li’s scaling laws that was pointed out in [21, 22] and previously explainedonly by full simulations.Eidmann’s scaling law (orange curves) that was developed for the lower temperature rangeof 100 < T
Rad < a)
20 40 60 80 100 D s [km/sec]100200300 T [ e V ] Semi-Analytic Model - τ =-0.05...0.3Li Scaling + Sim.Kauffman Scaling + Sim. Eidmann Scaling + Exp.Mishra Scaling + Sim t pulse =1nsec τ =0 τ =-0.05 τ =0.3 τ =0.05 (b)
20 40 60 80 100 D s [km/sec]100200300 T [ e V ] Semi-Analytic Model - τ =-0.05...0.3Li Scaling + Sim.Kauffman Scaling + Sim. Eidmann Scaling + Exp.Mishra Scaling + Sim t pulse =2nsec τ =0 τ =-0.05 τ =0.3 τ =0.05 FIG. 11. (Color online) (a) The radiation temperature T Rad as a function of the shock velocity D S for short pulse t pulse = 1nsec using the simple semi-analytic model for different values of − . τ . t pulse = 2nsec. τ = 0, and this is the reason that their scaling is lowerthan Kauffman’s. Not surprisingly, Mishra’s results (magenta) matches almost perfectly ourmodel using τ = 0, in Fig. 11(b) (long pulses).We also note that Nova’s “two-steps” temperature profile (see black/red curves in Fig. 2),creates a ‘kink’ in the distance traversed by the shock trajectory curve, yielding two differentshock-velocities [21, 28, 30, 33]. The simple analytic model reproduces this fact completely,separating the temperature profile to two pulses with different temperatures. Ghosh et al.presents some quantitative data [30], showing that the shock velocity changes from 35.4to 54.6km/sec, and the first step is characterized by an incident radiation temperature of T Rad ≈ τ ≈ − .
05 (this experimental result is less sensitive to the temperature profile’sduration).The simple model predicts that the drive temperature T Rad ( D S ) curve is a function ofboth the duration and the temporal profile of the drive temperature pulse. However, Li et al.claim that the T Rad ( D S ) curve is duration-dependent, with a difference of ≈ temporal profiles onthe scaling relation is negligible, for x-ray sources driven by 1nsec laser. This apparentcontradiction may be solved easily. The temporal profile of the SG-III experiment may beapproximated by τ ≈ . − .
15 and that of SG-II by τ ≈ .
35 (see Fig. 2 and Sec. II). Inthis regime of τ > . T Rad ( D S ) is indeed almost universal, and saturates, as can be seenin both Figs. 10 and 11. Moreover, in Fig. 12 which is taken from [21], the difference of ≈ ≈ ≈ τ > . τ s or different temporal shapes, would lead to amajor discrepancy due to the different temporal shapes. V. CONCLUSION
Ablative subsonic radiative heat waves, or the subsonic Marshak waves have been studiedfor three decades. The coupling between the hydrodynamic equations and the radiative heat24
IG. 12. (Color online) The drive temperature T Rad as a function of shock velocity in Li et al.simulations using SG-II and SG-III pulses. Kauffman’s (dashed) and Li’s (solid) scaling relationsare presented as well. Reproduced from [21], with the permission of AIP Publishing. transfer is crucial for modeling hohlraums, in the quest for indirect drive ICF [1, 2]. Thisfeature enables us to evaluate the radiation drive temperature T Rad inside the hohlraum,through the measurement of the velocity of the emitted shock wave D S that is developed ina well-characterized material, such as aluminium [19–21, 23, 24, 27].Recently, we have derived a basic theoretical study, yielding a full self-similar solutionfor the subsonic problem, patching two self-similar solutions, one for the heat region andone for the shock region [17]. We have expanded this basic model to include more complexmaterial behavior, i.e., EOS, since the EOS properties of the shock and the heat regions,are very different [18].This study takes the basic theoretical study of ablative subsonic radiative waves, and con-front it versus the results of the various experiments in which the actual relation betweenthe heat region (via T Rad ) and the shock region (via D S ) was measured. The different stud-ies usually modeled this problem with direct radiative-hydrodynamics simulations, yieldingdifferent scaling relations between T Rad and D S [19, 21, 27, 31] (see Sec. II). The currentstudy enables us to test this issue semi-analytically (via the self-similar solutions), and isaimed to understand the differences between the different scaling laws.The model is derived in details for aluminium in Sec. III, taking into account the trans-port effects via calibration of an opacity factor by IMC simulations in the heat region, and25he detailed Hugoniot relations [40], in the shock region. We have found that the T Rad ( D s )relation is not universal, and depends on the different features of the temporal behaviorof the temperature pulse, such as the duration t pulse , and its temporal profile (through τ ),with up to ≈ T Rad . This diversity is due to the different total energy thatis stored inside the aluminium sample for the different conditions of the profile’s parame-ters. This explains the difference between the different scaling laws found in the literature.Moreover, the simple model recovers the different experimental and simulation data, sep-arating short (Fig. 11(a)) and long (Fig. 11(b)) pulses, each for different temporal profiles( τ ). Specifically the model explains the difference between Kauffman’s and Li’s scaling laws,by simple analytic expressions.The simple model enables an estimate of T Rad ( D s ) for any future experiments using apower-law fit of the drive temperature temporal profile and the temperature pulse’s du-ration, using the expressions of Eqs. 26, for aluminium, with which the vast majority ofexperiments were performed. We plan to expand the research in the future for differentmaterials, depending on the advancement of the research and experimental program in thisfield. [1] J.D. Lindl, P. Amendt, R.L. Berger, S.G. Glendinning and S.H. Glenzer, Phys. Plas. , , 339(2004).[2] M.D. Rosen, Phys. Plas. , , 1803 (1996).[3] J.I. Castor, Radiation Hydrodynamics , Cambridge University Press (2004).[4] Zel’dovich, B.Ya., Raizer, P.Yu.,
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