Numerical simulations for the design of absolute equation-of-state
measurements by laser-driven shock waves
M. TEMPORAL(1)(*), S. ATZENI(2), D. BATANI(3), M. KOENIG(4) A. BENUZZI(4) and B. FARAL(4)
(1) INFN, Laboratori Nazionali di Legnaro - 35020 Legnaro (Padova), Italy (2) Associazione EURATOM-ENEA sulla Fusione, Centro Ricerche Frascati
C.P. 65, 00044 Frascati (Roma), Italy
(3) Dipartimento di Fisica, Università di Milano - Via Celoria 16, 20133, Milano
INFM, Sezione di Milano - Milano, Italy
(4) Laboratoire pour l’Utilisation des Laser Intenses, CNRS, Ecole Polytechnique
91128 Palaiseau, France
(ricevuto il 29 Maggio 1997; approvato il 14 Luglio 1997)
Summary. — A recently proposed experiment for the absolute measurement of the Equation of State (EOS) of solid materials in the 10–50 Mbar pressure range is analyzed by means of numerical simulations. In the experiment, an intense laser pulse drives a shock wave in a solid target. The shock velocity and the fluid velocity are measured simultaneously by rear side time-resolved imaging and by transverse X-radiography, respectively. An EOS point is then computed by using the Hugoniot equations. The target evolution is simulated by a two-dimensional radiation-hydrodynamics code; ad hoc developed post-processors then generate simulated diagnostic images. The simulations evidence important two-dimensional effects, related to the finite size of the laser spot and to lateral plasma expansion. The first one may hinder detection of the fluid motion, the second results in a decrease of the shock velocity with time (for constant intensity laser pulses). A target design is proposed which allows for the accurate measurement of the fluid velocity; the variation of the shock velocity can be limited by the choice of a suitably time-shaped laser pulse.
PACS 52.35 – Waves, oscillations, and instabilities in plasma. PACS 52.35.Tc – Shock waves.
PACS 62.50 – High-pressure and shock-wave effects in solids and liquids. PACS 52.50.Jm – Plasma production and heating by laser beams.
(*) SIF-ENEA fellow.
1. – Introduction
Intense laser pulses can generate shock waves with pressures in the 10–100 Mbar range in solid materials, and allow for equation-of-state (EOS) measurements of relevance to inertial confinement fusion and to astrophysics (see, e.g., the reviews by Anisimov et al. [1] and by More [2], and many reports of recent experiments [3-10]). Two approaches can be followed: the target material can be irradiated directly by the laser light (direct-drive) or by the thermal radiation (X-rays) produced by laser-heated matter (indirect-drive). Criteria for shock quality (planarity, stationarity, etc.) and for laser beam quality have been discussed in the literature [1, 2, 11].
The measurement of the EOS of the shocked materials exploits the well-known Hugoniot relations [12], expressing the conservation of mass, momentum and energy, respectively, at a shock front:
r 4r0 D D 2u , (1) P 4P01 r0Du , (2) E CE01 u2 2 , (3)
where r is the mass density, P is the pressure, E is the specific internal energy, D is the shock velocity, and u is the fluid velocity behind the shock; the subscripts 0 refer to the values before shock transit. According to eqs. (1)-(3), the state of the shocked matter (that is, its EOS) is completely determined by the knowledge of the pre-shock quantities and of two shock parameters, e.g. the velocities D and u, which are the most amenable to detection.
Actually, due to the difficulty of a simultaneous measurement of D and u in the same shocked material, most of the EOS measurements by laser-driven shock waves have so far been of relative nature, i.e. measurements of the EOS of a material A, using a material B (whose EOS is known) as a reference. They have been performed by the so-called impedance-matching technique [1, 12], which requires measuring the shock velocities simultaneously in the two materials. This has been done by launching a shock in a target containing appropriately structured layers of both materials. Both the direct [4, 6, 13] and the indirect [6] approach have been demonstrated. While indirect drive seems to allow for higher accuracy, direct drive exploits the available energy more efficiently (which has allowed to achieve pressures up to 40 Mbars with the use of lasers pulses of 100 J only [4, 6]).
Absolute EOS measurements, instead, require the measurement of D and u in the same material. An important result in this direction was achieved a few years ago by Hammel et al. [7], who showed that both the shock propagation and the fluid motion can be detected by transverse X-radiography. However, tracking the shock front requires the sample material to be transparent to the probe X-rays.
Recently, a direct-drive experiment has been proposed aiming at absolute EOS measurements in opaque materials [14]. The envisaged set-up is shown in fig. 1. The sample material is in the form of a stepped layer, supported by a layer of a material transparent to the X-rays, which is irradiated by the laser beam. The velocity D of the shock is measured by time-resolved imaging of the visible light emitted by the rear face
Fig. 1. – Sketch of the experimental set-up.
of the target; the fluid velocity is obtained by following (by transverse radiography) the motion of the interface between the two materials. Since, in fact, the two quantities u and D are measured at different places in the target, the feasibility of the proposed method requires uniform pre-shock target conditions as well as planar shocks of constant velocity.
In this paper we present a study based on 2D numerical simulations, aiming at the evaluation of some of the effects which could hinder the proposed measurement and/or limit their accuracy, and at contributing to the design of a suitable experimental set-up. Target evolution is computed by the Lagrangian radiation-hydrocode DUED, while the responses of the rear imaging and of the transverse radiography are simulated by appropriate post-processors. Details on models and numerical set-up are given in sect. 2. The simulations of a few configurations of interest are discussed in sect. 3. Two main obstacles to accurate measurements are evidenced; first, the non-planarity of the shocked surface due to unavoidable edge effects hinders the accurate measurement of the flow velocity; second, transverse plasma expansion results in worsening laser-to-shock coupling with time. We show that the first effect can be completely overcome by a suitable target design, while the second one can be at least partially contrasted by appropriately time shaping the laser pulse. Conclusions are drawn in sect. 4.
2. – Models and set-up
2.1. Fluid code. – The behaviour of the laser-irradiated targets is simulated by the two-dimensional (2D) Lagrangian fluid code DUED [15]. Cylindrical symmetry is assumed around the laser beam propagation axis. The present version of the code assumes a two-temperature model (distinct ion and electron temperatures), with classical transport coefficients and electron flux limitation. Matter properties are
described by an EOS [16] which is in good agreement with the well-known SESAME library [17]. Radiative transfer is treated by multigroup-flux limited diffusion [18]; usually about 20 appropriately spaced groups are used. Opacities are provided by the non-local-thermodynamic-equilibrium (non-LTE) version of the SNOP code [19]. Laser-matter interaction is described by a geometric optics approximation (ray tracing), taking into account plasma refractivity and inverse-bremsstrahlung absorption.
2.2. Laser and target. – The laser beam parameters have been chosen to approximate those of the pulses of a single beam of the frequency doubled Phebus Nd:glass laser (located at the CEA-Limeil Laboratory) [20], which will be used in the planned experiment (wavelength l 40.53 mm; pulse energy EL4 2.5 kJ; pulse duration Dt B2.5 ns). Kinoform Phase Plates [21] will be used to produce the required flat-intensity focal spot. In the following simulations the laser beam is assumed to have a time-independent, axially symmetric spatial profile (see below) with a flat-top with radius rF4 180 mm, and a Gaussian tail with full width at half maximum (FWHM) DrFWHM4 100 mm (see fig. 2). The laser power has a 2.5 ns flat-top, preceded and followed by Gaussian tails with FWHM of 300 ps; the peak intensity is I04 6 3 1014W/cm2.
As far as the target is concerned, since we are interested in a demonstration of the method, we consider aluminum (with r04 2.7 g/cm3) as the (optically thick) EOS sample material; this is motivated by the good knowledge of its thermodynamics and radiative properties. Of course, the method can be applied as well to other materials for which experimental EOS data are not available. The supporting (optically thin) layer is made of plastics (with CH composition, and r04 1.05 g/cm3). The plastic layer has thickness DZCH4 40 mm; the aluminum layer has thickness DZAl4 80 mm; the aluminum step has thickness DZs4 20 mm and radius Rs4 60 mm. The choice of such parameters follows motivations analogous to those discussed in the other recent experiments quoted above. Scaling of the expected pressure vs. laser and target parameters is also discussed elsewhere [6].
2.3. Diagnostics simulation. – The space- and time-dependent status of the target, as produced by the DUED code, is used as an input to two post-processors, simulating, respectively, the response of the visible-light streak-camera placed on the target rear and of transverse X-radiography. In both cases the analyses refer to an axial slice of the target (see fig. 2).
2.3.1. V i s i b l e a x i a l s t r e a k f o r s h o c k v e l o c i t y m e a s u r e m e n t . The visible streak simulator considers an axial section of the target and, at a set of selected times, solves the radiative transfer equation for a given photon energy, in the visible or near-uv spectrum (in the present simulations we have used hn 41.85 eV), along a set of lines parallel to the symmetry axis. The result is a 2D (r-t) matrix of the intensity exiting the target, approximating the signal recorded by the streak camera. The sampling intervals in time (10 ps) and space (5 mm) are of the same order of the foreseen instrumental resolution. From this diagnostics we evaluate the shock transit time through the Al step, as a difference between the breakthrough times from the step itself, t2, and from the Al basis, t1. We can then determine the average shock velocity in the step, D 4DZs/(t22 t1). Previous experiments have shown that errors as low as 5 % can be achieved [4]. They resulted from errors in the measurements of the
Fig. 2. – Target geometry, laser pulse spatial profile, and principle of the diagnostics for the measurement of the shock and fluid velocities. The image generated by the visible-light streak-camera provides the times of shock breakthrough from the aluminum basis and from the 20 mm aluminum step. The X-radiography detects the position of the Al-CH interface.
thickness of the target step, in the planarity of the shock, and in the calibration of the visible streak-camera, and from the finite resolution of the streak camera and from the width of the used slit. In the present case such errors can be further reduced to about 2 %, since the thicker step which will be used in the experiment reduces the relative errors in the measurements of both DZsand Dt.
2.3.2. T r a n s v e r s e X - r a d i o g r a p h y f o r f l u i d v e l o c i t y m e a s u r e m e n t . Transverse X-radiography employs an external, extended X-ray source (typically generated by laser-heated matter) with intense line emission in the energy range hn 4 5–7 keV. The post processor computes X-ray attenuation at selected times along a set of parallel lines, orthogonal to the symmetry axis (see fig. 2), thus generating a simulated 2D (z-t) radiographic image. Given the different opacities of the two materials, the opaque-bright interface will follow the motion of the material interface, and allow for the evaluation of the fluid velocity u. An essential requirement for a good measurement is, of course, the planarity of the shock (and hence of the interface). In case of a perfectly flat shock an error of about 8 % can be expected (mainly due to the spatial resolution of the diagnostic, i.e. to the pin-hole size).
3. – Numerical simulations
In order to get physical insight into the relevant phenomena, we first consider a rather idealized case, neglecting lateral expansion and edge effects (subsect. 3.1). We then study in a realistic geometry the target shown in the introduction, which turns out
100 50 0 50 100 -Z ( m)m R( m ) m
time=0 ns time=1 ns time=2 ns time=3 ns
0 100 0 100 0 100 0 100
Fig. 3. – Target evolution and shock wave propagation for the ideal case (no edge effects and no lateral expansion), illustrated by a sequence of 2D grey-scale mass density maps. Densities are represented on a linear gray-scale ranging from 0 (white) to 10 g/cm3 (black). The target is irradiated from the right-hand side, and the propagation of a nearly planar shock moving leftward is clearly observed. The ablated, low-density plasma is not shown.
to be inadequate for the detection of the interface motion and for the generation of a shock propagating at constant velocity (subsect. 3.2). A simple modification to the target, however, seems to allow for the desired measurement of u (paragraph 3.3.1); furthermore, by time-tailoring the laser power the temporal variation of the shock velocity D can be limited to acceptable values (paragraph 3.3.2). A further possible target configuration is also analyzed (subsect. 3.4).
3 2 1 0 0 20 40 60 80 100 120 140 Z ( m)m time (ns)
Fig. 4. – Simulated radiographic image for the same ideal case as in fig. 3. The figure also shows the on-axis positions of the shock front (dashed curve) and of the material interface (continuous curve) as computed from the fluid code DUED. It appears that the radiografic signal follows accurately the interface position.
200 100 0 100 200 -0 100 0 100 0 100 0 100 Z ( m)m R( m ) m
time=0 ns time=1 ns time=2 ns time=3 ns
Fig. 5. – Same as in fig. 3, but for the preliminary design, and with edge effects included. Notice the (X-ray opaque) curved shock zone generated by the edges of the laser (at r F180 mm).
3.1. Ideal case, no edge effects. – We have first simulated only a central portion of the target, r G120 mm, where the laser intensity is uniform. In addition, sliding-boundary conditions are imposed at r 4120 mm, so that no lateral expansion of the target occurs and edge effects are removed (see the target evolution shown in fig. 3). The X-radiography simulated image of this case is shown in fig. 4. (Here and in the following the transmitted intensity is normalised: black indicates maximum transparency, and white total absorption of the X-rays.) The shock reaches the Al-CH interface at tiC 1 ns. For t E ti the signal (grey-black interface) represents the shock motion, because the shocked plastics is more transparent than the unshocked material. For t Dti the sample can be divided into three zones: unshocked Al, shocked Al (between the shock front and the Al-CH interface) and shocked plastics. Since both the shocked Al and the unshocked Al are opaque with respect to the plastics, then for t Dti
4 3 2 1 0 0 20 40 60 80 100 120 140 time (ns) Z ( m)m
Fig. 6. – Same as fig. 4, but for the preliminary design. It is seen that the radiography follows the motion of the shock generated by the edges of the pulse, thus not allowing detection of the interface motion.
200 100 0 100 200 -Z ( m)m R( m ) m
time=0 ns time=1 ns time=2 ns time=3 ns
0 100 0 100 0 100 0 100
Fig. 7. – Same as fig. 3, but for the improved target design and the standard laser pulse. The radial size of the whole aluminum sample has been reduced to 120 mm, which is smaller than the radius of the laser plateau. The aluminum sample is immersed in a plastics cylinder so that the shock at the edges runs faster than the shock in the aluminum.
the radiographic signal detects and follows the interface motion. Figure 4 shows that interface motion is well identified; the simulated signal follows the interface motion accurately as computed by DUED (continuous line). The dashed line in the figure represents the position of the shock front.
It is seen that both the velocity D of the shock in the aluminum and the velocity u of the Al-CH interface are nearly constanty during the interval 1 GtG3 ns, which allows using eqs. (1) and (2) for evaluating the (constant) shock pressure. We have D C 47 mm/ns and u C30 mm/ns, from which we get PC38 Mbar and rC7.5 g/cm3 (in agreement with the values obtained in the fluid simulation).
4 3 2 1 0 0 25 50 75 100 125 time (ns) Z ( m)m
Fig. 8. – Same as fig. 4, but for the improved target design and the standard laser pulse. It is seen that the radiography now detects the interface motion accurately.
0 1 2 3 4 150 100 50 0 Z ( m), Velocity ( m/ns) mm time (ns)
Fig. 9. – For the improved target design and the standard laser pulse: points with error bar: position of the opaque-bright interface in the radiographic image of fig. 8; upper solid curve: fit to the radiographic data; dash-dotted curve: interface velocity u obtained by differentiating in time the fit of the radiographic data; solid curve: interface velocity on axis as computed by the fluid code DUED; dashed curve: shock velocity on axis as computed by DUED.
3.2. Preliminary design. – We now analyze a case in which the full target and laser beam are considered, with allowance for the lateral expansion of the target.
Figure 5 shows the initial target configuration and the density evolution at a few selected times. It appears that the interface is no more planar.
3.5 3 2.5 time (ns) -200 -100 0 100 200 R ( m)m
Fig. 10. – Simulated image generated by the rear visible streak-camera for the improved target
design and the standard laser pulse. From the shock breakthrough times, the shock transit time
through the Al step is evaluated (Dt 40.64 ns). The figure also shows the target cross-section (not to scale) and the radial shape of the laser pulse.
4 3 2 1 0 0 25 50 75 100 125 time (ns) Z ( m)m
Fig. 11. – Same as fig. 8, but for the target driven by a shaped pulse (see main text).
From the corresponding radiographic image, shown in fig. 6, it is seen that in this case one cannot detect the interface motion and hence the velocity u (see the large separation between the interface position on the z-axis, as evaluated by the code DUED, and the black-white interface in the radiography). This is due to the curvature of the material interface; the region corresponding to the laser edges moves at slower velocity than the central region and shadows the interface in the central, flat zone.
3.3. Improved design.
3.3.1. I m p r o v e d t a r g e t , s t a n d a r d p u l s e s h a p e . To overcome such a problem we make the radial size of the whole Al layer smaller than the radial laser plateau, in such a way that the shocked Al-CH interface keeps flat while accelerating axially. Moreover, we place same plastics around the Al sample, in correspondence to the laser beam edges (see fig. 7), so that the shock generated by the edges of the laser pulse run faster (propagating in CH) than the shock in the Al sample. In this manner no opaque material is present between the X-ray source and the shocked interface we want to follow. This is confirmed by the simulated radiographic image (fig. 8), which now allows for detection of the trajectory of the Al-CH interface. In fig. 9 we show such a trajectory, as determined by the radiography, and its time derivative, giving for t EtiC 1 ns the shock velocity in the plastics, and for t Dtithe interface velocity u. To obtain such velocities we have first fitted a discrete set of the radiographic data with a polynomial and then computed the derivative analytically. The figure shows good agreement between the radiographic data and the interface velocity on axis computed by DUED; it is clear, however, that u changes non-negligibly during the relevant time interval.
Figure 9 also shows the time evolution of the shock velocity D, making apparent a limitation of the set-up under consideration. Indeed the velocity D changes substantially in the time interval 1 GtG3 ns, decreasing from about 50 mm/ns to about 30 mm/ns. From the simulated visible-light streak image, shown in fig. 10, one deduces
0 1 2 3 4 150 100 50 0 Z ( m), Velocity ( m/ns) mm time (ns)
Fig. 12. – Same as fig. 9, but for the target driven by a shaped pulse (see main text).
an average shock velocity D C31 mm/ns, in the interval 2.7 GtG3.4 ns, corresponding to the shock transit through the Al step. It is obvious that under the present circumstances application of the Hugoniot equations to compute an EOS point can lead to rather large errors.
It is worth observing that the decelation of the shock illustrated by fig. 9 was not observed in the quasi-1D case studied in subsect. 3, and is probably to be attributed to 2D geometrical effects. Indeed, particularly towards the end of the pulse, the separation between the critical density and the ablation front (and hence between the critical density and the shock front) becomes comparable to the flat intensity portion of the spot size, so that the lateral expansion of the plasma becomes important.
The evolution could probably be made closer to 1D, and the above difficulties could be mitigated by the adoption of a few design changes. In the next paragraph we discuss the use of a time-tailored laser pulse, but other solutions could also be adopted, such as the inclusion of the target in a cylinder of (transparent) material to limit lateral expansion, the use of shorter laser wavelength (e.g., l 40.35 mm or l40.26 mm) or of indirect-drive, in order to reduce the distance between the power deposition zone and the shock front, and the use a target with a thinner Al layer.
3.3.2. I m p r o v e d t a r g e t , s h a p e d p u l s e . The decrease of the shock velocity (determined by a decrease of the driving pressure) can be contrasted by the use of a laser pulse with power increasing in time. For instance, we have considered a case with intensity rising linearly from 3 31014W/cm2
to 9 31014W/cm2
for 0.5 GtG3.5 ns. The simulation shows that the shock velocity is no more monotonically decreasing (see figs. 11 and 12); In particular, in the time interval during which the shock crosses the step the fluid simulations show that D 4 (4863) mm/ns, and the rear streak data give D C50 mm/ns. In the same time interval, uC31 mm/ns. These values correspond to pressure P C42 Mbar and density rC7 g/cm3. Although further optimization is
3 2.5 2 time (ns) -200 -100 0 100 200 R ( m)m
Fig. 13. – Simulated image generated by the rear streak camera for the wedged target showed in the lower portion of the figure. The shock motion can be followed in the interval between t 41.9 ns and t 43.2 ns.
required to increase the accuracy of the EOS data determination, this simulation confirms that means exist to overcome the limitation enlightened in the previous paragraph.
3.4. Wedged sample. – While the use of a step allows to determine the average shock velocity, by using a wedged target (fig. 13), it is possible to observe the shock motion continuously during the whole transit time. Such wedged targets might be used in the process of optimizing the laser pulse shape, prior to conducting EOS experiments on configurations such as that shown in the previous subsections. Just as an example, we show in fig. 13 the simulated rear visible streak image obtained for a target in which the rear of the Al layer is inclined by an angle a 418.43 deg with respect to the normal to the symmetry axis. Such an image provides us information on the shock motion during the last ns before breakthrough.
It is however to be observed that the accuracy of the information that can be obtained from these esperiments depends crucially on the planarity of the shock.
4. – Conclusions
The proposed experiment aims at measuring both an average shock velocity D and the instantaneous fluid velocity u in a laser-shocked sample, for the purpose of determining an EOS point.
The simulations evidence that essential requirements for such an experiment are shock planarity and a target design which allows radiography to discriminate the Al-CH interface. In addition, our study shows that a careful design is needed in order to assure that the shock propagates at nearly constant velocity through the target step. This point, in particular, deserves further analysis and optimization. Finally, we
observe that, although the code we have used includes multi-group radiative diffusion, we have not analyzed the so-called preheating. More work is then required to evaluate such an effect, and estimate the errors it might introduce in EOS measurements.
* * *
One of us (MT) wishes to thank Prof. R. A. RICCI for his continuous interest and support.
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