Investigations of equation of state of matter at extremely high pressures

Equation of state measurement using two frames shadowgraphy in mixed (Gold/Copper) step target

The mixture of two or more high-Z elements is generally utilized in high energy-density physics (HEDP) and inertial confinement fusion (ICF) experiments for enhancing the effective opacity. Apart from opacity, the mixing of materials also affects conductivity of materials, radiative transport, and multi-fluid flows. Therefore, the equations of state (EOS) studies of mixed materials under extreme conditions (pressures ~ tens of Mbars to Gbar and temperatures ~millions of degrees) is a subject of great interest in different fields of sciences including studies in ICF, astrophysics and other HEDP experiments. Such extreme pressure conditions can be simulated in laboratory using the high energy, high power (HEHP) lasers. An intense shock wave is generated in the thin target, when a material is irradiated by an HEHP laser pulse. The shock pressures in the foil target can only be indirectly estimated by measurements of shock velocities and particle velocities outside the thin targets and using the Hugoniot relations. The EOS of the materials can be determined by different techniques. One of them is, by simultaneously measuring the shock and particle velocities of foil target to determine the EOS. And other technique is based on the impedance match method and involves measuring only two shock velocities using a streak camera. We had proposed a technique for the EOS measurements using measurements of two particle velocities on a step target.

The EOS of a 0.5 μm step of mixed-z alloy of Au/Cu coated on 6.5 μm aluminum (Al) reference-foil was measured. The composition of the thin film is: Au, Cu mixed with 87.8% and 12.2% by weight respectively. The pump Nd:Glass high power laser beam is focused through a Fresnel bi-prism on a step target with the thin step (mixed Au/Cu) at one of the and the two shock waves are generated in the two regions of it; namely; a) Al foil region and b) region of the foil with the step of material whose EOS is to be determined. The experiments were performed on Al-Au/Cu step targets at different laser intensities. The EOS of the mixed material (Au/Cu) in step geometry is measured using the two measurements of the particle velocities along with the assumption that EOS of the material of the thin (reference) foil is known, and then EOS of the coated material on the foil can then be determined as is mentioned in previous study.

Figure 1: Schematic for step target

Experimental set-up
The experimental set-up is shown in Fig. 2. In this experiment, the two focal spots with ~1 mm separation were generated by splitting the heating beam using a Fressnel bi-prism. The laser energy in these two beams was made equal by appropriate adjustment of the bi-prism, using a CCD camera placed near the focal region behind filters with strong attenuation factor. The pump laser beams were incident on the Al-surface of step-target; such that the step regions on the rear side and without step region both are covered. The particle velocities in the two regions of the step target were measured by two frame shadowgraphy by generating two orthogonally polarized second harmonic probe beams by using a type-II KDP crystal from the tapped 1054 nm pump laser beam after 50 mm dia. Amplifier. A plano-convex lens (focal length, f = 5 cm) was used to achieve a magnification factor of ~19x for imaging the plasma plane. The orthogonally polarized probe components were separated by a polarizing beam splitter and reach CANON 1200D cameras as shown in the Fig.2 placed at the image plane.

Figure 2: A schematic of the experimental setup

When the high power laser (wavelength 1054nm and the pulse duration of 500ps) is incident on the foil target, it generates a strong compression wave in both the regions of the step target leading to production of strong- shock waves. The focal-spot of the heating beam was measured to be ~ 130 m. Using two beams shadowgraphy, we measured the particle velocities in both the regions. It is shown in the discussion below that the EOS of step can be determined from the particle velocity information in the two regions of the step target.

Results and discussion
The experiments were performed on Al-Au/Cu step targets at different laser intensities (~2.5 - 5 X 10¹³ W/cm²). Typical shadowgraphs of the two probe beams for aluminum foil and mixed coated region of the foil are shown in the Fig. (3).

Figure 3: (a) The shadowgraphs of the reference foil target (b) The shadowgraphs of the foil target at (4.5ns) and (c) The shadowgraphs of the foil target at (7.5ns) indicating the two regions of irradiation namely aluminum foil and mixed coated region

The particle velocity (u_p) in Al and mixed Z regions was measured by two frames shadowgraphy and the shock velocity was estimated. Since, the shock (u_s) and particle velocity (u_p) are related in the form u_s = c + s*u_p, where ‘c’ and ‘s’ are the material constants, where ‘c’ is the shock velocity at infinitesimally small particle velocity or sound velocity at low pressure.

Figure 4: shows the variation of intercept with different materials that is density (ρ)

And the Hugoniot equation P=ρ_ou_su_p was used for evaluating the shock pressure (P₁) from experimental data and the material constant for Al was taken from reported earlier 11. By knowing the shock pressure (P₁) & measured particle velocities (u_p^Al) in Al and particle velocities (u_p^step) in mixed target, the shock-pressure (P₂) transmitted in step due to impedance matching can measured using the equation (2). From the pressure and particle velocity, we have calculated the shock velocities in mixed target. The EOS of mixed (Au/Cu) target is unknown; hence, the value of intercept i.e. the ‘c’ parameter i.e. the velocity of shock in case of the mixed target was determined by the dependence of ‘c’ on density. Fig. 4 shows the variation of ‘c’ with density for different metals having different density reported in earlier. From the density of mixed target, the ‘c’ parameter was obtained from Fig. 5.

Figure 5: Variation of the shock and particle velocity of mixed (Au/Cu)

Fig. 5 shows the plot of calculated shock velocity in the mixed using equation (2) with the measured particle velocity. The value of intercept (‘c’ parameter) in the Fig. 5 was 0.328 X 106 cm/s. The measured data in Fig. 5 showed a scatter for the slope parameter (‘s’ parameter). The variation of ‘s’ from the scatter of the experimental data was obtained to be between 1.05-1.50. The least square fit gave the value of ‘s’ to be 1.19. Since EOS of mixed target is unknown and no other measurements exist, EOS will be verified with another technique using streak camera in near future and further with simulations.