Collision frequency measurement from optical reﬂectivity of laser indirect-driven CH/Al/diamond

– A collision frequency measurement from the optical reﬂectivity of laser indirect-driven CH/Al/diamond on the SG-10kJ laser facility is presented. The optical reﬂectivity and the Al/diamond interface velocity were measured simultaneously by the velocity interferometer. The aluminum rear surface density was deduced from the interface velocity by analyzing the wave interaction. The deduced sample state was compared with the simulation and quite good agreement was found. The electron collision frequency was deduced by ﬁtting the sample state to the optical reﬂectivity, and it is found that the experimental collision frequency agrees with a semi-empirical result within the error bar, but is larger than the simulated result based on the average-atom model with the hypernetted chain approximation.

Introduction.-The optical reflectivity, resulting from the interaction of the electromagnetic wave with matter, is an effective method to study characteristics of the matter in laser experiments.The reflectivity measurement was commonly used in the past to test the conductivity model by comparing the simulated reflectivity with experimental data [1][2][3].Benuzzi et al. [4] performed a preheating effect study in the laser-driven shock experiments, and the results indicated that the reflectivity measurement is more sensitive than the emissivity.An estimation of the preheating process was also made through the comparison between the experimental data and the calculated reflectivity, in which a collision frequency model was used.Hicks et al. [5] and Celliers et al. [6,7] studied the insulatormetal transform and electronic conduction through the reflectivity measurement, and a model in which reflectivity depends on dielectric parameter was used in these experiments.Forsman et al. [8] illustrated an approach to directly measure the conductivity of strongly coupled plasmas from the reflectivity and transmission with femtosecond lasers, and Widmann et al. [9] used this method to (a) E-mail: yjm70018@sina.cn(corresponding author) measure the electrical conductivity of warm dense gold directly in the experiment and provided further insights into the collision time with the assumption of nearly free electron behavior.Overall, the optical reflectivity is strongly dependent on electrical conductivity through the dielectric function [10].As is known, the electron collision frequency is related to the electrical conductivity.Therefore, the optical reflectivity measurement can be used to perform the study of the electron collision frequency.
The electron collision frequency is important in the coupling of the ultrashort laser pulse to the dense target, which involves the absorption of the laser pulse in the steep gradient plasma, the propagation of an electron heat wave propagating into the solid, and the generation of a shock wave as a consequence of the high pressure.It refers in the first place to the collisional absorption of laser light in the interaction layer, but also to the energy exchange between electrons and ions and to the thermal conductivity which is important for the propagation of the electron heat wave into the dense target.Since the material state ranges from cold solid to hot ideal plasma, the collision frequency is needed over a wide temperature and density.However, a unique theory for the whole range is not available and more sophisticated models, even molecular dynamics method, are less practicable for code implementation in the radiative hydrodynamic code [11][12][13][14].Generally, the collision frequency model used in the radiative hydrodynamic code is usually semi-empirical.For the hot plasma Spitzer's formula for the collision frequency is often used.However, in the limit of a cold solid at temperatures below the Fermi temperature the electrons are in a degenerate state.The collision frequency is no longer dependent on the electron temperature, but governed by the scattering of electrons by phonons or lattice vibrations.The semi-empirical models often apply existing results for the two limiting cases and join them smoothly in the transition range [15] with some approximation and hypothesis, and the result lacks direct verification.
In this paper, an electron collision frequency measurement from optical reflectivity of laser indirect-driven CH/Al/diamond is presented.The paper is organized as follows.The experimental arrangement and results are described firstly.Then, the density determination, the simulation of state evolution and the measurement of the electron collision frequency are presented.Prospects for future investigations are finally addressed.
Experimental arrangement and results.-The experiment was performed on the SG-10kJ laser facility.As shown in fig.1(a), eight pulse-shaped laser beams smoothed by continuous phase plates (CPP) were injected into the hohlraum [16] placed in an evacuated target chamber from two laser entrance holes (LEHs).Two disks of CH foam with density of 30 mg/cm 3 and thickness of 250 μm were put in the hohlraum to provide necessary isolations of the scattering laser and the hot high-Z plasma blown off from the hohlraum wall.A cylindrical shield was inserted at the central part of the hohlraum to shield the hard x-ray from the laser-target spot and at the bottom of the cylinder was a slot window of 400 × 500 μm 2 .A planar multi-layer sample, including 20 μm CH as ablator, 1 μm aluminum as shock-compressed sample and 200 μm diamond as transparent window, was attached to the slot.The diamond layer was used to protect the aluminum rear surface from free diffusion, which will result in a slow electron density gradient and affect the reflectivity measurement.The transmissivity of the diamond remains at unity until very high post shock pressure.The temporal evolution of the radiation temperature for the hohlraum measured from the LEH with flat-response x-ray diode (FXRD) detector [17] is shown in fig.1(c).The temporal profile of the radiation temperature measured from LEH has three steps, consistent with the temporal profile of the heating laser as shown in fig.1(b).
An imaging velocity interferometer system for any reflector (VISAR) with two different etalon branches [18] was placed facing the rear surface of the sample for recording the reflected streaked image.The reflected streaked image for the uncompressed sample is defined as static streaked image, as shown in fig.2(a).Similarly,  that for the compressed sample is defined as dynamic streaked image, as shown in fig.2(b).The fringes for the uncompressed sample are straight all the time.But for the compressed sample, the fringes are shifted and the intensities of the fringes become weak at first as the shock wave reaches the interface of aluminum and diamond.Then, the fringes miss as time goes on.Since the fringes miss at the later stage, there is no reflected surface forming in the diamond.Therefore, the reflected surface is the interface of aluminum and diamond.Moreover, the shift of the fringes in the streaked image is directly proportional to the velocity of the reflecting surface and related to the etalon.Therefore, by analyzing the shift of the fringes recorded by the two branches VISAR with the deconvolution technique for trip point of multi-shock fringes which is based on the flood algorithm [19], the interface velocity is 3.7±0.4km/s when the first shock wave reaches the rear surface of the aluminum.
The formation of the fringes is caused by the interference of the reflected probe lasers.Therefore, the intensity of these fringes reflects the reflectivity of the sample.Firstly, the probe laser is reflected a little partially by the rear surface of the diamond (R 0 ).Secondly, the probe laser may be absorbed when it is propagating in the diamond with transmissivity of T .Thirdly, the probe laser is reflected by the aluminum (R al ).Finally, the reflected probe laser may be absorbed by the diamond again.Moreover, the noise also contributes to the streaked image.Therefore, the counts of the streaked image recorded by VISAR are given as (2) Here, the subscript s represents the static streaked image and d the dynamic streaked image.Therefore, the normalized reflectivity of aluminum is The transmissivity of the diamond is unity while the diamond is uncompressed and compressed up to a certain level.As the shock pressure increases further, the transmissivity of the diamond will decrease to 0. It is difficult to determine what the transmissivity of the compressed diamond is before out the data analysis.Therefore, what can be extracted from streaked images directly is the normalized reflectivity of the whole sample: Here, the first term can be extracted from the streaked images, as shown in fig.2(c).The second term can be substituted by the laser profiles monitored by oscilloscope, as shown in fig.2(d).
The normalized reflectivity of the whole sample from the VISAR image is plotted in fig.3, and the uncertainty is about 20%.The reflectivity does not change before it sharply drops to 0.52 at 5.5 ns, which indicates that the preheating is inhibited effectively.Then the reflectivity remains unchanged for 0.4 ns.Finally, the reflectivity sharply drops to zero.
Result analyses and discussion.-Since the interface velocity has been obtained in the experiment, the rear surface aluminum density can be derived by analyzing the wave interaction.Then, the simulations for the state evolution of the sample are carried out.The simulated results agree well with the experimental data.Finally, the measurement of the electron collision frequency by fitting the reflectivity is presented.

Density determination.
When the shock wave travels in the uncompressed aluminum, the Rankine-Hugoniot conservation relations for mass and momentum are Here, D, P , u and V represent the shock velocity, pressure, particle velocity and specific volume, respectively.The subscript 0 represents the uncompressed aluminum and 1 the shock-compressed aluminum.Moreover, the shock velocity and the particle velocity of the shock-compressed aluminum satisfy a linear relation: Here, c al = 5.685, and λ al = 1.275, which are obtained based on the experimental shock velocity in a range between 10 and 28 km/s [20].
Similarly, when the shock wave transports in the uncompressed diamond, The subscript 0 represents the uncompressed diamond and 1 the shock-compressed diamond.Pavlovsk [21] gave a relationship between the shock velocity and the particle velocity of the shock-compressed diamond using an underground nuclear explosion at pressures from 1 Mbar to 5.9 Mbar: Here, c c = 12.16, λ c = 1.00.The impedance of diamond is higher than that of aluminum.Therefore, when the shock wave travels into diamond from aluminum, a reflected shock wave travels into aluminum from the interface and satisfies, The subscript 1 represents the shock-compressed aluminum and 2 the re-shocked aluminum.Moreover, the 2.1 ± 0.2 ρ al,2 = 1/V al,2 (g/cm 3 ) 5.0 ± 0.2 reflected shock curve of aluminum is calculated on the basis of the following equation: Here, P H is the pressure on the original Hugoniot of aluminum at the volume V = V al,2 .For present applications the Grüneisen Γ is usually determined from the relationship Γ/V = Γ 0 /V 0 .In this calculation, the Grüneisen parameter for aluminum, Γ 0 = 2.15, is used [22].At the interface of aluminum and diamond, continuity conditions are satisfied: The number of the unknown parameters is one more than the number of the equations.Therefore, getting anyone of the unknown parameters, these equations can be solved.As mentioned above, the interface velocity, u al,2 is 3.7 ± 0.4 km/s when the first shock wave reaches the interface.Therefore, these equations can be solved and the results are shown in table 1.

Simulations for the state evolution of the shockcompressed sample.
Firstly, the propagation of the xrays generated by the heating laser as shown in fig.1(b) in the hohlraum is simulated with the view factor code IRAD3D [23,24].The simulated results indicate that the laser intensity at the first step is so small that the x-rays is absorbed almost completely by the CH foam.However, the laser intensity at the second and third step is large enough so that the x-rays pass through the CH foam.Therefore, the radiation temperature at the target position has only two steps, corresponding to the second and third step of the heating laser, as shown in fig. 4.
Using the radiation source plotted in fig. 4 as input, both the density, pressure, particle velocity and temperature distributions of the whole sample were simulated with the hydrodynamic code MULTI-1D [25], as shown in fig.5(a)-(d).
Two shock waves driven by the radiation source transport into the aluminum at 5.7 ns and 6.1 ns, respectively.When the first shock wave reaches the interface of aluminum and diamond, a reflected shock wave travels into the aluminum from the interface.Moreover, the density and temperature of aluminum and diamond at the interface are discontinuous but their pressure and particle velocity are continuous.These results are consistent with the above analyses.The density, pressure and velocity of the aluminum rear surface when the first shock wave reaches the interface of aluminum and diamond are obtained and then compared with the experimental data.Good consistency can be found, as shown in fig.5(e)-(g).Figure 5(h) presents the simulated temperature distribution of the aluminum at the interface, and the temperature is about 0.5 eV when the first shock reaches the interface of aluminum and diamond.The uncertainty is about 0.2 eV, taking into account the possible deviations in the simulation results.

The measurement of the electron collision frequency.
The dielectric coefficient of aluminum can be given by the following formula: Here, ω L is the frequency of the probe laser, ω 2 pe = nee 2 ε0me is the plasma frequency, and ν is the electron collision frequency.
The reflectivity of the aluminum can be given by the Fresnel formula: Here, n d = 2.424 is the refractive index of the diamond.The transparency of the shock-compressed diamond has been studied [26] and the results indicate that the refractive index of the diamond remains almost unchanged when the shock pressure is 2.1 Mbar so that the uncompressed value of the refractive index can be used in this experiment.
Therefore, the reflectivity of aluminum is the function of plasma frequency and collision frequency.If the density and temperature are known, the plasma frequency can be given.Then, the reflectivity depends on the collision frequency only.The reflectivity of aluminum has been calculated through eqs.( 16) and (17), and the results indicate that there is a monotonous relationship between reflectivity and collision frequency when the density and temperature keeps unchanged.
In this experiment, when the first shock wave reaches the interface of aluminum and diamond, the normalized reflectivity of the whole sample is 0.52 ± 0.10.However, this reflectivity is the product of the aluminum reflectivity and the diamond transmissivity.The diamond transmissivity may be affected by the brittle fracture caused by the stresses exceeding 2 Mbar or the free electrons caused by the shock compression.If these factors led to loss of diamond optical transparency, as the shock wave travels in the diamond more and more diamond would be shock-compressed and then the diamond transmissivity would decrease with time.However, as shown in fig.3, the reflectivity of the sample remains unchanged for 0.4 ns after the shock wave reaches the interface of aluminum and diamond at 5.5 ns.This means that the diamond transmissivity remains unchanged during this 0.4 ns.It is inconsistent with the hypothesis that these factors would lead to loss of diamond optical transparency.Therefore, the diamond transmissivity is 1 when the shock wave reaches the interface of aluminum and diamond at 5.5 ns.Moreover, the previous study [26] indicates that Fig. 6: Comparison of the experimental collision frequency (red star) with the semi-empirical result when the density is fixed to 5.0 g/cm 3 (thick solid line) and the result from average-atom model with the hypernetted chain approximation (AAHNC) and Ziman formula (blue thick dotted line).The thin solid line is the result from the interpolation of νSpitzer and ν e-phonon , the dashed line the upper limit of the collision frequency given by the requirement λe > r0.
the transmissivity of the shock-compressed diamond is 1 when the shock pressure is 2.1 Mbar.As shown in table 1, the shock pressure of diamond is 2.1 ± 0.2 Mbar when the first shock wave reaches the interface.Therefore, the reflectivity of the aluminum is 0.52 ± 0.10 when the first shock wave reaches the interface.
The rear surface aluminum density has been derived by analyzing the wave interaction, which is 5.0 ± 0.2 g/cm 3 when the first shock wave reaches the interface.The temperature has not been determined in this experiment.However, the simulation, whose results agree well with the experimental data, indicates that the temperature of the aluminum at the interface is about 0.5 ± 0.2 eV when the first shock wave reaches the interface.Substituting the reflectivity, density and temperature into eqs.( 16) and ( 17), the collision frequency is obtained, which is about 8.14 +4.12 −2.56 × 10 15 s −1 .Figure 6 presents a comparison of the experimental collision frequency with the results given by a semi-empirical formula and a theoretical simulation, respectively.The semi-empirical result is an interpolation of ν Spitzer and ν e-phonon , and limited to the requirement λ e > r 0 , and the detail can be found in ref. [15].In the theoretical simulation, the structure is calculated by the average-atom model with the hypernetted chain approximation (AAHNC) [27,28] firstly, and then the collision frequency is calculated with a scattering theory [29].The experimental collision frequency is consistent with the semi-empirical result within the error bar, but larger than the simulated result from AAHNC.To compare with the results in the literature, the DC conductivity can be obtained from the collision frequency by the formula σ 0 = nee 2 meνei , which is 1.23 −0.41 +0.56 ×10 6 S/m while the collision frequency is 8.14 +4.12 −2.56 ×10 15 s −1 .In the literature [30], the experimental DC conductivity of the aluminum with solid density and T e = 0.2 ± 0.1 eV is 1.3 ± 0.7 × 10 6 S/m.The DC conductivity of the aluminum with solid density and T e = 0.2 eV given by the semi-empirical formula is about 0.8 × 10 6 S/m, consistent with experimental data within the error bar [30].The DC conductivity decreases with temperature, but increases with density [31].Therefore, the experimental result for DC conductivity of the aluminum with density 5.0 ± 0.2 g/cm 3 and T e = 0.5 ± 0.2 eV in this work should be in better agreement with the literature [30], which reports a smaller DC conductivity value than the simulations similarly [30,32].
Conclusion. -In summary, a measurement of collision frequency using optical reflectivity of the laser indirectdriven CH/Al/diamond is presented.The optical reflectivity and the Al/diamond interface velocity of the compressed sample were measured simultaneously by the VISAR.The aluminum rear surface density was deduced from the interface velocity by analyzing the wave interaction.A simulation of the sample state evolution is also performed and the simulated results agree well with the experimental data.The electron collision frequency is determined based on the reflectivity, density and temperature, and found to be consistent with the semi-empirical result within the error bar, but larger than the averageatom model with the hypernetted chain approximation.Further investigation will focus on the optimization of experimental setup.Step target will be used to ensure that the shock velocity and particle velocity can be measured simultaneously so that the density will be more objective.Additionally, temperature measurements will be performed.

Fig. 2 :
Fig. 2: (a) Static streaked image, (b) dynamic streaked image, (c) the counts extracted from the streaked images, (d) the temporal profiles of the probe lasers monitored by the oscilloscope.

Fig. 4 :
Fig. 4: The evolution the radiation temperature at the target position simulated by the view factor code IRAD3D.

Fig. 5 :
Fig. 5: (a)-(d) The simulated density, pressure, particle velocity, temperature distributions of the whole sample.(e)-(g) The comparison of the experimental data with the simulated density, pressure, particle velocity of the aluminum at the interface.(h) The simulated temperature of the aluminum at the interface.

Table 1 :
The shock-compressed state and the re-shocked state of the aluminum when the first shock wave reaches the interface.