Measurements of the equations of state and ... - APS Link Manager

PHYSICAL REVIEW E 82, 026401 共2010兲

Measurements of the equations of state and spectrum of nonideal xenon plasma under shock compression J. Zheng, Y. J. Gu, Z. Y. Chen, and Q. F. Chen* National Key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics, P.O. Box 919-102, Mianyang, Sichuan, People’s Republic of China 共Received 13 April 2010; published 3 August 2010兲 Experimental equations of state on generation of nonideal xenon plasma by intense shock wave compression was presented in the ranges of pressure of 2–16 GPa and temperature of 31–50 kK, and the xenon plasma with the nonideal coupling parameter ⌫ range from 0.6–2.1 was generated. The shock wave was produced using the flyer plate impact and accelerated up to ⬃6 km/ s with a two-stage light gas gun. Gaseous specimens were shocked from two initial pressures of 0.80 and 4.72 MPa at room temperature. Time-resolved spectral radiation histories were recorded by using a multiwavelength channel pyrometer. The transient spectra with the wavelength range of 460–700 nm were recorded by using a spectrometer to evaluate the shock temperature. Shock velocity was measured and particle velocity was determined by the impedance matching methods. The equations of state of xenon plasma and ionization degree have been discussed in terms of the self-consistent fluid variational theory. DOI: 10.1103/PhysRevE.82.026401

PACS number共s兲: 52.25.Kn, 62.50.⫺p, 51.30.⫹i, 78.40.Dw

I. INTRODUCTION

Warm dense matter is an active research area, bridging the traditional definitions of plasma physics and condensed matter physics 关1兴. Rare gases were widely studied in warm dense matter because of their simple electric structure as closed shell system 关2–5兴. Of all the rare gases, xenon is the one with a smaller first ionization potential and various experiments were performed measuring equations of state 共EOS兲 and electrical conductivity of xenon 关6–14兴. Liquid xenon has been compressed over the pressure range of 10– 130 GPa and temperature ranging from 5.3 to 29 kK with a two-stage light gas gun. The generator of shock wave of the hemispherical geometry was employed, and the shock temperature and pressure data were measured up to 33 kK and 230 GPa 关6兴. Compared with liquid xenon, gaseous xenon is the one for which the higher temperatures are implemented at the lower shock pressures. The Hugoniot measurements on xenon gas were made with the maximum pressure of 4 GPa at the temperature of 55 kK 关12兴. Also the insulator–to–metal transition under shock compression is one of attractive tasks in nonideal plasma. However, in the range of higher pressure the shock data for dense gaseous xenon are rare, and the measurement of shock temperature at above ⬃1 eV by means of the emission and absorption spectrum is an open question due to various reasons such as the effect on the absorption of the unshock gas in the shock wave front and the unknown emissivity of xenon plasma. Shock wave production of nonideal plasma has been performed using various techniques such as gas gun, laser, Z-pinch facilities, and high explosive. In this paper, shock waves generated by accelerated flyer plate impact with a two-stage light gas gun were used in our experiments to compress xenon gas, which was heated to incandescent plasma. In this way, the advantage of the shock-generated

*[email protected] 1539-3755/2010/82共2兲/026401共6兲

plasma is that density, pressure, and temperature are very uniform throughout the shocked volume of several cubic centimeters, and the shock states of gaseous xenon including the shock velocity, density, and pressure can be determined accurately. Also the shock temperature might be evaluated from the time-integrated spectra. The experimental data were discussed in terms of the self-consistent fluid variational theory 共SFVT兲关15,16兴 which is taken account of the interactions among the atoms, ions, and electrons in nonideal xenon plasma. II. EXPERIMENTAL DESIGN

The similar experimental setup has been already described in detail elsewhere 关2兴. The present design adds a spectrometer to measure the spectrum. So only a brief summary is given here. The xenon specimen was sealed in a cylinder box of 44 mm in diameter and 3 mm thick, which was bounded by a base plate and a sapphire window 共Al2O3兲. By means of the two-stage light gas gun, the tantalum flyer was accelerated up to a high velocity of 4.5–6.1 km/s which was measured by using a magnetoflyer velocity system 共MAVIS兲 with an accuracy of 0.5%. When the flyer impacted the base plate, a strong shock wave was generated and transmitted into gaseous xenon specimen. The xenon gas was compressed from different initial pressures P0 of 0.80 and 4.72 MPa at room temperature T0, respectively. Through the sapphire window and optical fibers the shock-induced spectral radiation transmitted into a seven-wavelength channels pyrometer 共PMT兲 with the wavelengths of 400, 450, 550, 650, 700, 750, and 800 nm, respectively. According to a photonelectron conversion system the output voltage signals were recorded by the digital oscillographs, and the spectral radiation histories of dense xenon were obtained. Meanwhile, the transient spectra of dense gaseous xenon under shock compression were determined by means of an optical multichannel analyzer 共OMA兲 and recorded by charge-coupled devices 共CCDs兲. The xenon plasma temperature and ionization pro-

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TABLE I. Material parameters used in the shock impedance matching calculations. 共Note: ␳0 is the density, C0 and S are the linear fit parameters with the relation US = C0 + SUP, and ␥0 is the Grüneisen parameter.兲 Materials

␳0 共g / cm3兲

C0 共km/s兲

S

␥0

Taa Al2024a 304 Steela Al2O3 b

16.65 2.784 7.87 3.985

3.293 5.33 4.58 8.74

1.307 1.34 1.49 0.957

1.6 2.0 2.2 1.32

a

Reference 关18兴. Reference 关2兴.

b

Ps = exp关␥0共␩ − ␩i兲兴

再

FIG. 1. 共Color online兲 The experimental spectral radiation histories recorded by pyrometer in shot GXe-3: 400 nm共0兲, 450 nm共−50兲, 550 nm共−100兲, 650 nm共−150兲, 700 nm共−200兲, 750 nm共−250兲, and 800 nm共−300兲. The numbers of parentheses are the shift of signal amplitude.

cesses could be derived from the recorded radiating spectra. When the gaseous specimen was prepressurized to several MPa, the bulge of the base-plate’s surface would happen. In order to keep planar approximately when impacting, the base-plate materials with 2024 Al and Fe-Cr-Ni alloy 共304 stainless steel兲 were chosen under different initial pressure conditions. In our experimental device 2024 Al was applied for pressure of 0.80 MPa, and 304 steel for 4.72 MPa. According to the measuring results, the maximum displacement of the base-plate’s surface was about 0.02 mm under the experimental initial pressure region. Four experimental shots were performed which consisted of three shots from initial pressure of 0.80 MPa and one from 4.72 MPa. The typical recording signals of the spectral radiation histories were shown in Fig. 1. The self-emission from the xenon gas provides a clear indication of shock arrival at the base-plate/specimen interface 共at time t0兲. From time t0 to t1 the rise time of the light emission from the specimen indicates that they become optically thick within tens of ns, short compared to the shock transit time. When the shock wave in the xenon gas arrives at the window, a bound of the signal is observed at time t2. Between time t1 and t2 the flat regions of the signal imply that the shock temperature is constant during the shock wave transits across the specimen. After time t2 a falling edge indicates that the window would be opaque gradually at high temperatures. According to the transmitting time ⌬t = t2 − t0 and the thickness d0 of the gaseous specimen chamber, the shock velocity US can be measured accurately. On the basis of the measured shock velocity and the flyer velocity W, the shock pressure PH and the particle velocity UP could be determined by the impedance matching methods 关17兴. The material parameters used in the shock impedance matching calculations were listed in Table I 关2,18兴. When the shock arrives at the baseplate/specimen interface, the shock wave in gaseous xenon is generated. The isentropic release path Ps of the base plate is given by 关2兴

⫻ Pi + ␳0C20

冕

␩

␩i

冎

1 + Sx − ␥0x exp关␥0共␩i − x兲兴dx , 共1 − Sx兲3 共1兲

where ␥0 is the Grüneisen parameter and Pi is the shock pressure of base plate before isentropic release. ␩ = 1 − Vs,b / V0 and ␩i = 1 − Vi,b / V0, where V0, Vi,b, and Vs,b are the specific volume at normal conditions, at the initial volume of the isentropic release, and by the end of the release process, respectively. In virtue of the shock adiabat of xenon specimen and the isentropic release of base plate, the shock Hugoniot state of gaseous xenon could be determined from the continuous conditions of the interface. In order to compare the isentropic release path of Eq. 共1兲 with the mirror reflection of the Hugoniot of Al 2024, as an example, the experimental shot GXe-3 was used to analyze the difference between two approximations. Figure 2 shows the different calculated results from shock impedance matching. It could be seen that the discrepancy was obvious and has a difference by ⬃4% between the isentropic release and mirror reflection paths, especially in the lower pressure range. If the

FIG. 2. 共Color online兲 The comparison of the isentropic release paths with the mirror reflection of the Hugoniot of Al 2024 for shot GXe-3.

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MEASUREMENTS OF THE EQUATIONS OF STATE AND…

FIG. 3. 共Color online兲 The comparison of xenon density with calculations and experiments at different temperatures.

mirror reflection was used, it should be corrected properly 关19兴. In the Rankine-Hugoniot equation the small density fluctuations of density ␳0 will result in the greater transferred errors of the shock pressure, so the specimen initial density is very important according to the momentum conservation of the shock wave front, P − P0 = ␳0 ⫻ D ⫻ u. Though the temperature and the pressure could be measured accurately, the calculated gaseous xenon densities are of rather difference with the experiential formulas such as ideal gas, van der Waals 共vdW兲关20兴, and Peng-Robinson 共PR兲 equations 关21兴. In order to determine the gas density, a special container which can bear the pressure range of several MPa has been employed. The container was connected with a pressure pump and a pressure sensor with an accuracy of 0.01 MPa. In the container the gaseous specimen was prepressurized by using a pressure pump, then the weight and volume of the specimen inside the container were measured accurately by a method of draining and the gas density could be obtained. Figure 3 shows the comparison of the experiments with calculations. Considering the prohibitive price of xenon, only three density points in pressures of 4.74, 6.22, and 20.14 MPa were measured. From Fig. 3 the calculated densities are almost the same within the lower pressure region. As the pressure elevates, the discrepancy appears. It’s clearly seen that the calculations start to deviate at the pressure range above 2 MPa where the densities with ideal gas equation are lower than others. When the pressure increases up to 5.8 MPa, the calculations with the PR equation are higher than

that with the vdW equation. With the pressure increases gradually, the distinctions among these calculations are quite greater. The calculated results with the PR equation are in agreement with the present experimental data and database developed by Vargaftik 关22兴. It is validated that the xenon initial density can be estimated by using the PR equation with the small corrections by experiments. Based on the Planck’s radiation law, the shock temperature may be inferred from the wavelength-dependent spectral radiance in Fig. 1. Nevertheless, it is very difficult for the seven spectral points obtained in the experiments to determine the temperature because the measuring range is far from the peak wavelength of the Planck distribution, and the wavelength-dependent emissivity and the absorption of unshock xenon gas in the shock wave front are unknown. Another reason is the difficulty of finding a suitable calibrated source, and the shocked xenon plasma with temperature of tens of thousands of Kelvins and plenty of line spectra is so much brighter than the calibrated source in our experiments adopted by a standard bromine-tungsten lamp with temperature of thousands of Kelvins, that the latter are unsuitable. Therefore, the shock temperature of xenon plasma could not be obtained from the recorded spectral radiance of PMT, and other ways should be taken into account to evaluate the shock temperature. We will try to estimate the temperature of xenon plasma from the time-integrated spectra in the next section.

III. RESULTS AND DISCUSSION

The experimental Hugoniot data of dense gaseous xenon were listed in Table II, which shows the shock pressure in the range of 2–16 GPa. Meanwhile, the measured results were analyzed in terms of the self-consistent fluid variational theory, which has been described in our previous work in detail. The SFVT has been successfully applied to determine the equations of state and the ionization of dense helium 关15兴, and xenon 关16兴. An important feature of the SFVT model is the introduction of corrections of ionization energy by fulfilling self-consistently when minimizing free energy in the condition of chemical equilibrium. The shock temperatures were calculated by means of this model in the corresponding experimental pressure region, and also listed in Table II. The electron Coulomb coupling parameter ⌫ range from 0.6–2.1 in our experimental conditions was generated and the formula could be written as 关23兴

TABLE II. Experimental Hugoniot data and calculated temperatures of gaseous xenon. 共Note: Tspec is the measured temperature from the radiating spectra, and Tcal is the calculated temperature.兲

Shot No. GXe-1 GXe-2 GXe-3 GXe-4

Flyer/Base-plate

P0 共MPa兲

␳0 共g / cm3兲

W 共km/s兲

US 共km/s兲

Up 共km/s兲

PH 共GPa兲

Tspec 共kK兲

Tcal 共kK兲

Ta/Al2024 Ta/Al2024 Ta/Al2024 Ta/304 Steel

0.80 0.80 0.80 4.72

0.044 0.043 0.043 0.429

4.576⫾ 0.023 5.137⫾ 0.026 6.094⫾ 0.030 4.910⫾ 0.025

7.948⫾ 0.121 8.716⫾ 0.094 10.230⫾ 0.129 6.684⫾ 0.036

6.672⫾ 0.159 7.456⫾ 0.105 8.772⫾ 0.111 5.591⫾ 0.022

2.35⫾ 0.09 2.80⫾ 0.07 3.87⫾ 0.09 16.04⫾ 0.15

46.3

39.4 43.6 50.4 31.0

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FIG. 4. 共Color online兲 The recorded radiating spectra of xenon plasma with the wavelength range of 460–700 nm. The solid lines were the measured results by the spectrometer. The dashed lines were fitted by the Planck curve.

⌫=

冉冊冉冊冉

Z 共Ze兲2 ⯝ 36 akBT 6

2

A 12

−1/3

␳m 6 10 g/cm3

冊冉冊 1/3

T

107 K

−1

, 共2兲

where Z = 1 for the cases of ionization of a single electron, e is electron charge, a is ion-sphere radius, kB denotes the Boltzmann constant, T is the shock temperature, A is the mass number, and ␳m is the mass density after shock compression. Figure 4 shows the radiating spectra of shock irradiance recorded by a spectrometer which has the resolving power of 0.175 nm and the wavelength range of 460–700 nm. It was one of the simultaneous diagnostics in our experiments. The time-integrated spectra were resulting from the plenty of resonance absorption lines superposed in continuous spectra. The absorption lines were due to the absorption of unshock xenon in the shock wave front. The possible electron transitions corresponding to the absorption lines are also presented in Fig. 4. The observed absorption lines are mainly the transitions arising from the onefold xenon ions 共Xe+兲, which indicates at least that the first ionization process of xenon would happen. At lower pressure the time-integrated spectra are the continuous spectra without obvious absorption lines 关see Fig. 4共a兲兴. The strongest absorption lines in Figs. 4共b兲–4共d兲 are 670.48, 589.33, and 521.07 nm in our observed wavelength region, which are related to the 5p4共 3P1兲5d – 5p4共 1D2兲6p, and 5p4共 3P2兲5d – 5p4共 3P2兲6p,

5p4共 3P0兲6s – 5p4共 3P0兲6p transitions, respectively. This indicates that the strongest absorption peak would shift to shorter wavelength with the increase of the densities and shock pressures of xenon. Though the experimental radiation spectra were of saturation owing to the overexposure in the shorter wavelength range of Fig. 4共b兲, the absorption lines with wavelength of about 589 nm were clearly observed, which also appeared in shot GXe-3 and GXe-4 关see Figs. 4共c兲 and 4共d兲兴. It implies the excellent shot-to-shot reproducibility. The absorption bands were observed nearby the wavelength of 468 and 516 nm in Fig. 4共c兲. The number of absorption lines arising from the onefold xenon ions increase with the increase of the densities and shock pressures. This indicates the increase of pressure ionization of gaseous xenon. Moreover, taking into account the corrections of the quantum efficiency of CCDs, the peak wavelength ␭peak can be obtained by fitting a Planck curve from the relative intensity of the time-integrated spectra. The approximate temperature was calculated by the Wien’s law T = 2.898⫻ 106 / ␭peak, where the peak wavelength measured in nanometers. The temperatures derived from the Planck curve of the extrapolated peak wavelength were 46.3 kK for shot GXe-1 and 28.3 kK for GXe-4, respectively. It has a difference by ⬃15% as the theoretical calculations except for shot GXe-3. The larger difference between temperatures estimated by the Wien’s law and calculated by SFVT model in experimental shot GXe-3 might result from extrapolating deviation of peak wavelength of the time-integrated spectra. The reliability of the estimated

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MEASUREMENTS OF THE EQUATIONS OF STATE AND…

FIG. 5. 共Color online兲 Comparison of experimental shock velocity with calculations at different initial pressures P0. The solid curve was calculated by the SFVT model at the initial pressure of 0.80 MPa. The dashed curve was fitted by the experimental data of liquid xenon.

FIG. 6. 共Color online兲 Comparisons of experimental Hugoniot with calculations of gaseous xenon 共Solid curves兲 by SFVT model at different initial pressures P0. The dashed curves were fitted by the experimental data of Fortov et al. and Gryaznov et al. 关12,24兴 IV. CONCLUSION

temperatures using the above method will be expected to verify by other theoretical models and experiments in future. Figure 5 shows the comparison of the experimental Hugoniot data with calculations of gaseous xenon by SFVT model as shock velocity versus particle velocity. The gaseous xenon data of Gryaznov et al. and Fortov et al. 关12,24兴, and the liquid xenon data of Nellis et al. and Urlin et al. 关7,11兴 were also plotted in Fig. 5. It is clearly seen that the experimental data with different initial pressures of 0.80 and 4.72 MPa in our work are in agreement with the calculated results. The SFVT model can also reproduce the data of Gryaznov et al. and Fortov et al. The experimental results of gaseous specimen shocked from different initial pressures are almost traced by a line. This indicates that the Hugoniot parameter C0 is less changed. Meanwhile, compared the calculated Us-Up relation of gaseous xenon 共C0 = 0.22 km/ s , S = 1.11, 3 ⬍ Up ⬍ 9 km/ s兲 with the fitting linear Us-Up relation of liquid xenon experiments 共C0 = 1.60 km/ s , S = 1.17, 1 ⬍ Up ⬍ 6 km/ s兲, they are parallel approximately. It indicates that the Hugoniot parameter S is not strongly dependent on the initial density ␳0 of dense xenon in the range of our consideration. Figure 6 shows experimental data and the calculated relationship between the shock pressure and the particle velocity by SFVT model. We can see that the experiments with initial pressure of 0.80 MPa are in agreement with the calculations, which are situated between the experiments with pressures of 0.3 and 1.0 MPa and close to the data with 1.0 MPa. This indicates that the SFVT model can reproduce experimental results in the pressures up to 16 GPa. Also it is clearly seen that the experimental data by Fortov et al. at 5.0 MPa 关24兴 is lower than our calculations. Through analyzing their experiments, we found that the initial xenon density 共⬃0.269 g / cm3兲 obtained from ideal gas equation by Fortov et al. is lower than the present xenon’s initial density 共⬃0.429 g / cm3兲 determined by experiments. This leads to lower Hugoniot pressures compared with our experiments and calculations.

The nonideal xenon plasma with volume of several cubic centimeters was generated from different initial pressures by means of the two-stage light gas gun. The Hugoniot data in the ranges of pressure 2–16 GPa and the temperature of 31–50 kK were presented in our work, which can be reproduced by the SFVT model. On the basis of the measured gaseous xenon densities which were in agreement with the calculations of PR equation in the pressure range within 20 MPa, the Hugoniot equations of state could be determined accurately. The radiating spectra of shock irradiance were recorded to evaluate the ionization processes and shock temperature. The temperature has a difference by ⬃15% as the theoretical calculations. The most common technique to determine temperature under shock wave compression is to record the spectral radiance emitted from the xenon plasma through the window with a multi-wavelength optical pyrometer and to use the Planck’s law to calculate the color temperature, However, the emissivity of the shocked gaseous xenon is unknown and the greybody hypothesis is failed due to the absorption of unshock xenon specimen in the shock wave front and the emissivity with the strong dependent of wavelength. These lead to uncertainties to estimate the true temperature. So the temperature derived from the resonance absorption lines will be our aim in future works.

ACKNOWLEDGMENTS

We are grateful to the gas gun, diagnostics, and device fabrication teams for laborious assistance. This work was supported by the National Natural Science Foundation of China 共Grant No. 10674120兲, by the Science and Technology Development Foundation of China, Academy of Engineering Physics 共Grant No. 2007A01002兲, and by the Foundation of National Key Laboratory of Shock Wave and Detonation Physics Research, China Academy of Engineering Physics 共Grant No. 9140C6712011003兲.

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