Equation of state of fluid helium at high temperatures and densities

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1 Science in China Ser. G Physics, Mechanics & Astronomy 005 Vol.48 No Equation of state of fluid helium at high temperatures and densities CAI Lingcang, CHEN Qifeng, GU Yunjun, ZHANG Ying, ZHOU Xianming & JING Fuqian Laboratory for Shock Wave and Detonation Physics Research, Institute of Fluid Physics, Mianyang 61900, China Received June 6, 005 Abstract Hugoniot curves and shock temperatures of gas helium with initial temperature 93 K and three initial pressures 0.6, 1., and 5.0 MPa were measured up to K using a two-stage light-gas gun and transient radiation pyrometer. It was found that the calculated Hugoniot EOS of gas helium at the same initial pressure using Saha equation with Debye-Hückel correction was in good agreement with the experimental data. The curve of the calculated shock wave velocity with the particle velocity of gas helium which is shocked from the initial pressure 5 MPa and temperature 93 K, i.e., the D~u relation, D = C 0 +λu (u < 10 km/s, λ = 1.3) in a low pressure region, is approximately parallel with the fitted D~u (λ = 1.36) of liquid helium from the experimental data of Nellis et al. Our calculations show that the Hugoniot parameter λ is independent of the initial density ρ 0. The D~u curves of gas helium will transfer to another one and approach a limiting value of compression when their temperature elevates to about K and the ionization degree of the shocked gas helium reaches Keywords: equation of state, helium, Hugoniot curves, Saha equation. DOI: / Helium, hydrogen, and their isotopes are the simplest monoatomic and diatomic molecules. It is relatively easy to describe their properties using the basic principles of quantum mechanics. In condensed matter physics, hydrogen and helium serve as the models for the condensed matter properties at extreme conditions so that both experimental and theoretical physicists pay much attention to the study of their properties [1], especially the insulator-metal transition of hydrogen []. The aim to study the equation of state of hydrogen, deuterium, helium, and their mixtures at high pressures and temperatures is to further understand the structural changes of the atoms and molecules of mixtures and the laws of interactions of inter-particle, which will be the developing trend of high pressure condensed matter physics extending to the extreme conditions and microcosmic field. As a main composition of Saturn and Jupiter, the equation of state of helium is needed for the structure model of the planetary. Nellis et al. measured the equa- Copyright by Science in China Press 005

2 696 Science in China Ser. G Physics, Mechanics & Astronomy 005 Vol.48 No tion of state of liquid helium with shock temperature up to 1000 K and pressure up to 56 GPa [3]. Pollian [4] measured the sound velocity when liquid helium was shock compressed up to 1 GPa. Barrat [5] investigated the isotopic shift in the melting curve of helium using a path integral Monte Carlo method and their result has been reproduced by the measurement of Loubery [6]. The equation of state and phase diagram of solid 4 He from single-crystal X-ray diffraction over a large pressure-temperature domain was also studied [7]. There is still no report on the study of the equation of state of gaseous helium. In this paper we present our investigation of the equation of state of dense gaseous helium shock compressed by a two-stage light-gas gun from environmental temperature and three initial pressures 0.6, 1. and 5.0 MPa. Spectral radiance histories from the shocked helium were recorded by using a pyrometer. Shock wave velocity was measured, particle velocity was determined by the method of shock impedance matching, and the data were discussed in terms of the Saha model. 1 Principle of experimentation 1.1 Experimental technique A layout of experimental arrangement is shown in Fig. 1. A two-stage gas gun accelerates a 93W (4.Ni.45Fe0.35-CoW) or tantalum flyer to 4 6 km/s. Impact of the flyer on the aluminum base-plate generates a planar shock wave which enters cylinder-like gas chamber and heats the gas to incandescent plasma. The fiber optic bundle collects the emitted light after passing through sapphire (Al O 3 ) window. The spectral radiance from the shocked dense helium is recorded using a seven-channel fiber optic-coupled pyrometer, which is used to determine the temperature of the gaseous sample and the shock wave velocity. The flyer velocity is measured by magneto-flyer velocity system and it determines the shock pressure and particle velocity of the base-plate/sample interface. Fig. 1. A layout of experimental arrangement. Copyright by Science in China Press 005

3 Equation of state of fluid helium at high temperatures and densities Multichannel transient optic pyrometer Multichannel transient optic pyrometer is a sort of system to record the signal amplitude of spectrum, and it is an equipment of measuring substance temperatures. Its measuring range is generally K and can extend to the lower and upper limits, 1500 and 0000 K, respectively. The physical principle of pyrometer is based on the theory of the thermal radiation equilibrium of Plank. Assuming that the thermal radiation equilibrium had been reached, the radiance of high-temperature matter could be described by a grey-body model as εc I ( ελ,, T) = εi ( λ, T) = λ 1 gre Pl 5 1 exp( C / λt) 1, (1) where I gre is the radiance of grey-body, C 1 and C are the first and second radiation coefficients, respectively (C 1 = W m Sr 1, C = m K), ε is the effective emissivity, λ is the wavelength, T is the temperature, and h is Plank s constant. In our analysis, we assumed that ε did not depend on wavelength and used a nonlinear least squares method to find the values of ε and T. 1.3 Performance of high-pressure gas chamber High-pressure gas chamber should be in vacuum before inflation and keep its original shape after inflation because the shock compression experiments are carried out at a definite initial pressure. In order to keep pure helium in the gas chamber, firstly it was pumped to approach its limiting pressure 0.1 Pa, and then washed more than 10 times using the high-pressure gas. We carried out an experiment of the leak hunting and compression resistance of the high-pressure gas chamber and its gas inlet/outlet pipes at the Institute of Matter Structure of China Academy of Engineering Physics, and measured the vacuum leak less than Pa m 3 /s. Fig. shows the measurements of displacement of central point on the base-plate with increasing pressures. The deformation of base-plate is less than 0.06 mm at the filling pressure up to 5 MPa, which is within the permitted range for our present experiments. The rear-plate of the gas chamber is a sapphire (Al O 3 ) window, through which the emitted light of plasma can be collected by the fiber optic bundle. Sapphire is a single crystal Al O 3 that is offered by Institute of Solid Physics of Sichuan Normal University. The glabrous degree of sapphire surface is superior to 0.05 μm (National standard GB31-83). The optical transmissivity γ s of sapphire (Al O 3 ) window, 8 mm in thickness, to the light waves with different wavelengths is shown in Table 1. It could be seen that the transmissivity is about 75% in the wavelength range of μm in the main. One performance parameter of the high-pressure gas chamber is the optical reflectance of the base-plate surface, which is measured by the China Academy of Measuring Technique. The static reflectance of Al base-plate surface in the wavelength range of

4 698 Science in China Ser. G Physics, Mechanics & Astronomy 005 Vol.48 No Fig.. Displacement of the central point of LY-1 base-plate vs. the filled gas pressures. Table 1 Optical transmissivity γ s of the Al O 3 window λ (nm) γ s (%) nm is basically about 80%. The method of experimental data processing A shock wave will be generated by the high-speed flyer impacting the aluminum base-plate. The measurable parameters are the impact velocity of the flyer W and the radiance of spectrum of gas plasma with time changing. The shock wave enters the gas chamber and compresses the gas, for which plasma will form due to high pressures and temperatures. Spectral radiance histories are recorded by a transient radiation pyrometer. The sketch of a typical record of spectral radiance history is shown in Fig. 3. t 0 and t 1 correspond to the time shock wave enters gas sample and arrives at sample/window interface, respectively. The strength of spectral radiance changes from weak to strong during shock transit (t 1 -t 0 ), and finally approaches the value of plateau h i (λ), which means that the shock-induced incandescent plasmas have approximately come to the thermal equilibrium. The second plateau appears after t 1 as the reshock wave recompresses the plasma. The following parameters can be obtained based on the radiance characteristic value of the recorded spectrum. (1) Since the thickness d of the gas sample is known before the experiment, the shock wave velocity D could be immediately obtained as Copyright by Science in China Press 005 D = d/( t t ). () 1 0

5 Equation of state of fluid helium at high temperatures and densities 699 Fig. 3. Schematic representation of spectral radiance history. () The shock temperature T H can be fitted by h i (λ). Subscript H refers to Hugoniot state..1 The determination of shock parameters of gas plasmas The density ρ H and the shock pressure P H of the shocked gas can be determined by the method of shock impedance matching from the measured shock wave velocity D of gas plasmas. The density ρ 0, coefficients C 0, γ 0 and λ of the flyer and base-plate material used in data processing are listed in Table. Table ρ 0, C 0, γ 0 and λ of the flyer and base-plate material used in data processing Parameter 93W [8] Ta [9] LY-1A1 [3] ρ 0 (g/cm 3 ) C 0 (km/s) λ γ After impact, two compressive shock waves are created: One travels backwards into the flyer with the particle velocity u f and shock pressure P H,f ;the other travels forwards into the base plate with particle velocity u b and shock pressure P H,b. The pressures in the flyer and base-plate could be written as [10] P = ρ [ C + λ ( u W)]( u W), (3) H, f 0 f 0 f f f f P = ρ ( C + λ u ) u, (4) H, b 0b 0b b b where W is the velocity of flyer. According to the pressure and particle velocity continuity conditions across the flyer/base-plate interface, i.e. u f = u b = u and P H,f = P H,b = P H, we can obtain the particle velocity u from eqs. (3) and (4) as follows b B ± B 4AC u =, (5) A

6 700 Science in China Ser. G Physics, Mechanics & Astronomy 005 Vol.48 No where A = ρ λ ρ λ, (6a) B C W 0f f 0b b = ρ0 f 0f ρ0fλf ρ0b 0b C, (6b) C = ρ W( C + λ W). (6c) 0f 0f f So shock wave velocity D can be solved using the Hugoniot relation D = C + λ u. (7) 0b Shock pressure P H of Al can be obtained from the conservation of momentum of shock wave as follows P H 0b b = ρ Du. (8) Then the Hugoniot data of the shocked gas can be obtained [11]. When the shock wave reaches the base-plate/sample interface, a release wave and a compressive shock are created: the former reflects from the interface into the base-plate; the latter enters into the gas sample. Thus the relation of shock wave of gas sample can be written as P H,gas = ρ Du. (9) 0,gas The particle velocity of gas will be solved from the pressure and particle velocity continuity condition across the gas sample/base-plate interface if the isentropic release path can be given. The shock pressure of gas can be calculated from eq. (9). The isentropic release path of Al can be obtained by the following equations. Assuming that it is appropriate to the Mie-Grüneisen EOS, it can be written as P s P γ H = ( E S E H V ), (10) where E, V, and γ are the internal energy, volume, and the Grüneisen parameter, respectively. The parameters in eq. (10) are given by the following equations E s Pd s, = V H H = ρ0c0 (1 ληh ) P η, 1 EH E0 = P H( V0 VH), H η H = 1 V, V γ = 0 (Experiential equation). V γ V Copyright by Science in China Press 005

7 Equation of state of fluid helium at high temperatures and densities 701 Substituting the above equations to eq. (10), the release pressure in base-plate can be calculated by η 1+ λx γ0x S 0 i i i η (1 λx) i P = exp[ γ ( η η )]{ P + ρ C exp[ γ ( η x)] dx}, (11) where γ 0 is the Grüneisen parameter, P i is the initial isentropic release pressure. The shocked state parameters of the gas sample, P H, ρ H can be calculated by combing eqs. (9) and (11) with the continuity condition of interface.. Determination of shock temperature of gas plasmas The fiber bundle was positioned near the window so that the light emitted from the shocked sample permeated its aperture angle in the experiments. The incident radiance energy of the fiber cross section of the i-th channel could be written as E exp = I S, (1) i where I i is the recorded radiance of the i-th channel. The ratio of the recorded plateau value of the digital oscillograph, h i (λ), to calibration signal amplitude, h c (λ), can be expressed as hi Ii π (1 cos θ ) =, (13) h B( λ) E ( λ) c The corresponding radiance of the i-th channel, I i can be calculated from eq. (13). Assuming that the light radiance of the shocked gas is appropriate to a grey-body model, the radiated temperature was determined by eq. (1). The shocked temperature T H and emissivity ε can be determined by measuring the radiance of two wavelengths for ideal grey-body. However, due to the uncertainties in the measurements, a large number of channels are required to accurately determine the spectrum radiance to reduce uncertainties in the data processing. Each observed channel of multichannel pyrometer actually corresponds to the measure scale of radiance energy in this wavelength (λ i ). Precisions in different measured channels are different. According to Planck radiance law, the measured radiance, I exp (ε, λ, T) for each channel and calibration value h c are different for the light source of identical temperature T. Therefore, factors should be considered that precision in every measured channel affects the optimization temperature T. The weighted least-squares method was used for fitting the I exp (ε, λ, T) data [11] 0 λ [ εi pl ( λ, T) Iexp ] χ =, (14) σ λ where I pl (λ, T) and I exp (ε, λ, T) are given by eqs. (1) and (13), respectively. The weight

8 70 Science in China Ser. G Physics, Mechanics & Astronomy 005 Vol.48 No of each channel is taken for the standard uncertainty. It is determined by the following formula σ = σ + σ + σ, (15) λ λ N λ k λ,k, t, 0 where is the error of calibration lamp, is the calibrated error of pyrometer, σ λ N, t σ λ,k0 σ λ,k is the observed error in experiments. A nonlinear least-squares method has been used to find the values of ε and T. The best parameters will be picked to minimize χ, i.e. χ ε, T = min. (16) The minimum of eq. (16) occurs where the derivative of χ with respect to two parameters ε and T vanishes. This condition yields the two equations [1] χ = 0, (17) ε χ T = 0. (18) ε and T can be solved by the normal eqs. (17) and(18). 3 Experimental and calculated results The Hugoniot data and shock temperatures for gaseous helium samples were measured, covering the pressure range of MPa and the temperature range of K with the initial conditions of P 0 = 0.6, 1., 5.0 MPa and room temperature. Fig. 4 is an example of the recorded output of one of the photomultipliers at 700 nm (shot Fig. 4. An example of the recorded output of one of the photomultipliers at 700 nm (shot He990419). Copyright by Science in China Press 005

9 Equation of state of fluid helium at high temperatures and densities 703 He990419). Based on the recorded signal brightness amplitude of spectral radiance histories, we obtained the experimental Hugoniots by data processing. As a comparison, the Hugoniot data measured by Nellis [3] along with the ones calculated using the Saha model are given in Fig. 5. In order to further analyze the effects of electronic excitation and ionization of helium on D~u curves, the calculated D~u and T~u curves of helium from an initial pressure 5 MPa using the Saha model with Debye-Hückel correction in the higher temperature domain are shown in detail in Fig. 6. The measured shock tempera- Fig. 5. The D~u curves of fluid helium with different initial densities.,, represent the experimental data with corresponding initial pressures of 0.6, 1., and 5 MPa, respectively., The calculated data with an initial pressure of 5 MPa;, the fitted curve from experimental data of gaseous helium with an initial pressure of 5 MPa (D = u km/s); , the fitted curve from experimental data of liquid helium [3] (D = u km/s). Fig. 6. The calculated D~u curve and T~u curve of gaseous helium in the high temperature range., The calculated shock temperatures;, the calculated D~u;, the fitted D~u from the calculations: D =.1+1.u km/s.

10 704 Science in China Ser. G Physics, Mechanics & Astronomy 005 Vol.48 No ture data, compared with the results calculated by Nellis [3] using the fluid variational perturbation theory, are shown in Fig. 7. Fig. 7. The T~D curves of fluid helium with different initial densities.,, represent the present experiments with the filled pressure of 0.6, 5 and 1. MPa, respectively., The calculated data with an initial pressure of 5 MPa., Nellis s experiments of liquid helium [3]. 4 Discussions and conclusions The calculated Hugoniot EOSs using Saha equation with Debye-Hückel correction from initial pressure 5 MPa show a good agreement with the experimental results in Figs. 5 and 7. The calculated D~u curve of gaseous helium with an initial pressure P 0 = 5.0 MPa and temperature T 0 = 93 K approximately parallels to the one of liquid helium measured by Nellis, which shows that the value of λ is independent of the initial density of helium when the temperature is not higher than K. When the temperature is higher than K at which the order of magnitude of degree of ionization is 10 3, the value of λ becomes instantaneously from 1.3 to 1., which relates to the compressive limit of gas helium. This phenomenon results from the rapidly increasing absorption of energy from the electronic excitation and ionization. The twelve measured points of (D, u) of helium with three initial pressures 0.6, 1. and 5.0 MPa and initial temperature 93 K are very close to the calculated D~u curves with an initial 5.0 MPa and the same temperature. This means that the absolute difference values in initial density ρ 0 of helium sample for the three initial pressures and same temperature are small in the present experiments so that the effect of the initial density ρ 0 on the Hugoniot parameter C 0 is very small. The relation between the initial density ρ 0 and the Hugoniot parameter C 0 can be approximately represented as [8] C ρ00 = C, ρ 0x 00 where 00 denotes reference system and 0x denotes the required measure system. If using 0x Copyright by Science in China Press 005

11 Equation of state of fluid helium at high temperatures and densities 705 the liquid helium as the reference system (C 00 = 0.71, ρ 00 = 0.13 g/cm 3 ), the Hugoniot parameter C 0 of the gas helium at 5.0 MPa and 93 K is 0.1, which is very close to the measured value of 0.7 (see Fig. 5). It is found that the interaction of helium atoms behind shock front below initial pressure 5 MPa may be neglected by analyzing the present experimental data. This indicates that the Saha equation with Debye-Hückel correction can well depict the thermodynamic properties of helium in the pressure temperature domain under present consideration. However, It is worth noticing that all the measured points of (T, D) of helium with three initial pressures 0.6, 1., and 5.0 MPa and initial temperature 93 K are systematically above the calculated T~D curve of liquid helium [3]. This means that the interaction of helium atoms behind shock front will increase with increasing initial density. This shows that in calculating the equation of state of helium above initial pressure 5 MPa using the Saha equation with Debye-Hückel correction, it is necessary to take into account the contributions of the interaction of helium atoms behind shock front. References 1. Nellis, W. J., Shock compression of deuterium near 100 GPa pressures, Phys. Rev. Lett., 00, 89(16): [DOI]. Collins, G. W., Da Silva, L. B., Celliers, P. et al., Measurements of the equation of state of deuterium at the fluid insulator-metal transition, Science, 1998, 81: [DOI] 3. Nellis, W. J., Holmes, N. C., Mitchell, A. C. et al., Shock compression of liquid helium to 56GPa (560 kbar), Phys. Rev. Lett., 1984, 53(13): [DOI] 4. Polian, A., Grimsditch, M., Elastic properties and density of helium up to 0 GPa, Europhys. Lett., 1986, (11): Barrat, J. L., Louberye, P., Klein, M. L., Isotopic shift in the melting curve of helium: A path integral Montel Carlo study, J. Chem. Phys., 1989, 90: [DOI] 6. Louberye, P., LeToullec, R., Pinceaux, J. P., High-pressure measurements of the isotopic shift in the melting curve of He, Phys. Rev. Lett., 199, 69: [DOI] 7. Louberye, P., LeToullec, R., Pinceaux, J. P., Equation of state and phase diagram of solid 4 He from singlecrystal X-ray diffraction over a large P-T domain, Phys. Rev. Lett., 1993, 71: 7 75.[DOI] 8. Wang, J. G., An accurate measurement technique for shock Hugoniot, Chin. J. High Pres. Phys. (in Chinese), 1995, 9(4): Mitchell, A. C., Nellis, W. J., Shock compression of aluminum, copper and tantalum, J. Appl. Phys., 1981, 5: [DOI] 10. Jing, F. Q., Introduction to the Experimental Equation Of State (in Chinese), Beijing: Science Press, 1999, Chen, Q. F., Cai, L. C., Jing, F. Q., The isotope effect of gaseous hydrogen under shock compression, Shock Waves, 003, 13: [DOI] 1. Tan, H., Shock remperaturature measurements for metals (Ⅰ), Chin. J. High Pres. Phys. (in Chinese), 1994, 8(4):

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