Recent Nova laser shock wave experiments on pre-compressed liquid deuterium (Da Silva, 1997; Collins et al., 1998) provided the first direct measurements of the high temperature equation of state of deuterium for pressures up to 330 GPa. It was found that deuterium has a significantly higher compressibility than predicted by the semi-empirical equation of state based on plasma many-body theory and lower pressure shock data (see SESAME model by Kerley (1983)). In an earlier series of experiments using the two-stage gas gun (Nellis et al., 1983; Holmes et al., 1995), pressures of up to 23 GPa were reached. The laser experiments are of particular importance to this work because they represent the only experimental data our PIMC simulation results can be directly compared to. The temperatures reached in gas gun experiments did not exceed 4400 K. Therefore a direct comparison with PIMC simulations is currently not feasible.
Shock wave experiments are an established technique (Zeldovich and Raizer, 1966) to
determine the equation of state at high pressures and temperature,
which has been applied to a wide range of materials including
aluminum, iron, and water. In the experiment, a driving force is
utilized to propel a pusher at constant velocity into a material
at predetermined initial conditions (
) as shown in
Fig. . The impact generates a planar shock wave, which
travels at the constant velocity with . The shock
compression drives the material to a point on the principle Hugoniot,
which is the locus of all final states characterized by ()
that can be achieved by a single shock wave passing through. Under the
assumption of an ideal shock (see below), the conservation laws of
mass, momentum, and energy require only the measurement of the
velocities of the pusher and the shock front in order to
obtain an absolute equation of state data point. Pressure and density
of the shock material are related to and by,
By an ideal shock, one means that a planar pusher is driven at constant velocity into the sample. The resulting shock wave is characterized by a planar shock front that travels at constant velocity during the measurement. Furthermore, one assumes that unshocked material remains at known initial conditions and not preheated as for example by x-rays created at the laser target interaction. Under these assumptions, one can determine the equation of state from the measured velocities and using Eqs. -.
In the recent laser shock experiments, a shock wave is propagating through a sample of pre-compressed liquid deuterium characterized by an initial state, (, , ) with and . In our calculations, we set to its exact value of per atom (Kolos and Wolniewicz, 1964) and because . Using the PIMC simulation results for and , we calculate from Eq. and then interpolate linearly at constant between the two densities corresponding to and to obtain a point on the Hugoniot in the plane. Results at confirm that the function is linear within the statistical errors. The PIMC data for , , and the Hugoniot are given in Tab. .
|1000000||53.79 (5)||245.7 (3)||66.85 (8)||245.3 (4)||0.7019 (1)||56.08 (5)|
|500000||25.98 (4)||113.2 (2)||32.13 (5)||111.9 (2)||0.7130 (1)||27.48 (4)|
|250000||12.12 (3)||45.7 (2)||14.91 (3)||44.3 (2)||0.7242 (1)||12.99 (2)|
|125000||5.29 (4)||11.5 (2)||6.66 (2)||11.0 (1)||0.7300 (3)||5.76 (2)|
|62500||2.28 (4)||-3.8 (2)||2.99 (4)||-3.8 (2)||0.733 (1)||2.54 (3)|
|31250||1.11 (6)||-9.9 (3)||1.58 (7)||-9.7 (3)||0.733 (3)||1.28 (5)|
|15625||0.54 (5)||-12.9 (3)||1.01 (5)||-12.0 (2)||0.721 (4)||0.68 (4)|
|10000||0.47 (5)||-13.6 (3)||0.80 (8)||-13.2 (4)||0.690 (7)||0.51 (5)|
In Fig. , we compare the effects of different approximations made in the PIMC simulations such as time step , number of pairs and the type of nodal restriction. For pressures above 3 Mbar, all these approximations have a very small effect. The reason is that PIMC simulation become increasingly accurate as temperature increases. The first noticeable difference occurs at , which corresponds to . At lower pressures, the differences become more and more pronounced. We have performed simulations with free particle nodes and for three different values of . Using a smaller time step makes the simulations computationally more demanding and it shifts the Hugoniot curves to lower densities. These differences come mainly from enforcing the nodal surfaces more accurately, which seems to be more relevant than the simultaneous improvements in the accuracy of the action , that is the time step is more constrained by the Fermi statistics than it is by the potential energy. We improved the efficiency of the algorithm by using a smaller time step for evaluating the Fermi action than the time step used for the potential action. Unless specified otherwise, we used . At even lower pressures not shown in Fig. , all of the Hugoniot curves with FP nodes turn around and go to low densities as expected.
As a next step, we replaced the FP nodes by VDM nodes. Those results show that the form of the nodes has a significant effect for below 2 Mbar. Using a smaller also shifts the curve to slightly lower densities. In the region where atoms and molecules are forming, it is plausible that VDM nodes are more accurate than free nodes because they can describe those states (see chapter 3). We also show a Hugoniot derived on the basis of the VDM alone (dashed line). These results are quite reasonable considering the approximations (Hartree-Fock) made in that calculation. Therefore, we consider the PIMC simulation with the smallest time step using VDM nodes () to be our most reliable Hugoniot. Going to bigger system sizes and using FP nodes also shows a shift towards lower densities.
Fig. compares the Hugoniot from laser shock wave experiments (Da Silva, 1997; Collins et al., 1998) with PIMC simulations (VDM nodes, ) and several theoretical approaches: SESAME model by Kerley (1983) (thin solid line), linear mixing model (dashed line) by Ross (1998), DFT-MD by Lenosky et al. (2000) (dash-dotted line), Padé approximation in the chemical picture (PACH) by Ebeling and Richert (1985a) (dotted line), and the work by Saumon and Chabrier (1992) (thin dash-dotted line).
The differences of the various PIMC curves in Fig. as well as in Fig. are small compared to the deviation from the experimental results by Da Silva (1997) and Collins et al. (1998). One finds that the corrections from Eq. have only a small effect on the Hugoniot. In the experiments, an increased compressibility with a maximum value of was found while PIMC predicts , only slightly higher than that given by the SESAME model. Only for , does our Hugoniot lie within experimental error bars. In this regime, the deviations in the PIMC and PACH Hugoniot are relatively small, less than in density. In the high pressure limit, the Hugoniot goes to the FP limit of 4-fold compression. This trend is also present in the experimental findings. For pressures below 1 Mbar, the PIMC Hugoniot goes back to lower densities and shows the expected tendency towards the experimental values from earlier gas gun work Nellis et al. (1983); Holmes et al. (1995) and lowest data points from Da Silva (1997); Collins et al. (1998). This trend can be studied best in the logarithmic graph shown in Fig , where we also included our lowest available pressure point on the Hugoniot, which was obtained from simulations with 32 pairs of electrons and deuterons and the time step and . Within the statistical error bars, the PIMC Hugoniot curve tends towards the results from gas gun experiments. For these low pressures, one also finds that the differences between PIMC and DFT-MD are also relatively small compared to the deviation from the laser shock data.
Using the PIMC equation of state, one can also determine the shock and
pusher velocity on the Hugoniot. From Eqs. and
, one finds,
Summarizing, one can say that PIMC simulations predict a slightly increased compressibility of compared to the SESAME model but they cannot reproduce the experimental findings of values of about . Further theoretical and experimental work will be needed to resolve this discrepancy.