Single crystal lithium fluoride (LiF), oriented [100], was shock loaded and subsequently shocklessly compressed in two experiments at the Z Machine. Velocimetry measurements were employed to obtain an impactor velocity, shock transit times, and in-situ particle velocities for LiF samples up to ∼1.8 mm thick. A dual thickness Lagrangian analysis was performed on the in-situ velocimetry data to obtain the mechanical response along the loading path of these experiments. An elastic response was observed on one experiment during initial shockless compression from 100 GPa before yielding. The relatively large thickness differences utilized for the dual sample analyses (up to ∼1.8 mm) combined with a relative timing accuracy of ∼0.2 ns resulted in an uncertainty of less than 1% on density and stress at ∼200 GPa peak loading on one experiment and <4% on peak loading at ∼330 GPa for another. The stress-density analyses from these experiments compare favorably with recent equation of state models for LiF.

Lithium Fluoride (LiF) is commonly utilized as a window material in dynamic compression experiments because it maintains transparency under shock1 and shockless2 loading to higher stresses than any other known window material. The mechanical response of LiF couples into analyses of dynamic compression experiments that employ this material as a window.3 High accuracy equation of state data for LiF are therefore desirable to minimize the propagation of errors arising from window response uncertainty. The mechanical and optical response of LiF has been analyzed in detail on the principal Hugoniot1,4–6 and the principal shockless compression2,7,8 (quasi-isentropic) path. However, the advent of controlled off-principal dynamic loading9–11 demands validation of LiF equation of state models in phase space regimes more directly relevant to these “off-principal” experiments.

Two experiments on Sandia's Z Machine12 were designed to interrogate the mechanical response of LiF under shock-ramp loading. The first experiment (Z2920) shocked LiF to ∼100 GPa and subsequently shocklessly compressed to ∼200 GPa. The second experiment (Z2974) shocked LiF to ∼45 GPa and shocklessly compressed to ∼300 GPa. Aluminum 6061-T6 tapered stripline panels, 14 mm wide at mid-height, were fabricated with a built-in flight gap of 0.75 mm (Z2920) or 0.2 mm (Z2974). Tapering of the panel width is employed to minimize axial variations in the current density on the power flow surface which arises due to three dimensional effects.13 The power flow surfaces, panel floor, and flight gap shelf were all diamond turned to a surface roughness of <400 Å. Single crystal LiF samples and windows with initial density ρ0 = 2.638 ± 0.002 g/cm3, oriented [100], were installed on the flight gap shelf within the panels. Lateral dimensions of the LiF were 12.75 mm × 10.0 mm for Z2920 and 8.6 mm × 6.85 mm for Z2974. A tailored, “double-ramp” pulse shape was designed to accelerate the panel floors to a ballistic velocity of ∼7.4 km/s (Z2920) or ∼3.9 km/s (Z2974), at which point the flight gap closes and the panels impact the LiF samples. A further rise in the current/magnetic field in the AK gap launches a shockless compression wave into the shocked LiF samples. A combination of velocimetry probes for Velocity Interferometer System for Any Reflector (VISAR) were employed to observe the aluminum panel floor (flyer) velocity, the aluminum/LiF velocity at and after the time of impact, and the in-situ apparent velocities of ∼1, ∼1.5, and ∼1.8 mm thick LiF-windowed LiF samples. The measurements of the velocity at the aluminum-LiF interface are at Lagrangian position zero within the LiF and are therefore described as “zero thickness” samples. LiF-windows have a 0.25 μm thickness aluminum reflector deposited on the surface; these windows are secured to LiF samples, ranging in thickness from ∼1 to ∼1.8 mm, with a ∼1 μm thick layer of AngstromBond epoxy. Velocity measurements of the LiF-windowed LiF samples provide an in-situ velocity measurement at finite thickness. Each sample pair contains a zero thickness and finite thickness in-situ velocity measurement; this pair of velocities may be directly utilized in a Lagrangian sound speed analysis. Figure 1(a) shows one of the two panels (the cathode) used for Z2920, and in (b), a cross-sectional view of the assembled panels with samples and probe positions. The panels used for Z2974 had a similar design, but each sample was placed in an individual counter bore with separately machined flight gaps. The individual counter-bore design resulted in flight gaps which were not exactly the same for each sample on Z2974 necessitating timing corrections to the measured data as described below.

Multiple velocity-per-fringe (vpf) sensitivities, ranging from 0.277 to 1.058 km/s/fringe, on each sample from the VISAR diagnostic were averaged. The resulting measured apparent velocities from Z2920 are shown in Figures 2(a) and 2(b); apparent velocities from Z2974 are plotted in Figures 2(c) and 2(d). Multiple vpf sensitivities at each measurement position eliminated fringe jump ambiguity at shock breakout. Acceleration of the aluminum panel floor is observed up to the time of impact on the zero thickness samples. The flyers were smoothly accelerated up to 7.453 ± 0.010 km/s and 7.354 ± 0.010 km/s for the top and bottom pairs on Z2920, respectively; on Z2974 flyers reached 3.928 ± 0.010 km/s and 3.895 ± 0.010 km/s for the top and bottom pairs, respectively. Subsequent to the time of impact, the VISAR tracks the apparent velocity of the flyer/sample interface: Lagrangian position zero within the LiF. A slight pull back in the velocity was observed prior to arrival of the primary ramp wave on Z2920 (see Figs. 2(a) and 2(b) from ∼3080 to 3120 ns). This pull back is due to a slight decrease in the driving magnetic pressure during the plateau in the pulse shape generated by the machine. The apparent in-situ velocities of the LiF-windowed LiF samples, which are a measurement at finite thickness LiF, show a shock jump followed by smooth acceleration up to peak velocity.

With an identical flight gap for each sample in a pair, the time difference from impact to shock breakout at the aluminum coatings combined with the known sample thicknesses provides a measure of the shock velocity (us). When the flight gaps are slightly different for each sample in a pair, as was the case for Z2974, the observed impact time must be adjusted for one of the samples. This is accomplished by integrating the observed flyer velocity, Figures 2(c) and 2(d) from ∼2800 to 2930 ns, and requiring the distance travelled to be equal to the measured flight gap associated with the LiF-windowed LiF sample. As an example, consider the apparent velocities in Figure 2(d): the zero-thickness sample had a mechanically measured flight gap of 190.0 ± 3.0 μm and the 1.035 mm thick sample had a mechanically measured flight gap of 177.4 ± 3.0 μm. The observed impact time of the zero-thickness sample was 2927.3 ns; we note that the implied flight gap of the zero thickness sample, found by integrating the measured velocity up to 2927.3 ns, is 190.4 μm in good agreement with the mechanical measurement before the shot. The flyer traversed a total of 177.4 μm at a time of 2923.9 ns; hence, a time correction of −3.4 ns must be applied to the apparent impact time for the zero thickness sample in the pair. We note that the flight gap correction is particle velocity dependent; the magnitude of the correction will decrease with increasing particle velocity during the ramp portion of the experiment.

The particle velocity dependence of the flight gap timing correction is found by running a set of hydrodynamic simulations of the experiment utilizing a common drive pressure and sample thickness, but varying the flight gap. A set of simulated velocities at the Al-LiF interface are generated which depend on the flight gap, aluminum equation of state, and weakly on the LiF equation of state. Flight gaps from 160 μm up to 240 μm were simulated; SESAME table 370014 was used for the aluminum equation of state and several models for the LiF equation of state, including SESAME table 7271v3,8,15 were utilized for the LiF response. The time difference to reach a given particle velocity due to varying the flight gap was found to be linear in flight gap. The magnitude of this correction varied from 3.4 ns at the Hugoniot state, down to 0.5 ns at peak velocity. The sensitivity of the magnitude of the time correction to the LiF equation of state was tested and found to be insignificant. A stiffening of the LiF equation of state in these simulations by 5% at 200 GPa resulted in a change in the timing correction of ∼0.2%, or ∼10 ps maximum. The reason the LiF response is largely uncoupled from the flight gap timing correction results from the correction being applied at zero thickness in the LiF; at this position, the waves have not traversed through any LiF. The acoustic impedance of LiF sets the sound speed in the aluminum at the interface, but the magnitude of this second order effect is so small as to effectively uncouple the LiF response from the timing correction. The timing correction is, however, very sensitive to the aluminum equation of state, which has an estimated uncertainty of <3%. For this reason, the Z2974 analysis is coupled to the aluminum equation of state; errors associated with the timing correction due to the uncertainty in the aluminum response are propagated through to the stress-density analysis of the LiF. The result is larger uncertainty associated with Z2974 relative to Z2920, which is a truly independent LiF response measurement.

The apparent velocities must also be corrected for index of refraction effects. The analyses included in this publication tested a non-linear index of refraction model for LiF1 as well as an additional temperature dependent refractive index model3,8 in order to gauge the effect of different index models on the analysis. The two analyses were essentially identical with less than <1% difference in stress density between the temperature dependent and non-linear temperature independent models. The true velocity immediately following impact is the particle velocity (up) associated with the shock Hugoniot state. This particle velocity, in combination with the shock velocity, constitutes an absolute Hugoniot measurement. The observed and calculated Hugoniot parameters are included in Table I.

After timing and index corrections to the apparent velocities, the Lagrangian sound velocity, CL(up), may be calculated directly from the observed in-situ particle velocities during the interval for which they correspond to a simple right going wave: CL(up) = Δxt(up), where Δx is the thickness difference between samples of a pair, and Δt(up) is the time difference for which each sample in a pair reaches particle velocity up. The uncertainty on the sound velocity is

δCL=CL{(δxΔx)2+2(δtΔt)2+(δup1Δtdup1dt)2+(δup2Δtdup2dt)2}1/2,
(1)

where δx is the thickness difference uncertainty, 3 μm; δup1 and δup2 are the uncertainties on velocity of each sample taken as 0.01 km/s; and dup1/dt and dup2/dt are the accelerations of each sample at velocity up1 and up2.

The Lagrangian sound velocity is equal to the shock velocity from zero velocity up to the particle velocity associated with the Hugoniot state. Figure 3 shows the measured Lagrangian sound velocities for each sample pair in both experiments. Both pairs in experiment Z2920 exhibit a decrease in sound velocity for the first ∼200 m/s following shock breakout consistent with quasi-elastic behavior. A plastic wave then follows up to peak loading. It is not possible to uniquely determine the shear stress supported at the Hugoniot condition or the yield strength from these data. However, the change in shear stress from the Hugoniot state to the point of yielding can be determined.16 Due to the pulse shape generated by the machine, a slight decrease in the driving magnetic pressure resulted in a pull-back in the velocity (between ∼3025 and 3120 ns, Figs. 2(a) and 2(b)) at the Al-LiF interface before arrival of the main compression wave resulting in relaxation of the LiF off the yield surface. The change in shear stress from the shock state up to yielding may therefore be interpreted as a lower bound on the yield strength. The change in shear stress, τ, is calculated from17 

Δτ=34ρ0εHε2(CL2CB2)dε.
(2)

Where CB is the bulk sound velocity, which is extrapolated back to the Hugoniot state, and ε is the engineering strain defined as ε = 1−ρ0/ρ. εH is the engineering strain at the Hugoniot state and ε2 is the strain at which the bulk and Lagrangian sound velocities are equal. The change in shear stress from the Hugoniot to yielding was nearly identical for both sample pairs at 1.5 ± 0.2 GPa. The strength of LiF at 100 GPa shock pressure is reported as 1.9 GPa18,19 or 1.2 GPa.7 If it is assumed that the pull-back observed in the velocity resulted in full relaxation which was then followed by quasi-elastic loading to the yield surface, the yield strength is then simply the change in shear stress observed, which falls within the range of reported yield strengths for LiF at this stress.

In contrast with Z2920, no velocity pull back was observed on the LiF samples in experiment Z2974 (Figs. 2(c) and 2(d)). This was due to a small but continuous increase in the driving magnetic field in the plateau region of the pulse shape, which is responsible for supporting the shock state before arrival of the ramp compression wave. Despite the lack of an observable pull back in the velocity of the LiF, the Lagrangian wave velocities (Figs. 3(c) and 3(d)) are suggestive of quasi-elastic behavior on initial compression from the shock state, although at much smaller magnitude relative to experiment Z2920. The change in shear stress from the Hugoniot state up to the point of yielding is 0.07 ± 0.1 GPa and 0.2 ± 0.1 GPa for the top and bottom pairs, respectively. A non-zero change in shear stress with no observed pull back in the velocities suggests that the Hugoniot state does not fall on the yield surface at this shock stress; i.e., there has been some relaxation back toward the hydrostat. The behavior has also been observed in ceramics20 and aluminum.16 

The observed Lagrangian velocities may be directly integrated to obtain the longitudinal stress, σx, and density, ρ, of LiF along the path of the experiments

σx=ρ00upCL(up)dup=ρ0usupH+ρ0upHupCL(up)dup,
(3)
ρ=ρ0[10updupCL(up)]1=ρ0[1{upHus+upHupdupCL(up)}]1,
(4)

where us is the shock velocity and upH is the particle velocity associated with the Hugoniot state. The uncertainty on the stress and density may be determined from

δσx=ρ00upδCL(up)dup=ρ0upHδus+ρ0upHupδCL(up)dup,
(5)
δρ=ρ2ρ00upδCL(up)(CL(up))2dup=ρ2ρ0{upHδusus2+upHupδCL(up)(CL(up))2dup}.
(6)

The resulting stress-density analysis from all sample pairs is compared to an equation of state model (Sesame 7271v3)15 in Figure 4. Version 3 (“v3”) of SESAME table 7271 is a modified version of 7271 which includes a slightly softer cold curve.8 In all cases, excellent agreement was found between the shock state observed in these experiments (Table I) and the Hugoniot from the SESAME table. Both sample pairs from experiment Z2920 exhibit an apparently stiffer response compared to the table; however, the table does not include a strength model. Supported shear stresses add to the hydrostatic pressure to produce a stress which is typically higher than the hydrostat. The change in shear stress observed for the LiF samples loaded in Z2920 (∼1.5–1.9 GPa) is consistent with the observed difference in stress/pressure between the experiment and table 7271v3 along the isentrope. However, even without shear stress subtraction, the analysis of Z2920 falls within the error of the table isentrope over the entire compression range up to 200 GPa. In experiment Z2974, where only a small change in shear stress was observed, the stress-density response of both sample pairs follows the table isentrope almost exactly.

New types of controlled loading experiments probe regions of phase space far off the principal Hugoniot and isentrope. Window materials, such as lithium fluoride, are commonly employed in these dynamic experiments to minimize the effects of free-surface wave interactions and tamp the sample to increase the spatially averaged pressure. The mechanical response of these windows, however, often couples into the analyses of the mechanical response of a windowed sample. High accuracy equation of state data are therefore needed for window materials to limit error propagation and aid in modelling and design of dynamic compression experiments. Two experiments utilizing the Z Machine were conducted on single crystal lithium fluoride far off the principal Hugoniot and isentrope in order to validate a commonly utilized equation of state model for this material. Data were obtained on ramp compression up to ∼200 GPa from a ∼100 GPa shock with an uncertainty of <1% in stress-density on experiment Z2920. A second experiment, Z2974, probed up to ∼330 GPa from an initial shock of ∼45 GPa with an uncertainty of <4% in stress-density. Both sets of data are in excellent agreement with SESAME table 7271v3. The design and analysis of dynamic experiments utilizing LiF as a window or sample material may utilize SESAME Table 7271v3 with confidence, even when the loading path is far off the principal Hugoniot or isentrope.

The authors thank the large team required to design, prepare, and execute experiments on the Z Machine. These experiments would not be possible without their dedicated efforts. Sandia National Laboratories is a multi-mission laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under Contract DE-AC04-94AL85000.

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